The Timbre of Musical Instruments: The Physics of Music Laboratory Manual Written by Derek Stadther and Dr. J. David Turner Furman University, Greenville, SC August 8, 2011
1. Introduction to AC Signals and the Oscilloscope- pg. 2 2. Standing Waves on a Stretched String- pg. 7 3. The Helmholtz Resonator- pg. 12 4. Standing Waves in an Open Pipe- pg.19 5. Standing Waves in a Pipe Closed at One End- pg. 26 6. Musical Scales- pg. 29 7. Lissajous Figures: A Visual Representation of Musical Harmony- pg. 37 8. An Introduction to Surface Resonances: Chladni Shapes- pg. 42 9. A Percussion Instrument Modeled by Simple Pipes- pg. 43 10. An Introduction to Organ Notes- pg. 48 11. Physics of the Violin- pg. 55 12. The Theremin: An Introduction to Synthetic Music- pg. 70
This lab manual was designed to guide and assistant students in discovering the interconnectedness of physics and music. It is specifically used in a class offered by Furman University titled The The Physics of Music. Creation of this manual was made possible by a Furman Advantage Research Grant. The authors hope that readers will enjoy this first edition of The Timbre of Musical Instruments. “
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1
Introduction to AC Signals and an d the Oscilloscope
Sound is characterized characterized by longitudinal pressure waves. These pressure waves are created by the back and forth movement of air molecules. Devices known as transducers can change a time varying quantity such as pressure into a voltage which is proportional to the pressure at every moment. A microphone and an associated amplifier are the transducers most often used to study audible sounds. In this lab we will study the voltage waveforms of various sounds using the oscilloscope. I. Introduction
The oscilloscope is a versatile lab instrument capable of recording and analyzing time varying voltages. In its simplest mode the scope provides a graph of voltage as a function of time. Most scopes these days have two or more channels. Each channel can be used to look at a different signal, although they will all share the same horizontal scale. The horizontal axis is time and the calibration is controlled by a knob labeled "sec/div". If this control is set to 1 ms (millisecond) per division, a 1000 Hz signal would go through a full cycle in one horizontal division (one centimeter on the screen). With 10 divisions across the screen, 10 cycles would be visible. The vertical scale of each channel is controlled with a knob labeled with its channel number under the heading of "volts/div". Note that each channel has an associated color. An AC signal with a 2 volt amplitude would go up and down two divisions if the vertical scale is set to 1 volt per division. II. Measurement DC Voltages
Turn on the scope. Push the trigger menu. Set to chan 1 and auto. This will ensure a reliable trace for this section. Push chan 1 menu and set mode to DC, and select the 10x setting for the probe. Adjust the volts per division to 1 V/div. Connect the probe to the battery. What happens to the trace?
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III. The Observation of Time Varying Signals
Connect a short length of wire to the channel 1 probe. Set the seconds per division control to 5 ms/div and the vertical scale to 0.1 v/div. Note that touching the bare wire increases the signal. Use the run/stop control to capture the waveform. The oscilloscope is especially useful for measuring time intervals Now press the cursor button and select the time cursor 1. Move cursor one to peak. Select cursor 2 and move it to the next peak Note time interval and associated frequency. This 60 Hz signal comes from power lines and transformers and composes much of the background electronic noise. Many cables are surrounded by a metallic shield to protect the signal from this noise. Any time you hear a hum of see a 60 Hz signal, look out for a bad connection to ground or the cable shield.
Connect the amplifier and microphone to the channel one input. Use the scope to verify the frequency stamped on several tuning forks. The trace below is from a fork stamped 320 Hz.
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IV. The trigger control
The run/stop control is useful in capturing a waveform from an extended sound. In many cases the sound may not be sustained as in percussion instruments, or we may have interest in the attack portion of the sound of any instrument. The trigger is operated in one of three modes Auto: Best for DC signal or unknown signals Normal: Best for smooth repetitive waveforms Single sweep: Best for non-repeating waveforms. This is the setting we will use often to look at the attack portion of musical notes.
Application: The Determination Determination of the Speed Speed of Sound Through Time of Flight.
Consider two microphones placed approximately 4 feet (125 cm) apart. With the speed of sound being about 1000 ft/s, it would take sound about 4 ms to travel the distance between microphones. The trace below shows the signals from two microphones located approximately 4 ft apart. Note that the time interval is 3.960. This trace represents the sound made by striking a piece of plywood with a hammer.
Connect two microphones to the inputs of your scope. Select the single sweep mode and measure the speed of sound.
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V. The FFT
Complex sounds can be composed of components of several frequencies. A technique called the "Fast Fourier Transform" is capable of determining the frequency and amplitude of the components. Consider the 320 Hz waveform seen below.
If the scope is set to the FFT mode the following is displayed.
Note the single peak at 320 Hz.
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Now consider the waveform shown below. This was taken from the 320 Hz fork being struck at the same time as a 480 Hz fork. Note that the difference tone (160Hz) is the most conspicuous aspect of the waveform. While there is no direct source of a 160 Hz tone, the ear will hear this frequency distinctly.
The FFT of the above above waveform is shown below. Note that the frequencies of the two forks are distinct, and the frequency of the difference tone is absent.
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Standing Waves on a Stretched String
I. Introduction
It is easily shown that traveling waves on a stretched string have the form
A sin(kx wt ) (1) 2
2
and f are the wavelength and frequency 2 f and and T respectively. The represents two possibilities. possibilities . The positive sign corresponds to waves moving in the negative x direction and the negative sign corresponds to waves moving in the positive x direction. where k
It is also easily shown that the propagation speed c is related to the frequency and wavelength through: c f
A uniform string can be characterized by a mass per unit length defined as .The propagation speed can also be shown to be related to the tension and mass per unit length: c
T
In this lab we will investigate the relationship between the frequency, wavelength and string tension for standing waves.
II. Discussion
Consider the case of a string fixed at both ends. Such a string is forced by its mountings to have a node at each end (to have 0) . A wave traveling to the end of the string will experience a closed end reflection and be reflected in the other direction with equal amplitude and opposite phase. Now consider a continuous wave leaving x=0 and reflecting off of the fixed end at x=L. It will have the same amplitude and frequency, but opposite sign in the term and some phase difference . For simplicity we will assume that the original traveling wave originates from x=0, traveling to the right. Thus the original wave moving to the right is given by:
7
R A sin(kx wt ) The reflected wave moving to the left will be given by:
L A sin(kx wt ) The net wave disturbance at any point will be the sum of these two waves:
net R L A[sin(kx t ) sin(kx wt )] Using the trigonometric identity 1 1 sin( A) sin( B) 2 sin[ ( A B)] cos[ ( A B)] 2 2
gives
net 2 A sin(kx ) cos[(wt )]] (2)
Since the string is constrained to be stationary at x=0 and x=L, locations. From the constraint at x=0 we see that since. sin( n ) 0 .
net must
be zero at these
must me an integer multiple of pi,
Applying the constraint at x=L demands kL be an integer multiple of pi as well. There two equations together imply that kL n where n is an integer. Substituting Substituti ng for k gives 2
L n
which leads to the condition: n( ) L . (3) 2
Thus standing waves will occur when the length of the string exactly equals an integer number of half wavelengths. This means that the string will vibrate with large net amplitude when the string is stimulated with frequencies that meet the condition given by (3). Notice that equation (2) implies that the string will have nodes separated by one half wavelength.
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In this exercise we will find the frequencies that allow a standing wave to exist with 5 through 10 loops.
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III. Procedure
Consider the arrangement shown below.
Note that the tension in the string is provided by 155g of weight. The left end of the string is vibrated by a sin wave current produced by the computer. The driving device is a modified speaker. The sine wave motion is accomplished by magnetic forces on a current carrying coil in a large magnetic field created by the permanent magnet at the bottom of the driver shown below.
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We can get an overview of the lab by sweeping through frequencies from 5 to 30 Hz and observing the string. Start the program STRING and observe the string. Note and record the frequencies where standing waves are evident. Also note the number of loopw. Repeat the sweep as need until the frequencies that produce 5-10 loops have been approximately determined. By hitting the spacebar during the frequency sweep, you can mark frequencies at which there are an integer number of standing waves present on the vibrating string. After the sweep has been completed carefully search for the resonant frequencies and be as accurate as possible. Once the resonant frequency has been determined, carefully measure the distance between nodes. Be sure to record your your answer to the nearest nearest millimeter. Compute the wave speed as the product of frequency and wavelength.
Data
number of loops
approx. frequency
resonant frequency
distance between nodes
5 6 7 8 9
Average wave speed __________ Predicted wave speed _________ % difference _________
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wave speed
The Helmholtz Resonance I. Introduction
Picture perhaps the simplest early instrument, the drum. A typical drum is a rounded cylinder of wood or metal that is covered at both ends with some slightly flexible material. The enclosed cavity of the drum allows sound, caused by striking one of the ends, to resonate inside of the drum. From early times, stringed instruments featured strings coupled to a resonant cavity, very similar in design to a drum. By connecting a long piece of wood, called the neck of the instrument, to the side of a drum-like enclosure and stretching strings across the resonant cavity and neck, a crude stringed instrument could be made. The resonant effect of a cavity with a neck was noted by Helmholtz and serves as a model from the resonant character of stringed instruments. Helmholtz constructed a series of resonators in the 1850s to study acoustics. A diagram of one of his resonators is shown below. Sound would enter the resonator through the hole at point “a” and the human ear would be placed at point “b.” Helmholtz even used sealing sealing wax to couple his ear as efficiently as possible to the resonator. The resonator has a characteristic resonant frequency which allows the resonator to act as an acoustic filter. The Helmholtz resonator can be understood by analogy to a mass connected to a spring. The mass on the end of a string demonstrates simple harmonic motion if the spring obeys Hook’s law. We begin with Newton’s second law with the force exerted by the spring being Hook’s law: F ma F kx
substituting for the acceleration a
dv dt
2
d x
2
dt 2
gives
kx
d x 2
dt
.
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2
Rearrangement gives
d x 2
dt
k m
x (1)
It can be easily shown that this has a solution given by: x A cos(wt ) (2)
where w
k m
(3)
and A and are the amplitude and phase respectively.
II. The Helmholtz Model
Consider the situation below.
A cavity of volume V is connected to a neck of cross sectional area A and length L. The air inside the neck serves as the mass and the air inside the cavity serves as the spring. The mass of the air in the neck is given by: m AL
where is the density of air. Imagine the air in the neck is pushed downward so as to displace this body of air a distance x. This results in a decrease in the volume of the gas in the cavity of Ax.
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The resulting compression increases the pressure in the cavity and provides a restoring force. The restoring force resulting from the increase in pressure, dp, is given by F dpA
where A is the cross sectional area of the neck and dp is the increase in pressure arising from the decrease in volume. The decrease in volume dV is given by dV Ax Ax .
From elementary thermodynamics it is easily shown that pV b , where b is a constant. Rearranging and taking the derivative gives
p bV dp
bV ( 1) (bV )V 1
dV
p V
and hence F dpA
p V
dVA dVA
pA2 x V
Substituting into Newton’s second law gives:
pA2 x V
2
AL
d x 2
dt
which gives 2
d x 2
dt
(
p
A )( ) x LV
From elementary thermodynamics it can be shown that the speed of sound is given by c
p
or
p
c2
Making this substitution gives:
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2
d x 2
dt
c 2 (
A
LV
) x
A Thus we find 2 c 2 ( ). LV
This gives the Helmholtz resonant frequency to be: f
w 2
c
A
2 LV
or f
c
A 1
2
V L
(4)
This result predicts that a plot of frequency against the reciprocal of L would be a straight line with a slope given by
c
A
2
V
and an intercept of zero.
Typical data are shown below.
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III. Experimental Method
In this activity the resonant cavity will be formed from some larger plumbing parts and the neck will be created with any of a variety of lengths of pipe of smaller diameter. This will allow investigation of the role of neck length in the resonant frequency.
16
The pressure oscillations inside the cavity will be monitored with a small microphone inserted in to the back of the resonant cavity.
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One interesting feature of this apparatus is the junction between the smaller diameter pipe and the resonant cavity. Vibrations inside the smaller pipe will take place as if the smaller pipe was isolated since the junction is an open end for vibrations traveling in that direction. For this reason we expect two modes of vibration. One mode of vibration will be the Helmholtz resonance, and the other will be the vibrations of the open pipe located at a higher frequency. A typical spectrum is shown below.
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It is interesting to compare the frequencies of the second peak in the spectrum to those of the open pipe. Note that the two agree very well at longer lengths and less well at shorter wavelengths. From this data it is evident that the second peak is easily explained as the resonance for the open at both ends for longer lengths. As the length gets shorter it is no longer a good approximation due to impedance issues at the junction.
IV. Procedure
Measurement of V (volume) Data table Acquire and analyze spectra for two peaks
pipe length ____ _____ ____ ____ _____ _____ _____ _____ _____ _____ frequency
_____ _____ _____ ____ _____ _____ _____ _____ _____ _____
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Standing Waves in an Open Pipe I. Introduction
Many instruments derive their unique sounds from standing waves similar to those that exist in a segment pipe open at both ends. All instruments in the brass and woodwind family, with the exception of the clarinet family, function as a pipe open at both ends. Woodwind Instruments (Reed Aerophones)- Flute (non-reed), Oboe, English Horn, Clarinet, Saxophone, Bassoon Brass Instruments- French Horn, Trumpet, Euphonium, Trombone, Tuba To understand how these instruments produce different pitches, one must realize that the above mentioned instruments are essentially a pipe open at both ends. If we assume that there will be an anti-node at each end (which is only approximately correct), the pipe will have a fundamental resonant frequency given by: f
v 2 L
(1)
where f is the fundamental resonant frequency, v is the speed of sound in air, and L is the length of the pipe. The purpose of all the buttons, slides, keys and valves found on these instruments are to change the length of the tube. For example, consider the trombone: A piece of tubing with a flair at one end and a mouthpiece at the other. The slide of the trombone enables the player to change the effective length of the instrument (L) thereby changing the fundamental resonant frequency (f).
http://meta.wikimedia.org/w http://meta.wikimedia.org/wiki/File:Britanni iki/File:Britannica_Trombone_Tenor ca_Trombone_Tenor.png .png (public domain picture)
With the slide all the way out, the resonant frequency is that of an E-natural (E1) and all the way in, B-flat (Bb1). The range of the trombone is generally generally from E2 to Bb4, so how can this be if the range of the fundamentals are from E1 to Bb1? Trombone players
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rarely play the fundamental frequency of a given position, but play pitches in the overtone series of the fundamental pitch. With seven steps from E1 to Bb1, and a unique overtone series for each step, the trombone can play all scales in many octaves. While the trombone varies L by use of a slide, other brass instruments accomplish the same thing by using valves. These valves do nothing more than to change the effective length of the pipe by routing the air flow through various smaller lengths of pipe.
In a pipe opened at both ends, a whole number of half-wavelengths always forms.
These diagrams represent the fundamental frequency, first overtone, and second overtone.
Going up an overtone always adds one more half-wavelength to the inside of the opened pipe.
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II. Discussion
This lab will investigate the relationship between the length of an organ pipe and its fundamental frequency. The pitch of an open pipe is only approximately related to its length through (1). A similar relationship exists for waves on a stretched string. The case of the string is simpler in that the nodes are always exactly at the end of the string. The case of standing waves in an air column is complicated by the fact that the pressure anti-node is located outside the end of the tube, contrary to the assumption that led to (1). We will denote this distance with the letter s.
Diagram The fundamental note is related to the length: 2
L2s
(2) where s is the distance between the end of the pipe and the antinode.
Since the speed of sound is related to the frequency and wavelength through c f (2) can be rewritten: L
c 1 2 f
2s (3) where c is the speed of sound.
Equation (3) can be written as: 1
f
2
4s
c
c
L
(4)
In Part I you will measure the resonant frequency for a number of pipes of varying lengths. This will allow a precise measurement of the speed of sound. Equation (4) has the form y mx b where the reciprocal of the frequency has the role of y and the length has the role of x. The slope will equal to the reciprocal of half the speed of sound and the intercept will be 4s/c. Typical data and fit are shown below. The slope gives 344 m/s for the speed of sound . This gives a value for s or 0.8 cm.
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In Part II you will investigate the relationship between the distance denoted as s above and the diameter of the pipe. Equation (3) can be rearranged to give: s
c 4 f
L 2
Many texts state that the relationship s and diameter D is given by: s=eD
where e is a constant between 0.3 and 0.4.
Use polyfit to find a value for e from your data. Typical data are shown below, with a value of e=0.270.
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Part I. Fundamental Tone as a Function of Tube length length
Procedure Set up the equipment. Take the first pipe and place one end approximately 10cm from the speaker and place the microphone directly outside of the other end of the pipe.
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“FreqencySpec3” program for running the experiment. The On the computer, open op en the “FreqencySpec3” three top panels contain the controls and the panels below contain a processed waveform of the data you are about to collect. In the top left panel, click the box labeled “DAC” to turn on the audio processing. Next set the parameters of the frequency ramp. Set the initial frequency (fi), final frequency (ff), the duration (milliseconds), and click “set”. The middle panel of the controls contain a slider which sets the volume of the frequency ramp which will be sent to the speaker. Turn this up to an appropriate level. The last panel sets input volume of the microphone. Only the first one will be used in this lab. With the controls set, click “Run” in the first control panel to do a test test run to make sure that everything is working properly. You should see the processed waveform appear in the lower panel. If you see that the signal is clipping, try either turning down the output of the frequency ramp or turning down the microphone input. Continue to re-run the ramp until the levels are acceptable. Measure the length of each pipe and record it into the table below. Run the frequency sweep. You should see the processed waveform appear in the viewing pane looking similar to the
example below. Locate the most prominent peak, this is the fundamental resonance of the length of pipe. Highlight this area by clicking and dragging the mouse then click selection zoom to focus the viewing window on this area. Click the point of the waveform where the amplitude is the highest and record the number in the above box labeled “Frequency at Cursor” in the right column of the table below. Repeat this for each length of pipe.
Measure the length of each pipe and record it into the data table. Next setup the equipment. Take the first pipe and place one end approximately 10cm from the speaker and place the microphone directly outside of the other end of the pipe.
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Part II. II. The Relationship Relationship Between Between “s” and the Pipe Radius
Measure and record the resonant frequency of each of the pipes in the data table below. Measure and record their diameters as well. Make a plot of the parameter s and a function of the pipe diameter. Perform a linear fit and calculate a value for e from the slope. Compare your value to the values often seen in the literature.
Part I Data
Length of Pipe
Resonant Frequency
(a list of Matlab commands can be found in the appendix)
slope = ________
temperature ________
measured speed of sound ______ speed of sound corrected for temperature ________ percent error _______ intercept = _________
pipe diameter ________
diameter
Part II Data ______ ______ _______ ________ _______
resonant freq
______ _______ _______ _______ _______
slope (value of e) ______ intercept ______
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Standing Waves in a Pipe Closed at One End I. Introduction
In a previous lab we considered the case of a pipe open at both ends. In this case we have a pipe that is closed at one end and open at the other. In the fundamental mode the pipe length is approximately one quarter of a wavelength. If the pipe is lengthened, resonances will occur each time an additional half wave fits in the pipe. The anti-node forms a distance denoted by s from the end of the tube. Thus the general expression for a resonance is given by L
2
n(
4
s) (1)
where n is an integer (0,1,2,3 etc) and is the wavelength. In this experiment the length of the pipe will be adjusted to find the resonances. Note that (1) has the form of y=mx+b, where n has the role of x and ( of b. Typical data are shown below.
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4
s) has the role
These three diagrams represent the fundament and its first and second overtones in an organ pipe which is closed at one end.
Notice that a node of the wave forms at the closed end of the pipe. There are also always an odd number of quarter-wavelengths.
Each overtone adds one more half-wavelength than the overtone immediately preceding it.
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II. Procedure
Set the signal generator to 1000 Hz. This means that the wavelength will be about 34.3 cm. Set up the microphone and to measure the intensity of the sound wave with an AC meter. The first resonance will be expected at about one fourth wavelength, with the others at half wavelength intervals. Note that the sixth resonance is close to the full one meter length.
III. Analysis
Fit your data and compute the wavelength from the slope. Use the frequency to compute the speed of sound. Compute the end correction from the intercept.
Data
resonance number distance
0 _____
1
2
_____
3
_____ _____
diameter of tube _________
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4 _____
Musical Scales
A musical scale is a way of dividing the audible frequency spectrum into discrete parts. Although the methods of division vary throughout different cultures, the octave is generally recognized as a fundamental division. This lab will examine 3 different methods of tuning/temperament used in the western musical system. I. Listen to the Scales
Throughout this lab, three different scales will be introduced, the Pythagorean, the Just, and the Equally Tempered Tempered scales. While comparing these scales is simply a matter of looking at numbers and comparing differing ratios, it is also helpful, as well as practical, to listen to these scales, since we are studying music, a form of expression using sound. Open the program “Scales(Overtones23).” At the top in the “Master Volume” Volume” box, put the slider labeled “Change Both” Both” between 60% and 75%. Also at the bottom left of the screen, change the slider labeled “Set “Set All Levels” to the same value. Next, beside the label “Fundamental,” type in a value in the box for the the root pitch in Hertz. Thedrop The drop down menu in the top left allows you to choose one of the three scales listed in this lab. Using the keys a,s,d,f,g,h,j,k, you can sound the pitches of the scale just like a simple piano keyboard. Multiple pitches may also be sounded at the same time to create chords.
II. Octaves
The piano keyboard has numerous “C's” of different frequency frequency.. Their “C“C-ness” is a result of octave equivalence. We perceive different frequencies related by factors of 2 as having an underlying unification and similarity. similarity. Imagine a mixed group of people singing a monophonic tune such as “happy birthday”. For any given note, the entire entire group usually sings in octaves since different people have different vocal ranges. Despite having different frequencies, each pitch of the tune is thought to be equivalent regardless of the octave differences. To find higher or lower octaves of a given frequency, simply multiply or divide by a factor of 2. Suppose 200Hz as a starting point and octaves are found as follows: 1/8 25Hz
¼ 50Hz
½ 100Hz
200Hz
2 400Hz
4 800Hz
III. Harmonic Series
The harmonic series, also known as the overtone series, provides an appropriate starting point for dividing the frequencies within the span of an octave. All harmonics of an arbitrary fundamental fundamental frequency frequency ( ) can be calculated calculated by the following equation:
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8 1,600Hz
where n= 1, 2, 3, … and
is the nth partial of
Calculating the harmonic series in terms of
55Hz
=55Hz (pitch “A”) gives
3 220Hz
110Hz
.
4 440Hz
5 880Hz
6 1,760Hz
and so on. Each step of the harmonic series is known as a harmonic, overtone, or partial. The fundamental can be called the “first harmonic,” twice the fundamental can be called the “second partial,” etc.
IV. IV. Basic Rules for Creating Scales
For any scale, all the notes comprising the scale must be in the same octave, between and 2 . Also, in most most scales, scales, all the the letter letter names in a given octave are used. For example, the order of pitches in the C scale is spelledspell ed- C, D, E, F, F, G, G, A, B, C. All seven notes, A-G, are used and the letter name of the fundamental is also the name of the octave. V. Using Intervals to Create a Scale
From the harmonic series, the second partial is twice the fundamental, and thus, the th octave. octave. The next next partial partial is is 3 and is is the inter interval val of of an octave octave+a +a perfect perfect 5 above . Divi Dividi ding ng 3 by 2, we brin bring g the the pitc pitch h with within in rang rangee of and and 2 .
th
Thus to obtain a perfect 5 (P5) up from
, simply multiply
by 3/2.
VI. Pythagorean Scale th
The Pythagorean scale is based on the interval of a perfect 5 (in ratio form, 3/2) as it is the smallest ratio of whole number integers after the unison (1/1) and octave (2/1). The th perfect 5 is the most consonant intervallic relationship between notes, excluding the octave. Beginning with an arbitrary fundamental frequency, we ascend in P5s. (Letter names are given as examples)
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7 3,520Hz
(please note that means to lower a pitch by one octave (multiple the frequency by 1/2), and lowers a note by two octaves (multiple by 1/4)) Notice that the above calculations are missing notes F and C-8va from the C-Major scale. Because the octave is defined as twice the fundamental frequency, C-8va is merely twice C. Here, because C is , C-8va is th
The interval from C up to G is a perfect 5 , however the interval from C d own to G is a th perfect 4 . Given that G is related to the fundamental by a ratio of , the interval of a th
perfect 4 (P4) can be derived by dividing the fundamental by 3/2 and then transposing it up an octave by multiplying by 2.
We now have all the ratios for constructing the Pythagorean scale. Arranging them in ascending order gives:
C
D
E
F
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G
A
B
C
Exercises:
1. Construct a Pythagorean major scale on scale degree 2 in terms of become scale degree 1 and will be
D
E
F#
G
A
B
above. D will
C#
2. Comparing the pitches from the the C-Major and D-Major scale, which ones are different? Ignore pitches that are not common to both scales.
3. Moving from C, 12 steps around the cycle of 5ths brings you to B#. Show that, in the Pythagorean scale, B# is not the same as C by ascending 12 5ths from C (use ). Bring the result into the range of by repeatedly dividing by 2 and write B# in terms of . th Round the coefficient of to 4 decimal places. (Hint- one 5 up from is )
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D
VII. Just Scale rd
The 3 parti partial al of of the the harmo harmonic nic serie seriess (3 , or down down an an octa octave ve,, 3/2 ) is suf suffic ficien ientt for for deriving the lowest ratios for scale degrees 5, 4, and 2, however scale degrees 3, 6, and 7 are modified in the just scale. th
Moving to higher partials can provide better ratios for scale degrees 3, 6, and 7. The 4 – 4/2 – 4/4 – 4/4 ) is merely partial (4 – 4/2 merely two octav octaves es above above the fundam fundamenta ental. l. However However the th 5 partial (5 ) is the next point of interest. By transposing transposing within range of and , rd the 3 scale degree of a major scale is derived in its most pure form.
Three major triads exist in a major scale: 1. Tonic (I-1 3 5) 2. Dominant (V-5 (V-5 7 2) and 3. Sub-dominant (IV-4 (IV-4 6 8). The Roman numerals denote which scale degree is the root of the chord. Notice that these three triads contain all scale degrees for a major scale. Now we construct the just scale by stacking intervals of both a major 3 to scale degrees 1, 4, and 5.
rd
and a perfect 5
th
Scale Deg Tonic SubDominant Dominant
In the above diagram, scale degree 2 in the first octave is empty. empty. So we transpose 2 in the second octave by dividing it by two. Simplifying the diagram we have the formula for constructing a just major scale for any given .
Scale Deg
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Exercises: th
1. Find the frequency of the 7 scale degree if
=243.2 Hz
2. Given = Pi, find the frequencies of a scale 5 octaves above the fundamental keeping 4 decimal places.
3. Two frequencies sounded simultaneously simultaneousl y have both an interval of a just major 3 a differential frequency of 55Hz. What are the two frequencies?
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rd
and
VIII. Equally Equa lly Tempered
The Pythagorean and Just scales can be problematic for fixed-pitch instruments when transposing from one key to another. another. Equal temperament attempts to mediate the problems of other tunings by dividing the octave so that the ratio of each semitone, or half-step, is constant. This allows each key to have the same semitone intervals, and key modulation can be done with freedom. The following equation, which forms a scale system commonly known as Twelve-Tone Equal Temperament, or 12-TET, provides the means to calculate the frequencies for all notes of a 12-tone octave. Where x is a constant ratio and n=1,2,3...13. Example Notes in C
Interval
C
Unison
C# / Db
Minor 2nd
D
Major 2nd
D# / Eb
Minor 3rd
E
Major 3rd
F
Perfect 4th
F# / Gb
Tritone
G
Perfect 5th
G# / Ab
Minor 6th
A
Major 6th
A# / Bb
Minor 7th
B
Major 7th
C - 8va
Octave
Next we find the value for x. We know that the fundamental fundamental and octave ( , related by and that
therefore
and dividing through by
then solving for x
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) are
With a value for x, a 12-tone equal temperament scale can be constructed from a single frequency ( ). Higher or lower lower octaves octaves are found found by either multiplying multiplying or dividing by a factor of 2.
Exercises:
1. = 420Hz. Find the frequency an octave plus a tritone higher using equal temperament.
2. Show the validity of enharmonic equivalence (i.e. C# is the same pitch as Db) in the equal temperament scale by ascending 12 5ths from and then repeatedly dividing by 2 to return to .
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Lissajous Figures: Figures: A Visual Representation Representation of Musical Harmony Harmony
It has long been recognized that certain tones produce a harmonious result when played together. These harmonious tones exist when the two tones have frequencies that are simple integer ratios. Octaves are 2/1, that is, two times the fundamental frequency, and the perfect fifth is 3/2, or one and a half times the fundamental. In this lab we will investigate a visual representation of these harmonies. This technique was first used by Lissajous himself. This work is mentioned in Helmholtz's famous text On the Sensations of Tone. A drawing of his first apparatus is shown below.
A strong magnifying lens is located at point L in the figure, on the arm of an electrically driven turning fork. A small, highly reflective object is located on the arm of a second oscillator oriented perpendicular to the first. Motion of either the source or the lens produces motion of the image seen by an observer looking through L. Not surprisingly the motion of the lens produces much the same effect as motion of the source Lissajous later improved his device to use perpendicular motions of a mirror to reflect a spot of light onto a screen, so the Lissajous figures could be seen by a number of people at once.
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Two electronic oscillator circuits generating sine wave signals can easily be used to produce interesting plots. One sine wave is used to generate the output which is the x value and the other generates the y value. Oscilloscope manufactures have long included “xy” mode in addition to the standard voltage vs time mode among oscilloscope features. Before the existence of frequency meters these figures were used to measure relative frequencies of two signals. In this lab we will use two independent signal generators. Since the generators do not include a phase control, one has to be added to view the figures in the most symmetrical form. Appendix X includes a diagram of the circuit and an explanation of its design.
Procedure
Turn on the oscilloscope and push the “display”. Select “xy” mode from from the options. Then hit the persistence button and set that to 2 seconds. (Note: for some of the more complex Lissajous figures, the persistence setting may have to be changed to 5 seconds or more) Turn on the two signal generators and set them to 100 Hz. Adjust the amplitudes and phase until a circle is displayed.
Now it is possible to see how sensitive these figures are to the precise frequencies involved. Set one generator to 101 Hz while leaving the other at 100. Note how rapidly the figure changes, compared to the stability of perfect relative pitch. Try 100.1 and 100 Hz. Even small differences matter. Now set the generators to be an octave apart (say 100 and 200). Adjust the amplitudes and phase until the figure has maximum symmetry.
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Below are examples of Lissajous figures from the harmonic series. Setting one sine wave generator to an integer multiple of the other generator produces very predictable figures. Try recreating these figures, and while doing so, notice the visual pattern that allows you to recognize at once a Lissajous curve of the overtone series.
1/1 Unison
2/1 2
nd
3/1
Partial
4/1
5/1
4 Partial
5 Partial
th
rd
3 Partial
10/1
th
th
10 Partial
Now, we will examine a different set of Lissajous figures, those whose intervallic ratios are between 1/1 and 2/1. As previously hinted at, 3/2, or the perfect fifth, is the simplest non-integer multiple harmonic ratio in existence. Using the sine wave generators, set one generator to 200 Hz and the other to 300 Hz. After adjusting the amplitudes and phase, notice how simple this figure is. Compare the 3/2 with other whole number ratios and take note of the varying complexities of their shapes. Some examples can be seen below.
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1/1
9/8
Fundamental
Major 2
5/4
6/5 nd
4/3 rd
3/2
Major 3
Perfect 4
8/5
5/3 th
Minor 6
rd
Minor 3
th
Perfect 5
th
2/1
Major 6
th
Octave
(This is not meant to be an exhaustive list of possible Lissajous figures and ratios, try some on your own.) Up until this point, all of the Lissajous figures we have been creating have been intervals from the Just Scale. However, the most common scale in modern music is the Equally
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Tempered Scale. Remember that the biggest weakness of Equally Tempered Scale is that it sacrifices the Just Scale’s pure intervals for the ability to modulate between keys. Let’s look at a soso-called “perfect fifth” interval in this tuning system. Set the sine wave generators generators to 220 Hz (pitch “A”) “A”) and 329.65 (pitch “E”). Also, set the persistence to one second. On a standard piano, these pitches are a fifth apart. Notice on the oscilloscope how much more unstable this figure is compared to a perfect 3/2 interval (the upper frequency would be changed to 330 Hz). A difference of only 0.35 Hz turns a perfect harmonically stable figure into an unstable, rotating figure that only offers brief th glimpses of the true perfect 5 Lissajous curve. Now we are going to examine the Equally Tempered major third (represented in the Just Scale by the ratio ratio 5/4). Set the generators to 440 Hz (pitch (pitch “A”) and 554.35 (pitch “C#”). Immediately it is clear that this figure is even less stable than the Equally Tempered fifth we looked at before. This major third differs from the Just major third (550 Hz) by 5.65 Hz. It is visually obvious that the Just and Equally Tempered major thirds are not even recognizable as similar intervals.
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An Introduction to Surface Resonances: Chladni Shapes
The sounds made by bowed string instruments are rich in overtones. In the course of his investigations of these sounds, Ernst Chladni (1756-1827) recognized that some of the many resonances evidenced by the violin could be explained as surface resonances. To show surface resonances Chladni scattered fine white sand over a surface while the edge of the surface was bowed. Under various circumstances a variety of patterns were observed and recorded. Under resonant conditions vibrations reflecting from the surface boundaries combine on the surface in a fashion that produces locations where nearly total destructive interference takes place. These locations can be points or lines, often referred to as nodes or nodal lines. The pattern forms because the grains of sand stop moving when they land on a nodal line or point. These days it is very easy to conduct observations of this kind. Vibrations from a modified audio loudspeaker are coupled to each shape at a point near the middle of the shape. Amplified sine waves are delivered to the speaker through an audio amplifier. The frequency of the sine waves is swept from zero to several thousand cycles per second. As the applied frequency is swept various patterns of nodal points and lines are briefly visible. The three videos found in "oscillator movies" show fine white sand on circle, square, and violin shapes. Note that the applied frequency is indicated on the meter.
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A Percussion Instrument Modeled by Simple Pipes
Musical instruments fall into four major classes: Chordophones- an instrument whose sound is created through the use of vibrating strings coupled to a resonant cavity (guitar, violin, banjo) Aerophones- instruments where notes are produced by pushing vibrating air through the body of the instrument (clarinet, trumpet, organ) Membranophones- a percussive instrument made by stretching a membrane/skin/drum head over the opening of a resonant cavity (drums, timpani, bongos) Idiophones- percussion instruments that are struck with their resultant musical sound coming from the vibrations of the object (xylophone, handbells, cymbals)
This lab deals with a drum like instrument made from the ordinary PVC pipe that is often used in plumbing. This instrument is quite similar in design and playing technique to percussive keyboard instruments such as the xylophone and marimba. However, the manner in which air moves through these pipes is quite similar to how organ pipes create tones. This seems to suggest that this PVC pipe instrument is a cross between an idiophone and an aerophone. Eight pipes were each tuned to individual notes making a D minor pentatonic scale (notes from low to high are: D, A, C, D, F, G, A, C). The instrument is shown below:
(The design is based on a drum shown at http://dennishavlena.com/PVC-hang.htm)
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The same software and set-up used in the previous open pipe lab can be used here as well. The response spectrum for the pipe tuned to G is shown below. It has the fundamental as well as many overtones.
The table below shows the lengths and fundamental resonant frequencies. The following plot shows the expected linear relationship between the reciprocal of length and resonant frequency.
Note
Length in mm
D A C D F G A C
2482 1609 1353 1208 1013 898 805 637
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Fundamental Resonant Frequency 73.264 Hz 110.499 Hz 130.799 Hz 146.18 Hz 173.392 Hz 196.126 Hz 215.985 Hz 260 Hz
Below is a typical wave packet created by striking the upper open end of a pipe with a solid, yet slightly pliable object. Both rubber spatulas and flip-flops will produce good sound results.
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The following are waveforms and Fourier transforms of the pitches made by three different pipes.
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Procedure
Connect the microphone and preamp to the oscilloscope. Compare the waveforms you obtain to those shown. Oscilloscope note: Set the scope to single sweep with a trigger level about half scale.
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An Introduction to Organ Notes
The organ has played a significant role in the history of music. Organ pipes fall into one of several categories. Some pipes make use of a stream of air to produce a sound while others rely on a vibrating reed. Some pipes are open at both ends while others are stopped (open only on one end). The voltage waveform for a pipe open at both ends and the associated FFT are shown below.
The rich overtones are responsible for the flat tops on the waveform. As discussed in the "Standing Waves in an Open Pipe" lab, we expect an integer number of half wavelengths in the pipe, and thus we expect all multiples of the fundamental in the overtone structure. Note that all overtone frequencies are reported as multiples of the fundamental.
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FFT's and waveforms for several other pipes are shown below. Melodia Pipe The Melodia’s characteristic sound comes from the manner in which the air forced through the pipe travels from the initial chamber it is pumped into (bottem of picture), through a row of small slits, and into the large resonance chamber of the pipe.
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Diapason Pipe Diapason style pipes have a design very similar to that of a flute and are typically very bright sounding.
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Trumpet Pipe The trumpet pipe has a vibrating metal "reed" which stimulates its voice. The complex waveform is responsible for the dramatic overtone structure.
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Gedeckt Pipe A Gedeckt pipe is very similar in construction to a Melodia pipe. The chief difference is that this Gedeckt pipe is stopped. A wooden plug is placed in the top of the pipe, closing the opening completely. Thus, this pipe acts as a pipe with only one open end and and only displays odd overtones.
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Procedure
Connect the microphone and preamp to the scope and compare the waveform you obtain to those above. Note the role that microphone location has on the quality of the waveform.
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Physics of the Violin
The first string instruments were almost surely a result of the sounds made by the bow used in hunting. Even early instruments had resonance chambers which produced a much louder sound and gave the instrument its voice. The strings on a musical instrument, especially the violin are very small and produce very little sound sound by themselves. Consider the example of the “thunder “thunder jug.”
This device has a spring like “stimulator” which is coupled to material similar to a drum head which is stretched over the top of a hollow pipe. The other end is cut at an angle to allow the cavity to resonate at many frequencies. When the spring is gently stroked a very loud “booming” sound is produced. Note that when the spring spring is de-coupled de-coupled from the cavity, little sound is produced. This is the fundamental principle behind instruments like the violin. The action of the bow stimulates the string to vibrate, and the violin body is mechanically coupled to the string.
Physics Interlude: Characteristic emission by atoms
The chemical element helium was discovered from the spectrum of the solar corona during a total eclipse of the sun. Large amounts of light at only very specific wavelengths were emitted from the atmosphere of the sun. We have since learned from quantum theory that each atomic system has a unique set of energy levels. When exposed to the broad spectrum of light from the sun, helium atoms can absorb energy at only certain wavelengths. Quantum theory explains this as a result of the atoms having a limited number of states with different energies. Only incident energy of just the right wavelength (and energy) can be absorbed. Once the energy is absorbed it can be emitted by the reverse process.
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The sounds made by the violin are analogous to the emissions by helium atoms. An incident spectrum of many frequencies results in an output at very specific wavelengths only. While this is a situation perfectly described by classical physics, the mathematical parallels result in very analogous behavior.
Physics of the Bowed String
The violin string has a small mass and is under considerable tension. If the string is pulled aside and released, the acceleration of the string near the point of distortion will be very large. The violin bow is almost always made from the tail hair of horses. The hair from living Siberian stallions is especially valued for this purpose. Just before use, fresh rosin is applied to the bow, making the bow somewhat sticky. This results in a high static coefficient of friction. As the bow is moved across a string its motion is characterized by two actions: sticking and slipping. The slipping action is most often initiated by vibrations already present on the string. Thus vibrations naturally present on the string encourage the bow to slip at one or more of these natural frequencies. The coupling of the natural motion of the strings with the motion of the bow results in periodic stimulation. stimulation. The combination of slipping and sticking results in a “saw tooth” motion of the string under the bow. A saw tooth wave form is shown below. The horizontal axis is time and the vertical axis is displacement from the equilibrium position.
The gently sloped portion of the motion is created while the string is “stuck” to the bow. When it slips it moves very rapidly back toward the equilibrium point. When the restoring force weakens as the string reaches the equilibrium point, the string again sticks to the bow. The motion toward the equilibrium position is so rapid that the slope of the waveform is nearly vertical during the slipping phase.
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Fourier’s Theorem
Fourier was able to prove that any periodic function can be reproduced from the sum of infinitely many other periodic functions. In most cases a function can be wellapproximated by the first few terms. Fourier’s Theorem can be written as: f ( x)
where
a0 2
N
[a n cos(nx) bn sin(nx)]
an
n 1
1
f ( x) cos(nx)dx
and bn
1
f ( x) sin(nx)dx
The series when N is the Fourier series. For finite N the series is approximately equal to the function
Consider the plot of f(x)=x . This shows that f(x) is an odd function in that f(-x)=-f(x). f(-x)=-f(x ). The product of f(x) and cosine is itself an odd function. Since the integral of an odd function over a symmetrical region is always zero, all of the a n are equal to zero. The b coefficients can be found as: bn
1
x sin(nx)dx 2
(1)
n
n 1
,
n 1
As more and more terms are included the Fourier sum begins becomes a better and better approximation of the original function. The original plots of the series with 1, 3, 5, and 30 terms are shown.
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The Fourier Transform
Consider a function that is the sum of sine waves, such at the plots shown above. A process called the Fourier transform allows a plot of power vs frequency for the function. For our bowed string this process changes an amplitude vs time into a function of power vs frequency. The transforms of the functions with 1, 3, 5, and 30 terms are shown below in both semi-logarithmic and linear scaling. It is no surprise that each term in the series can be associated with a specific peak in the power curve.
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(Note that all 30 terms on the final linear graph are not large enough in amplitude to be seen all at once. The higher the overtone, the smaller the dB rating.)
Physics Interlude: Voltage induced by motion in a magnetic field
Imagine a rod moving in a uniform magnetic field as shown below. Any charged particle moving perpendicular to a magnetic field experiences a force. This force is called the Lorentz force after a physicist of that name. For velocity and field perpendicular the force is given by: F qvB where q is the charge on the particle, v is the velocity, and B is the intensity of
the magnetic field.
The Lorentz force will cause charges to move along the length of the rod until an electric field is created that produces an equal and opposite force on the charges within the rod. Thus the effect of moving a rod through a magnetic field is to induce an electric field
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proportional to the velocity of the rod. The effect of an electric field acting over distance is a potential difference, an electrical voltage. In summation, a conductor moving perpendicular to a magnetic field will have an induced voltage which is proportional to the velocity of that conductor.
The measurement of string velocity
Since the violin string is metallic, a small but powerful magnet (a so called rare-earth magnet because the magnetic material is neodymium) placed below the string will produce a voltage along the length of the string that is proportional to the velocity of the string.
Note that the connections to the string are made in a fashion that does not disturb the motion of the string.
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As the string is bowed a voltage is produced that is proportional to the velocity. A typical voltage waveform is shown below. Note that this graph represents the derivative of the saw-tooth waveform. This is natural since the velocity is the derivative of the position.
Using the violin bow, run it across the strings in both directions. Observe up bow and down bow waveforms. Why are they reversed? Note also that the derivative of a sine function produces a cosine function; the velocity waveform will have the exact same frequency dependence as the position waveform. The Fourier transform of the string waveform is shown below. This is the power spectrum.
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Note that many overtones are present in the power spectrum. The violin bow is the stimulator of the resonant object that is the violin body (with strings attached). Note that the action of the strings flopping in the air produces very little sound. The bulk of the sound is produced by the violin body as a consequence of the coupling of the strings to the violin body through the bridge and sound post.
The following are graphs of overtones of different violin strings as well as the waveforms of these same strings.
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Physics Interlude: Observing the waveforms
The environment characteristic of a physics lab is one in which there is a great deal of changing magnetic fields, mostly from AC wiring and transformers. These extraneous fields can easily induce voltages that are as large as those we hope to study. Fortunately special amplifiers have been developed that reject this noise. The special amplifier used in this exercise could also be used to study the tiny voltages produced by the heart
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(EKG). This rejection is called common mode rejection because the signals are present on both conductors. Probes with wire shielding are also critical. Below is a picture of the amplifier used the above experiment.
Procedure
Connect the differential amplifier as shown in the photos above. Also connect the microphone to channel 2 of the scope. Compare your waveforms and Fourier transforms to those shown below.
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The Theremin: An Introduction to Synthetic Music
When President John Kennedy made the commitment to go to the moon a tremendous evolution began in the electronics industry. Integrated circuits with hundreds of thousands of parts on a single chip made all kind of complex circuits practical. But the history of electronic instruments goes back much farther. In the late 1920's radio had emerged as a profitable enterprise. Efforts to improve radio operation resulted in the development of a technique called heterodyning. The first AM radio using the technique was produced by RCA in 1928.
Many attribute the first major electronic instrument to Leon Theremin (1896-1993), a Russian and Soviet inventor. Besides the musical instrument that bears his name, he invented the first motion detector and the interlace technique still used today to improve video quality. While working to improve an electronic device developed for another purpose, Theremin noticed that the pitch of an interference tone between two radio frequency oscillators changed as his hand moved. Theremin developed a volume control circuit and then he had his instrument. By 1928 he had secured a patent and was giving public concerts. Radio frequency oscillators have a natural frequency that strongly depends of a small value of capacitance. The human hand has sufficient capacitance to affect oscillator frequency. Thus the pitch of the instrument is controlled by the proximity of the musician's hand to an antenna. Various effects such as vibrato can be produced by a variety of hand motions. Two radio frequency oscillators are adjusted so that they differ in frequency by a few hundreds of cycles. One oscillator has an antenna connected to its capacitor, so the human body can add to this capacitance by its proximity.
Shown below is the output of our Theremin and its FFT.
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Note the very large number of overtones. Since uncontrolled radio frequency oscillators naturally oscillate at the fundamental as well as many overtones, the superposition of the output from two oscillators will have many overtones as well.
It is interesting to compare the Theremin output to the output from a simple timing chip, the LM555. This chip can be connected in an astable fashion that turns the chip on and off at a rate controlled by two resistors and a capacitor. Below is a diagram of a circuit using the 555 chip. Note that the 100k resistor is variable.
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The output of this device will be a square wave with a longer "on" time than "off". Typical output and the FFT are shown below.
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The 555 chip’s output can easily drive a speaker. It is interesting interesting to note that when the output of the above diagram is connected to a speaker as shown below, the sound produced by the speaker has a differently shaped waveform and FFT.
Observe the effects that changing the value of C and the setting of the 100k resistor have on the sound.
Interlude: Theremin and "The thing"
Included among Theremin's many accomplishments was the construction of a cold war "bugging" device that allowed interception of secret conversations inside the American Embassy.
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The "Thing" (Source: http://www.forensicgenealogy.info/) The carved replica of the great seal of the United States shown above is a Trojan horse of sorts. It contains a listening device designed by Theremin. It was presented to the US ambassador by Soviet school children in 1946. The heart of the device was a hollow brass cavity which was coupled to the room through a hollow tube that ran to the eagle's open mouth. The device had no power source and no iron (it was fashioned from brass) so it was not easily detected even though the embassy was scanned for such devices often. The brass cavity was resonant with microwave radiation beamed at the embassy from a truck parked down the block. Sounds near the seal caused the brass chamber to vibrate. These vibrations slightly changed the resonant frequency of the cavity. As the cavity oscillated it was more or less resonant with the external microwaves and hence absorbed and re-radiated more or less of the incident microwave beam. When Air Force Pilot Gary Powers was shot down during a spying mission over the Soviet Union the Soviets introduced a motion in the UN condemning the US got spying. When the US produced this device as evidence of Soviet spying the motion was quickly defeated. The device is currently on display at the Spy Museum in Washington DC.
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