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{{Technical data page}}
The 5-limit parent comma for the '''diaschismic family''' is 2048/2025, the [[diaschisma|diaschisma]]. Its monzo is |11 -4 -2>, and flipping that yields <<2 -4 -11|| for the wedgie for 5-limit diaschismic, or '''srutal''', temperament. This tells us the period is half an octave, the 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|34edo]] is a good tuning choice, with [[46edo|46edo]], [[56edo|56edo]], [[58edo|58edo]] or [[80edo|80edo]] being other possibilities. Both [[12edo|12edo]] and [[22edo|22edo]] support it, and retuning them to a MOS of diaschismic gives two scale possibilities.
The '''diaschismic family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the diaschisma, [[2048/2025]].  


[[Tuning_Ranges_of_Regular_Temperaments|valid range]]: [600.000 to 720.000] (2 to 5)
== Diaschismic ==
{{Main| Diaschismic }}


nice range: [701.955, 706.843]
The [[period]] of diaschismic is half an [[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.


strict range: [701.955, 706.843]
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.


[[POTE_tuning|POTE generator]]: ~3/2 = 704.898
[[Subgroup]]: 2.3.5


Map: [<2 0 11|, <0 1 -2|]
[[Comma list]]: 2048/2025


EDOs: 34, 46, 80, 206c, 286bc
{{Mapping|legend=1| 2 0 11 | 0 1 -2 }}
: mapping generators: ~45/32, ~3


==Seven limit children==
[[Optimal tuning]]s:
The second comma of the [[Normal_lists|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. Diaschismic adds 2097152/2066715 to obtain 7-limit harmony by more complex methods, but with greater accuracy. Keen adds 2240/2187, echidna 1728/1715 and shrutar 245/243, the sensamagic comma. The pajara, diaschismic and keen keep the same 1/2 octave period and fifth generator, but shrutar has a generator of a quarter-tone (which can be taken as [[36/35|36/35]], the septimal quarter-tone) and echidna has a generator of 9/7. Adding 4375/4374 does no significant tuning damage, so for that we keep the 5-limit label srutal.
* [[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 }}


=Srutal=
[[Tuning ranges]]:
Commas: 2048/2025, 4375/4374
* [[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]


valid range: [703.448, 705.882] (58 to 34d)
{{Optimal ET sequence|legend=1| 10, 12, 22, 34, 46, 80, 206c, 286bc }}


nice range: [701.955, 706.843]
[[Badness]] (Sintel): 0.467


strict range: [703.448, 705.882]
=== 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]].


POTE generator: ~3/2 = 704.814
Those all keep the same half-octave period and fifth generator.  


Map: [<2 0 11 -42|, <0 1 -2 15|]
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.


Wedgie: <<2 -4 30 -11 42 81||
Temperaments discussed elsewhere include [[stearnsmic clan #Echidna|echidna]]. The rest are considered below.


EDOs: 46, 80, 126, 206cd, 332bcd
==== Subgroup extensions ====
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_2|#Subgroup extensions]]. The [[S-expression]]-based comma list of this temperament is {[[256/255|S16]], [[289/288|S17]]}.


Badness: 0.0915
More generally, one may note that since the fifth of diaschismic is sharp in any good tuning, it is compatible with the logic of [[parapyth]] in making ~13/11 a minor third and ~14/11 a major third, leading to the rank-3 temperament [[varda]], which multiple temps here are rank-2 tunings of, such as 17-limit diaschismic and srutal (and their obvious extensions to prime 23, na"naa' and srutaloo, both by mapping ~23/16 as an augmented fourth aka. tritone).


==11-limit==
== Septimal diaschismic ==
Commas: 176/175, 896/891, 1331/1323
{{Main| Diaschismic }}
{{See also| Srutal vs diaschismic }}


valid range: [704.348, 705.882] (46 to 34d)
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.


nice range: [701.955, 706.843]
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.


strict range: [704.348, 705.882]
[[Subgroup]]: 2.3.5.7


POTE generator: ~3/2 = 704.856
[[Comma list]]: 126/125, 2048/2025


Map: [&lt;2 0 11 -42 -28|, &lt;0 1 -2 15 11|]
{{Mapping|legend=1| 2 0 11 31 | 0 1 -2 -8 }}


EDOs: 46, 80, 126, 206cd
[[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 }}


Badness: 0.0353
[[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 tradeoff]]: ~3/2 = [701.955, 706.843]


==13-limit==
{{Optimal ET sequence|legend=1| 12, 34, 46, 58, 104c, 162c }}
Commas: 169/168, 176/175, 325/324, 364/363


valid range: [704.348, 705.882] (46 to 34d)
[[Badness]] (Sintel): 0.959


nice range: [701.955, 706.843]
=== 11-limit ===
Subgroup: 2.3.5.7.11


strict range: [704.348, 705.882]
Comma list: 126/125, 176/175, 896/891


POTE generator: ~3/2 = 704.881
{{Mapping|legend=0| 2 0 11 31 45 | 0 1 -2 -8 -12 }}


Map: [&lt;2 0 11 -42 -28 -18|, &lt;0 1 -2 15 11 8|]
Optimal tunings:  
* WE: ~45/32 = 599.4471{{c}}, ~3/2 = 703.0657{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 703.7996{{c}}


EDOs: 34d, 46, 80, 206cd, 286bcde
Tuning ranges:  
* 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]


Badness: 0.0253
{{Optimal ET sequence|legend=0| 12, 34e, 46, 58, 104c, 162ce }}


=Pajara=
Badness (Sintel): 0.828
Main article: [[pajara|Pajara]]


Pajara, with wedgie &lt;&lt;2 -4 -4 -11 -12 2|| is closely associated with 22et (not to mention [[Paul_Erlich|Paul Erlich]]) but other tunings are possible. The 1/2 octave period serves as both a [[10/7|10/7]] and a [[7/5|7/5]]. Aside from 22et, 34 with the val &lt;34 54 79 96| and 56 with the val &lt;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 12et and of common practice Western music in general, while retaining the distictiveness of a sharp fifth.
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


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.
Comma list: 126/125, 176/175, 196/195, 364/363


Commas: 50/49, 64/63
{{Mapping|legend=0| 2 0 11 31 45 55 | 0 1 -2 -8 -12 -15 }}


valid range: [700.000, 720.000] (12 to 10)
Optimal tunings:  
* WE: ~45/32 = 599.4451{{c}}, ~3/2 = 703.0528{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 703.7813{{c}}


nice range: [701.955, 715.587]
Tuning ranges:  
* 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]
* 15-odd-limit diamond tradeoff: ~3/2 = [701.955, 711.731]


strict range:  [701.955, 715.587]
{{Optimal ET sequence|legend=0| 12f, 34ef, 46, 58, 104c, 162cef }}


[[POTE_tuning|POTE generator]]: 707.048
Badness (Sintel): 0.782


Map: [&lt;2 0 11 12|, &lt;0 1 -2 -2|]
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


EDOs: 22, 34, 56
Comma list: 126/125, 136/135, 176/175, 196/195, 256/255


Badness: 0.0200
{{Mapping|legend=0| 2 0 11 31 45 55 5 | 0 1 -2 -8 -12 -15 1 }}


==11-limit==
Optimal tunings:
Commas: 50/49, 64/63, 99/98
* WE: ~17/12 = 599.6253{{c}}, ~3/2 = 703.3726{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 703.8520{{c}}


valid range: [700.000, 709.091] (12 to 22)
Tuning ranges:  
* 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]


nice range: [701.955, 715.587]
{{Optimal ET sequence|legend=0| 12f, 34ef, 46, 58, 104c }}


strict range: [701.955, 709.091]
Badness (Sintel): 0.837


[[POTE_tuning|POTE generator]]: 706.885
=== 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.  


Map: [&lt;2 0 11 12 26|, &lt;0 1 -2 -2 -6|]
Subgroup: 2.3.5.7.11.13.17.23


EDOs: 22, 34, 56, 146
Comma list: 126/125, 136/135, 176/175, 196/195, 231/230, 256/255


Badness: 0.0203
{{Mapping|legend=2| 2 0 11 31 45 55 5 63 | 0 1 -2 -8 -12 -15 1 -17 }}


==13-limit==
Optimal tunings:
Commas: 50/49, 64/63, 65/63, 99/98
* WE: ~17/12 = 599.6272{{c}}, ~3/2 = 703.4326{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 703.9093{{c}}


valid range: []
{{Optimal ET sequence|legend=0| 12i, 34efi, 46, 58i, 104ci }}


nice range: [701.955, 738.573]
Badness (Sintel): 0.882


strict range: []
== Pajara ==
{{Main| Pajara }}


POTE generator: ~3/2 = 708.919
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.


Map: [&lt;2 0 11 12 26 1|, &lt;0 1 -2 -2 -6 2|]
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.


EDOs: 12, 22
[[Subgroup]]: 2.3.5.7


Badness: 0.0276
[[Comma list]]: 50/49, 64/63


==Pajarous==
{{Mapping|legend=1| 2 0 11 12 | 0 1 -2 -2 }}
Commas: 50/49, 55/54, 64/63


valid range: 709.091 (22)
[[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 }}


nice range: [701.955, 715.803]
[[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 tradeoff]]: ~3/2 = [701.955, 715.587]


strict range: 709.091
{{Optimal ET sequence|legend=1| 10, 12, 22, 34d, 56d }}


POTE generator: ~3/2 = 709.578
[[Badness]] (Sintel): 0.507


Map: [&lt;2 0 11 12 -9|, &lt;0 1 -2 -2 5|]
=== 2.3.5.7.17 subgroup ===
Subgroup: 2.3.5.7.17


EDOs: 10, 12e, 22, 120bce, 142bce
Comma list: 50/49, 64/63, 85/84


Badness: 0.0283
{{Mapping|legend=0| 2 0 11 12 5 | 0 1 -2 -2 1 }}


===13-limit===
Optimal tunings:
Commas: 50/49, 55/54, 64/63, 65/63
* WE: ~7/5 = 599.053{{c}}, ~3/2 = 706.355{{c}}
* CWE: ~7/5 = 600.000{{c}}, ~3/2 = 707.607{{c}}


valid range: []
{{Optimal ET sequence|legend=0| 10, 12, 22, 56d }}


nice range: [701.955, 738.573]
Badness (Sintel): 0.438


strict range: []
=== 11-limit ===
Subgroup: 2.3.5.7.11


POTE generator: ~3/2 = 710.240
Comma list: 50/49, 64/63, 99/98


Map: [&lt;2 0 11 12 -9 1|, &lt;0 1 -2 -2 5 2|]
{{Mapping|legend=0| 2 0 11 12 26 | 0 1 -2 -2 -6 }}


EDOs: 10, 22, 54f, 76bdf
Optimal tunings:  
* WE: ~7/5 = 598.8485{{c}}, ~3/2 = 705.5285{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 707.1826{{c}}


Badness: 0.0252
Tuning ranges:
* 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]


==Pajaric==
{{Optimal ET sequence|legend=0| 10e, 12, 22, 34d, 56d }}
Commas: 45/44, 50/49, 56/55


POTE generator: ~3/2 = 705.524
Badness (Sintel): 0.673


Map: [&lt;2 0 11 12 7|, &lt;0 1 -2 -2 0|]
==== 2.3.5.7.11.17 subgroup ====
Subgroup: 2.3.5.7.11.17


EDOs: 10, 12, 22e, 34de
Comma list: 50/49, 64/63, 85/84, 99/98


Badness: 0.0238
{{Mapping|legend=0| 2 0 11 12 26 5 | 0 1 -2 -2 -6 1 }}


===13-limit===
Optimal tunings:
Commas: 40/39, 45/44, 50/49, 56/55
* WE: ~7/5 = 599.062{{c}}, ~3/2 = 706.095{{c}}
* CWE: ~7/5 = 600.000{{c}}, ~3/2 = 707.370{{c}}


POTE generator: ~3/2 = 707.442
{{Optimal ET sequence|legend=0| 10e, 12, 22, 34d, 56d }}


Map: [&lt;2 0 11 12 7 17|, &lt;0 1 -2 -2 0 -3|]
Badness (Sintel): 0.645


EDOs: 10, 12f, 22ef, 34def
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Badness: 0.0205
Comma list: 50/49, 64/63, 65/63, 99/98


==Pajaro==
{{Mapping|legend=0| 2 0 11 12 26 1 | 0 1 -2 -2 -6 2 }}
Commas: 40/39, 50/49, 55/54, 64/63


POTE generator ~3/2 = 710.818
Optimal tunings:
* WE: ~7/5 = 599.9732{{c}}, ~3/2 = 708.8873{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 708.9227{{c}}


Map: [&lt;2 0 11 12 -9 17|, &lt;0 1 -2 -2 5 -3|]
{{Optimal ET sequence|legend=0| 10e, 12, 22 }}


EDOs: 10, 22f, 32f, 54f
Badness (Sintel): 1.14


Badness: 0.0274
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


==Hemipaj==
Comma list: 50/49, 52/51, 64/63, 65/63, 99/98
Commas: 50/49, 64/63, 121/120


POTE generator: ~11/8 = 546.383
{{Mapping|legend=0| 2 0 11 12 26 1 5 | 0 1 -2 -2 -6 2 1 }}


Map: [&lt;2 1 9 10 8|, &lt;0 2 -4 -4 -1|]
Optimal tunings:  
* WE: ~7/5 = 599.8871{{c}}, ~3/2 = 708.6725{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 708.8176{{c}}


EDOs: 20, 22, 68d, 90d
{{Optimal ET sequence|legend=0| 10e, 12, 22 }}


Badness: 0.0389
Badness (Sintel): 1.06


=Diaschismic=
==== Pajarina ====
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 wedgie &lt;&lt;2 -4 -16 -11 -31 -26||, with a 1/2 period and a sharp fifth generator like pajara, but not so sharp, giving a more accurate but more complex temperament. [[58edo|58et]] provides an excellent tuning, but an alternative is to make [[7/4|7/4]] just by making the fifth 703.897 cents, as opposed to 703.448 cents for 58et.
Subgroup: 2.3.5.7.11.13


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.
Comma list: 50/49, 64/63, 78/77, 99/98


Commas: 126/125, 2048/2025
{{Mapping|legend=0| 2 0 11 12 26 36 | 0 1 -2 -2 -6 -9 }}


[[POTE_tuning|POTE generator]]: 703.681
Optimal tunings:  
* WE: ~7/5 = 598.7732{{c}}, ~3/2 = 704.6889{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 706.3950{{c}}


Map: [&lt;2 0 11 31|, &lt;0 1 -2 -8|]
{{Optimal ET sequence|legend=0| 12f, 22, 34d }}


EDOs: 46, 58, 104c, 162c
Badness (Sintel): 0.923


==11-limit==
===== 17-limit =====
Commas: 126/125, 176/175, 896/891
Subgroup: 2.3.5.7.11.13.17


[[POTE_tuning|POTE generator]]: 703.714
Comma list: 50/49, 64/63, 78/77, 85/84, 99/98


Map: [&lt;2 0 11 31 45|, &lt;0 1 -2 -8 -12|]
{{Mapping|legend=0| 2 0 11 12 26 36 5 | 0 1 -2 -2 -6 -9 1 }}


EDOs: 46, 58, 104c, 162ce
Optimal tunings:  
* WE: ~7/5 = 599.0204{{c}}, ~3/2 = 705.2572{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 706.5660{{c}}


==13-limit==
{{Optimal ET sequence|legend=0| 12f, 22, 34d }}
Commas: 126/125, 196/195, 364/363, 2048/2025


[[POTE_tuning|POTE generator]]: 703.704
Badness (Sintel): 0.936


Map: [&lt;2 0 11 31 45 55|, &lt;0 1 -2 -8 -12 -15|]
==== Pajarita ====
Subgroup: 2.3.5.7.11.13


EDOs: [[46edo|46]], [[58edo|58]], [[104edo|104c]], [[162edo|162cef]]
Comma list: 40/39, 50/49, 64/63, 66/65


==17-limit==
{{Mapping|legend=0| 2 0 11 12 26 17 | 0 1 -2 -2 -6 -3 }}
Commas: 126/125, 136/135, 176/175, 196/195, 256/255


[[POTE_tuning|POTE generator]]: 703.812
Optimal tunings:  
* WE: ~7/5 = 598.3048{{c}}, ~3/2 = 705.4512{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 707.9238{{c}}


Map: [&lt;2 0 11 31 45 55 5|, &lt;0 1 -2 -8 -12 -15 1|]
{{Optimal ET sequence|legend=0| 10e, 12f, 22f, 34dff }}


EDOs: 46, 58, 104c
Badness (Sintel): 0.937


=Keen=
===== 17-limit =====
Keen adds 875/864 as well as 2240/2187 to the set of commas, and has wedgie &lt;&lt;2 -4 18 -11 23 53||. It may also be described as the 22&amp;56 temperament. [[78edo|78et]] is a good tuning choice, and remains a good one in the 11-limit, where keen, &lt;&lt;2 -4 18 -12 ...||, is really more interesting, adding 100/99 and 385/384 to the commas.
Subgroup: 2.3.5.7.11.13.17


Commas: 2048/2025, 875/864
Comma list: 40/39, 50/49, 64/63, 66/65, 85/84


[[POTE_tuning|POTE generator]]: 707.571
{{Mapping|legend=0| 2 0 11 12 26 17 5 | 0 1 -2 -2 -6 -3 1 }}


Map: [&lt;2 0 11 -23|, &lt;0 1 -2 9|]
Optimal tunings:  
* WE: ~7/5 = 598.6103{{c}}, ~3/2 = 706.3076{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 708.2256{{c}}


EDOs: 22, 56, 78, 134b, 212b, 290b
{{Optimal ET sequence|legend=0| 10e, 12f, 22f }}


==11-limit==
Badness (Sintel): 0.968
Commas: 100/99, 385/384, 1232/1215


[[POTE_tuning|POTE generator]]: 707.609
=== Pajarous ===
Subgroup: 2.3.5.7.11


Map: [&lt;2 0 11 -23 26|, &lt;0 1 -2 9 -6|]
Comma list: 50/49, 55/54, 64/63


EDOs: 22, 56, 78, 212bf, 290bf
{{Mapping|legend=0| 2 0 11 12 -9 | 0 1 -2 -2 5 }}


=Bidia=
Optimal tunings:
Bidia adds 3136/3125 to the commas, splitting the period into 1/4 octave. It may be called the 12&amp;56 temperament.
* WE: ~7/5 = 599.4055{{c}}, ~3/2 = 708.8747{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 709.5508{{c}}


Commas: 2048/2025, 3136/3125
Tuning ranges:
* 11-odd-limit diamond monotone: ~3/2 = 709.091 (13\22)
* 11-odd-limit diamond tradeoff: ~3/2 = [701.955, 715.803]


POTE generator: ~3/2 = 705.364
{{Optimal ET sequence|legend=0| 10, 12e, 22, 120bce, 142bce }}


Map: [&lt;4 0 22 43|,&lt;0 1 -2 -5|]
Badness (Sintel): 0.937


Wedgie: &lt;&lt;4 -8 -20 -22 -43 -24||
==== 2.3.5.7.11.17 subgroup ====
Subgroup: 2.3.5.7.11.13.17


EDOs: 12, 56, 68, 80, 148d
Comma list: 50/49, 52/51, 55/54, 64/63, 65/63


Badness: 0.0565
{{Mapping|legend=0| 2 0 11 12 -9 1 5 | 0 1 -2 -2 5 2 1 }}


==11-limit==
Optimal tunings:
Commas: 176/175, 896/891, 1375/1372
* WE: ~7/5 = 599.408{{c}}, ~3/2 = 708.878{{c}}
* CWE: ~7/5 = 600.000{{c}}, ~3/2 = 709.544{{c}}


POTE generator: ~3/2 = 705.087
{{Optimal ET sequence|legend=0| 10, 12e, 22 }}


Map: [&lt;4 0 22 43 71|,&lt;0 1 -2 -5 -9|]
Badness (Sintel): 0.766


EDOs: 12, 68, 80
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Badness: 0.0402
Comma list: 50/49, 55/54, 64/63, 65/63


==13-limit==
{{Mapping|legend=0| 2 0 11 12 -9 1 | 0 1 -2 -2 5 2 }}
Commas: 176/175, 325/324, 640/637, 896/891


POTE generator: ~3/2 = 705.301
Optimal tunings:  
* WE: ~7/5 = 599.9064{{c}}, ~3/2 = 710.1289{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 710.2325{{c}}


Map: [&lt;4 0 22 43 71|,&lt;0 1 -2 -5 -9|]
{{Optimal ET sequence|legend=0| 10, 22 }}


EDOs: 12, 68, 80, 148d, 228bcd, 376bcdf
Badness (Sintel): 1.04


Badness: 0.0411
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


=Echidna=
Comma list: 50/49, 52/51, 55/54, 64/63, 65/63
Echidna adds 1728/1715 to the commas and takes 9/7 as a generator. It has a wedgie &lt;&lt;6 -12 10 -33 -1 57|| and may be called the 22&amp;58 temperament. [[58edo|58et]] or [[80edo|80et]] make for good tunings, or their vals can be add to &lt;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.
{{Mapping|legend=0| 2 0 11 12 -9 1 5 | 0 1 -2 -2 5 2 1 }}


Commas: 2048/2025, 1728/1715
Optimal tunings:  
* WE: ~7/5 = 599.8239{{c}}, ~3/2 = 710.0128{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 710.2067{{c}}


[[POTE_tuning|POTE generator]]: 434.856
{{Optimal ET sequence|legend=0| 10, 22, 54f, 76bdff }}


Map: [&lt;2 1 9 2|, &lt;0 3 -6 5|]
Badness (Sintel): 0.930


EDOs: 22, 58, 80, 138cd, 218cd
==== Pajaro ====
Subgroup: 2.3.5.7.11.13


Badness: 0.0580
Comma list: 40/39, 50/49, 55/54, 64/63


==11-limit==
{{Mapping|legend=0| 2 0 11 12 -9 17 | 0 1 -2 -2 5 -3 }}
Commas: 176/175, 896/891, 540/539


11-limit minimax
Optimal tunings:
* WE: ~7/5 = 598.8257{{c}}, ~3/2 = 709.4266{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 710.8414{{c}}


[|1 0 0 0 0&gt;, |7/4 0 0 1/4 -1/4&gt;, |2 0 0 -1/2 1/2&gt;,
{{Optimal ET sequence|legend=0| 10, 22f, 32f }}
|37/12 0 0 5/12 -5/12&gt;, |37/12 0 0 -7/12 7/12&gt;]


Eigenmonzos: 2, 11/7
Badness (Sintel): 1.13


Minimax generator: (224/11)^(1/12) = 434.792
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


[[POTE_tuning|POTE generator]]: 434.852
Comma list: 40/39, 50/49, 55/54, 64/63, 85/84


Map: [&lt;2 1 9 2 12|, &lt;0 3 -6 5 -7|]
{{Mapping|legend=0| 2 0 11 12 -9 17 5 | 0 1 -2 -2 5 -3 1 }}


EDOs: 22, 58, 80, 138cde, 218cde
Optimal tunings:  
* WE: ~7/5 = 598.8865{{c}}, ~3/2 = 709.5472{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 710.8704{{c}}


Badness: 0.0260
{{Optimal ET sequence|legend=0| 10, 22f, 32f }}


==13-limit==
Badness (Sintel): 1.01
Commas: 176/175, 351/350, 364/363, 540/539


[[POTE_tuning|POTE generator]]: 434.756
=== Pajaric ===
Subgroup: 2.3.5.7.11


Map: [&lt;2 1 9 2 12 19|, &lt;0 3 -6 5 -7 -16|]
Comma list: 45/44, 50/49, 56/55


EDOs: 22, 58, 80, 138cde
{{Mapping|legend=0| 2 0 11 12 7 | 0 1 -2 -2 0 }}


Badness: 0.0237
Optimal tunings:  
* WE: ~7/5 = 597.4807{{c}}, ~3/2 = 702.5616{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 706.0542{{c}}


==17-limit==
{{Optimal ET sequence|legend=0| 10, 12, 22e }}
Commas: 136/135, 176/175, 221/220, 256/255, 540/539


[[POTE_tuning|POTE generator]]: 434.816
Badness (Sintel): 0.787


Map: [&lt;2 1 9 2 12 19 6|, &lt;0 3 -6 5 -7 -16 3|]
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


EDOs: 22, 58, 80, 138cde
Comma list: 40/39, 45/44, 50/49, 56/55


Badness: 0.0203
{{Mapping|legend=0| 2 0 11 12 7 17 | 0 1 -2 -2 0 -3 }}


=Echidnic=
Optimal tunings:
Commas: 686/675, 1029/1024
* WE: ~7/5 = 597.1952{{c}}, ~3/2 = 704.1350{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 708.1989{{c}}


[[POTE_tuning|POTE generator]]: 234.492
{{Optimal ET sequence|legend=0| 10, 12f, 22ef }}


Map: [&lt;2 2 7 6|, &lt;0 3 -6 -1|]
Badness (Sintel): 0.845


EDOs: 10, 36, 46, 194bcd, 240bcd, 286bcd, 332bcd
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


Badness: 0.0722
Comma list: 34/33, 40/39, 45/44, 50/49, 56/55


==11-limit==
{{Mapping|legend=0| 2 0 11 12 7 17 5 | 0 1 -2 -2 0 -3 1 }}
Commas: 385/384, 441/440, 686/675


[[POTE_tuning|POTE generator]]: 235.096
Optimal tunings:  
* WE: ~7/5 = 597.6509{{c}}, ~3/2 = 705.7702{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 708.9719{{c}}


Map: [&lt;2 2 7 6 3|, &lt;0 3 -6 -1 10|]
{{Optimal ET sequence|legend=0| 10, 12f, 22ef }}


EDOs: 10, 46, 102, 148, 342bcd
Badness (Sintel): 0.896


Badness: 0.0451
=== Hemipaj ===
Subgroup: 2.3.5.7.11


==13-limit==
Comma list: 50/49, 64/63, 121/120
Commas: 91/90, 169/168, 385/384, 441/440


[[POTE_tuning|POTE generator]]: 235.088
{{Mapping|legend=0| 2 1 9 10 8 | 0 2 -4 -4 -1 }}


Map: [&lt;2 2 7 6 3 7|, &lt;0 3 -6 -1 10 1|]
: mapping generators: ~2, ~16/11


EDOs: 10, 46, 102, 148f, 194bcdf
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: 0.0289
{{Optimal ET sequence|legend=0| 2, 20, 22 }}


Compositions:
Badness (Sintel): 1.29


[http://untwelve.org/2011competition_audio/Kosmorsky-A_Stiff_Shot_of_Turpentine.mp3 http://untwelve.org/2011competition_audio/Kosmorsky-A_Stiff_Shot_of_Turpentine.mp3]
=== Hemifourths ===
Subgroup: 2.3.5.7.11


(the description says "lemba" which has a similar sale structure but different mapping for 5)
Comma list: 50/49, 64/63, 243/242


=Shrutar=
{{Mapping|legend=0| 2 0 11 12 -1 | 0 2 -4 -4 5 }}
Shrutar adds 245/243 to the commas, and also tempers out 6144/6125. With wedgie &lt;&lt;4 -8 14 -22 11 55||, 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|68edo]] makes for a good tuning, but another and excellent choice is a generator of 14^(1/7), making 7s just.
: mapping generators: ~2, ~55/32


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^(1/7) generator can again be used as tunings.
Optimal tunings:
* WE: ~7/5 = 597.6509{{c}}, ~55/32 = 950.8475{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~55/32 = 953.1172{{c}}


Commas: 2048/2025, 245/243
{{Optimal ET sequence|legend=0| 10, 24d, 34d }}


[[POTE_tuning|POTE generator]]: 52.811
Badness (Sintel): 1.62


Map: [&lt;2 1 9 -2|, &lt;0 2 -4 7|]
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


EDOs: 22, 46, 68, 182b, 250bc
Comma list: 50/49, 64/63, 78/77, 144/143


==11-limit==
{{Mapping|legend=0| 2 0 11 12 -1 9 | 0 2 -4 -4 5 -1 }}
Commas: 2048/2025, 245/243, 121/120


[[POTE_tuning|POTE generator]]: 52.680
Optimal tunings:  
* WE: ~7/5 = 598.6748{{c}}, ~26/15 = 950.9691{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~26/15 = 953.1052{{c}}


Map: [&lt;2 1 9 -2 8|, &lt;0 2 -4 7 -1|]
{{Optimal ET sequence|legend=0| 10, 24d, 34d }}


EDOs: 22, 46, 68, 114, 296bce, 410bce
Badness (Sintel): 1.19


==13-limit==
==== 17-limit ====
Commas: 121/120, 176/175, 196/195, 245/243
Subgroup: 2.3.5.7.11.13.17


POTE generator: ~28/27 = 52.654
Comma list: 50/49, 64/63, 78/77, 85/84, 144/143


Map: [&lt;2 1 9 -2 8 -10|, &lt;0 2 -4 7 -1 16|]
{{Mapping|legend=0| 2 0 11 12 -1 9 5 | 0 2 -4 -4 5 -1 2 }}


EDOs: 22, 24, 46, 68, 114
Optimal tunings:  
* WE: ~7/5 = 598.8411{{c}}, ~26/15 = 951.3687{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~26/15 = 953.2169{{c}}


Badness: 0.0281
{{Optimal ET sequence|legend=0| 10, 24d, 34d }}


==17-limit==
Badness (Sintel): 1.11
Commas: 121/120, 136/135, 154/153, 176/175, 196/195


POTE generator: ~28/27 = 52.647
== Srutal ==
{{See also| Srutal vs diaschismic }}


Map: [&lt;2 1 9 -2 8 -10 6|, &lt;0 2 -4 7 -1 16 2|]
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]].


EDOs: 22, 24, 46, 68, 114
[[Subgroup]]: 2.3.5.7


Badness: 0.0187
[[Comma list]]: 2048/2025, 4375/4374


==19-limit==
{{Mapping|legend=1| 2 0 11 -42 | 0 1 -2 15 }}
Commas: 121/120, 136/135, 154/153, 176/175, 196/195, 343/342


POTE generator: ~28/27 = 52.730
[[Optimal tuning]]s:
* [[WE]]: ~45/32 = 599.4046{{c}}, ~3/2 = 704.1150{{c}}
: [[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 }}


Map: [&lt;2 1 9 -2 8 -10 6 -10|, &lt;0 2 -4 7 -1 16 2 17|]
[[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 tradeoff]]: ~3/2 = [701.955, 706.843]


EDOs: 22, 24, 46, 68, 114, 182bef
{{Optimal ET sequence|legend=1| 34d, 46, 80, 126, 206cd, 332bcd }}


Badness: 0.0175
[[Badness]] (Sintel): 2.32


=Sruti=
=== 11-limit ===
Commas: 2048/2025, 19683/19600
Subgroup: 2.3.5.7.11


POTE generator: ~175/144 = 351.876
Comma list: 176/175, 896/891, 1331/1323


Map: [&lt;2 0 11 -15|, &lt;0 2 -4 13|]
{{Mapping|legend=0| 2 0 11 -42 -28 | 0 1 -2 15 11 }}


Wedgie: &lt;&lt;4 -8 26 -22 30 83||
Optimal tunings:  
* WE: ~45/32 = 599.4413{{c}}, ~3/2 = 704.1999{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 704.8017{{c}}


EDOs: 24, 34d, 58, 150cd, 208cd, 266cd
Tuning ranges:  
* 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]


Badness: 0.1174
{{Optimal ET sequence|legend=0| 34d, 46, 80, 126, 206cd }}


==11-limit==
Badness (Sintel): 1.17
Commas: 176/175, 243/242, 896/891


POTE generator: ~11/9 = 351.863
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Map: [&lt;2 0 11 -15 -1|, &lt;0 2 -4 13 5|]
Comma list: 169/168, 176/175, 325/324, 364/363


EDOs: 24, 34d, 58, 150cde, 208cde
{{Mapping|legend=0| 2 0 11 -42 -28 -18 | 0 1 -2 15 11 8 }}


Badness: 0.0415
Optimal tunings:  
* WE: ~45/32 = 599.5490{{c}}, ~3/2 = 704.3516{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 704.8347{{c}}


==13-limit==
Tuning ranges:
Commas: 144/143, 176/175, 351/350, 676/675
* 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]
* 15-odd-limit diamond tradeoff: ~3/2 = [701.955, 711.731]


POTE generator: ~11/9 = 351.886
{{Optimal ET sequence|legend=0| 34d, 46, 80 }}


Map: [&lt;2 0 11 -15 -1 9|, &lt;0 2 -4 13 5 -1|]
Badness (Sintel): 1.04


EDOs: 24, 34d, 58, 150cdef, 208cdef
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Badness: 0.0238
Comma list: 136/135, 169/168, 176/175, 221/220, 256/255


=Anguirus=
{{Mapping|legend=0| 2 0 11 -42 -28 -18 5 | 0 1 -2 15 11 8 1 }}
Commas: 49/48, 2048/2025


POTE generator: ~8/7 = 246.979
Optimal tunings:  
* WE: ~17/12 = 599.6459{{c}}, ~3/2 = 704.4237{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 704.8083{{c}}


Map: [&lt;2 0 11 4|, &lt;0 2 -4 1|]
Tuning ranges:  
* 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]


Wedgie: &lt;&lt;4 -8 2 -22 -8 27||
{{Optimal ET sequence|legend=0| 34d, 46, 80, 126 }}


EDOs: 10, 24, 34
Badness (Sintel): 0.947


Badness: 0.0780
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19


==11-limit==
Comma list: 136/135, 169/168, 176/175, 190/189, 221/220, 256/255
Commas: 49/48, 56/55, 243/242


POTE generator: ~8/7 = 247.816
{{Mapping|legend=0| 2 0 11 -42 -28 -18 5 -55 | 0 1 -2 15 11 8 1 20 }}


Map: [&lt;2 0 11 4 -1|, &lt;0 2 -4 1 5|]
Optimal tunings:  
* WE: ~17/12 = 599.6371{{c}}, ~3/2 = 704.4790{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 704.8745{{c}}


EDOs: 10, 24, 34, 58d, 92de
{{Optimal ET sequence|legend=0| 34dh, 46, 80 }}


Badness: 0.0493
Badness (Sintel): 1.04


==13-limit==
==== Srutaloo ====
Commas: 49/48 56/55 91/90 352/351
Srutaloo adds 576/575, 736/729 or 208/207, and rhymes with [[skidoo]].


POTE generator: ~8/7 = 247.691
Subgroup: 2.3.5.7.11.13.17.19.23


Map: [&lt;2 0 11 4 -1 9|, &lt;0 2 -4 1 5 -1|]
Comma list: 136/135, 169/168, 176/175, 190/189, 208/207, 221/220, 256/255


EDOs: 10, 24, 34, 58d, 92def
{{Mapping|legend=0| 2 0 11 -42 -28 -18 5 -55 -10 | 0 1 -2 15 11 8 1 20 6 }}


Badness: 0.0308
Optimal tunings:  
* WE: ~17/12 = 599.6690{{c}}, ~3/2 = 704.5098{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 704.8713{{c}}


=Shru=
{{Optimal ET sequence|legend=0| 34dh, 46, 80 }}
Commas: 392/375, 1323/1280


POTE generator: ~64/63 =  50.135
Badness (Sintel): 0.971


Map: [&lt;2 1 9 11|, &lt;0 2 -4 -5|]
===== 29-limit =====
Subgroup: 2.3.5.7.11.13.17.19.23.29


Wedgie: &lt;&lt;4 -8 -10 -22 -27 -1||
Comma list: 136/135, 169/168, 176/175, 190/189, 208/207, 221/220, 232/231, 256/255


EDOs: 22d, 24
{{Mapping|legend=0| 2 0 11 -42 -28 -18 5 -55 -10 -76 | 0 1 -2 15 11 8 1 20 6 27 }}


Badness: 0.1576
Optimal tunings:  
* WE: ~17/12 = 599.6664{{c}}, ~3/2 = 704.5138{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 704.8807{{c}}


==11-limit==
{{Optimal ET sequence|legend=0| 34dhj, 46, 80 }}
Commas: 56/55, 77/75, 1323/1280


POTE generator: ~64/63 = 50.130
Badness (Sintel): 1.10


Map: [&lt;2 1 9 11 8|, &lt;0 2 -4 -5 -1|]
===== 31-limit =====
Subgroup: 2.3.5.7.11.13.17.19.23.29.31


EDOs: 22d, 24
Comma list: 136/135, 169/168, 176/175, 190/189, 208/207, 217/216, 221/220, 232/231, 256/255


Badness: 0.0635
{{Mapping|legend=0| 2 0 11 -42 -28 -18 5 -55 -10 -76 48 | 0 1 -2 15 11 8 1 20 6 27 -12 }}


==13-limit==
Optimal tunings:
Commas: 56/55, 77/75, 105/104, 507/500
* WE: ~17/12 = 599.8115{{c}}, ~3/2 = 704.5958{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 704.8086{{c}}


POTE generator: ~64/63 = 50.535
{{Optimal ET sequence|legend=0| 46, 80, 126 }}


Map: [&lt;2 1 9 11 8 15|, &lt;0 2 -4 -5 -1 -7|]
Badness (Sintel): 1.44


EDOs: 24
== Keen ==
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.


Badness: 0.0457
[[Subgroup]]: 2.3.5.7
[[Category:diaschismic]]
 
[[Category:family]]
[[Comma list]]: 875/864, 2048/2025
[[Category:theory]]
 
[[Category:todo:add_definition]]
{{Mapping|legend=1| 2 0 11 -23 | 0 1 -2 9 }}
[[Category:todo:intro]]
 
[[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 }}
 
{{Optimal ET sequence|legend=1| 22, 56, 78, 134b }}
 
[[Badness]] (Sintel): 2.13
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 100/99, 385/384, 1232/1215
 
{{Mapping|legend=0| 2 0 11 -23 26 | 0 1 -2 9 -6 }}
 
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 ET sequence|legend=0| 22, 56, 78 }}
 
Badness (Sintel): 1.50
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 100/99, 105/104, 144/143, 1078/1053
 
{{Mapping|legend=0| 2 0 11 -23 26 -18 | 0 1 -2 9 -6 8 }}
 
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 ET sequence|legend=0| 22f, 34, 56f }}
 
Badness (Sintel): 1.85
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 100/99, 105/104, 119/117, 144/143, 154/153
 
{{Mapping|legend=0| 2 0 11 -23 26 -18 5 | 0 1 -2 9 -6 8 1}}
 
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 ET sequence|legend=0| 22f, 34, 56f }}
 
Badness (Sintel): 1.54
 
==== Keenic ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 91/90, 100/99, 352/351, 385/384
 
{{Mapping|legend=0| 2 0 11 -23 26 36 | 0 1 -2 9 -6 -9 }}
 
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 ET sequence|legend=0| 22, 34, 56 }}
 
Badness (Sintel): 1.67
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 91/90, 100/99, 136/135, 154/153, 256/255
 
{{Mapping|legend=0| 2 0 11 -23 26 36 5 | 0 1 -2 9 -6 -9 1 }}
 
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 ET sequence|legend=0| 22, 34, 56 }}
 
Badness (Sintel): 1.37
 
== Bidia ==
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
 
[[Comma list]]: 2048/2025, 3136/3125
 
{{Mapping|legend=1| 4 0 22 43 | 0 1 -2 -5 }}
: mapping generators: ~25/21, ~3
 
[[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 }}
 
{{Optimal ET sequence|legend=1| 12, …, 56, 68, 80, 148d }}
 
[[Badness]] (Sintel): 1.43
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 176/175, 896/891, 1375/1372
 
{{Mapping|legend=0| 4 0 22 43 71 | 0 1 -2 -5 -9 }}
 
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 ET sequence|legend=0| 12, 56e, 68, 80 }}
 
Badness (Sintel): 1.33
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 176/175, 325/324, 640/637, 896/891
 
{{Mapping|legend=0| 4 0 22 43 71 -36 | 0 1 -2 -5 -9 8 }}
 
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 ET sequence|legend=0| 12, 68, 80, 148d, 228bcd, 376bbcddf }}
 
Badness (Sintel): 1.70
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 136/135, 176/175, 256/255, 325/324, 640/637
 
{{Mapping|legend=0| 4 0 22 43 71 -36 10 | 0 1 -2 -5 -9 8 1 }}
 
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 ET sequence|legend=0| 12, 68, 80, 148d }}
 
Badness (Sintel): 1.46
 
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 136/135, 176/175, 190/189, 256/255, 325/324, 640/637
 
{{Mapping|legend=0| 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
 
{{Mapping|legend=0| 4 0 22 43 71 -36 10 17 -20 | 0 1 -2 -5 -9 8 1 0 6 }}
 
Optimal tunings:
* WE: ~19/16 = 299.7961{{c}}, ~3/2 = 704.8577{{c}}
* CWE: ~19/16 = 300.0000{{c}}, ~3/2 = 705.3413{{c}}
 
{{Optimal ET sequence|legend=0| 12, 68, 80, 148di }}
 
Badness (Sintel): 1.24
 
== 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.
 
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.
 
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.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 245/243, 2048/2025
 
{{Mapping|legend=1| 2 1 9 -2 | 0 2 -4 7 }}
: mapping generators: ~45/32, ~35/24
 
[[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 }}
 
{{Optimal ET sequence|legend=1| 22, 46, 68, 182b, 250bc }}
 
[[Badness]] (Sintel): 1.20
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 121/120, 176/175, 245/243
 
{{Mapping|legend=0| 2 1 9 -2 8 | 0 2 -4 7 -1 }}
 
Optimal tunings:
* WE: ~45/32 = 599.7721{{c}}, ~16/11 = 652.4321{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~16/11 = 652.6672{{c}}
 
{{Optimal ET sequence|legend=0| 22, 46, 68, 114 }}
 
Badness (Sintel): 0.876
 
=== 13-limit ===
Note that an alternative, simpler mapping for prime 13 exists at -7 gens, as in [[#Shru]]. This mapping is especially valuable when considering shrutar as a no-7's 17-limit temperament (22 & 24, supported by 46, 68f and 70), as the complexity from mapping 7 at +7 gens is one of the motivations for picking a +16 gen mapping for 13. This temperament isn't as accurate, but it is simple, and makes many [[MOS]]es, specifically 2L 2''n''s for ''n'' up to [[2L 20s|10]], followed by [[22L 2s]] for the 24-note MOS, with a quarter-tone generator equivalent to ~11/8.
 
Subgroup: 2.3.5.7.11.13
 
Comma list: 121/120, 176/175, 196/195, 245/243
 
{{Mapping|legend=0| 2 1 9 -2 8 -10 | 0 2 -4 7 -1 16 }}
 
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 ET sequence|legend=0| 22f, 46, 68, 114 }}
 
Badness (Sintel): 1.16
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 121/120, 136/135, 154/153, 176/175, 196/195
 
{{Mapping|legend=0| 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|legend=0| 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|legend=0| 2 1 9 -2 8 -10 6 -10 -4 | 0 2 -4 7 -1 16 2 17 12 }}
 
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 ET sequence|legend=0| 22fh, 46, 68, 114 }}
 
Badness (Sintel): 1.03
 
== Shru ==
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. It can be seen as one way of merging [[parapyth]] with [[diaschismic]] (and hence as a lower-accuracy rank 2 tuning of [[varda]]).
 
Note that shru is especially efficient as a no-7's 17-limit temperament, as its 7 is the highest damage, and without it, the other primes can be tuned more accurately.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 392/375, 1323/1280
 
{{Mapping|legend=1| 2 1 9 11 | 0 2 -4 -5 }}
: mapping generators: ~45/32, ~10/7
 
[[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 }}
 
{{Optimal ET sequence|legend=1| 2, 22d, 24 }}
 
[[Badness]] (Sintel): 3.99
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 56/55, 77/75, 1323/1280
 
{{Mapping|legend=0| 2 1 9 11 8 | 0 2 -4 -5 -1 }}
 
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 ET sequence|legend=0| 2, 22d, 24 }}
 
Badness (Sintel): 2.10
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 56/55, 77/75, 105/104, 507/500
 
{{Mapping|legend=0| 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 ==
{{Redirect|Sruti|the tuning concept in Indian music|Shruti}}
 
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|legend=0| 2 0 11 -15 -1 | 0 2 -4 13 5 }}
 
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 ET sequence|legend=0| 24, 34d, 58, 150cdee, 208ccddee, 266ccddeee }}
 
Badness (Sintel): 1.37
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 144/143, 176/175, 351/350, 676/675
 
{{Mapping|legend=0| 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}}
 
{{Optimal ET sequence|legend=0| 24, 34d, 58, 150cdeef, 208ccddeeff, 266ccddeeefff }}
 
Badness (Sintel): 0.983
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 136/135, 144/143, 170/169, 176/175, 221/220
 
{{Mapping|legend=0| 2 0 11 -15 -1 9 5 | 0 2 -4 13 5 -1 2 }}
 
Optimal tunings:
* WE: ~17/12 = 599.3003{{c}}, ~26/15 = 950.7465{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~26/15 = 951.8142{{c}}
 
{{Optimal ET sequence|legend=0| 24, 34d, 58 }}
 
Badness (Sintel): 1.05
 
== 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.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 49/48, 2048/2025
 
{{Mapping|legend=1| 2 0 11 4 | 0 2 -4 1 }}
: mapping generators: ~45/32, ~7/4
 
[[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 }}
 
{{Optimal ET sequence|legend=1| 10, 24, 34 }}
 
[[Badness]] (Sintel): 1.97
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 49/48, 56/55, 243/242
 
{{Mapping|legend=0| 2 0 11 4 -1 | 0 2 -4 1 5 }}
 
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 ET sequence|legend=0| 10, 24, 34 }}
 
Badness (Sintel): 1.63
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 49/48, 56/55, 91/90, 243/242
 
{{Mapping|legend=0| 2 0 11 4 -1 9 | 0 2 -4 1 5 -1 }}
 
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 ET sequence|legend=0| 10, 24, 34, 58d, 92ddef }}
 
Badness (Sintel): 1.27
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 49/48, 56/55, 91/90, 119/117, 154/153
 
{{Mapping|legend=0| 2 0 11 4 -1 9 5 | 0 2 -4 1 5 -1 2 }}
 
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 ET sequence|legend=0| 10, 24, 34 }}
 
Badness (Sintel): 1.10
 
== Echidnic ==
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.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 686/675, 1029/1024
 
{{Mapping|legend=1| 2 2 7 6 | 0 3 -6 -1 }}
: mapping generators: ~45/32, ~8/7
 
[[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 }}
 
{{Optimal ET sequence|legend=1| 10, 26c, 36, 46 }}
 
[[Badness]] (Sintel): 1.83
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 385/384, 441/440, 686/675
 
{{Mapping|legend=0| 2 2 7 6 3 | 0 3 -6 -1 10 }}
 
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 ET sequence|legend=0| 10, 36e, 46, 102, 148 }}
 
Badness (Sintel): 1.49
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 91/90, 169/168, 385/384, 441/440
 
{{Mapping|legend=0| 2 2 7 6 3 7 | 0 3 -6 -1 10 1 }}
 
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 ET sequence|legend=0| 10, 36e, 46, 102, 148f }}
 
Badness (Sintel): 1.19
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 91/90, 136/135, 154/153, 169/168, 256/255
 
{{Mapping|legend=0| 2 2 7 6 3 7 7 | 0 3 -6 -1 10 1 3 }}
 
Optimal tunings:
* WE: ~17/12 = 599.9571{{c}}, ~8/7 = 235.0709{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~8/7 = 235.0860{{c}}
 
{{Optimal ET sequence|legend=0| 10, 36e, 46, 102, 148f }}
 
Badness (Sintel): 0.983
 
; 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)
 
== Quadrasruta ==
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.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2048/2025, 2401/2400
 
{{Mapping|legend=1| 2 0 11 8 | 0 4 -8 -3 }}
: mapping generators: ~45/32, ~21/16
 
[[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 }}
 
{{Optimal ET sequence|legend=1| 10, …, 58, 68, 126, 446bbccd }}
 
[[Badness]] (Sintel): 1.86
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 176/175, 896/891, 2401/2400
 
{{Mapping|legend=0| 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|legend=0| 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|legend=0| 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|legend=0| 2 0 11 8 -1 | 0 4 -8 -3 10 }}
 
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 ET sequence|legend=0| 10, 48c, 58, 184cee, 242ccdeee }}
 
Badness (Sintel): 1.62
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 144/143, 196/195, 243/242, 676/675
 
{{Mapping|legend=0| 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|legend=0| 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 }}
 
This extension of 5-limit diaschismic adds prime 17 and which with respect to [[MVP archagall]] is able to express the harmonics [[75/1|75]] and [[85/1|85]] in their appropriate prime subgroup. It achieves this by equating [[85/64]] with [[4/3]] by tempering out their difference of [[256/255]] (S16). Therefore it also tempers out [[289/288]] (S17) and thus equates [[17/15]] with [[9/8]] due to tempering out [[136/135]] (S16⋅S17). It could be described as the 10 & 12 temperament with strong emphasis on 12edo being the better tuning on the 2.3.5.17 subgroup, implying ideal tunings of 34edo, 46edo or 80edo.
 
Subgroup: 2.3.5.17
 
Comma list: 136/135, 256/255
 
{{Mapping|legend=2| 2 0 11 5 | 0 1 -2 1 }}
: mapping generators: ~17/12, ~3
 
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 ET sequence|legend=0| 10, 12, 22, 34, 80, 114, 194bc }}
 
Badness (Sintel): 0.212
 
[[Category:Diaschismic family| ]] <!-- main article -->
[[Category:Temperament families]]
[[Category:Catalogs of rank-2 temperaments]]
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