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| == Temperaments == | | == 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 [[subgroup]]s leads to the [[schismatic family]] of temperaments. | | {{main|schismatic family}} |
| | | Tempering out this comma gives a [[5-limit]] microtemperament called [[schismic|schismatic, schismic or helmholtz]], which if extended to larger [[subgroup]]s leads to the [[schismatic family]] of temperaments. |
| === Nestoria ===
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| {{See also| No-sevens subgroup temperaments #Nestoria }}
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| 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 [[361/360|S19]]/([[1216/1215|S16/S18]])<sup>2</sup> and ([[513/512|S15/S20]])/([[1216/1215|S16/S18]]).
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| === Garibaldi ===
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| {{Main| Garibaldi }}
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| 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 [[225/224|S15]]/([[5120/5103|S8/S9]]).
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| ==== 2.3.5.7.19 subgroup ====
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| {{Main| Garibaldi }}
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| 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.
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| === 2.3.5.7.17 12 & 118 & 171 (unnamed) ===
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| As the schisma also equals [[57375/57344|S15/S16]] * [[1701/1700|S18/S20]], we can derive the extremely accurate 12 & 118 & 171 temperament:
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| [[Subgroup]]: 2.3.5.7.17
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| [[Comma list]]: 1701/1700, 32805/32768
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| {{mapping|legend=1| 1 0 15 0 -32 | 0 1 -8 0 21 | 0 0 0 1 1 }}
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| : mapping generators: ~2, ~3, ~7
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| [[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 701.7197, ~7/4 = 968.8307
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| {{Optimal ET sequence|legend=1| 12, 29, 41, 53, 106d, 118, 171, 472, 525, 643, 814, 985, 1799, 2324, 2495, 3138b, 3309bd, 4294bdg }}
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| ==== 2.3.5.7.17.19 12 & 118 & 171 (unnamed) ====
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| 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:
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| [[Subgroup]]: 2.3.5.7.17.19
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| [[Comma list]]: 361/360, 513/512, 1701/1700
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| {{mapping|legend=1| 1 0 15 0 -32 9 | 0 1 -8 0 21 -3 | 0 0 0 1 1 0 }}
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| : mapping generators: ~2, ~3, ~7
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| [[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 701.7053, ~7/4 = 968.9281
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| {{Optimal ET sequence|legend=1| 12, 29, 41, 53, 106d, 118, 171, 289h, 460hh }}
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| {{Todo| improve readability }}
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| === 2.3.5.41 53 & 65 (unnamed) ===
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| 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.
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| == Trivia == | | == Trivia == |
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.
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
