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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 == |