Meantone family: Difference between revisions
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{{main|Mohajira}} | {{main|Mohajira}} | ||
Mohajira really makes more sense as an 11-limit temperament. It has a generator of a neutral third, two of which make up a fifth, and which can be taken to represent 128/105. Mohajira tempers out 6144/6125, the porwell comma. [[31edo]] makes for an excellent (7-limit) mohajira tuning, with generator 9/31. It has a 7-note MOS with three larger steps and four smaller ones, going sLsLsLs. | Mohajira really makes more sense as an 11-limit temperament. It has a generator of a neutral third, two of which make up a fifth, and which can be taken to represent 128/105. Mohajira tempers out 6144/6125, the porwell comma. [[31edo]] makes for an excellent (7-limit) mohajira tuning, with generator 9/31. It has a 7-note MOS with three larger steps and four smaller ones, going sLsLsLs. | ||
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Mohajira can also be thought of, intuitively, as "meantone with quarter tones"; as is the 3/2 generator subdivided in half, so is the 25/24 chromatic semitone divided into two equal ~33/32 quarter tones (in the 11-limit). Within this paradigm, mohajira is the temperament that splits the 3/2 into two equal 11/9's, that splits the 6/5 into two equal 11/10's, that maps four 3/2's to 5/1, and that maps the interval one quarter tone flat of 16/9 to 7/4. | Mohajira can also be thought of, intuitively, as "meantone with quarter tones"; as is the 3/2 generator subdivided in half, so is the 25/24 chromatic semitone divided into two equal ~33/32 quarter tones (in the 11-limit). Within this paradigm, mohajira is the temperament that splits the 3/2 into two equal 11/9's, that splits the 6/5 into two equal 11/10's, that maps four 3/2's to 5/1, and that maps the interval one quarter tone flat of 16/9 to 7/4. | ||
Period: 1\1 | |||
[ | Optimal ([[POTE]]) generator: ~128/105 = 348.415 | ||
[[ | EDO generators: [[24edo|7\24]], [[31edo|9\31]], [[38edo|11\38]], [[55edo|16\55]] | ||
[[ | Scales (Scala files): [[Mohaha7]], [[Mohaha10]] | ||
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;"> | |||
<div style="line-height:1.6;">Technical data</div> | |||
<div class="mw-collapsible-content"> | |||
[[Comma]] list: 81/80, 6144/6125 | |||
Mapping: [{{val| 1 1 0 6 }}, {{val| 0 2 8 -11 }}] | |||
Mapping generators: ~2, ~128/105 | |||
[[Wedgie]]: {{wedgie| 2 8 -11 8 -23 -48 }} | |||
[[ | [[Minimax tuning]]: | ||
* [[7-odd-limit|7]]- and [[9-odd-limit]]: 1/4 comma | |||
: [{{Monzo| 1 0 0 0 }}, {{Monzo| 1 0 1/4 0 }}, {{Monzo| 0 0 1 0 }}, {{Monzo| 6 0 -11/8 0 }}] | |||
: [[Eigenmonzo]]s: 2, 5 | |||
Algebraic generator: Mohabis, real root of 3''x''<sup>3</sup> - 3''x''<sup>2</sup> - 1, 348.6067 cents. Corresponding recurrence converges quickly. | |||
{{Vals|legend=1| 7, 24, 31 | {{Vals|legend=1| 7, 24, 31 }} | ||
[[Badness]]: 0.0557 | [[Badness]]: 0.0557 | ||
</div></div> | |||
== 11-limit == | == 11-limit == | ||
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Optimal ([[POTE]]) generator: ~11/9 = 348.477 | Optimal ([[POTE]]) generator: ~11/9 = 348.477 | ||
EDO generators: [[24edo|7\24]], [[31edo|9\31]] | EDO generators: [[24edo|7\24]], [[31edo|9\31]], [[38edo|11\38]], [[55edo|16\55]] | ||
Scales (Scala files): | Scales (Scala files): [[Mohaha7]], [[Mohaha10]] | ||
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;"> | <div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;"> | ||
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: Eigenmonzos: 2, 5 | : Eigenmonzos: 2, 5 | ||
{{Vals|legend=1| 7, 24, 31 | {{Vals|legend=1| 7, 24, 31 }} | ||
Badness: 0.0261 | Badness: 0.0261 | ||
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== 13-limit == | == 13-limit == | ||
Period: 1\1 | |||
POTE generator: ~11/9 = 348.558 | Optimal ([[POTE]]) generator: ~11/9 = 348.558 | ||
EDO generators: [[24edo|7\24]], [[31edo|9\31]], [[38edo|11\38]], [[55edo|16\55]] | |||
Scales (Scala files): [[Mohaha7]], [[Mohaha10]] | |||
{{Vals|legend=1| 7, 24, 31 | <div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;"> | ||
<div style="line-height:1.6;">Technical data</div> | |||
<div class="mw-collapsible-content"> | |||
Comma list: 66/65, 81/80, 105/104, 121/120 | |||
Mapping: [{{val| 1 1 0 6 2 4 }}, {{val| 0 2 8 -11 5 -1 }}] | |||
Mapping generators: ~2, ~11/9 | |||
{{Vals|legend=1| 7, 24, 31 }} | |||
Badness: 0.0234 | Badness: 0.0234 | ||
</div></div> | |||
== 17-limit == | == 17-limit == | ||
Period: 1\1 | |||
POTE generator: ~11/9 = 348.736 | Optimal ([[POTE]]) generator: ~11/9 = 348.736 | ||
EDO generators: [[24edo|7\24]], [[31edo|9\31]], [[55edo|16\55]] | |||
Scales (Scala files): [[Mohaha7]], [[Mohaha10]] | |||
{{Vals|legend=1| 7, 24, 31, | <div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;"> | ||
<div style="line-height:1.6;">Technical data</div> | |||
<div class="mw-collapsible-content"> | |||
Comma list: 66/65, 81/80, 105/104, 121/120, 154/153 | |||
Mapping: [{{val| 1 1 0 6 2 4 7 }}, {{val| 0 2 8 -11 5 -1 -10 }}] | |||
Mapping generators: ~2, ~11/9 | |||
{{Vals|legend=1| 7, 24, 31, 86ef }} | |||
Badness: 0.0206 | Badness: 0.0206 | ||
</div></div> | |||
== 19-limit == | == 19-limit == | ||
Period: 1\1 | |||
POTE generator: ~11/9 = 348.810 | Optimal ([[POTE]]) generator: ~11/9 = 348.810 | ||
EDO generators: [[24edo|7\24]], [[31edo|9\31]], [[55edo|16\55]] | |||
Scales (Scala files): [[Mohaha7]], [[Mohaha10]] | |||
{{Vals|legend=1| 7, 24, 31, | <div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;"> | ||
<div style="line-height:1.6;">Technical data</div> | |||
<div class="mw-collapsible-content"> | |||
Comma list: 66/65, 77/76, 81/80, 96/95, 105/104, 153/152 | |||
Mapping: [{{val| 1 1 0 6 2 4 7 6 }}, {{val| 0 2 8 -11 5 -1 -10 -6 }}] | |||
Mapping generators: ~2, ~11/9 | |||
{{Vals|legend=1| 7, 24, 31, 55, 86efh }} | |||
Badness: 0.0173 | Badness: 0.0173 | ||
</div></div> | |||
= Dominant = | = Dominant = | ||