Commas and Temperament Units | Why is temperament necessary?
http://www-personal.umich.edu/~bpl/temper.html
( or
http://how.to/tune )
© Bradley Lehman,
9 April 2000
bpl@umich.edu
Latest update: 30-Jun-2005, to v2.2d. If you downloaded before that date (or if you're not sure), please re-download now. See the version review or the download instructions.
New 15-Feb-2005: Added the temperament published in my February 2005 Early Music article, "Bach's extraordinary temperament: our Rosetta Stone" (part 1). See also http://www.larips.com for details.
New 9-April-2000: Pie charts showing the relative amounts that fifths and major thirds are out of tune.
New 10-March-1999: A page of examples is now available. It gives a complete set of comparative charts for three representative temperaments: 1/4 Comma Meantone, Kirnberger 3 Well Temperament, and Equal Temperament.
(If you need a definition of commas and basic temperament strategy, before diving into the more technical explanations below, see this section.)
For example, in a well temperament such as Werckmeister III (illustrated below), one can see how modulation around the circle of fifths increases/decreases tension in the triads and sevenths. And in 1/4 comma meantone, the Bb-D-F-G# chord shows up as an almost pure dominant seventh of Eb, because of the pure Bb-D major third and the startling Bb-G# which almost coincides with a 7/4 minor 7th. (Try this chord on a 1/4 comma meantone organ!)
Tuning recipe by cycle of fifths (across) and major thirds (down):
1/4 Syntonic Comma Meantone
Comma | E | B | F# | C# | |||||
SC | (0) | (0) | (0) | (0) | |||||
SC | C | (- 1/4 ) | G | (- 1/4 ) | D | (- 1/4 ) | A | (- 1/4 ) | E |
SC | (...) | (0) | (0) | (0) | |||||
G# | Eb | Bb | F | ||||||
(0) | (...) | (...) | (...) | ||||||
E | (- 1/4 ) | B | (- 1/4 ) | F# | (- 1/4 ) | C# | (- 1/4 ) | G# | |
(0) | (0) | (0) | (0) | ||||||
C | G | D | A | ||||||
(...) | (0) | (0) | (0) | ||||||
G# | (...) | Eb | (- 1/4 ) | Bb | (- 1/4 ) | F | (- 1/4 ) | C | |
(0) | (...) | (...) | (...) | ||||||
E | B | F# | C# |
Beat Rate Chart (beats per second) | |||||||||||
1/4 Syntonic Comma Meantone, set from A=440 | |||||||||||
Master | Freq | Cents | Sm Min3 | Min3 | Maj3 | Per4 | Per5 | Min6 | Maj6 | Min7 | Maj7 |
C | 131.591 | 310.26 | 23.38 | -2.45 | 0.00 | 1.64 | -1.22 | -24.67 | 2.05 | -7.33 | -6.12 |
C# | 137.500 | 386.31 | 24.43 | -2.56 | 16.50 | 1.71 | -1.28 | 0.00 | 18.69 | -7.66 | 42.95 |
D | 147.123 | 503.42 | 26.14 | -2.74 | 0.00 | 1.83 | -1.37 | 0.00 | 2.29 | -8.20 | -6.84 |
Eb | 157.419 | 620.53 | 1.49 | -25.00 | 0.00 | -12.85 | -1.46 | -29.52 | 2.45 | -41.77 | -7.32 |
E | 164.488 | 696.58 | 29.23 | -3.06 | 0.00 | 2.05 | -1.53 | 0.00 | 2.56 | -9.17 | 51.38 |
F | 176.000 | 813.69 | 1.66 | -27.95 | 0.00 | 2.19 | -1.64 | -33.00 | 2.74 | -9.81 | -8.19 |
F# | 183.904 | 889.74 | 32.67 | -3.42 | 22.07 | 2.29 | -1.71 | 0.00 | 25.00 | -10.25 | 57.45 |
G | 196.774 | 1006.84 | 34.96 | -3.66 | 0.00 | 2.45 | -1.83 | 0.00 | 3.06 | -10.97 | -9.15 |
G# | 205.610 | 1082.89 | 36.53 | -3.83 | 24.67 | 2.56 | 12.85 | 0.00 | 27.95 | -11.46 | 64.23 |
A | 220.000 | 0.00 | 39.09 | -4.09 | 0.00 | 2.74 | -2.05 | 0.00 | 3.42 | -12.26 | -10.23 |
Bb | 235.397 | 117.11 | 2.22 | -37.38 | 0.00 | 2.93 | -2.19 | -44.14 | 3.66 | -62.46 | -10.95 |
B | 245.967 | 193.16 | 43.70 | -4.58 | 29.52 | 3.06 | -2.29 | 0.00 | 3.83 | -13.71 | 76.83 |
Middle C | 263.181 | (10.26) | 46.76 | -4.90 | 0.00 | 3.27 | -2.45 | -49.35 | 4.09 | -14.67 | -12.24 |
C# | 275.000 | -(13.69) | 48.86 | -5.12 | 33.00 | 3.42 | -2.56 | 0.00 | 37.38 | -15.33 | 85.90 |
D | 294.246 | (3.42) | 52.28 | -5.47 | 0.00 | 3.66 | -2.74 | 0.00 | 4.58 | -16.40 | -13.69 |
Eb | 314.838 | (20.53) | 2.97 | -49.99 | 0.00 | -25.69 | -2.93 | -59.03 | 4.90 | -83.55 | -14.64 |
E | 328.977 | -(3.42) | 58.45 | -6.12 | 0.00 | 4.09 | -3.06 | 0.00 | 5.12 | -18.33 | 102.76 |
F | 352.000 | (13.69) | 3.33 | -55.90 | 0.00 | 4.38 | -3.27 | -66.00 | 5.47 | -19.62 | -16.37 |
F# | 367.807 | -(10.26) | 65.35 | -6.84 | 44.14 | 4.58 | -3.42 | 0.00 | 49.99 | -20.50 | 114.89 |
G | 393.548 | (6.84) | 69.92 | -7.32 | 0.00 | 4.90 | -3.66 | 0.00 | 6.12 | -21.93 | -18.30 |
G# | 411.221 | -(17.11) | 73.06 | -7.65 | 49.35 | 5.12 | 25.69 | 0.00 | 55.90 | -22.92 | 128.45 |
A | 440.000 | (0.00) | 78.18 | -8.19 | 0.00 | 5.47 | -4.09 | 0.00 | 6.84 | -24.52 | -20.47 |
Bb | 470.793 | (17.11) | 4.45 | -74.76 | 0.00 | 5.86 | -4.38 | -88.27 | 7.32 | -124.93 | -21.90 |
B | 491.935 | -(6.84) | 87.40 | -9.15 | 59.03 | 6.12 | -4.58 | 0.00 | 7.65 | -27.41 | 153.67 |
C | 526.363 | (10.26) | 93.52 | -9.79 | 0.00 | 6.55 | -4.90 | -98.69 | 8.19 | -29.33 | -24.48 |
In each column the number indicates beats in the given interval above the master note; 0 is pure, positive number is wide, negative number is narrow | |||||||||||
(* difference in cents from equal temperament) |
Werckmeister III pie charts
Relative amounts that the major thirds are out of tune (from the named note to a major third above it):
Relative amounts that the fifths are out of tune (from the named note to a fifth above it):
Subjective definition of wolves:
Subjective Tolerability | ||||||||||||||||||
Werckmeister 3: 1/4 Ditonic Comma | ||||||||||||||||||
Master | best Min3 | Maj3 | Per4 | Per5 | best Min6 | best Maj6 | best Min7 | Maj7 | | | Maj Triad | Min Triad | Dim Triad | Mm7 | MM7 | mm7 | dim7 | As Neap | |
C | -9 | 2 | 0 | -5 | -9 | 2 | 0 | 5 | | | 2 | 9 | 9 | 1 | 4 | 4 | 5 | 8 | C |
Db | -6 | 9 | 0 | 0 | -6 | 9 | 5 | 26 | | | 9 | 6 | 5 | 7 | 19 | 6 | 7 | 6 | Db |
D | -4 | 4 | 5 | -5 | -2 | 6 | -5 | 12 | | | 3 | 3 | 6 | 4 | 8 | 4 | 6 | 2 | D |
Eb | -9 | 6 | 0 | 0 | -6 | 9 | 0 | 12 | | | 6 | 9 | 8 | 3 | 10 | 4 | 8 | 7 | Eb |
E | -4 | 6 | 0 | 0 | -2 | 6 | 5 | 19 | | | 6 | 2 | 5 | 6 | 14 | 4 | 6 | 3 | E |
F | -9 | 2 | 0 | 0 | -9 | 4 | 0 | 5 | | | 1 | 9 | 8 | 0 | 3 | 4 | 5 | 8 | F |
Gb | -6 | 9 | 5 | 0 | -2 | 9 | 5 | 26 | | | 9 | 6 | 4 | 7 | 19 | 6 | 7 | 4 | Gb |
G | -6 | 4 | 5 | -5 | -4 | 4 | 5 | 5 | | | 3 | 6 | 8 | 4 | 4 | 6 | 5 | 5 | G |
Ab | -6 | 9 | 0 | 0 | -6 | 9 | 0 | 19 | | | 9 | 6 | 6 | 3 | 15 | 3 | 8 | 6 | Ab |
A | -2 | 6 | 5 | 0 | 1 | 6 | -5 | 19 | | | 6 | 1 | 5 | 6 | 14 | 3 | 6 | 1 | A |
Bb | -9 | 4 | 0 | 0 | -9 | 6 | 0 | 5 | | | 2 | 9 | 8 | 1 | 4 | 4 | 7 | 8 | Bb |
B | -6 | 6 | 0 | -5 | -4 | 6 | 0 | 19 | | | 6 | 6 | 5 | 3 | 14 | 3 | 6 | 5 | B |
Arranged as circle of fifths: | ||||||||||||||||||
C | -9 | 2 | 0 | -5 | -9 | 2 | 0 | 5 | | | 2 | 9 | 9 | 1 | 4 | 4 | 5 | 8 | C |
G | -6 | 4 | 5 | -5 | -4 | 4 | 5 | 5 | | | 3 | 6 | 8 | 4 | 4 | 6 | 5 | 5 | G |
D | -4 | 4 | 5 | -5 | -2 | 6 | -5 | 12 | | | 3 | 3 | 6 | 4 | 8 | 4 | 6 | 2 | D |
A | -2 | 6 | 5 | 0 | 1 | 6 | -5 | 19 | | | 6 | 1 | 5 | 6 | 14 | 3 | 6 | 1 | A |
E | -4 | 6 | 0 | 0 | -2 | 6 | 5 | 19 | | | 6 | 2 | 5 | 6 | 14 | 4 | 6 | 3 | E |
B | -6 | 6 | 0 | -5 | -4 | 6 | 0 | 19 | | | 6 | 6 | 5 | 3 | 14 | 3 | 6 | 5 | B |
Gb | -6 | 9 | 5 | 0 | -2 | 9 | 5 | 26 | | | 9 | 6 | 4 | 7 | 19 | 6 | 7 | 4 | Gb |
Db | -6 | 9 | 0 | 0 | -6 | 9 | 5 | 26 | | | 9 | 6 | 5 | 7 | 19 | 6 | 7 | 6 | Db |
Ab | -6 | 9 | 0 | 0 | -6 | 9 | 0 | 19 | | | 9 | 6 | 6 | 3 | 15 | 3 | 8 | 6 | Ab |
Eb | -9 | 6 | 0 | 0 | -6 | 9 | 0 | 12 | | | 6 | 9 | 8 | 3 | 10 | 4 | 8 | 7 | Eb |
Bb | -9 | 4 | 0 | 0 | -9 | 6 | 0 | 5 | | | 2 | 9 | 8 | 1 | 4 | 4 | 7 | 8 | Bb |
F | -9 | 2 | 0 | 0 | -9 | 4 | 0 | 5 | | | 1 | 9 | 8 | 0 | 3 | 4 | 5 | 8 | F |
Subjective rules for overall tension of chords:
Harmony Chart: tension of common chords when playing in the given key | |||||||||||||||||
1/4 Syntonic Comma Meantone | |||||||||||||||||
Key | I | i | Neap | ii | iio | III | iii | IV | iv | V | V7 | v | VI | vi | VII | viio | viio7 |
C | 1 | 2 | 3 | 2 | 2 | 1 | 2 | 1 | 1 | 1 | 4 | 2 | 31 | 2 | 1 | 2 | 2 |
G | 1 | 2 | 6 | 2 | 2 | 1 | 2 | 1 | 2 | 1 | 4 | 2 | 1 | 2 | 1 | 2 | 1 |
D | 1 | 2 | 1 | 2 | 2 | 1 | 2 | 1 | 2 | 1 | 4 | 2 | 1 | 2 | 1 | 2 | 1 |
A | 1 | 2 | 1 | 2 | 2 | 1 | 2 | 1 | 2 | 1 | 4 | 2 | 1 | 2 | 1 | 2 | 1 |
E | 1 | 2 | 1 | 2 | 1 | 1 | 31 | 1 | 2 | 17 | 10 | 2 | 1 | 2 | 1 | 2 | 2 |
B | 17 | 2 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 17 | 10 | 2 | 1 | 31 | 1 | 2 | 2 |
F# | 17 | 2 | 1 | 31 | 1 | 1 | 1 | 17 | 2 | 17 | 10 | 2 | 1 | 1 | 1 | 2 | 2 |
C# | 17 | 2 | 1 | 1 | 2 | 1 | 1 | 17 | 2 | 31 | 16 | 31 | 1 | 1 | 17 | 2 | 2 |
G# | 31 | 31 | 1 | 1 | 2 | 17 | 2 | 17 | 2 | 1 | 1 | 1 | 1 | 1 | 17 | 2 | 2 |
Eb | 1 | 1 | 1 | 1 | 2 | 17 | 2 | 31 | 31 | 1 | 1 | 1 | 17 | 2 | 17 | 2 | 2 |
Bb | 1 | 1 | 5 | 2 | 2 | 17 | 2 | 1 | 1 | 1 | 4 | 1 | 17 | 2 | 31 | 2 | 2 |
F | 1 | 1 | 3 | 2 | 2 | 31 | 2 | 1 | 1 | 1 | 4 | 2 | 17 | 2 | 1 | 2 | 2 |
Harmony Chart: tension of common chords when playing in the given key | |||||||||||||||||
Werckmeister 3: 1/4 Ditonic Comma | |||||||||||||||||
Key | I | i | Neap | ii | iio | III | iii | IV | iv | V | V7 | v | VI | vi | VII | viio | viio7 |
C | 2 | 9 | 8 | 3 | 6 | 6 | 2 | 1 | 9 | 3 | 4 | 6 | 9 | 1 | 2 | 5 | 6 |
G | 3 | 6 | 8 | 1 | 6 | 2 | 6 | 2 | 9 | 3 | 4 | 3 | 6 | 2 | 1 | 4 | 7 |
D | 3 | 3 | 5 | 2 | 6 | 1 | 6 | 3 | 6 | 6 | 6 | 1 | 2 | 6 | 2 | 5 | 7 |
A | 6 | 1 | 2 | 6 | 6 | 2 | 6 | 3 | 3 | 6 | 6 | 2 | 1 | 6 | 3 | 6 | 8 |
E | 6 | 2 | 1 | 6 | 7 | 3 | 6 | 6 | 1 | 6 | 3 | 6 | 2 | 6 | 3 | 8 | 8 |
B | 6 | 6 | 3 | 6 | 7 | 3 | 9 | 6 | 2 | 9 | 7 | 6 | 3 | 6 | 6 | 8 | 7 |
Gb | 9 | 6 | 5 | 6 | 8 | 6 | 9 | 6 | 6 | 9 | 7 | 6 | 3 | 9 | 6 | 8 | 5 |
Db | 9 | 6 | 4 | 9 | 8 | 6 | 9 | 9 | 6 | 9 | 3 | 6 | 6 | 9 | 6 | 9 | 5 |
Ab | 9 | 6 | 6 | 9 | 7 | 6 | 9 | 9 | 6 | 6 | 3 | 9 | 6 | 9 | 9 | 8 | 5 |
Eb | 6 | 9 | 6 | 9 | 5 | 9 | 6 | 9 | 6 | 2 | 1 | 9 | 6 | 9 | 9 | 6 | 6 |
Bb | 2 | 9 | 7 | 9 | 5 | 9 | 3 | 6 | 9 | 1 | 0 | 9 | 9 | 6 | 9 | 5 | 6 |
F | 1 | 9 | 8 | 6 | 5 | 9 | 1 | 2 | 9 | 2 | 1 | 9 | 9 | 3 | 6 | 5 | 6 |
Harmony Chart: tension of common chords when playing in the given key | |||||||||||||||||
Equal Temperament | |||||||||||||||||
Key | I | i | Neap | ii | iio | III | iii | IV | iv | V | V7 | v | VI | vi | VII | viio | viio7 |
C | 3 | 6 | 5 | 6 | 6 | 3 | 6 | 3 | 6 | 3 | 3 | 6 | 3 | 6 | 3 | 6 | 6 |
G | 3 | 6 | 5 | 6 | 6 | 3 | 6 | 3 | 6 | 3 | 3 | 6 | 3 | 6 | 3 | 6 | 6 |
D | 3 | 6 | 5 | 6 | 6 | 3 | 6 | 3 | 6 | 3 | 3 | 6 | 3 | 6 | 3 | 6 | 6 |
A | 3 | 6 | 5 | 6 | 6 | 3 | 6 | 3 | 6 | 3 | 3 | 6 | 3 | 6 | 3 | 6 | 6 |
E | 3 | 6 | 5 | 6 | 6 | 3 | 6 | 3 | 6 | 3 | 3 | 6 | 3 | 6 | 3 | 6 | 6 |
B | 3 | 6 | 5 | 6 | 6 | 3 | 6 | 3 | 6 | 3 | 3 | 6 | 3 | 6 | 3 | 6 | 6 |
F# | 3 | 6 | 5 | 6 | 6 | 3 | 6 | 3 | 6 | 3 | 3 | 6 | 3 | 6 | 3 | 6 | 6 |
C# | 3 | 6 | 5 | 6 | 6 | 3 | 6 | 3 | 6 | 3 | 3 | 6 | 3 | 6 | 3 | 6 | 6 |
G# | 3 | 6 | 5 | 6 | 6 | 3 | 6 | 3 | 6 | 3 | 3 | 6 | 3 | 6 | 3 | 6 | 6 |
Eb | 3 | 6 | 5 | 6 | 6 | 3 | 6 | 3 | 6 | 3 | 3 | 6 | 3 | 6 | 3 | 6 | 6 |
Bb | 3 | 6 | 5 | 6 | 6 | 3 | 6 | 3 | 6 | 3 | 3 | 6 | 3 | 6 | 3 | 6 | 6 |
F | 3 | 6 | 5 | 6 | 6 | 3 | 6 | 3 | 6 | 3 | 3 | 6 | 3 | 6 | 3 | 6 | 6 |
Conditions of use: I developed this as a personal research tool, and offer it as such. Users are welcome to customize it for their own needs, as long as a modified version is not published or distributed elsewhere. If you modify it in a way which you think is a significant improvement for your needs, I would appreciate receiving a copy of your version by e-mail. (For example, would someone like to move the different analysis regions and charts to separate sheets, instead of having everything on one?)
If this is a helpful tool for you in compiling a published work or school project, or if you have another success story because of it (perhaps in tuning for a concert, or getting a better personal understanding of temperament), please send me a copy of your paper or anecdote; I'm interested to see what happens with this project. Thanks! Download the spreadsheet now.
If you are interested in trying this spreadsheet, but unable to download it properly from this website (it doesn't arrive to you at all), contact me at bpl@umich.edu and I can e-mail the .xls file to you as an attachment.
A TU is defined as 1/720th of the interval of a ditonic (Pythagorean) comma. That is, -720 TU must be distributed among the twelve fifths in a "circle of fifths" to remove the excess over seven octaves. (The ditonic comma is (3/2)^12 / 2^7; the logarithm of that amount is then for convenience called 720 Temperament Units.)
With this system, the other important comma (syntonic comma) works out to be almost exactly 660 TU, and the difference between the two commas (the "schisma") is therefore 60. All three of the numbers 720, 660, and 60 are easily divisible by 1, 2, 3, 4, 5, 6, and 12, the divisors used in describing most temperaments. Therefore, most temperaments can be described in TU's using only integer values, without use of a calculator.
The syntonic comma is the interval difference between the "Pythagorean" major third generated by four fifths ((3/2)^4) vs. the pure major third in that same octave (4*(5/4)). The syntonic comma is therefore ((3/2)^4) / (4*(5/4)), or 81/80. Because a pure major third is 660 TU's narrower than a Pythagorean major third, if one wants a pure major third one must distribute -660 TU among the four fifths that generate that major third around a "circle of fifths." This is of course incompatible with the goal of distributing only -720 TU around twelve fifths to make pure octaves. (If you've used up 660 of them already to make one major third pure, there are only 60 left for the other eight fifths, and therefore all the other major thirds can't be pure. In general: the more you take from each fifth (within reason), the better the resultant major thirds will be.)
Reconciliation of conflicting goals (quality of fifths and octaves, vs. quality of major thirds) is the basis of the art and science of temperament. (N.B.: minor thirds and other intervals are also considered in design and choice of temperaments, but most of the main temperament issues are with the treatment of fifths and major thirds.) In all twelve-note keyboard temperaments, considering all possible fifths and all possible major thirds (including the enharmonic shifts), fifths must be an average of 60 TU narrow, and major thirds must be an average of 420 TU wide.
Here are some of the most common basic strategies in developing temperaments. The strategies are not mutually compatible. Remember, you have -720 TU (as described above) to distribute among the twelve fifths:
See also the excellent "Tuning and Temperament Bibliography" (formerly hosted at Mills College, now moved to mirror sites).
Dr. Bradley Lehman, bpl@umich.edu
Visits since 10-Apr-97: