40ed10: Difference between revisions

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== Regular temperaments ==

40ED10 can also be thought of as a [[generator]] of the 2.3.5.17.19 [[Subgroup temperaments|subgroup temperament]] which tempers out 4624/4617, 6144/6137, and 6885/6859, which is a [[cluster temperament]] with 12 clusters of notes in an octave (''quintilischis'' temperament). This temperament is supported by {{~~Val list~~|12, 253, 265, 277, 289, 301, 313}}, and [[325edo|325]] EDOs.

40ED10 can also be thought of as a [[generator]] of the 2.3.5.17.19 [[Subgroup temperaments|subgroup temperament]] which tempers out 4624/4617, 6144/6137, and 6885/6859, which is a [[cluster temperament]] with 12 clusters of notes in an octave (''quintilischis'' temperament). This temperament is supported by {{Optimal ET sequence|12, 253, 265, 277, 289, 301, 313}}, and [[325edo|325]] EDOs.

Tempering out 400/399 (equating 20/19 and 21/20) leads ''[[Schismatic family #Quintilipyth|quintilipyth]]'' (12&253), and tempering out 476/475 (equating 19/17 with 28/25) leads ''[[Schismatic family #Quintaschis|quintaschis]]'' (12&289).

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Gencom mapping: [{{val|1 2 -1 5 4}}, {{val|0 -5 40 -11 3}}]

POTE generator: ~18/17 = 99.652

~~Vals:~~ 12, 253, 265, 277, 289

{{Optimal ET sequence|legend=1| 12, 253, 265, 277, 289 }}

RMS error: 0.1636 cents

; [[Schismatic family #Quintilipyth|Quintilipyth]] (12 & 253)

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Mapping: [{{val|1 2 -1 -4}}, {{val|0 -5 40 82}}]

POTE generator: ~625/588 = 99.625

~~Vals:~~ 12, 253, 265

{{Optimal ET sequence|legend=1| 12, 253, 265 }}

Badness: 0.253966

'''11-limit'''

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Mapping: [{{val|1 2 -1 -4 -7}}, {{val|0 -5 40 82 126}}]

POTE generator: ~35/33 = 99.616

~~Vals:~~ 12, 253, 265, 518c, 783cc

{{Optimal ET sequence|legend=1| 12, 253, 265, 518c, 783cc }}

Badness: 0.113044

'''13-limit'''

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Mapping: [{{val|1 2 -1 -4 -7 -9}}, {{val|0 -5 40 82 126 153}}]

POTE generator: ~35/33 = 99.612

~~Vals:~~ 12f, 253, 518c, 771cc

{{Optimal ET sequence|legend=1| 12f, 253, 518c, 771cc }}

Badness: 0.069127

'''17-limit'''

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Mapping: [{{val|1 2 -1 -4 -7 -9 5}}, {{val|0 -5 40 82 126 153 -11}}]

POTE generator: ~18/17 = 99.612

~~Vals:~~ 12f, 253, 518c, 771cc

{{Optimal ET sequence|legend=1| 12f, 253, 518c, 771cc }}

Badness: 0.045992

'''19-limit'''

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Mapping: [{{val|1 2 -1 -4 -7 -9 5 4}}, {{val|0 -5 40 82 126 153 -11 3}}]

POTE generator: ~18/17 = 99.615

~~Vals:~~ 12f, 253, 265, 518ch

{{Optimal ET sequence|legend=1| 12f, 253, 265, 518ch }}

Badness: 0.038155

; [[Schismatic family #Quintaschis|Quintaschis]] (12 & 289)

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Mapping: [{{val|1 2 -1 -5}}, {{val|0 -5 40 94}}]

POTE generator: ~200/189 = 99.664

~~Vals:~~ 12, 277d, 289

{{Optimal ET sequence|legend=1| 12, 277d, 289 }}

Badness: 0.132890

'''11-limit'''

Line 285:

Mapping: [{{val|1 2 -1 -5 -8}}, {{val|0 -5 40 94 138}}]

POTE generator: ~35/33 = 99.653

~~Vals:~~ 12, 277d, 289

{{Optimal ET sequence|legend=1| 12, 277d, 289 }}

Badness: 0.111477

'''13-limit'''

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Mapping: [{{val|1 2 -1 -5 -8 -11}}, {{val|0 -5 40 94 138 177}}]

POTE generator: ~35/33 = 99.658

~~Vals:~~ 12f, 277df, 289

{{Optimal ET sequence|legend=1| 12f, 277df, 289 }}

Badness: 0.074218

'''17-limit'''

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Mapping: [{{val|1 2 -1 -5 -8 -11 5}}, {{val|0 -5 40 94 138 177 -11}}]

POTE generator: ~18/17 = 99.656

~~Vals:~~ 12f, 277df, 289

{{Optimal ET sequence|legend=1| 12f, 277df, 289 }}

Badness: 0.050571

'''19-limit'''

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Mapping: [{{val|1 2 -1 -5 -8 -11 5 4}}, {{val|0 -5 40 94 138 177 -11 3}}]

POTE generator: ~18/17 = 99.659

~~Vals:~~ 12f, 277df, 289

{{Optimal ET sequence|legend=1| 12f, 277df, 289 }}

Badness: 0.042120

; [[Schismatic family #Quintaschis|Quintahelenic]] (12 & 301)

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Mapping: [{{val|1 2 -1 -5 -9}}, {{val|0 -5 40 94 150}}]

POTE generator: ~200/189 = 99.671

~~Vals:~~ 12, 289e, 301, 915, 1216ce

{{Optimal ET sequence|legend=1| 12, 289e, 301, 915, 1216ce }}

Badness: 0.082225

'''13-limit'''

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Mapping: [{{val|1 2 -1 -5 -9 -11}}, {{val|0 -5 40 94 150 177}}]

POTE generator: ~200/189 = 99.661

~~Vals:~~ 12f, 289e, 301

{{Optimal ET sequence|legend=1| 12f, 289e, 301 }}

Badness: 0.055570

'''17-limit'''

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Mapping: [{{val|1 2 -1 -5 -9 -11 5}}, {{val|0 -5 40 94 150 177 -11}}]

POTE generator: ~18/17 = 99.665

~~Vals:~~ 12f, 289e, 301

{{Optimal ET sequence|legend=1| 12f, 289e, 301 }}

Badness: 0.040412

'''19-limit'''

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Mapping: [{{val|1 2 -1 -5 -9 -11 5 4}}, {{val|0 -5 40 94 150 177 -11 3}}]

POTE generator: ~18/17 = 99.668

~~Vals:~~ 12f, 289e, 301

{{Optimal ET sequence|legend=1| 12f, 289e, 301 }}

Badness: 0.036840

; [[Schismatic family #Quintaschis|Quintahelenoid]] (12 & 301)

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Mapping: [{{val|1 2 -1 -5 -9 14}}, {{val|0 -5 40 94 150 -124}}]

POTE generator: ~200/189 = 99.672

~~Vals:~~ 12, 301, 614, 915

{{Optimal ET sequence|legend=1| 12, 301, 614, 915 }}

Badness: 0.066108

'''17-limit'''

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Mapping: [{{val|1 2 -1 -5 -9 14 5}}, {{val|0 -5 40 94 150 -124 -11}}]

POTE generator: ~18/17 = 99.671

~~Vals:~~ 12, 301, 915gg, 1216cegg

{{Optimal ET sequence|legend=1| 12, 301, 915gg, 1216cegg }}

Badness: 0.047908

'''19-limit'''

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Mapping: [{{val|1 2 -1 -5 -9 14 5 4}}, {{val|0 -5 40 94 150 -124 -11 3}}]

POTE generator: ~18/17 = 99.672

~~Vals:~~ 12, 301, 614gh, 915gghh

{{Optimal ET sequence|legend=1| 12, 301, 614gh, 915gghh }}

Badness: 0.039542

@@ Line 230: / Line 230: @@
 == Regular temperaments ==
-ED10 can also be thought of as a [[generator]] of the 2.3.5.17.19 [[Subgroup temperaments|subgroup temperament]] which tempers out 4624/4617, 6144/6137, and 6885/6859, which is a [[cluster temperament]] with 12 clusters of notes in an octave (''quintilischis'' temperament). This temperament is supported by {{Val list|12, 253, 265, 277, 289, 301, 313}}, and [[325edo|325]] EDOs.
+ED10 can also be thought of as a [[generator]] of the 2.3.5.17.19 [[Subgroup temperaments|subgroup temperament]] which tempers out 4624/4617, 6144/6137, and 6885/6859, which is a [[cluster temperament]] with 12 clusters of notes in an octave (''quintilischis'' temperament). This temperament is supported by {{Optimal ET sequence|12, 253, 265, 277, 289, 301, 313}}, and [[325edo|325]] EDOs.
 Tempering out 400/399 (equating 20/19 and 21/20) leads ''[[Schismatic family #Quintilipyth|quintilipyth]]'' (12&amp;253), and tempering out 476/475 (equating 19/17 with 28/25) leads ''[[Schismatic family #Quintaschis|quintaschis]]'' (12&amp;289).
@@ Line 241: / Line 241: @@
 Gencom mapping: [{{val|1 2 -1 5 4}}, {{val|0 -5 40 -11 3}}]<br>
 POTE generator: ~18/17 = 99.652<br>
-Vals: 12, 253, 265, 277, 289<br>
+{{Optimal ET sequence|legend=1| 12, 253, 265, 277, 289 }}<br>
 RMS error: 0.1636 cents<br><br>
 ; <font style="font-size: 1.15em">[[Schismatic family #Quintilipyth|Quintilipyth]] (12 &amp; 253)</font>
@@ Line 248: / Line 248: @@
 Mapping: [{{val|1 2 -1 -4}}, {{val|0 -5 40 82}}]<br>
 POTE generator: ~625/588 = 99.625<br>
-Vals: 12, 253, 265<br>
+{{Optimal ET sequence|legend=1| 12, 253, 265 }}<br>
 Badness: 0.253966<br><br>
 '''11-limit'''<br>
@@ Line 254: / Line 254: @@
 Mapping: [{{val|1 2 -1 -4 -7}}, {{val|0 -5 40 82 126}}]<br>
 POTE generator: ~35/33 = 99.616<br>
-Vals: 12, 253, 265, 518c, 783cc<br>
+{{Optimal ET sequence|legend=1| 12, 253, 265, 518c, 783cc }}<br>
 Badness: 0.113044<br><br>
 '''13-limit'''<br>
@@ Line 260: / Line 260: @@
 Mapping: [{{val|1 2 -1 -4 -7 -9}}, {{val|0 -5 40 82 126 153}}]<br>
 POTE generator: ~35/33 = 99.612<br>
-Vals: 12f, 253, 518c, 771cc<br>
+{{Optimal ET sequence|legend=1| 12f, 253, 518c, 771cc }}<br>
 Badness: 0.069127<br><br>
 '''17-limit'''<br>
@@ Line 266: / Line 266: @@
 Mapping: [{{val|1 2 -1 -4 -7 -9 5}}, {{val|0 -5 40 82 126 153 -11}}]<br>
 POTE generator: ~18/17 = 99.612<br>
-Vals: 12f, 253, 518c, 771cc<br>
+{{Optimal ET sequence|legend=1| 12f, 253, 518c, 771cc }}<br>
 Badness: 0.045992<br><br>
 '''19-limit'''<br>
@@ Line 272: / Line 272: @@
 Mapping: [{{val|1 2 -1 -4 -7 -9 5 4}}, {{val|0 -5 40 82 126 153 -11 3}}]<br>
 POTE generator: ~18/17 = 99.615<br>
-Vals: 12f, 253, 265, 518ch<br>
+{{Optimal ET sequence|legend=1| 12f, 253, 265, 518ch }}<br>
 Badness: 0.038155<br><br>
 ; <font style="font-size: 1.15em">[[Schismatic family #Quintaschis|Quintaschis]] (12 &amp; 289)</font>
@@ Line 279: / Line 279: @@
 Mapping: [{{val|1 2 -1 -5}}, {{val|0 -5 40 94}}]<br>
 POTE generator: ~200/189 = 99.664<br>
-Vals: 12, 277d, 289<br>
+{{Optimal ET sequence|legend=1| 12, 277d, 289 }}<br>
 Badness: 0.132890<br><br>
 '''11-limit'''<br>
@@ Line 285: / Line 285: @@
 Mapping: [{{val|1 2 -1 -5 -8}}, {{val|0 -5 40 94 138}}]<br>
 POTE generator: ~35/33 = 99.653<br>
-Vals: 12, 277d, 289<br>
+{{Optimal ET sequence|legend=1| 12, 277d, 289 }}<br>
 Badness: 0.111477<br><br>
 '''13-limit'''<br>
@@ Line 291: / Line 291: @@
 Mapping: [{{val|1 2 -1 -5 -8 -11}}, {{val|0 -5 40 94 138 177}}]<br>
 POTE generator: ~35/33 = 99.658<br>
-Vals: 12f, 277df, 289<br>
+{{Optimal ET sequence|legend=1| 12f, 277df, 289 }}<br>
 Badness: 0.074218<br><br>
 '''17-limit'''<br>
@@ Line 297: / Line 297: @@
 Mapping: [{{val|1 2 -1 -5 -8 -11 5}}, {{val|0 -5 40 94 138 177 -11}}]<br>
 POTE generator: ~18/17 = 99.656<br>
-Vals: 12f, 277df, 289<br>
+{{Optimal ET sequence|legend=1| 12f, 277df, 289 }}<br>
 Badness: 0.050571<br><br>
 '''19-limit'''<br>
@@ Line 303: / Line 303: @@
 Mapping: [{{val|1 2 -1 -5 -8 -11 5 4}}, {{val|0 -5 40 94 138 177 -11 3}}]<br>
 POTE generator: ~18/17 = 99.659<br>
-Vals: 12f, 277df, 289<br>
+{{Optimal ET sequence|legend=1| 12f, 277df, 289 }}<br>
 Badness: 0.042120<br><br>
 ; <font style="font-size: 1.15em">[[Schismatic family #Quintaschis|Quintahelenic]] (12 &amp; 301)</font>
@@ Line 310: / Line 310: @@
 Mapping: [{{val|1 2 -1 -5 -9}}, {{val|0 -5 40 94 150}}]<br>
 POTE generator: ~200/189 = 99.671<br>
-Vals: 12, 289e, 301, 915, 1216ce<br>
+{{Optimal ET sequence|legend=1| 12, 289e, 301, 915, 1216ce }}<br>
 Badness: 0.082225<br><br>
 '''13-limit'''<br>
@@ Line 316: / Line 316: @@
 Mapping: [{{val|1 2 -1 -5 -9 -11}}, {{val|0 -5 40 94 150 177}}]<br>
 POTE generator: ~200/189 = 99.661<br>
-Vals: 12f, 289e, 301<br>
+{{Optimal ET sequence|legend=1| 12f, 289e, 301 }}<br>
 Badness: 0.055570<br><br>
 '''17-limit'''<br>
@@ Line 322: / Line 322: @@
 Mapping: [{{val|1 2 -1 -5 -9 -11 5}}, {{val|0 -5 40 94 150 177 -11}}]<br>
 POTE generator: ~18/17 = 99.665<br>
-Vals: 12f, 289e, 301<br>
+{{Optimal ET sequence|legend=1| 12f, 289e, 301 }}<br>
 Badness: 0.040412<br><br>
 '''19-limit'''<br>
@@ Line 328: / Line 328: @@
 Mapping: [{{val|1 2 -1 -5 -9 -11 5 4}}, {{val|0 -5 40 94 150 177 -11 3}}]<br>
 POTE generator: ~18/17 = 99.668<br>
-Vals: 12f, 289e, 301<br>
+{{Optimal ET sequence|legend=1| 12f, 289e, 301 }}<br>
 Badness: 0.036840<br><br>
 ; <font style="font-size: 1.15em">[[Schismatic family #Quintaschis|Quintahelenoid]] (12 &amp; 301)</font>
@@ Line 335: / Line 335: @@
 Mapping: [{{val|1 2 -1 -5 -9 14}}, {{val|0 -5 40 94 150 -124}}]<br>
 POTE generator: ~200/189 = 99.672<br>
-Vals: 12, 301, 614, 915<br>
+{{Optimal ET sequence|legend=1| 12, 301, 614, 915 }}<br>
 Badness: 0.066108<br><br>
 '''17-limit'''<br>
@@ Line 341: / Line 341: @@
 Mapping: [{{val|1 2 -1 -5 -9 14 5}}, {{val|0 -5 40 94 150 -124 -11}}]<br>
 POTE generator: ~18/17 = 99.671<br>
-Vals: 12, 301, 915gg, 1216cegg<br>
+{{Optimal ET sequence|legend=1| 12, 301, 915gg, 1216cegg }}<br>
 Badness: 0.047908<br><br>
 '''19-limit'''<br>
@@ Line 347: / Line 347: @@
 Mapping: [{{val|1 2 -1 -5 -9 14 5 4}}, {{val|0 -5 40 94 150 -124 -11 3}}]<br>
 POTE generator: ~18/17 = 99.672<br>
-Vals: 12, 301, 614gh, 915gghh<br>
+{{Optimal ET sequence|legend=1| 12, 301, 614gh, 915gghh }}<br>
 Badness: 0.039542<br><br>