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The 5-limit parent comma for the '''diaschismic family''' is 2048/2025, the [[diaschisma]]. Its monzo is {{monzo| 11 -4 -2 }}, and flipping that yields {{multival| 2 -4 -11 }} for the wedgie for 5-limit '''diaschismic''', or '''srutal''', temperament. This tells us the period is half an octave, the [[Wikipedia: Greatest common divisor|GCD]] of 2 and -4, and that 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.
{{Technical data page}}
The '''diaschismic family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the diaschisma, [[2048/2025]].  


== Srutal (12&34, aka diaschismic) ==
== Diaschismic ==
{{Main| Diaschismic }}


Subgroup: 2.3.5
The [[period]] of diaschismic is half an [[2/1|octave]], and the [[generator]] is a fifth; the [[ploidacot]] is diploid monocot. 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.
 
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


[[Comma list]]: 2048/2025
[[Comma list]]: 2048/2025


[[Mapping]]: [{{val| 2 0 11 }}, {{val| 0 1 -2 }}]
{{Mapping|legend=1| 2 0 11 | 0 1 -2 }}
: mapping generators: ~45/32, ~3


[[POTE generator]]: ~3/2 = 704.898
[[Optimal tuning]]s:
* [[WE]]: ~45/32 = 599.4107{{c}}, ~3/2 = 704.2059{{c}}
: [[error map]]: {{val| -1.179 +1.072 +1.150 }}
* [[CWE]]: ~45/32 = 600.0000{{c}}, ~3/2 = 704.9585{{c}}
: error map: {{val| 0.000 +3.003 +3.769 }}


[[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]


{{Val list|legend=1| 10, 12, 22, 34, 46, 80, 206c, 286bc }}
{{Optimal ET sequence|legend=1| 10, 12, 22, 34, 46, 80, 206c, 286bc }}
 
[[Badness]] (Sintel): 0.467
 
=== Overview to extensions ===
==== 7-limit extensions ====
To get the 7-limit extensions, we add another comma:
* Septimal diaschismic adds [[126/125]], the starling comma, to obtain 7-limit harmony by more complex methods than pajara, but with greater accuracy.
* Pajara adds [[50/49]] or [[64/63]] and is a popular and well-known choice.
* Srutal adds [[4375/4374]], the ragisma, which is about as accurate as septimal diaschismic but has a much more complex mapping of 7.
* Keen adds [[875/864]].


[[Badness]]: 0.019915
Those all keep the same half-octave period and fifth generator.  


=== Seven limit extensions ===
Bidia adds [[3136/3125]], the hemimean comma, with a 1/4-octave period. Shrutar adds [[245/243]] and shru adds [[392/375]], with a quartertone generator. Sruti adds [[19683/19600]] and anguirus adds [[49/48]], with a neutral third or hemitwelfth generator. Those split the original generator in two. Echidna adds [[1728/1715]], the orwellisma, with a ~9/7 generator. Echidnic adds [[686/675]], the senga, with a ~8/7 generator. Those split the original generator in three. Finally, quadrasruta adds [[2401/2400]] and splits the original  generator in four.
The second comma of the [[Normal lists #Normal interval list|normal comma list]] defines which 7-limit family member we are looking at.  


* Pajara derives from [[64/63]] and is a popular and well-known choice.
==== Subgroup extensions ====
* Diaschismic adds 2097152/2066715 to obtain 7-limit harmony by more complex methods, but with greater accuracy.  
Since the diaschisma factors into ([[256/255]])<sup>2</sup>([[289/288]]) in the 17-limit, it extends naturally to the 2.3.5.17 subgroup as ''srutal archagall'', considered in [[#Subgroup extensions]]. The [[S-expression]]-based comma list of this temperament is {[[256/255|S16]], [[289/288|S17]]}.
* Srutal adds {{monzo| 21 -15 0 1 }}. It does no significant tuning damage, so for that we keep the 5-limit label srutal.  
* Keen adds 2240/2187.
* Bidia adds [[3136/3125]], the hemimean comma.
* Echidna adds [[1728/1715]], the orwellisma.
* Shrutar adds [[245/243]], the sensamagic comma.  


Pajara, diaschismic, srutal and keen keep the same half-octave period and fifth generator, but shrutar has a generator of a quarter-tone (which can be taken as [[36/35]], the septimal quarter-tone) and echidna has a generator of 9/7. Bidia has a quarter-octave period and a fifth generator.
== Septimal diaschismic ==
{{Main| Diaschismic }}
{{See also| Srutal vs diaschismic }}


== Srutal ==
A simpler characterization than the one given by the normal comma list is that septimal diaschismic adds [[126/125]] or [[5120/5103]] to the set of commas, and it can also be called {{nowrap| 46 & 58 }}. However described, septimal diaschismic has a 1/2-octave period and a sharp fifth generator like the 5-limit version, but not so sharp, giving a more accurate but more complex temperament. [[104edo]] with the 104c [[val]] provides an excellent tuning, which is close to tuning [[7/4]] just by making the fifth 703.897 cents.
{{see also| Srutal vs diaschismic }}


Subgroup: 2.3.5.7
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. This mapping can also be rationalized by [[parapyth]], which makes sense due to the sharp fifth, and prime 17 is found as in srutal archagall. If you want to explore higher-limit harmonies, diaschismic is certainly one excellent way to do it; [[mos]] scales 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.


[[Comma list]]: 2048/2025, 4375/4374
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val|2 0 11 -42}}, {{val|0 1 -2 15}}]
[[Comma list]]: 126/125, 2048/2025


{{Multival|legend=1| 2 -4 30 -11 42 81 }}
{{Mapping|legend=1| 2 0 11 31 | 0 1 -2 -8 }}


[[POTE generator]]: ~3/2 = 704.814
[[Optimal tuning]]s:
* [[WE]]: ~45/32 = 599.4449{{c}}, ~3/2 = 703.0299{{c}}
: [[error map]]: {{val| -1.110 -0.035 +3.740 -1.391 }}
* [[CWE]]: ~45/32 = 600.0000{{c}}, ~3/2 = 703.7739{{c}}
: error map: {{val| 0.000 +1.819 +6.138 +0.983 }}


[[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 = [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 = [703.448, 705.882]


{{Val list|legend=1| 34d, 46, 80, 126, 206cd, 332bcd }}
{{Optimal ET sequence|legend=1| 12, 34, 46, 58, 104c, 162c }}


[[Badness]]: 0.091504
[[Badness]] (Sintel): 0.959


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 176/175, 896/891, 1331/1323
Comma list: 126/125, 176/175, 896/891


Mapping: [{{val| 2 0 11 -42 -28 }}, {{val| 0 1 -2 15 11 }}]
Mapping: {{mapping| 2 0 11 31 45 | 0 1 -2 -8 -12 }}


POTE generator: ~3/2 = 704.856
Optimal tunings:  
* WE: ~45/32 = 599.4471{{c}}, ~3/2 = 703.0657{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 703.7996{{c}}


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 = [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 = [704.348, 705.882]


Optimal GPV sequence: {{Val list| 34d, 46, 80, 126, 206cd }}
{{Optimal ET sequence|legend=0| 12, 34e, 46, 58, 104c, 162ce }}


Badness: 0.035315
Badness (Sintel): 0.828


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 169/168, 176/175, 325/324, 364/363
Comma list: 126/125, 176/175, 196/195, 364/363


Mapping: [{{val| 2 0 11 -42 -28 -18 }}, {{val| 0 1 -2 15 11 8 }}]
Mapping: {{mapping| 2 0 11 31 45 55 | 0 1 -2 -8 -12 -15 }}


POTE generator: ~3/2 = 704.881
Optimal tunings:  
* WE: ~45/32 = 599.4451{{c}}, ~3/2 = 703.0528{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 703.7813{{c}}


Tuning ranges:  
Tuning ranges:  
* 13- and 15-odd-limit diamond monotone: ~3/2 = [704.348, 705.882] (27\46 to 20\34)
* 13- and 15-odd-limit diamond monotone: ~3/2 = [703.448, 704.348] (34\58 to 27\46)
* 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 GPV sequence: {{Val list| 34d, 46, 80, 206cd, 286bcde }}
{{Optimal ET sequence|legend=0| 12f, 34ef, 46, 58, 104c, 162cef }}


Badness: 0.025286
Badness (Sintel): 0.782


=== 17-limit ===
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 136/135, 169/168, 176/175, 221/220, 256/255
Comma list: 126/125, 136/135, 176/175, 196/195, 256/255


Mapping: [{{val| 2 0 11 -42 -28 -18 5 }}, {{val| 0 1 -2 15 11 8 1 }}]
Mapping: {{mapping| 2 0 11 31 45 55 5 | 0 1 -2 -8 -12 -15 1 }}


POTE generator: ~3/2 = 704.840
Optimal tunings:  
* WE: ~17/12 = 599.6253{{c}}, ~3/2 = 703.3726{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 703.8520{{c}}


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


Optimal GPV sequence: {{Val list| 34d, 46, 80, 126, 206cd }}
{{Optimal ET sequence|legend=0| 12f, 34ef, 46, 58, 104c }}


Badness: 0.018594
Badness (Sintel): 0.837


=== 19-limit ===
=== 2.3.5.7.11.13.17.23 subgroup (Na"Naa') ===
<b>Na"Naa'</b> is a remarkable subgroup temperament of {{nowrap| 46 & 58 }} with a prime harmonic of 23. It is yet to be found why it got this strange name.
 
Subgroup: 2.3.5.7.11.13.17.23
 
Comma list: 126/125, 136/135, 176/175, 196/195, 231/230, 256/255


Srutal and Shrutar have similar 19-limit  properties, tempering 190/189.
Subgroup-val mapping: {{mapping| 2 0 11 31 45 55 5 63 | 0 1 -2 -8 -12 -15 1 -17 }}


Comma list: 136/135, 169/168, 176/175, 190/189, 221/220, 256/255
Optimal tunings:  
* WE: ~17/12 = 599.6272{{c}}, ~3/2 = 703.4326{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 703.9093{{c}}


Mapping: [{{val| 2 0 11 -42 -28 -18 5 -55 }}, {{val| 0 1 -2 15 11 8 1 20}}]
{{Optimal ET sequence|legend=0| 12i, 34efi, 46, 58i, 104ci }}


Optimal GPV sequence: {{Val list| 34dh, 46, 80, 206cd }}
Badness (Sintel): 0.882


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


Pajara is closely associated with 22 EDO (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 22 EDO, 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 12 EDO 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 }} (34d) and 56 with the val {{val| 56 89 130 158 }} (56d) 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


[[Comma list]]: 50/49, 64/63
[[Comma list]]: 50/49, 64/63


[[Mapping]]: [{{val| 2 0 11 12 }}, {{val| 0 1 -2 -2 }}]
{{Mapping|legend=1| 2 0 11 12 | 0 1 -2 -2 }}
 
{{Multival|legend=1| 2 -4 -4 -11 -12 2 }}


[[POTE generator]]: ~3/2 = 707.048
[[Optimal tuning]]s:
* [[WE]]: ~7/5 = 598.8483{{c}}, ~3/2 = 705.6906{{c}}
: [[error map]]: {{val| -2.303 +1.432 -5.756 +10.580 }}
* [[CWE]]: ~7/5 = 600.0000{{c}}, ~3/2 = 707.3438{{c}}
: error map: {{val| 0.000 +5.389 -1.001 +16.487 }}


[[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]


{{Val list|legend=1| 10, 12, 22, 34d, 56d }}
{{Optimal ET sequence|legend=1| 10, 12, 22, 34d, 56d }}
 
[[Badness]] (Sintel): 0.507


[[Badness]]: 0.020033
=== 2.3.5.7.17 subgroup ===
Subgroup: 2.3.5.7.17
 
Comma list: 50/49, 64/63, 85/84
 
Mapping: {{mapping| 2 0 11 12 5 | 0 1 -2 -2 1 }}
 
Optimal tunings:
* WE: ~7/5 = 599.053{{c}}, ~3/2 = 706.355{{c}}
* CWE: ~7/5 = 600.000{{c}}, ~3/2 = 707.607{{c}}
 
{{Optimal ET sequence|legend=0| 10, 12, 22, 56d }}
 
Badness (Sintel): 0.438


=== 11-limit ===
=== 11-limit ===
Line 151: Line 195:
Comma list: 50/49, 64/63, 99/98
Comma list: 50/49, 64/63, 99/98


Mapping: [{{val| 2 0 11 12 26 }}, {{val| 0 1 -2 -2 -6 }}]
Mapping: {{mapping| 2 0 11 12 26 | 0 1 -2 -2 -6 }}


POTE generator: ~3/2 = 706.885
Optimal tunings:  
* WE: ~7/5 = 598.8485{{c}}, ~3/2 = 705.5285{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 707.1826{{c}}


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 GPV sequence: {{Val list| 10e, 12, 22, 34d, 56d }}
{{Optimal ET sequence|legend=0| 10e, 12, 22, 34d, 56d }}
 
Badness (Sintel): 0.673
 
==== 2.3.5.7.11.17 subgroup ====
Subgroup: 2.3.5.7.11.17


Badness: 0.020343
Comma list: 50/49, 64/63, 85/84, 99/98
 
Mapping: {{mapping| 2 0 11 12 26 5 | 0 1 -2 -2 -6 1 }}
 
Optimal tunings:
* WE: ~7/5 = 599.062{{c}}, ~3/2 = 706.095{{c}}
* CWE: ~7/5 = 600.000{{c}}, ~3/2 = 707.370{{c}}
 
{{Optimal ET sequence|legend=0| 10e, 12, 22, 34d, 56d }}
 
Badness (Sintel): 0.645


==== 13-limit ====
==== 13-limit ====
Line 169: Line 229:
Comma list: 50/49, 64/63, 65/63, 99/98
Comma list: 50/49, 64/63, 65/63, 99/98


Mapping: [{{val| 2 0 11 12 26 1 }}, {{val| 0 1 -2 -2 -6 2 }}]
Mapping: {{mapping| 2 0 11 12 26 1 | 0 1 -2 -2 -6 2 }}


POTE generator: ~3/2 = 708.919
Optimal tunings:  
* WE: ~7/5 = 599.9732{{c}}, ~3/2 = 708.8873{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 708.9227{{c}}


Optimal GPV sequence: {{Val list| 10e, 12, 22 }}
{{Optimal ET sequence|legend=0| 10e, 12, 22 }}


Badness: 0.027642
Badness (Sintel): 1.14


===== 17-limit =====
===== 17-limit =====
Line 182: Line 244:
Comma list: 50/49, 52/51, 64/63, 65/63, 99/98
Comma list: 50/49, 52/51, 64/63, 65/63, 99/98


Mapping: [{{val| 2 0 11 12 26 1 5 }}, {{val| 0 1 -2 -2 -6 2 1 }}]
Mapping: {{mapping| 2 0 11 12 26 1 5 | 0 1 -2 -2 -6 2 1 }}


POTE generator: ~3/2 = 708.806
Optimal tunings:  
* WE: ~7/5 = 599.8871{{c}}, ~3/2 = 708.6725{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 708.8176{{c}}


Optimal GPV sequence: {{Val list| 10e, 12, 22 }}
{{Optimal ET sequence|legend=0| 10e, 12, 22 }}


Badness: 0.020899
Badness (Sintel): 1.06


==== Pajarina ====
==== Pajarina ====
Line 195: Line 259:
Comma list: 50/49, 64/63, 78/77, 99/98
Comma list: 50/49, 64/63, 78/77, 99/98


Mapping: [{{val| 2 0 11 12 26 36 }}, {{val| 0 1 -2 -2 -6 -9 }}]
Mapping: {{mapping| 2 0 11 12 26 36 | 0 1 -2 -2 -6 -9 }}


POTE generator: ~3/2 = 706.133
Optimal tunings:  
* WE: ~7/5 = 598.7732{{c}}, ~3/2 = 704.6889{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 706.3950{{c}}


Optimal GPV sequence: {{Val list| 12f, 22, 34d }}
{{Optimal ET sequence|legend=0| 12f, 22, 34d }}


Badness: 0.022327
Badness (Sintel): 0.923


===== 17-limit =====
===== 17-limit =====
Line 208: Line 274:
Comma list: 50/49, 64/63, 78/77, 85/84, 99/98
Comma list: 50/49, 64/63, 78/77, 85/84, 99/98


Mapping: [{{val| 2 0 11 12 26 36 5 }}, {{val| 0 1 -2 -2 -6 -9 1 }}]
Mapping: {{mapping| 2 0 11 12 26 36 5 | 0 1 -2 -2 -6 -9 1 }}
 
Optimal tunings:
* WE: ~7/5 = 599.0204{{c}}, ~3/2 = 705.2572{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 706.5660{{c}}
 
{{Optimal ET sequence|legend=0| 12f, 22, 34d }}
 
Badness (Sintel): 0.936
 
==== Pajarita ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 40/39, 50/49, 64/63, 66/65
 
Mapping: {{mapping| 2 0 11 12 26 17 | 0 1 -2 -2 -6 -3 }}
 
Optimal tunings:
* WE: ~7/5 = 598.3048{{c}}, ~3/2 = 705.4512{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 707.9238{{c}}
 
{{Optimal ET sequence|legend=0| 10e, 12f, 22f, 34dff }}
 
Badness (Sintel): 0.937
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 40/39, 50/49, 64/63, 66/65, 85/84
 
Mapping: {{mapping| 2 0 11 12 26 17 5 | 0 1 -2 -2 -6 -3 1 }}


POTE generator: ~3/2 = 706.410
Optimal tunings:  
* WE: ~7/5 = 598.6103{{c}}, ~3/2 = 706.3076{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 708.2256{{c}}


Optimal GPV sequence: {{Val list| 12f, 22, 34d }}
{{Optimal ET sequence|legend=0| 10e, 12f, 22f }}


Badness: 0.018375
Badness (Sintel): 0.968


=== Pajarous ===
=== Pajarous ===
Line 221: Line 319:
Comma list: 50/49, 55/54, 64/63
Comma list: 50/49, 55/54, 64/63


Mapping: [{{val| 2 0 11 12 -9 }}, {{val| 0 1 -2 -2 5 }}]
Mapping: {{mapping| 2 0 11 12 -9 | 0 1 -2 -2 5 }}


POTE generator: ~3/2 = 709.578
Optimal tunings:  
* WE: ~7/5 = 599.4055{{c}}, ~3/2 = 708.8747{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 709.5508{{c}}


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 GPV sequence: {{Val list| 10, 12e, 22, 120bce, 142bce }}
{{Optimal ET sequence|legend=0| 10, 12e, 22, 120bce, 142bce }}
 
Badness (Sintel): 0.937


Badness: 0.028349
==== 2.3.5.7.11.17 subgroup ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 50/49, 52/51, 55/54, 64/63, 65/63
 
Mapping: {{mapping| 2 0 11 12 -9 1 5 | 0 1 -2 -2 5 2 1 }}
 
Optimal tunings:
* WE: ~7/5 = 599.408{{c}}, ~3/2 = 708.878{{c}}
* CWE: ~7/5 = 600.000{{c}}, ~3/2 = 709.544{{c}}
 
{{Optimal ET sequence|legend=0| 10, 12e, 22 }}
 
Badness (Sintel): 0.766


==== 13-limit ====
==== 13-limit ====
Line 239: Line 353:
Comma list: 50/49, 55/54, 64/63, 65/63
Comma list: 50/49, 55/54, 64/63, 65/63


Mapping: [{{val| 2 0 11 12 -9 1 }}, {{val| 0 1 -2 -2 5 2 }}]
Mapping: {{mapping| 2 0 11 12 -9 1 | 0 1 -2 -2 5 2 }}
 
Optimal tunings:
* WE: ~7/5 = 599.9064{{c}}, ~3/2 = 710.1289{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 710.2325{{c}}
 
{{Optimal ET sequence|legend=0| 10, 22 }}


POTE generator: ~3/2 = 710.240
Badness (Sintel): 1.04


Optimal GPV sequence: {{Val list| 10, 22, 54f, 76bdf }}
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


Badness: 0.025176
Comma list: 50/49, 52/51, 55/54, 64/63, 65/63
 
Mapping: {{mapping| 2 0 11 12 -9 1 5 | 0 1 -2 -2 5 2 1 }}
 
Optimal tunings:
* WE: ~7/5 = 599.8239{{c}}, ~3/2 = 710.0128{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 710.2067{{c}}
 
{{Optimal ET sequence|legend=0| 10, 22, 54f, 76bdff }}
 
Badness (Sintel): 0.930


==== Pajaro ====
==== Pajaro ====
Line 252: Line 383:
Comma list: 40/39, 50/49, 55/54, 64/63
Comma list: 40/39, 50/49, 55/54, 64/63


Mapping: [{{val| 2 0 11 12 -9 17 }}, {{val| 0 1 -2 -2 5 -3 }}]
Mapping: {{mapping| 2 0 11 12 -9 17 | 0 1 -2 -2 5 -3 }}
 
Optimal tunings:
* WE: ~7/5 = 598.8257{{c}}, ~3/2 = 709.4266{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 710.8414{{c}}
 
{{Optimal ET sequence|legend=0| 10, 22f, 32f }}
 
Badness (Sintel): 1.13
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 40/39, 50/49, 55/54, 64/63, 85/84
 
Mapping: {{mapping| 2 0 11 12 -9 17 5 | 0 1 -2 -2 5 -3 1 }}


POTE generator ~3/2 = 710.818
Optimal tunings:
* WE: ~7/5 = 598.8865{{c}}, ~3/2 = 709.5472{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 710.8704{{c}}


Optimal GPV sequence: {{Val list| 10, 22f, 32f, 54f }}
{{Optimal ET sequence|legend=0| 10, 22f, 32f }}


Badness: 0.027355
Badness (Sintel): 1.01


=== Pajaric ===
=== Pajaric ===
Line 265: Line 413:
Comma list: 45/44, 50/49, 56/55
Comma list: 45/44, 50/49, 56/55


Mapping: [{{val| 2 0 11 12 7 }}, {{val| 0 1 -2 -2 0 }}]
Mapping: {{mapping| 2 0 11 12 7 | 0 1 -2 -2 0 }}


POTE generator: ~3/2 = 705.524
Optimal tunings:  
* WE: ~7/5 = 597.4807{{c}}, ~3/2 = 702.5616{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 706.0542{{c}}


Optimal GPV sequence: {{Val list| 10, 12, 22e, 34de }}
{{Optimal ET sequence|legend=0| 10, 12, 22e }}


Badness: 0.023798
Badness (Sintel): 0.787


==== 13-limit ====
==== 13-limit ====
Line 278: Line 428:
Comma list: 40/39, 45/44, 50/49, 56/55
Comma list: 40/39, 45/44, 50/49, 56/55


Mapping: [{{val| 2 0 11 12 7 17 }}, {{val| 0 1 -2 -2 0 -3 }}]
Mapping: {{mapping| 2 0 11 12 7 17 | 0 1 -2 -2 0 -3 }}


POTE generator: ~3/2 = 707.442
Optimal tunings:  
* WE: ~7/5 = 597.1952{{c}}, ~3/2 = 704.1350{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 708.1989{{c}}


Optimal GPV sequence: {{Val list| 10, 12f, 22ef, 34def }}
{{Optimal ET sequence|legend=0| 10, 12f, 22ef }}


Badness: 0.020461
Badness (Sintel): 0.845
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 34/33, 40/39, 45/44, 50/49, 56/55
 
Mapping: {{mapping| 2 0 11 12 7 17 5 | 0 1 -2 -2 0 -3 1 }}
 
Optimal tunings:
* WE: ~7/5 = 597.6509{{c}}, ~3/2 = 705.7702{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 708.9719{{c}}
 
{{Optimal ET sequence|legend=0| 10, 12f, 22ef }}
 
Badness (Sintel): 0.896


=== Hemipaj ===
=== Hemipaj ===
Line 291: Line 458:
Comma list: 50/49, 64/63, 121/120  
Comma list: 50/49, 64/63, 121/120  


Mapping: [{{val| 2 1 9 10 8 }}, {{val| 0 2 -4 -4 -1 }}]
Mapping: {{mapping| 2 1 9 10 8 | 0 2 -4 -4 -1 }}


POTE generator: ~11/8 = 546.383
: mapping generators: ~2, ~16/11


Optimal GPV sequence: {{Val list| 20, 22, 68d, 90d }}
Optimal tunings:  
* WE: ~7/5 = 597.6509{{c}}, ~16/11 = 652.7788{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~16/11 = 653.7119{{c}}


Badness: 0.038890
{{Optimal ET sequence|legend=0| 2, 20, 22 }}
 
Badness (Sintel): 1.29


=== Hemifourths ===
=== Hemifourths ===
Line 304: Line 475:
Comma list: 50/49, 64/63, 243/242
Comma list: 50/49, 64/63, 243/242


Mapping: [{{val| 2 0 11 12 -1 }}, {{val| 0 2 -4 -4 5 }}]
Mapping: {{mapping| 2 0 11 12 -1 | 0 2 -4 -4 5 }}
: mapping generators: ~2, ~55/32


POTE generator: ~64/55 = 246.907
Optimal tunings:
* WE: ~7/5 = 597.6509{{c}}, ~55/32 = 950.8475{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~55/32 = 953.1172{{c}}


Optimal GPV sequence: {{Val list| 10, 24d, 34d }}
{{Optimal ET sequence|legend=0| 10, 24d, 34d }}


Badness: 0.048885
Badness (Sintel): 1.62


==== 13-limit ====
==== 13-limit ====
Line 317: Line 491:
Comma list: 50/49, 64/63, 78/77, 144/143
Comma list: 50/49, 64/63, 78/77, 144/143


Mapping: [{{val| 2 0 11 12 -1 9 }}, {{val| 0 2 -4 -4 5 -1 }}]
Mapping: {{mapping| 2 0 11 12 -1 9 | 0 2 -4 -4 5 -1 }}


POTE generator: ~15/13 = 246.926
Optimal tunings:  
* WE: ~7/5 = 598.6748{{c}}, ~26/15 = 950.9691{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~26/15 = 953.1052{{c}}


Optimal GPV sequence: {{Val list| 10, 24d, 34d }}
{{Optimal ET sequence|legend=0| 10, 24d, 34d }}


Badness: 0.028755
Badness (Sintel): 1.19


==== 17-limit ====
==== 17-limit ====
Line 330: Line 506:
Comma list: 50/49, 64/63, 78/77, 85/84, 144/143
Comma list: 50/49, 64/63, 78/77, 85/84, 144/143


Mapping: [{{val| 2 0 11 12 -1 9 5 }}, {{val| 0 2 -4 -4 5 -1 2 }}]
Mapping: {{mapping| 2 0 11 12 -1 9 5 | 0 2 -4 -4 5 -1 2 }}


POTE generator: ~15/13 = 246.790
Optimal tunings:  
* WE: ~7/5 = 598.8411{{c}}, ~26/15 = 951.3687{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~26/15 = 953.2169{{c}}


Optimal GPV sequence: {{Val list| 10, 24d, 34d }}
{{Optimal ET sequence|legend=0| 10, 24d, 34d }}


Badness: 0.021790
Badness (Sintel): 1.11


== Diaschismic ==
== Srutal ==
{{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&amp;58. However described, diaschismic has a 1/2 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.
Srutal can be described as the {{nowrap| 34d & 46 }} temperament, where 7/4 is located at 15 generator steps, or the double-augmented fifth (C–Gx). As such, it weakly extends [[leapfrog]]. 80edo and [[126edo]] are among the possible tunings. Srutal, shrutar and bidia have similar 19-limit properties, tempering out 190/189, related to rank-3 [[julius]].


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 two 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.
[[Subgroup]]: 2.3.5.7


Subgroup: 2.3.5.7
[[Comma list]]: 2048/2025, 4375/4374
 
[[Comma list]]: 126/125, 2048/2025


[[Mapping]]: [{{val| 2 0 11 31 }}, {{val| 0 1 -2 -8 }}]
{{Mapping|legend=1| 2 0 11 -42 | 0 1 -2 15 }}


{{Multival|legend=1| 2 -4 -16 -11 -31 -26 }}
[[Optimal tuning]]s:
 
* [[WE]]: ~45/32 = 599.4046{{c}}, ~3/2 = 704.1150{{c}}
[[POTE generator]]: ~3/2 = 703.681
: [[error map]]: {{val| -1.191 +0.969 +1.289 +0.044 }}
* [[CWE]]: ~45/32 = 600.0000{{c}}, ~3/2 = 704.7646{{c}}
: error map: {{val| 0.000 +2.810 +4.157 +2.643 }}


[[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 = [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 = [701.955, 705.882]


{{Val list|legend=1| 12, 46, 58, 104c, 162c }}
{{Optimal ET sequence|legend=1| 34d, 46, 80, 126, 206cd, 332bcd }}


[[Badness]]: 0.037914
[[Badness]] (Sintel): 2.32


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 126/125, 176/175, 896/891
Comma list: 176/175, 896/891, 1331/1323


Mapping: [{{val| 2 0 11 31 45 }}, {{val| 0 1 -2 -8 -12 }}]
Mapping: {{mapping| 2 0 11 -42 -28 | 0 1 -2 15 11 }}


POTE generator: ~3/2 = 703.714
Optimal tunings:  
* WE: ~45/32 = 599.4413{{c}}, ~3/2 = 704.1999{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 704.8017{{c}}


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 = [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 = [701.955, 704.348]


Optimal GPV sequence: {{Val list| 12, 46, 58, 104c, 162ce }}
{{Optimal ET sequence|legend=0| 34d, 46, 80, 126, 206cd }}


Badness: 0.025034
Badness (Sintel): 1.17


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 126/125, 176/175, 196/195, 364/363
Comma list: 169/168, 176/175, 325/324, 364/363


Mapping: [{{val| 2 0 11 31 45 55 }}, {{val| 0 1 -2 -8 -12 -15 }}]
Mapping: {{mapping| 2 0 11 -42 -28 -18 | 0 1 -2 15 11 8 }}


POTE generator: ~3/2 = 703.704
Optimal tunings:
* WE: ~45/32 = 599.5490{{c}}, ~3/2 = 704.3516{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 704.8347{{c}}


Tuning ranges:  
Tuning ranges:  
* 13- and 15-odd-limit diamond monotone: ~3/2 = [703.448, 704.348] (34\58 to 27\46)
* 13- and 15-odd-limit diamond monotone: ~3/2 = [704.348, 705.882] (27\46 to 20\34)
* 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 GPV sequence: {{Val list| 46, 58, 104c, 162cef }}
{{Optimal ET sequence|legend=0| 34d, 46, 80 }}


Badness: 0.018926
Badness (Sintel): 1.04


=== 17-limit ===
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 126/125, 136/135, 176/175, 196/195, 256/255
Comma list: 136/135, 169/168, 176/175, 221/220, 256/255


Mapping: [{{val| 2 0 11 31 45 55 5 }}, {{val| 0 1 -2 -8 -12 -15 1 }}]
Mapping: {{mapping| 2 0 11 -42 -28 -18 5 | 0 1 -2 15 11 8 1 }}


POTE generator: ~3/2 = 703.812
Optimal tunings:  
* WE: ~17/12 = 599.6459{{c}}, ~3/2 = 704.4237{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 704.8083{{c}}


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 = [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 = [703.448, 704.348]


Optimal GPV sequence: {{Val list| 46, 58, 104c }}
{{Optimal ET sequence|legend=0| 34d, 46, 80, 126 }}


Badness: 0.016425
Badness (Sintel): 0.947


==== Na"Naa' ====
=== 19-limit ===
<b>Na"Naa'</b> is a remarkable subgroup temperament of 46&amp;58 with a prime harmonic of 23.
Subgroup: 2.3.5.7.11.13.17.19


Subgroup: 2.3.5.7.11.13.17.23
Comma list: 136/135, 169/168, 176/175, 190/189, 221/220, 256/255
 
Mapping: {{mapping| 2 0 11 -42 -28 -18 5 -55 | 0 1 -2 15 11 8 1 20 }}
 
Optimal tunings:
* WE: ~17/12 = 599.6371{{c}}, ~3/2 = 704.4790{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 704.8745{{c}}
 
{{Optimal ET sequence|legend=0| 34dh, 46, 80 }}
 
Badness (Sintel): 1.04
 
==== Srutaloo ====
Srutaloo adds 576/575, 736/729 or 208/207, and rhymes with [[skidoo]].
 
Subgroup: 2.3.5.7.11.13.17.19.23
 
Comma list: 136/135, 169/168, 176/175, 190/189, 208/207, 221/220, 256/255
 
Mapping: {{mapping| 2 0 11 -42 -28 -18 5 -55 -10 | 0 1 -2 15 11 8 1 20 6 }}
 
Optimal tunings:
* WE: ~17/12 = 599.6690{{c}}, ~3/2 = 704.5098{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 704.8713{{c}}
 
{{Optimal ET sequence|legend=0| 34dh, 46, 80 }}
 
Badness (Sintel): 0.971
 
===== 29-limit =====
Subgroup: 2.3.5.7.11.13.17.19.23.29
 
Comma list: 136/135, 169/168, 176/175, 190/189, 208/207, 221/220, 232/231, 256/255
 
Mapping: {{mapping| 2 0 11 -42 -28 -18 5 -55 -10 -76 | 0 1 -2 15 11 8 1 20 6 27 }}
 
Optimal tunings:
* WE: ~17/12 = 599.6664{{c}}, ~3/2 = 704.5138{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 704.8807{{c}}
 
{{Optimal ET sequence|legend=0| 34dhj, 46, 80 }}
 
Badness (Sintel): 1.10
 
===== 31-limit =====
Subgroup: 2.3.5.7.11.13.17.19.23.29.31
 
Comma list: 136/135, 169/168, 176/175, 190/189, 208/207, 217/216, 221/220, 232/231, 256/255


[[Comma list]]: 126/125, 136/135, 176/175, 196/195, 231/230, 256/255
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|Sval mapping]]: [{{val| 2 0 11 31 45 55 5 63 }}, {{val| 0 1 -2 -8 -12 -15 1 -17 }}]
Optimal tunings:  
* WE: ~17/12 = 599.8115{{c}}, ~3/2 = 704.5958{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 704.8086{{c}}


[[POTE generator|POL2 generator]]: ~3/2 = 703.870
{{Optimal ET sequence|legend=0| 46, 80, 126 }}


{{Val list|legend=1| 46, 58i, 104ci }}
Badness (Sintel): 1.44


== 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 & 34 }} temperament. [[78edo]] is a good tuning choice, and remains a good one in the 11-limit, where the temperament is really more interesting, adding 100/99 and 385/384 to the list of commas.


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 875/864, 2048/2025
[[Comma list]]: 875/864, 2048/2025


[[Mapping]]: [{{val|2 0 11 -23}}, {{val|0 1 -2 9}}]
{{Mapping|legend=1| 2 0 11 -23 | 0 1 -2 9 }}


{{Multival|legend=1| 2 -4 18 -11 23 53 }}
[[Optimal tuning]]s:
* [[WE]]: ~45/32 = 599.6603{{c}}, ~3/2 = 707.1707{{c}}
: [[error map]]: {{val| -0.679 +4.536 -3.033 -2.591 }}
* [[CWE]]: ~45/32 = 600.0000{{c}}, ~3/2 = 707.5294{{c}}
: error map: {{val| 0.000 +5.574 -1.373 -1.061 }}


[[POTE generator]]: ~3/2 = 707.571
{{Optimal ET sequence|legend=1| 22, 56, 78, 134b }}


{{Val list|legend=1| 22, 56, 78, 134b, 212b, 290bb }}
[[Badness]] (Sintel): 2.13
 
[[Badness]]: 0.083971


=== 11-limit ===
=== 11-limit ===
Line 454: Line 685:
Comma list: 100/99, 385/384, 1232/1215
Comma list: 100/99, 385/384, 1232/1215


Mapping: [{{val|2 0 11 -23 26}}, {{val|0 1 -2 9 -6}}]
Mapping: {{mapping| 2 0 11 -23 26 | 0 1 -2 9 -6 }}


POTE generator: ~3/2 = 707.609
Optimal tunings:  
* WE: ~45/32 = 599.6286{{c}}, ~3/2 = 707.1712{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 707.5984{{c}}


Optimal GPV sequence: {{Val list| 22, 56, 78, 212be, 290bbe }}
{{Optimal ET sequence|legend=0| 22, 56, 78 }}


Badness: 0.045270
Badness (Sintel): 1.50


==== 13-limit ====
==== 13-limit ====
Line 467: Line 700:
Comma list: 100/99, 105/104, 144/143, 1078/1053
Comma list: 100/99, 105/104, 144/143, 1078/1053


Mapping: [{{val|2 0 11 -23 26 -18}}, {{val|0 1 -2 9 -6 8}}]
Mapping: {{mapping| 2 0 11 -23 26 -18 | 0 1 -2 9 -6 8 }}


POTE generator: ~3/2 = 707.167
Optimal tunings:  
* WE: ~45/32 = 599.3498{{c}}, ~3/2 = 706.4009{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 707.1309{{c}}


Optimal GPV sequence: {{Val list| 22f, 34, 56f }}
{{Optimal ET sequence|legend=0| 22f, 34, 56f }}


Badness: 0.044877
Badness (Sintel): 1.85


===== 17-limit =====
===== 17-limit =====
Line 480: Line 715:
Comma list: 100/99, 105/104, 119/117, 144/143, 154/153
Comma list: 100/99, 105/104, 119/117, 144/143, 154/153


Mapping: [{{val|2 0 11 -23 26 -18 5}}, {{val|0 1 -2 9 -6 8 1}}]
Mapping: {{mapping| 2 0 11 -23 26 -18 5 | 0 1 -2 9 -6 8 1}}


POTE generator: ~3/2 = 707.155
Optimal tunings:  
* WE: ~17/12 = 599.4053{{c}}, ~3/2 = 706.4544{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 707.1243{{c}}


Optimal GPV sequence: {{Val list| 22f, 34, 56f }}
{{Optimal ET sequence|legend=0| 22f, 34, 56f }}


Badness: 0.030297
Badness (Sintel): 1.54


==== Keenic ====
==== Keenic ====
Line 493: Line 730:
Comma list: 91/90, 100/99, 352/351, 385/384
Comma list: 91/90, 100/99, 352/351, 385/384


Mapping: [{{val|2 0 11 -23 26 36}}, {{val|0 1 -2 9 -6 -9}}]
Mapping: {{mapping| 2 0 11 -23 26 36 | 0 1 -2 9 -6 -9 }}


POTE generator: ~3/2 = 707.257
Optimal tunings:  
* WE: ~45/32 = 599.8547{{c}}, ~3/2 = 707.0858{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 707.2596{{c}}


Optimal GPV sequence: {{Val list| 22, 34, 56 }}
{{Optimal ET sequence|legend=0| 22, 34, 56 }}


Badness: 0.040351
Badness (Sintel): 1.67


===== 17-limit =====
===== 17-limit =====
Line 506: Line 745:
Comma list: 91/90, 100/99, 136/135, 154/153, 256/255
Comma list: 91/90, 100/99, 136/135, 154/153, 256/255


Mapping: [{{val|2 0 11 -23 26 36 5}}, {{val|0 1 -2 9 -6 -9 1}}]
Mapping: {{mapping| 2 0 11 -23 26 36 5 | 0 1 -2 9 -6 -9 1 }}


POTE generator: ~3/2 = 707.252
Optimal tunings:  
* WE: ~17/12 = 599.8338{{c}}, ~3/2 = 707.0558{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 707.2537{{c}}


Optimal GPV sequence: {{Val list| 22, 34, 56 }}
{{Optimal ET sequence|legend=0| 22, 34, 56 }}


Badness: 0.026917
Badness (Sintel): 1.37


== 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 & 68 }} temperament; its ploidacot is tetraploid monocot. Scales of bidia [[cluster temperament|cluster]] around [[12edo]], with a small residue left behind when three semitones exceed the quarter-octave period. This residue represents [[64/63]], and somewhat peculiarly, [[81/80]] is represented by ''two'' of these intervals.


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 2048/2025, 3136/3125
[[Comma list]]: 2048/2025, 3136/3125


[[Mapping]]: [{{val|4 0 22 43}}, {{val|0 1 -2 -5}}]
{{Mapping|legend=1| 4 0 22 43 | 0 1 -2 -5 }}
: mapping generators: ~25/21, ~3


{{Multival|legend=1|4 -8 -20 -22 -43 -24}}
[[Optimal tuning]]s:
* [[WE]]: ~25/21 = 299.6887{{c}}, ~3/2 = 704.6318{{c}}
: [[error map]]: {{val| -1.245 +1.432 +0.064 +0.854 }}
* [[CWE]]: ~25/21 = 300.0000{{c}}, ~3/2 = 705.5070{{c}}
: error map: {{val| 0.000 +3.552 +2.672 +3.639 }}


[[POTE generator]]: ~3/2 = 705.364
{{Optimal ET sequence|legend=1| 12, …, 56, 68, 80, 148d }}


{{Val list|legend=1| 12, 56, 68, 80, 148d }}
[[Badness]] (Sintel): 1.43
 
[[Badness]]: 0.056474


=== 11-limit ===
=== 11-limit ===
Line 536: Line 780:
Comma list: 176/175, 896/891, 1375/1372
Comma list: 176/175, 896/891, 1375/1372


Mapping: [{{val|4 0 22 43 71}}, {{val|0 1 -2 -5 -9}}]
Mapping: {{mapping| 4 0 22 43 71 | 0 1 -2 -5 -9 }}


POTE generator: ~3/2 = 705.087
Optimal tunings:  
* WE: ~25/21 = 299.6809{{c}}, ~3/2 = 704.3367{{c}}
* CWE: ~25/21 = 300.0000{{c}}, ~3/2 = 705.2170{{c}}


Optimal GPV sequence: {{Val list| 12, 68, 80 }}
{{Optimal ET sequence|legend=0| 12, 56e, 68, 80 }}


Badness: 0.040191
Badness (Sintel): 1.33


=== 13-limit ===
=== 13-limit ===
Line 549: Line 795:
Comma list: 176/175, 325/324, 640/637, 896/891
Comma list: 176/175, 325/324, 640/637, 896/891


Mapping: [{{val|4 0 22 43 71 -36}}, {{val|0 1 -2 -5 -9 8 }}]
Mapping: {{mapping| 4 0 22 43 71 -36 | 0 1 -2 -5 -9 8 }}


POTE generator: ~3/2 = 705.301
Optimal tunings:  
* WE: ~25/21 = 299.7538{{c}}, ~3/2 = 704.7222{{c}}
* CWE: ~25/21 = 300.0000{{c}}, ~3/2 = 705.3241{{c}}


Optimal GPV sequence: {{Val list| 12, 68, 80, 148d, 228bcd, 376bbcddf }}
{{Optimal ET sequence|legend=0| 12, 68, 80, 148d, 228bcd, 376bbcddf }}


Badness: 0.041137
Badness (Sintel): 1.70


=== 17-limit ===
=== 17-limit ===
Line 562: Line 810:
Comma list: 136/135, 176/175, 256/255, 325/324, 640/637
Comma list: 136/135, 176/175, 256/255, 325/324, 640/637


Mapping: [{{val|4 0 22 43 71 -36 10 }}, {{val|0 1 -2 -5 -9 8 1 }}]
Mapping: {{mapping| 4 0 22 43 71 -36 10 | 0 1 -2 -5 -9 8 1 }}


POTE generator: ~3/2 = 705.334
Optimal tunings:  
* WE: ~25/21 = 299.7883{{c}}, ~3/2 = 704.8365{{c}}
* CWE: ~25/21 = 300.0000{{c}}, ~3/2 = 705.3496{{c}}


Optimal GPV sequence: {{Val list| 12, 68, 80, 148d, 228bcd, 376bbcddf }}
{{Optimal ET sequence|legend=0| 12, 68, 80, 148d }}


Badness: 0.028631
Badness (Sintel): 1.46


=== 19-limit ===
=== 19-limit ===
Line 575: Line 825:
Comma list: 136/135, 176/175, 190/189, 256/255, 325/324, 640/637
Comma list: 136/135, 176/175, 190/189, 256/255, 325/324, 640/637


Mapping: [{{val|4 0 22 43 71 -36 10 17 }}, {{val|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 tunings:
* WE: ~19/16 = 299.7967{{c}}, ~3/2 = 704.8609{{c}}
* CWE: ~19/16 = 300.0000{{c}}, ~3/2 = 705.3519{{c}}
 
{{Optimal ET sequence|legend=0| 12, 68, 80, 148d }}
 
Badness (Sintel): 1.25
 
=== 23-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23
 
Comma list: 136/135, 176/175, 190/189, 253/252, 256/255, 325/324, 640/637


POTE generator: ~3/2 = 705.339
Mapping: {{mapping| 4 0 22 43 71 -36 10 17 -20 | 0 1 -2 -5 -9 8 1 0 6 }}


Optimal GPV sequence: {{Val list| 12, 68, 80, 148d, 376bbcddfh }}
Optimal tunings:
* WE: ~19/16 = 299.7961{{c}}, ~3/2 = 704.8577{{c}}
* CWE: ~19/16 = 300.0000{{c}}, ~3/2 = 705.3413{{c}}


Badness: 0.020590
{{Optimal ET sequence|legend=0| 12, 68, 80, 148di }}


== Echidna ==
Badness (Sintel): 1.24
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 add to {{val| 138 219 321 388 }}.


Echidna becomes more interesting when extended to be an 11-limit temperament by adding 176/175, 896/891 or 540/539 to the commas, where the same tunings can be used as before. It then is able to represent the entire 11-limit diamond to within about six cents of error, within a compass of 24 notes. The 28 note 2MOS gives scope for this, and the 36 note MOS much more.
== Shrutar ==
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. Its ploidacot is diploid alpha-dicot. [[68edo]] makes for a good tuning, but another excellent choice is a generator of 14<sup>(1/7)</sup>, making 7's just.


Subgroup: 2.3.5.7
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.


[[Comma list]]: 1728/1715, 2048/2025
Additionally, shrutar can employ the standard diaschismic mapping of prime 17, and most naturally represents the 2.3.5.7.11.17 subgroup temperament where 15:16:17:18 and 32:33:34:35:36 are equalized. Shrutar canonically maps primes 13, 19, and 23 as the 46 & 68 temperament; these mappings are significantly more complex and need finer tuning, however.


[[Mapping]]: [{{val|2 1 9 2}}, {{val|0 3 -6 5}}]
[[Subgroup]]: 2.3.5.7


{{Multival|legend=1| 6 -12 10 -33 -1 57 }}
[[Comma list]]: 245/243, 2048/2025


[[POTE generator]]: ~9/7 = 434.856
{{Mapping|legend=1| 2 1 9 -2 | 0 2 -4 7 }}
: mapping generators: ~45/32, ~35/24


{{Val list|legend=1| 22, 58, 80, 138cd, 218cd }}
[[Optimal tuning]]s:
* [[WE]]: ~45/32 = 599.5401{{c}}, ~35/24 = 652.3108{{c}}
: [[error map]]: {{val| -0.920 +2.207 +0.304 -1.730 }}
* [[CWE]]: ~45/32 = 600.0000{{c}}, ~35/24 = 652.7736{{c}}
: error map: {{val| 0.000 +3.592 +2.592 +0.589 }}


[[Badness]]: 0.058033
{{Optimal ET sequence|legend=1| 22, 46, 68, 182b, 250bc }}
 
[[Badness]] (Sintel): 1.20


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 176/175, 540/539, 896/891
Comma list: 121/120, 176/175, 245/243
 
Mapping: [{{val|2 1 9 2 12}}, {{val|0 3 -6 5 -7}}]


POTE generator: ~9/7 = 434.852
Mapping: {{mapping| 2 1 9 -2 8 | 0 2 -4 7 -1 }}


Minimax tuning:  
Optimal tunings:
* 11-odd-limit: ~9/7 = {{monzo|5/12 0 0 1/12 -1/12}}
* WE: ~45/32 = 599.7721{{c}}, ~16/11 = 652.4321{{c}}
: [{{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}}]
* CWE: ~45/32 = 600.0000{{c}}, ~16/11 = 652.6672{{c}}
: Eigenmonzos: 2, 11/7


Optimal GPV sequence: {{Val list| 22, 58, 80, 138cde, 218cde }}
{{Optimal ET sequence|legend=0| 22, 46, 68, 114 }}


Badness: 0.025987
Badness (Sintel): 0.876


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 176/175, 351/350, 364/363, 540/539
Comma list: 121/120, 176/175, 196/195, 245/243


Mapping: [{{val|2 1 9 2 12 19}}, {{val|0 3 -6 5 -7 -16}}]
Mapping: {{mapping| 2 1 9 -2 8 -10 | 0 2 -4 7 -1 16 }}


POTE generator: ~9/7 = 434.756
Optimal tunings:
* WE: ~45/32 = 599.7699{{c}}, ~16/11 = 652.4035{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~16/11 = 652.6374{{c}}


Optimal GPV sequence: {{Val list| 22, 58, 80, 138cde }}
{{Optimal ET sequence|legend=0| 22f, 46, 68, 114 }}


Badness: 0.023679
Badness (Sintel): 1.16


=== 17-limit ===
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 136/135, 176/175, 221/220, 256/255, 540/539
Comma list: 121/120, 136/135, 154/153, 176/175, 196/195
 
Mapping: {{mapping| 2 1 9 -2 8 -10 6 | 0 2 -4 7 -1 16 2 }}
 
Optimal tunings:
* WE: ~17/12 = 599.7995{{c}}, ~16/11 = 652.4287{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~16/11 = 652.6334{{c}}
 
{{Optimal ET sequence|legend=0| 22f, 46, 68, 114 }}
 
Badness (Sintel): 0.953
 
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 121/120, 136/135, 154/153, 176/175, 196/195, 343/342
 
Mapping: {{mapping| 2 1 9 -2 8 -10 6 -10 | 0 2 -4 7 -1 16 2 17 }}
 
Optimal tunings:
* WE: ~17/12 = 599.8060{{c}}, ~16/11 = 652.5190{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~16/11 = 652.7164{{c}}
 
{{Optimal ET sequence|legend=0| 22fh, 46, 68, 114, 182bef }}
 
Badness (Sintel): 1.07
 
==== 23-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23
 
Comma list: 121/120, 136/135, 154/153, 176/175, 196/195, 253/252, 343/342


Mapping: [{{val|2 1 9 2 12 19 6}}, {{val|0 3 -6 5 -7 -16 3}}]
Mapping: {{mapping| 2 1 9 -2 8 -10 6 -10 -4 | 0 2 -4 7 -1 16 2 17 12 }}


POTE generator: ~9/7 = 434.816
Optimal tunings:
* WE: ~17/12 = 599.7879{{c}}, ~16/11 = 652.4776{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~16/11 = 652.6926{{c}}


Optimal GPV sequence: {{Val list| 22, 58, 80, 138cde }}
{{Optimal ET sequence|legend=0| 22fh, 46, 68, 114 }}


Badness: 0.020273
Badness (Sintel): 1.03


== Echidnic ==
== Shru ==
Subgroup: 2.3.5.7
Shru tempers out 392/375 and slices the compound semitone into two generators of ~10/7. Its ploidacot is diploid alpha-dicot, the same as shrutar.
 
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 686/675, 1029/1024
[[Comma list]]: 392/375, 1323/1280


[[Mapping]]: [{{val|2 2 7 6}}, {{val|0 3 -6 -1}}]
{{Mapping|legend=1| 2 1 9 11 | 0 2 -4 -5 }}
: mapping generators: ~45/32, ~10/7


[[POTE generator]]: ~8/7 = 234.492
[[Optimal tuning]]s:
* [[WE]]: ~45/32 = 600.2519{{c}}, ~10/7 = 650.4083{{c}}
: [[error map]]: {{val| +0.504 -0.887 +14.321 -18.096 }}
* [[CWE]]: ~45/32 = 600.0000{{c}}, ~10/7 = 650.1017{{c}}
: error map: {{val| 0.000 -1.752 +13.279 -19.334 }}


{{Val list|legend=1| 10, 36, 46, 194bcd, 240bcd, 286bcd, 332bccdd }}
{{Optimal ET sequence|legend=1| 2, 22d, 24 }}


[[Badness]]: 0.072246
[[Badness]] (Sintel): 3.99


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 385/384, 441/440, 686/675
Comma list: 56/55, 77/75, 1323/1280


Mapping: [{{val|2 2 7 6 3}}, {{val|0 3 -6 -1 10}}]
Mapping: {{mapping| 2 1 9 11 8 | 0 2 -4 -5 -1 }}


POTE generator: ~8/7 = 235.096
Optimal tunings:
* WE: ~17/12 = 600.2356{{c}}, ~10/7 = 650.3856{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~10/7 = 650.1008{{c}}


Optimal GPV sequence: {{Val list| 10, 36e, 46, 102, 148, 342bcdd }}
{{Optimal ET sequence|legend=0| 2, 22d, 24 }}


Badness: 0.045127
Badness (Sintel): 2.10


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 91/90, 169/168, 385/384, 441/440
Comma list: 56/55, 77/75, 105/104, 507/500
 
Mapping: {{mapping| 2 1 9 11 8 15 | 0 2 -4 -5 -1 -7 }}
 
Optimal tunings:
* WE: ~45/32 = 599.9067{{c}}, ~10/7 = 649.4907{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~10/7 = 649.5950{{c}}
 
{{Optimal ET sequence|legend=0| 2, 24 }}
 
Badness (Sintel): 2.12
 
== Sruti ==
Sruti tempers out 19683/19600, setting itself up as a [[hemipyth]] temperament. It has the same semi-octave period as diaschismic, but the generator can be taken as a neutral third or a hemitwelfth. The temperament can be described as {{nowrap| 24 & 34d }}; its ploidacot is diploid dicot. [[58edo]] may be recommended as a tuning.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2048/2025, 19683/19600
 
{{Mapping|legend=1| 2 0 11 -15 | 0 2 -4 13 }}
: mapping generators: ~45/32, ~140/81
 
[[Optimal tuning]]s:
* [[WE]]: ~45/32 = 599.2764{{c}}, ~140/81 = 950.7284{{c}}
: [[error map]]: {{val| -1.447 -0.498 +2.813 +1.497 }}
* [[CWE]]: ~45/32 = 600.0000{{c}}, ~140/81 = 951.8227{{c}}
: error map: {{val| 0.000 +1.690 +6.395 +4.869 }}
 
{{Optimal ET sequence|legend=1| 24, 34d, 58, 150cd, 208ccdd, 266ccdd }}
 
[[Badness]] (Sintel): 2.97
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 176/175, 243/242, 896/891


Mapping: [{{val|2 2 7 6 3 7}}, {{val|0 3 -6 -1 10 1}}]
Mapping: {{mapping| 2 0 11 -15 -1 | 0 2 -4 13 5 }}


POTE generator: ~8/7 = 235.088
Optimal tunings:
* WE: ~45/32 = 599.1951{{c}}, ~121/70 = 950.5864{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~121/70 = 951.7972{{c}}


Optimal GPV sequence: {{Val list| 10, 46, 102, 148f, 194bcdf }}
{{Optimal ET sequence|legend=0| 24, 34d, 58, 150cdee, 208ccddee, 266ccddeee }}


Badness: 0.028874
Badness (Sintel): 1.37


=== 17-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13
 
Comma list: 144/143, 176/175, 351/350, 676/675
 
Mapping: {{mapping| 2 0 11 -15 -1 9 | 0 2 -4 13 5 -1 }}
 
Optimal tunings:
* WE: ~45/32 = 599.1479{{c}}, ~26/15 = 950.5337{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~26/15 = 951.8314{{c}}


Comma list: 91/90, 136/135, 154/153, 169/168, 256/255
{{Optimal ET sequence|legend=0| 24, 34d, 58, 150cdeef, 208ccddeeff, 266ccddeeefff }}


Mapping: [{{val|2 2 7 6 3 7 7}}, {{val|0 3 -6 -1 10 1 3}}]
Badness (Sintel): 0.983


POTE generator: ~8/7 = 235.088
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Optimal GPV sequence: {{Val list| 10, 46, 102, 148f, 194bcdf }}
Comma list: 136/135, 144/143, 170/169, 176/175, 221/220


Badness: 0.019304
Mapping: {{mapping| 2 0 11 -15 -1 9 5 | 0 2 -4 13 5 -1 2 }}


; Compositions
Optimal tunings:
* [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]]
* WE: ~17/12 = 599.3003{{c}}, ~26/15 = 950.7465{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~26/15 = 951.8142{{c}}


== Shrutar ==
{{Optimal ET sequence|legend=0| 24, 34d, 58 }}
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 and excellent choice is a generator of 14<sup>(1/7)</sup>, making 7s 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.
Badness (Sintel): 1.05


Subgroup: 2.3.5.7
== Anguirus ==
As another hemipyth temperament, anguirus tempers out 49/48. It can be described as the {{nowrap| 10 & 24 }} temperament; its ploidacot is diploid dicot, the same as sruti.  


[[Comma list]]: 245/243, 2048/2025
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val| 2 3 5 5 }}, {{val| 0 2 -4 7 }}]
[[Comma list]]: 49/48, 2048/2025


{{Multival|legend=1| 4 -8 14 -22 11 55 }}
{{Mapping|legend=1| 2 0 11 4 | 0 2 -4 1 }}
: mapping generators: ~45/32, ~7/4


[[POTE generator]]: ~36/35 = 52.811
[[Optimal tuning]]s:  
* [[WE]]: ~45/32 = 600.2758{{c}}, ~7/4 = 953.4593{{c}}
: [[error map]]: {{val| +0.552 +4.964 +2.883 -14.264 }}
* [[CWE]]: ~45/32 = 600.0000{{c}}, ~7/4 = 953.0188{{c}}
: error map: {{val| 0.000 +4.083 +1.611 -15.807 }}


{{Val list|legend=1| 22, 46, 68, 182b, 250bc }}
{{Optimal ET sequence|legend=1| 10, 24, 34 }}


[[Badness]]: 0.047377
[[Badness]] (Sintel): 1.97


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 121/120, 176/175, 245/243
Comma list: 49/48, 56/55, 243/242


Mapping: [{{val| 2 3 5 5 7 }}, {{val| 0 2 -4 7 -1 }}]
Mapping: {{mapping| 2 0 11 4 -1 | 0 2 -4 1 5 }}


POTE generator: ~33/32 = 52.680
Optimal tunings:
* WE: ~45/32 = 599.9250{{c}}, ~7/4 = 952.0646{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~7/4 = 952.1784{{c}}


Optimal GPV sequence: {{Val list| 22, 46, 68, 114, 296bce, 410bce }}
{{Optimal ET sequence|legend=0| 10, 24, 34 }}


Badness: 0.026489
Badness (Sintel): 1.63


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 121/120, 176/175, 196/195, 245/243
Comma list: 49/48, 56/55, 91/90, 243/242


Mapping: [{{val| 2 3 5 5 7 6 }}, {{val| 0 2 -4 7 -1 16 }}]
Mapping: {{mapping| 2 0 11 4 -1 9 | 0 2 -4 1 5 -1 }}


POTE generator: ~33/32 = 52.654
Optimal tunings:
* WE: ~45/32 = 599.7575{{c}}, ~7/4 = 951.9241{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~7/4 = 952.2980{{c}}


Optimal GPV sequence: {{Val list| 22f, 24f, 46, 68, 114 }}
{{Optimal ET sequence|legend=0| 10, 24, 34, 58d, 92ddef }}


Badness: 0.028057
Badness (Sintel): 1.27


=== 17-limit ===
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 121/120, 136/135, 154/153, 176/175, 196/195
Comma list: 49/48, 56/55, 91/90, 119/117, 154/153


Mapping: [{{val| 2 3 5 5 7 6 8 }}, {{val| 0 2 -4 7 -1 16 2 }}]
Mapping: {{mapping| 2 0 11 4 -1 9 5 | 0 2 -4 1 5 -1 2 }}


POTE generator: ~33/32 = 52.647
Optimal tunings:
* WE: ~17/12 = 599.7925{{c}}, ~7/4 = 952.0004{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~7/4 = 952.3178{{c}}


Optimal GPV sequence: {{Val list| 22f, 24f, 46, 68, 114 }}
{{Optimal ET sequence|legend=0| 10, 24, 34 }}


Badness: 0.018716
Badness (Sintel): 1.10


=== 19-limit ===
== Echidna ==
Subgroup: 2.3.5.7.11.13.17.19
Echidna adds 1728/1715 to the commas and takes 9/7 as a generator. It may be called the {{nowrap| 22 & 58 }} temperament; its ploidacot is diploid alpha-tricot. [[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.  


Comma list: 121/120, 136/135, 154/153, 176/175, 196/195, 343/342
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.


Mapping: [{{val| 2 3 5 5 7 6 8 7 }}, {{val| 0 2 -4 7 -1 16 2 17 }}]
The generator can be interpreted as 11/10, the period complement of 9/7, as a stack of 11/10 and 9/7 makes [[99/70]] which is extremely close to 600{{cent}} and is equal to it if we temper out [[9801/9800|S99]]. Three 11/10's then make a 4/3 (tempering out [[4000/3993|S10/S11]] thus making 10/9 and 12/11 equidistant from 11/10), implying a flat tuning of 4/3.


POTE generator: ~33/32 = 52.730
Like most srutal extensions, the 13- and 17-limit interpretations are possible by observing that since we have tempered out [[176/175]], tempering out [[351/350]] and [[352/351]] which sum to 176/175 is very elegant. In the 17-limit we can equate the half-octave with 17/12 and 24/17 and we can take advantage of the sharp fifth by combining echidna with [[srutal archagall]], leading to a particularly beautiful temperament (one that prefers a very slightly less sharp fifth than srutal archagall). This mapping of 13 and 17 is supported by the patent vals of the three main echidna edos of 22, 58 and 80, of which all except 22 are consistent in the [[17-odd-limit]].


Optimal GPV sequence: {{Val list| 22fh, 24fh, 46, 68, 114, 182bef }}
[[Subgroup]]: 2.3.5.7


Badness: 0.017540
[[Comma list]]: 1728/1715, 2048/2025
 
== Sruti ==
Subgroup: 2.3.5.7


[[Comma list]]: 2048/2025, 19683/19600
{{Mapping|legend=1| 2 1 9 2 | 0 3 -6 5 }}
: mapping generators: ~45/32, ~9/7


[[Mapping]]: [{{val| 2 0 11 -15 }}, {{val| 0 2 -4 13 }}]
[[Optimal tuning]]s:  
* [[WE]]: ~45/32 = 599.3056{{c}}, ~9/7 = 434.3524{{c}}
: [[error map]]: {{val| -1.389 +0.408 +1.322 +1.547 }}
* [[CWE]]: ~45/32 = 600.0000{{c}}, ~9/7 = 434.8327{{c}}
: error map: {{val| 0.000 +2.543 +4.690 +5.338 }}


{{Multival|legend=1| 4 -8 26 -22 30 83 }}
{{Optimal ET sequence|legend=1| 22, 58, 80, 138cd, 218cd }}


POTE generator: ~175/144 = 351.876
[[Badness]] (Sintel): 1.47
 
{{Val list|legend=1| 24, 34d, 58, 150cd, 208ccdd, 266ccdd }}
 
[[Badness]]: 0.117358


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 176/175, 243/242, 896/891
Comma list: 176/175, 540/539, 896/891


Mapping: [{{val| 2 0 11 -15 -1 }}, {{val| 0 2 -4 13 5 }}]
Mapping: {{mapping| 2 1 9 2 12 | 0 3 -6 5 -7 }}


POTE generator: ~11/9 = 351.863
Optimal tunings:
* WE: ~45/32 = 599.3085{{c}}, ~9/7 = 434.3511{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~9/7 = 434.8647{{c}}


Optimal GPV sequence: {{Val list| 24, 34d, 58 }}
Minimax tuning:  
* 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 }}]
: unchanged-interval (eigenmonzo) basis: 2.11/7


Badness: 0.041459
{{Optimal ET sequence|legend=0| 22, 58, 80, 138cde, 218cde }}
 
Badness (Sintel): 0.859


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 144/143, 176/175, 351/350, 676/675
Comma list: 176/175, 351/350, 364/363, 540/539


Mapping: [{{val| 2 0 11 -15 -1 9 }}, {{val| 0 2 -4 13 5 -1 }}]
Mapping: {{mapping| 2 1 9 2 12 19 | 0 3 -6 5 -7 -16 }}


POTE generator: ~11/9 = 351.886
Optimal tunings:
* WE: ~45/32 = 599.3397{{c}}, ~9/7 = 434.2772{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~9/7 = 434.7864{{c}}


Optimal GPV sequence: {{Val list| 24, 34d, 58, 150cdeef, 208ccddeeff }}
{{Optimal ET sequence|legend=0| 22, 36f, 58, 80, 138cde }}


Badness: 0.023791
Badness (Sintel): 0.978


=== 17-limit ===
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 136/135, 144/143, 170/169, 176/175, 221/220
Comma list: 136/135, 176/175, 221/220, 256/255, 540/539


Mapping: [{{val| 2 0 11 -15 -1 9 5 }}, {{val| 0 2 -4 13 5 -1 2 }}]
Mapping: {{mapping| 2 1 9 2 12 19 6 | 0 3 -6 5 -7 -16 3 }}


POTE generator: ~11/9 = 351.857
Optimal tunings:
* WE: ~45/32 = 599.4645{{c}}, ~9/7 = 434.4282{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~9/7 = 434.8340{{c}}


Optimal GPV sequence: {{Val list| 24, 34d, 58 }}
{{Optimal ET sequence|legend=0| 22, 36f, 58, 80, 138cde }}


Badness: 0.020536
Badness (Sintel): 1.03


== Anguirus ==
== Echidnic ==
Subgroup: 2.3.5.7
Echidnic tempers out 686/675 and [[1029/1024]]. It has the same semi-octave period as diaschismic, but slices the generator of a fifth into three ~8/7's. It can be described as the {{nowrap| 10 & 46 }} temperament; its ploidacot is diploid tricot.  


[[Comma list]]: 49/48, 2048/2025
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val| 2 0 11 4 }}, {{val| 0 2 -4 1 }}]
[[Comma list]]: 686/675, 1029/1024


{{Multival|legend=1| 4 -8 2 -22 -8 27 }}
{{Mapping|legend=1| 2 2 7 6 | 0 3 -6 -1 }}
: mapping generators: ~45/32, ~8/7


[[POTE generator]]: ~8/7 = 246.979
[[Optimal tuning]]s:
* [[WE]]: ~45/32 = 599.7208{{c}}, ~8/7 = 234.8330{{c}}
: [[error map]]: {{val| -0.558 +1.986 +2.733 -5.334 }}
* [[CWE]]: ~45/32 = 600.0000{{c}}, ~8/7 = 234.9539{{c}}
: error map: {{val| 0.000 +2.907 +3.963 -3.780 }}


{{Val list|legend=1| 10, 24, 34 }}
{{Optimal ET sequence|legend=1| 10, 26c, 36, 46 }}


[[Badness]]: 0.077955
[[Badness]] (Sintel): 1.83


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 49/48, 56/55, 243/242
Comma list: 385/384, 441/440, 686/675


Mapping: [{{val| 2 0 11 4 -1 }}, {{val| 0 2 -4 1 5 }}]
Mapping: {{mapping| 2 2 7 6 3 | 0 3 -6 -1 10 }}


POTE generator: ~8/7 = 247.816
Optimal tunings:
* WE: ~45/32 = 599.8022{{c}}, ~8/7 = 235.0185{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~8/7 = 235.0893{{c}}


Optimal GPV sequence: {{Val list| 10, 24, 34, 58d, 92de }}
{{Optimal ET sequence|legend=0| 10, 36e, 46, 102, 148 }}


Badness: 0.049253
Badness (Sintel): 1.49


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 49/48, 56/55, 91/90, 243/242
Comma list: 91/90, 169/168, 385/384, 441/440


Mapping: [{{val| 2 0 11 4 -1 9 }}, {{val| 0 2 -4 1 5 -1 }}]
Mapping: {{mapping| 2 2 7 6 3 7 | 0 3 -6 -1 10 1 }}


POTE generator: ~8/7 = 247.691
Optimal tunings:
* WE: ~45/32 = 599.9570{{c}}, ~8/7 = 235.0708{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~8/7 = 235.0862{{c}}


Optimal GPV sequence: {{Val list| 10, 24, 34, 58d, 92ddef }}
{{Optimal ET sequence|legend=0| 10, 36e, 46, 102, 148f }}


Badness: 0.030829
Badness (Sintel): 1.19


=== 17-limit ===
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 49/48, 56/55, 91/90, 119/117, 154/153
Comma list: 91/90, 136/135, 154/153, 169/168, 256/255
 
Mapping: {{mapping| 2 2 7 6 3 7 7 | 0 3 -6 -1 10 1 3 }}


Mapping: [{{val| 2 0 11 4 -1 9 5 }}, {{val| 0 2 -4 1 5 -1 2 }}]
Optimal tunings:
* WE: ~17/12 = 599.9571{{c}}, ~8/7 = 235.0709{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~8/7 = 235.0860{{c}}


POTE generator: ~8/7 = 247.670
{{Optimal ET sequence|legend=0| 10, 36e, 46, 102, 148f }}


Optimal GPV sequence: {{Val list| 10, 24, 34, 58d, 92ddef }}
Badness (Sintel): 0.983


Badness: 0.021796
; 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://www.youtube.com/watch?v=VsBXIvBZY6A ''56edo Track (Echidnic16 Scale)''] by [[Budjarn Lambeth]] (2025)


== Shru ==
== Quadrasruta ==
Subgroup: 2.3.5.7
Named by [[Xenllium]] in 2022, quadrasruta tempers out 2401/2400, the breedsma, and extends [[buzzard]]. It may be described as {{nowrap| 58 & 68 }}; its ploidacot is diploid alpha-tetracot. 126edo may be recommended as a tuning.  


[[Comma list]]: 392/375, 1323/1280
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val| 2 1 9 11 }}, {{val| 0 2 -4 -5 }}]
[[Comma list]]: 2048/2025, 2401/2400


{{Multival|legend=1| 4 -8 -10 -22 -27 -1 }}
{{Mapping|legend=1| 2 0 11 8 | 0 4 -8 -3 }}
: mapping generators: ~45/32, ~21/16


[[POTE generator]]: ~64/63 = 50.135
[[Optimal tuning]]s:  
* [[WE]]: ~45/32 = 599.4443{{c}}, ~21/16 = 475.7746{{c}}
: [[error map]]: {{val| -1.111 +1.143 +1.377 -0.595 }}
* [[CWE]]: ~45/32 = 600.0000{{c}}, ~21/16 = 476.2394{{c}}
: error map: {{val| 0.000 +3.003 +3.771 +2.456 }}


{{Val list|legend=1| 2, 22d, 24 }}
{{Optimal ET sequence|legend=1| 10, …, 58, 68, 126, 446bbccd }}


[[Badness]]: 0.157619
[[Badness]] (Sintel): 1.86


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 56/55, 77/75, 1323/1280
Comma list: 176/175, 896/891, 2401/2400
 
Mapping: {{mapping| 2 0 11 8 22 | 0 4 -8 -3 -19 }}
 
Optimal tunings:
* WE: ~45/32 = 599.4648{{c}}, ~21/16 = 475.6929{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~21/16 = 476.1507{{c}}
 
{{Optimal ET sequence|legend=0| 10e, …, 58, 126, 184c, 310bccde }}
 
Badness (Sintel): 1.62
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 176/175, 196/195, 512/507, 676/675
 
Mapping: {{mapping| 2 0 11 8 22 9 | 0 4 -8 -3 -19 -2 }}
 
Optimal tunings:
* WE: ~45/32 = 599.3787{{c}}, ~21/16 = 475.6065{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~21/16 = 476.1345{{c}}
 
{{Optimal ET sequence|legend=0| 10e, …, 58, 126f, 184cff }}
 
Badness (Sintel): 1.18
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 136/135, 170/169, 176/175, 196/195, 256/255
 
Mapping: {{mapping| 2 0 11 8 22 9 5 | 0 4 -8 -3 -19 -2 4 }}
 
Optimal tunings:
* WE: ~17/12 = 599.5077{{c}}, ~21/16 = 475.7713{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~21/16 = 476.1814{{c}}
 
{{Optimal ET sequence|legend=0| 10e, 58, 126f }}
 
Badness (Sintel): 1.21
 
=== Quadrafourths ===
Subgroup: 2.3.5.7.11
 
Comma list: 243/242, 441/440, 2048/2025


Mapping: [{{val| 2 1 9 11 8 }}, {{val| 0 2 -4 -5 -1 }}]
Mapping: {{mapping| 2 0 11 8 -1 | 0 4 -8 -3 10 }}


POTE generator: ~33/32 = 50.130
Optimal tunings:
* WE: ~45/32 = 599.2593{{c}}, ~21/16 = 475.4292{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~21/16 = 476.0088{{c}}


Optimal GPV sequence: {{Val list| 2, 22d, 24 }}
{{Optimal ET sequence|legend=0| 10, 48c, 58, 184cee, 242ccdeee }}


Badness: 0.063483
Badness (Sintel): 1.62


=== 13-limit ===
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 56/55, 77/75, 105/104, 507/500
Comma list: 144/143, 196/195, 243/242, 676/675
 
Mapping: {{mapping| 2 0 11 8 -1 9 | 0 4 -8 -3 10 -2 }}
 
Optimal tunings:
* WE: ~45/32 = 599.2147{{c}}, ~21/16 = 475.4052{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~21/16 = 476.0253{{c}}
 
{{Optimal ET sequence|legend=0| 10, 48c, 58, 126eef, 184ceeff, 242ccdeeeff }}
 
Badness (Sintel): 1.11
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 136/135, 144/143, 170/169, 196/195, 221/220
 
Mapping: {{mapping| 2 0 11 8 -1 9 5 | 0 4 -8 -3 10 -2 4 }}
 
Optimal tunings:
* WE: ~17/12 = 599.3353{{c}}, ~21/16 = 475.5495{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~21/16 = 476.0691{{c}}
 
{{Optimal ET sequence|legend=0| 10, 48c, 58 }}
 
Badness (Sintel): 1.13
 
== Subgroup extensions ==
=== Srutal archagall (2.3.5.17) ===
{{See also | Fiventeen }}
 
Subgroup: 2.3.5.17
 
Comma list: 136/135, 256/255


Mapping: [{{val| 2 1 9 11 8 15 }}, {{val| 0 2 -4 -5 -1 -7 }}]
Subgroup-val mapping: {{mapping| 2 0 11 5 | 0 1 -2 1 }}
: mapping generators: ~17/12, ~3


POTE generator: ~33/32 = 50.535
Optimal tunings:
* WE: ~45/32 = 599.5585{{c}}, ~3/2 = 704.6188{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 705.1356{{c}}


Optimal GPV sequence: {{Val list| 22df, 24 }}
{{Optimal ET sequence|legend=0| 10, 12, 22, 34, 80, 114, 194bc }}


Badness: 0.045731
Badness (Sintel): 0.212


[[Category:Regular temperament theory]]
[[Category:Temperament families]]
[[Category:Diaschismic family| ]] <!-- main article -->
[[Category:Diaschismic family| ]] <!-- main article -->
[[Category:Temperament family]]
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Diaschismic_family"