Diaschisma
| Interval information |
reduced subharmonic
2048/2025, the diaschisma, an interval of 19.553 cents, is the difference between four perfect fifths plus two major thirds and three octaves. Tempering it out leads to the diaschismic family of temperaments. It may also be defined as the difference between a Pythagorean minor seventh (16/9) and a just augmented sixth (225/128), as the difference between two classic diatonic semitones (16/15) and the major whole tone (9/8), that is, (9/8)/(16/15)2, or as the difference between the 5-limit tritone 45/32 and its enharmonic equivalent 64/45.
Example
parizek1 A comma pump progression that assumes that the diaschisma is tempered out (i.e. equates two notes that are separated by a diaschisma).
In the progression, the bassline moves as follows: D-(up 5/4)-F#-(down 4/3)-C#-(down 4/3)-G#-(up 5/4)-C-(up 4/3)-G-(up 3/2)-D (*). If we ignore octaves, the first three steps (cumulatively D to G#) moves us up by the tritone 45/32, and the last three steps (cumulatively G# to D) are the same moves as the first three, thus it moves us up by the tritone 45/32 a second time. In pure JI, since 45/32 is flat of 600c, each cycle of this progression (*) would shift the tonic down by the diaschisma. The fact that the D we come back to is exactly the same as the first D, indicates that the basic 5-limit intervals, 5/4 and 3/2, are adjusted, or tempered, such that a stack of two 45/32 tritones is sharpened up to the octave 2/1. In temperament contexts, we see this as equivalent to saying that their difference, which is (2/1) / (45/32)^2 = 2048/2025 is tempered out.
This also implies that there is an interval that is equal to exactly half of an octave‚ namely the tempered 45/32 tritone. Thus all edos (such as 12edo, 22edo, 34edo and 46edo) and MOS scale structures (such as the MOS scales of diaschismic and pajara) split the octave into two equal parts; in particular, all diaschismic edos are even-numbered edos.