This is a guide of advice and articles having to do with how to understand temperaments and fret placement for fretted string instruments -- in particular the lute. This page, its opinions and recommendations, and any mistakes or omissions are the sole responsibility of me. for clarifications or additions, please contact me via the Contact page of the Renaissance Cittern Site.
The following is a list of books and articles dealing with tuning and temperament with the specific concern of plucked string instruments such as the lute. for lute players wishing to discover the joys of meantone temperament with a minimum of theory, Damiani's chapter or Dolata's article, "Lute Tuning with Meantone Temperaments" are highly recommended and both contain tables of factors for theoretical fret placement. (But see note about differences in factor numbers below in "Tuning Advice.")
The following list is sorted chronologically, with older articles appearing first:
Another highly interesting and recommended book that 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 that addresses some of the criticisms Isacoff received for his book as well as his explanation of his views of temperaments.
It should be pointed out that exact mathematical values for factors for fret placement do differ slightly from article to article, perhaps in part due to the number of decimal places available on calculators used for calculations. Despite this discrepancy, the factors are within an acceptable range when one considers that precise adjustment of the 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 perception 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 tuning. A tuning menthod such as the following described by Stewart McCoy, and posted by Leonard Williams to the Lutenet (April, 2000), is recommended:
I think you should be able to manage to tune the lute well without a sophisticated 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 you want. The exact spacing 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 string doesn't effectively vibrate right up against the nut and bridge. So for a string-length of 60 cm, the vibrating length will possibly be somewhere in the region of 59.8 cm.
Although my tuning box can give me all sorts of fancy temperaments, I use it most of the time just to 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 note. But that 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 other instruments. Tuning to a' is literally the last thing you want to do with a lute in g'. So I tune the lower string of the 6th course first (using the box), followed by its octave. more
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 pitch. This discrepancy will affect all stoppable octave courses to some extent, but the fatter the 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 the 1st course to the 6th course, if need be using a harmonic at the 5th fret.
3) Next I tune the 2nd course (as always to the lower string of the 6th 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 open 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 the strings are FALSE and need replacing.
I play the 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 6th course. If need be, I use the harmonic at the 12th fret of the 6th 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 the 12th fret of the 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 spot 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 the lower string of the 6th 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 to 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 temperament you go for. the difficulties arise when you TRY to judge tuning with intervals like major thirds, because the ear allows for greater tolerance with thirds than it does with octaves and unisons.
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 | |||||||||
|---|---|---|---|---|---|---|---|---|---|
| fret | note | Pythagorean Tuning | 1/4 Comma | 1/5 Comma | 1/6 Comma | 1/8 Comma | 1/11 Comma (Equal Temperament) |
"396" | "Gerle" |
| 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 |
| n | G | 1200 | 1200 | 1200 | 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. |
| Major second | 204 c. | 200 c. | -4 c. |
| Minor third | 316 c. | 300 c. | -16 c. |
| Major 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. |
| Minor sixth | 814 c. | 800 c. | -14 . |
| Major sixth | 884 c. | 900 c. | +16 c. |
| Minor seventh | 996 c. | 1000 c. | +4 c. |
| Major seventh | 1088 c. | 1100 c. | +12 c. |
| Octave | 1200 c. | 1200 c. | 0 c. |
| * taken from "An Introduction to Tuning and Temperaments, Part II" by David Dolata | |||
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 piece of grid or graph paper, 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 temperaments prepared for a single lute: all one needs to do is use the templates to quickly switch fret position.
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 string at the nut to the location where the string passes under itself near the bridge. One may also wish, depending 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 perfect octave.
Last updated Sep. 10, 2005