Quarter-comma Meantone - Construction

Construction

In a meantone tuning, we have diatonic and chromatic semitones, with the diatonic semitone larger. In Pythagorean tuning, these correspond to the Pythagorean limma and the Pythagorean apotome, only now the apotome is larger. In any meantone or Pythagorean tuning, a whole tone is composed of one semitone of each kind, a major third is two whole tones and therefore consists of two semitones of each kind, a perfect fifth of meantone contains four diatonic and three chomatic semitones, and an octave seven diatonic and five chromatic semitones, it follows that:

• Five fifths down and three octaves up make up a diatonic semitone, so that the Pythagorean limma is tempered to a diatonic semitone.
• Two fifths up and an octave down make up a whole tone consisting of one diatonic and one chromatic semitone.
• Four fifths up and two octaves down make up a major third, consisting of two diatonic and two chromatic semitones, or in other words two whole tones.

Thus, in Pythagorean tuning, where sequences of just fifths (frequency ratio 3:2) and octaves are used to produce the other intervals, a whole tone is

and a major third is

An interval of a seventeenth, consisting of sixteen diatonic and twelve chromatic semitones, such as the interval from D4 to F6♯, can be equivalently obtained using either

• a stack of four fifths (e.g. D4—A4—E5—B5—F6♯), or
• a stack of two octaves and one major third (e.g. D4—D5—D6—F6♯).

This large interval of a seventeenth contains (5 + (5 − 1) + (5 − 1) + (5 − 1) = 20 − 3 = 17 staff positions). In Pythagorean tuning, the size of a seventeenth is defined using a stack of four justly tuned fifths (frequency ratio 3:2):

In quarter-comma meantone temperament, where a just major third (5:4) is required, a slightly narrower seventeenth is obtained by stacking two octaves (4:1) and a major third:

By definition, however, a seventeenth of the same size (5:1) must be obtained, even in quarter-comma meantone, by stacking four fifths. Since justly tuned fifths, such as those used in Pythagorean tuning, produce a slightly wider seventeenth, in quarter-comma meantone the fifths must be slightly flattened to meet this requirement. Letting x be the frequency ratio of the flattened fifth, it is desired that four fifths have a ratio of 5:1,

which implies that a fifth is

a whole tone, built by moving two fifths up and one octave down, is

and a diatonic semitone, built by moving three octaves up and five fifths down, is

Notice that, in quarter-comma meantone, the seventeenth is 81/80 times narrower than in Pythagorean tuning. This difference in size, equal to about 21.506 cents, is called the syntonic comma. This implies that the fifth is a quarter of a syntonic comma narrower than the justly tuned Pythagorean fifth. Namely, this system tunes the fifths in the ratio of

which is slightly smaller (or flatter) than the ratio of a justly tuned fifth:

The difference between these two sizes is a quarter of a syntonic comma:

In sum, this system tunes the major thirds to the just ratio of 5:4 (so, for instance, if A is tuned to 440 Hz, C♯' is tuned to 550 Hz), most of the whole tones (namely the major seconds) in the ratio, and most of the semitones (namely the diatonic semitones or minor seconds) in the ratio . This is achieved by tuning the seventeenth a syntonic comma flatter than the Pythagorean seventeenth, which implies tuning the fifth a quarter of a syntonic comma flatter than the just ratio of 3:2. It is this that gives the system its name of quarter-comma meantone.