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Meantone family: Difference between revisions

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= Dominant =
= Dominant =
[[Comma]]s: 36/35, 64/63


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]].


valid range: [700.000, 720.000] (12 to 5)
Period: 1\1


nice range: [694.786, 715.587]
Optimal ([[POTE]]) generator: ~3/2 = 701.573


strict range: [700.000, 715.587]
EDO generators: [[5edo|3\5]], [[7edo|4\7]], [[12edo|7\12]], [[17edo|10\17]]


[[POTE generator]]: 701.573
Scales (Scala files):  


Mapping generator: ~3
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
[[Comma]] list: 36/35, 64/63


Map: [&lt;1 0 -4 6|, &lt;0 1 4 -2|]
Mapping: [{{val| 1 0 -4 6 }}, {{val| 0 1 4 -2 }}]


[[Wedgie]]: &lt;&lt;1 4 -2 4 -6 -16||
Mapping generators: ~2, ~3


{{Vals|legend=1| 5, 7, 12, 17c, 29cd }}
[[Wedgie]]: {{wedgie| 1 4 -2 4 -6 -16 }}
 
[[Tuning ranges]]:
* valid range: [700.000, 720.000] (12 to 5)
* nice range: [694.786, 715.587]
* strict range: [700.000, 715.587]
 
{{Vals|legend=1| 5, 7, 12, 41cd, 53cdd, 65ccddd }}


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


== 11-limit ==
== 11-limit ==
Commas: 36/35, 64/63, 56/55


valid range: [700.000, 705.882] (12 to 17)
Period: 1\1
 
Optimal ([[POTE]]) generator: ~3/2 = 703.254
 
EDO generators: [[12edo|7\12]], [[17edo|10\17]]


nice range: [691.202, 715.587]
Scales (Scala files):  


strict range: [700.000, 705.882]
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
Comma list: 36/35, 56/55, 64/63


POTE generator: ~3/2 = 703.254
Mapping: [{{val| 1 0 -4 6 13 }}, {{val| 0 1 4 -2 -6 }}]


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


Map: [&lt;1 0 -4 6 13|, &lt;0 1 4 -2 -6|]
Tuning ranges:
* valid range: [700.000, 705.882] (12 to 17)
* nice range: [691.202, 715.587]
* strict range: [700.000, 705.882]


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


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


=== 13-limit ===
=== 13-limit ===
Commas: 36/35, 56/55, 64/63, 66/65


valid range: 705.882 (17)
Period: 1\1
 
Optimal ([[POTE]]) generator: ~3/2 = 703.636
 
EDO generators: [[12edo|7\12]], [[17edo|10\17]]
 
Scales (Scala files):


nice range: [691.202, 715.587]
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
Comma list: 36/35, 56/55, 64/63, 66/65


strict range:705.882
Mapping: [{{val| 1 0 -4 6 13 18 }}, {{val| 0 1 4 -2 -6 -9 }}]


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


Map: [&lt;1 0 -4 6 13 18|, &lt;0 1 4 -2 -6 -9|]
Tuning ranges:
* valid range: 705.882 (17)
* nice range: [691.202, 715.587]
* strict range: 705.882


{{Vals|legend=1| 12f, 17c, 29cdef }}
{{Vals|legend=1| 12f, 17c, 29cdef }}


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


=== Dominion ===
=== Dominion ===
Commas: 26/25, 36/35, 56/55, 64/63


POTE generator: ~3/2 = 704.905
Period: 1\1


Map: [&lt;1 0 -4 6 13 -9|, &lt;0 1 4 -2 -6 8|]
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">
Comma list: 26/25, 36/35, 56/55, 64/63
 
Mapping: [{{val| 1 0 -4 6 13 -9 }}, {{val| 0 1 4 -2 -6 8 }}]


{{Vals|legend=1| 5, 12, 17c, 46cde }}
{{Vals|legend=1| 5, 12, 17c, 46cde }}


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


== Domineering ==
== Domineering ==
Commas: 36/35, 45/44, 64/63


POTE generator: ~3/2 = 698.776
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">
Comma list: 36/35, 45/44, 64/63


Mapping generator: ~3
Mapping: [{{val| 1 0 -4 6 -6 }}, {{val| 0 1 4 -2 6 }}]


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


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


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


=== 13-limit ===
=== 13-limit ===
Commas: 36/35, 45/44, 52/49, 64/63


POTE generator: ~3/2 = 695.762
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">
Comma list: 36/35, 45/44, 52/49, 64/63


Mapping generator: ~3
Mapping: [{{val| 1 0 -4 6 -6 10 }}, {{val| 0 1 4 -2 6 -4 }}]


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


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


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


==== 17-limit ====
==== 17-limit ====
Commas: 36/35, 45/44, 51/49, 52/49, 64/63


POTE generator: ~3/2 = 696.115
Period: 1\1
 
Optimal ([[POTE]]) generator: ~3/2 = 696.115
 
EDO generators: [[7edo|4\7]], [[12edo|7\12]]
 
Scales (Scala files):


Mapping generator: ~3
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
Comma list: 36/35, 45/44, 51/49, 52/49, 64/63


Map: [&lt;1 0 -4 6 -6 10 12|, &lt;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


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


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


==== 19-limit ====
==== 19-limit ====
Commas: 36/35, 39/38, 45/44, 51/49, 52/49, 57/56


POTE generator: ~3/2 = 696.217
Period: 1\1


Mapping generator: ~3
Optimal ([[POTE]]) generator: ~3/2 = 696.217


Map: [&lt;1 0 -4 6 -6 10 12 9|, &lt;0 1 4 -2 6 -4 -5 -3|]
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">
Comma list: 36/35, 39/38, 45/44, 51/49, 52/49, 57/56
 
Mapping: [{{val| 1 0 -4 6 -6 10 12 9 }}, {{val| 0 1 4 -2 6 -4 -5 -3 }}]
 
Mapping generators: ~2, ~3


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


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


=== Dominatrix ===
=== Dominatrix ===
Commas: 27/26, 36/35, 45/44, 64/63


POTE generator: ~3/2 = 698.544
Period: 1\1


Mapping generator: ~3
Optimal ([[POTE]]) generator: ~3/2 = 698.544


Map: [&lt;1 0 -4 6 -6 -1|, &lt;0 1 4 -2 6 3|]
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">
Comma list: 27/26, 36/35, 45/44, 64/63
 
Mapping: [{{val| 1 0 -4 6 -6 -1 }}, {{val| 0 1 4 -2 6 3 }}]
 
Mapping generators: ~2, ~3


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


== Domination ==
== Domination ==
Commas: 36/35, 64/63, 77/75


POTE generator: ~3/2 = 705.004
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">
Comma list: 36/35, 64/63, 77/75


Mapping generator: ~3
Mapping: [{{val| 1 0 -4 6 -14 }}, {{val| 0 1 4 -2 11 }}]


Map: [&lt;1 0 -4 6 -14|, &lt;0 1 4 -2 11|]
Mapping generators: ~2, ~3


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


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


=== 13-limit ===
=== 13-limit ===
Commas: 26/25, 36/35, 64/63, 66/65


POTE generator: ~3/2 = 705.496
Period: 1\1
 
Optimal ([[POTE]]) generator: ~3/2 = 705.496
 
EDO generators: [[12edo|7\12]], [[17edo|10\17]]


Mapping generator: ~3
Scales (Scala files):  


Map: [&lt;1 0 -4 6 -14 -9|, &lt;0 1 4 -2 11 8|]
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
Comma list: 26/25, 36/35, 64/63, 66/65
 
Mapping: [{{val| 1 0 -4 6 -14 -9 }}, {{val| 0 1 4 -2 11 8 }}]
 
Mapping generators: ~2, ~3


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


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


== Arnold ==
== Arnold ==
Commas: 22/21, 33/32, 36/35


POTE generator: ~3/2 = 698.491
Period: 1\1
 
Optimal ([[POTE]]) generator: ~3/2 = 698.491
 
EDO generators: [[5edo|3\5]], [[7edo|4\7]], [[12edo|7\12]]


Mapping generator: ~3
Scales (Scala files):  


Map: [&lt;1 0 -4 6 5|, &lt;0 1 4 -2 -1|]
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
Comma list: 22/21, 33/32, 36/35
 
Mapping: [{{val| 1 0 -4 6 5 }}, {{val| 0 1 4 -2 -1 }}]
 
Mapping generators: ~2, ~3


{{Vals|legend=1| 5, 7, 12e }}
{{Vals|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">
Commas: 22/21, 27/26, 33/32, 36/35
Commas: 22/21, 27/26, 33/32, 36/35


POTE generator: ~3/2 = 696.743
Mapping: [{{val| 1 0 -4 6 5 -1 }}, {{val| 0 1 4 -2 3 }}]


Map: [&lt;1 0 -4 6 5 -1|, &lt;0 1 4 -2 -1 3|]
Mapping generators: ~2, ~3


{{Vals|legend=1| 5, 7, 12ef, 19def }}
{{Vals|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">
Commas: 22/21, 27/26, 33/32, 36/35, 51/49
Commas: 22/21, 27/26, 33/32, 36/35, 51/49


POTE generator: ~3/2 = 696.978
Mapping: [{{val| 1 0 -4 6 5 -1 12 }}, {{val| 0 1 4 -2 3 -5 }}]


Map: [&lt;1 0 -4 6 5 -1 12|, &lt;0 1 4 -2 -1 3 -5|]
Mapping generators: ~2, ~3


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


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


=== 19-limit ===
=== 19-limit ===
Period: 1\1
Optimal ([[POTE]]) generator: ~3/2 = 697.068
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">
Commas: 22/21, 27/26, 33/32, 36/35, 51/49, 57/56
Commas: 22/21, 27/26, 33/32, 36/35, 51/49, 57/56


POTE generator: ~3/2 = 697.068
Mapping: [{{val| 1 0 -4 6 5 -1 12 9 }}, {{val| 0 1 4 -2 3 -5 -3 }}]


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


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


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


= Sharptone =
= Sharptone =
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