Schisma: Difference between revisions
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 |
Cleanup; normalize mappings; review tunings and optimal ET sequences |
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{{Wikipedia| Schisma }} | {{Wikipedia| Schisma }} | ||
The '''schisma''', '''32805/32768''', is the difference between the [[Pythagorean comma]] and the [[syntonic comma]]. It is equal to ([[9/8]])<sup>4</sup>/([[8/5]]) and to ([[135/128]])/([[256/243]]) and also to ([[9/8]])<sup>3</sup>/([[64/45]]) | The '''schisma''', '''32805/32768''', is the difference between the [[Pythagorean comma]] and the [[syntonic comma]]. It is equal to ([[9/8]])<sup>4</sup>/([[8/5]]) and to ([[135/128]])/([[256/243]]) and also to ([[9/8]])<sup>3</sup>/([[64/45]]). | ||
== | == Temperaments == | ||
Tempering out this comma gives a [[5-limit]] microtemperament called [[schismatic family #Schismatic aka helmholtz|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 }} | |||
[[ | As the schisma is expressible as [[361/360|S19]]/([[1216/1215|S16/S18]])<sup>2</sup> and ([[513/512|S15/S20]])/([[1216/1215|S16/S18]]), we can derive the 12 & 53 temperament: | ||
=== Garibaldi === | |||
{{Main| Garibaldi }} | |||
As the schisma is also equal to [[225/224|S15]]/([[5120/5103|S8/S9]]), we can derive the 41&53 temperament: | As the schisma is also equal to [[225/224|S15]]/([[5120/5103|S8/S9]]), we can derive the 41&53 temperament: | ||
==== 2.3.5.7.19 subgroup ==== | |||
{{Main| Garibaldi }} | |||
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. | ||
=== 2.3.5.7.17 12 & 118 & 171 (unnamed) === | |||
As the schisma also equals [[57375/57344|S15/S16]] * [[1701/1700|S18/S20]], we can derive the extremely accurate 12&41 temperament: | |||
[[Subgroup]]: 2.3.5.7.17 | |||
[[Comma list]]: 1701/1700, 32805/32768 | |||
{{mapping|legend=1| 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|legend=1| 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 [[1216/1215|S16/S18]] we equate [[225/224|S15]] with [[400/399|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|legend=1| 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|legend=1| 12, 29, 41, 53, 106d, 118, 171, 289h, 460hh }} | |||
== Trivia == | == Trivia == | ||
Revision as of 13:22, 17 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).
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
As the schisma is expressible as S19/(S16/S18)2 and (S15/S20)/(S16/S18), we can derive the 12 & 53 temperament:
Garibaldi
As the schisma is also equal to S15/(S8/S9), we can derive the 41&53 temperament:
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&41 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
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.