Summary of Equal-Temperament, Meantone, and Well-Temperament Systems
The Basic Problem
If you tune twelve perfect* fifths upward, starting on C, you get a B-sharp:
C → G → D → A → E → B → F# → C# → G# → D# → A# → E# → B#
The problem is that the B# thus obtained is not a C seven octaves above the root C but 23.46 cents** sharp of that C. This discrepancy is called the Pythagorean comma.
An additional problem is that when the E thus tuned is dropped two octaves, the major third, C—E is wide from perfect by 21.5 cents (syntonic comma). Similar statements hold for G—B, D—F#, etc.
Note that tuning up a fifth is equivalent to tuning down a fourth and then up an octave. So, in actual practice, the above tuning sequence would utilize both fourths and fifths in order to group the tuned notes within one octave called the temperament octave. Therefore, in the following tuning schemes, which involve tuning a series of fifths upward or downward, assume that the resultant note is within an octave of the root tone.
Whenever an interval varies from perfect, beats are heard. When beats are more than a few per second, they become quite unpleasant to hear.
Next, tuning twelve perfect fifths downward, starting on C produces the following notes:
etc. ← G-flat ← D-flat ← A-flat ← E-flat ← B-flat ← F ← C
The problem now is that the G-flat thus obtained is 23.46 cents** flat from the F# from before. A similar statement can be made for all corresponding sharps and flats. That the sharps and corresponding flats differ by about 1/4 of a semitone means that one key lever cannot be used for both notes without problems.
The following systems deal with these problems in different ways and to varying degrees. Note that each method of tuning given here is for purposes of clarity, and tuning methods may vary in actual practice.
The goal is to close the Pythagorean comma by narrowing each of the twelve fifths by 1/12 of a Pythagorean comma, or 23.46 ÷ 12 = 1.96 cents. Since this narrowing is equally distributed in the four fifths required to tune E, the interval C—E is now wide by 21.5 – 4/12 of 23.46 = 13.68 cents. Because all fifths are tempered by the same amount (such a temperament is termed a regular temperament), all of the thirds are wide by 13.68 cents. As a result of this discrepancy and the fact that raising an interval by an octave doubles its beat rate, thirds in the higher registers beat at an unpleasantly high rate. In equal temperament, all keys sound the same, and there is no difference between corresponding sharps and flats (closed temperament). No intervals other than octaves are perfect.
The goal is to have perfect major thirds. As mentioned above, tuning four perfect fifths upward from C produces a major third C—E that is wide by 21.5 cents, so in order to produce a perfect major third C—E, these four fifths are each narrowed by one fourth of 21.5 cents = 5.375 cents:
C → G → D → A → E
Then B is tuned up a perfect major third from G, and F is tuned down a perfect major third from A. That takes care of the naturals. Next, sharps are tuned upward a perfect third, and flats are tuned downward a perfect third. For example, G# is tuned upward a perfect major third from E, and A-flat is tuned downward a perfect third from C. It can be shown that tuning a G# up am perfect third is equivalent to continuing to tune up from E by using fifths each narrowed by 5.375 cents.
E → B → F# → C# → G#
Similarly, tuning an A-flat down a perfect third is equivalent to continuing to tune down from C by using fifths all narrowed by 5.375 cents:
A-flat ← E-flat ← B-flat ← F ← C
The discrepancy between an associated sharp and flat then is 21.5(3) – 23.46 = 41 cents. Because this temperament is regular, all associated sharps and flats differ by 41 cents (sharps are 41 cents flatter than associated flats). Meantone is, therefore, not a closed temperament, and only certain keys are usable.
The notorious meantone wolf occurs in fifths such as C#—G#, where the upper accidental is tuned to A-flat instead of a G#. The resulting 41 – 5 = 36-cent discrepancy has been likened to the out-of-tune howling of wolves.
The goal is to have one perfect major third and to be able to play in all keys. In this particular version of well temperament, the third C—E will be perfect. We start out as in meantone, tuning up four fifths each narrowed by one fourth of 21.5 cents = 5.375 cents to produce a perfect major third C—E. That uses up 21.5 cents of a required amount of tempering of 23.46 cents. From there on, perfect fifths are tuned downward from C:
C-flat ← G-flat ← D-flat ← A-flat ← E-flat ← B-flat ← F ← C
Since there is only 23.46 – 21.5 = 1.96 cents left of tempering, C-flat will differ from B by 1.96 cents, and the interval E—C-flat will be a “fifth,” narrowed by 1.96 cents. This narrowing, called a schisma (the difference between the Pythagorean comma and the syntonic comma), is coincidentally very close to the narrowing of an equally tempered fifth. The result is that the discrepancy between C-flat and B is only 1.96 cents, as will also be the difference between any associated sharp and flat.
Well temperament is closed but irregular. So you can play in all keys, and each key sounds different. As you move away from the key of C (G, D, A, E, etc.), the beat rates of the major thirds increase. Close to the key of C, the beat rates are slow, but far from the key of C, the beat rates are much greater than in equal temperament. No intervals other than octaves, six fifths, and one major third are perfect.
*A perfect interval is one whose frequencies are in the ratio of small integers (e.g., a perfect fifth would involve a 3:2 frequency ratio). Such an interval does not beat.
**A cent is 1/100 of an equally tempered semitone.
©2011 by Robert Chuckrow