Schisma

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

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Schisma

Schismic temperaments derivable from its S-expressions

Nestoria

As the schisma is expressible as S19/(S16/S18)² and (S15/S20)/(S16/S18), we can derive the 12&53 temperament:

Subgroup: 2.3.5.19

Patent EDO tunings: 12, 17, 24, 29, 36, 41, 53, 65, 77, 82, 89, 94, 101, 106, 118, 130, 135, 142, 147, 154, 159, 171, 183, 195, 207, 219, 248, 260, 272

CTE generator: 701.684 ¢

Garibaldi

As the schisma is also equal to S15/(S8/S9), we can derive the 41&53 temperament:

Subgroup: 2.3.5.7

Patent EDO tunings: 12, 29, 41, 53, 82, 94, 106, 135, 147

CTE generator: 702.059 ¢

2.3.5.7.19[12&41] (garibaldi nestoria)

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.

Subgroup: 2.3.5.7.19

Patent EDO tunings: 12, 29, 41, 53, 82, 94, 106, 135, 147

Mapping: [⟨1 1 7 11 6], ⟨0 1 -8 -14 -3]]

CTE generator: 702.043 ¢

2.3.5.7.17[12&130&171] (unnamed)

As the schisma also equals S15/S16 * S18/S20, we can derive the extremely accurate 12&41 temperament:

Subgroup: 2.3.5.7.17

Patent EDO tunings < 300 (largest is 2548): 12, 29, 41, 53, 118, 130, 142, 159, 171, 183, 212, 224, 236, 289

Mapping: [⟨1 1 7 2 -9], ⟨0 1 -8 0 21], ⟨0 0 0 1 1]]

CTE generators: (2/1,) 3/2 = 701.72 ¢, 7/4 = 968.831 ¢

2.3.5.7.17.19[12&130&171] (unnamed Nestoria)

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

Patent EDO tunings: 12, 29, 41, 53, 118, 130, 142, 159, 171, 183

Mapping: [⟨1 1 7 2 -9 6], ⟨0 1 -8 0 21 -3], ⟨0 0 0 1 1 0]]

CTE generators: (2/1,) 3/2 = 701.705 ¢, 7/4 = 968.928 ¢

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.