Schisma
| Interval information |
reduced harmonic
(Shannon, [math]\sqrt{nd}[/math])
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
Nestoria tempers out 361/360 (S19) and 513/512 (S15/S20), 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 (S15) and 5120/5103 (S8/S9), 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 400/399 (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 sequence: 12, 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 sequence: 12, 29, 41, 53, 106d, 118, 171, 289h, 460hh
2.3.5.41 53 & 65 (unnamed)
The schisma can additionally split into two superparticular commas in the 41-limit: 32805/32768 = (1025/1024)*(6561/6560). Tempering both of these out provides a natural mapping for prime 41, if a little less practical than those for 19 or 7.
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.