Syntonic–limmic equivalence continuum: Difference between revisions
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== Ultrapyth (5-limit) == | == Ultrapyth (5-limit) == | ||
: ''For extensions, see [[Archytas clan #Ultrapyth]].'' | : ''For extensions, see [[Archytas clan #Ultrapyth]].'' | ||
The 5-limit version of ultrapyth tempers out the [[ultrapyth comma]]. It is generated by a perfect fifth. The interval class of 5 is found at +14 fifths as a double augmented unison (C–Cx). It corresponds to {{nowrap|''m'' {{=}} -1}} and {{nowrap|''n'' {{=}} 1/2}}. | |||
[[Subgroup]]: 2.3.5 | [[Subgroup]]: 2.3.5 | ||
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[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[CTE]]: ~2 = 1200.000, ~3/2 = 713.185 | |||
: [[error map]]: {{val| 0.000 +11.230 -1.722 }} | |||
* [[POTE]]: ~2 = 1200.000, ~3/2 = 713.829 | * [[POTE]]: ~2 = 1200.000, ~3/2 = 713.829 | ||
: error map: {{val| 0.000 +11.874 +7.289 }} | |||
{{Optimal ET sequence|legend=1| 5, 27c, 32, 37, 79bc, 116bbc }} | {{Optimal ET sequence|legend=1| 5, 27c, 32, 37, 79bc, 116bbc }} | ||
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== Rodan (5-limit) == | == Rodan (5-limit) == | ||
: ''For extensions, see [[Gamelismic clan #Rodan]].'' | : ''For extensions, see [[Gamelismic clan #Rodan]].'' | ||
The 5-limit version of rodan tempers out the [[rodan comma]], which is the difference between a stack of three [[729/640|retroptolemaic whole tones (729/640)]] and a perfect fifth (3/2). The only 7-limit extension that makes any sense to use is to add the gamelisma to the comma list. It corresponds to {{nowrap| ''n'' {{=}} 3 }}. | |||
[[Subgroup]]: 2.3.5 | [[Subgroup]]: 2.3.5 | ||
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[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[CTE]]: ~2 = 1200.000, ~729/640 = 234.457 | |||
: [[error map]]: {{val| 0.000 +1.417 -0.537 }} | |||
* [[POTE]]: ~2 = 1200.000, ~729/640 = 234.528 | * [[POTE]]: ~2 = 1200.000, ~729/640 = 234.528 | ||
: error map: {{val| 0.000 +1.629 +0.663 }} | |||
{{Optimal ET sequence|legend=1| 5, | {{Optimal ET sequence|legend=1| 5, …, 41, 46, 87, 220, 307 }} | ||
[[Badness]]: 0.168264 | [[Badness]]: 0.168264 | ||
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: ''For extensions, see [[Gamelismic clan #Gorgo]].'' | : ''For extensions, see [[Gamelismic clan #Gorgo]].'' | ||
Laconic tempers out [[2187/2000]], which is the difference between three [[10/9]]'s and a | Laconic tempers out [[2187/2000]], which is the difference between a stack of three [[10/9|ptolemaic whole tones (10/9)]]'s and a perfect fifth (3/2). Although a higher-error temperament, it does pop up enough in the low-numbered edos to be useful, most notably in [[16edo]] and [[21edo]]. The only 7-limit extension that makes any sense to use is to add the gamelisma to the comma list. It corresponds to {{nowrap| ''n'' {{=}} -3 }}. | ||
[[Subgroup]]: 2.3.5 | [[Subgroup]]: 2.3.5 | ||