Schisma: Difference between revisions
m when i talked to fumica, i couldnt get to the bottom of what was wrong with the page per se, so im going to revert and try to improve some issues i see to fit with Xenharmonic_Wiki:Notability_guidelines Tag: Undo |
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== Other intervals == | == Other intervals == | ||
Commas arising from the difference between a stack of Pythagorean intervals and other primes may also be called schismas. The difference between the [[Pythagorean comma]] and [[septimal comma]] is called the [[septimal schisma]]. Other examples are [[undevicesimal schisma]], [[Alpharabian schisma]] and [[tridecaschisma]]. | Commas arising from the difference between a stack of Pythagorean intervals and other primes may also be called schismas. The difference between the [[Pythagorean comma]] and [[septimal comma]] is called the [[septimal schisma]]. Other examples are [[undevicesimal schisma]], [[Alpharabian schisma]] and [[tridecaschisma]]. | ||
== Temperaments == | == Temperaments == | ||
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{{See also| No-sevens subgroup temperaments #Nestoria }} | {{See also| No-sevens subgroup temperaments #Nestoria }} | ||
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]]). | 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]]). This corresponds to making 19/16 the minor third and 24/19~19/15 the major third; a good tuning for this is [[65edo]], or if you prefer a more accurate [[19/16]], [[77edo]]. | ||
=== Garibaldi === | === Garibaldi === | ||
{{Main| Garibaldi }} | {{Main| Garibaldi }} | ||
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]]). | Garibaldi tempers out [[225/224]] (S15) and [[5120/5103]] ([[64/63|S8]]/[[81/80|S9]]), and can be described as the 41 & 53 temperament in the 7-limit that equates the two aforementioned commas (S8 = (8/7)/(9/8) = 64/63 and S9 = (9/8)/(10/9) = 81/80) into a general purpose comma reached at 12 fifths via (9/8)<sup>6 / (2/1). This is derived since the schisma is also equal to [[225/224|S15]]/([[5120/5103|S8/S9]]). | ||
==== 2.3.5.7.19 subgroup ==== | ==== 2.3.5.7.19 subgroup ==== | ||
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==== 2.3.5.7.17.19 12 & 118 & 171 (unnamed) ==== | ==== 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: | 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 (meaning the [[superparticular interval]]s in 14:15:16:17:18:19:20:21 are mapped equidistantly, so requiring a large edo tuning), leading to: | ||
[[Subgroup]]: 2.3.5.7.17.19 | [[Subgroup]]: 2.3.5.7.17.19 | ||
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{{Optimal ET sequence|legend=1| 12, 29, 41, 53, 106d, 118, 171, 289h, 460hh }} | {{Optimal ET sequence|legend=1| 12, 29, 41, 53, 106d, 118, 171, 289h, 460hh }} | ||
=== 2.3.5.41 53 & 65 (unnamed) === | === 2.3.5.41 53 & 65 (unnamed) === | ||
The schisma can additionally split into two superparticular commas in the 41-limit: 32805/32768 = | 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 microtemperament-accuracy mapping for prime 41 via tempering out [[6561/6560|S81]] = (81/80)/(82/81) (the second of the aforementioned commas) s.t any accurate schismic tuning (one with a very slightly flat 81/80) will have a good tuning for an otonal supermajor third [[41/32]] and a flat supermajor second (41/32)/(9/8) = [[41/36]]. | ||
== History and etymology == | == History and etymology == | ||