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Meantone family: Difference between revisions

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= Mothra =
= Mothra =
[[Comma]]s: 81/80, 1029/1024


Mothra splits the fifth into three 8/7 generators. It uses 1029/1024, the gamelisma, to accomplish this deed and also tempers out 1728/1715, the orwell comma. Using [[31edo]] with a generator of 6/31 is an excellent tuning choice. Once again something other than a MOS should be used as a scale to get the most out of mothra. In the 2.3.7-limit, mothra is identical to [[slendric]].
Mothra splits the fifth into three 8/7 generators. It uses [[1029/1024]], the gamelisma, to accomplish this deed and also tempers out [[1728/1715]], the orwell comma. Using [[31edo]] with a generator of 6/31 is an excellent tuning choice. Once again something other than a MOS should be used as a scale to get the most out of mothra. In the 2.3.7-limit, mothra is identical to [[slendric]].


Note that mothra can also be called cynder in the 7-limit, which can be a little confusing sometimes.
Note that mothra can also be called '''cynder''' in the 7-limit, which can be a little confusing sometimes.  


[[7-odd-limit|7]] and [[9-odd-limit|9-limit]] minimax 1/4 comma
Period: 1\1


[{{Monzo| 1 0 0 0 }}, {{Monzo| 1 0 1/4 0 }}, {{Monzo| 0 0 1 0 }}, {{Monzo| 3 0 -1/12 0 }}]
Optimal (POTE) generator: ~8/7 = 232.193


[[Eigenmonzo]]s: 2, 5
EDO generators: [[26edo|5\26]], [[31edo|6\31]], [[36edo|7\36]], [[57edo|11\57]], [[67edo|13\67]]


[[POTE generator]]: ~8/7 = 232.193
Scales (Scala files):


Mapping generator: ~8/7
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
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[[Comma]] list: 81/80, 1029/1024
 
[[Mapping]]: [{{val| 1 1 0 3 }}, {{val| 0 3 12 -1 }}]
 
Mapping generators: ~2, ~8/7


Algebraic generator: Rabrindanath, largest real root of ''x''<sup>8</sup> - 3''x''<sup>2</sup> + 1, or 232.0774 cents.
Algebraic generator: Rabrindanath, largest real root of ''x''<sup>8</sup> - 3''x''<sup>2</sup> + 1, or 232.0774 cents.


Map: [&lt;1 1 0 3|, &lt;0 3 12 -1|]
[[Wedgie]]: {{wedgie| 3 12 -1 12 -10 -36 }}


[[Generator]]s: 2, 8/7
[[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| 3 0 -1/12 0 }}]
: [[Eigenmonzo]]s: 2, 5


[[Wedgie]]: &lt;&lt;3 12 -1 12 -10 -36||
{{Vals|legend=1| 5, 26, 31 }}


{{Vals|legend=1| 5, 26, 31, 57, 88 }}
[[Badness]]: 0.0371


[[Badness]]: 0.0371
</div></div>


== 11-limit ==
== 11-limit ==
Commas: 81/80, 99/98, 385/384


POTE generator: ~8/7 = 232.031
Period: 1\1
 
Optimal (POTE) generator: ~8/7 = 232.031
 
EDO generators: [[26edo|5\26]], [[31edo|6\31]], [[57edo|11\57]]
 
Scales (Scala files):
 
<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, 99/98, 385/384


Mapping generator: ~8/7
Mapping: [{{val| 1 1 0 3 5 }}, {{val| 0 3 12 -1 -8 }}]


Map: [&lt;1 1 0 3 5|, &lt;0 3 12 -1 -8|]
Mapping generators: ~2, ~8/7


{{Vals|legend=1| 5, 26, 31, 57, 88, 150be, 181bee }}
{{Vals|legend=1| 5, 26, 31, 88, 150be, 181bee }}


Badness: 0.0256
Badness: 0.0256
</div></div>


=== 13-limit ===
=== 13-limit ===
Commas: 81/80, 99/98, 105/104, 144/143


POTE generator: ~8/7 = 231.811
Period: 1\1
 
Optimal (POTE) generator: ~8/7 = 231.811
 
EDO generators: [[26edo|5\26]], [[31edo|6\31]], [[57edo|11\57]]
 
Scales (Scala files):


Mapping generator: ~8/7
<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">


Map: [&lt;1 1 0 3 5 1|, &lt;0 3 12 -1 -8 14|]
Comma list: 81/80, 99/98, 105/104, 144/143
 
Mapping: [{{val| 1 1 0 3 5 1 }}, {{val| 0 3 12 -1 -8 14 }}]
 
Mapping generators: ~2, ~8/7


{{Vals|legend=1| 5, 26, 31, 57, 88 }}
{{Vals|legend=1| 5, 26, 31, 57, 88 }}


Badness: 0.0240
Badness: 0.0240
</div></div>


== Cynder ==
== Cynder ==
Commas: 45/44, 81/80, 1029/1024


POTE generator: ~8/7 = 231.317
Period: 1\1
 
Optimal (POTE) generator: ~8/7 = 231.317
 
EDO generators: [[26edo|5\26]], [[31edo|6\31]], [[57edo|11\57]]
 
Scales (Scala files):
 
<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: 45/44, 81/80, 1029/1024


Mapping generator: ~8/7
Mapping: [{{val| 1 1 0 3 0 }}, {{val| 0 3 12 -1 18 }}]


Map: [&lt;1 1 0 3 0|, &lt;0 3 12 -1 18|]
Mapping generators: ~2, ~8/7


{{Vals|legend=1| 5e, 26, 31e, 57e, 83bce }}
{{Vals|legend=1| 5e, 26, 57e, 83bce }}


Badness: 0.0557
Badness: 0.0557
</div></div>


=== 13-limit ===
=== 13-limit ===
Period: 1\1
Optimal (POTE) generator: ~8/7 = 232.293
EDO generators: [[26edo|5\26]], [[31edo|6\31]], [[57edo|11\57]]
Scales (Scala files):
<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">
Commas: 45/44, 78/77, 81/80, 640/637
Commas: 45/44, 78/77, 81/80, 640/637


POTE generator: ~8/7 = 231.293
Mapping: [{{val| 1 1 0 3 0 1 }}, {{val| 0 3 12 -1 18 14 }}]


Mapping generator: ~8/7
Mapping generators: ~2, ~8/7


Map: [&lt;1 1 0 3 0 1|, &lt;0 3 12 -1 18 14|]
{{Vals|legend=1| 5e, 26, 57e, 83bce }}


{{Vals|legend=1| 5e, 26, 31e, 57e, 83bce }}
Badness: 0.0341


Badness: 0.0341
</div></div>


== Mosura ==
== Mosura ==
Commas: 81/80, 176/175, 540/539


POTE generator: ~8/7 = 232.419
Period: 1\1
 
Optimal (POTE) generator: ~8/7 = 232.419
 
EDO generators: [[31edo|6\31]], [[36edo|7\36]], [[67edo|13\67]]
 
Scales (Scala files):
 
<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, 176/175, 540/539


Mapping generator: ~8/7
Mapping: [{{val| 1 1 0 3 -1 }}, {{val| 0 3 12 -1 23 }}]


Map: [&lt;1 1 0 3 -1|, &lt;0 3 12 -1 23|]
Mapping generators: ~2, ~8/7


{{Vals|legend=1| 31, 36, 67, 98, 129, 160be, 191bce, 222bce, 253bcee }}
{{Vals|legend=1| 31, 129, 160be, 191bce, 222bce, 253bcee }}


Badness: 0.0313
Badness: 0.0313
</div></div>


=== 13-limit ===
=== 13-limit ===
Commas: 81/80, 144/143, 176/175, 196/195


POTE generator: ~8/7 = 232.640
Period: 1\1


Mapping generator: ~8/7
Optimal (POTE) generator: ~8/7 = 232.640


Map: [&lt;1 1 0 3 -1 7|, &lt;0 3 12 -1 23 -17|]
EDO generators: [[31edo|6\31]], [[36edo|7\36]], [[67edo|13\67]]
 
Scales (Scala files):
 
<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, 144/143, 176/175, 196/195
 
Mapping: [{{val| 1 1 0 3 -1 7 }}, {{val| 0 3 12 -1 23 -17 }}]
 
Mapping generators: ~2, ~8/7


{{Vals|legend=1| 31, 36, 67, 98 }}
{{Vals|legend=1| 31, 36, 67, 98 }}


Badness: 0.0369
Badness: 0.0369
</div></div>


= Squares =
= Squares =
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