Temperaments and Tunings: A Guide for Lute Players This is a guide guide of advice advice and articles articles having to do with wit h how to understand t emperaments emperaments and fret p laceme lacement nt for fretted string instruments -- in particular the lute. This page, its opinions and recommendations, and any mist mistakes akes or omissions are the sole resp onsibility onsibility of me. for clarifi clarifications cations or additions, p lease lease contact me via the Contact page of page of t he Renaissance Renaissance Cittern Site. 1. 2. 3. 4. 5.
Books & Articles Tuning Advice Tab able le s of Cen C entt Value Value s Formula For mula for Convertin Converting g Ce nt Value Value s i nto Fret Factors Factors Links to Online Sources About Temperaments
1. Books & Articles The following following is is a list list of books and articles articles dealing with tuning and temperam temp erament ent with w ith t he sp ecific ecific concern concern of p lucked lucked string st ring instruments such s uch as the lute. for lute p layers wishing to discover the joy s of meantone meantone temperament with a minimum of theory, Damiani's chapter or Dolata's article, "Lute Tuning with Meantone Temp Temp eraments" eraments" are highly highly recommende recommended d and bot h contain cont ain tables of factors for theoretica t heoreticall fret p laceme lacement. nt. (But see note note about about differences in factor numbers below in "Tuning Advice.") The following list is sorted chronologically, with older articles appearing first: Dombois, Eugen. Eugen. "Corret and Easy Fret Placement." Placement." Journal Journal of the the Lute Societ Societyy of America America VI VI (1973), p p .30-33. Dombois, Eugen. Eugen. "Vari "Varieties eties of M eantone Temperament Temperament Realized Realized on the Lute." Journal Lute." Journal of the the Lute Society of America VII America VII (1974), (1974), pp p p .82-89. Lindley, Mark. Lutes, Mark. Lutes, Viol Violss and Temperaments. (Cambridge Temperaments. (Cambridge Univ. Press: 1984). Crum, Alison and Sonia Jackson. Jackson. "Ap p endix 1: Tuning (by Elizabeth Elizabeth Liddle)." Liddle)." Play Play the Viol. iol. (Oxford Univ. Press: 1989), pp.155-164. Dolata, David. "Lute Tuning with Harmonics: A Square Peg in a Round Hole." LSA Hole." LSA Quarterly (Feb. Quarterly (Feb. 1993), pp.7-9. Dolata, David. "Lute Tuning with Meantone Temperaments." LSA Temperaments." LSA Quarterly (Feb. Quarterly (Feb. 1993), 1993), pp p p .12-15. Robiczek, Bonnie. "A Method for Tuning the Lute." Lute Lute." Lute Societ Societyy of America America Newslet Newsletter ter , XVI, No. 1 (1981), reprinted in LSA in LSA Quarterly (May/Aug. Quarterly (May/Aug. 1993), pp.28-29. Dolata, David. "An Int roduction to Tuning and Temp Temp eraments." eraments." LSA LSA Quarterly, Quarterly, (M ay/August ay/August 1993), p p .30-31. [Note: cent values for Perfect 4th and Perfect 5th for 1/4 Comma Meantone in Table I are
reversed. Should read: Perfect 4th: 503.5 and Perfect 5th: 696.5. -A:H] Dolata, David. "The Secret of Tuning by Harmonics." GFA Soundboard (Spring 1993), pp.27-37, Dolata, David. "An Introduction to Tuning and Temperaments, Part II." LSA Quarterly" (Feb. 1994), pp.20-23. Damiani, Andrea. "Chapter 19: Temperament." Method for Renaissance Lute. (Ut Orpheus Edizioni, Bologna: 1999), pp.190-192. English Translation by Doc Rossi. [Book originally published in Italian.]
Another highly interesting and recommended book t hat looks at the debate between pure tuning versus meantone and equal temperaments is Temperament: How music became a battleground for the great minds of Western Civilization , by Stuart Isacoff. I would recommend getting the later edition of this book which contains an afterword t hat addresses some of the criticisms Isacoff received for his book as w ell as his explanation of his views of temperaments. top
2. Tuning Advice It should be p ointed out that exact mathematical values for factors for fret p lacement do differ slightly from article to article, perhaps in part due t o the number of decimal places available on calculators used for calculations. Desp ite this discrepancy, the factors are within an acceptable range when one considers that precise adjustment of t he frets is near impossible due to the nature of strings and their elasticity, variance in the height of action from instrument to instrument, and differences in percept ion of hearing from person to person. The mathematical and theoretical placement of frets is really just a guide to follow, after which one may decide to alter this fret or that in order to please one's own aesthetic sense. It should also be noted that for those who are interested in trying temperaments or tunings other than equal for the first time, it is recommended that one rethink one's method of t uning. A t uning menthod such as t he following described by Stewart M cCoy, and p osted by Leonard Williams to the Lutenet (Ap ril, 2000), is recommended: I think y ou should be able to manage to t une the lute well without a sop histicated tuning box. After all, the "old" guys didn't have one. The important thing is to be sure that you have the frets in the correct position for the temperament y ou want. The exact sp acing can be measured easily enough, of course. The important thing to bear in mind with measuring is to ensure that your calculations involve the vibrating length of the string, not the full length from nut to bridge, because the st ring doesn't effectively vibrate right up against the nut and bridge. So for a stringlength of 60 cm, the vibrating length will possibly be somewhere in the region of 59.8 cm. Although my t uning box can give me all sorts of fancy t emperaments, I use it most of the time just t o get me started. The rest I do by ear. If I have problems, and I just can't get the instrument in tune, then I will use the box for every not e. But t hat really is for emergency use only, when
panic is starting to set in. 1) I start by tuning the 6th course to G. The habit many musicians have of tuning to a' is a hangover from tuning violins and ot her instruments. Tuning to a' is literally the last thing you want to do with a lute in g'. So I tune the lower st ring of the 6t h course first (using the box), followed by its octave. Sometimes it helps to have the octave of the 6th course tuned very slightly on the flat side, because the differences of thickness between the strings causes discrepancies higher up the neck. When you press down say the 6th course at the 7th fret, the thinner string has further to go before it reaches the fingerboard, so it will be pulled down further and thus stretched further, increasing its p itch. This discrepancy will affect all stoppable octave courses to some extent, but the fatter t he string the more noticeable will be the discrepancy. Anyway, once tuned, that course stays put, and I tune everything else to the lower octave of it, come what may. 2) Next I tune t he 1st course to t he 6th course, if need be using a harmonic at the 5t h fret. 3) Next I tune t he 2nd course (as alway s to the lower string of the 6t h course) using a harmonic at the 7th fret. I aim to get the 2nd course the tiniest bit flat to that harmonic. I don't count beats like piano tuners do. That's unnecessarily sophisticated. The main thing is for the 2nd course not to be sharp to the harmonic on the 6th course. If I feel unsure, I check f2 against a1 (i.e. 5th fret 2nd course against op en 1st course), and a2 against h1. If it is impossible to get those two checks to work, the frets must be incorrectly placed, probably because of using an incorrect vibrating length, or t he strings are FALSE and need replacing. I p lay t he following chord, which must sound good before I can proceed: ___a___ ___a___ _______ _______ _______ ___a___
4) Now that 3 courses are well in tune, I tune the 4th course. I do this by tuning it at the 2nd fret, again to match the 6t h course. If need be, I use the harmonic at the 12th fret of the 6t h course. Again I sound a chord, which must sound well in tune: ___a___ ___a___ _______ ___c___ _______ ___a___
If it is not in tune, it must be the 4th course which needs fiddling with, because I know that the
other three were OK before. Next I check the 4th course against the 2nd, if need be using a harmonic at t he 12th fret of t he 4th course, but I can usually cope tuning notes an octave apart. _______ ___d___ _______ ___a___ _______ _______
5) Four down, two to go. The fifth course comes next. Actually I usually begin by tuning this course in 5ths to the 1st course, going for a slightly narrow fifth. That may do the trick, but even if I get it sp ot on (which is not often), I still have numerous checks. The main one is c5 against a2. I may also stick to my original plan of always checking against t he lower string of the 6t h course, so I check a5 against f6. I don't bother with a6 against h5. 6) Last comes the 3rd course, which is always the hardest to get right. That's why it's suicide to start tuning the lute t o a'. I begin by tuning a3 to c6. This should be an octave. Then I check a5 against d3. Then I test a few chords: ___a____a____c___ ___a____c____d___ ___c____d____d___ _____________a___ ________a________ ___a_____________
Last comes my ultimate check. If the following two chords sound well in tune, I know I've made it: ___c____d____ ___e____d____ ___f____f____ ___e____f____ ___c____f____ ________d____
The commonest difficulty I have at the very end is that the 3rd course is sometimes a bit on the sharp side. That's why I use those last two chords. If need be, I flatten the 3rd course very slightly, even though that may not give me a perfect octave with c6. It's a compromise which often needs making. One last point. If frets are correctly placed, and you always use octaves and unisons, it doesn't matter what t emperament y ou go for. the difficulties arise when y ou T RY to judge tuning with intervals like major thirds, because the ear allows for greater tolerance with t hirds than it does with octaves and unisons.
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3. Tables of Cent Value s Note: All values assume a standard " G" tuning with a wolf 5th between G# and Eb (an enharmonically spelled 5th). Most common fret placements for meantone temperaments are dentoed by an asterisk (*).
TABLE 1: Comparison of Cent Values for 1/4 Comma, 1/6 Comma, and Pythagorean Tuning Pure
1/4 Comma
1/6 Comma
Pythagorean
Major 3rd
386 c.
386 c.
393.3 c.
408 c.
Perfect 4th
498 c.
503.5 c.
501.6 c.
498 c.
Perfect 5th
702 c.
696.5 c.
698.3 c.
702 c.
TABLE 2: Cent Values for Various Temperaments and Tunings note
a [open]
G
0
0
0
0
0
0
0
0
b1 * b2
G# Ab
114 90
75.5 117.5
83.2 112
88.3 108.3
94.75 103.75
100
93 105
88.8 108
c
A
204
193
195.2
196.6
198.5
200
198
196.8
d1 * d2
A# Bb
318 214
268.5 310.5
278.4 307.2
285 305
293.25 302.25
300
291 303
285.6 304.8
* e1 e2
B Cb
408
386 428
390.4 419.2
393.3 413.3
397 406
400
396 408
393.6 412.8
f
C
498
503.5
502.4
501.6
500.75
500
501
501.6
g1 * g2
C# Db
612 588
579 621
585.6 614.4
590 610
595.5 604.5
600
594 606
590.4 609.6
h
D
702
696.5
697.6
698.3
699.25
700
699
698.4
i1 * i2
D# Eb
816 792
772 814
780.8 809.6
786.6 806.6
794 803
800
792 804
787.2 806.4
k
E
906
889.5
892.8
895
897.75
900
897
895.2
l1 * l2
E# F
1020 996
965 1007
976 1004.8
983.3 1003.3
992.5 1001.5
1000
990 1002
984 1003.2
* m1 m2
F# Gb
1110 1086
1082.5 1124.5
1088 1116.8
1091.6
1111.6
1100
1095 1107
1092 1111.2
fret
1/4 1/5 1/6 1/8 Comma Comma Comma Comma
1/11 Comma (Equal "396" "Gerle" Temperament)
Pythagorean Tuning
n
G
1200
1200
1200
1200
1200
TABLE 3: Comaprison of the sizes of Pure and Equal Intervals Interval
Pure
Equal
Difference
Chromatic semitone
90 c.
100 c.
+10 c.
Diatonic semitone
112 c.
100c.
-12 c.
M ajor second
204 c.
200 c.
-4 c.
M inor third
316 c.
300 c.
-16 c.
M ajor third
386 c.
400 c.
+24 c.
Fourth
498 c.
500 c.
+2 c.
Tritone
590 c.
600 c.
+10 c.
Fifth
702 c.
700 c.
-2 c.
M inor sixth
814 c.
800 c.
-14 .
M ajor sixth
884 c.
900 c.
+16 c.
M inor seventh
996 c.
1000 c.
+4 c.
M ajor seventh
1088 c.
1100 c.
+12 c.
Octave
1200 c.
1200 c.
0 c.
*
1200
1200
1200
*
taken from "An Introduction to Tuning and Temperaments, Part II" by David Dolata
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4. Formula for Converting Cent Values into Fret Factors In the equation below, y=factor and x=cent value.
y = 1 - 0.5 (x/1200) This formula can be used to convert cents into a fret factor for any kind of string instrument. Once you have determined the fret factors for all of the fret locations (based on cents) that you wish, simply multiply the vibrating string length by the factor in order to obtain the location of the fret as measured from the nut. It may be helpful to inscribe all of the fret locations onto a p iece of grid or graph p aper, then use this paper as a template which can be conveniently stored in one's lute case. In this way it is possible to have a number of different t emperaments prepared for a single lute: all one needs to do is use the t emplates to quickly switch fret p osition. It should be noted, however, that the actual vibrating string length may differ slightly from the lute's mensur: the vibrating length should be measured between the free end of the st ring at the nut t o the location where the string passes under itself near the bridge. One may also wish, dep ending upon the action of one's instrument, to make the vibrating string length slightly smaller than what is measured in order to account for "bending" of
the string when depressed. An "easy " way to calculate the "actual" vibrating string length to use for calculations is simply to measure the distance between the bridge-end of the nut and the center of the 12th fret (which is fixed on most instruments) -- assuming, of course, that this fret was originally placed in the correct position for a p erfect octave. top
5. Links to Online Sources About Temperaments (Just a start!) David van Ooijen's do-It-Yourself Article A History of Temperaments and Tunings The Just Intonation Network Tuning Forum/Digest
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Last updated Sep. 10, 2005
