Schisma

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Interval information
Ratio 32805/32768
Factorization 2-15 × 38 × 5
Monzo [-15 8 1
Size in cents 1.953721¢
Name schisma
Color name Ly-2, Layo comma
FJS name [math]\displaystyle{ \text{d}{-2}^{5} }[/math]
Special properties reduced,
reduced harmonic
Tenney height (log2 nd) 30.0016
Weil height (log2 max(n, d)) 30.0033
Wilson height (sopfr(nd)) 59
Harmonic entropy
(Shannon, [math]\displaystyle{ \sqrt{nd} }[/math])
~1.23468 bits
Comma size unnoticeable
Open this interval in xen-calc
English Wikipedia has an article on:

The schisma, 32805/32768, is a small interval about 2 cents. It arises as the difference between the Pythagorean comma and the syntonic comma. It is equal to (9/8)4/(8/5) and to (135/128)/(256/243) and also to (9/8)3/(64/45).

History and etymology

Schisma is a borrowing of Ancient Greek, meaning "split". The term was first used by Boethius (6th century), in his De institutione musica, using it to refer to half of the Pythagorean comma. The modern sense was introduced by Helmholtz' On the Sensations of Tone, in particular the translation by Alexander Ellis, where it is spelled skhisma. Since it is extremely close to the superparticular ratio 887/886 (2-1 443-1 887), it is used interchangably with this interval.

Temperaments

Tempering out this comma gives a 5-limit microtemperament called schismatic, schismic or helmholtz, which if extended to larger subgroups leads to the schismatic family of temperaments.

Other intervals

Commas arising from the difference between a stack of Pythagorean intervals and other primes may also be called schismas. The difference between the Pythagorean comma and septimal comma is called the septimal schisma. Other examples are undevicesimal schisma and Alpharabian schisma.

Trivia

The schisma explains how the greatly composite numbers 1048576 (220) and 104976 (184) look alike in decimal. The largest common power of two between these numbers is 25, (when 1049760 is written to equalize) and when reduced by that, 1049760/1048576 becomes 32805/32768.

See also