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Meantone family: Difference between revisions

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m →Septimal meantone: 19\31 > 18\31 (my mistake :))
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{{see also| Meantone vs meanpop }}
{{see also| Meantone vs meanpop }}


[[Comma]]s: 81/80, 126/125, 385/384
Period: 1\1


[[11-odd-limit|11-limit]] minimax 1/4 comma
Optimal ([[POTE]]) generator: ~3/2 = 696.434


[{{Monzo| 1 0 0 0 0 }}, {{Monzo| 1 0 1/4 0 0 }}, {{Monzo| 0 0 1 0 0 }},
EDO generators: [[19edo|11\19]], [[31edo|18\31]], [[50edo|29\50]]
{{Monzo| -3 0 5/2 0 0 }}, {{Monzo| 11 0 -13/4 0 0 }}]


[[Eigenmonzo]]s: 2, 5
Scales (Scala files):  


valid range: [694.737, 696.774] (19 to 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: 81/80, 126/125, 385/384


nice range: [691.202, 701.955]
Mapping: [{{val| 1 0 -4 -13 24 }}, {{val| 0 1 4 10 -13 }}]


strict range: [694.737, 696.774]
Mapping generator: ~2, ~3


[[POTE generator]]: 696.434
Minimax tuning:
* [[11-odd-limit]]: 1/4 comma
: [{{Monzo| 1 0 0 0 0 }}, {{Monzo| 1 0 1/4 0 0 }}, {{Monzo| 0 0 1 0 0 }}, {{Monzo| -3 0 5/2 0 0 }}, {{Monzo| 11 0 -13/4 0 0 }}]
: [[Eigenmonzo]]s: 2, 5


Mapping generator: ~3
Tuning ranges:  
* valid range: [694.737, 696.774] (19 to 31)
* nice range: [691.202, 701.955]
* strict range: [694.737, 696.774]


[[Algebraic generator]]: Cybozem; or else Radieubiz, the real root of 3''x''<sup>3</sup> + 6''x'' - 19. Unlike Cybozem, the recurrence for Radieubiz does not converge.
Algebraic generator: Cybozem; or else Radieubiz, the real root of 3''x''<sup>3</sup> + 6''x'' - 19. Unlike Cybozem, the recurrence for Radieubiz does not converge.
 
Map: [&lt;1 0 -4 -13 24|, &lt;0 1 4 10 -13|]


{{EDOs|legend=1| 12e, 19, 31, 81 }}
{{EDOs|legend=1| 12e, 19, 31, 81 }}


[[Badness]]: 0.0215
[[Badness]]: 0.0215
</div></div>


* [http://soonlabel.com/xenharmonic/archives/607 Scott Joplin's "The Entertainer" tuned into meanpop]{{Dead link}}
* [http://soonlabel.com/xenharmonic/archives/607 Scott Joplin's "The Entertainer" tuned into meanpop]{{Dead link}}
Line 420: Line 427:


=== 13-limit Meanpop ===
=== 13-limit Meanpop ===
Commas: 81/80, 105/104, 126/125, 144/143


valid range: [694.737, 696.774] (19 to 31)
Period: 1\1
 
Optimal ([[POTE]]) generator: ~3/2 = 696.211
 
EDO generators: [[19edo|11\19]], [[31edo|18\31]], [[50edo|29\50]]


nice range: [691.202, 701.955]
Scales (Scala files):  


strict range: [694.737, 696.774]
<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, 105/104, 126/125, 144/143


[[POTE generator]]: ~3/2 = 696.211
Mapping: [{{val| 1 0 -4 -13 24 -20 }}, {{val| 0 1 4 10 -13 15 }}]


Mapping generator: ~3
Mapping generator: ~2, ~3


Map: [&lt;1 0 -4 -13 24 -20|, &lt;0 1 4 10 -13 15|]
Tuning ranges:
* valid range: [694.737, 696.774] (19 to 31)
* nice range: [691.202, 701.955]
* strict range: [694.737, 696.774]


{{EDOs|legend=1| 12ef, 19, 31, 50, 81, 131bd, 212bbddf }}
{{EDOs|legend=1| 12ef, 19, 31, 50, 81, 131bd, 212bbddf }}


Badness: 0.0209
Badness: 0.0209
</div></div>


=== Meanplop ===
=== Meanplop ===
Commas: 65/64, 78/77, 81/80, 91/90


[[POTE generator]]: ~3/2 = 696.202
Period: 1\1


Mapping generator: ~3
Optimal ([[POTE]]) generator: ~3/2 = 696.202
 
EDO generators: [[19edo|11\19]], [[31edo|18\31]]
 
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: 65/64, 78/77, 81/80, 91/90
 
Mapping: [{{val| 1 0 -4 -13 24 10 }}, {{val| 0 1 4 10 -13 -4 }}]


Map: [&lt;1 0 -4 -13 24 10|, &lt;0 1 4 10 -13 -4|]
Mapping generators: ~2, ~3


{{EDOs|legend=1| 12e, 19, 31f, 50ff, 81fff }}
{{EDOs|legend=1| 12e, 19, 31f, 50ff, 81fff }}


Badness: 0.0277
Badness: 0.0277
</div></div>


== Meanenneadecal ==
== Meanenneadecal ==
Commas: 45/44, 56/55, 81/80


[[POTE generator]]: ~3/2 = 696.250
Period: 1\1
 
Optimal ([[POTE]]) generator: ~3/2 = 696.250
 
EDO generators: [[12edo|7\12]], [[19edo|11\19]]
 
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, 56/55, 81/80


Mapping generator: ~3
Mapping: [{{val| 1 0 -4 -13 -6 }}, {{val| 0 1 4 10 6 }}]


Map: [&lt;1 0 -4 -13 -6|, &lt;0 1 4 10 6|]
Mapping generators: ~2, ~3


{{EDOs|legend=1| 7d, 12, 19, 31e, 50ee }}
{{EDOs|legend=1| 7d, 12, 19, 31e, 50ee }}


Badness: 0.0214
Badness: 0.0214
</div></div>


=== 13-limit ===
=== 13-limit ===
Commas: 45/44, 56/55, 78/77, 81/80


[[POTE generator]]: ~3/2 = 696.146
Period: 1\1


Mapping generator: ~3
Optimal ([[POTE]]) generator: ~3/2 = 696.146
 
EDO generators: [[12edo|7\12]], [[19edo|11\19]]
 
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, 56/55, 78/77, 81/80
 
Mapping: [{{val| 1 0 -4 -13 -6 -20 }}, {{val| 0 1 4 10 6 15 }}]


Map: [&lt;1 0 -4 -13 -6 -20|, &lt;0 1 4 10 6 15|]
Mapping generators: ~2, ~3


{{EDOs|legend=1| 12f, 19, 31e, 50ee }}
{{EDOs|legend=1| 12f, 19, 31e, 50ee }}


Badness: 0.0212
Badness: 0.0212
</div></div>


=== Vincenzo ===
=== Vincenzo ===
Commas: 45/44, 56/55, 65/64, 81/80


[[POTE generator]]: ~3/2 = 695.060
Period: 1\1
 
Optimal ([[POTE]]) generator: ~3/2 = 695.060
 
EDO generators: [[12edo|7\12]], [[19edo|11\19]]
 
Scales (Scala files):


Mapping generator: ~3
<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, 56/55, 65/64, 81/80
 
Mapping: [{{val| 1 0 -4 -13 -6 10 }}, {{val| 0 1 4 10 6 -4 }}]


Map: [&lt;1 0 -4 -13 -6 10|, &lt;0 1 4 10 6 -4|]
Mapping generators: ~2, ~3


{{EDOs|legend=1| 7d, 12, 19 }}
{{EDOs|legend=1| 7d, 12, 19 }}


Badness: 0.0248
Badness: 0.0248
</div></div>


==== 17-limit ====
==== 17-limit ====
Commas: 45/44, 52/51, 56/55, 65/64, 81/80


[[POTE generator]]: ~3/2 = 695.858
Period: 1\1
 
Optimal ([[POTE]]) generator: ~3/2 = 695.858


Map: [&lt;1 0 -4 -13 -6 10 12|, &lt;0 1 4 10 6 -4 -5|]
EDO generators: [[12edo|7\12]], [[19edo|11\19]]
 
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, 52/51, 56/55, 65/64, 81/80
 
Mapping: [{{val| 1 0 -4 -13 -6 10 12 }}, {{val| 0 1 4 10 6 -4 -5 }}]
 
Mapping generators: ~2, ~3


{{EDOs|legend=1| 7d, 12, 19 }}
{{EDOs|legend=1| 7d, 12, 19 }}


Badness: 0.0255
Badness: 0.0255
</div></div>


==== 19-limit ====
==== 19-limit ====
Commas: 39/38, 45/44, 52/51, 56/55, 65/64, 81/80


[[POTE generator]]: ~3/2 = 696.131
Period: 1\1
 
Optimal ([[POTE]]) generator: ~3/2 = 696.131
 
EDO generators: [[12edo|7\12]], [[19edo|11\19]]
 
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: 39/38, 45/44, 52/51, 56/55, 65/64, 81/80


Map: [&lt;1 0 -4 -13 -6 10 12 9|, &lt;0 1 4 10 6 -4 -5 -3|]
Mapping: [{{val| 1 0 -4 -13 -6 10 12 9 }}, {{val| 0 1 4 10 6 -4 -5 -3 }}]
 
Mapping generators: ~2, ~3


{{EDOs|legend=1| 7d, 12, 19 }}
{{EDOs|legend=1| 7d, 12, 19 }}


Badness: 0.0223
Badness: 0.0223
</div></div>


==== 23-limit ====
==== 23-limit ====
Commas: 39/38, 45/44, 52/51, 56/55, 65/64, 69/68, 81/80


[[POTE generator]]: ~3/2 = 696.044
Period: 1\1
 
Optimal ([[POTE]]) generator: ~3/2 = 696.044
 
EDO generators: [[12edo|7\12]], [[19edo|11\19]]


Map: [&lt;1 0 -4 -13 -6 10 12 9 14|, &lt;0 1 4 10 6 -4 -5 -3 -6|]
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: 39/38, 45/44, 52/51, 56/55, 65/64, 69/68, 81/80
 
Mapping: [{{val| 1 0 -4 -13 -6 10 12 9 14 }}, {{val| 0 1 4 10 6 -4 -5 -3 -6 }}]
 
Mapping generators: ~2, ~3


{{EDOs|legend=1| 7d, 12, 19 }}
{{EDOs|legend=1| 7d, 12, 19 }}


Badness: 0.0201
Badness: 0.0201
</div></div>


==== 29-limit ====
==== 29-limit ====
Commas: 39/38, 45/44, 52/51, 56/55, 58/57, 65/64, 69/68, 81/80


POTE generator: ~3/2 = 695.913
Period: 1\1


Map: [&lt;1 0 -4 -13 -6 10 12 9 14 8|, &lt;0 1 4 10 6 -4 -5 -3 -6 -2|]
Optimal ([[POTE]]) generator: ~3/2 = 695.913
 
EDO generators: [[12edo|7\12]], [[19edo|11\19]]
 
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: 39/38, 45/44, 52/51, 56/55, 58/57, 65/64, 69/68, 81/80
 
Mapping: [{{val| 1 0 -4 -13 -6 10 12 9 14 8 }}, {{val| 0 1 4 10 6 -4 -5 -3 -6 -2 }}]
 
Mapping generators: ~2, ~3


{{EDOs|legend=1| 7d, 12, 19 }}
{{EDOs|legend=1| 7d, 12, 19 }}


Badness: 0.0182
Badness: 0.0182
</div></div>


==== 31-limit ====
==== 31-limit ====
Commas: 39/38, 45/44, 52/51, 56/55, 58/57, 65/64, 69/68, 81/80, 93/92


POTE generator: ~3/2 = 695.750
Period: 1\1
 
Optimal ([[POTE]]) generator: ~3/2 = 695.750
 
EDO generators: [[12edo|7\12]], [[19edo|11\19]]
 
Scales (Scala files):


Map: [&lt;1 0 -4 -13 -6 10 12 9 14 8 16|, &lt;0 1 4 10 6 -4 -5 -3 -6 -2 -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">
Comma list: 39/38, 45/44, 52/51, 56/55, 58/57, 65/64, 69/68, 81/80, 93/92
 
Mapping: [{{val| 1 0 -4 -13 -6 10 12 9 14 8 16 }}, {{val| 0 1 4 10 6 -4 -5 -3 -6 -2 -7 }}]
 
Mapping generators: ~2, ~3


{{EDOs|legend=1| 7d, 12, 19 }}
{{EDOs|legend=1| 7d, 12, 19 }}


Badness: 0.0171
Badness: 0.0171
</div></div>


==== 37-limit ====
==== 37-limit ====
Commas: 39/38, 45/44, 52/51, 56/55, 58/57, 65/64, 69/68, 75/74, 81/80, 93/92


POTE generator: ~3/2 = 695.603
Period: 1\1
 
Optimal ([[POTE]]) generator: ~3/2 = 695.603
 
EDO generators: [[12edo|7\12]], [[19edo|11\19]]
 
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: 39/38, 45/44, 52/51, 56/55, 58/57, 65/64, 69/68, 75/74, 81/80, 93/92
 
Mapping: [{{val| 1 0 -4 -13 -6 10 12 9 14 8 16 -9 }}, {{val| 0 1 4 10 6 -4 -5 -3 -6 -2 -7 9 }}]


Map: [&lt;1 0 -4 -13 -6 10 12 9 14 8 16 -9|, &lt;0 1 4 10 6 -4 -5 -3 -6 -2 -7 9|]
Mapping generators: ~2, ~3


{{EDOs|legend=1| 7d, 12, 19 }}
{{EDOs|legend=1| 7d, 12, 19 }}


Badness: 0.0161
Badness: 0.0161
</div></div>


==== 41-limit ====
==== 41-limit ====
Commas: 39/38, 45/44, 52/51, 56/55, 58/57, 65/64, 69/68, 75/74, 81/80, 93/92, 124/123


POTE generator: ~3/2 = 695.696
Period: 1\1
 
Optimal ([[POTE]]) generator: ~3/2 = 695.696
 
EDO generators: [[12edo|7\12]], [[19edo|11\19]]
 
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: 39/38, 45/44, 52/51, 56/55, 58/57, 65/64, 69/68, 75/74, 81/80, 93/92, 124/123


Map: [&lt;1 0 -4 -13 -6 10 12 9 14 8 16 -9 18|, &lt;0 1 4 10 6 -4 -5 -3 -6 -2 -7 9 -8|]
Mapping: [{{val| 1 0 -4 -13 -6 10 12 9 14 8 16 -9 18 }}, {{val| 0 1 4 10 6 -4 -5 -3 -6 -2 -7 9 -8 }}]
 
Mapping generators: ~2, ~3


{{EDOs|legend=1| 7d, 12, 19 }}
{{EDOs|legend=1| 7d, 12, 19 }}


Badness: 0.0154
Badness: 0.0154
</div></div>


==== 43-limit ====
==== 43-limit ====
Commas: 39/38, 45/44, 52/51, 56/55, 58/57, 65/64, 69/68, 75/74, 81/80, 86/85, 93/92, 124/123


POTE generator: ~3/2 = 695.688
Period: 1\1


Map: [&lt;1 0 -4 -13 -6 10 12 9 14 8 16 -9 18 7|, &lt;0 1 4 10 6 -4 -5 -3 -6 -2 -7 9 -8 -1|]
Optimal ([[POTE]]) generator: ~3/2 = 695.688
 
EDO generators: [[12edo|7\12]], [[19edo|11\19]]
 
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: 39/38, 45/44, 52/51, 56/55, 58/57, 65/64, 69/68, 75/74, 81/80, 86/85, 93/92, 124/123
 
Mapping: [{{val| 1 0 -4 -13 -6 10 12 9 14 8 16 -9 18 7 }}, {{val| 0 1 4 10 6 -4 -5 -3 -6 -2 -7 9 -8 -1 }}]
 
Mapping generators: ~2, ~3


{{EDOs|legend=1| 7d, 12, 19 }}
{{EDOs|legend=1| 7d, 12, 19 }}


Badness: 0.0139
Badness: 0.0139
</div></div>


==== 47-limit ====
==== 47-limit ====
Commas: 39/38, 45/44, 52/51, 56/55, 58/57, 65/64, 69/68, 75/74, 81/80, 86/85, 93/92, 95/94, 124/123


POTE generator: ~3/2 = 695.676
Period: 1\1


Map: [&lt;1 0 -4 -13 -6 10 12 9 14 8 16 -9 18 7 4|, &lt;0 1 4 10 6 -4 -5 -3 -6 -2 -7 9 -8 -1 1|]
Optimal ([[POTE]]) generator: ~3/2 = 695.676
 
EDO generators: [[12edo|7\12]], [[19edo|11\19]]
 
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: 39/38, 45/44, 52/51, 56/55, 58/57, 65/64, 69/68, 75/74, 81/80, 86/85, 93/92, 95/94, 124/123
 
Mapping: [{{val| 1 0 -4 -13 -6 10 12 9 14 8 16 -9 18 7 4 }}, {{val| 0 1 4 10 6 -4 -5 -3 -6 -2 -7 9 -8 -1 1 }}]
 
Mapping generators: ~2, ~3


{{EDOs|legend=1| 7d, 12, 19 }}
{{EDOs|legend=1| 7d, 12, 19 }}


Badness: 0.0138
Badness: 0.0138
</div></div>


=== Meanundec ===
=== Meanundec ===
Commas: 27/26, 40/39, 45/44, 56/55


POTE generator: ~3/2 = 697.254
Period: 1\1


Mapping generator: ~3
Optimal ([[POTE]]) generator: ~3/2 = 697.254
 
EDO generators: [[12edo|7\12]]
 
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: 27/26, 40/39, 45/44, 56/55
 
Mapping: [{{val| 1 0 -4 -13 -6 -1 }}, {{val| 0 1 4 10 6 3 }}]


Map: [&lt;1 0 -4 -13 -6 -1|, &lt;0 1 4 10 6 3|]
Mapping generators: ~2, ~3


{{EDOs|legend=1| 7d, 12f, 19f, 31eff }}
{{EDOs|legend=1| 7d, 12f, 19f, 31eff }}


Badness: 0.0242
Badness: 0.0242
</div></div>


== Meanundeci ==
== Meanundeci ==
Commas: 33/32, 55/54, 77/75
 
Period: 1\1
 
Optimal ([[POTE]]) generator: ~3/2 = 694.689
 
EDO generators: [[12edo|7\12]], [[19edo|11\19]]
 
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: 33/32, 55/54, 77/75


POTE generator: ~3/2 = 694.689
POTE generator: ~3/2 = 694.689
Mapping: [{{val|1 0 -4 -13 5 }}, {{val| 0 1 4 10 -1 }}]


Mapping generator: ~3
Mapping generator: ~3
Map: [&lt;1 0 -4 -13 5|, &lt;0 1 4 10 -1|]


{{EDOs|legend=1| 7d, 12e, 19e }}
{{EDOs|legend=1| 7d, 12e, 19e }}


Badness: 0.0315
Badness: 0.0315
</div></div>


=== 13-limit ===
=== 13-limit ===
Commas: 33/32, 55/54, 65/64, 77/75


POTE generator: ~3/2 = 694.764
Period: 1\1
 
Optimal ([[POTE]]) generator: ~3/2 = 694.764


Mapping generator: ~3
EDO generators: [[12edo|7\12]], [[19edo|11\19]]
 
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: 33/32, 55/54, 65/64, 77/75
 
Mapping: [{{val|1 0 -4 -13 5 10 }}, {{val| 0 1 4 10 -1 -4 }}]


Map: [&lt;1 0 -4 -13 5 10|, &lt;0 1 4 10 -1 -4|]
Mapping generators: ~2, ~3


{{EDOs|legend=1| 7d, 12e, 19e }}
{{EDOs|legend=1| 7d, 12e, 19e }}


Badness: 0.0263
Badness: 0.0263
</div></div>


= Flattone =
= Flattone =
Retrieved from "https://en.xen.wiki/w/Meantone_family"