Meantone family: Difference between revisions
→Septimal meantone: move certain things here from meantone article |
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===== 17-limit ===== | ===== 17-limit ===== | ||
This extension | This extension maps 17/16 to the minor second (C–D♭), and 19/16 to the minor third (C–E♭), suitable for a system generated by a mildly tempered fifth. | ||
Subgroup: 2.3.5.7.11.13.17 | Subgroup: 2.3.5.7.11.13.17 | ||
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=== Meanpop === | === Meanpop === | ||
{{See also| | {{See also| Huygens vs meanpop }} | ||
Meanpop<ref name="meantone & meanpop 2003"/><ref name="meantone & meanpop 2004"/> maps the 11/8 to the double-diminished fifth (C–G𝄫), and tridecimal meanpop maps the 13/8 to the double-augmented fifth (C–G𝄪), tempering out 144/143 like in grosstone. Note also 11/10 is a double-diminished third; 12/11~13/12, double-augmented unison; and 14/13, minor second. | Meanpop<ref name="meantone & meanpop 2003"/><ref name="meantone & meanpop 2004"/> maps the 11/8 to the double-diminished fifth (C–G𝄫), and tridecimal meanpop maps the 13/8 to the double-augmented fifth (C–G𝄪), tempering out 144/143 like in grosstone. Note also 11/10 is a double-diminished third; 12/11~13/12, double-augmented unison; and 14/13, minor second. | ||
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Badness (Sintel): 1.28 | Badness (Sintel): 1.28 | ||
=== Migration === | |||
See [[Rastmic clan #Migration|Rastmic clan]]. | |||
== Flattone == | == Flattone == | ||
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Badness (Sintel): 1.12 | Badness (Sintel): 1.12 | ||
=== 13-limit === | ==== 13-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
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Badness (Sintel): 0.920 | Badness (Sintel): 0.920 | ||
=== Ptolemy === | |||
See [[Rastmic clan #Ptolemy|Rastmic clan]]. | |||
== Dominant == | == Dominant == | ||
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Badness (Sintel): 0.864 | Badness (Sintel): 0.864 | ||
=== Neutrominant === | |||
See [[Rastmic clan #Neutrominant|Rastmic clan]]. | |||
== Flattertone == | == Flattertone == | ||
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[[Badness]] (Sintel): 2.67 | [[Badness]] (Sintel): 2.67 | ||
=== 11-limit === | |||
[[Subgroup]]: 2.3.5.7.11 | |||
[[Comma list]]: 81/80, 176/175, 7058/6875 | |||
{{Mapping|legend=1| 1 0 -4 -32 | 0 1 4 22 30}} | |||
[[Optimal tuning]]s: | |||
* [[WE]]: ~2 = 1199.816{{c}}, ~3/2 = 698.355{{c}} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 698.455{{c}} | |||
{{Optimal ET sequence|legend=1| 12, 43de, 55, 67 }} | |||
[[Badness]] (Sintel): 2.15 | |||
=== 13-limit === | |||
[[Subgroup]]: 2.3.5.7.11.13 | |||
[[Comma list]]: 81/80, 176/175, 196/195, 832/825 | |||
{{Mapping|legend=1| 1 0 -4 -32 -44 | 0 1 4 22 30}} | |||
[[Optimal tuning]]s: | |||
* [[WE]]: ~2 = 1199.788{{c}}, ~3/2 = 698.355{{c}} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 698.471{{c}} | |||
{{Optimal ET sequence|legend=1| 12f, 55f, 67 }} | |||
[[Badness]] (Sintel): 2.04 | |||
=== 17-limit === | |||
[[Subgroup]]: 2.3.5.7.11.13.17 | |||
[[Comma list]]: 81/80, 176/175, 189/197, 196/195, 832/825 | |||
{{Mapping|legend=1| 1 0 -4 -32 -44 12| 0 1 4 22 30 -5}} | |||
[[Optimal tuning]]s: | |||
* [[WE]]: ~2 = 1199.655{{c}}, ~3/2 = 698.295{{c}} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 698.488{{c}} | |||
{{Optimal ET sequence|legend=1| 12f, 55f, 67 }} | |||
[[Badness]] (Sintel): 1.98 | |||
=== 19-limit === | |||
[[Subgroup]]: 2.3.5.7.11.13.19 | |||
[[Comma list]]: 81/80, 96/95, 176/175, 189/187, 196/195, 832/825 | |||
{{Mapping|legend=1| 1 0 -4 -32 -44 12 9| 0 1 4 22 30 -5 -3}} | |||
[[Optimal tuning]]s: | |||
* [[WE]]: ~2 = 1199.371{{c}}, ~3/2 = 698.164{{c}} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 698.519{{c}} | |||
{{Optimal ET sequence|legend=1| 12f, 55f, 67 }} | |||
[[Badness]] (Sintel): 1.95 | |||
{{Todo|unify precision|review}} | |||
== Supermean == | == Supermean == | ||
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{{Main| Mohajira }} | {{Main| Mohajira }} | ||
Mohajira can be viewed as derived from mohaha which maps the interval half a [[chromatic semitone|chroma]] flat of the minor seventh to ~7/4 so that 7/4 is mapped to a semidiminished seventh (C–Bdb), although mohajira really makes more sense as an 11-limit temperament. It tempers out 6144/6125, the [[porwell comma]]. It can be described as {{nowrap| 24 & 31 }}; its ploidacot is dicot. [[31edo]] makes for an excellent mohajira tuning, with generator 9\31. | Mohajira can be viewed as derived from [[mohaha]] which maps the interval half a [[chromatic semitone|chroma]] flat of the minor seventh to ~7/4 so that 7/4 is mapped to a semidiminished seventh (C–Bdb), although mohajira really makes more sense as an 11-limit temperament. It tempers out 6144/6125, the [[porwell comma]]. It can be described as {{nowrap| 24 & 31 }}; its ploidacot is dicot. [[31edo]] makes for an excellent mohajira tuning, with generator 9\31. Note that while 24 + 31 = [[55edo]] doesn't apear in the optimal ET sequence, it is a [[patent val]] tuning and recommendable if you prefer a light meantone tempering. | ||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||