Schisma

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Interval information
Ratio 32805/32768
Factorization 2-15 × 38 × 5
Monzo [-15 8 1
Size in cents 1.9537208¢
Name schisma
Color name Ly-2, Layo comma
FJS name [math]\text{d}{-2}^{5}[/math]
Special properties reduced
Tenney height (log2 nd) 30.0016
Weil height (log2 max(n, d)) 30.0033
Wilson height (sopfr (nd)) 59
Harmonic entropy
(Shannon, [math]\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 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).

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.

Nestoria

See also: No-sevens subgroup temperaments #Nestoria

Nestoria tempers out 361/360 and 513/512, and can be described as the 12 & 53 temperament in the 2.3.5.19 subgroup. This is derived since the schisma is expressible as S19/(S16/S18)2 and (S15/S20)/(S16/S18).

Garibaldi

Garibaldi tempers out 225/224 and 5120/5103, and can be described as the 41 & 53 temperament in the 7-limit. This is derived since the schisma is also equal to S15/(S8/S9).

2.3.5.7.19 subgroup

Adding nestoria to garibaldi (tempering S20) results in an extremely elegant temperament which has all of the same patent tunings that garibaldi has but which includes a mapping for 19 through nestoria.

2.3.5.7.17 12 & 118 & 171 (unnamed)

As the schisma also equals S15/S16 * S18/S20, we can derive the extremely accurate 12 & 118 & 171 temperament:

Subgroup: 2.3.5.7.17

Comma list: 1701/1700, 32805/32768

Mapping[1 0 15 0 -32], 0 1 -8 0 21], 0 0 0 1 1]]

mapping generators: ~2, ~3, ~7

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.7197, ~7/4 = 968.8307

Optimal ET sequence12, 29, 41, 53, 106d, 118, 171, 472, 525, 643, 814, 985, 1799, 2324, 2495, 3138b, 3309bd, 4294bdg

2.3.5.7.17.19 12 & 118 & 171 (unnamed)

By tempering S16/S18 we equate S15 with S20 (tempering the other comma of Nestoria) because of S15~S16~S18~S20, leading to:

Subgroup: 2.3.5.7.17.19

Comma list: 361/360, 513/512, 1701/1700

Mapping[1 0 15 0 -32 9], 0 1 -8 0 21 -3], 0 0 0 1 1 0]]

mapping generators: ~2, ~3, ~7

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.7053, ~7/4 = 968.9281

Optimal ET sequence12, 29, 41, 53, 106d, 118, 171, 289h, 460hh

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