Guitar and violin acoustics

Dept of Speech, Music and Hearing

ACOUSTICS FOR VIOLIN AND GUITAR MAKERS

Erik Jansson

Fourth edition2002 http://www.speech.kth.se/music/acviguit4/part1.pdf

Index of chapters Preface/Chapter I

Sound and hearing

Chapter II

Resonance and resonators

Chapter III

Sound and the room

Chapter IV

Properties of the violin and guitar string

Chapter V

Vibration properties of the wood and tuning of violin plates

Chapter VI

The function, tone, and tonal quality of the guitar

Chapter VII

The function of the violin

Chapter VIII

The tone and tonal quality of the violin

Chapter IX

Sound examples and simple experimental material – under preparation

Webpage: http://www.speech.kth.se/music/acviguit4/index.html

PREFACE The aim of this compendium is to build an understanding for the acoustics of the guitar and the violin, and to explain how their different parts cooperate. This will be done with results obtained by research. I shall show what can be measured and what can be perceived by our ears. Further I shall hint at measuring procedures that the maker himself may develop. Finally I shall show how "standard" laboratory equipment can be used to measure characteristics of the violin and the guitar, which partly shall be used by the participants of this course. Much energy has been devoted to balance an informative presentation without too much complication. Hopefully this balance is appropriate for many readers. The material presented is a combination of well known acoustical facts, late research on the acoustics of the violin and the guitar and results of the latest results of the ongoing research at KTH. The material has been collected by some 30 years of research and co-operation with a large number of persons and institutions. At my KTH-home, the Department of Speech, Music and Hearing, Prof. Johan Sundberg, Dr Anders Askenfelt, our first Dr of Music Acoustics Jesus Alonso, Prof. Gunnar Fant and his successor Prof. Björn Granström, all friends and colleagues have helped and inspired my work, and especially Mrs. Gudrun Weiner-Rispe who has plotted a large number of the figures. Much material has been collected at the Department of Physics II, KTH and the Institute of Optical Research headed by Profs Erik Ingelstam and Klaus Biedermann together with my optical "colleagues" Drs Nils-Erik Molin and Leif Ek. Of the many makers I early came into contact with are Harry Sundin and Gunnar Mattson of the Stockholm Violin Maker's Club. Much work has been made in co-operation with Göran Levvin at AB Herman Carlsson Levin and Carlo and Birgit Hansen at Carlo Hansen Violinateljé AB. Lately much work has been made together with the Division of Experimental Mechanics at Luleå University of Technology and my early colleague Prof. Nils-Erik Molin, Prof. Lars Frydén and Benedykt Niewczyk, violin maker from Poznan Poland. Without all this help and co-operation, this information would never have been collected and summarised. In addition I would like to add two names: Prof. Arthur Benade and Dr Carleen Hutchins, who both have taught and inspired me a lot. Although much material is presented mainly from work by the author, it must be pointed out that there is much more collected by other persons. The manuscript has been read by Dr. Rodney Day which has resulted in considerable improvements, both presentation and language. I want to thank these persons and all other helpful persons not named. Misinterpretations and negligence to present important material by others is the fault of the author, who asks for forgiveness for his misbiased judgement. The course is limited to acoustical fundamentals for the violin and the guitar. It is presented with little or no mathematics. Additional valuable information can be found in the CAS Journal (earlier called Newsletter). For readers with a more general interest in musical acoustics I recommend "The Science of Musical Sounds" by Johan Sundberg (Academic Press, Inc. 1991 ), which should be a good supplement to this course. Much information is collected by Dr Hutchins in Benchmark Papers in Acoustics (Dowden, Hutchinson & Ross, Stroudsburg, Pennsylvania, vol. 5 1975, vol. 6 1976) and Research Papers in Violin Acoustics 1975-1993 (Acoustical Society of America through American Institute of Physics, 1996). The financial supports by above institutions, the Swedish Humanistic Science Research Council, Swedish Natural Research Council, Swedish Board for Technical Development, Carl Trygger Scientific Foundation, Wenner-Gren Scientific Foundation, and Swedish Institute are gratefully acknowledged. Stockholm 2002- Erik Jansson

Jansson: Acoustics for violin and guitar makers 1.2

The present compendium has been used teaching acoustics for violin and guitar makers by the author. The material contains in parts original information not generally known and therefore it has been translated into English and made public on the Internet. The written information can, however, be difficult to read for many makers. If questions, they can be directed to the author, [email protected], who hopefully can answer them. A simplified version may be put together after retirement, i.e. after 5 years if judged to be interesting. INTRODUCTION AND CONTENT FUNDAMENTALS OF ACOUSTICS: Chapter 1. SOUND AND HEARING First part: THE SOUND Second part: THE HEARING PROCESS Chapter 2. RESONANCE AND RESONATORS First part: RESONANCE Second part: RESONATORS Chapter 3. SOUND AND THE ROOM First part: FUNDAMENTAL ROOM ACOUSTICS Second part: THE SOUND IN THE ROOM APPLIED INSTRUMENT ACOUSTICS: Chapter 4. PROPERTIES OF THE VIOLIN AND THE GUITAR STRING First part: FUNDAMENTALS OF STRINGS Second part: PLAYED STRINGS AND SCALES Chapter 5. VIBRATION PROPERTIES OF THE WOOD AND TUNING OF VIOLIN PLATES First part: VIBRATION PROPERTIES OF THE WOOD Second part: TUNING OF THE TOP AND BACK PLATES OF THE VIOLIN Chapter 6. THE FUNCTION, TONE AND TONAL QUALITY OF THE GUITAR First part: THE FUNCTION OF THE GUITAR Second part: THE TONE AND TONAL QUALITY OF THE GUITAR Chapter 7. THE FUNCTION OF THE VIOLIN First part: PROPERTIES BUILT INTO THE VIOLIN Second part: INFLUENCE OF DIFFERENT PARTS Chapter 8. THE TONE AND TONAL QUALITY OF THE VIOLIN First part: FUNDAMENTALS OF THE VIOLIN TONE Second part: ACOUSTICAL QUALITY MEASURES Chapter 9. SOUND EXAMPLES AND SIMPLE EXPERIMENTAL MATERIAL.




Erik Jansson

Chapter I: Sound and Hearing

Fourth edition 2002


ACOUSTICS FOR VIOLIN AND GUITAR MAKERS Chapter 1 – Fundamentals of acoustics SOUND AND HEARING Part 1: THE SOUND 1.1. What is sound? 1.2. Sound and spectrum 1.2. Time history of the sound 1.4. Summary 1.5. Key words Part 2: THE HEARING PROCESS 1.6. The ear and the hearing process 1.7. Masking, critical bands, and roughness 1.8 Timbre and spectrum 1.9. Summary 1.10. Key words


Chapter 1. FUNDAMENTALS OF ACOUSTICS - SOUND AND HEARING First part: THE SOUND INTRODUCTION

In this first part of chapter 1 properties of sound will be presented. Fundamental measures such as frequency and level will be introduced. The measures will be developed into spectrum and time histories (oscillograms). Sounds as musical tones, single tones and noise will be described by their spectrum representation and by their time history representation. 1.1. WHAT IS SOUND? Sound is quick and microscopically small variations in the barometric air pressure. The variations are described by two fundamental acoustical measures FREQUENCY and LEVEL. FREQUENCY is related to the pitch, i.e., a low frequency corresponds to a low pitch and a high frequency to a high pitch. LEVEL relates to loudness, i.e. a low level corresponds to a weak sound and a high level to a loud sound. A simple example can illustrate that air pressure variations are sound, see Fig. 1.1. A loudspeaker emits sound. A loudspeaker placed with its opening directed upwards emits sound upward through the opening. A sheet of paper placed over the loudspeaker opening will vibrate and give a rattling sound. The loudspeaker membrane moves up and down, which will result in the following interaction with the paper sheet:

Figure 1.1. Cross-section of a loudspeaker box seen from the side, and stages 1 to 3 described in the text. The arrows mark the position of the loudspeaker membrane in relation to its equilibrium and the varying air pressure (the sound pressure) acting on the paper sheet, respectively.

1. At the equilibrium position of the loudspeaker membrane the air pressure is the same below as above the paper and the paper sheet is not moving.


2. When the loudspeaker membrane moves upwards, the air is compressed under the paper and the increased air pressure pushes the paper upwards. 3. When the membrane moves towards its lowest position the air pressure is lowered and the paper is pulled down towards the loudspeaker. The experiment is easily made and it shows that the vibrations of the paper sheet (a rattling sound and sensations of tickling in a hand held over the loudspeaker) are related to sound. When the loudspeaker emits strong sound the paper rattles strongly (strong hand tickling), and when the loudspeaker emits weak sound the paper rattles weakly (weak hand ticking). HOW LOUD IS THE SOUND – LEVEL? The strength of the sound, the sound pressure level is measured in dB (decibel soundpressure level – mathematically the level relative to a sound pressure of 0.00002 Pa). To give a coarse feeling for sound pressure levels three extreme examples are given in Table 1.1 and to give a more detailed feeling, some sound pressure levels typically met in daily life are given in Table 1.2. The barometric sound pressure, well above the threshold of pain, can only be obtained theoretically (mathematically). Table 1.1. Sound pressure levels – minimum to maximum and barometric pressure Faintest audible sound 0 dB Loudest (without pain) 120 dB Barometric pressure (760 mm Hg) 194 dB Table 1.2. Sound pressure levels of typical sounds Threshold of pain Disco Heavy traffic Conversation Library Rustling of leaves Threshold of hearing

120 dB 100 dB 80 dB 60 dB 40 dB 20 dB 0 dB

PITCH OF THE SOUND - OR FREQUENCY?

Frequency is a measure of how often an event (in the time history) is repeated per second and the unit of measurement is Hz (hertz). Often the given frequency unit is kHz (kilohertz), which is 1000 Hz in the same way as 1000 gram is 1 kilogram. Thus 2 kHz and 2 000 Hz are the same. Example 1.1. A metronome adjusted to 60 gives 60 beats per minute, which equals 1 per second and the frequency is 1 Hz. If the metronome is readjusted to 120 it gives 120 beats per minute, which equals 2 per second and the frequency is 2 Hz. Example 1.2. Sound is the vibration of air. If the membrane of a loudspeaker moves out and in 20 times a second, i.e., with the frequency 20 Hz, we can hear a very low tone. If the membrane moves in and out 440 times per second, we hear a tone of 440 Hz, i.e. the standard tuning A (A4).


The young, fresh human ear can hear sound approximately from 20 to 20 000 Hz, i.e., from 0.02 to 20 kHz. The frequency of the lowest note on the piano has a frequency of 27.5 Hz (A0 nominal value) and its highest note is 4.186 kHz (C8 nominal value). 1.2. SOUND AND SPECTRUM Let us look at pictures of the sounds. What do the sounds "look like"? Usually one draws the spectrogram of a musical note, shortly to be called the SPECTRUM. The spectrum is an acoustical snapshot of the note. The spectrum can be explained by means of Fig. 1.2. The horizontal scale - the frequency axis - of the spectrogram corresponds to the staves in the musical representation. A specific tone has a specific staff such as the note A of 220 Hz. In the spectrogram the tone is represented by a bar (a vertical line) at the frequency 220 Hz. The strength of the tone, strong (musical ff) corresponds to a high bar and weak (musical pp) corresponds to a low bar. The level of the tone can be read at the (vertical) level scale in dB and its frequency at the (horizontal) frequency scale in Hz. It has thus been shown that the frequency in Hz and level in dB can describe a TONE. The TONE is an acoustical building stone of the sound we hear. Example 1.3.a) The time signal (at least in Sweden) contains a tone of the frequency 1000 Hz, i.e., 1 kHz. The level of the tone can be adjusted by the volume control. b) The tuning forks with the standard tone give a tone with the frequency 440 Hz. The level is high in the beginning and decreases slowly. A stronger hit gives a higher level. Most musical tones do not have the simple representation shown so far. They consist of several tones, c.f. Fig. 1.3. The musical tone A 220 Hz in Fig. 1.3 is placed two staves below the five permanent ones (in guitar notation 110 Hz, a transposition to one octave lower). A more complete representation has been given on the staves to the left, i.e. a tone (the fundamental), a tone an octave higher (acoustically twice as high), a tone an octave plus a fifth higher (three times as high), a tone a double octave higher (four times as high), and a tone a double octave plus major third higher (five times as high). The played notes of the melody instruments such as violin, clarinet and trumpet are made up by several single tones as sketched in Fig 1.3. The spectrogram of a real musical tone consists of several bars, c.f. Fig. 1.3. The frequencies of the bars can be read at the horizontal scale. The levels of the different tones, the heights of the bars, can be read at the vertical scale. The different tones are called PARTIALS (harmonics) and the lowest partial is called the FUNDAMENTAL. A little closer look reveals that the frequency of the second tone is twice that of the fundamental, the frequency of the third partial is three times that of the fundamental, etc. The bars show the SPECTRUM of the musical tone. The partials are numbered 1, 2, 3 etc. and the partial frequencies are 2,3 etc times that of the fundamental. The frequency of the fundamental corresponds to the pitch and the levels of the partials correspond to the timbre. The spectrum of a sound is measured with a spectroscope. The spectrogram of the musical tone in Fig 1.3 show five partials (bars), which are placed at even distances. The partials are equally strong, i.e. of the same level, as the five bars have equal height.


Figure 1.2. Musical description and acoustical description of a tone.

Figure 1.3. Musical description of a real tone and acoustical description.


Figure 1.4. Time histories (left) and spectra (right) for a) a simple tone (upper plots), and b) a more real tone (lower plots) with four equally strong partials.

Example 1.4. A violinist plays the open A-string, i.e., a spectrum with the fundamental of 440 Hz. The second partial has the frequency 2 times the fundamental, i.e., 2 times 440 equals 880 Hz. In the same way the third, fourth and fifth partial frequencies can be calculated as 3, 4, and 5 times the fundamental frequency, respectively. The frequencies are 1320 Hz (1.32 kHz), 1760 Hz (1.76 kHz), and 2 200 Hz (2.2 kHz), respectively. 1.3. TIME HISTORY OF THE SOUND The simple tone with only one partial (= the fundamental) has a time history (oscillogram) and a spectrum corresponding to Fig. 1.4a. The time history repeats itself after equal time intervals. A spectrum with four equally strong partials has a time history corresponding to Fig. 1.4b (the simplest case). The time history repeats itself after equal time intervals here too. The time history is considerably more complicated though than that of one partial. A played tone does not look the same all the time. It may look as in Fig. 1.5. This played tone has a beginning, an extended constant (more or less) part and an end. The beginning is usually called the starting transient and corresponds musically to the attack. The end of the tone is usually called the ending transient and corresponds to the decay of the string vibrations. The time history is measured with an oscilloscope. Example 1.5. The motion of the tuning fork is very rapid but the tone of a tuning fork decays slowly, c.f. Fig. 1.6. It vibrates so rapidly but decays so slowly that it is not possible to plot in detail the sound pressure variations and the full ending transient. In the upper part of Fig. 1.6 the time history of the decaying motion has been drawn. Below, parts of the time history has been enlarged considerably, i.e., the central part shows the detailed time history at three instants. In the lower part of Fig. 1.6 the spectra corresponding to the three instants have been plotted. The figure shows that the tuning fork gives a simple tone at the different instants and a


single partial bar in the spectrum (c.f., Fig. 1.4a). The sound pressure decreases slowly and thus the level of the single partial decreases slowly.

Figure 1.5. Time history of a tone in principle: A the starting transient, B the constant (more or less) part, and C the ending transient (the oscillations are generally to close to be drawn, and are marked here by the shade).

Figure 1.6. The full time history of the tone of a tuning forks (an ending transient), detailed time histories and corresponding spectra.

Example 1.6. The pipe organ is an instrument, in which the starting transient is most important. It can be adjusted differently and can give different sounding qualities. Two different starting transients of the same pipe are shown in Fig. 1.7. The upper time history is somewhat more complex than the lower one. In a detailed time history one can read that the second partial (the octave shown as the fine pattern shade) dominates during the earliest part (0.02 to 0.03 seconds): Thereafter the first partial (the fundamental, the coarse pattern shade) starts to dominate. In the lower time history the first partial dominates the starting transient and just increases.


Figure 1.7. Time histories (shaded oscillograms) of starting transients of two pipe organ tones, the same pipe differently adjusted. In the upper frame the second partial dominates initially (dark area) the time history, in the lower frame the fundamental (the lowest partial) just grows (after Sundberg 1966).

Figure 1.8. Time history (oscillogram) of a played violin tone.

Usually technical people measure time in units of 0.001 seconds, which are called ms (milliseconds) compare m (meter) and mm (millimetres), 1 mm being 0.001 m). Example 1.7. A musical instrument such as the violin results in still more complicated time histories, c.f., Fig. 1.8. The time history is changing all the time in a specific way and monitored by the skilled musician. If one listens to the early, the middle, and the late part of the tone (by computer manipulations this can be done) one will hear that the loudness, timbre and also the pitch of the tone are different in the different parts. Timbre and pitch will be defined later in this chapter


Figure 1.9. Time history (left plots) and spectra (right plots) of a guitar tone immediately after attack, 0.2, and 0.4 seconds later.

Example 1.8. The guitar string is plucked and is thereafter left to decay freely. The time history of a guitar tone may look as in Fig. 1.9. It is rather irregular at the attack, but shows clearly repetitive (periodic) time histories at the later two instants. The time histories are more similar to that of Fig. 1.4 b (four partials) than that of Fig. 1.4a (one partial only). The spectra corresponding to the different instants also show that the guitar tone is made up by several partials. It shows 14 partials just after the attack, 9 after 0.2 seconds, and 7 after 0.4 seconds. The levels of the partials decrease differently. The highfrequency partials decay faster than the lowfrequency ones. Every partial can be thought of as produced by a tuning fork and the tuning forks of the different partials decay with different speeds. The shown guitar tone could be produced by fourteen different tuning forks and not by one (the line between the partial peaks is called the spectrum envelope).


Figure 1.10. Time history (left) and spectrum (right) of noise.

Figure 1.11. Time history and spectrum of an impulse.

NOISE AND IMPULSE SOUND In the violin-making world and in modern technology two different kinds of test sounds are used, i.e. IMPULSE sound and NOISE sound. NOISE is technically a sound that has irregular variations with time. Noise has the property that it contains partials at all frequencies and on average of the same level. A noise thus contains partials at 100 Hz, 101 Hz, 102 Hz, etc. It has also partials between 100 Hz and 101 Hz, i.e. at 100.1 Hz, 100,2 Hz .... 100.8 Hz, and 100.9 Hz. But between these partials there are also partials at 100,11 Hz, 100.12 Hz, etc. Between the latest partials there are also partials and so on. The noise is made up by a spectrum of infinitely densely packed partials, which on average are of equal strength (technically called white noise). At any frequency there is a partial, for instance at 100 Hz and 173.937 Hz. The time history for a noise sound may look as in Figure 1.10 , a completely irregular and wriggly curve. The spectrum looks as in the right part of the figure. The partials are so densely packed that single partials can not be seen and all partials are on the average equally strong.


Example 1.9. a) A television set emits noise when the transmission is turned off after the end of the last program. b) When a wooden surface is ground with sandpaper noise is obtained. An impulse is a single short sound (a spike) with a time history as shown in Fig. 1.10b. The impulse has partials of all frequencies (as the noise) but only at one instant. Example 1.10. a) When you hit a nail with a hammer, an impulse is obtained at the moment of the hammer impact. b) When you hit a violin top plate with you knuckle, an impulse is obtained at the moment of impact. Violinmakers use these two kinds of sounds to test the properties of a free top plate for instance. By letting a finger slide along the top plate surface a noise sound is obtained. Depending on the acoustical properties of the plate some partials of the noise are amplified. The maker can hear a sound of certain pitch and timbre. With some experimenting the maker can learn to use the procedure as a guiding test during the shaping process. The violinmaker can also use impulsive sound. By tapping (knocking) the free top plate with a knuckle, an impulsive sound is injected (the procedure is the same as hitting the tuning fork at a table edge). The response of the plate to the injected impulse give the resonances of the plate (each corresponding to a tuning fork giving different pitch, level and time decay). Thereby the pitches, levels and decays of the resonance vibrations reflect the properties of the plate and can guide the maker towards a wanted result. The maker cans optimise the tapping sound for specific purposes. By hitting with a soft "club" (the fingertip) the low resonances vibrations are favoured (longer duration of the impulse). The impulse spectrum is also changed. A softer club gives a spectrum with weaker partials at high frequencies. In tones from musical instruments there are also impulsive and noisy sounds, which can be important for the timbre. Example 1.11. a) When the violin bow is pulled over the string, a musical tone is generated, which also contains noise. b) Noise can be heard in the beginning of the irregular time history of the guitar tone, c.f., Fig. 1.9. When a left hand finger slides along a wrapped string, noise is generated. c) In wood wind instruments such as the clarinet, air turbulence causes noise in the same way as for a microphone outdoors in wind. Hitting a key gives impulsive sound. 1.4 SUMMARY: THE SOUND In this part I have shown what a played musical tone can look like. It consists of very small and rapid variations in the atmospheric pressure. The variations are as a rule complicated. The time history of the tone varies with timbre. In addition the played tones have a beginning and an end, i.e. a starting transient and an ending transient. The spectrum of a tone shows an instantaneous


view of the tone at one particular time. The spectrum varies from one part to another of the same tone. As a rule a spectrum representation is not typical for all tones of the played instrument. 1.5 KEY WORDS: Frequency, Hz, kHz, level, dB, spectrum, partials, fundamental, time history, ms, starting transients, ending transients, noise, and impulse.


Chapter 1. Second part: THE HEARING PROCESS INTRODUCTION In the second part of the first chapter the measured properties of sound will be connected to the perception by our hearing. First the function of our hearing organ is sketched, the sensitivity and the working range. Thereafter masking phenomena and the so-called critical bands of hearing are introduced. Finally timbre and spectrum of tones are discussed. 1.6. THE EAR AND THE HEARING PROCESS

Fig 1.12. Sketch of the ear.

The ear is a very complicated microphone with a built-in spectroscope, c.f., Fig. 1.12. The outer part of the ear or the pinna is the part we in everyday language call the ear. The pinna works as an antenna, picks up the sound, leads it via the auditory canal to the eardrum or tympanic membrane (corresponding to the microphone membrane). The vibrations of the eardrum are coupled via three small bones, the ossicles, to the oval window of the cochlea, c.f. Fig. 1.13a. The vibrations of the oval window set the upper liquid filled part of the cochlea, scala vestibuli, vibrating. The vibrations of the liquid will in its turn make the basilar membrane vibrate. Muscles connected to the ossicles adjust the sensitivity of hearing, i.e. these work as an automatic volume control. By adjusting the sensitivity, the hearing is protected against strong sounds. The


sensitivity adjustment takes, however, a short time during which the hearing is "unprotected". Therefore this adjustment can not protect our hearing against strong impulsive sounds, such as the firing of a gun. For very high sound levels it is also insufficient and ear protectors must be used.

Figure 1.13. a) Sketch of the function of the outer ear (upmost picture). The cochlea has the shape of a spiral with 2 and 3/4 turns, but has been drawn straight to give a clearer picture (after Hadding and Petersson). b) The displacement of the basilar membrane at different frequencies (frames below picture, after Denes and Pinson).

The basilar membrane is the part where the mechanical vibrations are transformed into electrical nerve pulses, see Fig. 1.13a. For single tones the basilar membrane gives maximum Jansson: Acoustics for violin and guitar makers 1.18

Figure 1.14. The working range of the hearing

vibrations at special positions depending on their frequencies. Low-frequency tones give a maximum far away from the oval window and high frequency tones close to the oval window see Fig. 1.13b. At the basilar membrane there are vibration sensitive cells, so called hair cells. It is the hair cells that react to the basil vibrations and transmit the nerve pulses. If we put our hand close to a loudspeaker we feel tickles, which are registered by the brain. If we put a part of our arm close to a loudspeaker it is received by the brain as tickling at a specific position of the arm. In a similar way the brain registers the positions of the basilar membrane vibrations. Different frequencies correspond to different places, which makes the basilar membrane act as a spectroscope. THE HEARING PROCESS The ear works as a complicated microphone as has been shown. The working range of hearing covers frequencies from 16 to 16 000 Hz (16 kHz), c.f., Fig. 1.14. This is a wide frequency range corresponding to 10 octaves (visible light has a much, much wider frequency range, from 390 000 000 000 000 to 750 000 000 000 000 Hz but this is less than a one-octave range in musical language). The dynamical working range of hearing is from -5 to 120 dB. The lower limit where a tone is just noticeable is called the HEARING THRESHOLD. The hearing threshold is different for different frequencies and is at the lowest at -5 dB at 2 kHz. The upper limit is called the PAIN THRESHOLD, because here the hearing sensation is changed into a sensation of pain (if a sound results in a pain sensation, stick a finger in each ear and leave the sounding area immediately). The threshold of pain is approximately at 120 dB. The maximal working range of level from -5 to +120 dB, i.e. close to 120 dB. A digital tape recorder is needed


to cover this range. The hearing threshold varies appreciably with frequency. At 30 Hz the hearing threshold is at 60 dB. Between 500 Hz and 5 000 Hz it is approximately at 0 dB. For frequencies above 5 000 Hz the hearing threshold increases rapidly. The threshold of pain can be regarded as independent of frequency and constant at 120 dB.

Figure 1.15. Levels of equal loudness (phone-curves)

The loudness of a tone depends both on its frequency and its level. Fig. 1.15 shows the level that a tone should have to give the same loudness at different frequencies. Levels of equal loudness have been plotted starting from 0, 20, 40 etc. dB at 1 000 Hz. The figure shows, that for frequencies below 500 Hz a fairly loud sound is necessary to be heard (the hearing threshold is at a fairly high level). Above the hearing threshold the lines of equal loudness are closely spaced, i.e. our hearing is very sensitive to increased sound level at low frequencies. The hearing threshold and the sensitivity decreases up to 500 Hz. Between 500 Hz and 5 000 Hz the equal loudness lines are approximately horizontal and parallel, i.e. the hearing is equally sensitive in this range and corresponds to the volume control of an amplifier. The importance of the equal loudness lines can be evaluated in a different way. At low sound levels the sensitivity of hearing is rather different at different frequency ranges, especially at low frequencies. At higher sound levels these differences become less prominent, the equal loudness lines become more horizontal. Thus the hearing properties can be summarised in the following way. Different sound levels give different sound impressions. High sound levels favour the low frequency sounds (the bass sounds).


Figure 1.16. Masked threshold of hearing from a) a simple tone of constant frequency but varying level and b) from a tone spectrum (from Sundberg)

1.7. MASKING CRITICAL BANDS AND ROUGHNESS Different frequencies give different positions of maximum deflections of the basilar membrane (cf., Fig. 1.13b). This fact explains two fundamental hearing phenomena, MASKING and ROUGHNESS.


The masking effect is that a sound may hide or mask, another, weaker sound. In the presence of a tone the hearing threshold (for an additional tone) is partially increased from the hearing threshold (in silence, the hearing threshold of one tone only). This so-called masked hearing threshold is 10 dB below the level of the first tone for another tone close in frequency. For frequencies below the masking tone, the hearing threshold soon reaches the threshold of hearing in silence, c.f., Fig. 1.16a. For frequencies above that of the first tone, the masked threshold is considerably higher than the hearing threshold in silence. For a tone as marked with a bar at 0.4 kHz (400 Hz) it is easily seen that the masked threshold (the full line leaning downwards to the right) lies considerably above the silence threshold up to 4 kHz. Towards lower frequencies the masked threshold is only higher to 0.2 kHz (200 Hz). The condition for a second tone to be noticed is that its level is above the masked threshold (it is not sufficient that its level is above the threshold in silence). The rule of thumb is that a loud tone masks (hides) a weak tones of higher frequency but not a weak tone of a lower frequency. Thus in tests with tap and noise sounds, one should avoid exciting resonances of lower frequencies than the one of interest. Resonances excited at frequencies above are less disturbing.

Figure 1.17. Analysis bands, so called critical bands of the hearing. The width of the critical bands in relation to the frequency (from Sundberg).

A played musical tone is not a single tone but a complete spectrum of partials. Therefore the masked threshold for the played tone is fairly complicated to estimate. Each partial will have its own masked threshold, c.f., Fig. 1.16b. The complete, the effective masked threshold of the partial spectrum is the highest partial threshold within each frequency range. Only the partials that reach above the effective masked threshold contribute to the hearing sensation of the partial spectrum. From the masked partial spectrum in Fig. 1.16b one can see that only the first, second, third, eighth and ninth partials contribute to the hearing sensation - the other partials are masked.


CRITICAL BANDS AND ROUGHNESS The second phenomenon that can be related to the deflection curves for the basilar membrane in Fig. 1.12b, is the roughness of a sound. If two partials are close in frequency, the two partials cannot be heard as separate. One can only hear them together, which gives the sound a sensation of roughness. The roughness sensation appears as the partials are closer spaced in frequency than a CRITICAL BAND of hearing. The widths at the different frequencies for the critical bands are shown in Fig. 1.17. For frequencies below 500 Hz the critical bands are 100 Hz wide but above they are 1/3 of an octave (approximately 25 %) wide.

Figure 1.18. Spectra of single tones and the single tones added.


Figure 1.19. Tone spectra with four and six partials respectively - the first spectrum sounds smooth, the second sounds rough.

Figure 1.20. Spectra of different vowels (only envelopes, c.f. Fig. 1.9 are drawn, from Fant)


Figure 1.21. Spectrum for a spoken vowel (o in often. dashed line) and sung (with full line, only envelopes drawn from Sundberg).

A simple listening exercise can be prepared as shown in Fig. 1.18. The single partials 1 through 4 give tones with little character and without roughness. If the partials are added a "tone" is heard, still without much character. In the spectrum of the four partials added one can still separate each of the four partials. Example 1.19. What will happen if we add many partials, can all partials still be distinguished? A little consideration gives that when the critical bandwidth and the frequency separation between partials match, then it should be difficult to separate the partials and roughness will occur. A few calculations show that the bandwidths and the partial separations for musical tones occur up to the fifth partial. Thus we should expect that a played tone with four partials has no roughness but a played tones with six has some roughness. A listening exercise as sketched by Fig. 1.19 will confirm that with six partials a sense of roughness is obtained. The critical bands of hearing can thus explain the sensation of roughness. Furthermore the musical dissonance can partly be explained by roughness phenomena. 1.8 TIMBRE AND SPECTRUM The partials of a spectrum results in a character called timbre (often represented as spectrum envelopes). We learn early to recognise differences in timbre of tonal sounds. The vowels of speech are as a matter of fact tonal sounds of different timbre. The timbre of a specific vowel is adjusted in an elegant way. In the vowel in "word" the partial levels follows the full line (i.e. the envelope) in the upper frame of Fig. 1.20. The same vowel is heard also if the fundamental frequency but not the envelope is changed. In the vowel in "cat" the partial levels (the envelope) follow the dashed line. In this case the vowel timbre is independent of the fundamental frequency too. The spectral difference between the two vowels is that the frequency of the leftmost hump has been shifted. By shifting the second leftmost hump the vowel "o" in shoot is shifted to the vowel "e” in three. The spectral humps are also used to give the sung vowels their timbre and character (not the fundamental frequency), c.f., Fig. 1.21. The male opera singer trains his voice to give strong partials in the range of maximum sensitivity of the hearing. It gives his voice a more brilliant timbre and it comes through the orchestra accompaniment better. This spectrum hump is generally called the singer's formant (a similar spectrum maxim is generally favourable for musical instruments). Jansson: Acoustics for violin and guitar makers 1.25

As a rule of thumb it can be said that a strong first partial (the fundamental) is necessary for a full sounding timbre. Many and strong high partials give the timbre a more brilliant but also more rough character. In everyday speech noise and impulse sounds are also included. The fricative consonants such as s, f and sh are filtered noise sounds. Stop consonants such as k, p and t are impulse excited sounds. The character of the fricatives and the stop sounds are formed by different adjustments of the speech organs. The consonant sounds give different timbres and sometimes also a feeling of pitch - similar to the tap and noise sounds used by the violinmaker. It must be pointed out that the sketched ear function is far from complete. Only most fundamental function has been introduced. 1.9. SUMMARY: THE HEARING PROCESS In the preceding part some of the properties of the hearing process have been presented. The working range of the hearing has been described. Examples on how we perceive sounds have been presented by using common speech sounds. Perception "units" such as pitch, loudness and masking have been introduced. The presented information on hearing could be used to predict well-known facts further, now not included. It is well known that playing louder, from mf to ff, is not only a turning up of the volume control, it corresponds mainly to an increase of the treble control. However, to show how this is done is outside the scope of the present compendium. 1.10. KEY WORDS: Measurable units - Frequency, level, spectrum, frequency and level of partials Perception - units - Pitch, loudness, timbre, hearing threshold, masking, and roughness.




Erik Jansson

Chapter II: Resonance and Resonators

Fourth edition 2002 http://www.speech.kth.se/music/acviguit4/part2.pdf


Sound and hearing

Chapter II


Chapter III

Sound and the room

Chapter IV


Chapter V


Chapter VI


Chapter VII


Chapter VIII


Chapter IX



ACOUSTICS FOR VIOLIN AND GUITAR MAKERS Chapter 2 – Fundamentals of Acoustics RESONANCE AND RESONATORS Part 1: RESONANCE 2.1. Resonance 2.2. Vibration sensitivity 2.3. The mechanical and acoustical measures of the resonator 2.4. Summary 2.5. Key words Part 2: RESONATORS 2.6. The hole-volume resonator 2.7. Complex resonators 2.8. Mesurements of resonances in bars, plates and shells 2.9. Summary 2.10. Key words


Chapter 2. FUNDAMENTALS OF ACOUSTICS - RESONANCE AND RESONATORS First part: RESONANCE INTRODUCTION

In chapter 1, I presented the fundamental properties of sound and how these properties can be measured. Fundamental hearing sensations were connected to measurable sound properties. In this, the second chapter the concept of RESONANCE and of RESONATORS will be introduced. Resonators are fundamental building blocks of the sound generating systems such as the violin and the guitar. The chapter starts with introducing the resonance and how a resonance is described. Thereafter vibration sensitivity (technical term mobility) of resonances is discussed and finally how mechanical and acoustical measures of a resonator are related. 2.1. RESONANCE A RESONATOR or a system of resonators gives one or several RESONANCES. The blown tone of a bottle is the result of a resonance and the bottle is the resonator. A practical property to measure a resonator is its frequency response (vibration sensitivity, techn term mobility). The response curve gives a picture of each resonance, i.e. its FREQUENCY (the peak frequency), its resonance LEVEL response (the peak height for the selected driving) and its BANDWIDTH (the peak width). These properties are related to the mechanical properties of the resonator; the MASS (weight), STIFFNESS (spring) and FRICTION. Often a resonator consists of a vibrating area, such as a violin top plate. Then the distribution of the vibrations are important. Positions of maximal vibrations are called ANTINODES and the positions of no vibrations are called NODES. Furthermore the decay (the reverberance) of a resonance can be important. The relations between these different measures and properties will be explained in this chapter. Let us first look at the guitar tone displayed in Fig. 1.9. The time history of the tone is smooth except the sharp wiggles soon after the attack. The tone is not made up by a single partial but a spectrum of partials, the levels of which slowly change. The frequency separation between the partials is constant and equals the frequency of the fundamental (the first partial). It is the level of the partials that give the guitar tone its character, the timbre. But why do we obtain a decaying spectrum of partials at these very frequencies? Let us cautiously move towards the answer by looking at a vibrating string. AN EXAMPLE OF RESONANCE: THE VIBRATING STRING

A large scale picture of the vibrating string can be presented by means of a long rubber band with one end fixed to a wall. The rubber band is held stretched by a hand at its free end. If the free end is slowly moved up and down, the whole band moves up and down in time with the hand. If the beat tempo, the frequency, is increased the string tends to swing out more between the fixed end and the hand end. At a specific frequency the band will swing up and down in a large bend between wall and hand, but still only requires small hand movements. The vibrations at the bends are maximum. If the beat frequency is increased further the bend will diminish and it is difficult to make the rubber band vibrate visibly. With still further increase of the beat frequency the rubber band will divide itself into two bends. At a specific beat frequency the Jansson: Acoustics for violin and guitar makers 2.3

vibrations at the bends are maximum. With further increased beat frequency the bends diminish and thereafter the rubber band will divide itself into three bends, etc. Let us repeat the same experiment with better control and use a small electrical vibrator attached to a rubber band with its other end fixed to a wall (a similar demonstration can be made using a string of a musical instrument). An electrical tone generator (oscillator) provides the electrical driving signal. The magnitude and the frequency of the driving signal are easy to adjust with high accuracy. The vibrator is replacing the hand. At low frequencies the rubber band again moves in phase with the motion of the vibrator, but with increasing frequency it starts bending out more and more in the middle. At a specific frequency, the first resonant frequency the bend becomes maximum. The position of maximum motion is called an ANTINODE. With further increase of the frequency the bend will first vanish and thereafter at the second resonant frequency, two maxima of motion will come out and the rubber band does not move in the middle. We have now two ANTINODES and in between a NODE. If we continue to increase the frequency the two antinodes vanish and at a higher frequency we will find three bends, antinodes, and between the antinodes, nodes. Further increase of the frequency will show vibrations with four, five, six etc. antinodes with three, four, five etc. nodes. We shall find that the maxima of vibrations are at at fixed frequencies. By measuring the frequencies we will find that the frequencies are 2, 3, 4, 5 times the frequency with maximum vibrations with only one bend. The experiment has shown that certain things happen at specific frequencies (the resonant frequencies) - the rubber band vibrations become maximum at specific positions (antinodes) and are zero at other positions (nodes). THE ORIGIN OF A RESONANCE

What is behind these peculiarities and what has that to do with the guitar tone? It is RESONANCES. The vibrating rubber band and also the vibrating string have resonances. But before I explain what is happening to the guitar string, let me explain the origin of a resonance, how its properties are described and are measured. The RESONANCE is an acoustical building brick of fundamental importance. To describe a resonance we need to answer the following questions: At what frequency is a resonance, the resonant frequency? How limited is this resonance in frequency, the bandwidth? How easily are the vibrations to excite, the vibration sensitivity at resonant frequency? How are the vibrations distributed, the modes of vibration (or operating deflection shapes)? A simple resonator is a small ball attached to a string of rubber bands, c.f., Fig. 2.1a. When the finger is slowly moved up and down the ball moves in the same way up and down (in time, in phase with the hand). If the beating frequency of the finger is increased the up and down motion of the ball increases more and more. At a certain beating frequency the ball motion becomes maximum. Further increase of the frequency will make the ball motion smaller and finally at high frequencies only the finger will move.


Figure 2.1. A simple resonance: a) a simple resonator - a ball (C) attached to a rubber band (B) and hung on a finger (A), and b) vibration sensitivity curve of the resonance - resonance frequency (RF), peak height (RL) and bandwidth (B measured 3 dB below the peak maximum).

The size of the ball vibrations (upand down motion) follow the vibration sensitivity curve plotted in Fig. 2.1b. The curve shows low vibration sensitivity at low and high frequencies. In between there is a peak, a resonance peak giving the resonant frequency (the peak frequency), the peak height giving the vibration sensitivity at the resonance and the peak width giving the bandwidth. The bandwidth of resonances vary much from a fraction of a Hz for the string to tenths of Hz for body resonances. The resonator consists of a mass (weight) - the ball - and a spring - the rubber band. The turbulence in the air surrounding the ball tries to brake its motion and provides friction. If the stiffness of the rubber band is increased the resonant frequency is increased. If the mass (weight) of the ball is increased the resonant frequency is decreased. If the friction (against the air) is increased by a thin plate under the ball, the width of the resonance peak is increased. It is difficult to move a finger up and down maintaining the same beat size and beat frequency. Therefore an arrangement as shown in Fig. 2.2 may be used. A loudspeaker gives a strong tone. By turning a knob on a tone generator the tone frequency is changed without changing the strength of the tone. The plate with attached rubber band is hung over the loudspeaker and the strong loudspeaker tone sets the plate into vibration. The frequency is slowly changed and at the same time the magnitude of the plate vibration is measured. The vibration size is plotted for each frequency and thereby the vibration sensitivity of the ball-rubber band resonator is obtained (in the real world this very experiment may be hard to conduct though – a suitable combination loudspeaker, plate and rubberband may be hard to find). Jansson: Acoustics for violin and guitar makers 2.5

Figure 2.2. Principle set-up for measuring the vibration sensitivity.

Figure 2.3. Specific vibration sensitivity and stiffness - bending with a force between a thick and a thin plank (applies to low frequencies).

2.2. VIBRATION SENSITIVITY VIBRATION SENSITIVITY ( techn term mobility) is a measure of how easily vibrations can be started and maintained. If one holds a swing, and pulls and pushes it to and fro with the same force then the swing starts to swing, i.e., it starts to vibrate. How much the swing vibrates is a measure of the vibration sensitivity. More mathematically expressed the is the ratio between vibration velocity and the driving force. Mathematical formulas describing the vibration sensitiivty (mobility) are given in the appendix of this chapter. At a specific push and pull frequency the swing vibration become maximum, i.e., the swing is vibrating at its resonant frequency. The SPECIFIC VIBRATION SENSITIVITY (specific mobility) is a measure on how easily a resonator (a swing, a string or a violin back plate) can be set into vibration, when its resonant properties are fully eliminated. Jansson: Acoustics for violin and guitar makers 2.6

Example 2.1. What is the specific vibration sensitivity of and how much does a plank bend, which is ten times stiffer compared to a second plank? The picture is drawn for static forces, but applies also to vibration forces below the resonant frequency of the system. Here only stiffness is involved, which is only applicable for low frequencies and the static case. The same force is applied in the two cases. By using formulas (see appendix) one finds that the stiffer plank is bent 1/10 of the second one. The difference in specific vibration sensitivity is squareroot of 1/10, which is approximately 1/3 and corresponds to -10 dB. Rule of thumb - 10 times stiffer results in a 10 dB lower level, c.f., Fig. 2.3. The relation is also true for vibration forces at frequencies well below the resonant frequency of the system.

Figure 2.4. Specific vibration sensitivity and mass (weight) - displacement by a force between a big motorboat and a small rubber boat (applicable at high frequencies, c.f., example 2.2).

Example 2.2. What is the specific vibration sensitivity and how much less does the ten times heavier motorboat move than the smaller rubberboat if pushed? The picture is drawn for a static force but applies to vibration forces above the resonant frequency of the system. A little calculation shows that the heavier boat obtains a motion of 1/10 of the smaller one. Furthermore the specific vibration sensitivity becomes the squareroot of 1/10, i.e. 1/3 corresponding to 10 dB lower specific vibration sensitivity for the heavy boat car compared to the small one. The relation is also true for vibration forces at frequencies well above the resonant frequency of the system


VIBRATION SENSITIVITY

FREQUENCY

Fig 2.5 The specific vibration sensitivity, the dashed line, can be low but resonance properties of the instrument body may increase the total vibration sensitivity, the full line, to give large interferance with the string motion and cause a “wolf tone”.

The results of examples 2.1 and 2.2 are "reasonable" - a stiff spring or a heavy mass (weight) is more difficult to set into vibration, and the specific mobilities are lower. Complications are added in the range of a resonant frequency. The resonant properties are superimposed, c.f., the formula predict larger vibrations close to the resonant frequency. The specific vibration sensitivity also gives an average (geometric mean) measure of how two vibrating systems may cooperate. If the specific vibration sensitivity is large (a string for instance) in one of the systems and small in the other (the body of a musical instrument), then the two systems will work fairly independently of each other, c.f. the strings on a violin or a guitar. If the specific vibration sensitivity with the superimposed resonance properties (c.f. Fig 2.5) are about the same of a string and of an instrument body, then the two systems will effect each other, c.f., the wolf tone of the cello.

Figure 2.6 A two-resonator system as example on vibration sensitivity and specific vibration sensitivity. Nailing a small block of wood to a bending board by means of a small hammer with bending handle and an axe as extra masssupport.


In common life one meets these practical problems such as nailing in weak boards, c.f., Fig. 2.6. The thin board is set into vibration, it will vibrate at its resonant frequency. The vibration sensitivity of the board can be decreased by adding mass with an axe, which greatly simplifies the nailing. The force from the hammer is transmitted by friction between the nail and the block to the board. The mass of the small hammer head together with the stiffness of the handle gives a second resonance. This resoance is much damped by the holding hand. In somewhat similar ways can the function of musical instruments be modelled as seen by the eyes of the physicist, although the musical instruments are much more complicated. The string of the guitar replaces the small hammer, the bridge replaces the block and nail, and the body replaces the board. A resonance is determined by the spring and the mass. At a specific frequency, the resonant frequency, the spring effect and the mass effect are of the same size. They counteract and the vibration sensitivity becomes maximimum. The effects of very large friction is also calculated. 2.3. THE MECHANICAL AND ACOUSTICAL MEASURES OF THE RESONATOR One very interesting question is: How much are the acoustical measures changed if one of the mechanical measures is changed? The changes we shall study correspond to doubling of the stiffness, mass and friction, c.f., Fig. 2.7. In the case shown in Fig 2.7a only the mass is changed (for simplicity two rubber bands and two weights are used as standard). If the mass is doubled (four weights) the resonance frequency decreases half an octave (from 500 to 350 Hz, i.e. 1/1.4 times). If the mass is halved (one weight) the resonance frequency is increased half an octave (from 500 to 700 Hz, i.e. 1.4 times). The peak heights (the levels) are approximately the same and the levels of these curves are the same at low frequencies (100 Hz for instance). At high frequencies the level (the vibration sensitivity) is larger for the smaller mass. If only the stiffness of our resonator is changed, then the following happens, see Fig. 2.7b. For doubled stiffness (four rubber bands) the resonance frequency increases half an octave and for half the stiffness ( one rubber band) the resonance frequency decreases half an octave. For high frequencies the level (the vibration sensitivity) is the same but for low frequencies it is higher for the smaller stiffness. The peak heights are little influenced by the stiffness changes. Thus we have found that the effect on the resonance frequency by a doubling of the mass is equivalent to halving the stiffness and vice versa. There is, however, one large difference. The doubling of the mass decreases the vibration sensitivity at high frequencies only, while halving of the stiffness increases the vibration sensitivity at low frequencies only. Finally if the friction is doubled and halved as indicated in Fig. 2.7c the peak height is decreased and increased by 6 dB, respectively. In addition the effect of very large friction, bandwidth 500 Hz being equal to the resonant frequency is calculated. For such large losses (large bandwidth) no clear resonance peak is obtained.


Suggested experiment: Start with three similar bars, use one as reference, add and take away a small weight from the second. Add and take away a stiffening rib from the third. Tap and listen - is the tap tone pitch (frequency) changing as one should expect. Further experiments can be made by thinning one of the bars, c.f., chapter 5.

Figure 2.7. Vibration sensitivity with different mechanical properties.


Figure 2.8. Vibration sensitivity for a vibrating system consisting of a single resonance (upper frame) with the resonant frequency RF, the height of the resonance peak RL, and the bandwidth B, and for a vibrating system consisting of several resonances (lower frame) where each resonance has it own resonance frequency, height and bandwidth.

ACOUSTICAL PROPERTIES OF A RESONANCE. MULTIRESONATOR SYSTEMS.

The properties of a resonance can be described by its vibration sensitivity curve i.e. a curve of the vibration sensitivity for every frequency. Usually such a curve is called a resonance curve or a frequency response curve, c.f., Fig. 2.8a. The frequency at the peak is the RESONANT FREQUENCY. The height of the peak is the PEAK LEVEL The width of the peak 3 dB below maximum is the BANDWIDTH (often the ratio of the resonant frequency divided by the bandwidth is given, the so called Q-factor). The acoustical properties of a resonance are Jansson: Acoustics for violin and guitar makers 2.11

described by the three measures RESONANT FREQUENCY, PEAK LEVEL and BANDWIDTH (Q-factor). The mechanical properties of a resonator are: 1. STIFFNESS such as the stiffness of a spring or the springiness of a rubber band. 2. MASS such as the weight of a ball. 3. FRICTION such as the air friction when the ball is moving. The different mechanical properties influence the resonance curve differently as has been shown in Fig. 2.7. More mass (weight) gives lower resonance frequency and lower levels at high frequencies, and higher stiffness a high resonance frequency and lower levels at low frequencies. The mass and the stiffness do not influence the bandwidth. The friction does however. Little friction gives a high level and narrow bandwidth and a high friction gives a low level and a large bandwidth. The friction influences only the resonance curve in the neighbourhood of the resonance frequency. To build up the vibrations at a resonance demands a specific time, which gives a specific starting characteristic. In the same way it takes some time for the resonance vibrations to vanish, which gives an ending characteristic. Both the starting and the ending time are set by the bandwidth of the resonance peak. The resonance curve for a single resonator can be measured and looks as in Fig. 2.8a, and gives a measure of the resonant frequency, the bandwidths and the level. The resonance curve for a multiple resonator system such as the guitar can look like the resonance curve 2.8b. Each peak corresponds to a resonance and has its own resonant frequency, bandwidth, and level as each resonance is made up its own stiffness, mass and friction. In general all three measures are different for the different resonances.

2.4 SUMMARY: RESONANCE In this first part of this chapter the concept of resonance has been introduced. The mechanical properties of a resonance system such as mass (weight), stiffness (springiness) and friction have been related to the acoustical properties of the resonance system, such as resonant frequency, bandwidth and level. The duration for starting and ending characteristics (transients) is determined by the bandwidth of the resonance. Furthermore a method to measure the acoustical properties of a resonance system has been sketched. 2.5 KEY WORDS: Resonance (eigenmode), resonant frequency, bandwidth and level, vibration sensitivity, starting time (start duration), ending time (end duration), mass (weight), stiffness (springiness), and friction.


CHAPTER 2. Second part: RESONATORS INTRODUCTION

In this part properties of resonators used in string instruments will be introduced. First simple resonators as the hole-volume (the Helmholtz) resonator and the string will be presented. Thereafter properties of complex resonators as the bar and the plate. Finally a simple way to measure the properties of plate resonantors as well as further complications as shape and arching will be introduced. 2.6. SIMPLE RESONATORS THE HOLE-VOLUME RESONATOR (RESONANT FREQUENCY) The simplest resonator encountered in musical instruments is an enclosed air volume with a hole. Often this resonance is called the Helmholtz resonance The resonator is called a Helmholtz resonator. The air volume works as a spring (stiffness) and the mass (the weight) is the air plug in the sound hole. Except for the constructional difference it works as the ballrubber band resonator, c.f. Fig. 2.1. A large volume and a hole with small cross section gives an air tone of low frequency. A simple example of a hole-volume resonator is a bottle, which is set into vibration by blowing over the neck opening. The air volume in the bottle works as the spring and the air plug in the bottle neck is the mass, see Fig. 2.9. When one gently blows over the neck opening the air plug is set into vibration and co-operates with the spring of the air volume in such a way that the air plug will vibrate in and out. These vibrations give the "bottle tone" one hears. In the musical instruments such as the guitar and the violin the inner air volume of the sound box is the spring and the air plugs in the sound holes are the vibrating masses. The resonant tone is often called the air tone.

spring mass

Figure 2.9. A simple resonator - hole-volume resonator (the Helmholtz resonator) - a bottle and its mechanical analogue.

The resonance frequency is determined by (area A/volume V see appendix), i.e. large sound holes and a small corpus volume give a high resonant frequency while small holes and a large Jansson: Acoustics for violin and guitar makers 2.13

volume give a low resonant frequency (formulas see appendix). The shape of the air volumes and sound holes are generally so complex that an accurate calculation of the air resonance frequency is difficult. For practical purposes it is simplest to measure the resonant frequency. The formulas suggests how changes of the size of volume and hole area alter the resonance frequency. THE STRING RESONATOR (FREQUENCIES - NODES - ANTINODES). Let us as the next example study the resonances of a stretched string. The vibrations may look like the upper part of Fig. 2.10. We have seen this earlier, see section 2.1. We have also seen that the string may vibrate as in the lower parts, i.e. the string has not one resonance but several. The resonances occur at different frequencies and the vibration modes look different for each resonance. The vibration mode must now be included in our description of a resonance. For all resonances sketched there are no vibrations at the end fastenings. In the first resonance (topmost) the string has its maximum of vibration in the middle of the string, i.e. there is an antinode in the middle. In the second resonance (the second topmost part) there are no vibrations in the middle, i.e. there is a node in the middle of the string. The maximum vibrations are one quarter string length from the ends, i.e. there are two antinodes at these positions. In the third resonance (the next lower part) there are two positions of no motion (excluding the fastening points), i.e.nodes and three positions of maximal vibrations, i.e. antinodes. In the fourth mode there are three interior positions of no vibration and four positions of maximum vibrations. At resonance the string divides itself into "subparts" by a number of nodes with antinodes in between.

Figure 2.10. Resonances of a stretched string.

We have thus made a simple summary of a string's way to vibrate at resonance, i.e. the maxima of vibrations, ANTINODES, and the minima of vibration, NODES, for the four Jansson: Acoustics for violin and guitar makers 2.14

lowest resonances. Often the vibration modes of an object are described by giving the nodes (nodal lines). Another way, which may be more informative is to use the antinodes. In Fig. 2.10. we see that the first resonance has one antinode, the second two, the third three…. the 17:th resonance 17 antinodes, etc.. Note that the number of nodes also increases as the resonance number increases. The vibration modes are thus fully determined by the fastening and the length of the string. The frequency is determined by the string length (l), the string mass (weight) and the string tension. Relations between resonance frequencies, positions of nodes and antinodes are presented in Table 2.1.

Figure 2.11. The vibrations of the second string resonance - the motion seen by the eye (top plot) and vibration shapes at consecutive instants (c.f., snapshots).

Table 2.1. The vibrating string - resonance frequencies, positions of nodes and antinodes. Jansson: Acoustics for violin and guitar makers 2.15

If the frequency of resonance no then the frequency of resoance no

1 2 3

is f1 f2 = 2 x f1 f3 = 3 x f1 .... 17 f17 = 17 x f1 i.e. the frequency of the n:th resonance equals n times the first one. If the length of the string is L then the nodes are at positions for resonance no 1 position 0 2 0 L/2 3 0 L/3 2L/3 ......... 17 0 L/17 2L/17 ……........ The antinodes are halfway between the nodes i.e. for resonance no 1 position L/2 2 L/4 3L/4 3 L/6 3L/6 5L/6 ........ 17 L/34, 3L/34, 5L/34, ........ …………33L/34

etc L L L L

etc.

etc.

The vibration shapes sketched in Fig. 2.9 are the extreme positions of the string during vibration.. The extreme positions of the second resonance looks like the upmost frame in Fig. 2.11. But the vibration shape is changing all the time. In the lower frames of Fig. 2.11, the real vibration shapes are sketched for consecutive instants (snapshots of string shapes). A resonance occurs at its own fixed frequency, when the resonator is driven (excited) at just that very frequency. At the pluck of a guitar string or at the hammer blow of the piano string, all the string resonances are set into vibration at once. The different string resonances will behave differently depending on their bandwidths for instance. Note the difference in the starting of the vibrations, plucking means pulling the string aside and then be left to vibrate freely (decay). In the piano the string is given a short push and is thereafter left to vibrate (decay) freely. 2.7 COMPLEX RESONATORS We have so far studied fairly simple resonators, the properties of which can be described by exact mathematical formulas. We shall now continue with more complex resonators for which no exact mathematical formulas can be derived.


Figure 2.12. Vibration modes for resonances of a bar with both ends free. THE BAR RESONATOR (FREQUENCIES, NODES, ANTINODES, VIBRATION MODES AND TAPPING TESTS).

Let us so turn our attention to the bar resonator and its resonances. We shall treat the free bar, i.e. a bar not fastened at the ends or elsewhere. The bar has many resonances. The vibration modes are sketched in Fig. 2.12. In the first resonance, the bar has two nodes, not at the ends but 22.4 % of the bar length from its ends. The vibrations are maximum at the ends and in the middle (about equal size), i.e. these are the positions of antinodes. For the second resonance there are three nodes at 13.2 %, from the ends and in the middle. The antinodes are at the ends and between the nodes. The following resonances have an increasing number of nodes, and the antinodes are in between and at the ends. Note that there are simple relations between the order number of the resonance and its number of nodes and antinodes. Resonant frequencies and positions of nodes are given in Table 2.2. Note that the relations between resonant frequencies and nodal positions are not as simple as for the string resonator. Resonant frequencies, position of nodes and position of antinodes can, however, be described with accurate (but complicated) mathematical formulas. The lowest resonant frequency is determined by the stiffness, by the mass (weight) and by the length. A little work with the formulas shows that the resonant frequency is proportional to the thickness of the bar (and thus to its mass too). What practical use follows from this theoretical backing? One good thing is that resonance frequencies can be obtained with tap tone testing. To do so one needs to know where to expect nodes and antinodes. For a bar it is simple. The length of the bar determines the position of nodes.


Table 2.2. Resonant frequencies and node positions of a free bar.

Resonance Frequency Nodes at no 1 f1 22.4% 77.6% 2 f2=2.76 x f1 (≈5/3)2) 13.2% 50% 86.8% 3 f3=5.40 x f1 (≈7/3)2 9.4% 35.6% 64.4% 90.6% 4 f4=8.93 x f1 (≈9/3)2 7.3% 27.7% 50% 72.3% 92.7% The antinodes are at the ends and in the middle between the nodal lines Take for instance resonance no 1 for a 50 cm long bar. Hold the bar lightly at 22.4% from one end, i.e. 50 x (22.4 / 100) = 11.2 cm from the end. The bar will not vibrate at this position for the first resonance and this is a good position for holding. Tap at the antinode halfway between the ends, and the optimum way of listening to resonance no 1 has been chosen. For further instructions, see paragraph 2.8 Measurement of resonances in bars and plates. The bandwidth of a resonance determines how long the tone can be heard. The pitch and the duration of the tap tone give measures of the frequency and the bandwidth of the resonance. In principle a scratching at the antinode can also be used. For a sufficiently sharp resonance, a weak tone with the pitch corresponding to that of the resonance is heard. A sharper resonance will give a better defined "scratch tone". 2.8. MEASUREMENT OF RESONANCES IN BARS, PLATES, AND SHELLS TAPPING TESTS FOR RESONANCES (EIGENTONES). The resonances of a bar (and a plate) can be sought and identified by applying the following rules: 1. HOLD lightly at a NODE. One should always hold at a node! 2. TAP at an ANTINODE. 3. To suppress a disturbing resonance, tap at the node of the disturbing resonance. 4. The author has found it convenient to hold the bar (or the plate) between the first finger and thumb of the left hand as lightly as possible. The left arm is held over the head with the bar hanging freely just outside the right ear. The tapping is done with the right hand, a finger tip, a knuckle or a nail (the higher resonance frequency the harder "hammer head" should be used). The tapping point is just outside the ear and it is shifted relative to the bar by moving the left arm up or down. The holding point is shifted by letting the bar slide a small amount between the first finger and thumb. It is often suitable to lean slightly forwards to ensure that the bar is hanging freely (nothing is allowed to touch the bar but the left hand first finger and thumb). The best way is, however, obtained with the so called Chladni method and a loudspeaker, c.f. Fig. 2.2. A loud tone of the loudspeaker sets the bar into vibration and small particles, such as coarse saw dust is sprinkled over the bar. The nodal lines are found in the following way: 1. the loudspeaker is placed under an expected antinode 2. the supports in the form of small pieces of foam plastic are placed under two expected nodal lines 3. the saw dust is sprinkled over the bar 4. the frequency of the loudspeaker tone is adjusted to a frequency at which the particles have the largest motion 5. the positions are sought where the saw dust collects, i.e., the positions of nodal lines Jansson: Acoustics for violin and guitar makers 2.18

6. the measurements are optimised by repositioning the supports at nodal lines and the loudspeaker under an antinode. The frequency of maximum vibration equals the resonance frequency and the lines where the saw dust collects are nodes (nodal lines). The positions where the saw dust starts moving are the antinodes. A coarse measure of the vibration sensitivity peak level can be obtained by watching how much the volume control of the amplifier must be turned up to start moving the saw dust at antinodes (a low volume means a strong resonance). THE PLATE RESONATOR (FREQUENCIES AND NODAL LINES). A plate has two main directions in which it may bend and therefore resonances are found in two directions. The nodal line patterns can for a free plate in principle be ordered as

Figure 2.13. Nodal lines (broken lines) for a rectangular plate with free edges (frequencies for a Sitka Spruce plate 110 g, 362 x 210 x 3.7 mm).

in Fig. 2.13. In the first row the resonances with zero vertical nodal lines are placed (two resonances with two and three horizontal nodal lines are sketched - compare the bar Fig. 2.12). In the second row the resonance with one vertical nodal line is placed, etc. In the first line there are resonances with zero horizontal nodal lines placed, in the second line with one Jansson: Acoustics for violin and guitar makers 2.19

horizontal nodal line, in the third line those with two horizontal nodal lines, etc. . Higher resonances have nodal lines which are combinations of the simplest ones. The antinodes are centred between the nodes in the inner part and along the edges. The resonance frequencies are determined by the thickness, length, width, and mass of the plate.

Figure 2.14. Vibration modes for a plate fastened along its edge compared to those of a string. Nodal lines along the edges and inside (broken-dotted lines). Lines of equal vibration are also plotted (full lines, bent and closed).

In Fig. 2.13 three measured resonances are marked together with their nodal lines and frequencies. A complication should be mentioned. When a plate is bent in one direction it will by itself bend in other ways at the same time. When the plate is bent down at the short edges of Fig. 2.13 the longer edges will bend slightly upwards. This means that the nodal line patterns Jansson: Acoustics for violin and guitar makers 2.20

sketched in Fig. 2.13 and 2.14 look somewhat different in the real world, especially the high frequency ones. THE INFLUENCE OF FASTENING AND THE SHAPE OF THE PLATES

The plates are, however, often fastened along the edges, and such a rectangular plate will give vibration modes as sketched in Fig. 2.14. The magnitude of the vibrations have been marked with lines of equal vibration. Observe the close analogy with the string in two directions. A1≈460Hz

Figure 2.15. (left) Vibration modes of a violin shaped rubber membrane - the first seven modes with nodal lines and antinodes indicated with a plus or minus sign , and (right) air modes of the violin body, A1 being the first..

The frequency and not only the vibration modes are much dependant on the fastening along the edges (or at the ends). Say that the first resonance frequency of a bar with free ends is 100 Hz. If the ends are clamped the same first resonance frequency is still obtained, but the nodal lines are moved to the ends. If the ends are fastened with hinges the frequency is lowered to 44 Hz. In the violin and the guitar the plate fastening is be somewhere between clamped and hinged.


The shape of the fastening also influences frequencies, nodal lines and antinodes. The influence of the nodal lines for a rubber membrane stretched over a set of ribs are shown in Fig. 2.15. The rubber membrane was set into vibration by a small vibrator and small cork fragments marked lines of no motion at the different resonance frequencies, i.e. nodal lines. It can be seen that the "waist" of the violin divides the membrane in two partly independent areas. The same kind of nodal patterns are found in the upper and lower part, c.f., 73 Hz and 90 Hz, and 106 and 127 Hz, respectively. The resonances of the aircavity show, similar patterns, see fig. 2.14 (right) . Note that resonance 1 (A1) is close to the main resonances of a violin. ARCHING (SHELLS) A violin top is not a flat plate but a slightly arched plate. Effects of the arching can be tried in a simple way by means of a playing card, c.f. Fig. 2.16. The card is held arched between the thumb and pointer. If the distance between the two fingers is varied the card will bend in and out in the middle. If one with the second hand increases or decreases the arch height of the card one feels with the first hand that the fingers on the sides of the card move together. The arching couples motions perpendicular to the card surface to motions in the plane of the card, especially along the edges. An arched plate is in physics called a shell and not a plate. To test the influence of arching a series of simple experiments were made. A rectangular spruce plate, 3 mm thick, 215 mm wide and 290 mm long (fibres along the plate and annular rings perpendicular to the surface) was selected. First the plate was bent by means of a string-loop across the plate at each of the nodal lines marked in Fig 2.13 (160 Hz). A wedge was pushed between each string and the plate giving a 6 mm arch (H, in Fig 2.16) and the the resonant frequency increased 50 % (increased stiffness) compared to flat. By locking the motion of the edges with a light clamp at each nodal line the resonant frequency increased another 50 %. Secondly the plate was bent by two string-loops along the nodal lines marked in Fig 2.13 (100Hz) and two wedges. The resonant frequency increased 60 %. Experiments with locking the edges by clamps introduced a new complication. The static clamping force gave a large influence on the dynamical properties. The experiments show that the arching has a large influence. In the top and back plates of the violin the arching is larger than in the experimental plate.

Figure 2.16. In an arched plate (arch height H) vibrations ∆y result also in vibrations ∆x.


2.9. SUMMARY:RESONATORS In this part four kinds of resonators have been described: the hole-volume resonator (the Helmholtz resonator), the stretched string, the bar and the plate. The resonant frequency of the hole-volume resonator is determined by the size of hole and volume - large hole and small volume give high resonant frequency. The resonant frequencies of the string are determined by the length, the mass (the weight) and the tension of the string - a short string with high tension and small mass results in high resonant frequencies. The frequency of the second, third, etc. resonances are two, three etc. times the frequency of the first resonance. The resonant frequencies of the bar and the plate are determined by the length, the width, thickness and mass. A short bar, and a short and narrow plate give high resonant frequencies. A thick and light bar and plate will have high resonant frequencies. In general there are no simple relations between the different resonant frequencies of the bar and plate, respectively. The effects of different fastenings have been introduced. Furthermore the vibration at resonance have been described by nodal lines and antinodes. Finally a simple way has been described on how to test resonant frequencies of bars and plates by tapping and listening. Furthermore a somewhat more advanced method by Chladni patterns, giving vibration patters as well as resonant frequencies. For the violin top and back the arching increases the stiffness of a plate in perpendicular to the arching. 2.10 KEYWORDS: Resonator, resonance, resonant frequency, bandwidth, decay time, vibration modes (vibration patterns), nodes, antinodes, hole-volume resonator, string resonator, bar resonator, plate resonator, free ends, hinged ends and clamped ends.


2.11 APPENDIX The resonant frequency of the hole-volume resonator is in principle A / V where A is area of hole and V is volume of cavity. The length with end corrections leff are considerably larger than the length l for thin-walled musical instrument, and resonant frequency = (c / 2π ) A /(leff V ) .

Formulas and some numerical values on the relation between acoustics properties and mechanical properties of a resonator In mathematical language the mobility (vibration sensitivity) for a resonance (vibration velocity/ vibration force) = = (2π f / SM ) × (1/ ( B / f o ) 2 + ( f / f o − f o / f ) 2 = =(specific mobility) × (resonance properties) (c.f. Fig. 2.5) In the formulas S is the spring of the resonator, M is its mass and B its bandwidth, f 0 is the resonance frequency = (1/ 2π ) S

M

and f is the frequency of evaluation.

Examples: 1a) If the mass is doubled or halved, what happens to the resonance frequency? A little calculation shows that the resonance frequency is lowered or increased approximately 6 semitone steps (from 500 to 350 or 500 to 700 Hz). 1b) If the stiffness is doubled or halved, what happens to the resonance frequency? A little calculation shows that the the results are opposite to those in 1a, i.e. the resonance frequency is lowered or increased approximately 6 semitone steps (500 to 700 Hz or 500 to 350 Hz). 2a) What is the time for 60 dB decay (the reverberation time) for the bandwidths 500, 25, 12.5 and 6.25 Hz? With some calculations it can be shown to be 0.0002, 0.088, 0.176, and 0.352 seconds, respectively. 2b) How much higher is the resonance peak higher level than the specific vibration sensitivity at 500 Hz for the three bandwidths? Again a little calculation shows the peak level is 0, 26, 32, and 38 dB above the specific vibration sensitivity, respectively.



ACOUSTICS FOR VIOLIN AND GUITAR MAKERS Erik Jansson

Chapter III: Sound and the Room



Sound and hearing

Chapter II


Chapter III

Sound and the room

Chapter IV


Chapter V


Chapter VI


Chapter VII


Chapter VIII


Chapter IX



ACOUSTICS FOR VIOLIN AND GUITAR MAKERS Chapter 3 – Fundamentals of Acoustics SOUND AND THE ROOM Part 1: FUNDAMENTAL ROOM ACOUSTICS 3.1 Propagation of sound 3.2 Sound radiation 3.3 Reception of sound 3.4. Summary 3.5. Key words Part 2: THE SOUND IN THE ROOM 3.6. The influence of the room 3.7 Properties for rooms 3.8 Acoustics of the podium 3.9. Summary 3.10. Key words

3.11 APPENDIX. Relations between sound pressure level and sound pressure.

Jansson: Acoustics for the violin and guitarmaker 3.2

Chapter 3. FUNDAMENTALS OF ACOUSTICS - SOUND AND THE ROOM First part: FUNDAMENTAL ROOM ACOUSTICS INTRODUCTION The sound is spread in a room roughly as sketched in Fig. 3.1. It is spread in all directions from the violin and reaches the ear of the player first. Soon thereafter the earliest reflected "sound ray" reaches the ear of the player. The direct "sound ray" from the violin first reaches the listener, followed soon after by reflected sound rays. It is the reflected sound that gives the characteristic sound of the room.

Figure 3.1. Spread of sound in a room.

In this part we shall talk about the sound and the room, i.e. the propagation of sound, musical instruments as sound sources, the ear as a sound receiver. We start with the Propagation of Sound, continues with Sound Radiation and concludes with Reception of Sound. 3.1 PROPAGATION OF SOUND (TIME AND LEVEL) TIME (distance and sound velocity) The sound travels through the air with a velocity of slightly less than 345 m/s (the sound velocity is somewhat dependant on temperature). The fact that time is needed for the sound to spread means that the listeners in the auditorium do not hear the tones at the same time as they are played. Furthermore they reach the different listeners at slightly different times. The played tones also reach the different members of the orchestra at different times. How much the tones are delayed by the distances is illustrated by the following examples. Fundamental formula: time x velocity = distance


Example 3.1. At a park concert one listener is sitting 5 m from the solo singer and another listener 100 m away on the lawn. How much later do the two listeners hear the tones than the singer? time x velocity = distance, i.e., time = distance / velocity The distance from mouth to ear of the singer is approx. 15 cm = 0.15 m i.e. time = (0.15 m) / (345 m/s) = 0.000 43 s. Thus the singer hears himself with a very short delay, in practice no delay. For the close listener, time = (5 m) / (345 m/s) = 0.015 s, and for the distant listener, time = (100 m) / (345 m/s) = 0.29 s. The example shows that fairly short distances gives measurable time delays from source to listener. Table 3.1. Time table for some distances time = distance /sound velocity (sound velocity 345 m/s) 1/1000 s = 1 ms (millisecond). DISTANCE m 1 2 5 10 15 20 30 40 50 100

TIME ms 3 6 14 29 43 58 87 116 145 i.e., distance increased 4 times 290 takes 4 times longer time

LEVEL (distance) The sound level decreases with the distance from the sound source. We shall treat the simple case of levels for spherical radiation in this part. The simplest case is equally strong radiation in all directions (spherical waves). In the appendix to this chapter there is a table, which can be used for "translation" between sound level and sound pressure. Formula: sound pressure = a constant / distance The formula says that the sound pressure decreases to half when the distance is doubled, which means that the sound level decreases 6 dB.


Example 3.2. Assume that a violin gives a sound level of 85 dB at the distance of 1 m. What is the level at the distances of 5 and 100 m respectively (assume spherical waves)? a) 5 m: The sound pressure amplitude at 5 m distance is 1/5 of the amplitude at 1 m. This can be translated by the appendix table, which shows that the ratio (1:5.0) corresponds to a level decrease of 14 dB. The sound level at 5 m is thus 85 - 14 = 71 dB sound level (SPL). The violin tones should be weak but easily heard. b) 100 m: The sound pressure amplitude at a distance of 100 m is 1/100 of that at 1 m, i.e., translated from 1:100 to -40 dB. The sound level at the distance 100 m becomes 85 - 40 = 45 dB. The violin tones will only be heard under quiet conditions. Example 3.3. If one stands at a distance of 10 m, how much closer should one walk to make the violin sound twice as loud? A level increase of 10 dB makes the sound twice as loud (rule of thumb). A 10 dB increase corresponds in the appendix table to an increase of the amplitude ratio of 3.2:1, i.e., the distance should be decreased to 10/3.2, which is close to 3 m. Note that twice as strong corresponds to a distance decrease to one third (not one half). Table 3.2. The level change at different distances relative to 1 m. Distance m 1 2 5 10 15 20 30 40 50 100

Level change dB -0 -6 -14 -20 -24 -26 -30 i.e., 3 times longer distance gives -32 10 dB lower level, and -34 10 times longer distance gives -40 20 dB lower level

For further calculations (translations) the reader is referred to the appendix. 3.2 SOUND RADIATION (spherical radiation, directional radiation, reflection, and size-frequency limit DG) Sound radiation is as a rule very complicated, but some simple rules of thumb can be made. The radiation properties are determined by two characteristics of the sound source: its size (area or diameter) and the frequency of the sound. It is the relation between the two measures that largely determines the radiation properties. Let us introduce a size/frequency measure DG to be able to simplify the properties of sound sources. The measure DG equals 100/frequency. For a diameter of the sound source


smaller than DG the sound is radiated with equal strength in all directions. A sound source with a diameter larger than DG gives a more directed and complex radiation. When sound hits an object, parts of the sound sneak round it, parts are spread around the object and parts are reflected. Again the same factors are involved, the size of the object and the frequency of the sound. At low frequencies most sound sneaks around the object and little is spread or reflected. With increasing frequency more and more of the sound is spread and reflected. At high frequencies most sound is reflected. We can again use the measure DG. A DG larger than 100/frequency gives mainly reflection and smaller DG gives little disturbance. Example 3.4. Can the head substantially influence the timbre of the singer's voice and if so from what frequency? At 1000 Hz DG = 100/1000 m = 10 cm. 500 100/500 20 cm The diameter of the head is slightly less than 20 cm, i.e., we should expect directional radiation above 500 Hz. Measurements support the prediction, see Fig. 3.2. The radiation is between 5 to 15 dB lower behind the head than in front for the range of appropriate frequencies, and thus the head influences the timbre.

Figure 3.2. Directional properties of the radiation around the human head for speech a) in the horizontal plane and b) in the vertical plane. The arrow marks the forward direction (the nose direction and the shaded areas level differences of less than 5 dB, after Kuttruff).

An object of certain size in comparison with the sound frequency will influence the sound radiation. This occurs when the size of the object D > DG (= 100/frequency). In table 3.3 some of these size-frequency limits are given.


Table 3.3. Size-frequency limit DG for some frequencies Frequency Hz 20 50 100 200 500 1 000 2 000 5 000 10 000 20 000 Example 3.5.

Diameter m 5 at low frequencies even large 2 objects give little influence 1 0.5 0.2 0.1 0.05 0.02 0.01 at high frequencies even small 0.005 objects give large influence

Do the traditional string instruments radiate equally strongly in all directions at the fundamental of the lowest string? A little arithmetic on the tuning, DG, and the instruments sizes gives: Instrument Violin Viola Cello Bass Guitar

Lowest fundamental DG G 196 Hz 0.56 m C 131 0.83 C 65.4 1.7 E 41.2 2.7 E 82.4 1.3

corpus length 0.36 m 0.43 0.75 1.15 0.48

i.e., the corpus length is much smaller than DG and the instruments radiate the lowest fundamental equally strong in all directions. Let us show examples on radiation of musical instruments and start with the simplest, the brass instruments. For the brass instruments and especially the trombone the following radiation properties are found. The sound is radiated via the bell in a wide angle (a wide lobe) at low frequencies, cf., Fig. 3.3. The angle (the lobe) becomes narrower with increasing frequency. The violin radiates somewhat similarly, in all directions for low frequencies, but the radiation is split into more complicated patterns for higher frequencies. A tendency for increased sound radiation from the top plate is found, cf., Fig. 3.4. The differences in radiation make spectra obtained in different directions look different. To demonstrate this, played music has been analysed with filters corresponding to the so called critical bands of hearing, cf., Fig. 3.5. It is easily seen that the spectra are different in the different directions, especially at high frequencies. The spectrum corresponding to radiation in the neck direction gives the highest levels, Jansson: Acoustics for the violin and guitarmaker 3.7

540 Hz

2000 – 6000 Hz

750 - 1100 Hz

7000 – 10 000 Hz

Figure 3.3. Main radiation directions for the trombone (to -3 dB limits, after Meyer).

200-500 Hz

1000-1250 Hz

550-700 Hz

1500 Hz

800 Hz

2000-5000 Hz

Figure 3.4. Main radiation directions for the violin (after Meyer).


Figure 3.5. Average spectra in different directions for the violin. In the lower parts spectra obtained in two directions are shown and in the upper part the difference between the two spectra together with the directions are plotted.

Figure 3.6. Sound level difference between the ear directed towards and away from the sound source (after Meyer and Neumannn, phones close to dB in the present application).

In the directions perpendicularly to the neck, i.e., the top turned towards or away from the auditorium, there are large level differences in the frequency range 2 - 3.5 kHz (the range of the singer's formant and of the violin BH-peak). Note that the last two cases correspond to the difference between first and second violin positions when seated to left and to the right of the conductor, respectively. When the first violins are placed to the left and the second violins to the right of the conductor the first violins turn the top plate towards the auditorium but the second violins the back plate. With this arrangement the first violins will give a more brilliant sound than the second violins. If the player does not want to be heard, he should turn his back towards the auditorium Jansson: Acoustics for the violin and guitarmaker 3.9

(note the obvious connection between vision and hearing). A practical interpretation is the following. A soloist, who really wants to be heard should aim with the violin neck at his listeners.

3.3. RECEPTION OF SOUND (head with ears, directional and room hearing) THE HEAD The ears (the pinnae) are on opposite sides of the head and slightly behind the middle. This arrangement gives a microphone system that can recognise directions by means of "spacial filtering". We should expect a marked influence on the sound by the head. Measurements also show that there are differences in sound level between the left and the right ears starting from 500 Hz. The level differences depend on the direction of the sound, cf., Fig. 3.6.

Figure 3.7. Average spectra in the left (full line) and right (dashed-dotted line) ear of the player and one ear of a nearby listener (dashed line). In the upper part are differences between left and right ears of player (full line) plotted together with the differences between left ear of the player and the listener ear (dashed line).

It is possible with simple experiments to prove that sounds with high frequency components are much more affected than low frequency ones. The s-sound contains high frequency components. If one person says an s-sound while turning around, the sound is much louder when facing the listener. A low frequency sound like an u-sound (u-sound as in do) will be much less dependant on direction (the experiments should be made outdoors with no reflecting walls nearby). Example 3.6. At what frequency should we expect the head to give directional effects for hearing? How long should it take for sound to travel from the left to the right ear? Jansson: Acoustics for the violin and guitarmaker 3.10

Say that the human head has a diameter of 15 cm. A little calculation shows that we should expect different levels in the left and the right ear from approximately 700 Hz. Say that the distance between the ears equals half the circumference of the head, i.e., 3.14 x radius = 25 cm (approximately) and with the sound velocity of 345 m/s a maximum time difference of 0.7 ms can be obtained. The player listens to his own instrument while playing and therefore a natural question is: How does the violin sound "look" in the two ears of the violin player? Measurements with small microphones inserted the ears of a player and the ears of a listener are shown in Fig. 3.7. The diagram shows that there are large differences between the left and right ears of the player, the left being approximately 10 dB above at 1.3 kHz (a replay with headphones sounds natural for a violin player but seems silent in the right ear of a non-player). The level for the listener was about 5 dB lower than the left ear of the player, somewhat more at high frequencies. THE PINNAE – DIRECTIONAL ANTENNAS The pinna also gives directional effects. Its diameter is about 4 cm, corresponding to a frequency limit for the directional effects of 3 kHz, which corresponds to the first resonance of the auditory canal. Above this frequency the pinna starts working as a direction sensitive antenna, and gives us possibilities to separate sound arriving from the front, back and above. THE BRAIN – THE COMPUTER Connected to the ears we have a powerful computer, the brain. By means of this computer we can store and compare sound impressions. The differences of arrival times can be used and are of great importance for the hearing impression. Time differences larger than 1 ms makes it possible to separate sound sources such as loudspeakers at the cinema. A time differences of 17 to 35 ms between your own instrument and that of your fellow players is optimal for playing together. A time difference between the direct sound and the first reflection of 50 to 80 ms, results in disturbing reverberation and echoes. The audio signals of music and speech include very fast changes within a hundredth of a second, while the reverberation time of a room is about 1 second. Experiments show that for the subjectively experienced reverberation the initial part is the most important, cf., Fig. 3.8. A listener uses only the first 160 ms and not a full second to experience the reverberation of a room. He or she can also use the first 10 dB decay but not the 60 dB decay of the classical reverberation time. Another important effect is the so-called precedence effect, which predicts that the reflected sound close after the direct sound is not disturbing. The reflected sound may even be 10 dB stronger than the direct sound and still gives little disturbance. As a Jansson: Acoustics for the violin and guitarmaker 3.11

rule of thumb one might say that delayed and added reflections within 50 ms give little disturbance. The effect may be used in large auditoriums for sound reinforcement with loudspeakers. A somewhat stronger loudspeaker signal than the direct sound increases the perceived sound level. If the loudspeaker signal is somewhat delayed relative to the direct sound, the impression is that the sound derives from the speaker, i.e., the directional impression is not distorted.

Figure 3.8. Experienced (subjective) reverberation time and initial reverberation time (160 ms reverberation) translated into traditional reverberation time (-60 dB, after Atal et al).

From the information presented on the precedence effect the following conclusions can be drawn. The initial part of the sound (the direct sound) gives the direction of the sound source. The reflected sounds arriving later increase the loudness but disturb the sound impression if arriving too late. The time ranges for different impressions are approximately: sound within the first ms is perceived as direct sound, time delays within 35 ms increase the loudness impression but time delays of 50 ms or more give disturbances. The phenomenon that the properties of sound vary with the direction relative to the listener's head may cause difficulties for a listener. The direction to a known person speaking (sound source) in the vertical plane can be localised by the hearing, c.f. Fig. 3.9, but if the listener does not recognise the voice of the speaker is it difficult to decide the direction to the speaker. Thus to juDGe safely the timbre of a sound source its direction in the vertical plane must be known.


100 %

50 % 50 %

Figure 3.9. Judgement of direction for a known voice (full, thick lines) and unknown voices (broken, thick lines, after Cremer and Müller).

3.4 SUMMARY: The sound spreads out with a velocity of 345 m/s. At low frequencies it is spread out with equal strength in all directions and the sound pressure level decreases 6 dB for a doubling of the distance. For increasing frequency the sound radiation becomes more directional. If the frequency is high (sound velocity divided by frequency is small in relation to the size of an object, the head for instance), then the sound is affected. The sound of a speaker is louder in front of the speaker than to the sides and back. A similar influence guides us to hear from which direction a sound comes. The pinnas of the ear act as direction sensitive antennas, increasing the direction sensitivity further. The brain acts as a computer, and adds the sound coming from the same direction if close in time but gives a disturbed sound impression if parts of the sound is delayed too much in time. The sound radiation of the musical instruments is approximately equally strong in all directions for low frequencies, but becomes more and more directional for higher frequencies. 3.5 KEY WORDS: Sound source (sound transmitter), sound receiver, frequency, size, non-directional and directional radiation, and hearing in room and directional hearing.


Chapter 3. Second part: THE SOUND IN THE ROOM INTRODUCTION The information presented below is mainly textbook knowledge supplemented with experience from a research project "Speech and Music in Rooms". We start with the influence of the room on the sound (reverberation time), summarises properties of typical rooms, and ends with acoustics of the concert podium. 3.4. THE INFLUENCE OF THE ROOM (reflection, reverberation time and reverberation distance).

absorption, resonances,

Figure 3.10. Time courses of a spoken message as registered by ear properties modelled by computer, a) at an ear of the speaker, and b) at an ear of a listener at a distance of 8 m in a lecture hall with poor acoustics.

In our project work a "computer ear" was developed. The computer ear contained known major properties of the human ear, the so called critical bands of hearing. The analysis of a spoken sentence is shown in Fig. 3.10. The two upmost wavy lines correspond to the sound level in the 2000 Hz critical band of the hearing, the two middle ones the sound level in the 500 Hz critical band. The lowest two wavy lines show the sound level without any filtering. The analysis was made of the same speech signal but obtained at the left ear of the speaker (it is shown in the upper lines, a, in the three


frames) and a listening position 8 m out in the lecture hall (shown by the lower lines, b, in the frames). The influence of the room is large. The sound level curves at the speaker have clear peaks and dips. At the listener the peaks are smoothed out

Figure 3.11. Direct sound (full line) and reflected sound (dashed lines, after Kuttruff).

Figure 3.12. Time course for reflections in a rectangular room of an impulsive sound (after Barron)

and the dips at 0.2 and 0.5 seconds are completely erased. Without the room (outdoors) the sound level curves should have looked the same at the speaker and at the listener, Jansson: Acoustics for the violin and guitarmaker 3.15

although the levels would have been lower at the listener. The room gives a higher sound level but smears out the sound. It can also be seen that the level of the disturbing reverberant sound increases all the time. The cause for this influence of the room will be discussed in the following. The walls of the room (including ceiling and floor) reflect the sound rays back into the room. By using ray tracing techniques, the magnitude of the sound reaching a listener is the combination of the direct sound and the sound reflected from the walls can be predicted as shown in Fig. 3.11. The sound distribution as a function of time, direct sound and reflections can also be calculated and is shown schematically in Fig. 3.12 for an impulsive sound.

Figure 3.13. Sound pulses in a lecture hall with poor acoustics (from Lundin). Radiated sound pulsse upper trace to the left and responses (reflectograms) in the lower trace. Low frequency (500 Hz) in the left figure and high frequency (2000 Hz) in the right (20 ms per division).

The different sound rays from source to listener are of different lengths - a longer path makes the sound come later and weaker. This means that the direct sound ray reaches the listener first and thereafter the different reflected sound rays. With increasing time the reflected rays come closer and closer and they smear out into a reverberant sound. Examples on how emitted sound pulses may be changed by a room to the listener are shown in Fig. 3.13. We see here that initially there is a weak pulse, thereafter it becomes stronger. Later there is a very complicated time history. The figure gives an example of how a room with poor acoustic properties could look. There is very little sound absorption in the air. At each wall reflection a considerable part of the sound energy is absorbed. The reverberation of a room is altered by selecting suitable wall materials. Large rooms give few reflections and thus long reverberation times. Small rooms give many reflections and short reverberation time even if the walls are hard. Thick and soft wall materials give strong absorption (thick carpets for instance) while hard materials such as glass plates give little absorption. Typical data for some materials and a person are given in Table 3.4.


Table 3.4. Absorption factor for different materials (from Kinsler and Frey). Material Brick wall Wooden panel Light draperies Heavy draperies Glass Plaster Concrete floor Wood floor One person

at 125 Hz at 500 Hz at 2000 Hz 0.02 0.03 0.05 0.10 0.10 0.08 0.04 0.11 0.30 0.10 0.50 0.82 0.04 0.05 0.05 0.04 0.05 0.05 0.01 0.02 0.02 0.06 0.06 0.06 0.5 m2 with absorption 1.0

A room with a large volume and little wall absorption gives strong resonances at very low frequencies. At higher frequencies the number of resonances become very large. This gives the room a very uneven frequency response curve. The unevenness of the frequency response curve is set by the random addition of several resonances and not by single resonances.

Figure 3.14. Sound level in a room as function a) of the frequency at a fixed position, and b) of the position at a fixed frequency (1 kHz after Kuttruff).


Figure 3.15. a) definition of reverberation time and b) reverberation distance of a room, horizontal line marks level of reverberant sound and the bent line marks level of the direct sound. The dashed vertical line marks the so called reverberation distance, i.e. the distance from the sound source at which the levels of the direct sound and the reverberation sound are equal.

Resonance density, pulse density and room volume are related in the following ways: A room with a small volume has few resonances within a specific frequency range but a large number of pulses arrive within a specific time interval (low resonance density but high pulse density). A room of large volume has a high resonance density but a low pulse density. In the large room the first reflected sound ray arrives much later than the direct sound ray. The high density of resonances of a room means that the sound levels of a room are very different from one frequency to another and from one position to another, see Fig. 3.14. This dependence of position makes it difficult to measure a representative curve for a guitar or a violin in an ordinary room. The sound is spread in the room in a very complicated way and a simple description of the sound is not possible. A statistical description must be used. The most common measures are the reverberation time and the reverberation distance (reverberation radius). The oldest and most common measure is the reverberation time, i.e., the time for a 60 dB level decrease after a sound has been shut off, c.f. Fig. 3.15a. The reverberation distance is the distance from the sound source where the direct sound and the reverberation sound are equally strong, c.f. Fig. 3.15b. The figure indicates that a 10 dB level increase is about the maximum that can be gained by going very close to a sound source (the directional properties of the sound source should be included in the reverberation distance, i.e. it is much longer in front of a trumpet than beside the same instrument). Large volume and lightly damped wall areas (the sum of floor, ceiling and walls) result in a long reverberation time and a short reverberation distance. Thus one can adjust both the reverberation time and the reverberation distance by using suitable


absorption materials on the walls. Outside the reverberation distance the sound level is independent of the distance to the sound source. In general the reverberation time and the reverberation distance are different for different frequencies. The reverberation time is typically long and the reverberation distance short at low frequencies and vice versa at high frequencies, c.f. Fig. 3.16. For different types of rooms a practice has been developed to choose different reverberation times, see Fig. 3.17. The number of persons in a room influence the reverberation time and distance considerably.

Figure 3.16 Measured a) reverberation time and b) reverberation distance for a lecture hall (E5 from Lundin)

The sound level in a room is determined by the strength of the source, the volume of the room and the absorption at the walls. Outside the reverberation distance the sound level is constant. Within the reverberation distance the sound level increases towards the sound source (c.f. Fig. 3.15b). Examples of the strength (sound power) from different sound sources are given in Table 3.5. For living rooms and class rooms a "silent" level below 35 dB is recommended and for the concert hall below 30 dB is recommended in Sweden. Table 3.5. Sound power for some musical sound sources and sound pressure level at a distance of 1 m (after Sundberg). The corresponding sound level (*) has been calculated at a distance of 1 m although this is rather unrealistic for the symphony orchestra. Bass singer Clarinet Trumpet Trombone Symphony orchestra (max)

0.03 W 0.05 0.3 6 60

94 dB at 1 m 98 104 117 127 (*)


Figure 3.17. Optimum reverberation times for different rooms (1000 feet3 equals 92.9 m3 from Beranek).

3.7 PROPERTIES FOR ROOMS (reverberation, resonances, pulse density etc) Let us define three rooms to get a grasp of the properties introduced in this chapter. The three rooms are a) a typical living room with a floor area 3 x 5 m2 and a wall height of 2.4 m, b) a small lecture hall (class room) 6 x 8 m2 floor and 3 m height, and c) a large concert hall 25 x 35 m2 and 13 m, respectively. Example 3.7 What are the reverberation times of the three rooms empty? Typical values of the reverberation time to aim at in the design are 0.5, 0.8 and 1.5 s, respectively for our three rooms. The formulas for calculating reverberation time show that the volume and the total wall area including floor and ceiling are the main parameters. For the three rooms the volumes are 36 m3, 150 m3, and 11000 m3, respectively. The total wall areas are 70 m2, 200 m2, and 3300 m2 Starting from the given numbers the reverberation time for the three rooms can be calculated. First the wall absorption is calculated. Let us assume that all walls have wooden panel with an absorption coefficient of α = 0.1, c.f. Table 3.4. The reverberation time T is calculated with the formula T = 0.16 x V /(S x α), where V is the room volume and S is the wall area. A little calculation shows that the reverberation times could be estimated to be 0.8 s, 1.2 s, and 5.3 s, respectively. Thus the small room can be expected to have a reasonable reverberation time, a little too long for the medium sized room, and a catastrophe for the concert hall. For the large rooms it is necessary to have wall absorption materials.


Example 3.8 What are the reverberation distances in the three rooms? The reverberation distance can also be calculated but by a somewhat more complicated formula (square root of the volume divided by 300 times the reverberation time). The reverberation distances can thus be estimated to 0.4 m, 0.6 m, and 2.6 m, respectively. This means that one must sit very close to the sound source to obtain dominating direct sound (typical values for real rooms are 0.5 m, 0.8 m, and 5 m). Example 3.9 SPL in reverberant field The sound level in the room is another interesting measure to calculate. We use a bass singer and the three rooms with reverberation times of 0.5, 0.8 and 1.5 s, respectively. A little calculation shows that the sound levels (outside the reverberance distance) become 102 dB SPL, 98 dB SPL, and 86 dB SPL, respectively. Thus it seems unwise to sing loudly in a small room since 65 dB SPL is sufficient for good communication. For comparison it should be mentioned that the sound levels for the guitar and violin are considerably lower than for the bass singer. For the guitar measurements have shown that the levels vary between 35 and 70 dB SPL and for the violin between 45 and 60 dB (Spelda). Thus the pp-levels seem a bit low to be heard well, which also is in agreement with common experience. Example 3.10 What are the effects of changes in the three rooms on reverberation time, reverberation distance and sound level? What is the influence of a thick carpet on the floor? Table 3.4 gives α = 0.3 as a reasonable absorption and a little calculation shows that the reverberation times become 0.6 s, 0.8 s, and 3.4 s, respectively, i.e. a thick carpet may give considerable influence on the room acoustics. What is the influence of persons in the rooms? Let us assume that the three rooms typically include 3 persons, 30 persons, and 1400 persons, respectively. One person corresponds to 0.5 m2 absorption area, which gives the reverberation times 0.7 s, 0.7 s, and 1.7 s, respectively, i.e. we find that a room full with people gives a large influence on the room acoustics (The sound level decreases a moderate 5 dB). Example 3.11 What are the relations between resonances and pulses in a room? Pulses and resonances? The sound rays, their reflections, the delay and number of pulses within a short time interval is a way to describe a room. For the three rooms the time delay between the direct sound ray and the first reflected sound ray, and the number sound rays reflected past a listener within the first 1/10 second can be calculated for the three rooms. It is assumed that the first reflection is against the ceiling (thick carpet on the floor) and that we are interested in the conditions for a singer 1.80 m tall. Jansson: Acoustics for the violin and guitarmaker 3.21

Using the given measures shows that the first reflected sound ray will arrive 3 ms, 7 ms and 68 ms, respectively after the direct sound ray. Within the first 0.1 seconds 4800 pulses, 1150 pulses, and 14 pulses, respectively will have passed. We can thus see that in the small room the sound comes back quickly and many times within a short time. In the large hall the opposite conditions occur. It takes long time for the first reflected sound ray to arrive and there are few pulses within the first 0.1 seconds. Another way to describe a room is by means of its resonances, the frequencies of the resonances and the density of resonances. A little calculation shows that the lowest resonant frequency is at 35 Hz, 22 Hz, and 5 Hz, respectively, Below 100 Hz the rooms have 7, 50 and 3700 resonances respectively. The numbers clearly show that the small room has a high first resonance frequency and few resonances while the concert hall has a very low first resonance frequency and a large number of resonances below 100 Hz. With increased damping of the room the pulses and resonances become less clear but the numbers remain the same. It has been shown that when the number of resonances are densely spaced compared to their bandwidths, then the room properties are determined by the density of the resonances and the reverberation time of the room. There exists a lower limit fg above which the room properties are so determined. Let us assume that the reverberation times for the three rooms are 0.5 seconds, 0.8 seconds and 1.5 seconds respectively. By using the formula for fg we find that fg equals 224 Hz, 139 Hz and 21 Hz, respectively. Thus the frequency limit is below the important frequency range for sound communication. 3.8 ACOUSTICS OF THE PODIUM (time delays, distance to walls and sound levels) The sound from the instruments of the fellow musicians is very important to be able to play together. Experiments have shown that the travel time for the sound from the fellow musician's instrument should if possible be within 0.03 s. The sound travels with a velocity of 345 m/s, which corresponds to 10 m in 0.03 s. The musicians of an orchestra thus should sit within a distance of 10 m from each others. For the symphony orchestra it is not possible to meet such a condition . Also the sounds that are reflected by reflectors lowered from the ceiling should give a travelling distance of less than 10 m. Some examples from Swedish investigations on acoustics for the symphony orchestra are presented in the following. Example 3.12. How long time does it take for the sound to travel between the players in the Stockholm Concert Hall?


The travelling time for the sound can easily be calculated from the size of the podium ( time = (distance m)/ (345 m/s) ), c.f. Example 3.1 (or estimated from Table 3.1). The podium is approximately 17 m wide, 12 m deep and has reflectors 7 m above. The width measure gives a travelling time of 0.05 s, the depth 0.035 s and 0.06 s along the diagonal. The sound path via the ceiling reflectors varies from 0.02 to 0.07 s. The distances to the conductor position are more favourable: 0.025 s to the side walls, 0.04 s to the farthest corner and a maximum of 0.05 s via the ceiling reflectors. This means that the sound paths and the corresponding delays between extreme positions are too long, but are more reasonable for a central position. The example 3.12 shows that fairly short distances give measurable time delays from source to listener. As a rule of thumb one may say that time differences of 0.01 s give noticeable sound effects. The position of the instrument groups in the orchestra can thus be most important for the players. The reverberation time is the most traditional measure of the acoustics of a room. Long reverberation time gives loud but non distinct sound. The reverberation of a sound disturbs the sound starting later. The reverberation time is set by the sound absorption at walls, ceilings and floors. On the podium the musicians of the orchestra should provide most of the absorption. From the reverberation time the reverberation distance also can be calculated, i.e., the distance within which the direct sound is stronger than the reverberation sound. Within this distance the sound level increases the closer one sits to the sound source. Example 3.13. What is the reverberation distance in the Stockholm Concert Hall? The reverberation time for the concert hall is approximately 2 s (below 2 kHz), which gives a reverberation distance of 5 m. This means that the direct sound dominates within 5 m. The musical instruments radiate the sound with different strengths in different directions, especially at high frequencies. The bell of the brass instruments act as a directional antenna and give a stronger sound straight ahead. Such a property affects the reverberation distance. For the trombone the reverberation distance should increase from approximately 7 m at 500 Hz to 20 m at 2500 Hz. A similar increase in reverberation distance should be expected for the trumpet from 900 Hz to 3500 Hz. This means that for high partial tones the "direct sound" will dominate "straight ahead" over the whole podium, while for the low partial tones the direct sound will dominate over a shorter distance. As a part of a series of experiments with the Stockholm Philharmonic Orchestra microphones were placed close to musicians with loud instruments: at the French horns before the timpanis (position A), at violas in front of oboes (position B), at bassoons in front of trumpets (position C) and at the second violins besides the piccolo (position D), Jansson: Acoustics for the violin and guitarmaker 3.23

see Fig. 3.18. The recorded music shows that the piccolo is very dominant, and that even the brass is difficult to hear "through" the orchestra at fff level.

Figure 3.18. Position of instruments in the symphony orchestra and measurement microphones A, B, C, and D (slagverk=percussion, Pk=timpanies, cor= French horns, trä = woodwinds).

Figure 3.19. Example of the time course of the sound level (level - vertical scale and time - horizontal).

The sound level in the symphony orchestra varies much, see Fig. 3.19. The results of an investigation (Jansson and Karlsson) gave the following average levels. A small orchestra with a soloist has a typical average level of 75 dB(A), a large part of the symphonic music give average levels of 85 dB(A), but the "heavy" music gives 90 dB(A) and 95 dB(A) in "normally" and "heavily exposed" orchestral positions, respectively (dB(A) is the sound level with a filter that should approximate the sensitivity of the ear). One can conclude that the average sound level of the symphony music is likely to be Jansson: Acoustics for the violin and guitarmaker 3.24

about 85 dB(A), but sometimes a little higher. The level from position to position in the orchestra is normally within 5 dB. The peak levels found were below 125 dB. This means that the orchestra members are exposed to sound levels that constitute a risk for hearing loss (when industrial noise limit criteria are applied). The conclusions from the evaluation of two large investigations (Karlsson et al, and Axelsson and Lindgren) were, however, the following: 1) The sound levels are such that there may be a risk for hearing losses, especially at "heavily exposed" positions within the symphony orchestra. 2) Audiograms for the musicians show, however, that 75 % of them have normal hearing. Among the remaining 25 % there are tendencies to worse hearing levels. No proof was found that some orchestral positions are more dangerous than others. 3.9 SUMMARY: THE SOUND IN THE ROOM The propagation of sound in rooms can be described by means of sound rays in analogy with light rays in a room with mirrors on the walls. The sound ray directly from the instrument to the listener, the direct sound, gives the directional perception. The sound reflected off the walls increases the sound level outside a specific distance from the instrument, the reverberation distance. The reflected sound also gives the character of the room including the reverberation time. A listener should select a position to give direct sound (he or she should see the musicians) at a suitable distance to add the right amount of the room character. Small rooms can give disturbing resonances at low frequencies and large room disturbing echoes. The ratio room volume to total absorbing wall area (including also ceiling and floor) is the most important "acoustical" measures. This ratio gives the reverberation time, reverberation distance and the sound level in the room. 3.10 KEY WORDS: Sound rays, direct sound, reverberation sound, reverberation time, reverberation distance, absorption factor.


3.11 APPENDIX. RELATIONS BETWEEN SOUND PRESSURE LEVEL AND SOUND PRESSURE Generally the strength of a sound is given in sound pressure level, dB. But in many cases the strength is given in sound pressure. The relation between sound pressure P and the sound pressure level L is given by the relation L = 20 log ( P / Po ), where Po is the reference for sound pressure = 0.000 02 Pa ( Pa is Pascal = N/m2 ). Differences in sound pressure level corresponds to ratios in sound pressure. To convert differences of sound pressure level to ratios of sound pressure (and vice versa) the following transformation table has been calculated Table 3.9: Relations between sound pressure level and sound pressure

Level difference dB 0 + 1 (-1) + 2 (-2) + 3 etc +4 +5 +6 +7 +8 +9 +10 Level difference dB + 20 (-20) + 40 (-40) + 60 etc + 80 +100 +120

Sound pressure ratio 1:1 1.1:1 (1:1.1) 1.3:1 (1:1.3) 1.4:1 etc 1.6:1 1.8:1 2.0:1 2.2:1 2.5:1 2.8:1 3.2:1

Level difference dB

Sound pressure ratio

+11 (-11) +12 (-12) +13 etc +14 +15 +16 +17 +18 +19 +20

3.5:1 (1:3.5) 4.0:1 (1:4.0) 4.5:1 etc 5.0:1 5.6:1 6.3:1 7.1:1 7.9:1 8.9:1 10:1

Sound pressure ratio 10:1 (1:10) 100:1 (1:100) 1 000:1 (1:1000)etc 10 000:1 100 000:1 1 000 000:1

Example 3.20: Which sound pressure corresponds to 67 dB sound pressure level?

+67 dB = 60 + 7 dB i.e. (1 000:1) x (2.2:1) = 2 200:1 and 2 200 x 0.000 02 = 0.044 Pa Example 3.21: Which sound pressure corresponds to -26 dB sound pressure level?

-26 dB = -20 - 6 dB i.e. (1:10) x (1:2) = 1:20 and 0.000 02 x (1:20) = 0.000 001 Pa Example 3.22: Which sound pressure level corresponds to the sound pressure 3.6 Pa?

3.6 Pa i.e. (3.6:0.000 02) = 180 000 = 1.8 x (100 000:1) and 5 + 100 dB = 105 dB



ACOUSTICS FOR VIOLIN AND GUITAR MAKERS Erik Jansson

Chapter IV: Properties of the Violin and the Guitar String



Sound and hearing

Chapter II


Chapter III

Sound and the room

Chapter IV


Chapter V


Chapter VI


Chapter VII


Chapter VIII


Chapter IX



ACOUSTICS FOR VIOLIN AND GUITAR MAKERS Chapter 4 – Applied Acoustics PROPERTIES OF THE VIOLIN AND THE GUITAR STRING Part 1: FUNDAMENTALS OF STRINGS 4.1. Fundamental string theory 4.2. Homogeneous strings 4.3. Wound strings 4.4. Summary 4.5. Key words Part 2: PLAYED STRINGS AND SCALES 4.6. Real strings 4.7. Played string in principle 4.8. Gesture of the bow and scales. 4.9. Summary 4.10. Key words 4.11. APPENDIX: Relations frequency and tone position


Chapter 4. APPLIED INSTRUMENT ACOUSTICS - PROPERTIES OF THE VIOLIN AND THE GUITAR STRING First part: FUNDAMENTALS OF STRINGS INTRODUCTION In the first part of this chapter the acoustical fundamentals of the string is presented. First fundamental string theory is introduced, resonances, vibration sensitivity and their relations to mechanical properties of the string. Thereafter the properties of string materials are given and finally the effects of winding on a string. 4.1 FUNDAMENTAL STRING THEORY (resonance, resonant frequency, bandwidth, nodes, antinodes, vibration sensitivity (mobility), tension, mass, specific specific vibration sensitivity (specific mobility)). ACOUSTICAL PROPERTIES The acoustical properties of a resonator can be measured in terms of vibration sensitivity (mobility). For the simple resonator a resonance curve is obtained with a peak like in Fig. 4.1. The peak and the shape of the complete curve is fully determined by three measures, the position, the height, and the width of the peak i.e., the RESONANT FREQUENCY, the PEAK LEVEL, and the BANDWIDTH respectively, see Fig. 4.1. Often another level measure is interesting, i.e. the specific vibration sensitivity (specific mobility). The specific vibration sensitivity can often be used as a material constant. The peak level can be calculated from the resonant frequency, the bandwidth, and the specific vibration sensitivity. The bandwidth is also a measure of how long it takes a constant driving force to bring the system into equilibrium. Thus the bandwidth predicts the length of the starting transient.

Figure 4.1. Acoustical properties of a resonance: vibration sensitivity response with resonance frequency RF, bandwidth B, peak level RN and specific vibration sensitivity SV.


Figure 4.2. Resonances of the string: modes of vibration, their nodes and antinodes

The string has several resonances, see Fig. 4.2. The vibrations are distributed in specific ways. For the ideal string the resonant frequencies (the partials) are 2x, 3x, 4x, etc. the frequency of the lowest resonance, the fundamental tone. At each resonance the string vibrates in a different mode, see Fig. 4.2. In the lowest resonance the string has one antinode, and nodes at its ends, in the second resonance it has two antinodes with nodes in the middle and at the ends, etc. The vibration sensitivity curve of a string displays a number of peaks, see Fig. 4.3. Each peak has its own frequency, the resonant frequency, its own bandwidth, and its own peak height. The peak height indicates the strength of each string tone (partial, resonance) and the bandwidth indicates its reverberation time. The specific vibration sensitivity is a measure of how the string will cooperate with the instrument.

Figure 4.3. Resonances of the string: every resonance peak has its own frequency, bandwidth and peak height.


SPECIFIC VIBRATION SENSITIVITY - DRIVING FORCE The specific vibration sensitivity is a fundamental measure of how sensitive a resonator, a string or a plate is to vibration forces. But, if one wants to study the sensitivity to vibrations in detail also the resonance properties such as frequency and bandwidth must be included. The specific vibration sensitivity (and the resonant properties) is a measure how vibrations are transferred, from a string to a violin for instance.

FIGURE. 4.4. Force transfer between two resonator systems a) stiffness controlled and b) mass controlled

Example 4.1: If you are standing on a thick plank and want to bend a board in the ceiling (for nailing it, see Fig. 4.4) then a) it is easy if the plank is stiff and the board soft, but b) it is difficult if the plank is soft and the board is stiff. Thus it is not only how strong you are but also the strength of the support you are pushing from that is important. If you push and pull repeatedly in the board in case a) the board will bend much and the plank little and b) the board will bend little and the plank much. (compare with the formula for specific vibration sensitivity - the stiffness corresponds to the spring in Fig. 2.8). Example 4.2: If you are pushing a small dinghy away from a yacht then the dinghy is given a considerable motion but the yacht a small one. If you push and pull alternatively the dinghy moves much and the yacht little. The motions are independent of whether you are in the dinghy or in the yacht (compare with the formula for specific vibration sensitivity - the weights of the boats corresponds to the mass in Fig. 2.8). We can see a certain relation between the motions in the two cases, board-plank and dinghy-yacht, i.e., vibrations are efficiently transferred from a stiff plank with low vibration sensitivity to a soft board with high vibration sensitivity. The same relation applies to the two boats. From a heavy boat with low vibration sensitivity vibrations are efficiently transferred to a light boat with high vibration sensitivity. In the opposite direction the transfer works inefficiently; it is difficult to transfer vibrations from an object with high vibration sensitivity to one with low vibration sensitivity. The transfer of


vibrations is dependent on both the force applied and the ratios of the mobilities of the two objects. Let us accept with the sketched background that the specific vibration sensitivity of the transmitter and the receiver determines the vibration transfer. Example 4.3. The output connectors of the amplifier should have the same labelling as the loudspeaker - 4 ohms to 4 ohms or 8 ohms to 8 ohms. Example 4.4: For a car motor to work efficiently an adaptation must be used - the gear box. Example 4.5: A string has a high specific vibration sensitivity and a violin or a guitar has a low one - thus the string will vibrate much and the violin or the guitar little. But the closer to equality the ratio of string vibration sensitivity to the violin or guitar vibration sensitivity are, the more sound comes out of the instrument. If the mobilities are equal a "wolf tone" is obtained. MECHANICAL PROPERTIES The properties of the simple resonator are determined by its mechanical properties, i.e. the mass (the weight), the stiffness and the total friction. For the multiresonator system like the string, material constants are obtained by the mass (weight) per meter of the string, the string tension (the stiffness), and the internal friction. PRACTICAL FORMULAS FOR SMALL CHANGES

In the practical cases one chooses between different strings and thus it is interesting to estimate how different mechanical properties influence the main acoustical properties. Two simple formulas can be derived: the resonant frequency change (in percent) = - length change (in percent) + 1/2 x tension change (in percent) - 1/2 x mass change (in percent). the specific vibration sensitivity change (in percent) = - 1/2 x tension change (in percent) - 1/2 x mass change (in percent) The formulas point out two very important facts which are presented in the examples below: Example 4.6. What is the consequence of the relation between the resonant frequency and mechanical properties like weight per meter and string tension? In most musical instruments the fundamental frequency and the string length are set by the tuning of the instrument and its construction. Only the string tension and the mass per meter of the string can be selected. Therefore the ratio between the tension and the mass per meter must be constant in order to keep the fundamental frequency as required. The practical formula says the same in another way - if the mass is increased by 10 % the tension also must be increased by 10 %. Result: for many


practical applications the string mass per meter is the fundamental measure. Example

4.7.: What is the consequence of the relations between specific vibration sensitivity and mechanical properties? Again we start from the fact that most musical instruments have a given string length and tuning. This means that only the string tension and the string weight per meter can be chosen. The tuning condition sets a specific ratio of tension to weight. The formula for the specific vibration sensitivity says that the specific vibration sensitivity is proportional to 1/weight and again the weight per meter is the most important measure for the string. Result: it is sufficient to know the weight per meter, thereby the specific vibration sensitivity of the string is given.

Example 4.8.: If we start from normal conditions and thereafter change a mechanical property at a time, how are the acoustical properties changed? a) if the string tension is increased 2 % => the frequency is increased 1 %, i.e., from 440 to 444 Hz the specific vibration sensitivity decreases 1% (the sound level increases 0.1 dB) b) if the weight is increased 10% but the tuning is kept => the string tension must be increased 10% and the specific vibration sensitivity is lowered 20 % (the sound level increases 2 dB) c) if the string length is increased 5 % but the tuning is kept. => the string tension must be increased 10% and the vibration sensitivity decreases 5 % (the sound level is increased 0.5 dB) Example 4.9.: Say that a resonance (partial) of a guitar string decays 10 dB in a second. What is the bandwidth of the string resonance? A little calculation shows that this reverberation corresponds to a bandwidth B of 0.3 Hz, which is a very sharp resonance peak. 4.2 HOMOGENEOUS STRINGS (mass, tension, tensile strength, elasticity modulus, typical string tension, thickness, inharmonicity, and partials) A common wish is to have a string with strong tone and a high fundamental frequency. How can this be obtained? In principle the string should be made short, heavy and tightened up to a high tension. But the heavy string favours tuning to low frequencies and the light one to high frequencies and therefore a compromise must be made. In the theoretical introduction we have already shown that the weight of the string is the most important and "automatically" determines the string tension. It is therefore


interesting to know the density (weight per cubic meter) for materials, see table 4.1. For string materials we are in the heavy range, see table 4.2 for typical string materials. Table 4.1. Densities of common materials Air Water Lead

1.2 kg/m3 1 000 " 11 000 "

Table 4.2. Density of typical string material Steel Gut Silk Nylon

7 700 kg/m3 1 300 " 1 300 " 1 200 "

Example 4.10. How much thicker should a gut string be to give the same tension as a steel string for the same pitch? A little calculation on weight, tension and fundamental frequency shows that the gut string must be 2.5 times thicker. Example 4.11. How much do we loose in specific vibration sensitivity when we use a gut string as thick as a steel string which we tune to the same frequency? The density for steel is 6 times that of gut which means that the gut string tension becomes 1/6 of the steel string tension. This means that the specific vibration sensitivity increases 16 dB, and the sound level decreases 16 dB (this is a very large decrease). The strength of a string is tested by tensile tests. In a measurement apparatus the string tension (load) is increased and the resulting lengthening (strain) is measured. The tension is increased until the string breaks. Thereby a load-strain diagram is obtained, see Fig. 4.5a. The first part of the curve is a straight line. If the pulling force is disconnected within a certain range (up to the limit P) the string regains its original length. The strain within this range is elastic. At the limit P the elastic strain turns into a plastic strain and a remaining lengthening is left after the tension has been disconnected, se Fig. 4.5b. If the string is stretched until it breaks, the fracture limit B, a measure is obtained of the maximum tension possible.


Figure 4.5 a) Typical tension-lengthening (load-strain) diagram and b) diagram of plastic lengthening.

Strings can not be stretched to any limit. First the string tension is increased, thereafter it is stretched and finally it breaks. The breaking limit B sets is the maximum frequency the string can be tuned up to but the elastic limit P should be the maximum useful limit. The breaking limit for some materials are given in table 4.3. Table 4.3. Fracture limit for typical string material (107 Pa = 1 "kg/mm2") Steel 2 100-2 600 x 106 Pa Gut 320 - 460 " " Silk 350 - 550 " " Nylon 600 - 750 " " i.e. a steel string can be stretched 3 to 4 times that of the gut, silk and nylon materials. The string should not be stretched to the breaking limit. Suitable limits are given for loading are given in table 4.4. The table says that strings are loaded to 50 % of the breaking limit. Thus the steel string can still be tensioned 3 to 4 times more than the gut, silk and nylon strings. Table 4.4. Suitable tension in percent of the fracture limit. Steel 40 - 75 % Gut and fibres 35 - 70 %


ON INHARMONIC PARTIALS

(This paragraph is accidentally too mathematical-technical. Do not read it if difficult) Table 4.5. Elasticity modulus E (typical) and inharmonicity factor I for different string tensions E x109 Pa 0.5 Imax 0.75 Imax 1.0 Imax Steel 220 56 38 28 Gut 5.5-6.5 9.4 6.3 4.7 Silk 5.0-6.0 7.5 5.0 3.8 Nylon 4.5-5.5 4.5 3.0 2.3 Steel has a high inharmonicity factor and must therefore be given a high tension (the piano string has so high a tension that the string can be broken by a hard touch). Example 4.12.What is the tension of the E-string of the violin if made by steel with the diameter 0.25 mm? A violin string has a free length of close to 325 mm = 0.325 m, diameter 0.25 mm = 0.00025 m and the density of steel is 7 800 kg/m3, which results in a weight of 7 800 x 3.14 x (diameter/2)2 = 0.000383 kg/m. The frequency is 660 Hz and a little calculation gives the string tension 70 N ("7.1 kg"). Example 4.13 If the A, D and G strings are made of steel and all should have the same tension, how thick should they be made? For the same tension and the same material it turns out that the diameter should be a constant divided by frequency. The violin is tuned in fifths which means that the string diameters should be increased successively by a factor 3/2 and the A string (3/2) x 0.25 = 0.38 mm, the D string (3/2) x (3/2) x 0.25 = 0.56 mm, and the G string (3/2) x (3/2) x (3/2) x 0.25= 0.84 mm Example 4.14: What kind of inharmonicity should we expect from the first and the third strings of a guitar if they are made of nylon and with diameter 0.71 and 1.03 mm and with tension 77 and 52 N. A little calculation shows that the strings are tensioned to 30 and 10 % of the fracture limit and the inharmonicity factor is approximately 8 and 16 respectively. If we multiply with (d/L)2 (1.19 and 2.5 x 1 000 000) and use the formula we find that the third string gives an inharmonicity 7 times that of the first string (note that the calculation example gives a good demonstration but can not be expected to be accurate). The examples show that one should have a high tension to obtain a low inharmonicity, or to make a heavier string without increasing the stiffness, i.e. by additional winding.


4.3. WOUND STRINGS (mass and material for winding)

Figure 4.6. Different types of string windings

To obtain strings with strong tone, i.e., heavy strings without making them very stiff, the strings are made in several parts.The core is fairly thin and has little stiffness. Thereafter this core is covered with one or several layers of windings and a wound string is obtained. The winding can be made in several ways, see Fig. 4.6. The winding is made in such a way that the string stiffness is little increased. By means of suitable winding materials, the string is made to the desired weight, see table 4.6 Example 4.15: The G-string of the violin has a diameter of 0.8 mm, of which 0.2 mm is a silver winding. What is the density in kg/m3 and what is the string tension? a) a little calculation shows that the string density is 4 700 kg/m3 b) the frequency of the open string is 196 Hz which together with some calculation gives the string tension as 38 N ("3.9 kg"). Table 4.6.Density for typical winding materials. Aluminium 2 700 kg/m3 Steel 7 700 " Copper 8 900 " Silver 11 000 " Gold 19 000 " Wolfram 19 000 " Example 4.16: What will the density of the string become if wound with gold instead of silver? Table 4.6 shows that the density for aluminium is 2 x Gut, for steel it is 6 x, for silver 8 x and for gold and wolfram (tungsten) it is 15 x. Thus the string mass could be greatly increased by winding with little increase of Jansson: Acoustics for the violin and guitarmaker 4.11

thickness", i.e., the string weight has been doubled but the diameter has not been changed. 4.4: SUMMARY: FUNDAMENTALS OF STRINGS In this part, fundamental string theory has been summarised, i.e., the relations between acoustical properties such as resonance frequencies and vibration sensitivity, and mechanical properties such as the mass (weight), length, and tension of the string have been given. The influence of the string diameter on the string inharmonicity has been demonstrated. Finally the mass increase by winding of a string without making it stiffer has been shown. 4.5: KEY WORDS: Resonant frequency, string length, string tension, string mass (weight), modes of vibration, antinodes, nodes, vibration sensitivity, string diameter, wound strings, fracture limit, and elasticity modulus.


Chapter 4. Second part: PLAYED STRINGS AND SCALES INTRODUCTION In the second part of this chapter we are approaching the real world of played strings. First properties of real strings are presented. Thereafter the fundamentals of the plucked and the bowed string is introduced, effects of bow gestures and played scales. In the appendix the relations between tone frequency (in Hz) and tone position (name and deviation in cents from nominal value). 4.6 REAL STRINGS As pointed out in the theory part it is sufficient to weigh a string to be able to calculate its most important properties. The weights of the strings and not their diameters should be given on the string envelopes. For very thin strings extremely sensitive scales are needed and it may be easier to measure the fundamental frequency of a string when tensioned with a well defined force, see Fig. 4.7. Measured frequencies can be recalculated into string weights. Measured string weights are given in table 4.7 for violin strings (Pickering found also that some strings had rather high inharmonicity and large bandwidth) and these are given in table 4.8 for guitar strings. Furthermore an example on inharmonicity for violin strings is given in table 4.9.

Figure 4.7. Two ways to determine the main properties of a string are a) by measuring the mass (weight) with a precision scale, or b) by measuring the frequency with a well defined tension (L is the string length and D a vibration sensitive detector).

Table 4.7. Measured properties of violin strings (after Pickering) Fundamental frequency (Hz) Diameter (mm) min-max Tension (N approx "kg/10") min- max Weight (mass) (g/m) min-max Spec vibration sensitivity (m/Ns) min-max

E-string 660

A-string 440

D-string 294

G-string 196

0.249-0.264

0.452-0.701

0.671-0.914

0.790-0.833

72.25-84.01

48.89-63.51

34.76-61.73

35.43-49.92

0.381-0.443

0.579-0.752

0.924-1.641

2.115-2.799

5.18-6.02

4.57-5.94

3.14-5.58

2.59-3.65


Table 4.8. Measured properties of guitar strings String Fundamental frequency (Hz) Diameter (mm) Tension (N approx "kg/10") Weight (mass) (g/m) Spec vibration sensitivity (m/Ns)

E

H

G

D

A

E

330 0.70 76.7 0.417 5.59

247 0.83 61.1 0.593 5.25

196 1.03 57.9 0.892 4.33

147 0.75 74.5 2.04 2.56

110 0.93 74.5 3.45 1.97

82.4 1.07 61.2 5.33 1.97

Table 4.9. Inharmonicity (cent*) and bandwidth (BD) measured for four violin strings STRÄNG (Hz) E 656.5 A 440.2 D 296.0 G 196.6

F1 / BD (cent / Hz) 0 0.27 0 0.32 0 0.29 0 0.16

F2 / BD2 (cent / Hz) 0 0.62 1 0.41 -2 0.79 -1 0.35

F3 / BD3 (cent / Hz) 0 0.50 2 0.56 0 1.1 1 0.72

F6 / BD6 (cent / Hz) 1 0.75 0 1.3 5 3.5 9 1.9

F10 / BD10 (cent / Hz) 4 1.5 1 2.3 17 13.1 19 8.3

*Cent is a measure of deviation from correct nominal value in hundreds of a semi-tone step (see section 4.8).

B P

FORCE

BRIDGE VIBRATION-FORCE

TIME

Fig 4.8 Time history (in principle) for the plucked string, bridge B and plucking position P (after Hall) and bridge forces. The different vibration shapes and corresponding bridge forces are indicated by points in the bridge force curve.


4.7 PLAYED STRING IN PRINCIPLE (time history, shape of deflection, spectra, the plucked and the bowed string) The plucked string has in principle the following time history. The string is pulled out of equilibrium and makes a triangle as showed in the upper left corner of Fig. 4.8. When the string is released it will develop a complicated vibration as sketched in the left column. It can be seen that the string is divided into three straight parts. The string angle in relation to the bridge determines the excitation force by which the string will shake the bridge. If one looks closely at the shape of the string one can see that the string only has two different angles at the bridge, which results in half the maximum force upwards in the first case and the maximum force downwards in the second case. The vibration force will show a time history as shown in right part of Fig. 4.8. It should be noted that the time history shows a downward pulse for one third of the time, i.e., the same relation as the one to three division of the string at the plucking position.

Figure 4.9 Position of plucking and excitation strength of resonance (in principle). Left upper part the lowest four resonances, right upper the corresponding partials, down left the initial shape of the string at plucking for different plucking positions and down right the resulting spectra for the different plucking positions.


FINGERTIP

NAIL

STIFF STRING

LEVEL

SOFT STRING

FREQUENCY

Figure 4.10 Influence of different pluckings and of different strings

The relation between position of plucking and the level of partials have been sketched in fig 4.9 When plucked in the middle, the string is initially displaced in form of a triangle with two sides alike. For a string resonance to be excited it must not be plucked at one of its nodal points. Thus we can understand that for a plucking in the middle the first partial, the fundamental is set into vibration, but not the second, the third is set into vibration, but not the fourth etc.. Thereby we obtain a spectrum like in the uppermost part of the lower right frame. If we choose to pluck at a third of the string length the third, the sixth etc. partials will be missing. If we pluck at a tenth of the string length the tenth,the twentieth partial, etc. will not be excited. The player has also other possibilities to influence the properties of the played tones, see Fig. 4.10. A soft "plectrum" like the fingertip will give a smoother bend of the string and thus weaker high partials. A stiff string can not be bend sharply and gives therefore a tone with weaker high partials.


PLUCKING AT l/2

PLUCKING AT l/10

LEVEL dB

PLUCKING AT l/2

Resonance no

TIME HISTORY

SPECTRUM

Figure 4.11 Time history and spectrum of string force on bridge for different plucking distances from the bridge.

Different plucking positions give different time histories for the force from the string to the bridge, and thus different vibrations of the bridge. A qualitative explanation was given in Fig. 4.9. As a matter of fact also a varying level for the different partials will be obtained, in principle as in Fig. 4.11.


NUT

BOW

BRIDGE

STRING MOTION AT BOW

STRING FORCE AT BRIDGE

STRING SHAPE

SPECTRUM

Figure 4.12 Time history for a bowed string etc. (after Hall)

Similar relations are also valid for the bowed string, see Fig. 4.12. When the bow is pulled over the string, the string is displaced as in plucking. The string is pulled aside but tries to regain its straight line. After a while the force to regain equilibrium becomes so large that the string pulls itself loose from the bow hair and moves towards a straight line. After some time the string motion is stopped and the string fastens to the bow hair again, is pulled aside once more, pulls loose, etc.. The excitation here corresponds to a plucking once each fundamental period of the string. Thereby a saw tooth shaped time history is obtained. The nature makes this repeated plucking happen once every period as long as the violin player handles his bow in a correct manner. The ratio of string-lengths, bow to bridge and bow to nut, is the same as the duration ratio of the two straight parts of the saw tooth time history. It should be mentioned that twisting motions of the string (torsion motions) are important and that temperature effects may also play a role.


4.8 GESTURE OF THE BOW AND SCALES GESTURES OF THE BOW

At our department Dr. Askenfelt has investigated the bow gestures and articulation. With a special experimental apparatus he can simultaneously measure 1) the fundamental frequency of the played tone (pitch), 2) the bow position at the string (contact point), and 3) the bow pressure (the bow force). The three parameters are registered simultaneously as the violin is played. The bow force was found to be between 0.15 and 1.5 N (corresponds to a weight of 15 to 150 g). The velocity of the bow is between 0.04 m/s and 3 m/s (3 m/s is approximately 8 mph). Examples of different ways of bowing are shown in Fig. 4.13. In the upper frame we can see examples of legato playing, and from top to bottom, music, bow force, bow position, and time respectively. It can be seen that the violin player decreases the bow force just before changing to a new note and keeps the low bow force at the beginning of the following note. In martellato and the sforzando playing it is found that the violin player adjusts the bowing force and velocity. In the martellato case the the velocity is constant (the bow position is a horizontal line) and the bow force is released just before the bow reaches the tip. In the sforzando case the bow is pressed against the string with a high force in the beginning, the bow is pulled quickly over the string, the bow force is released and the bow velocity has diminished to a third. Furthermore it was found that the normal and the reversed way of playing crescendo-diminuendo gives large differences in the time history of bow force. From research in psychology it is known that different feelings are reflected in differences in gestures, articulation. An example is shown in Fig. 4.14 for violin playing. Four bars from the Beethoven violin concerto were played softly and aggressively as shown in Fig. 4.14. The figure shows that in the aggressive version the player chose to vary the bow force rather abruptly.


BOW FORCE BOW POSITION

LEGATO

TIP FROG

DETACHÉ

TIP FROG

TIME

STACCATO

BOW POSITION

BOW FORCE

BOW POSITION

BOW FORCE

TIME

TIP FROG

TIME

Figure 4.13 "Bow pressure" (bow force) and contact point (bow position) for different types of bowing (from Askenfelt).


BOW FORCE BOW POSITION

TIP FROG

TIME Figure 4.14 "Bow pressure" and contact point for two different types of bowing of four bars of the Beethoven violin concerto (aggressively and smoothly full lines and broken lines respectively, from Askenfelt) THE EQUALLY TEMPERED SCALE AND PLAYED SCALES (definition of the scale, frequency and cent) The scale generally used for tuning keyboard instruments and in playing is the equally tempered scale. This scale uses a division of the octave into 12 equal steps called semitones. Each step corresponds to an h times higher frequency and

12 steps h x h x h x h x h ................ i.e.

h12 = 2

and h =

12

xhxh = 2

2 = 1.059

A semitone step thus corresponds to slightly short of a 6 % increase in frequency. Often the frequency of a played tone should be given with higher accuracy than a semitone step. For this purpose every semitone step is divided in 100 equal steps c. . 100 cent steps = 1 semitone step i.e. c x c x c x c x ...... x c x c = i.e. c100 =

12

2

and c =

1200

12

2

2 = 1.00058.

The step c is called a cent and corresponds to 6/100 %. Furthermore, for practical use a numbering system which gives the lowest useful octave the number 0 has been devised. This octave goes from C0 to C1 and covers the frequency range from 16 to 32 Hz and notes in the octave are labelled with "zeroes". The next octave goes from C1 to C2 and notes are labelled with a "one" etc. The tuning A = 440 Hz is labelled A4 ( to avoid


misunderstandings one can write A4 (440 Hz), as the introduced labelling does not agree with any of the ten most common notations used in musical scale systems. Further the introduced system only uses sharp signs, i.e. C#, D#, F# etc. For tuning of free plates this division in single cent steps is too detailed. Therefore a centfrequency table has been calculated in 20 cent steps, i.e., steps 1.012 times the lower frequency. Twenty cent (20 cent) is thus 12/10 % i.e., slightly above 1 %. This table can be found in the appendix 4.11 RELATIONS BETWEEN FREQUENCY AND TONE, table 4.10,starting at the 0th octave C= (16.4 Hz) ending at the 7th octave C8 (4186 Hz). Every octave, from the 0th to the 7th., has its own table.There are thus 8 tables starting with the 0th octave and ending with the 7th octave. The top line of every table gives the tone names and the first line with numbers the cent values 0 and corresponding frequencies in Hz. The second line gives the tones + 20 cent and corresponding frequency in Hz. The table continues with +40, +60 and +100 cent. The bottom line gives the tone names again but now at +100 cent, i.e. a semintone step higher. The column to the right give the cent values starting from the bottom line, i.e. –0 to –100 cent. Example 4.16: 440 Hz = tuning A = A4. Example 4.17. The note C2 has the frequency 65.4 Hz. Example 4.18. The frequency 119 Hz corresponds to the note A# in the second octave +40 cent (upper line and left column) or if preferred the note B in the second octave -60 cent (lower line and right column). Let me also present some measurements on scales and played frequencies. In Fig. 4.15 the relations are shown between three theoretically defined scales. As a rule of thumb one hears differences outside ±5 cent, which have been marked with two horizontal dotted lines. Along the horizontal scale the note numbers in the octave have been marked, where 0 is the lowest tone of the octave and 12 is the highest. A look shows that the differences as a rule are small between the three scales, but in some cases the differences are noticeable ±15 cent. For the construction purpose of a musical instrument the conclusion can be drawn that one should be able to play the equally tempered scale ±15 % variations.


DEVIATION CENT

SCALE TONE NO

DEVIATION CENT

Figure 4.15 Relations between the equally tempered scale (horizontal line), the pure scale of just intonation and the Pythagorean scale.

SCALE TONE NO Figure 4.16 Average of intervals used in woodwind quartet playing, students in tutti (triangles), in solo (crosses), teachers in tutti (squares) and in solo (circles) (from Mason).

Let me so show the differences that can be met in actual playing, see Fig. 4.16. The experiments showed that the students as a rule played considerably above the "tuning pitch". The teachers also tend to be high but as a rule less than +5 cent. In solo performance without accompaniment the oboist in Fig. 4.17 played even ±20 cent outside a arbitrarily chosen reference tone.


cent +50 +20 0 -20 -50 G#3

C4

E4

G#4

C5

E5

G#5 C6

E6

Figure 4.17 Deviations from the equally tempered scale used in unaccompanied solo playing (after Sundberg).

The important conclusion from the presented material is that the played scales do not follow simple mathematical formulas - one plays neither in the Pythagorean scale, nor the pure scale nor the equal tempered scale. It seems rather that the played tone frequency is a part of the musical speech and one chooses the frequency approximately as the mathematically defined scales but with minor deviations for the effects one wants to stress. When playing several instruments together the possibilities of selection for the single musician is strictly limited to avoid unwanted sound effects.

4.9 SUMMARY: PLAYED STRINGS AND SCALES In this part mechanical and acoustical properties have been presented for violin and guitar strings. Further the principles for the vibrations of the strings have been sketched for differences in plucking and bowing of the string. The gestures with the bow makes it possible for the player to "talk" with the violin. Finally the equally tempered scale has been described, and how the scales are used in playing and how frequency can be measured. 4.10 KEY WORDS: Plucked string, bowed string, time history spectrum, equally tempered scale, cent, bow velocity, bow force (bow pressure), position.


4.11 APPENDIX OCTAVE 0 TONE C C# cent Hz Hz +0 16.4 17.3 +20 16.5 17.5 +40 16.7 17.7

Table 4A.1 RELATIONS FREQUENCY AND TONE POSITION OCTAVE 0 D D# E F F# G G# A A# H TONE Hz Hz Hz Hz Hz Hz Hz Hz Hz Hz cent 18.4 19.4 20.6 21.8 23.1 24.5 26.0 27.5 29.1 30.9 -100 18.6 19.7 20.8 22.1 23.4 24.8 26.3 27.8 29.5 31.2 -80 18.8 19.9 21.1 22.3 23.7 25.1 26.6 28.1 29.8 31.6 -60

+60 16.9 17.9 19.0 20.1 21.3 +80 17.1 18.1 19.2 20.4 21.6 +100 17.3 18.4 19.4 20.6 21.8 TONE C# D D# E F

22.6 22.9 23.1 F#

23.9 24.2 24.5 G

OCTAVE 1 TONE C C# D D# E F F# +0 32.7 34.6 36.7 37.9 41.2 43.7 46.2 +20 33.1 35.1 37.1 39.3 41.7 44.2 46.8 +40 33.5 35.5 37.6 39.8 42.2 44.7 47.3

25.4 25.7 26.0 G#

+60 +80 +100 TONE

C# D D# E 69.3 73.4 77.8 82.4 70.1 74.3 78.7 83.4 70.9 75.1 79.6 84.3

67.7 71.7 76.0 80.5 85.3 68.5 72.6 76.9 81.5 86.3 69.3 73.4 77.8 82.4 87.3 C# D D# E F

OCTAVE 3 TONE C C# D D# E 0 131 139 147 156 165 +20 132 140 149 157 167 +40 134 142 150 159 169 +60 135 143 +80 137 145 +100 139 147 TONE C# D

30.2 32.0 -40 30.5 32.3 -20 30.9 32.7 0 H C TONE

OCTAVE 1 G G# A A# H TONE 49.0 51.9 55.0 58.3 61.7 -100 49.6 52.5 55.6 58.9 62.5 -80 50.1 53.1 56.3 59.6 63.2 -60

+60 33.9 35.9 38.0 40.3 42.7 45.2 47.9 50.7 +80 34.2 36.3 38.4 40.7 43.2 45.7 48.4 51.3 +100 34.6 36.7 38.9 41.2 43.7 46.2 49.0 51.9 TONE C# D D# E F F# G G# OCTAVE 2 TONE C 0 65.4 +20 66.2 +40 66.9

26.9 28.5 27.2 28.8 27.5 29.1 A A#

53.7 56.9 54.4 57.6 55.0 58.3 A A#

60.3 63.9 -40 61.0 64.7 -20 61.7 65.4 0 H C TONE

OCTAVE 2 F F# G G# A A# H TONE 87.3 92.5 98.0 104 110 117 123 -100 88.3 93.6 99.1 105 111 118 125 -80 89.3 94.7 100 106 113 119 126 -60 90.4 91.4 92.5 F# F 175 177 179

95.8 101 96.9 103 98.0 104 G G#

107 114 109 115 110 117 A A#

121 128 -40 122 129 -20 123 131 0 H C TONE

OCTAVE 3 F# G G# A A# H TONE 185 196 208 220 233 247 -100 187 198 210 223 236 250 -80 189 201 213 225 239 253 -60

152 161 171 181 192 203 154 163 173 183 194 205 156 165 175 185 196 208 D# E F F# G G#

215 228 217 230 220 233 A A#

241 244 247 H


256 -40 259 -20 262 0 C TONE

OCTAVE 4 TONE C C# D D# E F F# G G# 0 262 277 294 311 330 349 370 392 415 +20 265 280 297 315 333 353 374 397 420 +40 268 284 301 318 337 357 379 401 425

A 440 445 450

+60 271 287 304 322 341 362 383 406 430 +80 274 290 308 326 345 366 387 411 435 +100 277 294 311 330 349 370 392 415 440 TONE C# D D# E F F# G G# A

456 483 461 488 466 494 A# H

OCTAVE 5 TONE C 0 523 +20 529 +40 535 +60 +80 +100 TONE

C# 554 561 567

D 587 594 601

D# 622 629 637

OCTAVE 4 A# H TONE 466 494 -100 472 500 -80 477 505 -60 511 -40 517 -20 523 0 C TONE

OCTAVE 5 E F F# G G# A A# H TONE 659 698 740 784 831 880 932 988 -100 667 707 749 793 840 890 943 999 -80 675 715 757 802 850 901 954 1011 -60

542 574 608 644 683 548 581 615 652 690 554 587 622 659 698 C# D D# E F

723 766 812 731 775 821 740 784 831 F# G G#

860 870 880 A

911 965 922 976 932 988 A# H

1023 -40 1034 -20 1047 0 C TONE

OCTAVE 6 TONE C C# D D# E F F# 0 1047 1109 1175 1245 1319 13971480 +20 1059 1122 1188 1259 1334 1413 1497 +40 1071 1135 1202 1274 1349 1430 1515

OCTAVE 6 G G# A A# H TONE 1568 1661 1760 1865 1976 -100 1586 1681 1780 1886 1998 -80 1605 1700 1801 1908 2022 -60

+60 1083 1148 1216 1288 1365 +80 1096 1161 1230 1303 1381 +100 1109 1175 1245 1319 1397 TONE C# D D# E F

1623 1642 1661 G#

1446 1532 1463 1550 1480 1568 F# G

1720 1740 1760 A

1822 1843 1865 A#

1930 2045 -40 1953 2069 -20 1976 2093 0 H C TONE

OCTAVE 7 TONE C C# D D# E F F# 0 2093 2217 2349 2489 2637 2794 2960 +20 2117 2243 2377 2518 2668 2826 2994 +40 2142 2269 2404 2547 2699 2859 3029

OCTAVE 7 G G# A A# H TONE 3136 3322 3520 3729 3951 -100 3176 3361 3561 3773 3997 -80 3209 3400 3602 3816 4043 -60

+60 2167 2296 2432 2577 2630 +80 2192 2322 2460 2607 2762 +100 2217 2349 2489 2637 2794 TONE C# D D# E F

3247 3284 3322 G#

2892 3064 2926 3100 2960 3136 F# G

3440 3480 3520 A

3644 3861 4090 -40 3686 3906 4138 -20 3729 3951 4186 0 A# H C TONE




Erik Jansson

Chapter V: Vibration Properties of the Wood and Tuning of Violin Plates



Sound and hearing

Chapter II


Chapter III

Sound and the room

Chapter IV


Chapter V


Chapter VI


Chapter VII


Chapter VIII


Chapter IX



ACOUSTICS FOR VIOLIN AND GUITAR MAKERS Chapter 5 – Applied Acoustics VIBRATION PROPERTIES OF THE WOOD AND TUNING OF VIOLIN PLATES Part 1: VIBRATION PROPERTIES OF THE WOOD 5.1. Fundamental theory 5.2. Measurement methods 5.3. Acoustical and mechanical properties of wood 5.4. Summary 5.5. Key words Part 2: TUNING OF THE TOP AND BACK PLATES OF THE VIOLIN 5.6. Fundamental principles for tuning 5.7. Typical properties of free violin top and back plates 5.8. On practical tuning of violin plates 5.9. Summary 5.10. Key words 5.11 APPENDIX: Formulas and calculations 5.12 References


Chapter 5. VIBRATION PROPERTIES OF THE WOOD AND TUNING OF VIOLIN PLATES First part: VIBRATION PROPERTIES OF THE WOOD INTRODUCTION In this first part of chapter 5 fundamental vibration properties of wood will be introduced. First fundamental theory is presented. It is followed by presenting measurement methods and ends with giving vibration (mechanical, acoustical) properties of wood. 5.1 FUNDAMENTAL THEORY We have previously seen how properties like stiffness, mass and internal friction determines the vibration properties of a plate. We shall try to sort out how one should design a violin top to give it wanted acoustical (vibration) properties. The properties we shall design a top plate to have are 1) specific nodal line patterns and 2) specific resonant frequencies. In order to give a better understanding for the problems related to top and back plates and possible solutions we shall again summarise some vibration basics concerning the properties of resonances of bars and plates. Thereafter follows measurement methods of the resonance properties (resonant frequency, level, bandwidth and nodal line pattern), their use for deduction of mechanical vibration properties of wood (moduli of elasticity and internal friction) and some data of different wood species.

Figure 5.1 Definition of acoustical (vibration) measures at a resonance peak.

PROPERTIES OF A SINGLE RESONANCE (resonant frequency, level, bandwidth, and specific vibration sensitivity (specific mobility)) The vibration sensitivity at the driving point is a measure of how easily a guitar, a violin or a violin plate is set into vibration. The driving-point vibration sensitivity varies with frequency. At low and high frequencies the driving-point vibration sensitivity is low for the simple system with only one resonance (as the rubber band yo-yo). At an intermediate frequency the driving-point vibration sensitivity is maximum and the vibration sensitivity shows a resonance peak, see fig. 5.1. In a diagram of the driving-point vibration sensitivity as function of frequency three acoustical (vibration) properties can be read: the frequency of the resonance peak (RF Hz), the level at resonant


frequency (RL dB) and the bandwidth of the resonance (B Hz, i.e. the width 3 dB below the peak value).With the resonance removed the specific vibration sensitivity (SV, i.e. peak value minus 20log[RF/B]) is obtained. Inversely if we know that we have a simple system with one resonance only, the three properties; frequency at, bandwidth of, and level of the resonance peak are sufficient to predict (plot) the behaviour at all frequencies. Thus the curve for a simple resonance can be described by the three measures without any loss of information, c.f. fig. 5.1. The decay of resonance vibrations (after turning off the driving) is determined by the bandwidth of the resonance. The vibration sensitivity of the mechanical system is the combination of specific vibration sensitivity and resonance properties.

Figure 5.2 Predicted VIBRATION SENSITIVITY curve for a wooden plate (in principle).

MECHANICAL PROPERTIES (mass, stiffness, and friction) The simple resonance (as the rubber band yo-yo) has three mechanical parts: 1) a spring (the rubber band), 2) a mass (the weight), and 3) friction (the air friction when the weight is moving). Between the mechanical and the vibration properties there are mathematical relations as presented in section 5.11 appendix. VIBRATION (ACOUSTICAL) PROPERTIES OF A MULTIRESONANCE SYSTEM (vibration modes at resonance, nodes and antinodes) If we replace the simple resonator (the rubber band yo-yo) with a thin plate and measure the drivingpoint vibration sensitivity we shall obtain a large number of resonance peaks. Each resonance is described by the frequency, the level and the bandwidth of each peak, see fig. 5.2. The complete driving-point vibration sensitivity curve can be predicted (plotted) from the three measures of all resonances. In fact, a violin plate is a multiresonance system with a large number of peaks. What the driving point sensitivity curve does not reveal is that the plate vibrates in a different way for each resonance. At a given resonant frequency, different points of the plate vibrate with different magnitudes varying between a maximum and no vibrations. This vibration pattern (vibration mode)


must be added to the three vibration sensitivity measures for a complete description of the vibration (acoustical) properties of the plate. The positions of maximum vibrations are called antinodes and the lines of no motion are called nodal lines. BAR AND PLATE PROPERTIES Nodal lines for a bar or a plate can be measured with a loudspeaker and particles sprinkled over the bar or plate. Small pieces of foam plastic are placed under "suspected" nodal lines, see fig 5.3. The position of the supports are adjusted to the real nodal lines after the measurement has been started. Nodal lines of a thin rectangular spruce plate are shown in fig 5.3b. Nodal lines were first visualised by Ernst Chladni, and another name of the nodal line patterns are Chladni patterns. A rectangular plate can in some of its lower modes be regarded as made of broad bar along the grains or a broad bar across the grains, see fig. 5.4. In the plate there is a certain degree of coupling between vibrations spread along the two directions.

Figure 5.3. Vibration distribution for the resonances of a) a bar with free ends (an antinode marked with triangle and a node with circles) and b) a rectangular plate with free edges (dashed lines are nodal lines, Sitka Spruce 362 x 210 x 3.7 mm and 110 g).


Figure 5.4. Fundamental vibration modes of a) a free bar, b) a free plate across and along with no coupling, and c) a free plate across and along with coupling.

Figure 5.5. Dimension of top plate blank for test of material properties; for violin: 385 x 215 x (20 in the middle, 7.5 mm along edges) and for viola: 450 x 260 x (25 in the middle , 78 mm along edges) mm - proposal worked out by Gunnar Mattsson, Stockholm's violin makers club.


Figure 5.6. Positions of holding (at circle H, full lines for the best but somewhat more difficult position to find, dashed lines for the position simple to find) and tap positions for tapping (triangles T) for a blank (from Molin et al, 1988).

5.2 MEASUREMENT METHODS (resonant frequencies, antinodes and nodal lines) In the process of experimenting, standard measures have been proposed for blanks, see fig. 5.5. If these measures of blanks are used mechanical material properties can be measured by tapping. The tap tone related to the closest tone of a piano, for instance. METHOD OF TAPPING AND LISTENING (resonant frequencies). The resonances of a bar or a plate can detected by taping and listening using the two main rules, c.f. fig. 5.6.and 5.7.

Figure 5.7. Suitable way to hold a bar or a violin plate for tapping of a resonance. Hold lightly at a node and tap closely to an antinode. Only the thumb, the index finger and the knuckle (at the moment of tapping) are allowed to touch the bar or plate.

1. Hold lightly at a nodal line (one must hold at a nodal line). 2. Tap close to an antinode. The method can be refined to obtain safer and more information by applying additional rules 3 and 4:


3. To suppress an interfering resonance - tap at a nodal line and hold close to an antinode of the resonance to be suppressed. Note that one must hold at a nodal line of the resonance being investigated. 4. With some practice nodal lines and antinodes can be localised in two different ways: I) Start holding at a nodal line and tap at an antinode. Vary the holding point but not the tapping point. Mark the holding points where the tap tone of the resonance can be heard the best. Connect the holding points with lines. If correctly done, the lines mark nodal lines. Many "holding" points are needed. II) Hold at a nodal line and tap at many different points. Mark the points where the tap tone of the resonance cannot be heard. Connect these points and the nodal lines have been marked. Many "dead" points are needed. Method I should be more reliable but method II faster. The author has found it convenient to hold the plate between the thumb and the index finger of the left hand, c.f. fig. 5.7. The arm is held over the head as shown in the fig. 5.7 close to the right ear. The plate is tapped with a finger tip (low frequency tap tones), a knuckle or a nail (high frequency tap tones) level with the ear. The left arm is lowered or raised to position the tapping point close to an antinode at the right ear. By determining the frequencies of the resonances in fig. 5.6 the violin maker can determine the twisting, the longitudinal and the cross stiffness of the wood of a blank. Ultrasound methods can also be used but the method presented has the advantage that no extra apparatus is needed.

Figure 5.8. Set-up for measurement of Chladni patterns of the three lowest resonances (a, b, and c) of the plate P. The supports are marked with squares S, and the loudspeaker with circles L. Schematic diagram for measurement system (d, from Molin et al 1988)


Figure 5.9. Measurement of vibration sensitivity: a) the plate P is hung in a frame F with rubber bands RB and the measurement sond RT is fastened with wax in a corner. The measurement sond consists of a contact microphone and a magnet. By means of an electric coil a vibration force is obtained. b) Vibration sensitivity for a blank measured as shown in a (from Molin et al, 1988).

METHOD OF CHLADNI PATTERNS (resonant frequencies, nodes and antinodes). Chladni methods to investigate vibrations of violin plates were probably first used by Felix Savart. For this purpose he simplified the geometry of the violin to a flat triangular body. Lately the method has been reintroduced and further developed by Beldie and Hutchins. They have investigated properties of arched violin plates. The method is the following. The plate to be tested is placed over a loudspeaker in a table, c.f. fig. 5.8. The plate is lifted up a cm or two above the table with four pieces of plastic foam. Small particles, saw dust, tea leaves or similar, are sprinkled over the plate. By means of an audio oscillator and a powerful amplifier (about 15 W) the loudspeaker gives a loud tone. The frequency of the tone a adjusted until the particles jump the most. We have then found a resonance and the frequency of the loudspeaker tone is the frequency of the plate resonance. The positions of maximum jumps of the particles mark the antinodes. The particles will soon move and collect at the nodal lines. The supporting plastic foam pieces should be positioned at nodal lines and the loudspeaker at an antinode for the best results. Some experimenting may be necessary to find such best positions. Thus the method gives resonance frequencies and nodal lines, but with a little extra observation also antinodes. LABORATORY MEASUREMENT METHODS FOR VIBRATION SENSITIVITY AND VIBRATION MODES. The vibration sensitivity of a vibrating plate can be measured in the acoustical laboratory by a method developed by Jansson and Alonso, c.f. fig. 5.9 The object is hung in rubber bands and the driving and measuring sond is attached. The vibration sensitivity is obtained by slowly varying the driving

Jansson: Akustik för fiol och gitarrbyggare sid. 5.9

frequency (c.f. glissando) and by simultaneously measuring the resulting vibration amplitude. A diagram of the vibration sensitivity as function of frequency can be plotted within a couple of minutes (c.f. fig. 5.2). The resonance peaks give accurately the frequencies, the levels and the bandwidth at the resonances. Optical methods have been developed by Biedermann, Ek and Molin to measure vibrations. The advantage with the optical methods is that the measurements are made contact free, i.e. no weights attached and disturbing the plate vibrations. The last apparatus in this development line is the Vibravision. With a case containing a special optical interferometer, a laser and a TV-camera the vibrations are made visible on a TV-monitor. The method gives quickly and accurately the vibration amplitudes of the whole surface. Furthermore resonant frequencies are easily obtained and with some work bandwidths and vibration levels. The method is much more sensitive than the Chladni method.

Figure 5.10. Bars for test of material properties .

5.3 ACOUSTICAL AND MECHANICAL PROPERTIES friction)

OF WOOD (mass, stiffness, internal

Table 5.1. Density (kg/m3, mean from Haines, 2000). Spruce 460 kg/m3 Maple (Czech) 620 Cedar (red) 390 Mahogany (Khaya) 550 Rosewood (Indian) 730 (Brazilian) 830 Ebony 1200*


Table 5.2. Stiffness (elasticity modulus GPa, 1 GPa is 100 "kg/mm2" from Haines, 2000). shear longitudinal radial Spruce 0.84 GPa 15 GPa 0.76 GPa Maple (Czech) 1.7* 10 2.0 Cedar (red) 0.65 9.1 0.72 Mahogany (Khaya) 0.63 12 0.90 Rosewood (Indian) 2.2 13 2.4 (Brazilian) 3.0 16 2.8 * for German subset no data for Czech one, GPa stand for billions of Pa (Pascal). Table 5.3. Internal friction in the form of bandwidths at low frequencies (200 - 1000 Hz) and at high frequencies (10 000 - 15 000 Hz) for longitudinal and radial bars (bandwidth B Hz at frequency F kHz, B = F x log decrement x 3.14, calculated from Haines' data). B longitudinal Spruce 5.2 239 Maple (Czech) 5.5 226 Cedar (red) 5.6 173 Mahogany (Khaya) 3.8 321 Rosewood (Indian) 2.0 351 (Brazilian) 1.9 116

4.6

B radial 6.6 232 493 4.0 146 4.2 402 3.9 203 3.0 141

TEST BARS AND WOOD PROPERTIES A wooden plate contains the three mechanical properties - it "tries" to unfold when folded, i.e. it contains a spring, it has a specific mass (weight) and friction. It is interesting to know (how the three mechanical properties mass, stiffness and internal friction) determine the vibration (acoustical) properties of the plate. One way to test these properties is by means of test bars, see fig. 5.10. Often the dimensions of the bars are selected to 100 mm length, 10 mm width, and 3 mm thickness. The bars are cut along the grain (longitudinal bars) and across (radial bars, c.f. fig. 5.9). The first resonant frequency of the bar is a measure of its elastic properties. The elasticity modulus can be calculated from the resonant frequency, the geometry and the mass of the bar. In principle the frequency can be identified by the tap tone method, except from the fact that the tap tone is very short and weak. The Chladni method is however more suitable. The bar to be tested is laid on top of two plastic supports shaped as wedges. The wedges are positioned 22.4% of the bar length from its ends for the first (lowest frequency) vibration mode. The loudspeaker sets the bar in vibration and the tone frequency is adjusted to maximize the motion of the particles, i.e. to the lowest resonance frequency. With laboratory methods the internal friction and the twisting stiffness of the wood can also be measured. Examples of elasticity modulus for vibrations along, across, and twisting along are given in Tables 5.1-5.3. PLATES AND WOOD PROPERTIES A rectangular plate can sometimes be thought of as a bar of great width, both across and along the grain, c.f. fig. 5.4. The plate will thus get the same nodal patterns as the corresponding bars across


and along the grain. In a real plate there is coupling between the vibrations along and across the grain and the nodal lines will generally be bent, c.f. fig. 5.4c. Resonance frequencies of the plate can be obtained by the tap tone method. The resonance frequencies, the nodal lines and the antinodes are easily obtained with the Chladni method.

Figure 5.11. Typical nodal line patterns for spruce and maple - for the maple plate the frequency order is reversed compared to the spruce platefor the higher two modes shown (from Beldie, 1965).

Such measurements may give the result shown in fig. 5.11. The lowest resonance has two nodal lines forming a cross. In the second, the plate vibrates as a "wide bar" across the grains and in the third resonance as a "wide bar" along the grains. If the plate material is changed to maple the second and third resonance shift positions of nodal lines. This means that the first vibration mode of the plate is a twisting motion. In the following two modes the plate vibrates mainly in two bending motions across and along the grain for the spruce plate and vice versa for the maple plate. With resonance frequencies, length, width and thickness known together with the density the three elastic moduli can be calculated. This can be done with the tap tone method in principle. It may though be difficult to find suitable sizes for plates to obtain suitable tap tone frequencies without destroying the material for building.


Figure 5.12. Numerical experiments showing that the resonance frequency no 1 is determined by the shear modulus Gl (lowest frame), resonance frequency no 2 by the longitudinal elastic modulus El (middle frame) and resonance frequency no 3 by the transversal elastic modulus Et (topmost frame, from Molin et al, 1988).


We have experimented with blanks for top and back plates. By means of the Chladni method, the resonance frequencies, the nodal lines and the antinodes can be recorded and if the vibration sensitivity is recorded the internal friction can be measured in addition. We have started with a standard size for blanks, see fig. 5.5. For the blanks we will in principle obtain the same order of nodal lines as sketched in fig. 5.11b (for the evenly thick maple plate). By means of computer calculations (finite element calculations) we have proved that the three resonances are mainly determined by a single and separate material property. The first resonance is mainly determined by torsion stiffness (the shear modulus), the second by the longitudinal stiffness (the longitudinal elastic modulus), and the third by the radial stiffness (the radial elastic modulus), c.f. fig. 5.12. The computer calculations have also given simple formulas to calculate the material properties if the test blank is close in size to our defined standard, i.e. the length and width measures should be within 1 mm, the thickness within 0.5 mm. If so the density of the wood becomes the blank weight divided by the blank volume (the volume is length x half width x (edge thickness + center thickness) ). The elastic moduli can be calculated as follows: modulus for shear longitudinal radial

17 x 1000 x weight x frekv2 Pa 89 " " " " 4.4 " " " "

For the shear modulus the frequency of the first resonance should be used, for the longitudinal the frequency of the second resonance, and for the radial modulus the frequency of the third resonance. 5.4. SUMMARY: VIBRATION PROPERTIES OF THE WOOD After a repetition of properties of resonances, methods to measure these properties have been presented: the tapping method, c.f. fig. 5.7, the Chladni method and by measuring vibration sensitivity. Material properties for tone wood have been measured and how these influence the resonance properties. Finally methods for determining material properties have been described, which can be used in the maker's workshop. 5.5 KEY WORDS: Resonant frequency, vibration sensitivity, specific vibration sensitivity, bandwidth, nodal lines, antinodes, mass (weight), stiffness, friction, and elasticity (Young's) modulus.


Chapter 5. Second part: TUNING OF THE TOP AND BACK PLATES OF THE VIOLIN INTRODUCTION In this second part some results are presented that can be useful for the maker. First the fundamental principles for tuning followed by typical data of free violin top and back plates, and finally advices on practical tuning of violin plates. 5.6. FUNDAMENTAL PRINCIPLES FOR TUNING (frequencies, bandwidths and nodal lines) There is no simple answer either from the violin maker or the researcher on how a violin top plate should be tuned. Therefore the reader should not expect to find such answers in this work. We shall, however, look at some published works and compare the results with reasonable predictions, and with the results of our own research. Dr. Carleen Hutchins has for many years worked with such methods and is the most prominent maker-researcher of today. The results presented can be used by the maker (both the violin and the guitar maker) to give clues in his own research to improve his or her instruments. TUNING OF A BAR (frequencies, bandwidths, and nodal lines) The mechanical properties of the bar are stiffness, mass and internal friction. When wood is removed from the bar both the mass and the stiffness is reduced. As the stiffness is more sensitive to the thickness (proportional to the thickness to the third power) than the mass (proportional to the thickness), then the resonant frequency (proportional to the thickness) decreases when the bar is thinned by a constant amount over the whole length. J. Alonso made series of experiments to investigate the principles of bar tuning. A number of bars were made and their properties were measured. Thereafter wood was removed at well defined positions with a milling machine. Nodal line shifts and resonant frequency shifts were measured, see fig. 5.13. The experimental results gave the following rules: 1) The resonance frequency is lowered in all cases except from thinning close to a free end. 2) The nodal lines are shifted the most but not much for thinning close to a nodal line. The closest nodal line is shifted towards the near end, and the other nodal line is shifted less but in the same direction. 3) The bandwidths were little influenced For the bar this implies the main tuning rules: a) the resonance frequency is lowered by thinning between the nodal lines, b) nodal lines can only be shifted a little by thinning at nodal lines.


Figure 5.13 Frequency and nodal line shifts for thinning of a bar (from J. Alonso).

OUTLINE OF THE PLATES (Frequencies) : Beldie (1969) made some experiments on the resonant frequencies and nodal lines of violin plates. A flat rectangular spruce plate had the resonance frequencies and nodal line patterns shown in fig. 5.14 upper two lines. When the outline of a top plate is made, the Chladni patterns are changed and the resonant frequencies are shifted. The frequency of the first mode is increased (mass is mainly removed at antinodes, i.e. at the corners). The second resonant frequency has increased which should be expected as the width of the plate is decreased. The frequency of the third resonance is decreased, which also should be expected as the C-bouts give a plate of less width at the position of maximum bending for the corresponding bar. For the last two cases detailed information on the nodal lines are needed for more definite conclusions. Thus our main point here is: the outline of a violin plate has influence on its resonant frequencies.


Figure 5.14.Nodal lines and resonant frequencies for a rectangular spruce plate (360 x 212 x 3 mm with the elasticity modulii 8.9 Gpa and 0.52 GPa and the density 410 kg/m3), flat top plate with contour as a violin, and arched top plate with bass bar (from Beldie, 1965).

Figure 5.15 Arch a) across and b) along the grains.

THE ARCH OF THE PLATES (Frequencies and nodal lines) The arch height of a top or a back plate is fairly small. It is less than 1/5 of the smallest width (1/5 of 10 cm is 2 cm). The plates should therefore be regarded as somewhat arched plates (in technical terms so called "shallow shells").


Figure 5.16. From a blank to a finished top plate - nodal lines and frequencies for a) the blank, b) 8 mm thick rectangle with internal and external violin arching , c) as b but somewhat shorter and narrower upper part, d) 4 mm in the upper and lower parts, and 5.1 mm in the middle, and e) the finished top plate 2.8 mm thick (from Jansson, Alonso Moral and J. Niewczyk, 1988).

The radius of curvature across the violin plate is of the magnitude 0.2 m and that along the plate 1.5 m. The difference in radii implies that the arching will increase the cross stiffness more than the length stiffness of the violin plate. Experiments were made with two arched rectangular plates, c.f. fig. 5.15 by Jakub Niewczyk, with cross and length radii, respectively, as given above. Thereby it was found that 1) the first resonance had the nodal lines in a cross as for the rectangular plate 2) the plate become much stiffer than the corresponding flat plate for bending across the arch but about the same for bending along the arch. 3) the influence of length/width together with grain orientations was large, i.e. the so called edge effects are large. For the plates with violin shaped outline it is not so simple to make detailed comparisons. Assuming that the nodal lines are the same for the flat and the arched plates in fig. 5.14, this can however be done. One finds that the first and second resonances increase considerably with the arching (approximately 50 %). From the third resonance of the flat plate and to the fifth of the arched top plate the resonant frequency is much increased (approximately 200 %). The effect of arching can


be "translated" in thickness. This means that the 50 % increase in resonant frequency from the length arching corresponds to 50 % increase in thickness, i.e. an increase in thickness from 3 to 4.5. In the same way the cross arching corresponds to a 200 % increase, i.e. thickness increase from 3 to 9 mm. Beldie's experiments imply a large influence by the arching on the top and back plate stiffness. BLANKS, TOPS AND BACKS (Frequencies,

arching and thickness) The development of the nodal lines of the finished violin top plate from those of the blank have been investigated by Jakub Niewczyk and the author, see fig. 5.16. Starting with the blank we find two nodal lines forming a cross for the first resonance. This nodal cross remains through all following steps of adjustments. The second resonance has two horizontal nodal lines initially but changes to those typical for the top plate in violin arched and 8 mm thick but still rectangular (step b). This nodal pattern remains thereafter. Note that although the nodal lines changed from step a to b the antinodes (at the edges between the nodal lines) remained and throughout the whole experiment. The third resonance has in the step b one straight and one curved nodal line. The pattern but upside down was found in step c. In step d (the plate thickness varied from 4 to 5 mm) the fifth violin plate resonance, the ring mode was found. The ring mode became more rectangular with further thinning. This experiment implies that when the plate is worked to a certain thickness related to the arching, then the ring mode is obtained. In step d the third resonance of the violin plate was found (which is the same as the fourth of the blank). In the experiments the main features of the nodal patterns were little shifted with the thinning in the different areas. The fifth mode was the most sensitive to thinning.

5.7 TYPICAL PROPERTIES OF FREE VIOLIN TOP AND BACK PLATES DATA FOR BLANKS Experiments have been made with blanks of the standard dimensions (see fig. 5.5) to determine the relations between material properties and vibration properties. Ten top blanks and ten back blanks (not necessarily typical material properties) gave the first three resonance frequencies which fell in the ranges (tone names within the following brackets). By means of the formulas given in the appendix below the vibration properties were estimated. For the top-plate blanks the frequencies were 270-333 Hz (C4-E4), 471-577 Hz (A#4-D5), and 596-727 Hz (D5-F#5). The masses (weights) were 445-545 g. For the back-plate blanks the frequencies were 326-352 Hz (E-F), 474-540 Hz (A#4C#5), and 682-770 Hz (E5-G5). The masses were 630-710 g. The nodal lines varied moderately. Only in one case of a top-plate blank a large deviation was found.


Figure 5.17 Typical resonant frequencies, quality factors and nodal line patterns for top and back plates (L stands for vibration level measured in the very experimental arrangement)


CLADNI PATTERNS (resonance

frequencies, bandwidths, and nodal lines) The resonance vibrations have been recorded of free top and back plates by the Chladni method, see fig.5.17. The three resonances can easily be recorded and can be used as guidance in making. In the first resonance, c.f. fig. 5.17, both the top and the back plate have two nodal lines forming a cross. The positions of maximal vibrations, antinodes, are at the edges between the nodal lines. Diagonally positioned antinodes vibrate in phase. The antinode below or beside vibrate in antiphase (i.e. a twisting mode). The second resonance has maximum vibrations also along the edges and between the nodal lines, but now in centre of the upper and lower edges, and at the C-bouts. The nodal lines of the typical top plate are shaped as two vertical brackets )( and the nodal lines of the back plates of two horizontal but bent nodal lines. Note that the vibration maxima are in the same positions both for the top and the back plates but the nodal lines look rather different. The third resonance, which in reality is the fifth resonance has a vibration maximum in the centre and others along the edges. The nodal line tends to close and is therefore called the ring mode. For the back plate the node is closed but for the top plate it opens at the C-bouts. The typical nodal line patterns shown in fig. 5.17, were obtained from measurements of 14 top and back plates, which had not been tuned. These nodal patterns can be compared with those of the blanks. Thereby one finds that the first resonance has the same nodes for the blanks, the top and the back plates. The second resonances of the blanks and of the back plates look similar. If one compares positions of maximal motions one finds that these positions are similar for the second resonances of the blanks, of the top, and of the back plates. There are also similarities for the third blank resonances and for the ring modes of the top and back plates; maximal vibrations in centre and at edges (at corners). The first, second and fifth resonances of the free plates could be used to copy vibration properties of free plates. Carleen Hutchins has successfully used this method and it will be presented in the next section. The implied similarities between blank properties and plate properties must not be taken as evidence that the blank modes and the plate modes are the same. The relations are far more complicated. The ranges of resonant frequencies and bandwidths (Q-factors) are given in fig. 5.17. From theory and measured nodal lines some general predictions can be made. In mode no 1 the plate is twisting around the crossing points of the nodal lines. Thus reducing the thickness around the nodal cross should give a large influence. In mode no 2 there is bending both longitudinally and transversally with the main bending in the middle and thinning here should give large influence. In mode no 5 the plate is bending both longitudinally and transversally, and is twisting. Thinning off the edges and off the middle should give the largest influence on this mode.


Figure 5.18. Nodal lines for resonances 2 and 5 (average of five top plates) a) without f-holes (point-dashed lines), with fholes (dashed lines) and with bassbar (full lines) and b) with ribs (vibration mode 2 with dashed lines and two resonances approximately as no 5 with full lines and cross lines, respectively, in cooperation with Hansen).

Table 5.4: Frequency shifts and average frequencies for five violin top plates (Hansen) frequency change with f-holes plus bassbar plus ribs

mode no 1 -7% 8% -14 %

no 2 - 11 % +4% + 11 %

resonant frequencies with f-holes and bassbar

89 Hz

150 Hz

no 5 - 10 % + 13 % -25 resp -18 % 342 Hz

F-HOLES AND BASSBAR

Two large additional changes are made to the top plate. The first one is the cutting of the f-holes and the second is the instalment of the bassbar. Experiments were made together with Birgit and Carlo Hansen, who made five top and five back plates of the same geometry with f-holes and bassbar. The shifts of nodal lines and resonance frequencies were measured and the average measures are given in fig. 5.18 and Table 5.4. The experiments show that the f-holes lowered all three frequencies and the second and fifth the most (this should be expected from the experience of the author). The bassbar influence the cross stiffness little but the stiffness of resonance no 1 and 5 considerably.


When tuning plates it is mainly the vibration properties such as resonant frequencies and modes of vibration (nodal lines and antinodes) that are adjusted. It seems also to be advantageous to be able to adjust the bandwidths. The possibilities for the maker are that he can choose properties of his wood material, the arch height of the plates and their thickness distributions. In addition the violin maker has some freedom with the design of the bassbar and the f-holes, and the guitar maker with stiffening ribs and the bridge design. The tuning of violin plates is most common and will only be discussed in the following. For five top plates the nodal lines were investigated in detail, see fig. 5.18 and table 5.4. Thereby it was found that the nodal lines of the resonances no 2 and no 5 cut each other in closely the same positions. The positions were not noticeably changed by f-holes or bassbar and were also the same in five back plates.

Figure 5.19 Suitable holding and tapping positions for tests of free top and back plates. Hold at circle (full line for the best but somewhat more difficult place to find , dashed circle for the position simple to find but not the best) and tapping positions (triangles) for a free top plate and back plate (the correct holding point should be within 10 mm of the given measures).


5.8 ON PRACTICAL TUNING OF VIOLIN PLATES GENERAL

The effects of changing material properties and sizes of the plates have been demonstrated. Note that the order of nodal line patterns may change for very large changes of properties, i.e. the sizes of the test blanks must not deviate much from standard. This shift of order implies that it is the resonant frequencies and not the nodal pattern that are the most important. Therefore the frequencies of the blanks obtained bý tapping, c.f. Figs 5.6 and 5.19 are very important The results are supported by experiments with blanks employing the Chladni methods, vibration sensitivity and optical measurements.

Figure 5.20 General rules for tuning of the second and fifth resonance of the top plate (from Hutchins, 1983).

Figure 5.21 General rules for tuning of the second and fifth resonance of the back plate (from Hutchins, 1983).

By means of the tap tone method the violin maker can obtain the resonant frequencies of the three resonances of a blank, c.f. fig. 5.6. He or she holds the blank at a nodal line of the wanted resonance and taps close to an antinode, c.f. fig. 5.6. By keeping a standard size close to ours, the mass, and


the three resonance frequencies give a record of the wood properties. By using blanks of our sizes the technically defined elastic parameters can be calculated. We hope that violin makers will start to use this method and thereby be able to define blank properties suitable for their making and also to give clues to choice of arching and thickness of plates. CARLEEN HUTCHINS' RULES - MORE DETAILED TUNING Carleen Hutchins has much experience with tuning helped by Chladni patterns and has given some directions for use of the second and the fifth resonances. One should begin with a top plate without f-holes and bassbar that is a little too thick, c.f. the leftmost sketch in fig. 5.20. Thinning at the edge and a few centimetres inside lowers the frequencies of the two resonances by approximately the same amount. If the plate is thinned in the regions just above or below the f-holes the frequency of resonance no 2 is lowered more than that of resonance no 5. Thin from broken line towards the line of dots. It should be possible to tune resonance no 5 with the bassbar. For the back plate it is suggested to start from a thickness distribution as the leftmost sketch of fig. 5.21 Thinning along the edges as shown in the following sketch result in the frequency of resonance no 5 being lowered more than that of resonance no 2. Thinning inside the corners lowers the frequencies of resonances no 2 and 5 by the same amount. If the back is thinned towards the centre then the frequency of resonance no 2 is lowered more than that of resonance no 5. Start by thinning from the edges and towards the centre 3 to 2.5, 3.5 to 3 etc. and make the thickness distribution more circular in the centre. NIEWCZYK AND JANSSON’S TUNING RULES Table 5.5 Frequency shifts for thinning in different areas (-3/-9 means that a thinning on the outside (5.1 to 4.0 mm) gave a frequency shift of -3 % and a thinning on the inside (4.0 to 2.8 mm) gave a frequency shift of -9 % (areas defined in table figure).

Frequency shift rel. previous step for resonance Thinning in area 1

no 1

no 2

no 5

-3/0

-1/-1

-1/-1

2

-2/-2

-2/-6

-3/-9

3

-8/-8

-9/-11

-4/-1

4

-6/-11

-11/-9

-1/-4

In the later steps of the J. Niewczyk experiments (fig. 5.16) the thining were made in such a way that comparisons could be made with the presented experience by Hutchins. Thereby it was found that when the whole plate was thinned, then the frequencies of resonance no 1 and 2 were lowered 70 % of


the thickness decrease, while that of the resonance no 5 was only lowered 50 %. For a flat plate the resonance frequencies are lowered 100 % of the thickness reduction. Thus it can be concluded that the arching has a large influence and mainly on resonance no 5. The effects on frequencies of thinning in the different steps showed that the two outmost areas influenced mainly resonance no 5 and the two central ones mainly resonances no 1 and 2, i.e. in agreement with Hutchins' findings, c.f. Table 5.5. The results are also in agreement with theoretical predictions and finite-element calculations.

Figure 5.22 Nodal lines and tuning of the bass bar. Shadings mark diffuse nodal regions. Start in a) with a too high bassbar (17.2 mm and 313 Hz), in b) bassbar lowered to 14.4 mm (308 Hz) and ending in c) bassbar height 12.8 mm and 307 Hz (from Bissinger and Hutchins, 1976).

THEORETICAL PREDICTIONS Calculations with finite element methods predicted that the resonances no 1 and 2 should be the most sensitive to the thickness and that resonance no 5 is the one most sensitive to the arching. Furthermore the calculations indicated that the resonances 1 and 2 are the ones most sensitive for thinning in areas 3 and 4 but the resonance no 5 is most sensitive for thinning outside the middle (areas 2 and 3). The presented information agrees on that the central thickness influences mode no 2 more than mode no 5 and that the bassbar influences mode no 5 more than mode no 2 (table 5.4). BASSBAR TUNING The Chladni method can be used to help with adjusting the bassbar, see fig. 5.22, which shows the Chladni patterns for three stages of adjustment. With the very strong bassbar the nodal lines of resonance no 5 do not pass the bassbar towards the nearby edge. The top plate is so stiff that it does not want to vibrate in across the bassbar. By thinning and cutting down the bassbar the resonant frequency is lowered approximately 6 Hz and at least a tendency for a ring mode is obtained. By


further successive thinning and lowering the ring mode starts appearing. The changes are large for the nodal lines on the right side (as shown in the figure) but small on the left side. The last adjustment on the upper and middle parts of the bassbar seems mainly to "fold" the lower nodal line up to the wanted cutting point along the edge close to the bassbar.

Figure 5.23 Resonance no 2 and no 5 in the back plate a) with well balanced thickness distribution, the back b) with the upper nodal line spread in the upper part (upper part too thick compared to the lower part) and back plate c) with the upper nodal line of resonance 5 cutting the edges (plate too stiff between c bouts, from Hutchins 1989).

WHAT THE NODAL LINES CAN TELL US If the nodal lines differ from the expected one, this means that the distribution of stiffeness and mass in the plate deviate from the “right” one. If we learned the “language” spoken by the nodal lines, they tell us what is “wrong” or “erroneous”. Two examples of erroneously tuned back plates have been given by Carleen Hutchins, see fig. 5.23. Only second and fifth modes are discussed. The Chladni patterns of the second resonance look as if there is a central nodal line. This is, however, only an area of little motion and no nodal line. Thus the


second resonance has two nodal lines, the upper one shaped as a v and the lower one shaped as an upside down v. The fifth resonance has an almost closed ring node. The plate in b) has an upper part too thick compared to the lower one, which the distorted second resonance shows towards the second resonance of the blank, i.e. a much too thick plate, but the fifth resonance indicates somewhere between step d and e in fig. 5.16, i.e. a thin plate). With the central part too thick, the plate in c), the upper nodal lines of the ring "spill over" the edges (comparison with the Niewczyk experiments suggest that the back qualitatively is somewhere between steps b and d, i.e. a much too thick plate). Thus there is at least a partial agreement between our experiments and the examples by Hutchins. The deviations may stem from the fact that we are comparing a spruce top plate with a maple back plate. Increasing the cross stiffness much, with the ribs for instance, gives the top plate two modes corresponding to the third mode of step b the way it is shown and the upside down version. c.f. fig. 5.18b. The ring mode divides into two resonances.

5.9 SUMMARY: TUNING OF THE TOP AND THE BACK PLATE This chapter has been more of a research character than earlier, as we are at present in the area of research. It is difficult to give accurate rules for the best properties of a free top and back plate. Therefore typical properties have been presented together with principles for tuning, i.e. principles for how the resonances can be affected. 5.10 KEY WORDS Resonance, frequency, nodal lines, resonance no 1, 2 and 5, the ring mode


5.11 APPENDIX FORMULAS AND CALCULATIONS As a preparation for the physics experiments that the inclined violin maker might wish to carry out, the formulas and calculations for the determination of material properties are given below. For these calculations we introduce the bar length l, thickness t, and width b. Furthermore we introduce the elasticity modulus E, the mass M and the density ρ and the π = 3.14. FORMULAS: a) Density of the wood the bar is made of:

ρ=

M blh

b) First resonant frequency of a bar with free ends:

1 E F = h ⋅ l2 ⋅

ρ

(exact formula F = 1.028 ⋅ )

c) If the first resonant frequency, weight and dimensions of the bar were measured, one could calculate the elasticity modulus E = ρ

l4 1 l3 2 2 ⋅ F = M F h2 b h3

d) Specific vibration sensitivity SV =

1 blh 1 = M 2π F SM

e) Peak level RN = 20 ⋅ log(

F ⋅ SV ) B

e-πBt . If the bandwidth B was measured, one could calculate the decay time T60 of ln 1000 1 the free vibration T60 = * π B Note, the value of T60 is valid only for the bar or plate whose resoance delivered the B value. f) Reverberation

NUMERICAL EXAMPLES To give a feeling for material properties, usefulness of methods and formulas, some numerical examples are presented in the following. Example 5.1. The density of wood can be calculated by dividing the mass by the volume. A spruce bar weighs 1.4 g and a maple bar 2.0 g. Both are 3 mm thick, 10 mm wide, and 100 mm long. What is the density of the spruce and the maple, respectively?


Solution: with equation a Mass of the spruce bar 1.4 g = 1.4/1000 kg the volume is 100 mm x 10 mm x 3 mm = = 100/1000 m x 10/1000 m x 3/1000 m = 3/1000 000 m3 density (mass in kg / volume in m3) = = (1.4/1000 kg)/(3/1000 000 m3) = 466 kg/ m3 Calculated in the same way gives the density of the maple is 666 kg/m3. Example 5.2. The elasticity modulus of wood can be calculated from weight, resonance frequency, length, thickness, and width of a bar. Two bars of spruce cut along and across the grain (see fig. 5.9) have the resonance frequencies 1710 and 390 Hz, respectively. Corresponding values for similar maple bars are 1200 and 540 Hz, respectively. The bars have the same weights as in example 1. What are the elasticity modulus in the four cases? Solution: A little calculation from the frequency formula above (the c equation ) gives the elasticity modulus as spruce bar along (longitudinal) 15.2 GPa spruce bar across (radial) 0.79 " maple bar along (longitudinal) 10.1 " maple bar across (radial) 2.2 " (15.2 GPa = 15.2 x 1000 000 000 Pa = 15 200 000 000 000 Pa) Example 5.3. The internal friction is measured in form of bandwidths. Examples of values for the four bars (Example 5.2) are given below in the first column. Solution: The formula for the decay gives a decay time (-60 dB) of the values listed below in the second column spruce bar along (longitudinal) spruce bar across (radial) maple bar along (longitudinal) maple bar across (radial)

11.4 Hz 0.19 s 8.5 " 0.26 " 14.5 " 0.15 " 9.6 " 0.23 "

The decay times 0.15-0.23 sec are very short compared to that of a room, approx. 1 sec.


5.12. REFERENCES: 1. Carleen Hutchins: “The Acoustics of Violin Plates”, Sci Amer 245 no 4 1981, pp. 126-135. 2. Carleen Hutchins: “Plate Tuning for the Violin Maker”, Catgut Acoust Soc Newsletter no 39 1983, pp. 25-32. 3. Carleen Hutchins: “Note for the violin maker in free plate mode tuning and plate stiffnesses”, CAS Journl, Vol.1, No.3, May 1989. p. 25. 4. Erik Jansson, Jesus Alonso Moral and Jakub Niewczyk, Experiments with Free Violin Plates, J Catgut Acoust Soc 1 no 2 1988, pp. 2-6. 5. Nils-Erik Molin, Lars-Erik Lindgren and Erik Jansson: Parameters of Violin Plates and Their Influence on the Plate Modes, J Acoust Soc Amer 83, 1988, pp. 281-291. 6. Erik Jansson and Nils-Erik Molin: On Tuning of Free Violin Plates, J Catgut Acoust Soc 1 no 3 1989, pp. 27-30. 7. George Bissinger and Carleen Hutchins: “Tuning the Bass Bar in a violin Plate”, CAS Newsletter No.26, Nov.1976, pp. 10-12. 8. Ion Paul Beldie: “Chladni patterns and Resonant Frequencies of violin Plates“, InstrumentenbauZeitschrift, Vol.23, Feb.1969, No.2, pp.168-174 (in German). 9. Ion Paul Beldie, Measuremet of Resonant Frequencies of Violin Plates in the Tuning Process”, Industria Lemnului, Vol.16, No.4, 1965, pp. 141-147. (in Romanian) 10. Daniel Haines: “The essential mechanical properties of wood prepared for musical instruments”, CAS Journal, Vol.4, No.2, Nov.2000, pp.20-32. 11. Franz Jahnel:“ Die Gitarre und ihr Bau“ (in German), Verlag Ds Musikinstrument, Frankfurt am Main, 2nd edition 1973.




Erik Jansson

Chapter VI: The Function, Tone, and Tonal Quality of the Guitar



Sound and hearing

Chapter II


Chapter III

Sound and the room

Chapter IV


Chapter V


Chapter VI


Chapter VII


Chapter VIII


Chapter IX



ACOUSTICS FOR VIOLIN AND GUITAR MAKERS Chapter 6 – Applied Instrument Acoustics THE FUNCTION, TONE AND TONAL QUALITY OF THE GUITAR First part: THE FUNCTION OF THE GUITAR 6.1 Construction of the guitar 6.2 Resonances of the guitar 6.3 The function of the guitar 6.4. Summary 6.5. Key words Second part: THE TONE AND TONAL QUALITY OF THE GUITAR 6.6 Quality determining properties 6.7 Construction and acoustical properties 6.8 How to make the best guitar 6.9. Summary 6.10. Key words

Jansson: Acoustics for violin and guitar makers p 6.2

Chapter 6. THE FUNCTION, TONE AND TONAL QUALITY OF THE GUITAR First part: THE FUNCTION OF THE GUITAR INTRODUCTION As a guide for this chapter on the acoustics of the guitar six questions have been set up: 1. What is a guitar? 2. What properties are built into the guitar? 3. How does the guitar work? 4. What is quality? 5. How do the different construction details influence the acoustical properties? 6. How to make the best guitar? The questions are answered as well as possible in the following. But let us first look a little on the acoustical fundamentals for the function of the guitar. And as the guitar string cannot be separated from the guitar, this introduction contains a short repetition on the acoustics of the guitar string. A guitar tone contains a spectrum of partials. One effect of the number of partials is the timbre effect of smoothness versus roughness. For a spectrum of one to four partials all played tones sound smooth. If the number of partials are increased to six the played tone attains some roughness. The perceived difference smooth to rough can be explained by properties of our hearing. This is an example of how we can predict perceived qualities from acoustic measurements. But let us look further into the tone of the guitar. Three diagrams, the snapshots of spectra of a note of the sixth string (an octave above the open string fundamental 165 Hz) are shown in Fig. 6.1. The three diagrams show the spectrum just after plucking, 0.2 and 0.4 sec later. We see here that the uppermost "snapshot" contains 14 visible partials of different levels. The second spectrum contains 9 partials and the third 7 partials. If we look a little closer, we can see that the level for the different partials decrease at different speed, the lower five partials decrease approximately 3 dB in 0.2 sec, the following 5 partials 10 dB and the higher four 15 dB. A guitar sound can be synthesised, but not by only synthesising the partial spectrum (of the string) in Fig. 6.1. Care must also be taken of the partials of the guitar body. Let me point out another detail. If we look a little closer we can see that the sixth and eleventh partials have very low levels - two obvious minima are to be found here. I shall later return to these minima and explain what they derive from. The player can select point of plucking along the string, direction of plucking and way of plucking. But each selection will give a different spectrum and a different character of the played tone.


Figure 6.1 Analysis of a real guitar tone

Figure 6.2. Effects of different plucking directions.


Figure 6.3. The production of the guitar tone (in principle)

We have seen that a musical tone consists not only of one single frequency component but a number of frequency components, partials. Every partial corresponds to a specific mode of the string vibration, a string resonance. The lowest partial, the fundamental corresponds to a resonance with nodes at the ends and an antinode in the middle. In the second partial the string vibrates with nodes in the ends and in the middle. Halfway between the nodes there are antinodes. The two vibrations can be demonstrated by looking at the guitar string. To enhance the second resonance the octave flageolet is played. The following fundamental relation is valid: A partial can not be excited at the positions of its nodes. The minima in Fig. 6.1 correspond to plucking close to the nodes of the sixth and 11th partials. Another plucking effect can be used, see Fig. 6.2. If the string is plucked perpendicular to the top plate a strong but short tone is obtained, in principle as in the uppermost frame. If the string is plucked parallel with the plate a long but weak tone is obtained. A more normal plucking at an angle directed towards or away from the top results in a guitar tone as sketched in the lowest frame and is best described as consisting of two parts. In the first part the vibrations perpendicular to the top dominate and in the later part the vibrations parallel with the plate. In an intermediate phase both parts contribute which result in a soft transfer from initial to the later part. 6.1 CONSTRUCTION OF THE GUITAR The guitar consists of six strings stretched over a wooden box (the corpus) elongated with a neck. When a string is plucked the string starts to vibrate. The vibrations are coupled to the body via the bridge and the vibrations of the body set the surrounding air into vibration, which gives the guitar tone we hear, c.f. Fig. 6.3. Let us look a little closer at the construction of the guitar, see Fig. 6.4. The back plate, has strong cross bars. There are also such strong cross bars underneath the top plate under the fingerboard and just on the opposite sides of the sound hole. This favours the top plate vibrations in the lower part of the top plate, below the sound hole. Therefore we somewhat arbitrarily call this part the "the free top plate part". Two examples of the construction of two guitar top plates are shown somewhat more in detail in Fig. 6.5. Again we see the strong cross bars under the fingerboard and one strong cross bar on the


other side of the sound hole. The "free" part is made stiffer by the transverse bridge, sometimes combined with extra stiffening underneath the top, and mainly lengthways by thin braces. Often slightly slanted, an extra crossbar can be found at the sound hole, see Fig. 6.5b.

Figure 6.4. The construction of the guitar - an example (after Sloane)

Figure 6.5. Examples of different constructions of the guitar top plate: common simple construction (left) and typical handmade Spanish construction (right).


Figure 6.6. Typical measures for the classical guitar.

The measurements of a guitar vary moderately, see Fig. 6.6. The neck length equals half the string length. Guitars may look different but still the external measures are rather similar. Note that the bridge is positioned approximately halfway between the sound hole and the lower end, i.e. approximately in the middle of the "free top plate part". The width and length of the "free


part" are approximately the same. The bracing system may vary considerably, positions, number and dimensions. A most interesting question is: What do these differences do to the tone?

Figure 6.7. Function of the guitar, in principle (after Christensen)

Figure 6.8. For comparison with different loudspeaker enclosures a) open baffle, b) open back baffle, c) closed baffle, and d) bass reflex baffle (after Cohen).

6.2 RESONANCES OF THE GUITAR The fundamental construction of the guitar is thus an air volume enclosed by walls, see fig 6.7 top. The top has a part , the "free part" which is more easily set into vibration than the other parts. In the top plate there is also a large hole, the sound hole. The construction is basically the same as a rather advanced loudspeaker construction, the bass reflex enclosure, c.f. Fig. 6.8. The properties of the guitar, this several hundred years old bass reflex enclosure, are mainly determined by the top plate and the air volume resonances. But the guitar has a back plate too, which can vibrate, i.e. we have the much more complicated system, top plus volume plus back resonances to take care of. Experiments on this complex system have shown (c.f. Fig. 6.7 bottom) that the back can supply prominent resonances to the guitar body.


Figure 6.9. Top plate resonances recorded with hologram interferometry. The closed rings (fringes) connects lines of equal vibration in the same way as lines of equal altitude are connected in a topographical map. Resonant frequencies at a) 185 Hz, b) 285 Hz, c) 460 Hz, and d) 510 Hz, and e) 645 Hz. (guitar top made by G. Bolin, interferograms by Molin and Stetson).

It has been shown that the two resonances are present by vibrating the guitar with a small vibrator or a loud speaker. As the frequency of the vibrations are slowly increased, similar to glissando, one can clearly hear the so called Helmholtz resonance or A0 and thereafter the first top plate resonance. If the glissando is continued to higher frequencies higher resonances will be heard. The vibration patterns of the higher resonances can be demonstrated with a rubber membrane stretched over a set of sides for the guitar. Small pieces of cork are spread over the membrane. The membrane is set into vibration by a small vibrator. For a glissando the "first top" resonance will first be seen thereafter the second, the third etc. All show up at a specific frequency, their resonant frequency and they all show different patterns. The nodal patterns remind us of the patterns found for a real guitar top (measured with hologram interferometry, see Fig.. 6.9)


Figure 6.10. Vibrations of a guitar body with neck (by E Meinel)

The guitar neck may also vibrate. The neck vibrations can cooperate with the body vibrations, c.f. Fig.. 6.10. In Fig. 6.9 it is shown that the lowest resonance of a guitar top plate has one vibration antinode (a "hill"). The following resonance has two antinodes and a vertical nodal line in between. The third resonance has two antinodes with a horizontal nodal line at the bridge, the fourth has three vibration antinodes with two vertical nodal lines, and the fifth has four antinodes with two nodal lines in a cross through the bridge. We have thus shown that there are many resonances in the free part of the top plate. These resonances are important for the tonal quality. It can be seen that except for the first top plate resonance, the vibrations tend to be small at the bridge and large outside the bridge. In addition we have earlier shown that the guitar tone consists of several partials, which are important for the tonal quality. If the plate resonances are compared to the string resonances, similarities are to be found regarding the position of antinodes and position of antinodes, see Fig. 6.11. The top plate vibrations can be said to be a combination of string vibrations along and across the top plate.


Figure 6.11. String and plate resonances.

Figure 6.12. Air resonances at 370 , 540, 760, 980 and 1000 Hz - positions of pressure antinodes (thick lines), pressure nodes (dashed lines), white and shadowed areas mark phase of motion - the same marking for in phase and different marking for antiphase.


Figure 6.13. The production of the guitar tone including the feedback (in principle)

An air volume like the one of the guitar has many resonances too, see Fig. 6.12. These air resonances can play an important role and co-operate with plate modes. To be able to use the construction of a guitar to an optimum a monitoring of the co-operation between resonances is important. The lowest resonance is especially important. The lowest air resonance can be demonstrated with a small loudspeaker inside the guitar. With a glissando tone a strong maximum is heard at the frequency of the air resonance, but also higher air modes may be heard. 6.3 THE FUNCTION OF THE GUITAR Let us look more in detail at the co-operation between the strings and the guitar body, see Fig. 6.13 marked by the arrows pointing to the right. On plucking, the string is set into vibration and thereby the top plate too. But the top-plate vibrations can also react to the string vibrations. This reaction (feedback) has been marked with an arrow pointing to the left. The main road is though from left to right. Too much reaction gives a wolf note - too little means a weak tone (the electric guitar without amplifier).


Figure 6.14. Deformations of the guitar top plate Measured (with hologram interferometry, double exposure) for different string forces: parallel with the top plate along the bridge (force 1 N, upper left), perpendicular to the top plate (force 0.5 N, upper right), along a string (force 2 N, lower left) and torsion of a string (one revolution in the middle of the third string, lower right).

Let us continue a little theoretically, to show how one can use the resonances to explain what happens in the guitar body. We have earlier (in chapter 2 on resonances), explained the relations between resonance vibrations, normal modes, frequencies and bandwidths. For resonances of the top plate the same relations are valid. The resonances can be heard by tapping the top plate with damped strings.


Figure 6.15. Definition of initial level and initial level fall.

The string and the guitar's top plate co-operate in the following way, c.f. Fig. 6.14. The string is pulled from equilibrium in different ways and gives rise to three partial forces. The partial force parallel with the plate gives a deformation reminding us of resonance no 2, c.f. Fig. 6.9. The partial force perpendicular to the top plate gives a deformation corresponding to resonance no 1. The increased tension of the string at the plucking gives a deformation corresponding to resonance no 3. It can be seen that the top plate is most sensitive to the forces directed perpendicular to the top plate and the least sensitive to the forces directed along the string. This indicates that the string vibrations perpendicular to the top plate are the most important for setting the plate into vibration. An experiment to find how the guitar influences the tone was conducted by Graham Caldersmith and the author. For the experiment two measures were defined: initial level, i.e. the level just after plucking, and initial level fall, i.e. the decay of the level during the first second, c.f. Fig. 6.15. Three chromatic scales on each string were recorded. Let us see how the initial level varies with different pitches, frequencies for single partials (acoustically the analysis must divided into single partials as partials of different frequencies are treated differently). We can see that the initial levels marked with the different bars mark a softly bending line, see Fig. 6.16a, with maxima and minima. We continue by looking at the initial level fall at different frequencies for the single partials (here also the analysis must be made of single partials for the same reason as given above), see Fig. 6.16b. We can here see that the initial level fall marked with the different bars again mark a softly bending line with minima and maxima. In Fig. 6.16c a recording is shown of the sound resulting from driving the bridge with a small vibrator and a "glissando tone" (the frequency response), which should predict the level of partials at different frequencies (pitches). We can see that the level of radiation as a rule varies greatly from one frequency to another - we can see marked peaks with valleys in between. What does this mean? Let us try to find some relations!


Figure 6.16. From measurements on a guitar a) initial levels, b) initial level falls, c) sound radiation (in one direction), and d) vibration sensitivity.

First trial: If we place Fig. 6.16a on top of Fig. 6.16b no relations are found (the two softly bending curves indicated by the two set of bars do not follow each others). Second Trial: If we place Fig. 6.16b up-side-down on top of Fig. 6.16a, we can see that the two sets of bars indicate a similar curve. This implies that the initial level and the inverse of the initial level fall tend to follow the same curve. We have found a relation between initial level and initial level fall. Third trial: Place now Fig. 6.16c on top of the Fig. 6.16a and the up-side-down Fig. 6.16b. Again we see something. The sound radiation response predicts fairly well the initial level and the upside-down initial level fall. This means that the properties of the guitar tone can be tied to the properties of the guitar body. This is nice but not as surprising as it may look. The sound radiation peaks overestimates the initial levels but underestimates the initial level falls, though.


Figure 6.17. Force from string vibrations and sound.

Figure 6.18. Time history of sound levels for the same tone of a high quality guitar (HQ) and a medium quality guitar (MQ).

It is very difficult to measure the sound radiation like that shown in Fig. 6.16c and it demands an anechoic chamber. The same information is obtained by measuring the vibration sensitivity, see Fig. 6.16d. To measure the vibration sensitivity has the great advantage that it could be made in an ordinary room. Thus the string vibrations predict the properties of a guitar, see Fig. 6.17. This is demonstrated with the analysis of the tone of a high quality guitar. We see that the sound level curve and the force level curve follow the same course. Further it can be seen that the start levels are the same for the string forces but that the quickly decaying tone gives a higher initial sound level. I believe much of the secret with the quality of a guitar is shown with the presented material.


In Fig. 6.18 we can see that a tone with high initial level may in spite of faster decay give a tone of longer duration also for a fairly low level of the reverberant sound in the room. This indicates that one guitar can be good for fast loud music but less good for slow soft music. 6.4 SUMMARY - THE FUNCTION OF THE GUITAR In this part the construction of the guitar has been described, its function and its built-in properties. Properties of played tones have been related to properties of the guitar. 6.5 KEY WORDS: Resonances, frequency responses and vibration sensitivity, initial level and initial level fall.


Chapter 6. Second part: THE TONE AND TONAL QUALITY OF THE GUITAR 6.6 QUALITY DETERMINING PROPERTIES The results from three large investigations, one by Jürgen Meyer, one by Bernard Richardson and one by the author form the sources of information to the second part of this chapter. The author, in his investigation on guitars and their quality, asked: what is the most important properties of a concert guitar. The answers showed that the "tonal strength" of the guitar should be the most important. When asked, all guitar players answered that volume, carrying power, strength etc. are desirable. Tone length and timbre should be the second most important, as six gave answers relating to these properties. Further results are presented in table 6.1. Table 6.1. The most important properties of the guitar (3 of 9 means that three of nine guitar players gave this property as important)

1. Attack 3 of 9 2. Carrying 9 of 9 3 Dynamic range 3 of 9 4. Eveness 5 of 9 5. Timbre 6 of 9 6. Timbre plasticity 4 of 9 7. Length of tone 6 of 9. Further an informal listening test (with trained listeners) with five guitars was conducted, which gave the results presented in table 6.2. The three handmade Spanish guitars (no 1 to 3) were perceived as better than the factory made Swedish ones (no 4 and 5) both in music performed live and recorded. The order was slightly changed between the Spanish ones though. Table 6.2. Quality order of five guitars (how well do the guitars sound, thirteen and nine guitar students SMH)?

rank 1 2 3 4 5

music recorded music live guitar no 2 guitar no 3 1 1 3 2 4 4 5 5

RESULT: TRAINED LISTENERS ARE LIKELY TO JUDGE THE QUALITY OF A GUITAR WELL AT A LISTENING TEST. Tones from the open strings and their octaves were recorded on tape for guitar 1 and 4 for listening tests. The tones were presented in pairs for the subjects and the subjects were asked to judge which tone was the strongest, sounded the best and was the longest. The results are summarised in Fig. 6.18. The tests showed that it was rather simple to make the judgements and


that non-trained listeners gave judgements with good agreements. The results for the tone lengths are somewhat unpredicted as the tones were physically of identical length by "electronic cutting". RESULT: AT COMPARISONS OF TWO TONES DIRECTLY AFTER EACH OTHER, ALSO NON-TRAINED LISTENERS CAN EASILY JUDGE DIFFERENCES.

Figure 6.18. Results from three simple listening tests with tones of two guitars: How much stronger is (18 guitar students), how much better timbre has (5 members of Dept Speech Communication and Music Acoustics), and how much longer tone has (7 department members) guitar 1 than guitar 4?

Figure 6.19. Long time average spectra for a high quality (1) and a medium quality (4) guitar.

Tones from recorded music was analysed in form of average spectra over a long time, see Fig. 6.19. Thereby it was found that the better guitar had a higher level up to 3 000 Hz. Within this frequency range the less good guitar had a noticeably lower level below 200 Hz and above 400 Hz. Meyer has conducted listening tests and measured properties for 15 guitars. The listening test with recorded music showed that the quality order between the guitar depends somewhat on the played music. The variations are moderate though. The guitars seem to group into three quality classes, see Fig. 6.20.


For the 15 guitars also the sound radiation was measured. Three resonances (corresponding to our A0, T1 and T3) were analysed in detail. Thereby it was found

Figure 6.20. Quality ordering of guitars A through P (1 is the highest and 0 the lowest quality) for different pieces of music no 1-4, no 4-6 and all 1-6 (from Meyer).


Figure 6.21. Average spectra at filtering experiment. The total areas below the curves show non filtered, the shadowed area the high pass filtered and the white area the low pass filtered average spectrum.

that the third resonance (our T3) was the most important: it should have a high level and a narrow bandwidth. Further it was found that the second resonance (T1) should have a high level. The first resonance (A0) should have a high level too. In addition A0 should have a large bandwidth. Meyer also analysed the levels averaged over wide frequency bands. He found that the level should be high between 80 and 1000 Hz to give a full sound, the level should be high between 1 000 and 3 000 Hz to give brilliance and clarity (a level that is too high gives a harsh tone though). For the playing of chords a high level above 3 000 Hz is good. This high level affects


the tone just after plucking. A suitably high level in this range gives a clear and incisive tone but a level that is too high gives a too harsh tone. The importance of different frequency regions can be studied with filtering and listening experiments, see Fig. 6.21. The filtering to give only components above 4 kHz demonstrate that there is only plucking sounds here. The sound filtered to give components only below 2 kHz sounds dull and hollow. With the sound filtered to below 500 Hz dull and "revelling" tones with indistinct attacks are heard. The sound filtered to above 500 Hz sounds clear but very thin. RESULT: TOO STRONG PARTIALS AT LOW FREQUENCIES GIVE DULL, HOLLOW TONES WITH INDISTINCT ATTACKS. THE HIGH PARTIALS GIVE THE TONE CLARITY, ESPECIALLY AT THE ATTACKS, BUT GIVES A THIN TONE IF THE LOWEST PARTIALS ARE TOO WEAK. THUS A BALANCE OF SEVERAL FACTORS SHOULD GIVE THE BEST INSTRUMENT. 6.7 CONSTRUCTION AND ACOUSTICAL PROPERTIES For the guitar maker it is valuable to know how different construction details influence different properties of the guitar. Therefore the vibration sensitivity was measured for a guitar top plate after several steps of construction, see Fig. 6.22. The figure shows that the vibration sensitivity is little changed, possibly it is increased somewhat at high frequencies by the braces and increased somewhat at low frequencies after the top is glued to the sides. The bridge influences the vibration sensitivity strongly. The vibration sensitivity decreases much when the bridge is glued on. From this fact we can draw the conclusion that the bridge should be one of the most important construction elements of the top plate. In a following step, a thinning of the top plate edge gave a pronounced influence but the width of the glue joint gave little. The experiments show that the bridge make the vibration sensitivity decrease the most; approximately 5 dB at 200 Hz, 10 dB at 500 Hz and more than 20 dB above 1 000 Hz. Meyer has experimented with different shapes of the bridge and found that a bridge without "wings" gave the best result. The resonance frequencies for the top plate at different steps of assembly with sides and back, see Fig. 6.23. Again it is to be found that the large changes occur when the bridge is glued to the top (only in this step are most connecting lines sloping, not vertical).


Figure 6.22a. Vibration sensitivity for a top plate in different steps of production: uppermost frame) vibration sensitivity averaged for free top without bracing (dotted lines), for free top plates with bracing (dashed lines), and top plates with bracing and sides (full lines); (second frame) vibration sensitivity for free top without bracing, (third frame) for free top plates with bracing , and (lowest frame) top plates with bracing and sides lines.


Figure 6.22b. Vibration sensitivity for a top plate in a guitar body in different steps of production: summary (uppermost frame) of top without bridge (dashed lines) and with bridge (full lines), (second frame) vibration sensitivity without plates with bracing and sides but without bridge, and (lowest frame) as above but now for top plate with bridge (c.f. Fig. 6.22a).


Figure 6.23 The shift of resonant frequencies at different construction steps for a guitar body, from the top glued to the ribs (the uppermost line) to the completed guitar (the lowest line). Circles mark resonant frequencies and the vertical lines (mainly) connect the same modes in the different steps (after Richardson and Roberts).

Figure 6.24. The bridge of typical factory made guitar and of handmade Spanish guitar.

RESULT: According to the measurements presented the most important part of the guitar body is the bridge. As the bridge is likely to be one of the most important parts of the guitar let us look a little more in detail at the bridge. A guitar bridge looks as shown in Fig. 6.24, i.e. a high middle part to which the strings are fastened and over which the strings are stretched. The bridge has two lower "wings" on the sides. Factory made bridges seem to be considerably higher and more rigid than handmade Spanish bridges.


Figure 6.25. The bending of the top plate along the bridge line for different forces: for forces parallel (1 N) with the bridge and perpendicular (0.5 N) to the top (uppermost left frame - triangles and squares respectively), perpendicular (0.5 N) to the top plate on treble side (second left frame), and perpendicular to the top plate on the bass side (lowest left frame), c.f. Fig. 6.13. The bending of the top plate with and without the bridge for the mode with four antinodes in the bridge line, and for the mode with six antinodes (right frames after Richardson and Roberts).

When the guitar string is plucked, it is pulled aside and thereafter left to vibrate freely. When the string is pulled aside it will exert forces on the bridge and on the top plate. Thereby the top plate is deformed as is shown in the left side of Fig. 6.25. Only when a middle string is plucked perpendicular to the top plate, the top plate is deformed symmetrically. When plucked perpendicularly to the top plate besides the central line, the top plate tends to flip up at the nearby bridge "wing". This is one demonstration of the cross-stiffening effect by the bridge. The bridge has a marked stiffening effect on the top plate and thus affects the vibrations greatly, see Fig. 6.25 right part. For the two resonances with deflections as shown in the figure their resonant frequencies increase from 439 Hz to 622 Hz, and from 666 Hz to 1029 Hz, which corresponds an increase in plate thickness from 3 to 4.5 mm. For a heavy bridge the frequency of the first top plate resonance may decrease - the weight can give a larger contribution than the stiffness increase.


RESULT:

THE BRIDGE GIVES A WEIGHT INCREASE BUT ABOVE ALL A STIFFNESS INCREASE OF THE TOP PLATE. FOR LOW FREQUENCIES THE WEIGHT EFFECT MAY DOMINATE (A0 AND T1). FOR HIGHER FREQUENCIES THE STIFFENING EFFECT DOMINATES AND IS VERY LARGE.

Figure 6.26. Calculated modes for a guitar top plate ordered after the number of antinodes (marked with plus and minus signs for phases of vibration and nodal lines as point dashed lines (after Richardson and Roberts).

A better knowledge of how other construction details influence the properties of the bridge is necessary. Richardson and Roberts have therefore conducted a large series of computer calculations (with finite element modelling) for a guitar top plate fastened at its edge. The top plate for which the calculations were made is sketched in Fig. 6.5a. As a start the plate thickness was set to 2.9 mm, the cross bars were 14 mm high and 5 mm wide, and the braces were 4 or 5 mm high and 5 mm wide. There was no bridge on the top. Calculated vibration modes are presented in Fig. 6.26. The modes are ordered as sketched in Fig. 2.13, i.e. after the number of antinodes horizontally and vertically.


Figure 6.27. The effect of the crossbars and top plate stiffness on the resonant frequencies (from Richardson and Roberts).

Richardson and Roberts say that the wood stiffness varies the most across the grains and therefore the effect of a 50 % reduction in this stiffness was calculated, see Fig. 6.27. The calculations predicts a dominating frequency shift of up to half this value (25 %) for the resonances mainly oriented across the grains (modes with antinodes horizontally), which is in agreement with theoretical prediction. In a second step the influence of the cross bar height was calculated. The calculations show that the crossbars should give a large influence at least for the low resonances which are oriented mainly longitudinally (modes with few antinodes in the vertical direction). Similar computer calculations were made for different thickness and brace heights, see Fig. 6.28. The figure shows that a thickness reduction from 3.1 to 2.1 mm results in a large influence on the resonance frequencies. The measurements by the author shows that a reduction of plate thickness influences the level of the vibration sensitivity rather little and that the thinning along the edge does the most. In guitar making the braces (the fan bars) are usually considered to play a major role and a large amount of different arrangements can be found.. It is therefore of great interest to calculate the influence of the braces as Richardson and Roberts have done. Their calculations show, however, that the braces influence the resonant frequencies of the top plate little, not as much as changes in the top plate thickness. Possibly the braces give a slightly higher vibration sensitivity level than without braces for frequencies above 500 Hz (from the authors experiments).


Figure 6.28. Influence of top plate thickness and height of braces on the resonant frequencies - vertical lines indicate no change, leaning lines a change (from Richardson and Roberts).

Meyer has found in his extensive investigations using mirror symmetrical and fan shaped bracings that a maximum of seven braces should be used, they should be positioned far away from the centre line except for one brace along the centre. Increased number of braces improves the evenness of the response at the price of a weaker tone. RESULT: A THINNER TOP PLATE GIVES NOTICEABLY LOWER RESONANT FREQUENCIES BUT THE LEVEL IS LITTLE INFLUENCED. THE THICKNESS ALONG THE EDGE SEEMS TO BE THE MOST IMPORTANT. THE BRACES (THE FAN BRACING) SEEMS RATHER TO BE A FINE ADJUSTMENT. 6.8 HOW TO MAKE THE BEST GUITAR? What answer can we find to this most important question? A recipe can unfortunately not be given, the question is much too difficult. The information presented gives, however, some clues to what is likely to be the best choice for experimenting in optimising a construction or a design. In answer to our question of desirable properties for a concert guitar, we found that a strong, long tone with colour were the most wanted properties. Analysis of played guitar tones showed that a higher level was also perceived as advantageous. A balance between different frequency ranges does seem to be needed: sufficient level between 80 and 1 000 Hz for a full tone, a high level


between 1 000 - 3 000 Hz for brilliance, the right level in the initial part for frequencies above 3 000 Hz for an incisive tone. Too high a level in the low frequency range gives a dull tone and too high a level in the high frequency range gives a clear but thin tone. Listening tests have shown that a stronger tone sounds longer without necessarily being so. The guitar is not only top, sides, back and a volume. For an optimal design co-operation between these and other parts are demanded. Antoine Chaigne has shown with synthesized guitar tones that the dominating resonances in the vibration sensitivity curve can influence the perceived quality, i.e. more than 15 resonances below 1 000 Hz. This means that the maker cannot hope to be able to tune all noticeable resonances separately. The experiments and calculations presented have shown that the bridge has the largest influence on resonant frequencies and vibration sensitivity levels. The thickness of the top plate seems to be the second most important, especially along the edges, and the width of the fastening is less important than the thickness. The number of braces should be few and they should be fairly strong. Many braces give eveness but at the price of a weaker tone. Small internal friction in sides, back and neck seems also to be desirable. For low frequencies single resonances play a major role and the maker must at least intuitively be able to tune some resonances relative to others ( but not to the same frequency). The resonance T1 should be tuned in relation to the resonance A0, and the resonance T3 in relation to the resonance A1. Good properties at high frequencies should be determined rather by the general construction than the tuning of single resonances. MAIN RESULT: THE EXPERIMENTS SUGGEST THE FOLLOWING DIFFERENT PARTS - BRIDGE, TOP PLATE THICKNESS AND CROSS BARS.

ORDER OF IMPORTANCE FOR

The conclusion from the (experimental) experience of the author is thus to start with a fairly rigid but light construction (back, sides, neck and cross bars), a fairly thin top plate the properties of which primarily are adjusted by means of the bridge and thereafter edge thickness and braces. A good knowledge of the possibilities of bridge adjustments should be most valuable. In the opinion of the author a high handicraft skill and a good feeling for the material must be developed before experimenting after the sketched lines. 6.9 SUMMARY - THE TONE AND TONAL QUALITY OF THE GUITAR In this part we have tried to relate tonal properties to resonance properties. Further the effect of different construction details on the resonant properties have been described. Finally the information have been summarised in suggestions for profitable areas of experimenting in new constructions. 6.10 KEY WORDS Carrying power, timbre, duration, resonant frequencies, resonance levels, bridge, top plate thickness, cross bars and braces.



ACOUSTICS FOR VIOLIN AND GUITAR MAKERS ERIK JANSSON

Chapter VII: The Function of the Violin



Sound and hearing

Chapter II


Chapter III

Sound and the room

Chapter IV


Chapter V


Chapter VI


Chapter VII


Chapter VIII


Chapter IX



ACOUSTICS FOR VIOLIN AND GUITAR MAKERS Chapter 7. - Applied acoustics THE FUNCTION OF THE VIOLIN Part 1: PROPERTIES BUILT INTO THE VIOLIN 7.1. Construction of the violin 7.2. Resonances of the complete violin body 7.3. Resonances of the free top plate, the free back plate and the asembled body 7.4. Summary 7.5. Key words

Part 2: INFLUENCE OF DIFFERENT PARTS 7.6. The sound post 7.7. The bridge and the varnish 7.8. Vibration sensitivity and radiated sound 7.9. Summary 7.10. Key words

Jansson: Akustik för fiol- och gitarrbyggare sid 7.2

Chapter 7. APPLIED ACOUSTICS - THE FUNCTION OF THE VIOLIN First part: PROPERTIES BUILT INTO THE VIOLIN INTRODUCTION We shall now talk about the function of the violin and especially about the properties, which are built into the violin body. First I shall give a short description of the construction of the violin and how I measure its function. As an example a typical measurement of a Stradivarius violin is given. The properties built into the violin will be described by means of the acoustical building blocks, the resonances, their vibrations and the vibration sensitivity (mobility) of the assembled violin body. Previously in chapter 2, resonances have been discussed, in chapter 5 properties of the wood and the tuning of resonances. We have also talked about the fundamental function of the guitar body in chapter 6. The knowledge gained previously shall be used in explaining the function of the violin body. Thus we shall start with a description of the construction of the violin. Thereafter we shall continue with resonances of the complete violin body. Finally we shall discuss resonances of the free top plate, the free back plate, and the assembled body 7.1 CONSTRUCTION OF THE VIOLIN The violin can be regarded as having two main parts (as the guitar): 1) the acoustic amplifier, the violin body, and 2) the strings and the devices for holding the strings such as the neck with tuning pegs, the tailpiece, and the bridge. The main outline of a violin looks is sketched in Fig. 7.1. The different measures vary little, especially the measures from nut to top plate edge (130 mm) and that from the nut to the bridge (327 mm). These measures are necessary, otherwise the violin does not "feel" right for playing. The bridge is positioned at the notches of the f-holes, at 195 mm from the upper top plate edge (on a line between the notches in the f-holes). Note that the bridge position thereby is approximately halfway between the upper and lower edges of the body (195/356). The waist (the Cbouts) is necessary for the playing of the first and 4th strings (the bow must not touch the edges). Hereby the arching of the top plate and the bridge height play important roles too. The outward bends above and below the C-bouts are called the upper bouts and the lower bouts, respectively.

Figure 7.1 Typical geometrical measures of a violin in mm

Figure 7.2 Typical weights of violin parts


The total weight of a violin including chin rest has been found to be typically 450 g, cf. Fig. 7.2. The 450 g is, however, probably a little high, and a total weight of 400 g seems reasonable to aim at. The top plate, the ribs and the back plate make up for a total of half the weight. Chin rest, tailpiece and fingerboard make up a third of the total weight. The acoustical amplifier, the body, is made of three main parts, c.f., Fig. 7.3. The top plate is made of spruce and made arched (a shell), has two f-holes, and is strengthened by the bassbar and supported by the soundpost. The edges of the top plate are glued to the maple ribs with six blocks, the upper block, the lower block and the four corner blocks. To the other side of the ribs the back plate of maple (also made arched, a shell) is glued. The sound post is squeezed in between the top and back plate, and its position can be adjusted if necessary.

Figure 7.3 Outline of the violin seen from top, and sections seen from the side and the bottom end; bridge B, f holes F, top plate TP, ribs R, sound post SP, back plate BP, and bass bar BB. The main function of the violin is the following. The player sets the string into vibration by means of the bow. The vibrating string produces a vibration force on the bridge, which is transmitted via the bridge to the top plate and thereafter to the complete violin body. The vibration force sets the complete body into motion. The vibrations of the violin body set the surrounding air into vibration, which results in the violin tone we hear. The acoustical amplifier, the violin body gives two types of amplification. First the vibrations of the strings (with small area) result in the vibrations of the body walls (with large area). The violin body acts as an effective sound radiator for sound relative to that of the strings. The phenomenon is the same as stirring a cup of coffee - it is difficult with a tooth pick but with a spoon it is no problem. Secondly the resonances of the violin body give extra amplification at specific frequencies.


Figure 7.4. Force transformation of the bridge. A horizontal force in transformed into a couple of Vertical forces at the bridge feet.

Figure 7.5. Measuring the acosutical properties, bridge vibration sensitivity of a violin by means of impulse excitation (D string damping, H impulse hammer, M magnet, and C electrical coil).

Figure 7.6. Bridge vibration sensitivity of a Stradivarius violin ( "suspected" most important quality measures P1, P2 and BH).


A Stradivarius violin is generally assumed to be the best when it comes to violin quality. A natural question arises: How does a Stradivarius violin look acosutically? What is it that gives the acoustical quality? How can one make a similar violin? We shall work step by step trying to find answers to these questions. To find the answer to the first question the acoustical properties of a Stradivarius violin must be measured. The measurents must give a meaningful frequency response without any danger for the violin. Just to hold an art-object (and utility article) like a violin insured for several millions of Swedish crowns can make you nervous. At our department we have developed the following method of measurements which meets the demands, see fig. 7.5. 1) The violin is placed horisontally on top of two feltcovered supports. The supporting points are in a plane and the “lower supporting” approaches the supporting in playing. 2) The resonant vibrations of the strings are damped to avoid extra complications (in the last version damp wedges of rubber between neighboring strings are used except for the A- and E-string damped by cloth against the fingerboard). In the soundproduction chain from string vibrations via bridge vibrations, body vibrations and sounding tone it is only the bridge that offers a meaningful and easily available measurement point. The bridge is furthermore a part frequently replaced. 3) A small ”supermagnet 3/100 g is waxed at the bridge corner and the bridge vibrations (velocity) generates a voltage in a small electrical coil at 1.5 mm distance. The bridge (and the violin resonances) are excited by a small impulse force hammer in a pendulum arragement. By using a mechnanical impulse problems with electrical disturbances at the coil are avoided.The hammer has a built-in force tranducer. How does a Stradivarius violin look acoustically? Well, as the vibration sensitivity curve in fig. 7.6. The upper frame shows the level curve (i.e. vibrations divided by driving force) at each frequency and the lower frame shows the phase (the time lag between excitation force and resulting vibrations) also at each frequency. The Stradivarius violin belongs to the Järnåker foundation At the first glance the curves look very complicated and not at all as the ”friendly” curves within ±3dB usually shown for loudspeakers and microphones. From measurerments of other Stradivarius violins we know that the curves are typical. In cooperation with the Järnåker foundation (Dr. John Huber) we have been able to measure the vibration sensitivity of about 25 violins of soloist quality. From the measurements of these violins we have found three properties as ”suspected” for quality in the first approximation, see fig. 7.6. These are the peaks P1, P2 and the ”Bridge Hill” BH. P1 is between 400 and 500 Hz, P2 beween 500 and 600 Hz and the BH, here a gently smoothed (not peaky) hill with a maximum between 2 and 3 kHz. The Stradivarius violin has also a peak at about 1 kHz not included in our first set of suspected quality parameters. The level of the P2 peak is higher than the P1 peak. The BH maximum is level with the P1 peak. At P1 and P2 there are large and clear phase changes. At the BH maximum there is a smooth phase change (not a step) but still close to 180 degrees. Our working hypothesis is that P1, P2 and the BH are the main parameters to determine the quality of a violin (in technical terms we are measuring mobility, which will show single resonances effectively).


Figure 7.7. Top plate vibrations at the fundamental resonance of a violin photographed by means of hologram interferometry (rigidly clamped at upper corner blocks and chin rest, from Jansson, Molin and Sundin).

7.2 RESONANCES OF THE COMPLETE VIOLIN BODY The violin body works somewhat differently than the guitar body but has still the same fundamental function. The top-plate vibrations at the fundamental resonance of a violin clamped at three points are shown in Fig. 7.7. The figure shows three major effects: 1. The vibrations dominate in the left part, i.e., on the bass bar side and close to the bridge foot at the bass bar side. 2. The soundpost acts as a support and gives a nodal line in the top plate at the nearby bridge foot. 3. The vibrations have maximum amplitude (antinode) at the edge of the left f-hole, i.e., the f-hole efficiently cuts a part of the top plate free from its edges.

Figure 7.8. Typical top and back plate vibrations (left pair) - the first air resonance at 275 Hz (A0, the Helmholtz resonance, the air tone) and (right pair) - the fundamental resonance at 460 Hz introduced above in Fig. 7.7 (from Alonso Moral and Jansson).


Thus the bridge in this resonance rocks around its foot at the sound post side. It is a seesaw motion of the top plate, which is supported by the soundpost. A small shift of the soundpost in the left or right direction results in a large change in the seesaw properties of the top plate. In fig 7.8 simultaneous vibrations of the top and the back plate are shown. At the frequency of at 460 Hz mainly the top but also the back plate is moving. The motions of the two plates are at one moment towards each other and at the next away from each other. Thus the violin body swells at one moment but shrinks in the following. At the lower resonance A0, the Helmholtz resonance, the two plates move mainly in the same way. The A0 resonance works as an empty plastic bottle - when squeezed, air is pressed out and when released air is sucked into the bottle. The motion of air is maximum at the resonant frequency of the hole-volume resonator.

Figure 7.9. Examples of resonances limited to the back plate (from Alonso Moral and Jansson).

But similarly to the guitar there are higher resonances both in the top and the back plate of the violin. In fig 7.9 higher back-plate resonances are shown. The three leftmost resonances are made up by two, three and four antinodes in the lower part. The following resonance has three antinodes along the body and the other two combinations of antinodes along and across the plate. Each of these resonances can give amplification effects.

Figure 7.10. One-dimensional body resonances (from Alonso Moral and Jansson).

But there are also other kind of resonances, cf. fig 7.10. In the lowest resonance at 185 Hz, the whole violin vibrates as a bar with antinodes the tailpiece end, the scroll and in the center. The resonance has thus two nodal lines one close to the Jansson: Akustik för fiol- och gitarrbyggare sid 7.8

bridge and another one in the upper part of the neck. In a following resonance at 285 Hz the violin was found to work in a similar way, but now the neck with the finger board are the main vibrating parts. Modal analysis has shown that there is an additional nodal line in the position of the upper block for this resonance. There are further resonances in which the top and back plates move as a unit, as the upper and lower sides of a thick plate but presently the author regards these as of second order importance and has chosen to omit these.

Figure 7.11. Typical mobility curve with vibration modes indicated at some of the lower resonance peaks (from Alonso Moral and Jansson)

As was said earlier (chapter 4 second part) the bowed string drives the bridge with a force of sawtooth shape. This means that forces from the string are mainly transferred in form of repetitive impulses. The effect of an impulsive force have been investigated in co-operation with the Luleå University of Technology (Molin, Wåhlin and Jansson). The result of such measurements are shown in fig 7.12 the left part. The weight of a small pendulum was arranged to hit the bridge in parallel with the upper edge of the bridge. The deformation of the violin body was measured at different times shortly after the hitting. In the interferogram top left we can see that the top is deformed at both bridge feet and via the sound post also the back plate. In the cross-section figure bottom left the deformation has been sketched (much exaggerated). The motion of the top and back plates is rather different for the impulsive force than the motion of the 460 Hz resonances in Fig. 7.7. For the impulsive force the top plate moves in oppsite directions at the bridge feet. For the constant-frequency driving the top plate motion is mainly limited to the bass bar side and the sound post acts as a fixed fulcrum under the seesaw motion of the top plate.


FIGURE 7.12 Deformation of the top plate 0.125 ms after an impulse hits the bridge along its upper edge (top left), the vibration deformation of 460-H-resonance (top right), sketch of deformations (much exaggerated to show clearly) of the violin seen in cross-section for the impulse excitation (bottom left) and extreme positions in the 460-Hz- resonance (bottom right, from Jansson and Molin).

7.3. RESONANCES OF THE FREE TOP PLATE, THE FREE BACK PLATE, AND THE ASSEMBLED BODY The properties of the acoustical building blocks, the resonances, can vary considerably. The properties of resonances measured in a large number of different violins are shown in fig 7.13. Furthermore the resonant frequencies, and the bandwidths are given. It should be pointed out that there are other ways to analyse the resonances of a violin, modal analysis for instance. This method has been used by Marshall and the vibration patterns found deviates in some cases from the presented ones. These deviations implies that one must be very cautious when measuring and interpreting vibration modes. It is correct to say that the main character of our 460-Hz-mode is in agreement with the result of the modal analysis measurements. Recent measurement with our optical methods have confirmed previous results and therefore I have chosen to present only our results.


Figure 7.13. Summary of resonant frequencies, tones, and bandwidths for the lowest air mode A0 (270 Hz), the 460-Hz-resonance (average here 420 Hz), a “thick plate” resonance (average here505 Hz), and the “BH-resonance” (average 2.7 kHz), assembled violins, from Alonso Moral and Jansson).

J. Alonso Moral has conducted a well controlled experimental series with tops, backs and ribs which were made under Carleen Hutchins’ control. Alonso Moral selected three tops, three backs and three sets of ribs - one set soft, one average and one stiff. Thereafter they were combined in a total of 12 violins in an experimetally efficient way so that he could evaluate the influence of the top, back and ribs independently. The results of his experiments are shown in fig 7.14 and in table 7.1. In the analysis the ring mode, i.e. the free plate mode number five, was chosen to represent the properties of the free plate. The influence of the ribs was found to be small. In the fig 7.14 is shown how the properties of the assembled violins were influenced by the properties of the free top and free back plates. The figure is rather complex and a very detailed description is needed for a complete understanding, but the most important parts can be understood by the following simplified description.


The figure have three parts. Each part contains four frames. In the upper left frame the relation between frequencies of the free top plates (their fifth resonances, the ring modes) and those of the A0 of the assembled violin is plotted. In the upper right frame the relation between levels of the free top plates (their fifth resonances, the ring modes for a specific driving) and those of the A0 of the assembled violin is plotted. In the lower left frame the relation between frequencies of the back top plates (their fifth resonances, the ring modes) and those of the A0 of the assembled violin is plotted. In the lower right frame the relation between levels of the back plates (their fifth resonances, the ring modes at a specific driving) and those of the A0 of the assembled violin is plotted. In the middle part the four frames show the relations between free plate properties and those of the 460 Hz modes and in the lower part the four frames frames show the relations between free plate properties and those of the thick plate mode at 500 Hz. The crosses in Fig. 7.14 mark measurement points and the straight lines show the relation between the assembled violin properties and the free plate properties. If the line is leaning 45o there is a strong dependence. If the line is leaning less or more there is a weaker dependence. A horizontal (or vertical) line means no dependence. The relation between the A0 resonance and the free plate resonances are shown in fig 7.14, top part figure. We see that there is no dependence between the frequencies, the lines are closely horizontal. Between the levels of the free top and the assembled violin A0 there is a strong dependence, the line is leaning close to 45o. This means that a high resonance peak of the free top plate resonance resulted in a high peak for the A0 resonance. For the back plate there is a moderate dependence, i.e. a lower level for the free back plate peak should result in a slightly higher A0 peak.


FIGURE 7.14 TOP PART WITH FOUR FRAMES: (upper left frame) Relation frequency shifts free top plate mode five to frequency shifts of violin resonance A0, (upper right frame) relation level shifts free top plate mode five to level shifts of violin A0, (lower left frame) relation frequency shifts free back plate mode five to frequency shifts of violin resonance A0 (lower right frame) relation level shifts free back plate mode five to level shifts of violin A0, MIDDLE PART – FOUR FRAMES, the same relations for free plate modes and the 460 Hz mode, and LOWER PART, the same relations for free plate modes and for a thick plate mode (from Alonso Moral).


Corresponding analysis for the assembled “460 Hz resonance” showed that it was strongly dependent on the properties of the free top plate, see fig 7.14 middle part. A free top with high mobility level and a high resonance frequency resulted in a high resonance frequency and a high peak level for the assembled “460 Hz resonance”. The back plate did not influence the properties of the assembled “460 Hz. resonance” In the same way it is shown that the resonant frequency of thick plate resonance is weakly dependent on the free top and back plate frequencies, see fig 7.14 lower part. The levels of the free plates influenced the levels of the thick plate resonance peak somewhat more. A higher level of the top plate gave a higher level, while a higher free back plate resonance gave a lower level thick plate resonance peak. TABLE 7.1. The dependence of A0, “460-Hz”, and “Thick plate” on the properties of the free top and back plates otherwise as Fig 7.13 (1 for maximum dependence, large or smaller than 1 less dependence, from Alonso Moral). RESONANCE A0 “460 Hz” “Thick plate”

TOP Frequency/Level 0/+1.3 -0.6/+0.7 +0.2/+0.5

BACK Frequency/Level 0/-0.3 0/0 0.2/-0.4

ADJUSTMENT OF TOP AND BACK PLATE THICKNESSES

Generally, poor violins have plates which are too thick. Therefore they are often thinned to improve the tone quality. The result of 8 violins before and after such adjustment is shown in fig. 7.15. It can be seen that it is mainly the levels of the peaks that have changed. Only the frequency of the 460 Hz resonance has been shifted.

FIGURE 7.15. Shifts of resonant frequencies and mobility with tonal adjustment (8 violins), filled circles with single line crosses mark average and spread before and open circles with doubled line crosses mark average and spread after adjustment.

THE IMPORTANCE OF F-HOLES AND THE THICKNESS A series of experiments were conducted to test the importance of the f-holes and the thickness distribution of the top plate. The f-holes gives the A0 resonance but also the middle part of the violin top is allowed to move more freely. From the tuning of Jansson: Akustik för fiol- och gitarrbyggare sid 7.14

free tops and backs we have learnt to influence these properties, but still we cannot say from this how the assembled violin properties are affected. To examine the influence of the f-holes and the thinning of the top plate, three experimental violins were made. The first violin had its sound holes in the ribs. Step by step a slit was sawn along the f-hole positions, see fig 7.16a. After every step the violin was tested by playing and the mobility was measured, see table 7.2 and fig 7.16a. The test playings and the measurements could be made very fast as the work was made with the strings tuned all way through. Thereby the tonal quality changes could be judged accurately and no uncertainties were introduced by variations in the atmospheric conditions.

TABLE 7.2. Comments by test player at the cutting of f-holes, c.f. fig 7.16a.

STEP 0 1 2 3 4

QUALITY SHIFT moderate small large large

TONE QUALITY very bad, bass especially bad but now a violin tone better but shrill much better, no catastrophy now slightly damped but longer tone

The results of these experiments showed that the long part of the f-holes, along the fibres (1), should have less influence than the relatively short cuts across the fibres (2, 3 and 4 – the vibration sensitivity curves are normal, but not as “clean” as the Stradivarius ones).


TABLE 7.3 Comments by test player at the thinning of top plate, c.f. fig 7.16b

STEP 0 1 2 3 4 5 6

QUALITY SHIFT moderate small moderate moderate moderate small

TONE QUALITY full tone with substance stronger tone, better timbre worse and bad better, now a musical instrument better, timbre slightly Italian better, more power -

The second and third violins were made with 0.5 mm too thick top plates. The tops were thinned step by step on the outside. The violins were tested by playing and the mobility was again measured after each step. The fingerboard was screwed to the neck and could rapidly be removed. As the thinnings were made on the outside of the top plate, again several steps could be made rather fast, in some cases even without detuning of the strings. The results for the third violin are summarised in table 7.3 and fig 7.16b. The results of the experiments (violin 2 and violin 3 taken together) showed that a thinning in area 6 gave a small change while a thinning in areas 4 and 5 gave noticeable changes. The results indicates that the tuning of the "ring mode" (cf Fig. 5.20 and 5.21) of the free plate is a reasonable and important procedure as the same areas seems to beare the most important ones for the assembled violin. Another interesting observation was made. Violin 1 (fig 7.16a) had a marked BH (the hill between 2 and 3 kHz in the mobility curve) while violin 3 (fig 7.16b) had not. Why - is just now a hot question. Jansson: Akustik för fiol- och gitarrbyggare sid 7.16

Another interesting observation was made during the experiments. The E- and Astrings were fair in the beginning but the D- and G-strings were awfully bad. After a few steps the G-strings improved. Somewhat later the D-string improved. A few steps more the D-string improved but the G-string became less good. Thus the balance between G- and D-strings indicates when a violin is the best. No change of strings were made. It is believed though that the result is independent of strings.

Figure 7.17. Influence on the P2-peak by back plate stiffness and soundpost position

The properties of the Stradivarius violin, fig. 7.6 (and soloist violins, fig. 7.12), indicates that the P2-peak is very important (poor violins have in general a weak P2 peak and the holding for playing damps P1 more than P2, cf Fig. 8.9). Experiments with impulse excitation, fig 7.12 indicates that the back plate and the sound post in combination makes this peak Earlier experiments indicated that the back plate is most important for the P2 peak. Therefore new experiments were made with a too thin back plate stiffened by external cross bars. The results are shown in fig. 7.17 The figure indicates that the back plate stiffness sets the frequency - lowest for 0% height and highest for 100% height. The figure also indicates that the soundpost position sets the peak height – f for shift closer to the f-hole and m for closer to the centerline. A softer back plate gives a lower frequency. The peak height is increased by moving the soundpost towards the centerline. 7.4 SUMMARY - PROPERTIES OF THE VIOLIN BODY The fundamental properties of the violin body in form of resonances have been demonstrated in the top plate, in the back plate, the bridge and the complete assembled body. At low frequencies the resonances of the whole body dominate except for the 460 Hz resonance and the air resonance. With increasing frequency a large number of top and back plate resonances will dominate the body vibrations of the violin. At high frequencies the bridge will give a major contribution. The sound post gives a large asymmetry at low frequencies and the f-holes give in Jansson: Akustik för fiol- och gitarrbyggare sid 7.17

addition to the air tone an effective release of a part of the top plate from the stiffening effects of the ribs at low frequencies. 7.5 KEY WORDS: Top plate resonances, back plate resonances, body resonances and the bridge hill (the BH-peak).


Chapter 7 Second part: INFLUENCE OF DIFFERENT PARTS INTRODUCTION

In this part we shall introduce the influence of different parts, We start with the sound post, a very small part but acoustically a most important part. We continue with the bridge resonance and the varnish both their influence acoustically somewhat exaggerated Finally the vibration sensitivity (mobility) and radiated sound of very good violins will be presented.

Figure 7.18. Vibration patterns at peaks P1 resp P2, with soundpost (upper row),and without soundpost (lower row, both with constant frequency driving, Saldner et al).

7.6. THE SOUNDPOST In fig. 7.18 it could be seen that the violin without soundpost is symmetric in both top and back except a moderate asymmetry for the top in P2.With sound post all vibration patterns are asymmetic except the back at P2. The vibration pattern of the back plate is moderately changed by the sound post. The soundpost introduces a much larger asymmetry than the bassbar. The soundpost enforces a line of little motion (the frequency of the resonances are also changed with the sound post). The vibration sensitivity (the mobility) of the bridge is very dependant of the sound post, see fig. 7.19. Without the soundpost a resonance dominates at approx 550 Hz. With sound post both P1 and P2 can be found but now the BH dominates in this setting. The frequencies of BH and the P2 seems to be little changed by the sound post but the levels considerably. The position of the soundpost affects the timbre of a violin. When the soundpost is placed closer the bridge the timbre is sharper, further away it is softer. When the Jansson: Akustik för fiol- och gitarrbyggare sid 7.19

soundpost shifted towards the centerline the timbre turns darker and when the soundpost is moved towards the nearby f-hole the timbre becomes lighter. This can also be concluded from shifts in vibration sensitivity of a violin, see fig 7.20. In these experiments the soundpost was moved a large distance (± 5 mm) to guarantee that the influence of soundpost position dominated over minor uncontrollable variations of the violin properties. It is easily seen that with the soundpost closer to the bridge the BH-peak was attenuated, but it increased with the soundpost moved away. The level of the P2-peak increased much (+ 10 dB) when the soundpost was moved towards the centerline and decreased when the soundpost was moved away from the centerline (-10 dB, two soundposts of different lengths, but still stiff compared to thetop plate, must be used in this experiment). The resonant frequencies were less clearly shifted. The experiment indicates that P2 is most important for low frequencies and BH for high.

Figure 7.19. Vibration sensitivity (level and phase for a violin (St Niewczyk 1992), with and without soundpost (impulse driving).


Figure 7.20. Soundpost position, vibration sensitivity and timbre (impulse driving Niewczyk and Jansson

7.7. THE BRIDGE AND THE VARNISH In fig. 7.4 the fundamental function of the bridge was introduced – a transformation of a horisontal force from the string to a couple of vertical forces at the bridge feet. The function of the bridge is similar to the case with a horizontal push of a person which results in a couple of forces – vertical - between the feet and floor. But the bridge is not completely rigid but has a built-in resonance. This corresponds to a motion of the human body above the waist, see fig.7.21. Corresponding bridge resonance is at 3 kHz with the bridge on a non-moving support.

Figure 7.21. Illustration of the bridge resonance.


Figure 7.22.. Left curve vibration sensitivity of a bridge on a Stradivari violin and right curve the same bridge on a new violin (St Niewczyk 1992)

Figure 7.23. Tuning of the bridge resonance, remove material at a and the resonant frequency is increased, remove material at b and the resonant frequency is decreased.

A natural question is. How does the BH look for a good old violin? The difference between a new and an old soloist violin is shown in fig. 7.22. The Stradivarius violin has a gentle BH-peak but the new violin a pointed BH-peak allthough the same bridge was set on the two violins. The peak properties are thus much dependant of the violin body and not the the bridge only. It was decided to keep BH label for the hill being a combination of Body Hill and Bridge Hill. The BH hides something more than the resonance of the violin bridge. But what that is the question? A definite answer is still missing but it seems to be related to top plate properties in combination of the bridge. Except from the minor pointed peak in the broad BH-peak the resonance of the violin bridge seem to be of minor importance, see fig. 7.24. A normal bridge with a built-in resonance at 3 kHz and a special bridge with its resonance at 7.6 kHz gives in large the same BH peak. The resonance of the violin bridge can easily be tuned, see fig 7.23. The indicated rules are as can be expected from fundamental physics (cf chapter 2).


-

Figure 7.24. The BH and the bridge – normal bridge 3.0 kHz resonant frequency (thin line) with cursor circle and special bridge without “waist” 7.6 kHz resonant frequency (thick line) – vibration sensitivity on a violin (St Niewczyk, impulse excitation).

Figure 7.25 A violin - unvarnished and varnished, respectively (sweep-tone measurement).


A popular saying is that the Stradivarius varnish is the secret of Stradivarius. This is generally not believed by the makers through. The vibration sensitivity of a violin before and after varnishing is shown in fig 7.25. The violin varnish was applied much too thickly. The peaks of the curves were more pointed before the varnishing. The other changes should stem from adjustments of soundpost and bridge. Informal testplaying before and after varnishing indicated that the varnish made the violin sound more dull. 7.8. VIBRATION SENSITIVITY AND RADIATED SOUND Dünnwald has developed a special driving system close to ideal to measure radiation. He has measured radiation properties of a large number of violins. Typical results are shown in fig 7.28 for three different groups of violins, see fig. 7.26. In the upper frame the radiation curves of 10 old Italian violins have been drawn on top of one another. In the middle frame curves of 10 master violin have been drawn and in the lowest frame 10 factory made violins. In the curves we see the air resonance peak to the left for all violins. For the old Italian violins we can see two clear peaks between 400 and 600 Hz. For the other two groups of violins these two peaks are not clear, only traces can be seen.. The large differences are at higher frequencies though, i.e. above 700 Hz. The old Italian show a clear ”up-hill” to a maximum at 2.5 kHz and thereafter a clear ”down-hill”. The master violins have in large a constant level above 700 Hz, while the factory violins have a maximum at 1700 Hz and thereafter a ”downhill”. Our way to measure the vibration sensitivity of the bridge is practical but do our curves predict the radiated sound? A comparison of the Dünnwald curves with our vibration sensitivity curves show obvious similarities – P2 higher than P1 and a maximum with gentle BH-peak, cf fig 7.27. Similarities at other frequencies can also be found, A0 level (between 200 and 300 Hz). Stradivarius has never been surpassed as a violin maker. Is the varnish the secret? Let us first look at P1 and P2. Answer: No P1 and P2 are mainly set by the top, the back, and the wood properties. Standard varnishing should be of minor importance here. What about the high-frequency criterion, the ”bridge hill”? This peak is to a large extent determined by the stiffness of the top plate at the bridge feet. The ageing of the wood under influence of the varnish may be important here.


Relative Sound Level (25 dB range for each frame) Figure 7.26. Sound radiation (transmission driving to sound) for different types of violins (sweeptone measurements, from Dünnwald).


Figure 7.27. Sound radiation of 10 old talian violins (Dünnwald) and vibrations sensitivity of the bridges of 10 soloist violins (Jansson and Molin).

7.9. SUMMARY: In this part the influence of the free top plate and the free back plate on the assembled instrument been discussed. Furthermore the influence of the top plate, the thickness, the f-holes and ”tonal adjustment” been summarized. 7.10. KEY WORDS: Resonances and different violin parts




Erik Jansson

Chapter VIII: The Tone and Tonal Quality of the Violin



Sound and hearing

Chapter II


Chapter III

Sound and the room

Chapter IV


Chapter V


Chapter VI


Chapter VII


Chapter VIII


Chapter IX



ACOUSTICS FOR VIOLIN AND GUITAR MAKERS Chapter 8 – Applied Acoustics FUNDAMENTALS OF THE VIOLIN TONE Part 1: THE TONE OF THE VIOLIN 8.1. Fundamentals of bow-string interaction 8.2. Bow- string-tone. 8.3 Tone spectrum 8.4. Summary 8.5. Key words Part 2: ACOUSTICAL QUALITY MEASURES 8.6 The Catgut Acoustical Society and Carleen Hutchins 8.7 Peakiness and findings of J Alonso Moral 8.8. Averaged spectra (LTAS) and time function (WPT) 8.9. Summary 8.10. Key words

Jansson: Acoustics for violin and guitar makers page 8.2

Chapter 8. THE TONE AND TONAL QUALITY OF THE VIOLIN First part: FUNDAMENTALS OF THE VIOLIN TONE INTRODUCTION In this part fundamentals of the tone production of the violin will be presented, i.e. the road from bow via the violin to the tone radiated in the room. It starts with describing the interaction between the violin bow and the string. It continues with presenting how bowing parameters affect the violin tone. Finally the spectrum of the tone and limitations set by playing, by holding and by radiation properties of the violin are presented. 8.1. FUNDAMENTALS OF BOW-STRING INTERACTION The tone of the violin is generated by the bow pulled perpendicularly across the string, see Fig. 8.1. The bow hair is in contact with the string at a certain distance from the bridge, at the contact point. The bow is pressed against the string with a certain force, the so-called "bow pressure". The bow is pulled across the string at a certain velocity, the bow velocity.

Figure 8.1. Physical bowing parameters, bow force (bow pressure), contact point and bow velocity.

The bow is pressed against the string with a carefully selected bow force (bow pressure). When the bow is pulled across the string, the bow hair is attached to the string at the start. The string is moved with the same velocity as the bow, i.e. the bow velocity. This phase of motion of the string is called the stick phase. When the string has been pulled sufficiently far from equilibrium, it is torn free from the rosined bow hair and slides quickly back. This phase of motion is called the slip phase. The string slips past equilibrium, is caught by the bow hair and a new stick phase starts. The procedure is repeated periodically, i.e. stick-slip-stick-slip-stick etc. The summed duration of a stick and a slip phase is constant and sets the fundamental frequency of the played tone. During the stick phase the string motion is slow and during the slip phase it is fast. The string vibration under the bow, stick, slip, stick, slip, etc., is typically as shown in the upper part of Fig. 8.2. This vibration of the string results in a sideways varying angle, small but still an angle variation at the Jansson: Acoustics for violin and guitar makers page 8.3

bridge. The angle variations at the bridge results in a saw tooth force at the bridge, i.e. the Helmholtz-movement, a sawtooth curve. The saw-tooth force has a spectrum as shown in the bottom of Fig.. 8.2.

Figure 8.2. String motion at contact point, angle variations of string at bridge and string forces at the bridge (all three seen from above), and tone spectrum of bridge force. At low frequencies, i.e. a long period time compared to the time window of analysis, a step is obtained at the slip. Thereafter in the stick phase the force of the string is close to zero until the next slipphase. This time function is the best record of the force signal and can be thought of as whip lashes repeated with constant time intervals, see Fig. 8.3. At high frequencies, i.e. short period time relative the time window of analysis, the forces of the string give a constant spectrum. The spectral description is very good. The partials of the spectrum decreases to 1/2 (6 dB) at every frequency doubling (octave).


Figure 8.3. (Upper part) At very low frequencies a step of the force is caused at every slip and thereafter slowly varying amplitude. A short time window gives a flat pulse spectrum at the step, no spectrum signal between steps, and (lower part) with a time window long compared to the period of the force signal, the normal case, a tone spectrum is obtained. 8.2. BOW-STRING-TONE A violin player selects the bowing parameters with high accuracy, consciously and unconsciously. The player selects bow velocity, contact point bow-string (bridgebow distance), the bow position and the force the bow is pressed against the string (bow pressure), see Fig. 8.4. By means of the selection the "right" tone is obtained. The selection is continuously changed for the best result.

Figure 8.4. A violin string is played with a bow. The player controls bow velocity, bow position, bow distance to the bridge (contact point), and the force pressing the bow against the string (the “bow pressure”). But the violin player can not select the bowing parameters freely. For a specific value of the bow velocity the "bow pressure" must be selected within a permitted


working range to obtain a proper violin tone, a Helmholtz motion, see Fig. 8.5. Playing close to the bridge gives a brilliant but a tone difficult to control.

Figure 8.5. Allowed working range (shaded area) for “bow pressure” (the force pressing the bow against the string) at different bow contact points (after Schelleng 1973). . But a player can play soft or loud. This is done mainly by increasing the "bow pressure", see Fig.. 8.6. The spectra show that increasing from pp to mf it is mainly an amplification of the partials. But with playing mf to ff it is mainly the strength of the higher partials that is increased. This is a typical feature of our traditional musical instruments. Playing louder is not a simple amplification. It is also an increase of the strength of the highfrequency relative the lowfrequency partials. One can hear if an instrument is played soft or loud independent of the loudness control of the amplifier of the record player.

Figure 8.6. Spectral differences between pp, mf, and ff. Jansson: Acoustics for violin and guitar makers page 8.6

When a car is started it accelerates up to the wanted velocity. Without an initial acceleration no velocity can be obtained. The same is true for playing with a bow. Initially the bow must be accelerated up to the wanted velocity. As the tone start is crucial for a good tone the player must monitor the acceleration with high accuracy for the perfect tone start, see Fig.. 8.7. The tolerance is only 5/100 sec for the prolonged start and 9/100 sec for multiple slip start, if the tone start should be heard as perfect. Starting with a low acceleration demands a low and very well controlled bow force (“bow pressure”).

Figure 8.7. Typical allowed working range for perfect tone attack (range 2). Range 1 results in non-perfect "prolonged" attacks and range 3 in non-perfect "multiple slip" attacks (after Guettler 2002). 8.3 TONE SPECTRUM Let us again look at a comparison of radiation, of 10 old Italian violins, and the bridge vibration sensitivity, of 10 soloist violins, i.e. two sets of the highest quality of violins, Fig. 8.8. The similarities in spite of the two different ways of measuring are obvious, i.e. in the P1, P2 and BH levels and frequency ranges. Unfortunately there excists no measurement of radiation and bridge vibration sensitivity of the same set of top class violins. Still it seems fair to draw conclusions, also for radiation, of violins tonal properties from their vibration sensitivities of their bridges. The vibration sensitivity at the bridge of a violin can be measured under different "boundary conditions" without external disturbances. The vibration sensitivity can be measured for a violin freely suspended and held for playing, respectively (magnet coil replaced with a miniature accelerometer waxed to the bridge). If the sound radiation is measured in playing both the musician and the holding influences the result. A comparison of a violin measured in our standard way and held rather hard for playing is shown in Fig. 8.9. The P1 and P2 peaks are much influenced by the holding. Influence is moderate at higher frequencies. A closer look shows that the


P1 peak is almost lost in the holding for playing. This is a further support that the P2 represents the main resonance of the violin.

Figure 8.8. Sound radiation (sound level, after Dünnwald) and vibration sensitivity of bridge (mobility, after Jansson) of soloist violins.

Figure 8.9. Vibration sensitivity of a violin free on supports and held for playing – violinLeon Bernardel 1909 with cursor on P2. Jansson: Acoustics for violin and guitar makers page 8.8

8.4 SUMMARY The “correctly” bowed string results in a Helmholtz motion at the bowing point, a short slip-phase and a long stick-phase. The player must monitor the contact point, the "bow pressure" (bow force), and the bow velocity with high accuracy, especially with playing close to the bridge. For a perfect tonal start the bow force and the bow acceleration must be accurately selected. Playing ff results in stronger high partials and results in a perceived louder tone. 8.5 KEY WORDS Slip, stick, contact point, bow velocity, bow acceleration, and bow pressure (bow force).


Chapter 8. second part: ACOUSTICAL QUALITY MEASURES INTRODUCTION In this part properties of the played violin, its tone and its quality are introduced. First the Catgut Acoustical Society is presented together with some major findings, especially by Carleen Hutchins. Secondly results of tonal quality and violin response properties, details as peakiness of the violin response curves by Mathews and course properties as findings by Alonso Moral. Finally two methods of investigating spectral properties (LTAS) and time properties (WPT), respectively, of played, violin tones are presented.

SOUND LEVEL

8.6. CATGUT ACOUSTICAL SOCIETY AND CARLEEN HUTCHINS Around Frederick Saunders, a physics professor at Harvard in USA, a group of violin enthusiasts grew up and they called themselves facetiously, the Catgut Acoustical Society. Today this group has grown out to some 800 members all over the world. One of the methods used by Saunders was maximum sound level curves, i.e. the sound level of each note a semitone apart played as loud as possible over the complete range of the violin (these curves were somewhat unfortunately called loudness curves, which with today's standard terminology is misleading). In Fig. 8.10 such sound level curves are shown for the average of five and for two single Stradivarius violins (from Saunders: The Mechanical Action of Violins, J Acoust Soc. Am, October 1937). The air resonance A0 was found at C sharp ca 260 Hz and probably also the P1, P2 and BH as marked in Fig. 8.10. Saunders concluded there is "main wood" resonance is just above the open A-string frequency.

A0

P1? P2?

-- BH ?--

FREQUENCY (Hz) Figure 8.10. Sound level curves obtained from played tones (top frame) average of five Stradivarius violins and (lower two frames) two of the Stradivarius violins separate (from Saunders).


Starting from the two resonant frequencies for A0 and for the "main wood resonance" being close to the frequency of the middle two open strings of the violin, Saunders, Schelleng and Hutchins designed a new family of violins, "The New Violin Octet". The ordinary violin was acoustically rescaled into two new treble instruments and into five lower tuned instruments. The instruments were made possible to play by readjusting lengths, volumes and wood thickness. In the octet the ordinary violin turned out to be too weak and therefore a new "Mezzo violin" with a stronger tone was designed. This was obtained by increasing the size of the top and back plates and decreasing the height of the ribs. The instruments are most interesting from a timbre point of view as they are "acoustically normalised" to the working range of the instruments. Especially the vertical viola, which has a timbre and tone volume and which can be used as a solo instrument in the viola range (Hutchins: Founding a Family of Fiddles, Physics Today, February 1967).

Figure 8.11. The new violin octet constructed Saunders, Schelleng and Carleen Hutchins and built by Hutchins - string tunings and relative sizes for the big bass, the small bass, the new cello (baritone), the tenor violin, the vertical viola (Alto), the mezzo violin, the soprano violin and the treble violin (after Hutchins). Carleen Hutchins has lately published results from investigations with violins and violas of a new method (Hutchins: A Measurable Controlling Factor in the Tone and Playing Qualities of Violins, J. Catgut Acoust Soc, November 1989). By means of a small sound emitter inserted through one of the f-holes into one of the lower bouts of the air volume and a small microphone through the other f-hole in the other lower bout, the vibration sensitivity of the air cavity has been measured, c.f. Fig. 8.3. The measurement is technically simple to make with no influence on the violin. The vibration sensitivity has been measured for a number of violins and has been related to their quality. A strong relation was found between the frequency separation between two resonances, A1 and B1, and important quality properties of a violin, see table 8.1 (A1 is the air resonance similar to A1 for the guitar, see Fig. 6.11, and B1+ corresponds to the resonance at our peak P2). Jansson: Acoustics for violin and guitar makers page 8.11

Figure 8.12. Examples of vibration sensitivity curves with the new test method: The frequency separation between peak A1 (at 485 Hz) and valley B1 (at 540 Hz) have been marked with a triangle and the separation measured in Hz (after Hutchins).

Table 8.1. Frequency separation A1-B1 and typical tonal character for corresponding instrument (from Hutchins, summary). Frequency discrepancy Hz > 100 70-80 60-70 50-60

Tonal character very rough tone bright and carrying Bright, fine and carrying solo instrument

Frequency discrepancy Hz 40-50 30-40 20-30 < 20

Tonal character solo instrument easy to play, not carrying soft tone soft and weak tone

Hutchins has also found that it can be advantageous to tune the free part of the fingerboard of the frequency of the A0 resonance. This can be done by listening to A0 with damped fingerboard and thereafter tune the fingerboard to the same frequency with, damped A0. Unfortunately there is no published set of data on a large number of violins suitable to test the Hutchins criteria on. 8.7. PEAKINESS AND FINDINGS OF J. ALONSO MORAL The vibration sensitivity curve for a violin has many peaks and valleys. The importance of the peakiness has been investigated with an electronic experimental violin, see Fig. 8.13 (Mathews and Kohut: Electronic Simulation of Violin Resonances, J Acoust Soc Am vol 53 no 6 1973). Thereby it was found that an even frequency response like in the upper frame gives a peculiar insensitive violin tone, but a moderate peakiness as in in the middle frame gives a more violin sounding tone. Large unevenness as in the lowest frame gives a hollow tone. Thus it seems that a violin should have the right amount of peakiness. Later experiments with the same electronic violin have shown that a high level in the bridge hill range is very important. A similar type of electronic viola has been made for conventional use, see Fig. 8.14 (Gorrill: A Viola with Electronically Synthesised Resonances, Catgut Accost Soc. Newsletter, no 24, Nov. 1975). A viola with damped top and back plate was


LEVEL

FREQUENCY (Hz)

LEVEL

Figure 8.13. Different peakiness used in timbre experiments (from Mathews and Kohut).

FREQUENCY (Hz) Figure 8.14. Vibration sensitivity (mobility) curve for an electronic viola (from Gorrill). provided with a pickup system similar to the one used in Fig. 8.13. An electronic filter replaces the vibrating body and the sound is radiated via a loudspeaker in the Jansson: Acoustics for violin and guitar makers page 8.13

back of the viola. The electronic filter gives peakiness and broad hills similarly to the vibration sensitivity of the viola, see Fig. 8.14. The instrument works well. According to Gorrill the viola is not very sensitive to how it is adjusted in solo performance and for playing together with piano. In the string quartet, it is very important that the right amount of peakiness is adjusted for a good tonal quality. Further the overall sound level must not be increased much, because this will make the viola unacceptable in the traditional quartet music (the composers have evidently tailored their music to fit the fairly weak viola tone and a removal of this weakness gives the instrument limited use).

LEFT

RIGHT

Figure 8.15. Measurement positions for vibration sensitivity, left and right. ACOUSTICAL MEASUREMENTS

As a summary of our early investigations at KTH, the main parts of an investigation by J Alonso Moral are given (FIOL-80) with the important quality determining parameters for the violins The results should be regarded as a step on the way to determine the quality of a violin and not the final solution. If a violin is set into vibration and the vibrations are measured at the driving point, a measure of the vibration sensitivity (mobility) of the violin is obtained. In this investigation the vibration sensitivity was measured at two positions on the bridge, one at the G-string and the other at the E-string, see Fig. 8.15. The vibration sensitivity can be measured with a tone of specific frequency (pitch) and the vibration sensitivity at that very frequency is obtained. If the frequency is slowly changed from 50 Hz to 10 kHz, then we can measure the vibration sensitivity of the most important frequency range. This can be made automatically with electronic devices and curves like the ones in Fig. 8.16a are obtained. This violin was used as an example on a very good violin (it was lent to us before FIOL-80 and the vibration sensitivity was only measured on the left side). Thus the vibration sensitivity for 25 violins was measured in this investigation. Along the vertical axis of the diagram the vibration sensitivity can be read for the frequencies along the horizontal axis. To avoid the influence from vibrating strings, the strings were damped with pieces of foam plastic against the fingerboard. The violins were hung in rubberbands and were thus isolated from external vibrations and resonances of the holding structures.


For the investigation, the tonal quality of violins submitted to the amateur violin makers´ exhibition FIOL-80, were put at our disposal. Violins were selected according to table 8.2. Table 8.2. Violins selected from the FIOL-80 exhibition to the acoustical investigations. number of violins

tonal quality points

Class

10 7 7

72-62 60-50 48-32

I II III

IMPORTANT ACOUSTICAL PROPERTIES ACCORDING TO MEASUREMENTS

By looking at the vibration sensitivity (mobility) curve for the violin with Andrea Guarnerius label, Fig. 8.16a, five peaks labelled A0, T1, C3, C4 and F are found. This good violin has three strong and clean peaks marked T1, C3 and C4. The level of these peaks we shall refer to as acoustical property 1. Further we can see that these peaks are of similar height. The similarity in peak height for these peaks should be referred to as acoustical property 2. Other properties seemingly favouring the quality of a violin are indicated by the best violin of the exhibition in Fig. 8.16b, in contrast to the not so good violin, Fig. 8.16c. The curve for driving at the left side (full line) and the curve for driving to the right (dashed line) follow each other for the good violin above 1 kHz but not for the less good one (for frequencies below 1 kHz. The vibration sensitivity is lower for driving to the right than to the bass side). The similarity in course above 1500 Hz for these two curves we refer to as acoustical property 3.


VIBRATION SENSITIVITY (10 dB/div)

FREQUENCY (Hz) Figure 8.16. . Vibration sensitivity of a) a very good violin (measured earlier and only at the bass side, F Ruggieri with label Andrea Guarnerius), b) the best violin of FIOL-80, and c) a less good violin of FIOL-80. The vibration sensitivity measured at the bass side left is marked with a full line and at the treble side with a dashed line (from Alonso Moral) From the same diagrams (Figs 8.16b and 8.16c) a more marked level increase can be found for the better violin for the driving at the left side. This acoustical property we refer to as acoustical property 4. ACOUSTICAL QUALITY POINTS AND TONAL QUALITY POINTS

In the preceding paragraph four acoustical properties were introduced, which are likely to provide a measurement of the quality of a violin. These properties were "weighted" by the acoustical measurements and compared with quality points given by the test players. The result for the 24 violins from FIOL-80 is summarised in the following. The acoustical property 1 was weighted by the average height for the T1, C3 and C4 peaks for driving to the left. High level gives high points and this property should correspond to a strong tone at low frequencies. The property correlates well with the test players tonal quality points (correlation coefficient 0.53).


Property 2 was weighted by the summed level deviation of the three peaks from their average. Small deviations give high points for this property. High points should mean that the violin is evenly excited at low frequencies. It should correspond to tonal evenness between notes in the low register of the violin (correlation coefficient 0.23 between property 2 and tonal quality points). Property 3 was weighted by the similarity in level between 1500 and 3000 Hz for driving to the left and to the right. Small deviations give a high point. The property may predict the tonal evenness between different strings (correlation coefficient 0.17 between property 3 and tonal quality points). Property 4 was weighted by the slope of the vibration sensitivity curves from 1500 to 3000 Hz, for driving to the left. A steep slope should result in a high point (correlation coefficient 0.38 between this acoustical quality point and the tonal quality points). ACOUSTICAL EVALUATION

The relations between calculated acoustical quality points and the tonal quality points of the test players are shown in Fig. 8.17. There is a good agreement with the acoustical quality points and the tonal quality points, except for five violins. The correlation between the acoustical quality points and the tonal quality points, the five deviating violins excluded, is good (correlation coefficient better than 0.92).

ACOUSTICAL QUALIITY POINTS

Thus we have seen that the quality points derived from the selected acoustical properties should be useful to predict quality of violins. This means that the vibration sensitivity curves should measure important acoustical properties and that the quality of a violin may be predicted from its vibration sensitivity curve. Unfortunately, the very good violin, carefully selected by one of the top Swedish violin players, was measured earlier and only at the bass side, an earlier standard used.

TONAL QUALIY POINTS

Figure 8.17. Relation between tonal quality points and acoustical quality points – circles with numbers for well fitting violins, and squares with numbers for not well fitting violins. The F Ruggieri violin marked with AG and the different tonal quality classes with I, II, and III. Jansson: Acoustics for violin and guitar makers page 8.17

AVERAGE LEVEL AVERAGE LEVEL

FREQUIENCY (kHz)

FREQUIENCY (kHz)

Figure 8.18 Long-time spectra and played music (full lines the noted sample 2, 10 and 20 s, respectively, dashed lines 100 s).

8.8. AVERAGED SPECTRA (LTAS) AND TIME FUNCTION (WPT) It is well recognised that already after a few notes the player obtains a fair picture of how a violin sounds. This is difficult to understand from physical point of view as all partial tone spectra look different, even from the same violin. One way to attack this problem is to filter the sound in a way that corresponds to our hearing and average over a long time, long time average spectra (LTAS). In Fig. 8.18 it is shown that the long time average spectra quickly give a "constant" picture. Long time average spectra for 100 s is shown with a dashed line in the frames. Such spectrum for 2 s, the full line in upper left frame shows some resemblance with the 100 s one, the spectrum for 10 s and for 20 s show almost a perfect reproducibility of the 100 s spectrum, see the following two frames in Fig. 8.18. Also, with long time spectra groups of violins can be compared, see Fig. 8.19. Full lines here correspond to the average spectra of 8 good violins, and the dashed line represents the average of 7 less good violins. In the upper part of the frame the difference between the two groups are shown. High levels for frequencies below 800 Hz, low levels around 1300 Hz, high at 2500 Hz and low level above 3500 Hz seems to be good.


AVERAGE LEVEL 10 dB/div

Figure 8.19. Long-time spectra and quality. good quality (full line) and poor quality (dotted line). Difference good-poor quality top curve. (observe the three "frequency scales" BARK, kHz and tone, from Gabrielsson and Jansson).

With long time spectra and filtering the properties of the sound can also be compared with the perception of the same sound, c.f. Fig. 8.20. Above 8 kHz there is little sound energy, which thus should influence the timbre of the violin little. Also partial levels above 4 kHz should give a moderate influence. The partials above 2 kHz should, however, give a considerable influence on the timbre in agreement with the picture shown in the middle frame (the shadowed area is beginning to become rather large). The main influence should however come from partials below 2 kHz. When you hear only partials below 1 kHz the tone is hollow with indistinct tonal attacks (a little of this effect can be caused by the filter). With more and stronger high frequency components, the tone becomes less hollow with more clear attacks. Too strong partials at very high frequencies give a rattling sound. If all partials below 1 kHz are removed it is still perceived as a violin although violins with such a thin timbre do not exist. If all partials above 1 kHz are removed one cannot recognise the tone as coming from a violin. A balance between different frequency regions is thus important.


Figure 8.20. Long-time spectra and the influence of different frequency ranges on timbre (the Bark kHz relation is given in Fig. 8.19).

Also with long time spectra the properties of different strings can be compared and can be characterised, see Fig. 8.21. The long time spectra of each of the four violin strings are shown in the lower part of the frame. It can be seen that except for the lowest hill the four curves closely resemble each other. Further it can be seen that the G-string has the lowest level at high frequencies and that the partials vanish at 18 Bark. The levels became increasingly higher at high frequencies for the D, A and E-string. This is shown even more clearly in the upper part of the figure where the difference in spectrum for all four strings and the single strings have been plotted.

Figure 8.21. Long-time spectra and tones of different strings (the Bark - kHz relation is given in Fig. 8.19, from Alonso and Jansson).


TIME FUNCTION - WPT A simple synthesis experiment was made in which the initiating impulses and four resonators are modelled as function of time, see Fig. 8.22. Result of the synthesis is shown in Fig. 8.23. In the first line a period of the recorded tone and the reverberation of the high-frequency resonance ("the bridge hill"). In the second line the response to P1 and P1 plus "bridge hill". The synthesis is now much better. Synthesis with P1, P2 and "bridge hill", the bottom line, makes fairly good synthesis - although the high frequency ripples in the end of the period is missing. The 1000 Hz resonator changes the waveform somewhat but not much. Listening indicates that the levels at the resonant frequencies and not the frequencies are the most important. Furthermore it showed that the high frequency resonances dominates the perceived timbre.

Figure 8.22. Tone synthesis in the time domain – sketch of synthesis program implemented in ALADDIN.

Figure 8.23 Result of tone synthesis in the time domain (upper row) one period of a played tone and synthesis of BH-vibrations, respectively, (middle row) P1vibrations and P1- plus BH-vibrations, respectively and (bottom row) P1- plus P2plus BH-vibrations, and P1-, plus P2-, plus 1000Hz- and BH-vibrations, respectively.


WPT-plot of a played tone BRIDGE VIBRATION PERIOD NUMBER TIME WITHIN PERIOD %

Figure 8.24. Time history of a violin tone (bridge vibrations, WPT-plot invented and developed by F. Le Coustumer). As a part of his "technical training" Frédéric Le Coustumer developed the WPT diagram, i.e. the Wave shape of a Period as a function of Time and the period number, in a three dimensional drawing, see Fig. 8.24. The plot is of bridge vibrations. To begin with the first period is plotted from 0 % to 100 % of period time.. Thereafter the second period is plotted just behind the first (one step increase in period number). The plotting is repeated over and over. It starts every period from the x-value (0 %) and ends at x-value (100 %). The vibration is presented as the zvalue, the vertical axis. In the WPT-diagram of the bridge vibrations of a played tone is shown in Fig. 8.24. The WPT-plot shows mountain ridges separated by parallel valleys. Initially the impulse of the slip phase, the high leftmost ridge, is seen followed by a highfrequency ripple, three minor ridges, a lowfrequency hump, a broad and not so sharp ridge, and finally highfrequency ripple, two minor ridges, before the periods end (100 %). The vibrations of later periods are larger than the initial ones but the shape remains, the ridges and valleys are similar but more and more clear. The period shapes are representative and varies little from period to period. 8.9. SUMMARY P2 (B1+) is the main resonance at low frequencies, but the BH should be at least equally important. A specific peakiness is favourable. Synthesis in the time domain is informative. The peakiness of the violin response curve is something positive. High levels at low frequencies, low just above 1 kHz and high between 2 and 3 kHz and low above 4 kHz are favourable for the violin tone. 8.10. KEY WORDS P1, P2 and BH. Frequency ranges, peakiness, resonance peaks and level.




Erik Jansson

Chapter IX: Sound Examples and Simple Experimental Material

Fourth edition 2002

ACOUSTICS FOR VIOLIN AND GUITAR MAKERS Chapter 9 – Acoustics Supplement SOUND EXAMPLES AND SIMPLE EXPERIMENTAL MATERIAL Part 1: SOUND EXAMPLES 9.1. Sound examples related to the guitar 9.2. Sound examples related to the violin 9.3. Sound example related to the orchestra 9.4. Summary

Part 2: SIMPLE EXPERIMENTAL MATERIAL 9.5. Magnet-coil transducers 9.6. Optical transducers 9.7. Computer analysis 9.8. Summary


Chapter 9. ACOUSTICS SUPPLEMENT First part: SOUND EXAMPLES INTRODUCTION

In this chapter, part 1, I shall present sound examples used in lecturing. For simplicity a summarising text, the related figures with new numbers (but the old number in parenthesis) are given. To start each sound example a "button" is given to be pressed. The examples starts with those related to the guitar, continues with those related to the violin and ends with a demonstration of the sound within the symphony orchestra. 9.1. SOUND EXAMPLES RELATED TO THE GUITAR

Figure 9.1 (old 1.18). Spectra of single tones and the single tones added to a harmonic spectrum.


Press here for sound example 1 Sound example 1. Partials and harmonic spectrum: First four single tones (partials), tone 1, 2, 3, and 4 twice. Secondly tone 1, tones 1 + 2, tones 1+2+3, and tones 1+2+3+4 (i.e. the tones (partials) separate and added in harmonic spectra, as musical tones).

A simple listening exercise, sound example 1, can be prepared as shown in Fig. 9.1. The single partials 1 through 4 give tones with little character and without roughness. If the partials are added a "tone" is heard, still without much character. In the spectrum of the four partials added one can still separate each of the four partials.

Figure 9.2 (old 1.19). Tone spectra with four and six partials respectively - the first spectrum sounds smooth, the second sounds rough.

Press here for sound example 2 Sound example 2. Smoothness, roughness, and critical bands of hearing. First a harmonic spectrum with 4 partials three times. Secondly a harmonic spectrum with 6 partials three times. Thirdly the full example repeated.

What will happen if we add many partials, can all partials still be distinguished? A little consideration gives that when the critical bandwidth and the frequency separation between partials match, then it should be difficult to separate the partials and roughness will occur. A few calculations show that the bandwidths and the partial separations for musical tones occur up to the fifth partial. Thus we should expect that a played tone with four partials has no roughness but a played tones with six partials has some roughness. A listening exercise, sound example 2, as sketched by Fig. 9.2 will confirm that with six partials a sense of roughness is obtained. The critical bands of hearing can thus explain the sensation of roughness. Furthermore the musical dissonance can partly be explained by roughness phenomena.


A guitar sound can be synthesised, but not by only synthesising the partial spectrum (of the string). Care must also be taken of the partials of the guitar body, the body sound. The body sound is weak but important - listen to sound example 3.

Press here for sound example 3 Sound example 3. Tone spectrum with body sound three times, tone spectrum only three times, body sound only three times, and tone spectrum with body sound three times.

Figure 9.3 (old 4.9) Position of plucking and excitation strength of string resonance (in principle). Left upper part the lowest four resonances, right upper the corresponding partials, down left the initial shape of the string at plucking for different plucking positions and down right the resulting spectra for the different plucking positions.

Press here for sound example 4 Sound example 4. Tone spectrum and point of plucking of the string, three times at a) half of, b) a third of, c) a sixth of, and d) a tenth of the string length, and the full example a second time.


The relation between position of plucking and the level of string partials have been sketched in fig 9.3 . When plucked in the middle, the string is initially displaced in form of a triangle with two sides alike. For a string resonance to be excited it must not be plucked at one of its nodal points. Thus we can understand that for a plucking in the middle the first partial, the fundamental is set into vibration, but not the second, the third is set into vibration, but not the fourth etc.. Thereby we obtain a spectrum like in the uppermost part of the lower right frame. If we choose to pluck at a third of the string length the third, the sixth etc. partials will be missing. If we pluck at a tenth of the string length the tenth, the twentieth partial, etc. will not be excited.

Figure 9.4 (old 6.21). Average spectra at a filtering experiment. The total areas below the curves show non-filtered, the shadowed area the high pass filtered and the white area the low pass filtered average spectrum.

Press here for sound example 5 Jansson: Acoustics for violin and guitar makers 9.6

Sound example 5. The guitar and importance of different frequency bands. From Lagrima by Tarrega with the fist part broad band, the second part low-pass filtered, third part broadband, the fourth part high-pass filtered, and finally the fifth part broadband. The filter limit is set at 4 kHz the first time, 2 kHz the second time, and 500 Hz the third time.

The importance of different frequency regions (bands) can be studied with filtering and listening experiments, see Fig. 6.21. The filtering to give only components above 4 kHz demonstrate that there is only plucking sounds here. The sound filtered to give components only below 2 kHz sounds dull and hollow. With the sound filtered to below 500 Hz dull and "revelling" tones with indistinct attacks are heard. The sound filtered to above 500 Hz sounds clear but very thin. TOO STRONG PARTIALS AT LOW FREQUENCIES GIVE DULL, HOLLOW TONES WITH INDISTINCT ATTACKS. THE HIGH PARTIALS GIVE THE TONE CLARITY, ESPECIALLY AT THE ATTACKS, BUT GIVES A THIN TONE IF THE LOWEST PARTIALS ARE TOO WEAK. THUS A BALANCE OF SEVERAL FACTORS SHOULD GIVE THE BEST INSTRUMENT. 9.2. SOUND EXAMPLES RELATED TO THE VIOLIN

Figure 9.5 (old 1.8). Time history (oscillogram) of a played violin tone.

Press here for sound example 6 Sound example 6. The violin tone and time – three times full tone, thereafter a small portion of the initial part stretched, of the middle part stretched, and the final part stretched, repeated twice and finally the full tone three times.

A musical instrument such as the violin results in still more complicated time histories, c.f., Fig. 9.5. The time history is changing all the time in a specific way and monitored by the skilled musician. If one listens to the early, the middle, and the late part of the tone (by computer manipulations this can be done) one will hear that the loudness, timbre and also the pitch of the tone are different in the different parts.


Figure 9.6 (old 8.20, the relation pitch-frequency see fig 8.19). Long-time spectra and the influence of different frequency ranges on timbre.

Press here for sound example 7 Sound example 7. The violin and the importance of different frequency bands. From a Brandenburg concerto by Bach with the fist part broadband, the second part low-pass filtered, third part broadband, the fourth part high-pass filtered, and finally the fifth part broadband. The filter limit is set at 8 kHz first time (diagram not shown in this case), 4 kHz the second time, 2 kHz the third time, and 1 kHz the fourth time.

With long-time-average spectra and filtering the properties of the sound can also be compared with the perception of the same sound, c.f. Fig. 9.6. Above 8 kHz there is little sound energy, which thus should influence the timbre of the violin little. Also partial levels above 4 kHz should give a moderate influence. The partials above 2 kHz should, however, give a considerable influence on the timbre in agreement with the picture shown in the corresponding frame (the shaded area is beginning to become rather large). The main influence should however come from partials below 2 kHz. When you hear only partials below 1 kHz the tone is hollow with indistinct attacks (a small part of this effect can be caused by the filtering though). With more and stronger high frequency components, the tone becomes less hollow with more clear attacks. Too strong partials at very high frequencies give a rattling sound. If all partials below 1 kHz are removed it is still perceived as a violin although violins with such a thin timbre do not exist. If all partials above 1 kHz are removed one cannot recognise the tone as coming from a violin. A balance between different frequency regions is thus very important.


Figure 9.7 (old 3.7). Average spectra in the left (full line) and right (dashed-dotted line) ear of the player and one ear of a nearby listener (dashed line). In the upper part are differences between left and right ears of player (full line) plotted together with the differences between left ear of the player and the listener ear (dashed line).

Press here for sound example 8 Sound example 8. A violin recorded simultaneously in the player’s left and right ears simultaneously (stereo recording preferably replayed with head phones).

The ears (the pinnae) are on opposite sides of the head and slightly behind the middle. This arrangement gives a microphone system that can recognise directions by means of "spatial filtering". We should expect a marked influence on the sound by the head. Measurements also show that there are differences in sound level between the left and the right ears starting from 500 Hz. The player listens to his own instrument while playing and therefore a natural question is: How does the violin sound "look" in the two ears of the violin player? Measurements with small microphones inserted the ears of a player and the ears of a listener are shown in Fig. 9.7. The diagram shows that there are large differences between the left and right ears of the player, the left being approximately 10 dB above at 1.3 kHz (a replay with headphones sounds natural for a violin player but seems silent in the right ear of a non-player). The level for the listener was about 5 dB lower than the left ear of the player, somewhat more at high frequencies.


Figure 9.8 (old 3.5). Average spectra in different directions for the violin. In the lower parts spectra obtained in two directions are shown and in the upper part the difference between the two spectra together with the directions are plotted.

Press here for sound example 9 Sound example 9. A violin recorded (music from a Brandenburg concerto by Bach) in four different directions a) on the right hand side and in front, b) on the right hand side and behind, and c) on the right hand side and on the left hand side of the violin player.

The differences in radiation make spectra obtained in different directions look different. To demonstrate this, played music has been analysed with filters corresponding to the so- called critical bands of hearing, cf., Fig. 9.8. It is easily seen that the spectra are different in the different directions, especially at high frequencies. The spectrum corresponding to radiation in the neck direction gives the highest levels, In the directions perpendicularly to the neck, i.e., the top turned towards or away from the auditorium, there are large level differences in the frequency range 2 - 3.5 kHz (the range of the singer's formant and of the violin BH-hill). Note that the last two cases correspond to the difference between first and second violin positions when seated to left and to the right of the conductor, respectively. When the first violins are placed to the left and the second violins to the right of the conductor the first violins turn the top plate towards the auditorium but the second violins the back plate. With this arrangement the first violins will give a more brilliant sound than the second violins. If the player does not want to be heard, he should turn his back towards the auditorium (note the obvious connection between vision and hearing). A practical interpretation is the following. A soloist, who really wants to be heard, should aim with the violin neck at his listeners.


9.3. SOUND EXAMPLES RELATED TO THE ORCHESTRA

Figure 9.9 (old 3.18). Position of instruments in the symphony orchestra and measurement microphones A, B, C, and D (Slagverk=percussion, Pk=timpanies, Cor= French horns, Trä = woodwinds).

Press here for sound example 10 Sound example 10. Sound within the symphony orchestra – recording of a part of Scheherazade by RimskijKorsakov with microphones in four different positions in a symphony orchestra a) at the tympanis, b) in the viola section, c) in front of trumpets, and d) in the second-violin section at the piccolo.

As a part of a series of experiments with the Stockholm Philharmonic Orchestra microphones were placed close to musicians with loud instruments: at the French horns beside the tympanis (position A), at violas in front of oboes (position B), at bassoons in front of trumpets (position C) and at the second violins besides the piccolo (position D), see Fig. 9.9. The recorded music shows that the piccolo is very dominant, and that even the brass is difficult to hear "through" the orchestra at fff level. 9.4. SUMMARY –SOUND EXAMPLES In the physical-technical world we have tools to measure and describe sound. In the musical world we use our ears and brain to analyse and describe sound. To give the reader an introduction to the sound we hear and how it is described in the physical-technical world, recorded sound examples are given together with their physical analysis. First relations between partials and spectra, smooth spectra and spectra of roughness. In the specific guitar examples, the importance also of weak body sound, plucking position and the importance of different frequency regions are shown. In the violin examples variations within a violin tone, the importance of different frequency regions, the spectra in the ears of a player, and radiation in different directions of the player. The last example demonstrates how different the sound within the symphony orchestra can be.


CHAPTER 9. Second part: SIMPLE EXPERIMENTAL MATERIAL INTRODUCTION

In this part principles for simple vibration measurement transducers are presented. First simple transducers using a magnet and an electrical coil. Secondly optical sound using ready-made optical heads is introduced. Finally a very short introduction to an analysis tool which employs a personal computer, PC (or MAC). The tool, an analysis program can be loaded down free and opens many opportunities for experimenting. 9.5. MAGNET-COIL TRANSDUCERS New magnetic materials have made new simple and powerful transducers possible. The samarium- cobalt magnets could today be bought in the open market, typically at places offering electronic hardware. The magnets are very strong relative their weight and rather stable without special arrangements. The principle for the transducers is presented in fig 9.10. The magnet is fastened to the object (the violin or the guitar) and an electrical coil is placed coaxially without touching.

Figure 9.10.

Principle elements of the coil-magnet transducer.

The coil-magnet transducer can be used as a vibration pickup transducer, see fig. 9.11. The magnet is waxed to the vibrating top plate is vibrating and thus the magnet is vibrating too. The coil is held fix and a voltage corresponding to the vibration magnitude (velocity) is generated ( cf vibration measurements fig 7.5). If the procedure is reversed a powerful vibration driving-transducer is obtained. An electrical current of constant amplitude but varying frequency gives a driving force of constant force but varying frequency, i.e. a very handy driving transducer for a defined frequency.

Figure 9.11.

A coil-magnet transducer as a vibration driving-transducer or vibration pickup-transducer.


The coil-magnet transducer can also be used in a contact-microphone, see fig. 9.12. The coil is fasted to the vibrating plate (by wax). Inside the coli on a soft spring, foam plastic for example, the magnet is glued. The coil will again give a voltage corresponding to the plate vibrations (velocity).

Figure 9.12.

A coil-magnet transducer as a contact microphone.

9.6. OPTICAL TRANSDUCERS Today, ready-made optical transducers can also be be bought in the open market, typically at places offering electronic hardware. The optical transducers were probably constructed to register hole or no-hole on paper tapes or cards for computers. There are two kinds of transducers, reflexion transducers, see fig 9.13 and ttransmission transducers, se fig 9.14. The transducers needs simple supporting electronic circuits.

Figure 9.13.

Principle of an optical reflexion transducer.


The reflexion transducer registers reflected light. If there is a hole no light is reflected. If there is no hole, the flat surface reflects the light. By placing a vibrating surface very close to the transducer no light is reflected to the light-sensitive element. With the vibrating surface (white is good) at a small distance a part of emitted light is reflected to the light sensitive element, see fig 9.13. If the surface at this distance is vibrating, the amount of light reaching the light sensitive element varies and gives a measure of the vibrations (distance, displacement).

Figure 9.14.

Principle of an optical reading fork used for recording string vibrations.

The transmission transducer records light passing through the hole. No hole means no light and a hole means light. With only a part of the light passing through the hole, the lightsensitive element will give a voltage proportional to the amount of light reaching the element. This principle can be used to demonstrate string vibrations. An optical fork is waxed to the violin bridge. Its position is adjusted so the G-string partly covers the light to the light sensitive element. The amount of light now reaching the element records the string position relative the bridge. If the string vibrates, then the signal from the light sensitive element records the string vibrations. The method can be used to demonstrate string vibrations while playing a violin with bow. --


9.7. COMPUTER ANALYSIS (PC or Macintosh)

MICROPHONE

L R

M Figure 9.15. Typical arrangement for signal analysis with a PC (or a Macintosh) – microphone as transducer, PC (Mac) with its screen to show results and mouseM with left and right buttons to monitor the analysis..

DOWNLOADING WAVESURFER Open www-page http://www.speech.kth.se/software/ http://www.speech.kth.se/music/downloads/smptool/)

(another

very

useful

www-page

Download Wavesurfer click at Downloadpage . Choose for instance Binary release for Windows file download, save, wavesurfer, save in WinZip File, save Wavesurfer as a WinZip file Unzip, follow instructions and store unzipped file, gives wavesurfer .exe on hard disc,. WAVESURFER EXAMPLES AS STARTERS FOR USE For the a more complete description see “WaveSurfer manual”. Click at WaveSurfer manual” on the monitor screen below the summary presentation of the WaveSurfer.


1. Instructions for recording start wavesurfer, connect microphone, press red button in the wavebar-window, record the sound, press black button to stop recording, store as file one, repeat store as file two, repeat store as file three, etc. (The same procedure as with a tape recorder), exit by clicking at X button.

Figure 9.16. Window with wave bar

2. Example for analysis – Waveforms (cf fig 9.5). With wave surfer started, open, file (file Windows logon for example), choose configuration at standard, OK, click right mouse button (CR on waveline in the window with wave cf fig. 9.16), click left mouse button (CL) on create pane, CL Waveform, press left button and mark are range by sliding the mouse, CL at magnifying glasses above the waveform, repeat until suitable waveform has been cut out and enlarged, cf fig. 9.17. Click right mouse button (CR on wave line), CR create pane, time axis – now the time axis is created in fig 9.17.

Figure 9.17. Window with zoomed waveform with time axis.


3. Example for analysis – frequency responses (cf fig 7.5 and 7.6)

With wavesurfer started, open, file (file tap), OK, click right mouse button (CR on waveline in the window with wave cf fig. 9.16), click left mouse button (CL) on create pane, CL spectrogram (gives spectrogram cf. fig. 9.18), CR (on spectrum), properties (set suitable freq range, for example 500 Hz, cf fig. 9.19 ), CR, create pane, spectrum section (set FFT points tex 8192, reference –60dB, and range 50 dB, cf fig. 9.20), seek frequencies with cursor in spectrum section.

Figure 9.18. Window with time-zoomed waveform.

Figure 9.19. Window with properties to select for spectrograms (upper frequency limit = cut spectrogram at)


Figure 9.20. Window with frequency response (spectrum Section Plot) peak at 206 Hz and – 28.4 dB- move cursor lines to the interesting frequency. .

4. Example for analysis – tone spectra (cf fig 9.1 and 9.2) With wave surfer started, open, file (file three for example), click right mouse button (CR on waveline in the window with wave cf fig. 9.16), click left mouse button (CL) on create pane, CL spectrogram (gives spectrogram cf. fig. 9.21), CR (on spectrum), properties (set suitable freq range, for example 500 Hz, cf fig. 9.18 ), CR, create pane, spectrum section (set FFT points tex 8192, cf fig. 9.22), seek frequencies with cursor in spectrum section (the vertical lines result from the low fundamental frequency).


Figure 9.21. Window with tone spectrogram (Spectrum)

Figure 9.22. Window with tone spectrogram (Spectrum Section Plot)


9.8. SUMMARY – SIMPLE EXPERIMENTAL MATERIAL Two simple and powerful vibration measurement transducers have been presented. First the coilmagnet transducer, secondly the optical transducer. The coil-magnet transducers need some coilwindings but are therafter ready to use. Parts for the optical transducers can be bought ready – only some supporting eectronics must be built. The computer program Wavesurfer can loaded down free and can be used on most personal computer systems. The experimenter can develop his own analysis.


Guitar and violin acoustics

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