Diaschisma
Ratio | 2048/2025 |
Factorization | 2^{11} × 3^{-4} × 5^{-2} |
Monzo | [11 -4 -2⟩ |
Size in cents | 19.552569¢ |
Name | diaschisma |
Color name | sgg2, sagugu 2nd |
FJS name | [math]\text{d2}_{5,5}[/math] |
Special properties | reduced, reduced subharmonic |
Tenney height (log_{2} n⋅d) | 21.9837 |
Weil height (max(n, d)) | 2048 |
Benedetti height (n⋅d) | 4147200 |
Harmonic entropy (Shannon, [math]\sqrt{n\cdot d}[/math]) |
~3.64661 bits |
Comma size | small |
S-expression | S16^{2} × S17 |
open this interval in xen-calc |
2048/2025, the diaschisma, an interval of 19.553 cents, is the difference between four just perfect fifths plus two just major thirds and three octaves. 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 octave complement 64/45.
Temperaments
Tempering it out leads to the diaschismic family of temperaments.
Significance
parizek1 A comma pump progression that requires the diaschisma to be 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 (down 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;
- the last three steps (cumulatively G# to D) are the same moves as the first three, moving 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, which is (2/1) / (45/32)^{2} = 2048/2025. The fact that the D we come back to is exactly the same as the first D, indicates that that their difference, the diaschisma, is tempered out. To carry out this tempering-out (assuming octaves are kept pure), 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.
This also tells us that if a system tempers out the diaschisma, it has 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) that temper out the diaschisma split the octave into two equal parts; in particular, all diaschismic edos are even-numbered edos.
See also
- Diaschismic family, the rank-2 temperament family where it is tempered out
- Diaschismic rank three family, the rank-3 temperament family where it is tempered out
- Small comma