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# More Complicated Temperaments

Before equal temperament was generally adopted, other schemes were used which allowed music to be played in all twelve keys with acceptable results, even if each key had its own unique flavor.

One example of this is the Valotti temperament.

In this temperament, the notes from F through B on the circle of fifths are tuned in relation to each other according to a 1/6-comma meantone temperament. However, for reasons to be explained shortly, the comma used in these schemes needs to be understood as a Pythagorean comma, rather than a syntonic comma.

Thus, from F to C, the ratio of frequencies is 2 to (3 times the sixth root of 524288/531441), and this applies from C to G, from G to D, from D to A, from A to E, and from E to B.

Going from B back to F, however, the remaining fifths are tuned in relation to each other according to the Pythagorean temperament. So, from B to F#, from F# to C#, and so on, the ratio of frequencies is exactly 2:3.

Because six of the fifths were tuned according to the 1/6-comma meantone temperament, going around the circle of fifths one has 12 Pythagorean fifths, plus exactly one factor (the sixth root taken to the sixth power) of the Pythagorean comma. Therefore, while it is not the case that all the fifths are exactly the same, the circle of fifths does close on itself, allowing reasonably acceptable results in any key.

In terms of the Pythagorean comma, equal temperament, therefore, is a 1/12-comma tuning. A syntonic comma is approximately 0.9167212309 Pythagorean commas, and is very close to 11/12 of a Pythagorean comma.

Another temperament of this kind is the Kellner temperament. In this temperament, the series of fifths from C to G to D to A to E, as well as the fifth from B to F#, are depressed by 1/5 of a Pythagorean comma, with all remaining fifths tuned as perfect fifths.

Recent research by Dr. Bradley Lehman suggests that Bach used, for his Well-Tempered Clavier, not equal temperament as formerly thought, but a temperament of this general kind which is claimed to give very pleasing results, each key having a distinct flavor, but none sounding badly out of tune.

In this temperament, the following fifths are depressed by 1/6 comma:

F to C to G to D to A to E.

The following fifths are tuned as perfect fifths:

E to B to F# to C#.

The following fifths are depressed by 1/12 comma, that is, tuned in the same relationship as used for equal temperament:

C# to G# to D# to A#.

Finally, the fifth from A#, or B flat, to F, is (surprisingly, at least to me) sharpened, rather than depressed, by 1/12 comma from the Pythagorean perfect fifth.

Dr. Lehman also notes a temperament devised by George Andreas Sorge in 1758 which closely resembled this temperament of Bach, but which avoids the sharpening of one of the fifths. In this temperament:

The fifth from F to C is a Pythagorean fifth.

These fifths are depressed by 1/6 comma:

C to G to D to A.

The fifth from A to E is depressed by 1/12 comma.

The fifth from E to B is a Pythagorean fifth.

These fifths are depressed by 1/12 comma:

B to F# to C#.

The fifth from C# to G# is a Pythagorean fifth.

These fifths are depressed by 1/12 comma:

G# to D# to A# to F.

What is the goal of these types of temperaments?

The reason that the quarter-comma meantone tuning was popular for pipe organs was that an accurate major third was considered more important than an accurate perfect fifth.

Since the Pythagorean comma is slightly larger than a syntonic comma, one might consider depressing the fifths by 1/5 of a Pythagorean comma. But if the minor thirds between F and A, C and E, and G and B are all to sound the same, one has seven notes, and hence six intervals of a fifth, to deal with. The most obvious compromise is the Valotti tuning, where the fifths from F through B are all depressed by 1/6 of a Pythagorean comma, the other half of the cycle being Pythagorean, so that the major thirds belonging to the more remote parts of the scale are allowed to be sharp.

However, that might result in the distant portions of the scale falling into an unacceptably high level of error. Thus, if one makes a very limited sacrifice of allowing some additional sharpness of the minor thirds in the main part of the scale, one has the opportunity to control things better further out.

A simple scheme of this kind might run like this:

These fifths are Pythagorean:

B flat to F to C.

These fifths are depressed by 1/6 comma:

C to G to D to A to E.

These fifths are Pythagorean:

E to F# to C#.

These fifths are depressed by 1/12 comma:

C# to G# to D# to A#

Some temperaments by Neidhart, as well as an earlier one by Sorge, cited by Lehman, are somewhat similar to this, but distribute the interals in a different way.

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