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Diaschismic family: Difference between revisions

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{{Technical data page}}
{{Technical data page}}
The 5-limit parent comma for the '''diaschismic family''' is 2048/2025, the [[diaschisma]]. The period is half an octave, and the generator is a fifth. Three periods gives 1800 cents, and decreasing this by two fifths gives the major third. [[34edo]] is a good tuning choice, with [[46edo]], [[56edo]], [[58edo]], or [[80edo]] being other possibilities. Both [[12edo]] and [[22edo]] support it, and retuning them to a MOS of diaschismic gives two scale possibilities.
The [[5-limit]] parent [[comma]] for the '''diaschismic family''' of [[regular temperament|temperaments]] is 2048/2025, the [[diaschisma]]. The [[period]] is half an [[2/1|octave]], and the [[generator]] is a fifth. Three periods gives 1800 cents, and decreasing this by two fifths gives the major third. [[34edo]] is a good tuning choice, with [[46edo]], [[56edo]], [[58edo]], or [[80edo]] being other possibilities. Both [[12edo]] and [[22edo]] support it, and retuning them to a MOS of diaschismic gives two scale possibilities.


== Diaschismic ==
== Diaschismic ==
{{Main| Diaschismic }}
{{Main| Diaschismic }}


This temperament is also known as '''srutal''' in the 5-limit, but that name more strictly speaking refers to the [[Diaschismic family#Srutal|34d&46 extension]] to the [[7-limit]] that adds [[4375/4374]] to the comma list.
This temperament is also known as '''srutal''' in the 5-limit, but that name more strictly speaking refers to the [[#Srutal|34d & 46 extension]] to the [[7-limit]] that adds [[4375/4374]] to the comma list.


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
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: mapping generators: ~45/32, ~3
: mapping generators: ~45/32, ~3


[[Optimal tuning]] ([[POTE]]): ~45/32 = 1\2, ~3/2 = 704.898
[[Optimal tuning]] ([[POTE]]): ~45/32 = 600.000, ~3/2 = 704.898


[[Tuning ranges]]:  
[[Tuning ranges]]:  
* 5-odd-limit [[diamond monotone]]: ~3/2 = [600.000 to 720.000] (1\2 to 6\10)
* [[5-odd-limit]] [[diamond monotone]]: ~3/2 = [600.000 to 720.000] (1\2 to 6\10)
* 5-odd-limit [[diamond tradeoff]]: ~3/2 = [701.955, 706.843]
* 5-odd-limit [[diamond tradeoff]]: ~3/2 = [701.955, 706.843]
* 5-odd-limit diamond monotone and tradeoff: ~3/2 = [701.955, 706.843]


{{Optimal ET sequence|legend=1| 10, 12, 22, 34, 46, 80, 206c, 286bc }}
{{Optimal ET sequence|legend=1| 10, 12, 22, 34, 46, 80, 206c, 286bc }}


[[Badness]]: 0.019915
[[Badness]] (Smith): 0.019915


=== Srutal archagall ===
=== Srutal archagall ===
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: mapping generators: ~17/12, ~3
: mapping generators: ~17/12, ~3


Optimal tuning (CTE): ~17/12 = 1\2, ~3/2 = 705.1272
Optimal tuning (CTE): ~17/12 = 600.000, ~3/2 = 705.1272


Optimal ET sequence: {{Optimal ET sequence| 10, 12, 22, 34, 80, 114, 194bc }}
{{Optimal ET sequence|legend=0| 10, 12, 22, 34, 80, 114, 194bc }}


Badness: 0.00575
Badness (Smith): 0.00575


=== Overview to extensions ===
=== Overview to extensions ===
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{{See also| Srutal vs diaschismic }}
{{See also| Srutal vs diaschismic }}


A simpler characterization than the one given by the normal comma list is that diaschismic adds [[126/125]] or [[5120/5103]] to the set of commas, and it can also be called 46 & 58. However described, diaschismic has a 1/2-octave period and a sharp fifth generator like pajara, but not so sharp, giving a more accurate but more complex temperament. [[58edo]] provides an excellent tuning, but an alternative is to make [[7/4]] just by making the fifth 703.897 cents, as opposed to 703.448 cents for 58edo.
A simpler characterization than the one given by the normal comma list is that diaschismic adds [[126/125]] or [[5120/5103]] to the set of commas, and it can also be called {{nowrap| 46 & 58 }}. However described, diaschismic has a 1/2-octave period and a sharp fifth generator like pajara, but not so sharp, giving a more accurate but more complex temperament. [[58edo]] provides an excellent tuning, but an alternative is to make [[7/4]] just by making the fifth 703.897 cents, as opposed to 703.448 cents for 58edo.


Diaschismic extends naturally to the 17-limit, for which the same tunings may be used, making it one of the most important of the higher-limit rank-2 temperaments. Adding the 11-limit adds the commas 176/175, 896/891 and 441/440. The 13-limit yields 196/195, 351/350, and 364/363; the 17-limit adds 136/135, 221/220, and 442/441. If you want to explore higher-limit harmonies, diaschismic is certainly one excellent way to do it; Mos of 34 notes and even more the 46-note mos will encompass very great deal of it. Of course 46 or 58 equal provide alternatives which in many ways are similar, particularly in the case of 58.
Diaschismic extends naturally to the 17-limit, for which the same tunings may be used, making it one of the most important of the higher-limit rank-2 temperaments. Adding the 11-limit adds the commas 176/175, 896/891 and 441/440. The 13-limit yields 196/195, 351/350, and 364/363; the 17-limit adds 136/135, 221/220, and 442/441. If you want to explore higher-limit harmonies, diaschismic is certainly one excellent way to do it; Mos of 34 notes and even more the 46-note mos will encompass very great deal of it. Of course 46 or 58 equal provide alternatives which in many ways are similar, particularly in the case of 58.
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{{Multival|legend=1| 2 -4 -16 -11 -31 -26 }}
{{Multival|legend=1| 2 -4 -16 -11 -31 -26 }}


[[Optimal tuning]] ([[POTE]]): ~45/32 = 1\2, ~3/2 = 703.681
[[Optimal tuning]] ([[POTE]]): ~45/32 = 600.000, ~3/2 = 703.681


[[Tuning ranges]]:  
[[Tuning ranges]]:  
* 7- and 9-odd-limit [[diamond monotone]]: ~3/2 = [700.000, 705.882] (7\12 to 20\34)
* 7- and 9-odd-limit [[diamond monotone]]: ~3/2 = [700.000, 705.882] (7\12 to 20\34)
* 7- and 9-odd-limit [[diamond tradeoff]]: ~3/2 = [701.955, 706.843]
* 7- and 9-odd-limit [[diamond tradeoff]]: ~3/2 = [701.955, 706.843]
* 7- and 9-odd-limit diamond monotone and tradeoff: ~3/2 = [701.955, 705.882]


{{Optimal ET sequence|legend=1| 12, 46, 58, 104c, 162c }}
{{Optimal ET sequence|legend=1| 12, 46, 58, 104c, 162c }}


[[Badness]]: 0.037914
[[Badness]] (Smith): 0.037914


=== 11-limit ===
=== 11-limit ===
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Mapping: {{mapping| 2 0 11 31 45 | 0 1 -2 -8 -12 }}
Mapping: {{mapping| 2 0 11 31 45 | 0 1 -2 -8 -12 }}


Optimal tuning (POTE): ~45/32 = 1\2, ~3/2 = 703.714
Optimal tuning (POTE): ~45/32 = 600.000, ~3/2 = 703.714


Tuning ranges:  
Tuning ranges:  
* 11-odd-limit diamond monotone: ~3/2 = [700.000, 704.348] (7\12 to 27\46)
* 11-odd-limit diamond monotone: ~3/2 = [700.000, 704.348] (7\12 to 27\46)
* 11-odd-limit diamond tradeoff: ~3/2 = [701.955, 706.843]
* 11-odd-limit diamond tradeoff: ~3/2 = [701.955, 706.843]
* 11-odd-limit diamond monotone and tradeoff: ~3/2 = [701.955, 704.348]


{{Optimal ET sequence|legend=1| 12, 46, 58, 104c, 162ce }}
{{Optimal ET sequence|legend=0| 12, 46, 58, 104c, 162ce }}


Badness: 0.025034
Badness (Smith): 0.025034


=== 13-limit ===
=== 13-limit ===
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Mapping: {{mapping| 2 0 11 31 45 55 | 0 1 -2 -8 -12 -15 }}
Mapping: {{mapping| 2 0 11 31 45 55 | 0 1 -2 -8 -12 -15 }}


Optimal tuning (POTE): ~45/32 = 1\2, ~3/2 = 703.704
Optimal tuning (POTE): ~45/32 = 600.000, ~3/2 = 703.704


Tuning ranges:  
Tuning ranges:  
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* 13-odd-limit diamond tradeoff: ~3/2 = [701.955, 706.843]
* 13-odd-limit diamond tradeoff: ~3/2 = [701.955, 706.843]
* 15-odd-limit diamond tradeoff: ~3/2 = [701.955, 711.731]
* 15-odd-limit diamond tradeoff: ~3/2 = [701.955, 711.731]
* 13- and 15-odd-limit diamond monotone and tradeoff: ~3/2 = [703.448, 704.348]


{{Optimal ET sequence|legend=1| 46, 58, 104c, 162cef }}
{{Optimal ET sequence|legend=0| 46, 58, 104c, 162cef }}


Badness: 0.018926
Badness (Smith): 0.018926


=== 17-limit ===
=== 17-limit ===
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Mapping: {{mapping| 2 0 11 31 45 55 5 | 0 1 -2 -8 -12 -15 1 }}
Mapping: {{mapping| 2 0 11 31 45 55 5 | 0 1 -2 -8 -12 -15 1 }}


Optimal tuning (POTE): ~17/12 = 1\2, ~3/2 = 703.812
Optimal tuning (POTE): ~17/12 = 600.000, ~3/2 = 703.812


Tuning ranges:  
Tuning ranges:  
* 17-odd-limit diamond monotone: ~3/2 = [703.448, 704.348] (34\58 to 27\46)
* 17-odd-limit diamond monotone: ~3/2 = [703.448, 704.348] (34\58 to 27\46)
* 17-odd-limit diamond tradeoff: ~3/2 = [698.955, 711.731]
* 17-odd-limit diamond tradeoff: ~3/2 = [698.955, 711.731]
* 17-odd-limit diamond monotone and tradeoff: ~3/2 = [703.448, 704.348]


{{Optimal ET sequence|legend=1| 46, 58, 104c }}
{{Optimal ET sequence|legend=0| 46, 58, 104c }}


Badness: 0.016425
Badness (Smith): 0.016425


=== No-19s 23-limit (Na"Naa') ===
=== No-19s 23-limit (Na"Naa') ===
<b>Na"Naa'</b> is a remarkable subgroup temperament of 46&amp;58 with a prime harmonic of 23.
<b>Na"Naa'</b> is a remarkable subgroup temperament of {{nowrap| 46 & 58 }} with a prime harmonic of 23.


Subgroup: 2.3.5.7.11.13.17.23
Subgroup: 2.3.5.7.11.13.17.23
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Sval mapping: {{mapping| 2 0 11 31 45 55 5 63 | 0 1 -2 -8 -12 -15 1 -17 }}
Sval mapping: {{mapping| 2 0 11 31 45 55 5 63 | 0 1 -2 -8 -12 -15 1 -17 }}


Optimal tuning (POTE): ~17/12 = 1\2, ~3/2 = 703.870
Optimal tuning (POTE): ~17/12 = 600.000, ~3/2 = 703.870


{{Optimal ET sequence|legend=1| 46, 58i, 104ci }}
{{Optimal ET sequence|legend=0| 46, 58i, 104ci }}


== Pajara ==
== Pajara ==
{{Main| Pajara }}
{{Main| Pajara }}


Pajara is closely associated with 22edo (not to mention [[Paul Erlich]]) but other tunings are possible. The 1/2-octave period serves as both a [[10/7]] and a [[7/5]]. Aside from 22edo, 34 with the val {{val| 34 54 79 96 }} and 56 with the val {{val| 56 89 130 158 }} are are interesting alternatives, with more accpetable fifths, and a tetrad which is more clearly a dominant seventh. As such, they are closer to the tuning of 12edo and of common practice Western music in general, while retaining the distictiveness of a sharp fifth.
Pajara is closely associated with 22edo (not to mention [[Paul Erlich]]) but other tunings are possible. The 1/2-octave period serves as both a [[10/7]] and a [[7/5]]. Aside from 22edo, 34 with the val {{val| 34 54 79 96 }} and 56 with the val {{val| 56 89 130 158 }} are interesting alternatives, with more acceptable fifths, and a tetrad which is more clearly a dominant seventh. As such, they are closer to the tuning of 12edo and of common practice Western music in general, while retaining the distictiveness of a sharp fifth.


Pajara extends nicely to an 11-limit version, for which the 56 tuning can be used, but a good alternative is to make the major thirds pure by setting the fifth to be 706.843 cents. Now 99/98, 100/99, 176/175 and 896/891 are being tempered out.
Pajara extends nicely to an 11-limit version, for which the 56edo tuning can be used, but a good alternative is to make the major thirds pure by setting the fifth to be 706.843 cents. Now 99/98, 100/99, 176/175 and 896/891 are being tempered out.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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{{Multival|legend=1| 2 -4 -4 -11 -12 2 }}
{{Multival|legend=1| 2 -4 -4 -11 -12 2 }}


[[Optimal tuning]] ([[POTE]]): ~7/5 = 1\2, ~3/2 = 707.048
[[Optimal tuning]] ([[POTE]]): ~7/5 = 600.000, ~3/2 = 707.048


[[Tuning ranges]]:
[[Tuning ranges]]:
* 7- and 9-odd-limit [[diamond monotone]]: ~3/2 = [700.000, 720.000] (7\12 to 6\10)
* 7- and 9-odd-limit [[diamond monotone]]: ~3/2 = [700.000, 720.000] (7\12 to 6\10)
* 7- and 9-odd-limit [[diamond tradeoff]]: ~3/2 = [701.955, 715.587]
* 7- and 9-odd-limit [[diamond tradeoff]]: ~3/2 = [701.955, 715.587]
* 7- and 9-odd-limit diamond monotone and tradeoff: ~3/2 = [701.955, 715.587]


{{Optimal ET sequence|legend=1| 10, 12, 22, 34d, 56d }}
{{Optimal ET sequence|legend=1| 10, 12, 22, 34d, 56d }}


[[Badness]]: 0.020033
[[Badness]] (Smith): 0.020033


=== 11-limit ===
=== 11-limit ===
Line 186: Line 180:
Mapping: {{mapping| 2 0 11 12 26 | 0 1 -2 -2 -6 }}
Mapping: {{mapping| 2 0 11 12 26 | 0 1 -2 -2 -6 }}


{{Multival|legend=1| 2 -4 -4 -12 -11 -12 -26 2 -14 -20 }}
Wegie: {{multival| 2 -4 -4 -12 -11 -12 -26 2 -14 -20 }}


Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 706.885
Optimal tuning (POTE): ~7/5 = 600.000, ~3/2 = 706.885


Tuning ranges:
Tuning ranges:
* 11-odd-limit diamond monotone: ~3/2 = [700.000, 709.091] (7\12 to 13\22)
* 11-odd-limit diamond monotone: ~3/2 = [700.000, 709.091] (7\12 to 13\22)
* 11-odd-limit diamond tradeoff: ~3/2 = [701.955, 715.587]
* 11-odd-limit diamond tradeoff: ~3/2 = [701.955, 715.587]
* 11-odd-limit diamond monotone and tradeoff: ~3/2 = [701.955, 709.091]


{{Optimal ET sequence|legend=1| 10e, 12, 22, 34d, 56d }}
{{Optimal ET sequence|legend=0| 10e, 12, 22, 34d, 56d }}


Badness: 0.020343
Badness (Smith): 0.020343


==== 13-limit ====
==== 13-limit ====
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Mapping: {{mapping| 2 0 11 12 26 1 | 0 1 -2 -2 -6 2 }}
Mapping: {{mapping| 2 0 11 12 26 1 | 0 1 -2 -2 -6 2 }}


Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 708.919
Optimal tuning (POTE): ~7/5 = 600.000, ~3/2 = 708.919


{{Optimal ET sequence|legend=1| 10e, 12, 22 }}
{{Optimal ET sequence|legend=0| 10e, 12, 22 }}


Badness: 0.027642
Badness (Smith): 0.027642


===== 17-limit =====
===== 17-limit =====
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Mapping: {{mapping| 2 0 11 12 26 1 5 | 0 1 -2 -2 -6 2 1 }}
Mapping: {{mapping| 2 0 11 12 26 1 5 | 0 1 -2 -2 -6 2 1 }}


Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 708.806
Optimal tuning (POTE): ~7/5 = 600.000, ~3/2 = 708.806


{{Optimal ET sequence|legend=1| 10e, 12, 22 }}
{{Optimal ET sequence|legend=0| 10e, 12, 22 }}


Badness: 0.020899
Badness (Smith): 0.020899


==== Pajarina ====
==== Pajarina ====
Line 232: Line 225:
Mapping: {{mapping| 2 0 11 12 26 36 | 0 1 -2 -2 -6 -9 }}
Mapping: {{mapping| 2 0 11 12 26 36 | 0 1 -2 -2 -6 -9 }}


Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 706.133
Optimal tuning (POTE): ~7/5 = 600.000, ~3/2 = 706.133


{{Optimal ET sequence|legend=1| 12f, 22, 34d }}
{{Optimal ET sequence|legend=0| 12f, 22, 34d }}


Badness: 0.022327
Badness (Smith): 0.022327


===== 17-limit =====
===== 17-limit =====
Line 245: Line 238:
Mapping: {{mapping| 2 0 11 12 26 36 5 | 0 1 -2 -2 -6 -9 1 }}
Mapping: {{mapping| 2 0 11 12 26 36 5 | 0 1 -2 -2 -6 -9 1 }}


Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 706.410
Optimal tuning (POTE): ~7/5 = 600.000, ~3/2 = 706.410


{{Optimal ET sequence|legend=1| 12f, 22, 34d }}
{{Optimal ET sequence|legend=0| 12f, 22, 34d }}


Badness: 0.018375
Badness (Smith): 0.018375


==== Pajarita ====
==== Pajarita ====
Line 258: Line 251:
Mapping: {{mapping| 2 0 11 12 26 17 | 0 1 -2 -2 -6 -3 }}
Mapping: {{mapping| 2 0 11 12 26 17 | 0 1 -2 -2 -6 -3 }}


Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 707.450
Optimal tuning (POTE): ~7/5 = 600.000, ~3/2 = 707.450


{{Optimal ET sequence|legend=1| 10e, 12f, 22f }}
{{Optimal ET sequence|legend=0| 10e, 12f, 22f }}


Badness: 0.022677
Badness (Smith): 0.022677


===== 17-limit =====
===== 17-limit =====
Line 271: Line 264:
Mapping: {{mapping| 2 0 11 12 26 17 5 | 0 1 -2 -2 -6 -3 1 }}
Mapping: {{mapping| 2 0 11 12 26 17 5 | 0 1 -2 -2 -6 -3 1 }}


Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 707.947
Optimal tuning (POTE): ~7/5 = 600.000, ~3/2 = 707.947


{{Optimal ET sequence|legend=1| 10e, 12f, 22f }}
{{Optimal ET sequence|legend=0| 10e, 12f, 22f }}


Badness: 0.019007
Badness (Smith): 0.019007


=== Pajarous ===
=== Pajarous ===
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Mapping: {{mapping| 2 0 11 12 -9 | 0 1 -2 -2 5 }}
Mapping: {{mapping| 2 0 11 12 -9 | 0 1 -2 -2 5 }}


Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 709.578
Optimal tuning (POTE): ~7/5 = 600.000, ~3/2 = 709.578


Tuning ranges:
Tuning ranges:
* 11-odd-limit diamond monotone: ~3/2 = 709.091 (13\22)
* 11-odd-limit diamond monotone: ~3/2 = 709.091 (13\22)
* 11-odd-limit diamond tradeoff: ~3/2 = [701.955, 715.803]
* 11-odd-limit diamond tradeoff: ~3/2 = [701.955, 715.803]
* 11-odd-limit diamond monotone and tradeoff: ~3/2 = 709.091


{{Optimal ET sequence|legend=1| 10, 12e, 22, 120bce, 142bce }}
{{Optimal ET sequence|legend=0| 10, 12e, 22, 120bce, 142bce }}


Badness: 0.028349
Badness (Smith): 0.028349


==== 13-limit ====
==== 13-limit ====
Line 302: Line 294:
Mapping: {{mapping| 2 0 11 12 -9 1 | 0 1 -2 -2 5 2 }}
Mapping: {{mapping| 2 0 11 12 -9 1 | 0 1 -2 -2 5 2 }}


Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 710.240
Optimal tuning (POTE): ~7/5 = 600.000, ~3/2 = 710.240


{{Optimal ET sequence|legend=1| 10, 22, 54f, 76bdff }}
{{Optimal ET sequence|legend=0| 10, 22, 54f, 76bdff }}


Badness: 0.025176
Badness (Smith): 0.025176


===== 17-limit =====
===== 17-limit =====
Line 315: Line 307:
Mapping: {{mapping| 2 0 11 12 -9 1 5 | 0 1 -2 -2 5 2 1 }}
Mapping: {{mapping| 2 0 11 12 -9 1 5 | 0 1 -2 -2 5 2 1 }}


Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 710.221
Optimal tuning (POTE): ~7/5 = 600.000, ~3/2 = 710.221


{{Optimal ET sequence|legend=1| 10, 22, 54f, 76bdff }}
{{Optimal ET sequence|legend=0| 10, 22, 54f, 76bdff }}


Badness: 0.018249
Badness (Smith): 0.018249


==== Pajaro ====
==== Pajaro ====
Line 328: Line 320:
Mapping: {{mapping| 2 0 11 12 -9 17 | 0 1 -2 -2 5 -3 }}
Mapping: {{mapping| 2 0 11 12 -9 17 | 0 1 -2 -2 5 -3 }}


Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 710.818
Optimal tuning (POTE): ~7/5 = 600.000, ~3/2 = 710.818


{{Optimal ET sequence|legend=1| 10, 22f, 32f, 54ff }}
{{Optimal ET sequence|legend=0| 10, 22f, 32f, 54ff }}


Badness: 0.027355
Badness (Smith): 0.027355


===== 17-limit =====
===== 17-limit =====
Line 341: Line 333:
Mapping: {{mapping| 2 0 11 12 -9 17 5 | 0 1 -2 -2 5 -3 1 }}
Mapping: {{mapping| 2 0 11 12 -9 17 5 | 0 1 -2 -2 5 -3 1 }}


Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 710.866
Optimal tuning (POTE): ~7/5 = 600.000, ~3/2 = 710.866


{{Optimal ET sequence|legend=1| 10, 22f, 32f, 54ff }}
{{Optimal ET sequence|legend=0| 10, 22f, 32f, 54ff }}


Badness: 0.019844
Badness (Smith): 0.019844


=== Pajaric ===
=== Pajaric ===
Line 354: Line 346:
Mapping: {{mapping| 2 0 11 12 7 | 0 1 -2 -2 0 }}
Mapping: {{mapping| 2 0 11 12 7 | 0 1 -2 -2 0 }}


Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 705.524
Optimal tuning (POTE): ~7/5 = 600.000, ~3/2 = 705.524


{{Optimal ET sequence|legend=1| 10, 12, 22e, 34dee }}
{{Optimal ET sequence|legend=0| 10, 12, 22e, 34dee }}


Badness: 0.023798
Badness (Smith): 0.023798


==== 13-limit ====
==== 13-limit ====
Line 367: Line 359:
Mapping: {{mapping| 2 0 11 12 7 17 | 0 1 -2 -2 0 -3 }}
Mapping: {{mapping| 2 0 11 12 7 17 | 0 1 -2 -2 0 -3 }}


Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 707.442
Optimal tuning (POTE): ~7/5 = 600.000, ~3/2 = 707.442


{{Optimal ET sequence|legend=1| 10, 12f, 22ef }}
{{Optimal ET sequence|legend=0| 10, 12f, 22ef }}


Badness: 0.020461
Badness (Smith): 0.020461


==== 17-limit ====
==== 17-limit ====
Line 380: Line 372:
Mapping: {{mapping| 2 0 11 12 7 17 5 | 0 1 -2 -2 0 -3 1 }}
Mapping: {{mapping| 2 0 11 12 7 17 5 | 0 1 -2 -2 0 -3 1 }}


Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 708.544
Optimal tuning (POTE): ~7/5 = 600.000, ~3/2 = 708.544


{{Optimal ET sequence|legend=1| 10, 12f, 22ef }}
{{Optimal ET sequence|legend=0| 10, 12f, 22ef }}


Badness: 0.017592
Badness (Smith): 0.017592


=== Hemipaj ===
=== Hemipaj ===
Line 393: Line 385:
Mapping: {{mapping| 2 1 9 10 8 | 0 2 -4 -4 -1 }}
Mapping: {{mapping| 2 1 9 10 8 | 0 2 -4 -4 -1 }}


Optimal tuning (POTE): ~7/5 = 1\2, ~11/8 = 546.383
Optimal tuning (POTE): ~7/5 = 600.000, ~11/8 = 546.383


{{Optimal ET sequence|legend=1| 20, 22, 68d, 90d }}
{{Optimal ET sequence|legend=0| 20, 22, 68d, 90d }}


Badness: 0.038890
Badness (Smith): 0.038890


=== Hemifourths ===
=== Hemifourths ===
Line 406: Line 398:
Mapping: {{mapping| 2 0 11 12 -1 | 0 2 -4 -4 5 }}
Mapping: {{mapping| 2 0 11 12 -1 | 0 2 -4 -4 5 }}


Optimal tuning (POTE): ~7/5 = 1\2, ~55/32 = 953.093
Optimal tuning (POTE): ~7/5 = 600.000, ~55/32 = 953.093


{{Optimal ET sequence|legend=1| 10, 24d, 34d }}
{{Optimal ET sequence|legend=0| 10, 24d, 34d }}


Badness: 0.048885
Badness (Smith): 0.048885


==== 13-limit ====
==== 13-limit ====
Line 419: Line 411:
Mapping: {{mapping| 2 0 11 12 -1 9 | 0 2 -4 -4 5 -1 }}
Mapping: {{mapping| 2 0 11 12 -1 9 | 0 2 -4 -4 5 -1 }}


Optimal tuning (POTE): ~7/5 = 1\2, ~26/15 = 953.074
Optimal tuning (POTE): ~7/5 = 600.000, ~26/15 = 953.074


{{Optimal ET sequence|legend=1| 10, 24d, 34d }}
{{Optimal ET sequence|legend=0| 10, 24d, 34d }}


Badness: 0.028755
Badness (Smith): 0.028755


==== 17-limit ====
==== 17-limit ====
Line 432: Line 424:
Mapping: {{mapping| 2 0 11 12 -1 9 5 | 0 2 -4 -4 5 -1 2 }}
Mapping: {{mapping| 2 0 11 12 -1 9 5 | 0 2 -4 -4 5 -1 2 }}


Optimal tuning (POTE): ~7/5 = 1\2, ~26/15 = 953.210
Optimal tuning (POTE): ~7/5 = 600.000, ~26/15 = 953.210


{{Optimal ET sequence|legend=1| 10, 24d, 34d }}
{{Optimal ET sequence|legend=0| 10, 24d, 34d }}
 
Badness: 0.021790


Badness (Smith): 0.021790


== Srutal ==
== Srutal ==
Line 450: Line 441:
{{Multival|legend=1| 2 -4 30 -11 42 81 }}
{{Multival|legend=1| 2 -4 30 -11 42 81 }}


[[Optimal tuning]] ([[POTE]]): ~45/32 = 1\2, ~3/2 = 704.814
[[Optimal tuning]] ([[POTE]]): ~45/32 = 600.000, ~3/2 = 704.814


[[Tuning ranges]]:  
[[Tuning ranges]]:  
* 7- and 9-odd-limit [[diamond monotone]]: ~3/2 = [703.448, 705.882] (34\58 to 20\34)
* 7- and 9-odd-limit [[diamond monotone]]: ~3/2 = [703.448, 705.882] (34\58 to 20\34)
* 7- and 9-odd-limit [[diamond tradeoff]]: ~3/2 = [701.955, 706.843]
* 7- and 9-odd-limit [[diamond tradeoff]]: ~3/2 = [701.955, 706.843]
* 7- and 9-odd-limit diamond monotone and tradeoff: ~3/2 = [703.448, 705.882]


{{Optimal ET sequence|legend=1| 34d, 46, 80, 126, 206cd, 332bcd }}
{{Optimal ET sequence|legend=1| 34d, 46, 80, 126, 206cd, 332bcd }}


[[Badness]]: 0.091504
[[Badness]] (Smith): 0.091504


=== 11-limit ===
=== 11-limit ===
Line 468: Line 458:
Mapping: {{mapping| 2 0 11 -42 -28 | 0 1 -2 15 11 }}
Mapping: {{mapping| 2 0 11 -42 -28 | 0 1 -2 15 11 }}


Optimal tuning (POTE): ~45/32 = 1\2, ~3/2 = 704.856
Optimal tuning (POTE): ~45/32 = 600.000, ~3/2 = 704.856


Tuning ranges:  
Tuning ranges:  
* 11-odd-limit diamond monotone: ~3/2 = [704.348, 705.882] (27\46 to 20\34)
* 11-odd-limit diamond monotone: ~3/2 = [704.348, 705.882] (27\46 to 20\34)
* 11-odd-limit diamond tradeoff: ~3/2 = [701.955, 706.843]
* 11-odd-limit diamond tradeoff: ~3/2 = [701.955, 706.843]
* 11-odd-limit diamond monotone and tradeoff: ~3/2 = [704.348, 705.882]


{{Optimal ET sequence|legend=1| 34d, 46, 80, 126, 206cd }}
{{Optimal ET sequence|legend=0| 34d, 46, 80, 126, 206cd }}


Badness: 0.035315
Badness (Smith): 0.035315


=== 13-limit ===
=== 13-limit ===
Line 486: Line 475:
Mapping: {{mapping| 2 0 11 -42 -28 -18 | 0 1 -2 15 11 8 }}
Mapping: {{mapping| 2 0 11 -42 -28 -18 | 0 1 -2 15 11 8 }}


Optimal tuning (POTE): ~45/32 = 1\2, ~3/2 = 704.881
Optimal tuning (POTE): ~45/32 = 600.000, ~3/2 = 704.881


Tuning ranges:  
Tuning ranges:  
Line 492: Line 481:
* 13-odd-limit diamond tradeoff: ~3/2 = [701.955, 706.843]
* 13-odd-limit diamond tradeoff: ~3/2 = [701.955, 706.843]
* 15-odd-limit diamond tradeoff: ~3/2 = [701.955, 711.731]
* 15-odd-limit diamond tradeoff: ~3/2 = [701.955, 711.731]
* 13- and 15-odd-limit diamond monotone and tradeoff: ~3/2 = [704.348, 705.882]


{{Optimal ET sequence|legend=1| 34d, 46, 80, 206cd, 286bcde }}
{{Optimal ET sequence|legend=0| 34d, 46, 80, 206cd, 286bcde }}


Badness: 0.025286
Badness (Smith): 0.025286


=== 17-limit ===
=== 17-limit ===
Line 510: Line 498:
* 17-odd-limit diamond monotone: ~3/2 = [704.348, 705.882] (27\46 to 20\34)
* 17-odd-limit diamond monotone: ~3/2 = [704.348, 705.882] (27\46 to 20\34)
* 17-odd-limit diamond tradeoff: ~3/2 = [698.955, 711.731]
* 17-odd-limit diamond tradeoff: ~3/2 = [698.955, 711.731]
* 17-odd-limit diamond monotone and tradeoff: ~3/2 = [704.348, 705.882]


{{Optimal ET sequence|legend=1| 34d, 46, 80, 126, 206cd }}
{{Optimal ET sequence|legend=0| 34d, 46, 80, 126, 206cd }}


Badness: 0.018594
Badness (Smith): 0.018594


=== 19-limit ===
=== 19-limit ===
Srutal, shrutar and bidia have similar 19-limit properties, tempering 190/189, related rank-3 [[julius]].
Srutal, shrutar and bidia have similar 19-limit properties, tempering out 190/189, related to rank-3 [[julius]].


Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13.17.19
Line 525: Line 512:
Mapping: {{mapping| 2 0 11 -42 -28 -18 5 -55 | 0 1 -2 15 11 8 1 20 }}
Mapping: {{mapping| 2 0 11 -42 -28 -18 5 -55 | 0 1 -2 15 11 8 1 20 }}


Optimal tuning (POTE): ~17/12 = 1\2, ~3/2 = 704.905
Optimal tuning (POTE): ~17/12 = 600.000, ~3/2 = 704.905


{{Optimal ET sequence|legend=1| 34dh, 46, 80, 206cd }}
{{Optimal ET sequence|legend=0| 34dh, 46, 80, 206cd }}


Badness: 0.017063
Badness (Smith): 0.017063


==== Srutaloo ====
==== Srutaloo ====
Srutaloo adds 576/575, 736/729 or 208/207, rhymes with [[Skidoo]].
Srutaloo adds 576/575, 736/729 or 208/207, and rhymes with [[skidoo]].


Subgroup: 2.3.5.7.11.13.17.19.23
Subgroup: 2.3.5.7.11.13.17.19.23
Line 540: Line 527:
Mapping: {{mapping| 2 0 11 -42 -28 -18 5 -55 -10 | 0 1 -2 15 11 8 1 20 6 }}
Mapping: {{mapping| 2 0 11 -42 -28 -18 5 -55 -10 | 0 1 -2 15 11 8 1 20 6 }}


Optimal tuning (POTE): ~17/12 = 1\2, ~3/2 = 704.899
Optimal tuning (POTE): ~17/12 = 600.000, ~3/2 = 704.899


{{Optimal ET sequence|legend=1| 34dh, 46, 80, 206cd }}
{{Optimal ET sequence|legend=0| 34dh, 46, 80, 206cd }}


Badness: 0.013555
Badness (Smith): 0.013555


===== 29-limit =====
===== 29-limit =====
Line 553: Line 540:
Mapping: {{mapping| 2 0 11 -42 -28 -18 5 -55 -10 -76 | 0 1 -2 15 11 8 1 20 6 27 }}
Mapping: {{mapping| 2 0 11 -42 -28 -18 5 -55 -10 -76 | 0 1 -2 15 11 8 1 20 6 27 }}


Optimal tuning (POTE): ~17/12 = 1\2, ~3/2 = 704.906
Optimal tuning (POTE): ~17/12 = 600.000, ~3/2 = 704.906


{{Optimal ET sequence|legend=1| 34dhj, 46, 80, 206cd }}
{{Optimal ET sequence|legend=0| 34dhj, 46, 80, 206cd }}


Badness: 0.013203
Badness (Smith): 0.013203


===== 31-limit =====
===== 31-limit =====
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 567: Line 553:
Mapping: {{mapping| 2 0 11 -42 -28 -18 5 -55 -10 -76 48 | 0 1 -2 15 11 8 1 20 6 27 -12 }}
Mapping: {{mapping| 2 0 11 -42 -28 -18 5 -55 -10 -76 48 | 0 1 -2 15 11 8 1 20 6 27 -12 }}


Optimal tuning (POTE): ~17/12 = 1\2, ~3/2 = 704.817
Optimal tuning (POTE): ~17/12 = 600.000, ~3/2 = 704.817


{{Optimal ET sequence|legend=1| 46, 80, 126 }}
{{Optimal ET sequence|legend=0| 46, 80, 126 }}


Badness: 0.015073
Badness (Smith): 0.015073


== Keen ==
== Keen ==
Keen adds 875/864 as well as 2240/2187 to the set of commas. It may also be described as the 22 &amp; 56 temperament. [[78edo]] is a good tuning choice, and remains a good one in the 11-limit, where keen, {{multival| 2 -4 18 -12 … }}, is really more interesting, adding 100/99 and 385/384 to the commas.
Keen adds 875/864 as well as 2240/2187 to the set of commas. It may also be described as the {{nowrap| 22 & 56 }} temperament. [[78edo]] is a good tuning choice, and remains a good one in the 11-limit, where keen, {{multival| 2 -4 18 -12 … }}, is really more interesting, adding 100/99 and 385/384 to the commas.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 584: Line 570:
{{Multival|legend=1| 2 -4 18 -11 23 53 }}
{{Multival|legend=1| 2 -4 18 -11 23 53 }}


[[Optimal tuning]] ([[POTE]]): ~45/32 = 1\2, ~3/2 = 707.571
[[Optimal tuning]] ([[POTE]]): ~45/32 = 600.000, ~3/2 = 707.571


{{Optimal ET sequence|legend=1| 22, 56, 78, 134b, 212b, 290bb }}
{{Optimal ET sequence|legend=1| 22, 56, 78, 134b, 212b, 290bb }}


[[Badness]]: 0.083971
[[Badness]] (Smith): 0.083971


=== 11-limit ===
=== 11-limit ===
Line 597: Line 583:
Mapping: {{mapping| 2 0 11 -23 26 | 0 1 -2 9 -6 }}
Mapping: {{mapping| 2 0 11 -23 26 | 0 1 -2 9 -6 }}


Optimal tuning (POTE): ~45/32 = 1\2, ~3/2 = 707.609
Optimal tuning (POTE): ~45/32 = 600.000, ~3/2 = 707.609


{{Optimal ET sequence|legend=1| 22, 56, 78, 212be, 290bbe }}
{{Optimal ET sequence|legend=0| 22, 56, 78, 212be, 290bbe }}


Badness: 0.045270
Badness (Smith): 0.045270


==== 13-limit ====
==== 13-limit ====
Line 610: Line 596:
Mapping: {{mapping| 2 0 11 -23 26 -18 | 0 1 -2 9 -6 8 }}
Mapping: {{mapping| 2 0 11 -23 26 -18 | 0 1 -2 9 -6 8 }}


Optimal tuning (POTE): ~45/32 = 1\2, ~3/2 = 707.167
Optimal tuning (POTE): ~45/32 = 600.000, ~3/2 = 707.167


{{Optimal ET sequence|legend=1| 22f, 34, 56f }}
{{Optimal ET sequence|legend=0| 22f, 34, 56f }}


Badness: 0.044877
Badness (Smith): 0.044877


===== 17-limit =====
===== 17-limit =====
Line 623: Line 609:
Mapping: {{mapping| 2 0 11 -23 26 -18 5 | 0 1 -2 9 -6 8 1}}
Mapping: {{mapping| 2 0 11 -23 26 -18 5 | 0 1 -2 9 -6 8 1}}


Optimal tuning (POTE): ~17/12 = 1\2, ~3/2 = 707.155
Optimal tuning (POTE): ~17/12 = 600.000, ~3/2 = 707.155


{{Optimal ET sequence|legend=1| 22f, 34, 56f }}
{{Optimal ET sequence|legend=0| 22f, 34, 56f }}


Badness: 0.030297
Badness (Smith): 0.030297


==== Keenic ====
==== Keenic ====
Line 636: Line 622:
Mapping: {{mapping| 2 0 11 -23 26 36 | 0 1 -2 9 -6 -9 }}
Mapping: {{mapping| 2 0 11 -23 26 36 | 0 1 -2 9 -6 -9 }}


Optimal tuning (POTE): ~45/32 = 1\2, ~3/2 = 707.257
Optimal tuning (POTE): ~45/32 = 600.000, ~3/2 = 707.257


{{Optimal ET sequence|legend=1| 22, 34, 56 }}
{{Optimal ET sequence|legend=0| 22, 34, 56 }}


Badness: 0.040351
Badness (Smith): 0.040351


===== 17-limit =====
===== 17-limit =====
Line 649: Line 635:
Mapping: {{mapping| 2 0 11 -23 26 36 5 | 0 1 -2 9 -6 -9 1 }}
Mapping: {{mapping| 2 0 11 -23 26 36 5 | 0 1 -2 9 -6 -9 1 }}


Optimal tuning (POTE): ~17/12 = 1\2, ~3/2 = 707.252
Optimal tuning (POTE): ~17/12 = 600.000, ~3/2 = 707.252


{{Optimal ET sequence|legend=1| 22, 34, 56 }}
{{Optimal ET sequence|legend=0| 22, 34, 56 }}


Badness: 0.026917
Badness (Smith): 0.026917


== Bidia ==
== Bidia ==
Bidia adds [[3136/3125]] to the commas, splitting the period into 1/4 octave. It may be called the 12 &amp; 56 temperament.
Bidia adds [[3136/3125]] to the commas, splitting the period into 1/4 octave. It may be called the {{nowrap| 12 & 56 }} temperament.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 666: Line 652:
{{Multival|legend=1| 4 -8 -20 -22 -43 -24 }}
{{Multival|legend=1| 4 -8 -20 -22 -43 -24 }}


[[Optimal tuning]] ([[POTE]]): ~25/21 = 1\4, ~3/2 = 705.364
[[Optimal tuning]] ([[POTE]]): ~25/21 = 300.000, ~3/2 = 705.364


{{Optimal ET sequence|legend=1| 12, 56, 68, 80, 148d }}
{{Optimal ET sequence|legend=1| 12, 56, 68, 80, 148d }}


[[Badness]]: 0.056474
[[Badness]] (Smith): 0.056474


=== 11-limit ===
=== 11-limit ===
Line 679: Line 665:
Mapping: {{mapping| 4 0 22 43 71 | 0 1 -2 -5 -9 }}
Mapping: {{mapping| 4 0 22 43 71 | 0 1 -2 -5 -9 }}


Optimal tuning (POTE): ~25/21 = 1\4, ~3/2 = 705.087
Optimal tuning (POTE): ~25/21 = 300.000, ~3/2 = 705.087


{{Optimal ET sequence|legend=1| 12, 68, 80 }}
{{Optimal ET sequence|legend=0| 12, 68, 80 }}


Badness: 0.040191
Badness (Smith): 0.040191


=== 13-limit ===
=== 13-limit ===
Line 692: Line 678:
Mapping: {{mapping| 4 0 22 43 71 -36 | 0 1 -2 -5 -9 8 }}
Mapping: {{mapping| 4 0 22 43 71 -36 | 0 1 -2 -5 -9 8 }}


Optimal tuning (POTE): ~25/21 = 1\4, ~3/2 = 705.301
Optimal tuning (POTE): ~25/21 = 300.000, ~3/2 = 705.301


{{Optimal ET sequence|legend=1| 12, 68, 80, 148d, 228bcd, 376bbcddf }}
{{Optimal ET sequence|legend=0| 12, 68, 80, 148d, 228bcd, 376bbcddf }}


Badness: 0.041137
Badness (Smith): 0.041137


=== 17-limit ===
=== 17-limit ===
Line 705: Line 691:
Mapping: {{mapping| 4 0 22 43 71 -36 10 | 0 1 -2 -5 -9 8 1 }}
Mapping: {{mapping| 4 0 22 43 71 -36 10 | 0 1 -2 -5 -9 8 1 }}


Optimal tuning (POTE): ~25/21 = 1\4, ~3/2 = 705.334
Optimal tuning (POTE): ~25/21 = 300.000, ~3/2 = 705.334


{{Optimal ET sequence|legend=1| 12, 68, 80, 148d, 228bcd, 376bbcddf }}
{{Optimal ET sequence|legend=0| 12, 68, 80, 148d, 228bcd, 376bbcddf }}


Badness: 0.028631
Badness (Smith): 0.028631


=== 19-limit ===
=== 19-limit ===
Line 718: Line 704:
Mapping: {{mapping| 4 0 22 43 71 -36 10 17 | 0 1 -2 -5 -9 8 1 0 }}
Mapping: {{mapping| 4 0 22 43 71 -36 10 17 | 0 1 -2 -5 -9 8 1 0 }}


Optimal tuning (POTE): ~19/16 = 1\4, ~3/2 = 705.339
Optimal tuning (POTE): ~19/16 = 300.000, ~3/2 = 705.339


{{Optimal ET sequence|legend=1| 12, 68, 80, 148d, 376bbcddfh }}
{{Optimal ET sequence|legend=0| 12, 68, 80, 148d, 376bbcddfh }}


Badness: 0.020590
Badness (Smith): 0.020590


=== 23-limit ===
=== 23-limit ===
Line 732: Line 718:


Optimal tunings:
Optimal tunings:
* [[TE]]: ~19/16 = 299.797 (~2 = 1199.188), ~3 = 1904.048 (~3/2 = 704.860)
* [[TE]]: ~19/16 = 299.797, ~3/2 = 704.860
* [[CWE]]: ~19/16 = 1\4, ~3/2 = 705.341  
* [[CWE]]: ~19/16 = 300.000, ~3/2 = 705.341  
* [[POTE]]: ~19/16 = 1\4, ~3/2 = 705.337
* [[POTE]]: ~19/16 = 300.000, ~3/2 = 705.337


{{Optimal ET sequence|legend=1| 12, 68, 80, 148di }}
{{Optimal ET sequence|legend=0| 12, 68, 80, 148di }}


Badness: 0.017301
Badness (Smith): 0.017301


== Echidna ==
== Echidna ==
Echidna adds 1728/1715 to the commas and takes 9/7 as a generator. It may be called the 22 &amp; 58 temperament. [[58edo]] or [[80edo]] make for good tunings, or their vals can be added to {{val| 138 219 321 388 }} (138cde). In most of the tunings it has a significantly sharp 7/4 which some prefer.  
Echidna adds 1728/1715 to the commas and takes 9/7 as a generator. It may be called the {{nowrap| 22 & 58 }} temperament. [[58edo]] or [[80edo]] make for good tunings, or their vals can be added to {{val| 138 219 321 388 }} (138cde). In most of the tunings it has a significantly sharp 7/4 which some prefer.  


Echidna becomes more interesting when extended to be an 11-limit temperament by adding 176/175, 540/539 or 896/891 to the commas, where the same tunings can be used as before. It then is able to represent the entire 11-odd-limit diamond to within about six cents of error, within a compass of 24 notes. The 22-note 2mos gives scope for this, and the 36-note mos much more. Better yet, it is related to three important 11-limit edos: 22edo, a trivial tuning, is the smallest consistent in the 11-odd-limit, corresponding to the merge of this temperament with [[hedgehog]]; [[58edo]] is the smallest tuning that is distinctly consistent in the 11-odd-limit and [[80edo]] is the third smallest distinctly consistent in the 11-odd-limit.  
Echidna becomes more interesting when extended to be an 11-limit temperament by adding 176/175, 540/539 or 896/891 to the commas, where the same tunings can be used as before. It then is able to represent the entire 11-odd-limit diamond to within about six cents of error, within a compass of 24 notes. The 22-note 2mos gives scope for this, and the 36-note mos much more. Better yet, it is related to three important 11-limit edos: 22edo, a trivial tuning, is the smallest consistent in the 11-odd-limit, corresponding to the merge of this temperament with [[hedgehog]]; [[58edo]] is the smallest tuning that is distinctly consistent in the 11-odd-limit and [[80edo]] is the third smallest distinctly consistent in the 11-odd-limit.  
Line 757: Line 743:
{{Multival|legend=1| 6 -12 10 -33 -1 57 }}
{{Multival|legend=1| 6 -12 10 -33 -1 57 }}


[[Optimal tuning]] ([[POTE]]): ~45/32 = 1\2, ~9/7 = 434.856
[[Optimal tuning]] ([[POTE]]): ~45/32 = 600.000, ~9/7 = 434.856


{{Optimal ET sequence|legend=1| 22, 58, 80, 138cd, 218cd }}
{{Optimal ET sequence|legend=1| 22, 58, 80, 138cd, 218cd }}


[[Badness]]: 0.058033
[[Badness]] (Smith): 0.058033


=== 11-limit ===
=== 11-limit ===
Line 771: Line 757:
Mapping: {{mapping| 2 1 9 2 12 | 0 3 -6 5 -7 }}
Mapping: {{mapping| 2 1 9 2 12 | 0 3 -6 5 -7 }}


Optimal tuning (POTE): ~45/32 = 1\2, ~9/7 = 434.852
Optimal tuning (POTE): ~45/32 = 600.000, ~9/7 = 434.852


Minimax tuning:  
Minimax tuning:  
* 11-odd-limit: ~9/7 = {{monzo| 5/12 0 0 1/12 -1/12 }}
* 11-odd-limit: ~9/7 = {{monzo| 5/12 0 0 1/12 -1/12 }}
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 7/4 0 0 1/4 -1/4 }}, {{monzo| 2 0 0 -1/2 1/2 }}, {{monzo| 37/12 0 0 5/12 -5/12 }}, {{monzo| 37/12 0 0 -7/12 7/12 }}]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 7/4 0 0 1/4 -1/4 }}, {{monzo| 2 0 0 -1/2 1/2 }}, {{monzo| 37/12 0 0 5/12 -5/12 }}, {{monzo| 37/12 0 0 -7/12 7/12 }}]
: Eigenmonzo (unchanged-interval) basis: 2.11/7
: eigenmonzo (unchanged-interval) basis: 2.11/7


{{Optimal ET sequence|legend=1| 22, 58, 80, 138cde, 218cde }}
{{Optimal ET sequence|legend=0| 22, 58, 80, 138cde, 218cde }}


Badness: 0.025987
Badness (Smith): 0.025987


=== 13-limit ===
=== 13-limit ===
Line 789: Line 775:
Mapping: {{mapping| 2 1 9 2 12 19 | 0 3 -6 5 -7 -16 }}
Mapping: {{mapping| 2 1 9 2 12 19 | 0 3 -6 5 -7 -16 }}


Optimal tuning (POTE): ~45/32 = 1\2, ~9/7 = 434.756
Optimal tuning (POTE): ~45/32 = 600.000, ~9/7 = 434.756


{{Optimal ET sequence|legend=1| 22, 58, 80, 138cde }}
{{Optimal ET sequence|legend=0| 22, 58, 80, 138cde }}


Badness: 0.023679
Badness (Smith): 0.023679


=== 17-limit ===
=== 17-limit ===
Line 802: Line 788:
Mapping: {{mapping| 2 1 9 2 12 19 6 | 0 3 -6 5 -7 -16 3 }}
Mapping: {{mapping| 2 1 9 2 12 19 6 | 0 3 -6 5 -7 -16 3 }}


Optimal tuning (POTE): ~17/12 = 1\2, ~9/7 = 434.816
Optimal tuning (POTE): ~17/12 = 600.000, ~9/7 = 434.816


{{Optimal ET sequence|legend=1| 22, 58, 80, 138cde }}
{{Optimal ET sequence|legend=0| 22, 58, 80, 138cde }}


Badness: 0.020273
Badness (Smith): 0.020273


== Echidnic ==
== Echidnic ==
Line 817: Line 803:
{{Multival|legend=1| 6 -12 -2 -33 -20 29 }}
{{Multival|legend=1| 6 -12 -2 -33 -20 29 }}


[[Optimal tuning]] ([[POTE]]): ~45/32 = 1\2, ~8/7 = 234.492
[[Optimal tuning]] ([[POTE]]): ~45/32 = 600.000, ~8/7 = 234.492


{{Optimal ET sequence|legend=1| 10, 36, 46, 194bcd, 240bcd, 286bcd, 332bccdd }}
{{Optimal ET sequence|legend=1| 10, 36, 46, 194bcd, 240bcd, 286bcd, 332bccdd }}


[[Badness]]: 0.072246
[[Badness]] (Smith): 0.072246


=== 11-limit ===
=== 11-limit ===
Line 830: Line 816:
Mapping: {{mapping| 2 2 7 6 3 | 0 3 -6 -1 10 }}
Mapping: {{mapping| 2 2 7 6 3 | 0 3 -6 -1 10 }}


Optimal tuning (POTE): ~45/32 = 1\2, ~8/7 = 235.096
Optimal tuning (POTE): ~45/32 = 600.000, ~8/7 = 235.096


{{Optimal ET sequence|legend=1| 10, 36e, 46, 102, 148, 342bcdd }}
{{Optimal ET sequence|legend=0| 10, 36e, 46, 102, 148, 342bcdd }}


Badness: 0.045127
Badness (Smith): 0.045127


=== 13-limit ===
=== 13-limit ===
Line 843: Line 829:
Mapping: {{mapping| 2 2 7 6 3 7 | 0 3 -6 -1 10 1 }}
Mapping: {{mapping| 2 2 7 6 3 7 | 0 3 -6 -1 10 1 }}


Optimal tuning (POTE): ~45/32 = 1\2, ~8/7 = 235.088
Optimal tuning (POTE): ~45/32 = 600.000, ~8/7 = 235.088


{{Optimal ET sequence|legend=1| 10, 46, 102, 148f, 194bcdf }}
{{Optimal ET sequence|legend=0| 10, 46, 102, 148f, 194bcdf }}


Badness: 0.028874
Badness (Smith): 0.028874


=== 17-limit ===
=== 17-limit ===
Line 856: Line 842:
Mapping: {{mapping| 2 2 7 6 3 7 7 | 0 3 -6 -1 10 1 3 }}
Mapping: {{mapping| 2 2 7 6 3 7 7 | 0 3 -6 -1 10 1 3 }}


Optimal tuning (POTE): ~17/12 = 1\2, ~8/7 = 235.088
Optimal tuning (POTE): ~17/12 = 600.000, ~8/7 = 235.088


{{Optimal ET sequence|legend=1| 10, 46, 102, 148f, 194bcdf }}
{{Optimal ET sequence|legend=0| 10, 46, 102, 148f, 194bcdf }}


Badness: 0.019304
Badness (Smith): 0.019304


; Compositions
; Music
* [https://untwelve.org/competition/2011 ''A Stiff Shot of Turpentine''] [https://untwelve.org/static/audio/competition/2011/Kosmorsky-A_Stiff_Shot_of_Turpentine.mp3 play] by [[Peter Kosmorsky]]
* [https://untwelve.org/competition/2011 ''A Stiff Shot of Turpentine''] [https://untwelve.org/static/audio/competition/2011/Kosmorsky-A_Stiff_Shot_of_Turpentine.mp3 play] by [[Peter Kosmorsky]]
* [https://www.youtube.com/watch?v=VsBXIvBZY6A ''56edo Track (Echidnic16 Scale)''] by [[Budjarn Lambeth]] (2025)
* [https://www.youtube.com/watch?v=VsBXIvBZY6A ''56edo Track (Echidnic16 Scale)''] by [[Budjarn Lambeth]] (2025)


== Shrutar ==
== Shrutar ==
Shrutar adds 245/243 to the commas, and also tempers out 6144/6125. It can also be described as 22&amp;46. Its generator can be taken as either 36/35 or 35/24; the latter is interesting since along with 15/14 and 21/20, it connects opposite sides of a hexany. [[68edo]] makes for a good tuning, but another excellent choice is a generator of 14<sup>(1/7)</sup>, making 7's just.
Shrutar adds 245/243 to the commas, and also tempers out 6144/6125. It can also be described as {{nowrap| 22 & 46 }}. Its generator can be taken as either 36/35 or 35/24; the latter is interesting since along with 15/14 and 21/20, it connects opposite sides of a hexany. [[68edo]] makes for a good tuning, but another excellent choice is a generator of 14<sup>(1/7)</sup>, making 7's just.


By adding 121/120 or 176/175 to the commas, shrutar can be extended to the 11-limit, which loses a bit of accuracy, but picks up low-complexity 11-limit harmony, making shrutar quite an interesting 11-limit system. 68, 114 or a 14<sup>(1/7)</sup> generator can again be used as tunings.
By adding 121/120 or 176/175 to the commas, shrutar can be extended to the 11-limit, which loses a bit of accuracy, but picks up low-complexity 11-limit harmony, making shrutar quite an interesting 11-limit system. 68, 114 or a 14<sup>(1/7)</sup> generator can again be used as tunings.
Line 879: Line 865:
{{Multival|legend=1| 4 -8 14 -22 11 55 }}
{{Multival|legend=1| 4 -8 14 -22 11 55 }}


[[Optimal tuning]] ([[POTE]]): ~45/32 = 1\2, ~35/24 = 652.811
[[Optimal tuning]] ([[POTE]]): ~45/32 = 600.000, ~35/24 = 652.811


{{Optimal ET sequence|legend=1| 22, 46, 68, 182b, 250bc }}
{{Optimal ET sequence|legend=1| 22, 46, 68, 182b, 250bc }}


[[Badness]]: 0.189510
[[Badness]] (Smith): 0.189510


=== 11-limit ===
=== 11-limit ===
Line 892: Line 878:
Mapping: {{mapping| 2 1 9 -2 8 | 0 2 -4 7 -1 }}
Mapping: {{mapping| 2 1 9 -2 8 | 0 2 -4 7 -1 }}


Optimal tuning (POTE): ~45/32 = 1\2, ~16/11 = 652.680
Optimal tuning (POTE): ~45/32 = 600.000, ~16/11 = 652.680


{{Optimal ET sequence|legend=1| 22, 46, 68, 114, 296bce, 410bce }}
{{Optimal ET sequence|legend=0| 22, 46, 68, 114, 296bce, 410bce }}


Badness: 0.084098
Badness (Smith): 0.084098


=== 13-limit ===
=== 13-limit ===
Line 905: Line 891:
Mapping: {{mapping| 2 1 9 -2 8 -10 | 0 2 -4 7 -1 16 }}
Mapping: {{mapping| 2 1 9 -2 8 -10 | 0 2 -4 7 -1 16 }}


Optimal tuning (POTE): ~45/32 = 1\2, ~16/11 = 652.654
Optimal tuning (POTE): ~45/32 = 600.000, ~16/11 = 652.654


{{Optimal ET sequence|legend=1| 22f, 24f, 46, 68, 114 }}
{{Optimal ET sequence|legend=0| 22f, 24f, 46, 68, 114 }}


Badness: 0.079358
Badness (Smith): 0.079358


=== 17-limit ===
=== 17-limit ===
Line 918: Line 904:
Mapping: {{mapping| 2 1 9 -2 8 -10 6 | 0 2 -4 7 -1 16 2 }}
Mapping: {{mapping| 2 1 9 -2 8 -10 6 | 0 2 -4 7 -1 16 2 }}


Optimal tuning (POTE): ~17/12 = 1\2, ~16/11 = 652.647
Optimal tuning (POTE): ~17/12 = 600.000, ~16/11 = 652.647


{{Optimal ET sequence|legend=1| 22f, 24f, 46, 68, 114 }}
{{Optimal ET sequence|legend=0| 22f, 24f, 46, 68, 114 }}


Badness: 0.049392
Badness (Smith): 0.049392


=== 19-limit ===
=== 19-limit ===
Line 931: Line 917:
Mapping: {{mapping| 2 1 9 -2 8 -10 6 -10 | 0 2 -4 7 -1 16 2 17 }}
Mapping: {{mapping| 2 1 9 -2 8 -10 6 -10 | 0 2 -4 7 -1 16 2 17 }}


Optimal tuning (POTE): ~17/12 = 1\2, ~16/11 = 652.730
Optimal tuning (POTE): ~17/12 = 600.000, ~16/11 = 652.730


{{Optimal ET sequence|legend=1| 22fh, 24fh, 46, 68, 114, 182bef }}
{{Optimal ET sequence|legend=0| 22fh, 24fh, 46, 68, 114, 182bef }}


Badness: 0.044197
Badness (Smith): 0.044197


=== 23-limit ===
=== 23-limit ===
Line 944: Line 930:
Mapping: {{mapping| 2 1 9 -2 8 -10 6 -10 -4 | 0 2 -4 7 -1 16 2 17 12 }}
Mapping: {{mapping| 2 1 9 -2 8 -10 6 -10 -4 | 0 2 -4 7 -1 16 2 17 12 }}


Optimal tuning (POTE): ~17/12 = 1\2, ~16/11 = 652.708
Optimal tuning (POTE): ~17/12 = 600.000, ~16/11 = 652.708


{{Optimal ET sequence|legend=1| 22fh, 46, 68, 114 }}
{{Optimal ET sequence|legend=0| 22fh, 46, 68, 114 }}


Badness: 0.035137
Badness (Smith): 0.035137


== Sruti ==
== Sruti ==
Line 959: Line 945:
{{Multival|legend=1| 4 -8 26 -22 30 83 }}
{{Multival|legend=1| 4 -8 26 -22 30 83 }}


[[Optimal tuning]] ([[POTE]]): ~45/32 = 1\2, ~140/81 = 951.876
[[Optimal tuning]] ([[POTE]]): ~45/32 = 600.000, ~140/81 = 951.876


{{Optimal ET sequence|legend=1| 24, 34d, 58, 150cd, 208ccdd, 266ccdd }}
{{Optimal ET sequence|legend=1| 24, 34d, 58, 150cd, 208ccdd, 266ccdd }}


[[Badness]]: 0.117358
[[Badness]] (Smith): 0.117358


=== 11-limit ===
=== 11-limit ===
Line 972: Line 958:
Mapping: {{mapping| 2 0 11 -15 -1 | 0 2 -4 13 5 }}
Mapping: {{mapping| 2 0 11 -15 -1 | 0 2 -4 13 5 }}


Optimal tuning (POTE): ~45/32 = 1\2, ~121/70 = 951.863
Optimal tuning (POTE): ~45/32 = 600.000, ~121/70 = 951.863


{{Optimal ET sequence|legend=1| 24, 34d, 58 }}
{{Optimal ET sequence|legend=0| 24, 34d, 58 }}


Badness: 0.041459
Badness (Smith): 0.041459


=== 13-limit ===
=== 13-limit ===
Line 985: Line 971:
Mapping: {{mapping| 2 0 11 -15 -1 9 | 0 2 -4 13 5 -1 }}
Mapping: {{mapping| 2 0 11 -15 -1 9 | 0 2 -4 13 5 -1 }}


Optimal tuning (POTE): ~45/32 = 1\2, ~26/15 = 951.886
Optimal tuning (POTE): ~45/32 = 600.000, ~26/15 = 951.886


{{Optimal ET sequence|legend=1| 24, 34d, 58, 150cdeef, 208ccddeeff }}
{{Optimal ET sequence|legend=0| 24, 34d, 58, 150cdeef, 208ccddeeff }}


Badness: 0.023791
Badness (Smith): 0.023791


=== 17-limit ===
=== 17-limit ===
Line 998: Line 984:
Mapping: {{mapping| 2 0 11 -15 -1 9 5 | 0 2 -4 13 5 -1 2 }}
Mapping: {{mapping| 2 0 11 -15 -1 9 5 | 0 2 -4 13 5 -1 2 }}


Optimal tuning (POTE): ~17/12 = 1\2, ~26/15 = 951.857
Optimal tuning (POTE): ~17/12 = 600.000, ~26/15 = 951.857


{{Optimal ET sequence|legend=1| 24, 34d, 58 }}
{{Optimal ET sequence|legend=0| 24, 34d, 58 }}


Badness: 0.020536
Badness (Smith): 0.020536


== Anguirus ==
== Anguirus ==
Line 1,013: Line 999:
{{Multival|legend=1| 4 -8 2 -22 -8 27 }}
{{Multival|legend=1| 4 -8 2 -22 -8 27 }}


[[Optimal tuning]] ([[POTE]]): ~45/32 = 1\2, ~7/4 = 953.021
[[Optimal tuning]] ([[POTE]]): ~45/32 = 600.000, ~7/4 = 953.021


{{Optimal ET sequence|legend=1| 10, 24, 34 }}
{{Optimal ET sequence|legend=1| 10, 24, 34 }}


[[Badness]]: 0.077955
[[Badness]] (Smith): 0.077955


=== 11-limit ===
=== 11-limit ===
Line 1,026: Line 1,012:
Mapping: {{mapping| 2 0 11 4 -1 | 0 2 -4 1 5 }}
Mapping: {{mapping| 2 0 11 4 -1 | 0 2 -4 1 5 }}


Optimal tuning (POTE): ~45/32 = 1\2, ~7/4 = 952.184
Optimal tuning (POTE): ~45/32 = 600.000, ~7/4 = 952.184


{{Optimal ET sequence|legend=1| 10, 24, 34, 58d, 92de }}
{{Optimal ET sequence|legend=0| 10, 24, 34, 58d, 92de }}


Badness: 0.049253
Badness (Smith): 0.049253


=== 13-limit ===
=== 13-limit ===
Line 1,039: Line 1,025:
Mapping: {{mapping| 2 0 11 4 -1 9 | 0 2 -4 1 5 -1 }}
Mapping: {{mapping| 2 0 11 4 -1 9 | 0 2 -4 1 5 -1 }}


Optimal tuning (POTE): ~45/32 = 1\2, ~7/4 = 952.309
Optimal tuning (POTE): ~45/32 = 600.000, ~7/4 = 952.309


{{Optimal ET sequence|legend=1| 10, 24, 34, 58d, 92ddef }}
{{Optimal ET sequence|legend=0| 10, 24, 34, 58d, 92ddef }}


Badness: 0.030829
Badness (Smith): 0.030829


=== 17-limit ===
=== 17-limit ===
Line 1,052: Line 1,038:
Mapping: {{mapping| 2 0 11 4 -1 9 5 | 0 2 -4 1 5 -1 2 }}
Mapping: {{mapping| 2 0 11 4 -1 9 5 | 0 2 -4 1 5 -1 2 }}


Optimal tuning (POTE): ~17/12 = 1\2, ~7/4 = 952.330
Optimal tuning (POTE): ~17/12 = 600.000, ~7/4 = 952.330


{{Optimal ET sequence|legend=1| 10, 24, 34, 58d, 92ddef }}
{{Optimal ET sequence|legend=0| 10, 24, 34, 58d, 92ddef }}


Badness: 0.021796
Badness (Smith): 0.021796


== Shru ==
== Shru ==
Line 1,067: Line 1,053:
{{Multival|legend=1| 4 -8 -10 -22 -27 -1 }}
{{Multival|legend=1| 4 -8 -10 -22 -27 -1 }}


[[Optimal tuning]] ([[POTE]]): ~45/32 = 1\2, ~10/7 = 650.135
[[Optimal tuning]] ([[POTE]]): ~45/32 = 600.000, ~10/7 = 650.135


{{Optimal ET sequence|legend=1| 2, 22d, 24 }}
{{Optimal ET sequence|legend=1| 2, 22d, 24 }}


[[Badness]]: 0.157619
[[Badness]] (Smith): 0.157619


=== 11-limit ===
=== 11-limit ===
Line 1,080: Line 1,066:
Mapping: {{mapping| 2 1 9 11 8 | 0 2 -4 -5 -1 }}
Mapping: {{mapping| 2 1 9 11 8 | 0 2 -4 -5 -1 }}


Optimal tuning (POTE): ~45/32 = 1\2, ~10/7 = 650.130
Optimal tuning (POTE): ~45/32 = 600.000, ~10/7 = 650.130


{{Optimal ET sequence|legend=1| 2, 22d, 24 }}
{{Optimal ET sequence|legend=0| 2, 22d, 24 }}


Badness: 0.063483
Badness (Smith): 0.063483


=== 13-limit ===
=== 13-limit ===
Line 1,093: Line 1,079:
Mapping: {{mapping| 2 1 9 11 8 15 | 0 2 -4 -5 -1 -7 }}
Mapping: {{mapping| 2 1 9 11 8 15 | 0 2 -4 -5 -1 -7 }}


Optimal tuning (POTE): ~45/32 = 1\2, ~10/7 = 650.535
Optimal tuning (POTE): ~45/32 = 600.000, ~10/7 = 650.535


{{Optimal ET sequence|legend=1| 22df, 24 }}
{{Optimal ET sequence|legend=0| 22df, 24 }}


Badness: 0.045731
Badness (Smith): 0.045731


== Quadrasruta ==
== Quadrasruta ==
Line 1,108: Line 1,094:
{{Multival|legend=1| 8 -16 -6 -44 -32 31 }}
{{Multival|legend=1| 8 -16 -6 -44 -32 31 }}


[[Optimal tuning]] ([[POTE]]): ~45/32 = 1\2, ~21/16 = 476.216
[[Optimal tuning]] ([[POTE]]): ~45/32 = 600.000, ~21/16 = 476.216


{{Optimal ET sequence|legend=1| 10, 38c, 48c, 58, 68, 126 }}
{{Optimal ET sequence|legend=1| 10, 38c, 48c, 58, 68, 126 }}


[[Badness]]: 0.073569
[[Badness]] (Smith): 0.073569


=== 11-limit ===
=== 11-limit ===
Line 1,121: Line 1,107:
Mapping: {{mapping| 2 0 11 8 22 | 0 4 -8 -3 -19 }}
Mapping: {{mapping| 2 0 11 8 22 | 0 4 -8 -3 -19 }}


Optimal tuning (POTE): ~45/32 = 1\2, ~21/16 = 476.118
Optimal tuning (POTE): ~45/32 = 600.000, ~21/16 = 476.118


{{Optimal ET sequence|legend=1| 58, 126, 184c, 310bccde }}
{{Optimal ET sequence|legend=0| 58, 126, 184c, 310bccde }}


Badness: 0.049018
Badness (Smith): 0.049018


==== 13-limit ====
==== 13-limit ====
Line 1,134: Line 1,120:
Mapping: {{mapping| 2 0 11 8 22 9 | 0 4 -8 -3 -19 -2 }}
Mapping: {{mapping| 2 0 11 8 22 9 | 0 4 -8 -3 -19 -2 }}


Optimal tuning (POTE): ~45/32 = 1\2, ~21/16 = 476.099
Optimal tuning (POTE): ~45/32 = 600.000, ~21/16 = 476.099


{{Optimal ET sequence|legend=1| 58, 126f, 184cff }}
{{Optimal ET sequence|legend=0| 58, 126f, 184cff }}


Badness: 0.028463
Badness (Smith): 0.028463


==== 17-limit ====
==== 17-limit ====
Line 1,147: Line 1,133:
Mapping: {{mapping| 2 0 11 8 22 9 5 | 0 4 -8 -3 -19 -2 4 }}
Mapping: {{mapping| 2 0 11 8 22 9 5 | 0 4 -8 -3 -19 -2 4 }}


Optimal tuning (POTE): ~17/12 = 1\2, ~21/16 = 476.162
Optimal tuning (POTE): ~17/12 = 600.000, ~21/16 = 476.162


{{Optimal ET sequence|legend=1| 58, 126f }}
{{Optimal ET sequence|legend=0| 58, 126f }}


Badness: 0.023820
Badness (Smith): 0.023820


=== Quadrafourths ===
=== Quadrafourths ===
Line 1,160: Line 1,146:
Mapping: {{mapping| 2 0 11 8 -1 | 0 4 -8 -3 10 }}
Mapping: {{mapping| 2 0 11 8 -1 | 0 4 -8 -3 10 }}


Optimal tuning (POTE): ~45/32 = 1\2, ~21/16 = 476.017
Optimal tuning (POTE): ~45/32 = 600.000, ~21/16 = 476.017


{{Optimal ET sequence|legend=1| 10, 38c, 48c, 58 }}
{{Optimal ET sequence|legend=0| 10, 38c, 48c, 58 }}


Badness: 0.049114
Badness (Smith): 0.049114


==== 13-limit ====
==== 13-limit ====
Line 1,173: Line 1,159:
Mapping: {{mapping| 2 0 11 8 -1 9 | 0 4 -8 -3 10 -2 }}
Mapping: {{mapping| 2 0 11 8 -1 9 | 0 4 -8 -3 10 -2 }}


Optimal tuning (POTE): ~45/32 = 1\2, ~21/16 = 476.028
Optimal tuning (POTE): ~45/32 = 600.000, ~21/16 = 476.028


{{Optimal ET sequence|legend=1| 10, 38c, 48c, 58 }}
{{Optimal ET sequence|legend=0| 10, 38c, 48c, 58 }}


Badness: 0.026743
Badness (Smith): 0.026743


==== 17-limit ====
==== 17-limit ====
Line 1,186: Line 1,172:
Mapping: {{mapping| 2 0 11 8 -1 9 5 | 0 4 -8 -3 10 -2 4 }}
Mapping: {{mapping| 2 0 11 8 -1 9 5 | 0 4 -8 -3 10 -2 4 }}


Optimal tuning (POTE): ~17/12 = 1\2, ~21/16 = 476.077
Optimal tuning (POTE): ~17/12 = 600.000, ~21/16 = 476.077


{{Optimal ET sequence|legend=1| 10, 38c, 48c, 58, 126eef, 184ceeff }}
{{Optimal ET sequence|legend=0| 10, 38c, 48c, 58, 126eef, 184ceeff }}


Badness: 0.022239
Badness (Smith): 0.022239


[[Category:Temperament families]]
[[Category:Temperament families]]
Retrieved from "https://en.xen.wiki/w/Diaschismic_family"