Schisma: Difference between revisions
→Schismic temperaments derivable from its S-expressions: add reduced generator mappings for the other novel temperaments documented on this page |
m →2.3.5.7.19[53&147] (garibaldi nestoria): i dont remember why i wrote it as 53 & 147 when both x31eq and sintel's temp finder give it as 12&41 and when this is also the intuitive simple description that suggests 53 as a tuning |
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[[CTE]] generator: 702.059{{cent}} | [[CTE]] generator: 702.059{{cent}} | ||
==== 2.3.5.7.19[ | ==== 2.3.5.7.19[12&41] (garibaldi nestoria) ==== | ||
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. | 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. | ||
Revision as of 06:46, 3 February 2024
| Interval information |
reduced harmonic
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.
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
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
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]]
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.