Diaschisma

Interval information

Ratio	2048/2025
Factorization	211 × 3-4 × 5-2
Monzo	[11 -4 -2⟩
Size in cents	19.55257¢
Name	diaschisma
Color name	sgg2, sagugu 2nd, Sagugu comma
FJS name	[math]\displaystyle{ \text{d2}_{5,5} }[/math]
Special properties	reduced, reduced subharmonic
Tenney height (log2 nd)	21.9837
Weil height (log2 max(n, d))	22
Wilson height (sopfr(nd))	44
Harmonic entropy (Shannon, [math]\displaystyle{ \sqrt{nd} }[/math])	~3.69964 bits
Comma size	small
S-expression	S162 × S17
Open this interval in xen-calc

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Diaschisma

2048/2025, the diaschisma, a comma of 19.553 cents, is the size of a pythagorean comma minus two schismas, from which it derives its name. It may also be defined as the difference between four just perfect fifths plus two just major thirds and three octaves, 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, (16/15)²/(9/8), 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. See Diaschismic family for the rank-2 temperament family where it is tempered out, especially Srutal archagall which takes advantage of this comma's relation to 256/255 and 289/288 to make it as efficient and natural as possible. See Diaschismic rank three family for the rank-3 temperament family where it is tempered out.

Significance

Pařízek's diaschisma pump play xenpaper – a comma pump progression that requires the diaschisma to be tempered out (i.e. equates two notes that are separated by a diaschisma).

Pařízek's diaschisma comma pump example in JI (notated with HEJI) and 12edo.

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.

Etymology

The modern sense of the term is due to Hermann von Helmholtz and Alexander John Ellis in 1875 when the English translation of Sensations of Tone was first published.

2048/2025 was earlier referred to as the “diminished comma” and “comma minor” by Jean-Philippe Rameau (1683-1764). However in modern (1875 onwards) music theory the term “diaschisma” is almost always used.

There have been other intervals besides 2048/2025 that were called “diaschisma” in the Ancient Greek, Roman and medieval periods, however those alternate meanings of the word fell out of use centuries ago.

See also

• Small comma