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Meantone family: Difference between revisions

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= Meantone (12&19, 2.3.5) =
= Meantone (12&19, 2.3.5) =
{{main| Meantone }}
{{main| Meantone }}
Period: 1\1
Optimal ([[POTE]]) generator: ~3/2 = 696.239
EDO generators: [[12edo|7\12]], [[19edo|11\19]], [[31edo|18\31]], [[43edo|25\43]], [[50edo|29\50]]
Scales (Scala files): [[Meantone5]], [[Meantone7]], [[Meantone12]]
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Interval table (7-note MOS, 2.3.5.7 POTE tuning)</div>
<div class="mw-collapsible-content">
{| class="wikitable right-1 right-2 sortable"
|+
|-
! #
! Cents<ref>octave-reduced</ref>
! class="unsortable"| Approximate ratios<ref>2.3.5, odd limit ≤ 27</ref>
|-
| 0
| 0.00
| 1/1
|-
| 1
| 696.2
| 3/2
|-
| 2
| 192.5
| 9/8, 10/9
|-
| 3
| 888.7
| 5/3
|-
| 4
| 385.0
| 5/4
|-
| 5
| 1081.2
| 15/8
|-
| 6
| 577.4
| 25/18
|}
<references/></div></div>
<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">


Subgroup: 2.3.5
Subgroup: 2.3.5
Line 71: Line 19:
Mapping generators: ~2, ~3
Mapping generators: ~2, ~3


[[Wedgie]]: {{wedgie| 1 4 4 }}
{{multival|legend=1| 1 4 4 }}
 
[[POTE generator]]: ~3/2 = 696.239


[[Tuning ranges]]:  
[[Tuning ranges]]:  
Line 82: Line 32:
[[Badness]]: 0.00736
[[Badness]]: 0.00736


</div></div>
Scales: [[meantone5]], [[meantone7]], [[meantone12]]


== Seven-limit extensions ==
== Seven-limit extensions ==
Line 103: Line 53:
{{see also| Wikipedia: Septimal meantone temperament }}
{{see also| Wikipedia: Septimal meantone temperament }}


The [[7/4]] of septimal meantone is the augmented sixth, C-A#, and other septimal intervals are [[7/6]], C-D#, the augmented second, [[7/5]], C-F#, the tritone, and [[21/16]], C-E#, the augmented third. Septimal meantone also tempers out the common 7-limit comma [[225/224]] and is in fact can be defined as the 7-limit temperament that tempers out 81/80 and 225/224.
The [[7/4]] of septimal meantone is the augmented sixth, C-A#, and other septimal intervals are [[7/6]], C-D#, the augmented second, [[7/5]], C-F#, the tritone, and [[21/16]], C-E#, the augmented third. Septimal meantone tempers out the common 7-limit commas [[126/125]] and [[225/224]] and in fact can be defined as the 7-limit temperament that tempers out any two of 81/80, 126/125 and 225/224.
 
Period: 1\1
 
Optimal ([[POTE]]) generator: ~3/2 = 696.495
 
EDO generators: [[12edo|7\12]], [[19edo|11\19]], [[31edo|18\31]], [[43edo|25\43]], [[50edo|29\50]]
 
Scales (Scala files): [[Meantone5]], [[Meantone7]], [[Meantone12]]
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Interval table (12-note MOS, 2.3.5.7 POTE tuning)</div>
<div class="mw-collapsible-content">
 
{| class="wikitable right-1 right-2 sortable"
|+
|-
! #
! Cents<ref>octave-reduced</ref>
! class="unsortable"| Approximate ratios<ref>2.3.5.7, odd limit ≤ 27</ref>
|-
| 0
| 0.00
| 1/1
|-
| 1
| 696.5
| 3/2
|-
| 2
| 193.0
| 9/8, 10/9
|-
| 3
| 889.5
| 5/3
|-
| 4
| 386.0
| 5/4
|-
| 5
| 1082.5
| 15/8, 28/15
|-
| 6
| 579.0
| 7/5
|-
| 7
| 75.5
| 21/20, 25/24, 28/27
|-
| 8
| 772.0
| 14/9, 25/16
|-
| 9
| 268.5
| 7/6
|-
| 10
| 965.0
| 7/4
|-
| 11
| 461.4
| 21/16
|}
<references/></div></div>
 
<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">


Subgroup: 2.3.5.7
Subgroup: 2.3.5.7
Line 186: Line 63:
Mapping generators: ~2, ~3
Mapping generators: ~2, ~3


[[Wedgie]]: {{wedgie| 1 4 10 4 13 12 }}
{{Multival|legend=1| 1 4 10 4 13 12 }}
 
[[POTE generator]]: ~3/2 = 696.495


[[Minimax tuning]]:  
[[Minimax tuning]]:  
Line 194: Line 73:


[[Tuning ranges]]:  
[[Tuning ranges]]:  
* valid range: [694.737, 700.000] (19 to 12)
* valid range: [694.737, 700.000] (11\19 to 7\12)
* nice range: [694.786, 701.955]
* nice range: [694.786, 701.955]
* strict range: [694.786, 700.000]
* strict range: [694.786, 700.000]


Algebraic generator: Cybozem, the real root of 15''x''<sup>3</sup> - 10''x''<sup>2</sup> - 18, which comes to 503.4257 cents. The recurrence converges quickly.
[[Algebraic generator]]: Cybozem, the real root of 15''x''<sup>3</sup> - 10''x''<sup>2</sup> - 18, 503.4257 cents. The recurrence converges quickly.


{{Val list|legend=1| 12, 19, 31, 81, 112b, 143b }}
{{Val list|legend=1| 12, 19, 31, 81, 112b, 143b }}
Line 204: Line 83:
[[Badness]]: 0.0137
[[Badness]]: 0.0137


</div></div>
Scales: [[meantone5]], [[meantone7]], [[meantone12]]


== Bimeantone ==
== Bimeantone ==
11/8 is mapped to half octave minus the [[meantone diesis]].  
11/8 is mapped to half octave minus the [[meantone diesis]].  
Period: 1\2
Optimal ([[POTE]]) generator: ~3/2 = 696.016
EDO generators: [[38edo|22\38]], [[50edo|29\50]]
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">


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 228: Line 95:


Mapping generators: ~63/44, ~3
Mapping generators: ~63/44, ~3
POTE generator: ~3/2 = 696.016


{{Val list|legend=1| 12, 26de, 38d, 50 }}
{{Val list|legend=1| 12, 26de, 38d, 50 }}


Badness: 0.0381
Badness: 0.0381
</div></div>


=== 13-limit ===
=== 13-limit ===
Period: 1\2
Optimal ([[POTE]]) generator: ~3/2 = 695.836
EDO generators: [[38edo|22\38]], [[50edo|29\50]]
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">


Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13
Line 256: Line 111:


Mapping generators: ~55/39, ~3
Mapping generators: ~55/39, ~3
POTE generator: ~3/2 = 695.836


{{Val list|legend=1| 12f, 26deff, 38df, 50 }}
{{Val list|legend=1| 12f, 26deff, 38df, 50 }}


Badness: 0.0288
Badness: 0.0288
</div></div>


== Unidecimal meantone aka Huygens ==
== Unidecimal meantone aka Huygens ==
{{see also| Meantone vs meanpop }}
{{see also| Meantone vs meanpop }}


Period: 1\1
Subgroup: 2.3.5.7.11
 
Optimal ([[POTE]]) generator: ~3/2 = 696.967
 
EDO generators: [[31edo|18\31]], [[43edo|25\43]]
 
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">
 
Subgroup: 2.3.5.7.11


Comma list: 81/80, 126/125, 99/98
Comma list: 81/80, 126/125, 99/98
Line 285: Line 128:


Mapping generators: ~2, ~3
Mapping generators: ~2, ~3
POTE generator: ~3/2 = 696.967


Minimax tuning:  
Minimax tuning:  
Line 296: Line 141:
* strict range: [696.774, 700.000]
* strict range: [696.774, 700.000]


[[Algebraic generator]]: Traverse, the positive real root of ''x''<sup>4</sup> + 2''x'' - 13, or 696.9529 cents.
Algebraic generator: Traverse, the positive real root of ''x''<sup>4</sup> + 2''x'' - 13, or 696.9529 cents.


{{Val list|legend=1| 12, 19e, 31, 105, 136b, 167be, 198be }}
{{Val list|legend=1| 12, 19e, 31, 105, 136b, 167be, 198be }}
Line 302: Line 147:
Badness: 0.0170
Badness: 0.0170


</div></div>
; Music
 
* [http://micro.soonlabel.com/gene_ward_smith/Others/Meneghin/Claudi-Meneghin-Twinkle-canon-74-edo.mp3 Twinkle canon – 74 edo] by [http://soonlabel.com/xenharmonic/archives/573 Claudi Meneghin]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Meneghin/Claudi-Meneghin-Twinkle-canon-74-edo.mp3 Twinkle canon – 74 edo] by [http://soonlabel.com/xenharmonic/archives/573 Claudi Meneghin]


=== Tridecimal meantone ===
=== Tridecimal meantone ===
Period: 1\1
Optimal ([[POTE]]) generator: ~3/2 = 696.642
EDO generators: [[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">


Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13
Line 327: Line 159:


Mapping generators: ~2, ~3
Mapping generators: ~2, ~3
POTE generator: ~3/2 = 696.642


{{Val list|legend=1| 12f, 19e, 31 }}
{{Val list|legend=1| 12f, 19e, 31 }}


Badness: 0.0180
Badness: 0.0180
</div></div>


=== Grosstone ===
=== Grosstone ===
Period: 1\1
Optimal ([[POTE]]) generator: ~3/2 = 697.264
EDO generators: [[31edo|18\31]], [[43edo|25\43]]
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">


Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13
Line 355: Line 175:


Mapping generators: ~2, ~3
Mapping generators: ~2, ~3
POTE generator: ~3/2 = 697.264


Tuning ranges:  
Tuning ranges:  
Line 362: Line 184:


{{Val list|legend=1| 12, 19ef, 31, 43, 74 }}
{{Val list|legend=1| 12, 19ef, 31, 43, 74 }}
Badness: 0.0259
</div></div>


=== Meridetone ===
=== Meridetone ===
Period: 1\1
Optimal ([[POTE]]) generator: ~3/2 = 697.529
EDO generators: [[43edo|25\43]]
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">


Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13
Line 388: Line 194:


Mapping generators: ~2, ~3
Mapping generators: ~2, ~3
POTE generator: ~3/2 = 697.529


{{Val list|legend=1| 12f, 31f, 43 }}
{{Val list|legend=1| 12f, 31f, 43 }}


Badness: 0.0264
Badness: 0.0264
</div></div>


=== Hemimeantone ===
=== Hemimeantone ===


Period: 1\1
Subgroup: 2.3.5.7.11.13
 
Optimal ([[POTE]]) generator: ~15/13 = 250.304
 
EDO generators: [[43edo|9\43]], [[62edo|13\62]]
 
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">
 
Subgroup: 2.3.5.7.11.13


Comma list: 81/80, 99/98, 126/125, 169/168
Comma list: 81/80, 99/98, 126/125, 169/168
Line 416: Line 210:


Mapping generators: ~2, ~26/15
Mapping generators: ~2, ~26/15
POTE generator: ~15/13 = 250.304


{{Val list|legend=1| 19e, 43, 62, 167bef }}
{{Val list|legend=1| 19e, 43, 62, 167bef }}


Badness: 0.0314
Badness: 0.0314
</div></div>


== Meanpop ==
== Meanpop ==
{{see also| Meantone vs meanpop }}
{{see also| Meantone vs meanpop }}
Period: 1\1
Optimal ([[POTE]]) generator: ~3/2 = 696.434
EDO generators: [[19edo|11\19]], [[31edo|18\31]], [[50edo|29\50]]
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">


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 445: Line 227:


Mapping generator: ~2, ~3
Mapping generator: ~2, ~3
POTE generator: ~3/2 = 696.434


Minimax tuning:  
Minimax tuning:  
Line 462: Line 246:
[[Badness]]: 0.0215
[[Badness]]: 0.0215


</div></div>
; Music
 
* [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}}
* [http://micro.soonlabel.com/gene_ward_smith/Others/Meneghin/Claudi-Meneghin-Twinkle-canon-50-edo.mp3 Twinkle canon – 50 edo] by [http://soonlabel.com/xenharmonic/archives/573 Claudi Meneghin]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Meneghin/Claudi-Meneghin-Twinkle-canon-50-edo.mp3 Twinkle canon – 50 edo] by [http://soonlabel.com/xenharmonic/archives/573 Claudi Meneghin]


=== 13-limit Meanpop ===
=== 13-limit Meanpop ===
Period: 1\1
Optimal ([[POTE]]) generator: ~3/2 = 696.211
EDO generators: [[19edo|11\19]], [[31edo|18\31]], [[50edo|29\50]]
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">


Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13
Line 488: Line 259:


Mapping generator: ~2, ~3
Mapping generator: ~2, ~3
POTE generator: ~3/2 = 696.211


Tuning ranges:  
Tuning ranges:  
Line 497: Line 270:


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


=== Meanplop ===
=== Meanplop ===
Period: 1\1
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">


Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13
Line 521: Line 280:


Mapping generators: ~2, ~3
Mapping generators: ~2, ~3
POTE generator: ~3/2 = 696.202


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


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


== Meanenneadecal ==
== Meanenneadecal ==


Period: 1\1
Subgroup: 2.3.5.7.11


Optimal ([[POTE]]) generator: ~3/2 = 696.250
Comma list: 45/44, 56/55, 81/80
 
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">
 
Subgroup: 2.3.5.7.11
 
Comma list: 45/44, 56/55, 81/80


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


Mapping generators: ~2, ~3
Mapping generators: ~2, ~3
POTE generator: ~3/2 = 696.250


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


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


=== 13-limit ===
=== 13-limit ===
Period: 1\1
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">


Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13
Line 577: Line 312:


Mapping generators: ~2, ~3
Mapping generators: ~2, ~3
POTE generator: ~3/2 = 696.146


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


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


=== Vincenzo ===
=== Vincenzo ===
Period: 1\1
Optimal ([[POTE]]) generator: ~3/2 = 695.060
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">


Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13
Line 605: Line 328:


Mapping generators: ~2, ~3
Mapping generators: ~2, ~3
POTE generator: ~3/2 = 695.060


{{Val list|legend=1| 7d, 12, 19 }}
{{Val list|legend=1| 7d, 12, 19 }}


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


==== 17-limit ====
==== 17-limit ====
Period: 1\1
Optimal ([[POTE]]) generator: ~3/2 = 695.858
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">


Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17
Line 633: Line 344:


Mapping generators: ~2, ~3
Mapping generators: ~2, ~3
POTE generator: ~3/2 = 695.858


{{Val list|legend=1| 7d, 12, 19 }}
{{Val list|legend=1| 7d, 12, 19 }}


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


==== 19-limit ====
==== 19-limit ====


Period: 1\1
Subgroup: 2.3.5.7.11.13.17.19


Optimal ([[POTE]]) generator: ~3/2 = 696.131
Comma list: 39/38, 45/44, 52/51, 56/55, 65/64, 81/80
 
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">
 
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 39/38, 45/44, 52/51, 56/55, 65/64, 81/80


Mapping: [{{val| 1 0 -4 -13 -6 10 12 9 }}, {{val| 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
Mapping generators: ~2, ~3
POTE generator: ~3/2 = 696.131


{{Val list|legend=1| 7d, 12, 19 }}
{{Val list|legend=1| 7d, 12, 19 }}


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


==== 23-limit ====
==== 23-limit ====
Period: 1\1
Optimal ([[POTE]]) generator: ~3/2 = 696.044
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">


Subgroup: 2.3.5.7.11.13.17.19.23
Subgroup: 2.3.5.7.11.13.17.19.23
Line 689: Line 376:


Mapping generators: ~2, ~3
Mapping generators: ~2, ~3
POTE generator: ~3/2 = 696.044


{{Val list|legend=1| 7d, 12, 19 }}
{{Val list|legend=1| 7d, 12, 19 }}


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


==== 29-limit ====
==== 29-limit ====
Period: 1\1
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">


Subgroup: 2.3.5.7.11.13.17.19.23.29
Subgroup: 2.3.5.7.11.13.17.19.23.29
Line 717: Line 392:


Mapping generators: ~2, ~3
Mapping generators: ~2, ~3
POTE generator: ~3/2 = 695.913


{{Val list|legend=1| 7d, 12, 19 }}
{{Val list|legend=1| 7d, 12, 19 }}


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


==== 31-limit ====
==== 31-limit ====
Period: 1\1
Optimal ([[POTE]]) generator: ~3/2 = 695.750
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">


Subgroup: 2.3.5.7.11.13.17.19.23.29.31
Subgroup: 2.3.5.7.11.13.17.19.23.29.31
Line 745: Line 408:


Mapping generators: ~2, ~3
Mapping generators: ~2, ~3
POTE generator: ~3/2 = 695.750


{{Val list|legend=1| 7d, 12, 19 }}
{{Val list|legend=1| 7d, 12, 19 }}


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


==== 37-limit ====
==== 37-limit ====


Period: 1\1
Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37


Optimal ([[POTE]]) generator: ~3/2 = 695.603
Comma list: 39/38, 45/44, 52/51, 56/55, 58/57, 65/64, 69/68, 75/74, 81/80, 93/92
 
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">
 
Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37
 
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 }}]
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 }}]


Mapping generators: ~2, ~3
Mapping generators: ~2, ~3
POTE generator: ~3/2 = 695.603


{{Val list|legend=1| 7d, 12, 19 }}
{{Val list|legend=1| 7d, 12, 19 }}


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


==== 41-limit ====
==== 41-limit ====
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">


Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37.41
Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37.41
Line 801: Line 440:


Mapping generators: ~2, ~3
Mapping generators: ~2, ~3
POTE generator: ~3/2 = 695.696


{{Val list|legend=1| 7d, 12, 19 }}
{{Val list|legend=1| 7d, 12, 19 }}


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


==== 43-limit ====
==== 43-limit ====
Period: 1\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">


Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37.41.43
Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37.41.43
Line 829: Line 456:


Mapping generators: ~2, ~3
Mapping generators: ~2, ~3
POTE generator: ~3/2 = 695.688


{{Val list|legend=1| 7d, 12, 19 }}
{{Val list|legend=1| 7d, 12, 19 }}


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


==== 47-limit ====
==== 47-limit ====
Period: 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">


Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37.41.43.47
Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37.41.43.47
Line 857: Line 472:


Mapping generators: ~2, ~3
Mapping generators: ~2, ~3
POTE generator: ~3/2 = 695.676


{{Val list|legend=1| 7d, 12, 19 }}
{{Val list|legend=1| 7d, 12, 19 }}


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


=== Meanundec ===
=== Meanundec ===


Period: 1\1
Subgroup: 2.3.5.7.11.13


Optimal ([[POTE]]) generator: ~3/2 = 697.254
Comma list: 27/26, 40/39, 45/44, 56/55
 
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">
 
Subgroup: 2.3.5.7.11.13
 
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 }}]
Mapping: [{{val| 1 0 -4 -13 -6 -1 }}, {{val| 0 1 4 10 6 3 }}]


Mapping generators: ~2, ~3
Mapping generators: ~2, ~3
POTE generator: ~3/2 = 697.254


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


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


== Meanundeci ==
== Meanundeci ==
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">


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 910: Line 501:
Comma list: 33/32, 55/54, 77/75
Comma list: 33/32, 55/54, 77/75


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


Mapping: [{{val|1 0 -4 -13 5 }}, {{val| 0 1 4 10 -1 }}]
Mapping generators: ~2, ~3


Mapping generator: ~3
POTE generator: ~3/2 = 694.689


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


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


=== 13-limit ===
=== 13-limit ===
Period: 1\1
Optimal ([[POTE]]) generator: ~3/2 = 694.764
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">


Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13
Line 943: Line 520:


Mapping generators: ~2, ~3
Mapping generators: ~2, ~3
POTE generator: ~3/2 = 694.764


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


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


= Flattone =
= Flattone =
In flattone, 9 generator steps of 4/3 get to the interval class for 7, meaning that [[7/4]] is a diminished seventh interval (C-Bbb). Other intervals are [[7/6]], a diminished third (C-Ebb), and [[7/5]], a doubly diminshed fifth (C-Gbb). Good tunings for flattone are [[26edo]], [[45edo]] and [[64edo]].
In flattone, 9 generator steps of 4/3 get to the interval class for 7, meaning that [[7/4]] is a diminished seventh interval (C-Bbb). Other intervals are [[7/6]], a diminished third (C-Ebb), and [[7/5]], a doubly diminshed fifth (C-Gbb). Good tunings for flattone are [[26edo]], [[45edo]] and [[64edo]].


Period: 1\1
Subgroup: 2.3.5.7


Optimal ([[POTE]]) generator: ~3/2 = 693.779
[[Comma list]]: 81/80, 525/512


EDO generators: [[19edo|11\19]], [[26edo|15\26]], [[45edo|26\45]], [[64edo|37\64]]
[[Mapping]]: [{{val|1 0 -4 17}}, {{val|0 1 4 -9}}]


Scales (Scala files): [[Flattone12]]
Mapping generators: ~2, ~3


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
{{Multival|legend=1| 1 4 -9 4 -17 -32 }}
<div style="line-height:1.6;">Interval table (12-note MOS, 2.3.5.7 POTE tuning)</div>
<div class="mw-collapsible-content">


{| class="wikitable right-1 right-2 sortable"
[[POTE generator]]: ~3/2 = 693.779
|+
|-
! #
! Cents<ref>octave-reduced</ref>
! class="unsortable"| Approximate ratios<ref>2.3.5.7, odd limit ≤ 27. JI readings in parentheses are outside the subgroup but are supported by the defining EDOs.</ref>
|-
| 0
| 0.00
| 1/1
|-
| 1
| 693.8
| 3/2
|-
| 2
| 187.6
| 9/8, 10/9
|-
| 3
| 881.3
| 5/3
|-
| 4
| 375.1
| 5/4, (16/13), (11/9)
|-
| 5
| 1068.9
| 15/8, (24/13), (11/6)
|-
| 6
| 562.7
| (18/13), (11/8)
|-
| 7
| 56.5
|
|-
| 8
| 750.2
| (20/13)
|-
| 9
| 244.0
| 8/7
|-
| 10
| 937.8
| 12/7
|-
| 11
| 431.6
| 9/7
|}
<references/></div></div>


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
[[Minimax tuning]]:  
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
 
Subgroup: 2.3.5.7
 
[[Comma list]]: 81/80, 525/512
 
[[Mapping]]: [{{val|1 0 -4 17}}, {{val|0 1 4 -9}}]
 
Mapping generators: ~2, ~3
 
[[Wedgie]]: {{wedgie|1 4 -9 4 -17 -32}}
 
[[Minimax tuning]]:  
* [[7-odd-limit]]
* [[7-odd-limit]]
: [{{Monzo| 1 0 0 0 }}, {{Monzo| 21/13 0 1/13 -1/13 }}, {{Monzo| 32/13 0 4/13 -4/13 }}, {{Monzo| 32/13 0 -9/13 9/13 }}]
: [{{Monzo| 1 0 0 0 }}, {{Monzo| 21/13 0 1/13 -1/13 }}, {{Monzo| 32/13 0 4/13 -4/13 }}, {{Monzo| 32/13 0 -9/13 9/13 }}]
Line 1,055: Line 561:
[[Badness]]: 0.0386
[[Badness]]: 0.0386


</div></div>
Scales: [[flattone12]]


== 11-limit ==
== 11-limit ==
Period: 1\1
Optimal ([[POTE]]) generator: ~3/2 = 693.126
EDO generators: [[19edo|11\19]], [[26edo|15\26]], [[45edo|26\45]], [[64edo|37\64]]
Scales (Scala files): [[Flattone12]]
<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">


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 1,078: Line 572:


Mapping generators: ~2, ~3
Mapping generators: ~2, ~3
POTE generator: ~3/2 = 693.126


Tuning ranges:  
Tuning ranges:  
Line 1,088: Line 584:
Badness: 0.0338
Badness: 0.0338


</div></div>
Scales: [[flattone12]]


== 13-limit ==
== 13-limit ==
Period: 1\1
Optimal ([[POTE]]) generator: ~3/2 = 693.058
EDO generators: [[19edo|11\19]], [[26edo|15\26]], [[45edo|26\45]], [[64edo|37\64]]
Scales (Scala files): [[Flattone12]]
<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">


Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13
Line 1,111: Line 595:


Mapping generators: ~2, ~3
Mapping generators: ~2, ~3
POTE generator: ~3/2 = 693.058


Tuning ranges:  
Tuning ranges:  
Line 1,121: Line 607:
Badness: 0.0223
Badness: 0.0223


</div></div>
Scales: [[flattone12]]


= Godzilla =
= Godzilla =
Line 1,129: Line 615:


Godzilla tempers out 49/48, equating 8/7 with 7/6. Two of the step-and-a-quarter intervals these represent give a fourth, and so step-and-a-quarter generators generate godzilla. [[19edo]] is close to being the optimal generator tuning; hence it can be more or less equated with taking 4\19 as a generator. MOS are of 5, 9, or 14 notes.
Godzilla tempers out 49/48, equating 8/7 with 7/6. Two of the step-and-a-quarter intervals these represent give a fourth, and so step-and-a-quarter generators generate godzilla. [[19edo]] is close to being the optimal generator tuning; hence it can be more or less equated with taking 4\19 as a generator. MOS are of 5, 9, or 14 notes.
Period: 1\1
Optimal ([[POTE]]) generator: ~8/7 = 252.635
EDO generators: [[14edo|3\14]], [[19edo|4\19]], [[24edo|5\24]], [[33edo|7\33]], [[43edo|9\43]]
Scales (Scala files):
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Interval table (9-note MOS, 2.3.5.7 POTE tuning)</div>
<div class="mw-collapsible-content">
{| class="wikitable right-1 right-2 sortable"
|+
|-
! #
! Cents <ref>octave-reduced</ref>
! class="unsortable"| Approximate ratios<ref>2.3.5.7, odd limit ≤ 27. JI readings in parentheses are outside the subgroup but are supported by the defining EDOs.</ref>
|-
| 0
| 0.00
| 1/1
|-
| 1
| 252.6
| 7/6, 8/7, (15/13)
|-
| 2
| 505.3
| 4/3
|-
| 3
| 757.9
| 14/9, (20/13)
|-
| 4
| 1010.5
| 9/5, 16/9
|-
| 5
| 63.2
| 21/20, 28/27
|-
| 6
| 315.8
| 6/5
|-
| 7
| 568.4
| 7/5, (18/13)
|-
| 8
| 821.1
| 8/5
|}
<references/></div></div>
<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">


Subgroup: 2.3.5.7
Subgroup: 2.3.5.7
Line 1,199: Line 624:
Mapping generators: ~2, ~7/4
Mapping generators: ~2, ~7/4


[[Wedgie]]: {{wedgie| 2 8 1 8 -4 -20 }}
{{Multival|legend=1| 2 8 1 8 -4 -20 }}
 
[[POTE generator]]: ~8/7 = 252.635


[[Tuning ranges]]:  
[[Tuning ranges]]:  
Line 1,209: Line 636:


[[Badness]]: 0.0267
[[Badness]]: 0.0267
</div></div>


== 11-limit ==
== 11-limit ==
Period: 1\1
Optimal ([[POTE]]) generator: ~8/7 = 254.027
EDO generators: [[14edo|3\14]], [[19edo|4\19]], [[33edo|7\33]]
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">


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 1,233: Line 646:


Mapping generators: ~2, ~7/4
Mapping generators: ~2, ~7/4
POTE generator: ~8/7 = 254.027


Tuning ranges:  
Tuning ranges:  
Line 1,242: Line 657:


Badness: 0.0290
Badness: 0.0290
</div></div>


=== 13-limit ===
=== 13-limit ===
Period: 1\1
Optimal ([[POTE]]) generator: ~8/7 = 253.603
EDO generators: [[19edo|4\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">


Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13
Line 1,266: Line 667:


Mapping generators: ~2, ~7/4
Mapping generators: ~2, ~7/4
POTE generator: ~8/7 = 253.603


Tuning ranges:  
Tuning ranges:  
Line 1,275: Line 678:


Badness: 0.0225
Badness: 0.0225
</div></div>


== Semafour ==
== Semafour ==
Period: 1\1
Optimal ([[POTE]]) generator: ~8/7 = 254.042
EDO generators: [[14edo|3\14]], [[19edo|4\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">


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 1,299: Line 688:


Mapping generators: ~2, ~7/4
Mapping generators: ~2, ~7/4
POTE generator: ~8/7 = 254.042


{{Val list|legend=1| 14c, 19e, 33cdee }}
{{Val list|legend=1| 14c, 19e, 33cdee }}


Badness: 0.0285
Badness: 0.0285
</div></div>


== Varan ==
== Varan ==
Period: 1\1
Optimal ([[POTE]]) generator: ~8/7 = 251.079
EDO generators: [[19edo|4\19]], [[24edo|5\24]], [[43edo|9\43]]
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">


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 1,327: Line 704:


Mapping generators: ~2, ~7/4
Mapping generators: ~2, ~7/4
POTE generator: ~8/7 = 251.079


{{Val list|legend=1| 19e, 24, 43de }}
{{Val list|legend=1| 19e, 24, 43de }}


Badness: 0.0396
Badness: 0.0396
</div></div>


=== 13-limit ===
=== 13-limit ===


Period: 1\1
Subgroup: 2.3.5.7.11.13


Optimal ([[POTE]]) generator: ~8/7 = 251.165
Comma list: 49/48, 66/65, 77/75, 81/80


EDO generators: [[19edo|4\19]], [[24edo|5\24]], [[43edo|9\43]]
Mapping: [{{val| 1 0 -4 2 -10 -5 }}, {{val| 0 2 8 1 17 11 }}]


Scales (Scala files):
Mapping generators: ~2, ~7/4


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
POTE generator: ~8/7 = 251.165
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
 
Subgroup: 2.3.5.7.11.13
 
Comma list: 49/48, 66/65, 77/75, 81/80
 
Mapping: [{{val| 1 0 -4 2 -10 -5 }}, {{val| 0 2 8 1 17 11 }}]
 
Mapping generators: ~2, ~7/4


{{Val list|legend=1| 19e, 24, 43de }}
{{Val list|legend=1| 19e, 24, 43de }}


Badness: 0.0257
Badness: 0.0257
</div></div>


== Baragon ==
== Baragon ==
Period: 1\1
Optimal ([[POTE]]) generator: ~8/7 = 251.173
EDO generators: [[19edo|4\19]], [[24edo|5\24]], [[43edo|9\43]]
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">


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 1,383: Line 736:


Mapping generators: ~2, ~7/4
Mapping generators: ~2, ~7/4
POTE generator: ~8/7 = 251.173


{{Val list|legend=1| 19, 24, 43d }}
{{Val list|legend=1| 19, 24, 43d }}


Badness: 0.0357
Badness: 0.0357
</div></div>


= Dominant =
= Dominant =


The interval class for 7 is obtained from two fourths in succession, so that 7/4 is a minor seventh. The 7/6 interval is, like 6/5, now a minor third, and 7/5 is a diminished fifth. An excellent tuning for dominant is [[12edo]], but it also works well with the Pythagorean tuning of pure [[3/2]] fifths, and with [[29edo]], [[41edo]], or [[53edo]].
The interval class for 7 is obtained from two fourths in succession, so that 7/4 is a minor seventh. The 7/6 interval is, like 6/5, now a minor third, and 7/5 is a diminished fifth. An excellent tuning for dominant is [[12edo]], but it also works well with the Pythagorean tuning of pure [[3/2]] fifths, and with [[29edo]], [[41edo]], or [[53edo]].
Period: 1\1
Optimal ([[POTE]]) generator: ~3/2 = 701.573
EDO generators: [[5edo|3\5]], [[7edo|4\7]], [[12edo|7\12]], [[17edo|10\17]]
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">


Subgroup: 2.3.5.7
Subgroup: 2.3.5.7
Line 1,410: Line 751:
[[Comma list]]: 36/35, 64/63
[[Comma list]]: 36/35, 64/63


Mapping: [{{val| 1 0 -4 6 }}, {{val| 0 1 4 -2 }}]
[[Mapping]]: [{{val| 1 0 -4 6 }}, {{val| 0 1 4 -2 }}]


Mapping generators: ~2, ~3
Mapping generators: ~2, ~3


[[Wedgie]]: {{wedgie| 1 4 -2 4 -6 -16 }}
{{Multival|legend=1| 1 4 -2 4 -6 -16 }}
 
[[POTE generator]]: ~3/2 = 701.573


[[Tuning ranges]]:  
[[Tuning ranges]]:  
Line 1,424: Line 767:


[[Badness]]: 0.0207
[[Badness]]: 0.0207
</div></div>


== 11-limit ==
== 11-limit ==
Period: 1\1
Optimal ([[POTE]]) generator: ~3/2 = 703.254
EDO generators: [[12edo|7\12]], [[17edo|10\17]]
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">


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 1,453: Line 782:
* nice range: [691.202, 715.587]
* nice range: [691.202, 715.587]
* strict range: [700.000, 705.882]
* strict range: [700.000, 705.882]
POTE generator: ~3/2 = 703.254


{{Val list|legend=1| 5, 12, 17c, 29cde }}
{{Val list|legend=1| 5, 12, 17c, 29cde }}


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


=== 13-limit ===
=== 13-limit ===


Period: 1\1
Subgroup: 2.3.5.7.11.13


Optimal ([[POTE]]) generator: ~3/2 = 703.636
Comma list: 36/35, 56/55, 64/63, 66/65
 
EDO generators: [[12edo|7\12]], [[17edo|10\17]]
 
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">
 
Subgroup: 2.3.5.7.11.13
 
Comma list: 36/35, 56/55, 64/63, 66/65


Mapping: [{{val| 1 0 -4 6 13 18 }}, {{val| 0 1 4 -2 -6 -9 }}]
Mapping: [{{val| 1 0 -4 6 13 18 }}, {{val| 0 1 4 -2 -6 -9 }}]


Mapping generators: ~2, ~3
Mapping generators: ~2, ~3
POTE generator: ~3/2 = 703.636


Tuning ranges:  
Tuning ranges:  
Line 1,490: Line 809:


Badness: 0.0241
Badness: 0.0241
</div></div>


=== Dominion ===
=== Dominion ===
Period: 1\1
Optimal ([[POTE]]) generator: ~3/2 = 704.905
EDO generators: [[17edo|10\17]]
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">


Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13
Line 1,514: Line 819:


{{Val list|legend=1| 5, 12, 17c, 46cde }}
{{Val list|legend=1| 5, 12, 17c, 46cde }}
POTE generator: ~3/2 = 704.905


Badness: 0.0273
Badness: 0.0273
</div></div>


== Domineering ==
== Domineering ==
Period: 1\1
Optimal ([[POTE]]) generator: ~3/2 = 698.776
EDO generators: [[7edo|4\7]], [[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">


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 1,540: Line 833:


Mapping generators: ~2, ~3
Mapping generators: ~2, ~3
POTE generator: ~3/2 = 698.776


{{Val list|legend=1| 5e, 7, 12, 19d, 43de }}
{{Val list|legend=1| 5e, 7, 12, 19d, 43de }}


Badness: 0.0220
Badness: 0.0220
</div></div>


=== 13-limit ===
=== 13-limit ===
Period: 1\1
Optimal ([[POTE]]) generator: ~3/2 = 695.762
EDO generators: [[7edo|4\7]], [[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">


Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13
Line 1,568: Line 849:


Mapping generators: ~2, ~3
Mapping generators: ~2, ~3
POTE generator: ~3/2 = 695.762


{{Val list|legend=1| 5ef, 7, 12, 19d, 31def }}
{{Val list|legend=1| 5ef, 7, 12, 19d, 31def }}


Badness: 0.0270
Badness: 0.0270
</div></div>


==== 17-limit ====
==== 17-limit ====


Period: 1\1
Subgroup: 2.3.5.7.11.13.17


Optimal ([[POTE]]) generator: ~3/2 = 696.115
Comma list: 36/35, 45/44, 51/49, 52/49, 64/63
 
EDO generators: [[7edo|4\7]], [[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">
 
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 36/35, 45/44, 51/49, 52/49, 64/63


Mapping: [{{val| 1 0 -4 6 -6 10 12 }}, {{val| 0 1 4 -2 6 -4 -5 }}]
Mapping: [{{val| 1 0 -4 6 -6 10 12 }}, {{val| 0 1 4 -2 6 -4 -5 }}]


Mapping generators: ~2, ~3
Mapping generators: ~2, ~3
POTE generator: ~3/2 = 696.115


{{Val list|legend=1| 5ef, 7, 12, 19d, 31def }}
{{Val list|legend=1| 5ef, 7, 12, 19d, 31def }}


Badness: 0.0245
Badness: 0.0245
</div></div>


==== 19-limit ====
==== 19-limit ====
Period: 1\1
Optimal ([[POTE]]) generator: ~3/2 = 696.217
EDO generators: [[7edo|4\7]], [[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">


Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13.17.19
Line 1,624: Line 881:


Mapping generators: ~2, ~3
Mapping generators: ~2, ~3
POTE generator: ~3/2 = 696.217


{{Val list|legend=1| 5ef, 7, 12, 19d, 31def }}
{{Val list|legend=1| 5ef, 7, 12, 19d, 31def }}


Badness: 0.0204
Badness: 0.0204
</div></div>


=== Dominatrix ===
=== Dominatrix ===
Period: 1\1
Optimal ([[POTE]]) generator: ~3/2 = 698.544
EDO generators: [[7edo|4\7]], [[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">


Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13
Line 1,652: Line 897:


Mapping generators: ~2, ~3
Mapping generators: ~2, ~3
POTE generator: ~3/2 = 698.544


{{Val list|legend=1| 5e, 7, 12f, 19df }}
{{Val list|legend=1| 5e, 7, 12f, 19df }}
</div></div>


== Domination ==
== Domination ==
Period: 1\1
Optimal ([[POTE]]) generator: ~3/2 = 705.004
EDO generators: [[12edo|7\12]], [[17edo|10\17]]
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">


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 1,678: Line 911:


Mapping generators: ~2, ~3
Mapping generators: ~2, ~3
POTE generator: ~3/2 = 705.004


{{Val list|legend=1| 5e, 12e, 17c, 46cd }}
{{Val list|legend=1| 5e, 12e, 17c, 46cd }}


Badness: 0.0366
Badness: 0.0366
</div></div>


=== 13-limit ===
=== 13-limit ===


Period: 1\1
Subgroup: 2.3.5.7.11.13


Optimal ([[POTE]]) generator: ~3/2 = 705.496
Comma list: 26/25, 36/35, 64/63, 66/65
 
EDO generators: [[12edo|7\12]], [[17edo|10\17]]
 
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">
 
Subgroup: 2.3.5.7.11.13
 
Comma list: 26/25, 36/35, 64/63, 66/65


Mapping: [{{val| 1 0 -4 6 -14 -9 }}, {{val| 0 1 4 -2 11 8 }}]
Mapping: [{{val| 1 0 -4 6 -14 -9 }}, {{val| 0 1 4 -2 11 8 }}]


Mapping generators: ~2, ~3
Mapping generators: ~2, ~3
POTE generator: ~3/2 = 705.496


{{Val list|legend=1| 5e, 12e, 17c }}
{{Val list|legend=1| 5e, 12e, 17c }}


Badness: 0.0274
Badness: 0.0274
</div></div>


== Arnold ==
== Arnold ==
Period: 1\1
Optimal ([[POTE]]) generator: ~3/2 = 698.491
EDO generators: [[5edo|3\5]], [[7edo|4\7]], [[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">


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 1,734: Line 943:


Mapping generators: ~2, ~3
Mapping generators: ~2, ~3
POTE generator: ~3/2 = 698.491


{{Val list|legend=1| 5, 7, 12e }}
{{Val list|legend=1| 5, 7, 12e }}


Badness: 0.0261
Badness: 0.0261
</div></div>


=== 13-limit ===
=== 13-limit ===
Period: 1\1
Optimal ([[POTE]]) generator: ~3/2 = 696.743
EDO generators: [[5edo|3\5]], [[7edo|4\7]], [[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">


Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13
Line 1,762: Line 959:


Mapping generators: ~2, ~3
Mapping generators: ~2, ~3
POTE generator: ~3/2 = 696.743


{{Val list|legend=1| 5, 7, 12ef, 19def }}
{{Val list|legend=1| 5, 7, 12ef, 19def }}


Badness: 0.0233
Badness: 0.0233
</div></div>


=== 17-limit ===
=== 17-limit ===
Period: 1\1
Optimal ([[POTE]]) generator: ~3/2 = 696.978
EDO generators: [[5edo|3\5]], [[7edo|4\7]], [[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">


Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17
Line 1,790: Line 975:


Mapping generators: ~2, ~3
Mapping generators: ~2, ~3
POTE generator: ~3/2 = 696.978


{{Val list|legend=1| 5, 7, 12ef, 19def }}
{{Val list|legend=1| 5, 7, 12ef, 19def }}


Badness: 0.0245
Badness: 0.0245
</div></div>


=== 19-limit ===
=== 19-limit ===


Period: 1\1
Subgroup: 2.3.5.7.11.13.17.19


Optimal ([[POTE]]) generator: ~3/2 = 697.068
Comma list: 22/21, 27/26, 33/32, 36/35, 51/49, 57/56
 
EDO generators: [[5edo|3\5]], [[7edo|4\7]], [[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">
 
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 22/21, 27/26, 33/32, 36/35, 51/49, 57/56


Mapping: [{{val| 1 0 -4 6 5 -1 12 9 }}, {{val| 0 1 4 -2 3 -5 -3 }}]
Mapping: [{{val| 1 0 -4 6 5 -1 12 9 }}, {{val| 0 1 4 -2 3 -5 -3 }}]


Mapping generators: ~2, ~3
Mapping generators: ~2, ~3
POTE generator: ~3/2 = 697.068


{{Val list|legend=1| 5, 7, 12ef, 19def }}
{{Val list|legend=1| 5, 7, 12ef, 19def }}


Badness: 0.0211
Badness: 0.0211
</div></div>


= Sharptone =
= Sharptone =


Sharptone is a low-accuracy temperament tempering out 21/20 and 28/27. In sharptone, a 7/4 is a major sixth, a 7/6 a whole tone, and a 7/5 a fourth. Genuinely septimal sounding harmony therefore cannot be expected, but it can be used to translate, more or less, 7-limit JI into 5-limit meantone. [[12edo]] tuning does sharptone about as well as such a thing can be done, of course not in its patent val.
Sharptone is a low-accuracy temperament tempering out 21/20 and 28/27. In sharptone, a 7/4 is a major sixth, a 7/6 a whole tone, and a 7/5 a fourth. Genuinely septimal sounding harmony therefore cannot be expected, but it can be used to translate, more or less, 7-limit JI into 5-limit meantone. [[12edo]] tuning does sharptone about as well as such a thing can be done, of course not in its patent val.
Period: 1\1
Optimal ([[POTE]]) generator: ~3/2 = 700.140
EDO generators: [[5edo|3\5]], [[7edo|4\7]], [[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">


Subgroup: 2.3.5.7
Subgroup: 2.3.5.7
Line 1,845: Line 1,006:
[[Comma list]]: 21/20, 28/27
[[Comma list]]: 21/20, 28/27


Mapping: [{{val| 1 0 -4 -2 }}, {{val| 0 1 4 3 }}]
[[Mapping]]: [{{val| 1 0 -4 -2 }}, {{val| 0 1 4 3 }}]


Mapping generators: ~2, ~3
Mapping generators: ~2, ~3


[[Wedgie]]: {{wedgie| 1 4 3 4 2 -4 }}
{{Multival|legend=1| 1 4 3 4 2 -4 }}
 
[[POTE generator]]: ~3/2 = 700.140


{{Val list|legend=1| 5, 7d, 12d }}
{{Val list|legend=1| 5, 7d, 12d }}


[[Badness]]: 0.0248
[[Badness]]: 0.0248
</div></div>


== Meanertone ==
== Meanertone ==
Period: 1\1
Optimal ([[POTE]]) generator: ~3/2 = 696.615
EDO generators: [[5edo|3\5]], [[7edo|4\7]], [[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">


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 1,876: Line 1,025:


Mapping: [{{val| 1 0 -4 -2 5 }}, {{val| 0 1 4 3 -1 }}]
Mapping: [{{val| 1 0 -4 -2 5 }}, {{val| 0 1 4 3 -1 }}]
POTE generator: ~3/2 = 696.615


{{Val list|legend=1| 5, 7d, 12de }}
{{Val list|legend=1| 5, 7d, 12de }}


Badness: 0.0252
Badness: 0.0252
</div></div>


= Plutus =
= Plutus =
Period: 1\1
Optimal ([[POTE]]) generator: ~3/2 = 682.895
EDO generators: [[7edo|4\7]]
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">


Subgroup: 2.3.5.7
Subgroup: 2.3.5.7


Comma list: 15/14, 81/80
[[Comma list]]: 15/14, 81/80


Mapping: [{{val| 1 0 -4 -5 }}, {{val| 0 1 4 5 }}]
[[Mapping]]: [{{val| 1 0 -4 -5 }}, {{val| 0 1 4 5 }}]


Mapping generators: ~2, ~3
Mapping generators: ~2, ~3


Wedgie: {{wedgie| 1 4 5 4 5 0 }}
{{Multival|legend=1| 1 4 5 4 5 0 }}
 
[[POTE generator]]: ~3/2 = 682.895


{{Val list|legend=1| 7, 37bcccdd, 44bccccdd }}
{{Val list|legend=1| 7, 37bcccdd, 44bccccdd }}


Badness: 0.0453
[[Badness]]: 0.0453
 
</div></div>


== 11-limit ==
== 11-limit ==


Period: 1\1
Subgroup: 2.3.5.7.11


Optimal ([[POTE]]) generator: ~3/2 = 685.234
Comma list: 15/14, 22/21, 81/80
 
EDO generators: [[7edo|4\7]]
 
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">
 
Subgroup: 2.3.5.7.11
 
Comma list: 15/14, 22/21, 81/80


Mapping: [{{val| 1 0 -4 -5 -6 }}, {{val| 0 1 4 5 6 }}]
Mapping: [{{val| 1 0 -4 -5 -6 }}, {{val| 0 1 4 5 6 }}]


Mapping generators: ~2, ~3
Mapping generators: ~2, ~3
POTE generator: ~3/2 = 685.234


{{Val list|legend=1| 5de, 7 }}
{{Val list|legend=1| 5de, 7 }}


Badness: 0.0325
Badness: 0.0325
</div></div>


= Supermean =
= Supermean =


Period: 1\1
Subgroup: 2.3.5.7
 
Optimal ([[POTE]]) generator: ~3/2 = 704.889
 
EDO generators: [[12edo|7\12]], [[17edo|10\17]], [[29edo|17\29]]
 
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">


Subgroup: 2.3.5.7
[[Comma list]]: 81/80, 672/625


Comma list: 81/80, 672/625
[[Mapping]]: [{{val| 1 0 -4 -21 }}, {{val| 0 1 4 15 }}]


Mapping: [{{val| 1 0 -4 -21 }}, {{val| 0 1 4 15 }}]
[[POTE generator]]: ~3/2 = 704.889


{{Val list|legend=1| 5d, 12d, 17c, 29c }}
{{Val list|legend=1| 5d, 12d, 17c, 29c }}


Badness: 0.1342
[[Badness]]: 0.1342
 
</div></div>


== 11-limit ==
== 11-limit ==
Period: 1\1
Optimal ([[POTE]]) generator: ~3/2 = 705.096
EDO generators: [[12edo|7\12]], [[17edo|10\17]], [[29edo|17\29]]
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">


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 1,986: Line 1,087:


Mapping: [{{val| 1 0 -4 -21 -14 }}, {{val| 0 1 4 15 11 }}]
Mapping: [{{val| 1 0 -4 -21 -14 }}, {{val| 0 1 4 15 11 }}]
POTE generator: ~3/2 = 705.096


{{Val list|legend=1| 5de, 12de, 17c, 29c }}
{{Val list|legend=1| 5de, 12de, 17c, 29c }}


Badness: 0.0633
Badness: 0.0633
</div></div>


== 13-limit ==
== 13-limit ==
Period: 1\1
Optimal ([[POTE]]) generator: ~3/2 = 705.094
EDO generators: [[12edo|7\12]], [[17edo|10\17]], [[29edo|17\29]]
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">


Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13
Line 2,012: Line 1,101:


Mapping: [{{val| 1 0 -4 -21 -14 -9 }}, {{val| 0 1 4 15 11 8 }}]
Mapping: [{{val| 1 0 -4 -21 -14 -9 }}, {{val| 0 1 4 15 11 8 }}]
POTE generator: ~3/2 = 705.094


{{Val list|legend=1| 5de, 12de, 17c, 29c }}
{{Val list|legend=1| 5de, 12de, 17c, 29c }}
</div></div>


= Injera =
= Injera =
Line 2,023: Line 1,112:
[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_3091.html#3091 Origin of the name]
[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_3091.html#3091 Origin of the name]


Period: 1\2
Subgroup: 2.3.5.7


Optimal ([[POTE]]) generator: ~3/2 = 694.375
[[Comma list]]: 50/49, 81/80


EDO generators: [[12edo|7\12]], [[14edo|8\14]], [[26edo|15\26]], [[38edo|22\38]]
[[Mapping]]: [{{val| 2 0 -8 -7 }}, {{val| 0 1 4 4 }}]


Scales (Scala files):  
Mapping generators: ~7/5, ~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">
 
Subgroup: 2.3.5.7
 
[[Comma list]]: 50/49, 81/80
 
Mapping: [{{val| 2 0 -8 -7 }}, {{val| 0 1 4 4 }}]


Mapping generators: ~7/5, ~3
[[POTE generator]]: ~3/2 = 694.375


[[Tuning ranges]]:  
[[Tuning ranges]]:  
Line 2,048: Line 1,127:
* strict range: [688.957, 700.000]
* strict range: [688.957, 700.000]


[[Wedgie]]: {{wedgie| 2 8 8 8 7 -4 }}
{{Multival|legend=1| 2 8 8 8 7 -4 }}


{{Val list|legend=1| 12, 26, 38, 102bcd, 140bccd, 178bbccdd }}
{{Val list|legend=1| 12, 26, 38, 102bcd, 140bccd, 178bbccdd }}


[[Badness]]: 0.0311
[[Badness]]: 0.0311
</div></div>


; Music
; Music
Line 2,060: Line 1,137:


== 11-limit ==
== 11-limit ==
Period: 1\2
Optimal ([[POTE]]) generator: ~3/2 = 692.840
EDO generators: [[12edo|7\12]], [[14edo|8\14]], [[26edo|15\26]], [[38edo|22\38]]
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">


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 2,080: Line 1,145:


Mapping generators: ~7/5, ~3
Mapping generators: ~7/5, ~3
POTE generator: ~3/2 = 692.840


Tuning ranges:  
Tuning ranges:  
Line 2,089: Line 1,156:


Badness: 0.0231
Badness: 0.0231
</div></div>


=== 13-limit ===
=== 13-limit ===
Period: 1\2
Optimal ([[POTE]]) generator: ~3/2 = 692.673
EDO generators: [[12edo|7\12]], [[14edo|8\14]], [[26edo|15\26]], [[38edo|22\38]]
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">


Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13
Line 2,113: Line 1,166:


Mapping generators: ~7/5, ~3
Mapping generators: ~7/5, ~3
POTE generator: ~3/2 = 692.673


Tuning ranges:  
Tuning ranges:  
Line 2,122: Line 1,177:


Badness: 0.0216
Badness: 0.0216
</div></div>


=== Enjera ===
=== Enjera ===
Period: 1\2
Optimal ([[POTE]]) generator: ~3/2 = 694.121
EDO generators: [[12edo|7\12]], [[14edo|8\14]], [[26edo|15\26]]
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">


Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13
Line 2,146: Line 1,187:


Mapping generators: ~7/5, ~3
Mapping generators: ~7/5, ~3
POTE generator: ~3/2 = 694.121


{{Val list|legend=1| 12f, 14c, 26f, 38eff }}
{{Val list|legend=1| 12f, 14c, 26f, 38eff }}


Badness: 0.0265
Badness: 0.0265
</div></div>


== Injerous ==
== Injerous ==


Period: 1\2
Subgroup: 2.3.5.7.11


Optimal ([[POTE]]) generator: ~3/2 = 690.548
Comma list: 33/32, 50/49, 55/54
 
EDO generators: [[12edo|7\12]], [[14edo|8\14]]
 
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">
 
Subgroup: 2.3.5.7.11
 
Comma list: 33/32, 50/49, 55/54


Mapping: [{{val| 2 0 -8 -7 10 }}, {{val| 0 1 4 4 -1 }}]
Mapping: [{{val| 2 0 -8 -7 10 }}, {{val| 0 1 4 4 -1 }}]


Mapping generators: ~7/5, ~3
Mapping generators: ~7/5, ~3
POTE generator: ~3/2 = 690.548


{{Val list|legend=1| 12e, 14c, 26e, 40cee }}
{{Val list|legend=1| 12e, 14c, 26e, 40cee }}


Badness: 0.0386
Badness: 0.0386
</div></div>


== Lahoh ==
== Lahoh ==
Period: 1\2
Optimal ([[POTE]]) generator: ~3/2 = 699.001
EDO generators: [[12edo|7\12]], [[14edo|8\14]]
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">


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 2,202: Line 1,219:


Mapping generators: ~7/5, ~3
Mapping generators: ~7/5, ~3
POTE generator: ~3/2 = 699.001


{{Val list|legend=1| 12, 14ce }}
{{Val list|legend=1| 12, 14ce }}


Badness: 0.0431
Badness: 0.0431
</div></div>


= Mohaha =
= Mohaha =
Line 2,213: Line 1,230:


Mohaha is the 2.3.5.11 subgroup temperament with a generator of a neutral third, two of which make up a fifth, and which can be taken to represent 11/9. Mohaha can 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 2.3.5.11 subgroup). Within this paradigm, mohaha 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, and that maps four 3/2's to 5/1. It has a 7-note MOS with three larger steps and four smaller ones, going sLsLsLs.
Mohaha is the 2.3.5.11 subgroup temperament with a generator of a neutral third, two of which make up a fifth, and which can be taken to represent 11/9. Mohaha can 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 2.3.5.11 subgroup). Within this paradigm, mohaha 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, and that maps four 3/2's to 5/1. It has a 7-note MOS with three larger steps and four smaller ones, going sLsLsLs.
Period: 1\1
Optimal ([[POTE]]) generator: ~11/9 = 348.0938
EDO generators: [[17edo|5\17]], [[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;">Interval table (10-note MOS, 2.3.5.11 POTE tuning)</div>
<div class="mw-collapsible-content">
{| class="wikitable right-1 right-2 sortable"
|+
|-
! #
! Cents<ref>octave-reduced</ref>
! class="unsortable"| Approximate ratios<ref>2.3.5.11, odd limit ≤ 27. JI readings in parentheses are outside the subgroup but are supported by the defining EDOs.</ref>
|-
| 0
| 0.00
| 1/1
|-
| 1
| 348.1
| 11/9
|-
| 2
| 696.2
| 3/2
|-
| 3
| 1044.3
| 11/6
|-
| 4
| 192.4
| 9/8
|-
| 5
| 540.5
| 11/8, 15/11
|-
| 6
| 888.6
| 5/3
|-
| 7
| 36.7
|
|-
| 8
| 384.8
| 5/4
|-
| 9
| 732.8
| (32/21)
|}
<references/></div></div>
<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">


Subgroup: 2.3.5.11
Subgroup: 2.3.5.11


Comma list: 81/80, 121/120
[[Comma list]]: 81/80, 121/120


Sval mapping: [{{val|1 1 0 2}}, {{val|0 2 8 5}}]
[[Sval]] [[mapping]]: [{{val|1 1 0 2}}, {{val|0 2 8 5}}]


Sval mapping generators: ~2, ~11/9
Sval mapping generators: ~2, ~11/9
Line 2,289: Line 1,241:
Gencom mapping: [{{val|1 1 0 0 2}}, {{val|0 2 8 0 5}}]
Gencom mapping: [{{val|1 1 0 0 2}}, {{val|0 2 8 0 5}}]


Gencom: [2 11/9; 81/80 121/120]
[[Gencom]]: [2 11/9; 81/80 121/120]
 
[[POTE generator]]: ~11/9 = 348.0938


{{Val list|legend=1| 7, 17c, 24, 31, 100e, 131bee }}
{{Val list|legend=1| 7, 17c, 24, 31, 100e, 131bee }}


Badness: 0.0261
[[Badness]]: 0.0261
</div></div>
 
Scales: [[mohaha7]], [[mohaha10]]


== Mohoho ==
== Mohoho ==
Period: 1\1
Optimal ([[POTE]]) generator: ~11/9 = 348.9155
EDO generators: [[17edo|5\17]], [[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">


Subgroup: 2.3.5.11.13
Subgroup: 2.3.5.11.13
Line 2,321: Line 1,264:


Gencom: [2 11/9; 66/65 81/80 121/120]
Gencom: [2 11/9; 66/65 81/80 121/120]
POTE generator: ~11/9 = 348.9155


{{Val list|legend=1| 7, 17c, 24, 31, 55, 86ef, 141ceff }}
{{Val list|legend=1| 7, 17c, 24, 31, 55, 86ef, 141ceff }}


Badness: 0.0261
Badness: 0.0261
</div></div>
 
Scales: [[mohaha7]], [[mohaha10]]


= Mohajira =
= Mohajira =
Line 2,334: Line 1,280:


== 7-limit ==
== 7-limit ==
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">


Subgroup: 2.3.5.7
Subgroup: 2.3.5.7
Line 2,354: Line 1,289:
Mapping generators: ~2, ~128/105
Mapping generators: ~2, ~128/105


[[Wedgie]]: {{wedgie| 2 8 -11 8 -23 -48 }}
{{Multival|legend=1| 2 8 -11 8 -23 -48 }}
 
[[POTE generator]]: ~128/105 = 348.415


[[Minimax tuning]]:  
[[Minimax tuning]]:  
Line 2,361: Line 1,298:
: [[Eigenmonzo]]s: 2, 5
: [[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.
[[Algebraic generator]]: Mohabis, real root of 3''x''<sup>3</sup> - 3''x''<sup>2</sup> - 1, 348.6067 cents. Corresponding recurrence converges quickly.


{{Val list|legend=1| 7, 24, 31 }}
{{Val list|legend=1| 7, 24, 31 }}
Line 2,367: Line 1,304:
[[Badness]]: 0.0557
[[Badness]]: 0.0557


</div></div>
Scales: [[mohaha7]], [[mohaha10]]


== 11-limit ==
== 11-limit ==
Period: 1\1
Optimal ([[POTE]]) generator: ~11/9 = 348.477
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">


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 2,389: Line 1,315:


Mapping generators: ~2, ~11/9
Mapping generators: ~2, ~11/9
POTE generator: ~11/9 = 348.477


Minimax tuning:  
Minimax tuning:  
Line 2,399: Line 1,327:
Badness: 0.0261
Badness: 0.0261


</div></div>
Scales: [[mohaha7]], [[mohaha10]]


== 13-limit ==
== 13-limit ==
Period: 1\1
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]]
<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">


Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13
Line 2,421: Line 1,338:


Mapping generators: ~2, ~11/9
Mapping generators: ~2, ~11/9
POTE generator: ~11/9 = 348.558


{{Val list|legend=1| 7, 24, 31 }}
{{Val list|legend=1| 7, 24, 31 }}
Line 2,426: Line 1,345:
Badness: 0.0234
Badness: 0.0234


</div></div>
Scales: [[mohaha7]], [[mohaha10]]


== 17-limit ==
== 17-limit ==
Period: 1\1
Optimal ([[POTE]]) generator: ~11/9 = 348.736
EDO generators: [[24edo|7\24]], [[31edo|9\31]], [[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">


Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17
Line 2,448: Line 1,356:


Mapping generators: ~2, ~11/9
Mapping generators: ~2, ~11/9
POTE generator: ~11/9 = 348.736


{{Val list|legend=1| 7, 24, 31, 86ef }}
{{Val list|legend=1| 7, 24, 31, 86ef }}
Line 2,453: Line 1,363:
Badness: 0.0206
Badness: 0.0206


</div></div>
Scales: [[mohaha7]], [[mohaha10]]


== 19-limit ==
== 19-limit ==
Period: 1\1
Optimal ([[POTE]]) generator: ~11/9 = 348.810
EDO generators: [[24edo|7\24]], [[31edo|9\31]], [[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">


Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13.17.19
Line 2,475: Line 1,374:


Mapping generators: ~2, ~11/9
Mapping generators: ~2, ~11/9
POTE generator: ~11/9 = 348.810


{{Val list|legend=1| 7, 24, 31, 55, 86efh }}
{{Val list|legend=1| 7, 24, 31, 55, 86efh }}
Line 2,480: Line 1,381:
Badness: 0.0173
Badness: 0.0173


</div></div>
Scales: [[mohaha7]], [[mohaha10]]


= Mohamaq =
= Mohamaq =
== 7-limit ==
== 7-limit ==
Period: 1\1
Optimal ([[POTE]]) generator: ~25/21 = 350.586
EDO generators: [[17edo|5\17]], [[24edo|7\24]]
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">


Subgroup: 2.3.5.7
Subgroup: 2.3.5.7
Line 2,503: Line 1,393:


Mapping generators: ~2, ~25/21
Mapping generators: ~2, ~25/21
POTE generator: ~25/21 = 350.586


{{Val list|legend=1| 17c, 24, 65c, 89cd }}
{{Val list|legend=1| 17c, 24, 65c, 89cd }}
Line 2,508: Line 1,400:
Badness: 0.0777
Badness: 0.0777


</div></div>
Scales: [[mohaha7]], [[mohaha10]]


== 11-limit ==
== 11-limit ==


Period: 1\1
Subgroup: 2.3.5.7.11
 
Optimal ([[POTE]]) generator: ~11/9 = 350.565
 
EDO generators: [[17edo|5\17]], [[24edo|7\24]]
 
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">
 
Subgroup: 2.3.5.7.11


Comma list: 56/55, 77/75, 243/242
Comma list: 56/55, 77/75, 243/242
Line 2,531: Line 1,411:


Mapping generators: ~2, ~11/9
Mapping generators: ~2, ~11/9
POTE generator: ~11/9 = 350.565


{{Val list|legend=1| 17c, 24, 65c, 89cd }}
{{Val list|legend=1| 17c, 24, 65c, 89cd }}
Line 2,536: Line 1,418:
Badness: 0.0362
Badness: 0.0362


</div></div>
Scales: [[mohaha7]], [[mohaha10]]


== 13-limit ==
== 13-limit ==
Period: 1\1
Optimal ([[POTE]]) generator: ~11/9 = 350.745
EDO generators: [[17edo|5\17]], [[24edo|7\24]]
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">


Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13
Line 2,559: Line 1,429:


Mapping generators: ~2, ~11/9
Mapping generators: ~2, ~11/9
POTE generator: ~11/9 = 350.745


{{Val list|legend=1| 17c, 24, 41c, 65c }}
{{Val list|legend=1| 17c, 24, 41c, 65c }}
Line 2,564: Line 1,436:
Badness: 0.0287
Badness: 0.0287


</div></div>
Scales: [[mohaha7]], [[mohaha10]]


= Migration =
= Migration =
Migration takes [[#Septimal meantone]] mapping of 7 and [[#Mohaha]] mapping of 11.  
Migration takes [[#Septimal meantone]] mapping of 7 and [[#Mohaha]] mapping of 11.  
Period: 1\1
Optimal ([[POTE]]) generator: ~11/9 = 348.182
EDO generators: [[24edo|7\24]], [[31edo|9\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">


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 2,588: Line 1,448:


Mapping generators: ~2, ~11/9
Mapping generators: ~2, ~11/9
POTE generator: ~11/9 = 348.182


{{Val list|legend=1| 7d, 24d, 31, 100de, 131bdee, 162bdee }}
{{Val list|legend=1| 7d, 24d, 31, 100de, 131bdee, 162bdee }}


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


== 13-limit ==
== 13-limit ==
Period: 1\1
Optimal ([[POTE]]) generator: ~11/9 = 348.490
EDO generators: [[24edo|7\24]], [[31edo|9\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">


Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13
Line 2,616: Line 1,464:


Mapping generators: ~2, ~11/9
Mapping generators: ~2, ~11/9
POTE generator: ~11/9 = 348.490


{{Val list|legend=1| 7d, 24d, 31, 55d }}
{{Val list|legend=1| 7d, 24d, 31, 55d }}


Badness: 0.0281
Badness: 0.0281
</div></div>


= Ptolemy =
= Ptolemy =
Ptolemy takes [[#Flattone]] mapping of 7 and [[#Mohaha]] mapping of 11.  
Ptolemy takes [[#Flattone]] mapping of 7 and [[#Mohaha]] mapping of 11.  
Period: 1\1
Optimal ([[POTE]]) generator: ~11/9 = 346.922
EDO generators: [[38edo|11\38]], [[45edo|13\45]]
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">


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 2,643: Line 1,479:


Mapping: [{{val| 1 1 0 8 2 }}, {{val| 0 2 8 -18 5 }}]
Mapping: [{{val| 1 1 0 8 2 }}, {{val| 0 2 8 -18 5 }}]
POTE generator: ~11/9 = 346.922


{{Val list|legend=1| 7, 31dd, 38d, 45e, 83bcddee }}
{{Val list|legend=1| 7, 31dd, 38d, 45e, 83bcddee }}


Badness: 0.0588
Badness: 0.0588
</div></div>


== 13-limit ==
== 13-limit ==
Period: 1\1
Optimal ([[POTE]]) generator: ~11/9 = 346.910
EDO generators: [[38edo|11\38]], [[45edo|13\45]]
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">


Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13
Line 2,669: Line 1,493:


Mapping: [{{val| 1 1 0 8 2 6 }}, {{val| 0 2 8 -18 5 -8 }}]
Mapping: [{{val| 1 1 0 8 2 6 }}, {{val| 0 2 8 -18 5 -8 }}]
POTE generator: ~11/9 = 346.910


{{Val list|legend=1| 7, 31ddf, 38df, 45ef, 83bcddeeff }}
{{Val list|legend=1| 7, 31ddf, 38df, 45ef, 83bcddeeff }}


Badness: 0.0343
Badness: 0.0343
</div></div>


= Maqamic =
= Maqamic =
Line 2,682: Line 1,506:


Maqamic takes [[#Dominant]] mapping of 7 and [[#Mohaha]] mapping of 11, so it is [[36/35]] that vanishes instead of [[176/175]] as in mohajira. It makes the most sense if viewed as an adaptive temperament, whereby 7/4 and 9/5 simply share an equivalence class in the resulting scales, but don't need to share a particular tempered "middle-of-the-road" intonation.
Maqamic takes [[#Dominant]] mapping of 7 and [[#Mohaha]] mapping of 11, so it is [[36/35]] that vanishes instead of [[176/175]] as in mohajira. It makes the most sense if viewed as an adaptive temperament, whereby 7/4 and 9/5 simply share an equivalence class in the resulting scales, but don't need to share a particular tempered "middle-of-the-road" intonation.
Period: 1\1
Optimal ([[POTE]]) generator: ~11/9 = 350.934
EDO generators: [[7edo|2\7]], [[17edo|5\17]], [[24edo|7\24]]
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">


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 2,702: Line 1,514:


Mapping generators: ~2, ~11/9
Mapping generators: ~2, ~11/9
POTE generator: ~11/9 = 350.934


{{Val list|legend=1| 7, 17c, 24d, 41cd }}
{{Val list|legend=1| 7, 17c, 24d, 41cd }}
</div></div>


== 13-limit ==
== 13-limit ==
Period: 1\1
Optimal ([[POTE]]) generator: ~11/9 = 350.816
EDO generators: [[7edo|2\7]], [[17edo|5\17]], [[24edo|7\24]]
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">


Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13
Line 2,728: Line 1,528:


Mapping generators: ~2, ~11/9
Mapping generators: ~2, ~11/9
POTE generator: ~11/9 = 350.816


{{Val list|legend=1| 7, 17c, 24d, 41cd }}
{{Val list|legend=1| 7, 17c, 24d, 41cd }}
</div></div>


= Orphic =
= Orphic =
Period: 1\2
Optimal (POTE) generator: ~7/6 = 275.794
EDO generators: [[26edo|6\26]], [[48edo|11\48]]
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">


Subgroup: 2.3.5.7
Subgroup: 2.3.5.7
Line 2,755: Line 1,543:
Mapping generators: ~2401/1728, ~343/288
Mapping generators: ~2401/1728, ~343/288


[[Wedgie]]: {{wedgie| 8 32 6 32 -13 -76 }}
{{Multival|legend=1| 8 32 6 32 -13 -76 }}
 
POTE generator: ~7/6 = 275.794


{{Val list|legend=1| 26, 48c, 74, 174bd, 248bd }}
{{Val list|legend=1| 26, 48c, 74, 174bd, 248bd }}


[[Badness]]: 0.2588
[[Badness]]: 0.2588
</div></div>


== 11-limit ==
== 11-limit ==


Period: 1\2
Subgroup: 2.3.5.7.11


Optimal (POTE) generator: ~7/6 = 275.762
Comma list: 81/80, 99/98, 73728/73205
 
EDO generators: [[26edo|6\26]], [[48edo|11\48]]
 
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">
 
Subgroup: 2.3.5.7.11
 
Comma list: 81/80, 99/98, 73728/73205


Mapping: [{{val| 2 1 -4 4 8 }}, {{val|0 4 16 3 -2 }}]
Mapping: [{{val| 2 1 -4 4 8 }}, {{val|0 4 16 3 -2 }}]


Mapping generators: ~363/256, ~77/64
Mapping generators: ~363/256, ~77/64
POTE generator: ~7/6 = 275.762


{{Val list|legend=1| 26, 48c, 74, 248bd, 322bd }}
{{Val list|legend=1| 26, 48c, 74, 248bd, 322bd }}


Badness: 0.1015
Badness: 0.1015
</div></div>


== 13-limit ==
== 13-limit ==
Period: 1\2
Optimal (POTE) generator: ~7/6 = 275.774
EDO generators: [[26edo|6\26]], [[48edo|11\48]]
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">


Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13
Line 2,812: Line 1,576:


Mapping generators: ~55/39, ~63/52
Mapping generators: ~55/39, ~63/52
POTE generator: ~7/6 = 275.774


{{Val list|legend=1| 26, 48c, 74, 174bd, 248bd, 322bd }}
{{Val list|legend=1| 26, 48c, 74, 174bd, 248bd, 322bd }}


Badness: 0.0535
Badness: 0.0535
</div></div>


= Mothra =
= Mothra =
Line 2,824: Line 1,588:


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.  
Period: 1\1
Optimal (POTE) generator: ~8/7 = 232.193
EDO generators: [[26edo|5\26]], [[31edo|6\31]], [[36edo|7\36]], [[57edo|11\57]], [[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">


Subgroup: 2.3.5.7
Subgroup: 2.3.5.7
Line 2,845: Line 1,597:
Mapping generators: ~2, ~8/7
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.
[[POTE generator]]: ~8/7 = 232.193
 
[[Algebraic generator]]: Rabrindanath, largest real root of ''x''<sup>8</sup> - 3''x''<sup>2</sup> + 1, or 232.0774 cents.


[[Wedgie]]: {{wedgie| 3 12 -1 12 -10 -36 }}
[[Wedgie]]: {{wedgie| 3 12 -1 12 -10 -36 }}
Line 2,857: Line 1,611:


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


== 11-limit ==
== 11-limit ==
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">


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 2,881: Line 1,621:


Mapping generators: ~2, ~8/7
Mapping generators: ~2, ~8/7
POTE generator: ~8/7 = 232.031


{{Val list|legend=1| 5, 26, 31, 88, 150be, 181bee }}
{{Val list|legend=1| 5, 26, 31, 88, 150be, 181bee }}


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


=== 13-limit ===
=== 13-limit ===


Period: 1\1
Subgroup: 2.3.5.7.11.13


Optimal (POTE) generator: ~8/7 = 231.811
Comma list: 81/80, 99/98, 105/104, 144/143
 
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">
 
Subgroup: 2.3.5.7.11.13
 
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: [{{val| 1 1 0 3 5 1 }}, {{val| 0 3 12 -1 -8 14 }}]


Mapping generators: ~2, ~8/7
Mapping generators: ~2, ~8/7
POTE generator: ~8/7 = 231.811


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


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


== Cynder ==
== Cynder ==
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">


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 2,937: Line 1,653:


Mapping generators: ~2, ~8/7
Mapping generators: ~2, ~8/7
POTE generator: ~8/7 = 231.317


{{Val list|legend=1| 5e, 26, 57e, 83bce }}
{{Val list|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">


Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13
Line 2,965: Line 1,669:


Mapping generators: ~2, ~8/7
Mapping generators: ~2, ~8/7
POTE generator: ~8/7 = 232.293


{{Val list|legend=1| 5e, 26, 57e, 83bce }}
{{Val list|legend=1| 5e, 26, 57e, 83bce }}


Badness: 0.0341
Badness: 0.0341
</div></div>


== Mosura ==
== Mosura ==
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">


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 2,993: Line 1,685:


Mapping generators: ~2, ~8/7
Mapping generators: ~2, ~8/7
POTE generator: ~8/7 = 232.419


{{Val list|legend=1| 31, 129, 160be, 191bce, 222bce, 253bcee }}
{{Val list|legend=1| 31, 129, 160be, 191bce, 222bce, 253bcee }}


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


=== 13-limit ===
=== 13-limit ===


Period: 1\1
Subgroup: 2.3.5.7.11.13


Optimal (POTE) generator: ~8/7 = 232.640
Comma list: 81/80, 144/143, 176/175, 196/195
 
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">
 
Subgroup: 2.3.5.7.11.13
 
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: [{{val| 1 1 0 3 -1 7 }}, {{val| 0 3 12 -1 23 -17 }}]


Mapping generators: ~2, ~8/7
Mapping generators: ~2, ~8/7
POTE generator: ~8/7 = 232.640


{{Val list|legend=1| 31, 36, 67, 98 }}
{{Val list|legend=1| 31, 36, 67, 98 }}


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


= Squares =
= Squares =


Squares splits the interval of an eleventh, or 8/3, into four supermajor third ([[9/7]]) intervals, and uses it for a generator. [[31edo]], with a generator of 11/31, makes for a good squares tuning, with 8, 11, and 14 note MOS available. Squares tempers out [[2401/2400]], the breedsma, as well as [[2430/2401]].
Squares splits the interval of an eleventh, or 8/3, into four supermajor third ([[9/7]]) intervals, and uses it for a generator. [[31edo]], with a generator of 11/31, makes for a good squares tuning, with 8, 11, and 14 note MOS available. Squares tempers out [[2401/2400]], the breedsma, as well as [[2430/2401]].
Period: 1\1
Optimal (POTE) generator: ~9/7 = 425.942
EDO generators: [[14edo|5\14]], [[17edo|6\17]], [[31edo|11\31]]
Scales (Scala files): [[Skwares8]], [[Skwares11]], [[Skwares14]]
<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">


Subgroup: 2.3.5.7
Subgroup: 2.3.5.7
Line 3,052: Line 1,720:
Mapping generators: ~2, ~9/7
Mapping generators: ~2, ~9/7


Minimax tuning:  
[[POTE generator]]: ~9/7 = 425.942
 
[[Minimax tuning]]:  
* 7- and 9-odd-limit: 1/4 comma
* 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| 3/2 0 9/16 0 }}]
: [{{Monzo| 1 0 0 0 }}, {{Monzo| 1 0 1/4 0 }}, {{Monzo| 0 0 1 0 }}, {{Monzo| 3/2 0 9/16 0 }}]
: [[Eigenmonzo]]s: 2, 5
: [[Eigenmonzo]]s: 2, 5


Algebraic generator: Sceptre2, the positive root of 9''x''<sup>2</sup> + ''x'' - 16, or (sqrt (577) - 1)/18, which is 425.9311 cents.
[[Algebraic generator]]: Sceptre2, the positive root of 9''x''<sup>2</sup> + ''x'' - 16, or (sqrt (577) - 1)/18, which is 425.9311 cents.


{{Val list|legend=1| 14c, 17c, 31 }}
{{Val list|legend=1| 14c, 17c, 31 }}
Line 3,063: Line 1,733:
[[Badness]]: 0.0460
[[Badness]]: 0.0460


</div></div>
Scales: [[skwares8]], [[skwares11]], [[skwares14]]


; Music:
; Music:
Line 3,069: Line 1,739:


== 11-limit ==
== 11-limit ==
Period: 1\1
Optimal (POTE) generator: ~9/7 = 425.957
EDO generators: [[14edo|5\14]], [[17edo|6\17]], [[31edo|11\31]]
Scales (Scala files): [[Skwares8]], [[Skwares11]], [[Skwares14]]
<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">


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 3,089: Line 1,747:


Mapping generators: ~2, ~9/7
Mapping generators: ~2, ~9/7
POTE generator: ~9/7 = 425.957


{{Val list|legend=1| 14c, 17c, 31 }}
{{Val list|legend=1| 14c, 17c, 31 }}
Line 3,094: Line 1,754:
Badness: 0.0216
Badness: 0.0216


</div></div>
Scales: [[skwares8]], [[skwares11]], [[skwares14]]


=== 13-limit ===
=== 13-limit ===
Period: 1\1
Optimal (POTE) generator: ~9/7 = 425.550
EDO generators: [[14edo|5\14]], [[17edo|6\17]], [[31edo|11\31]]
Scales (Scala files): [[Skwares8]], [[Skwares11]], [[Skwares14]]
<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">


Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13
Line 3,117: Line 1,765:


Mapping generators: ~2, ~9/7
Mapping generators: ~2, ~9/7
POTE generator: ~9/7 = 425.550


{{Val list|legend=1| 14c, 17c, 31, 79cf }}
{{Val list|legend=1| 14c, 17c, 31, 79cf }}
Line 3,122: Line 1,772:
Badness: 0.0255
Badness: 0.0255


</div></div>
Scales: [[skwares8]], [[skwares11]], [[skwares14]]


=== Agora ===
=== Agora ===


Period: 1\1
Subgroup: 2.3.5.7.11.13


Optimal (POTE) generator: ~9/7 = 426.276
Comma list: 81/80, 99/98, 105/104, 121/120


EDO generators: [[14edo|5\14]], [[31edo|11\31]], [[45edo|16\45]]
Mapping: [{{val| 1 3 8 6 7 14 }}, {{val| 0 -4 -16 -9 -10 -29 }}]


Scales (Scala files): [[Skwares8]], [[Skwares11]], [[Skwares14]]
Mapping generators: ~2, ~9/7
 
<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">
 
Subgroup: 2.3.5.7.11.13
 
Comma list: 81/80, 99/98, 105/104, 121/120
 
Mapping: [{{val| 1 3 8 6 7 14 }}, {{val| 0 -4 -16 -9 -10 -29 }}]


Mapping generators: ~2, ~9/7
POTE generator: ~9/7 = 426.276


{{Val list|legend=1| 14cf, 31, 45ef, 76e }}
{{Val list|legend=1| 14cf, 31, 45ef, 76e }}
Line 3,150: Line 1,790:
Badness: 0.0245
Badness: 0.0245


</div></div>
Scales: [[skwares8]], [[skwares11]], [[skwares14]]


==== 17-limit ====
==== 17-limit ====
Period: 1\1
Optimal (POTE) generator: ~9/7 = 426.187
EDO generators: [[14edo|5\14]], [[31edo|11\31]], [[45edo|16\45]]
Scales (Scala files): [[Skwares8]], [[Skwares11]], [[Skwares14]]
<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">


Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17
Line 3,173: Line 1,801:


Mapping generators: ~2, ~9/7
Mapping generators: ~2, ~9/7
POTE generator: ~9/7 = 426.187


{{Val list|legend=1| 14cf, 31, 76e }}
{{Val list|legend=1| 14cf, 31, 76e }}


</div></div>
Scales: [[skwares8]], [[skwares11]], [[skwares14]]


==== 19-limit ====
==== 19-limit ====
Period: 1\1
Optimal (POTE) generator: ~9/7 = 426.225
EDO generators: [[14edo|5\14]], [[31edo|11\31]], [[45edo|16\45]]
Scales (Scala files): [[Skwares8]], [[Skwares11]], [[Skwares14]]
<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">


Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13.17.19
Line 3,199: Line 1,817:


Mapping generators: ~2, ~9/7
Mapping generators: ~2, ~9/7
POTE generator: ~9/7 = 426.225


{{Val list|legend=1| 14cf, 31, 76e }}
{{Val list|legend=1| 14cf, 31, 76e }}


</div></div>
Scales: [[skwares8]], [[skwares11]], [[skwares14]]


== Cuboctahedra ==
== Cuboctahedra ==
Period: 1\1
Optimal (POTE) generator: ~9/7 = 425.993
EDO generators: [[31edo|11\31]], [[45edo|16\45]]
Scales (Scala files): [[Skwares8]], [[Skwares11]], [[Skwares14]]
<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">


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 3,225: Line 1,833:


Mapping generators: ~2, ~9/7
Mapping generators: ~2, ~9/7
POTE generator: ~9/7 = 425.993


{{Val list|legend=1| 14ce, 17ce, 31, 107b }}
{{Val list|legend=1| 14ce, 17ce, 31, 107b }}


[[Badness]]: 0.0568
Badness: 0.0568


</div></div>
Scales: [[skwares8]], [[skwares11]], [[skwares14]]


= Liese =
= Liese =
Line 3,236: Line 1,846:


Liese splits the twelfth interval of 3/1 into three generators of 10/7, using the comma 1029/1000. It also tempers out 686/675, the senga. [[74edo]] makes for a good liese tuning, though [[19edo]] can be used. The tuning is well-supplied with MOS: 7, 9, 11, 13, 15, 17, 19, 36, 55.
Liese splits the twelfth interval of 3/1 into three generators of 10/7, using the comma 1029/1000. It also tempers out 686/675, the senga. [[74edo]] makes for a good liese tuning, though [[19edo]] can be used. The tuning is well-supplied with MOS: 7, 9, 11, 13, 15, 17, 19, 36, 55.
Period: 1\1
Optimal (POTE) generator: ~10/7 = 632.406
EDO generators: [[17edo|9\17]], [[19edo|10\19]], [[36edo|19\36]], [[55edo|29\55]]
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">


Subgroup: 2.3.5.7
Subgroup: 2.3.5.7
Line 3,253: Line 1,851:
[[Comma list]]: 81/80, 686/675
[[Comma list]]: 81/80, 686/675


Mapping: [{{val| 1 0 -4 -3 }}, {{val| 0 3 12 11 }}]
[[Mapping]]: [{{val| 1 0 -4 -3 }}, {{val| 0 3 12 11 }}]


Mapping generators: ~2, ~10/7
Mapping generators: ~2, ~10/7
[[POTE generator]]: ~10/7 = 632.406


Minimax tuning:  
Minimax tuning:  
Line 3,262: Line 1,862:
: [[Eigenmonzo]]s: 2, 5
: [[Eigenmonzo]]s: 2, 5


Algebraic generator: Radix, the real root of ''x''<sup>5</sup> - 2''x''<sup>4</sup> + 2''x''<sup>3</sup> - 2''x''<sup>2</sup> + 2''x'' - 2, also a root of ''x''<sup>6</sup> - ''x''<sup>5</sup> - 2. The recurrence converges.
[[Algebraic generator]]: Radix, the real root of ''x''<sup>5</sup> - 2''x''<sup>4</sup> + 2''x''<sup>3</sup> - 2''x''<sup>2</sup> + 2''x'' - 2, also a root of ''x''<sup>6</sup> - ''x''<sup>5</sup> - 2. The recurrence converges.


{{Val list|legend=1| 17c, 19, 55, 74d }}
{{Val list|legend=1| 17c, 19, 55, 74d }}


[[Badness]]: 0.0467
[[Badness]]: 0.0467
</div></div>


== Liesel ==
== Liesel ==
Period: 1\1
Optimal (POTE) generator: ~10/7 = 633.073
EDO generators: [[17edo|9\17]], [[19edo|10\19]], [[36edo|19\36]]
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">


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 3,291: Line 1,877:


Mapping generators: ~2, ~10/7
Mapping generators: ~2, ~10/7
POTE generator: ~10/7 = 633.073


{{Val list|legend=1| 17c, 19, 36, 91cee }}
{{Val list|legend=1| 17c, 19, 36, 91cee }}


Badness: 0.0407
Badness: 0.0407
</div></div>


=== 13-limit ===
=== 13-limit ===
Liesel is a very natural 13-limit tuning, given the generator is so near 13/9.
Liesel is a very natural 13-limit tuning, given the generator is so near 13/9.
Period: 1\1
Optimal (POTE) generator: ~10/7 = 633.042
EDO generators: [[17edo|9\17]], [[19edo|10\19]], [[36edo|19\36]]
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">


Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13
Line 3,320: Line 1,894:


Mapping generators: ~2, ~10/7
Mapping generators: ~2, ~10/7
POTE generator: ~10/7 = 633.042


{{Val list|legend=1| 17c, 19, 36, 91ceef }}
{{Val list|legend=1| 17c, 19, 36, 91ceef }}


Badness: 0.0273
Badness: 0.0273
</div></div>


== Elisa ==
== Elisa ==
Period: 1\1
Optimal (POTE) generator: ~10/7 = 633.061
EDO generators: [[17edo|9\17]], [[19edo|10\19]], [[36edo|19\36]]
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">


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 3,348: Line 1,910:


Mapping generators: ~2, ~10/7
Mapping generators: ~2, ~10/7
POTE generator: ~10/7 = 633.061


{{Val list|legend=1| 17c, 19e, 36e }}
{{Val list|legend=1| 17c, 19e, 36e }}


Badness: 0.0416
Badness: 0.0416
</div></div>


== Lisa ==
== Lisa ==
Period: 1\1
Optimal (POTE) generator: ~10/7 = 631.370
EDO generators: [[19edo|10\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">


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 3,376: Line 1,926:


Mapping generators: ~2, ~10/7
Mapping generators: ~2, ~10/7
POTE generator: ~10/7 = 631.370


{{Val list|legend=1| 17cee, 19 }}
{{Val list|legend=1| 17cee, 19 }}


Badness: 0.0548
Badness: 0.0548
</div></div>


=== 13-limit ===
=== 13-limit ===


Period: 1\1
Subgroup: 2.3.5.7.11.13


Optimal (POTE) generator: ~10/7 = 631.221
Comma list: 45/44, 81/80, 91/88, 147/143
 
EDO generators: [[19edo|10\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">
 
Subgroup: 2.3.5.7.11.13
 
Comma list: 45/44, 81/80, 91/88, 147/143


Mapping: [{{val| 1 0 -4 -3 -6 0 }}, {{val| 0 3 12 11 -1 18 7 }}]
Mapping: [{{val| 1 0 -4 -3 -6 0 }}, {{val| 0 3 12 11 -1 18 7 }}]


Mapping generators: ~2, ~10/7
Mapping generators: ~2, ~10/7
POTE generator: ~10/7 = 631.221


{{Val list|legend=1| 17cee, 19 }}
{{Val list|legend=1| 17cee, 19 }}


Badness: 0.0361
Badness: 0.0361
</div></div>


= Jerome =
= Jerome =
Jerome is related to [[20ed5|Hieronymus' tuning]]; the Hieronymus generator is 5<sup>1/20</sup>, or 139.316 cents. While the generator represents both 13/12 and 12/11, the POTE and Hieronymus generators are close to 13/12 in size.
Jerome is related to [[20ed5|Hieronymus' tuning]]; the Hieronymus generator is 5<sup>1/20</sup>, or 139.316 cents. While the generator represents both 13/12 and 12/11, the POTE and Hieronymus generators are close to 13/12 in size.
Period: 1\1
Optimal (POTE) generator: ~54/49 = 139.343
EDO generators: [[17edo|2\17]], [[26edo|3\26]], [[43edo|5\43]]
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">


Subgroup: 2.3.5.7
Subgroup: 2.3.5.7
Line 3,430: Line 1,956:
[[Comma list]]: 81/80, 17280/16807
[[Comma list]]: 81/80, 17280/16807


Mapping: [{{val| 1 1 0 2 }}, {{val| 0 5 20 7 }}]
[[Mapping]]: [{{val| 1 1 0 2 }}, {{val| 0 5 20 7 }}]


Mapping generators: ~2, ~54/49
Mapping generators: ~2, ~54/49


Wedgie: {{wedgie| 5 30 7 20 -3 -40 }}
{{Multival|legend=1| 5 30 7 20 -3 -40 }}
 
[[POTE generator]]: ~54/49 = 139.343


{{Val list|legend=1| 17c, 26, 43, 69, 112bd }}
{{Val list|legend=1| 17c, 26, 43, 69, 112bd }}


Badness: 0.1087
[[Badness]]: 0.1087
 
</div></div>


== 11-limit ==
== 11-limit ==
Period: 1\1
Optimal (POTE) generator: ~12/11 = 139.428
EDO generators: [[17edo|2\17]], [[26edo|3\26]], [[43edo|5\43]]
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">


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 3,463: Line 1,977:


Mapping generators: ~2, ~12/11
Mapping generators: ~2, ~12/11
POTE generator: ~12/11 = 139.428


{{Val list|legend=1| 17c, 26, 43, 69 }}
{{Val list|legend=1| 17c, 26, 43, 69 }}


Badness: 0.0479
Badness: 0.0479
</div></div>


== 13-limit ==
== 13-limit ==
Period: 1\1
Optimal (POTE) generator: ~12/11 = 139.387
EDO generators: [[17edo|2\17]], [[26edo|3\26]], [[43edo|5\43]]
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">


Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13
Line 3,491: Line 1,993:


Mapping generators: ~2, ~12/11
Mapping generators: ~2, ~12/11
POTE generator: ~12/11 = 139.387


{{Val list|legend=1| 17c, 26, 43, 69 }}
{{Val list|legend=1| 17c, 26, 43, 69 }}


Badness: 0.0293
Badness: 0.0293
</div></div>


== 17-limit ==
== 17-limit ==


Period: 1\1
Subgroup: 2.3.5.7.11.13.17


Optimal (POTE) generator: ~12/11 = 139.362
Comma list: 78/77, 81/80, 99/98, 144/143, 189/187
 
EDO generators: [[26edo|3\26]], [[43edo|5\43]]
 
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">
 
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 78/77, 81/80, 99/98, 144/143, 189/187


Mapping: [{{val| 1 1 0 2 3 3 2 }}, {{val| 0 5 20 7 4 6 18 }}]
Mapping: [{{val| 1 1 0 2 3 3 2 }}, {{val| 0 5 20 7 4 6 18 }}]


Mapping generators: ~2, ~12/11
Mapping generators: ~2, ~12/11
POTE generator: ~12/11 = 139.362


{{Val list|legend=1| 17cg, 26, 43, 69 }}
{{Val list|legend=1| 17cg, 26, 43, 69 }}


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


== 19-limit ==
== 19-limit ==
Period: 1\1
Optimal (POTE) generator: ~12/11 = 139.313
EDO generators: [[26edo|3\26]], [[43edo|5\43]]
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">


Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13.17.19
Line 3,547: Line 2,025:


Mapping generators: ~2, ~12/11
Mapping generators: ~2, ~12/11
POTE generator: ~12/11 = 139.313


{{Val list|legend=1| 17cgh, 26, 43, 69 }}
{{Val list|legend=1| 17cgh, 26, 43, 69 }}


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


= Meanmag =
= Meanmag =
Period: 1\19
Optimal (POTE) generator: ~8/7 = 238.396
EDO generators: [[38edo|7\38]], [[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">


Subgroup: 2.3.5.7
Subgroup: 2.3.5.7
Line 3,577: Line 2,043:


[[Wedgie]]: {{wedgie| 0 0 19 0 30 44 }}
[[Wedgie]]: {{wedgie| 0 0 19 0 30 44 }}
[[POTE generator]]: ~8/7 = 238.396


{{Val list|legend=1| 19, 38, 57, 76, 95bc }}
{{Val list|legend=1| 19, 38, 57, 76, 95bc }}


Badness: 0.0770
[[Badness]]: 0.0770
 
</div></div>


= Undevigintone =
= Undevigintone =
Period: 1\19
Optimal (POTE) generator: ~11/8 = 538.047
EDO generators: [[38edo|17\38]]
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">


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 3,604: Line 2,059:


Mapping generators: ~21/20, ~11
Mapping generators: ~21/20, ~11
[[POTE generator]]: ~11/8 = 538.047


{{Val list|legend=1| 19, 38d }}
{{Val list|legend=1| 19, 38d }}


Badness: 0.0364
Badness: 0.0364
</div></div>


== 13-limit ==
== 13-limit ==
Period: 1\19
Optimal (POTE) generator: ~11/8 = 537.061
EDO generators: [[38edo|17\38]]
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">


Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13
Line 3,631: Line 2,075:


Mapping generators: ~21/20, ~11
Mapping generators: ~21/20, ~11
POTE generator: ~11/8 = 537.061


{{Val list|legend=1| 19, 38d }}
{{Val list|legend=1| 19, 38d }}


Badness: 0.0229
Badness: 0.0229
</div></div>


= Cloudtone =
= Cloudtone =
Line 3,642: Line 2,086:


== 7-limit ==
== 7-limit ==
Period: 1\5
Optimal ([[POTE]]) generator: ~3/2 = 695.720
EDO generators: [[45edo|1\45]], [[50edo|1\50]]
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">


Subgroup: 2.3.5.7
Subgroup: 2.3.5.7
Line 3,661: Line 2,094:


{{Multival|legend=1| 5 20 0 20 -14 -56 }}
{{Multival|legend=1| 5 20 0 20 -14 -56 }}
[[POTE generator]]: ~3/2 = 695.720


{{Val list|legend=1| 5, 45, 50, 95bcd, 145bcdd, 195bbcdd }}
{{Val list|legend=1| 5, 45, 50, 95bcd, 145bcdd, 195bbcdd }}


[[Badness]]: 0.102256
[[Badness]]: 0.102256
</div></div>


== 11-limit ==
== 11-limit ==
Period: 1\5
Optimal (POTE) generator: ~3/2 = 696.536
EDO generators: [[50edo|1\50]], [[55edo|1\55]]
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">


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 3,687: Line 2,108:


Mapping: [{{val| 5 0 -20 14 41 }}, {{val| 0 1 4 0 -3 }}]
Mapping: [{{val| 5 0 -20 14 41 }}, {{val| 0 1 4 0 -3 }}]
POTE generator: ~3/2 = 696.536


{{Val list|legend=1| 5, 50, 55, 105d, 155bdd, 205bddd }}
{{Val list|legend=1| 5, 50, 55, 105d, 155bdd, 205bddd }}


Badness: 0.070378
Badness: 0.070378
</div></div>


== 13-limit ==
== 13-limit ==
Period: 1\5
Optimal (POTE) generator: ~3/2 = 696.162
EDO generators: [[50edo|1\50]], [[55edo|1\55]]
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">


Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13
Line 3,712: Line 2,122:


Mapping: [{{val| 5 0 -20 14 41 -21 }}, {{val| 0 1 4 0 -3 5 }}]
Mapping: [{{val| 5 0 -20 14 41 -21 }}, {{val| 0 1 4 0 -3 5 }}]
POTE generator: ~3/2 = 696.162


{{Val list|legend=1| 5, 45f, 50, 55 }}
{{Val list|legend=1| 5, 45f, 50, 55 }}


Badness: 0.048829
Badness: 0.048829
</div></div>


[[Category:Theory]]
[[Category:Theory]]
Retrieved from "https://en.xen.wiki/w/Meantone_family"