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__FORCETOC__
{{Technical data page}}
The '''diaschismic family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the diaschisma, [[2048/2025]].


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


[[Tuning_Ranges_of_Regular_Temperaments|valid range]]: [600.000 to 720.000] (2 to 5)
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.


nice 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.


strict range: [701.955, 706.843]
[[Subgroup]]: 2.3.5


[[POTE_tuning|POTE generator]]: ~3/2 = 704.898
[[Comma list]]: 2048/2025


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


EDOs: 34, 46, 80, 206c, 286bc
[[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 }}


==Seven limit children==
[[Tuning ranges]]:
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.
* [[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]


=Srutal=
{{Optimal ET sequence|legend=1| 10, 12, 22, 34, 46, 80, 206c, 286bc }}
Commas: 2048/2025, 4375/4374


valid range: [703.448, 705.882] (58 to 34d)
[[Badness]] (Sintel): 0.467


nice range: [701.955, 706.843]
=== 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]].


strict range: [703.448, 705.882]
Those all keep the same half-octave period and fifth generator.  


POTE generator: ~3/2 = 704.814
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.  


Map: [<2 0 11 -42|, <0 1 -2 15|]
Temperaments discussed elsewhere include [[stearnsmic clan #Echidna|echidna]]. The rest are considered below.


Wedgie: <<2 -4 30 -11 42 81||
==== 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]]}.


EDOs: 46, 80, 126, 206cd, 332bcd
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).


Badness: 0.0915
== Septimal diaschismic ==
{{Main| Diaschismic }}
{{See also| Srutal vs diaschismic }}


==11-limit==
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.
Commas: 176/175, 896/891, 1331/1323


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


nice range: [701.955, 706.843]
[[Subgroup]]: 2.3.5.7


strict range: [704.348, 705.882]
[[Comma list]]: 126/125, 2048/2025


POTE generator: ~3/2 = 704.856
{{Mapping|legend=1| 2 0 11 31 | 0 1 -2 -8 }}


Map: [&lt;2 0 11 -42 -28|, &lt;0 1 -2 15 11|]
[[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 }}


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


Badness: 0.0353
{{Optimal ET sequence|legend=1| 12, 34, 46, 58, 104c, 162c }}


==13-limit==
[[Badness]] (Sintel): 0.959
Commas: 169/168, 176/175, 325/324, 364/363


valid range: [704.348, 705.882] (46 to 34d)
=== 11-limit ===
Subgroup: 2.3.5.7.11


nice range: [701.955, 706.843]
Comma list: 126/125, 176/175, 896/891


strict range: [704.348, 705.882]
{{Mapping|legend=0| 2 0 11 31 45 | 0 1 -2 -8 -12 }}


POTE generator: ~3/2 = 704.881
Optimal tunings:  
* WE: ~45/32 = 599.4471{{c}}, ~3/2 = 703.0657{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 703.7996{{c}}


Map: [&lt;2 0 11 -42 -28 -18|, &lt;0 1 -2 15 11 8|]
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]


EDOs: 34d, 46, 80, 206cd, 286bcde
{{Optimal ET sequence|legend=0| 12, 34e, 46, 58, 104c, 162ce }}


Badness: 0.0253
Badness (Sintel): 0.828


=Pajara=
=== 13-limit ===
Main article: [[pajara|Pajara]]
Subgroup: 2.3.5.7.11.13


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


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.
{{Mapping|legend=0| 2 0 11 31 45 55 | 0 1 -2 -8 -12 -15 }}


Commas: 50/49, 64/63
Optimal tunings:  
* WE: ~45/32 = 599.4451{{c}}, ~3/2 = 703.0528{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 703.7813{{c}}


valid range: [700.000, 720.000] (12 to 10)
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]


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


strict range: [701.955, 715.587]
Badness (Sintel): 0.782


[[POTE_tuning|POTE generator]]: 707.048
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Map: [&lt;2 0 11 12|, &lt;0 1 -2 -2|]
Comma list: 126/125, 136/135, 176/175, 196/195, 256/255


EDOs: 22, 34, 56
{{Mapping|legend=0| 2 0 11 31 45 55 5 | 0 1 -2 -8 -12 -15 1 }}


Badness: 0.0200
Optimal tunings:  
* WE: ~17/12 = 599.6253{{c}}, ~3/2 = 703.3726{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 703.8520{{c}}


==11-limit==
Tuning ranges:
Commas: 50/49, 64/63, 99/98
* 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]


valid range: [700.000, 709.091] (12 to 22)
{{Optimal ET sequence|legend=0| 12f, 34ef, 46, 58, 104c }}


nice range: [701.955, 715.587]
Badness (Sintel): 0.837


strict range: [701.955, 709.091]
=== 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.  


[[POTE_tuning|POTE generator]]: 706.885
Subgroup: 2.3.5.7.11.13.17.23


Map: [&lt;2 0 11 12 26|, &lt;0 1 -2 -2 -6|]
Comma list: 126/125, 136/135, 176/175, 196/195, 231/230, 256/255


EDOs: 22, 34, 56, 146
{{Mapping|legend=2| 2 0 11 31 45 55 5 63 | 0 1 -2 -8 -12 -15 1 -17 }}


Badness: 0.0203
Optimal tunings:  
* WE: ~17/12 = 599.6272{{c}}, ~3/2 = 703.4326{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 703.9093{{c}}


==13-limit==
{{Optimal ET sequence|legend=0| 12i, 34efi, 46, 58i, 104ci }}
Commas: 50/49, 64/63, 65/63, 99/98


valid range: []
Badness (Sintel): 0.882


nice range: [701.955, 738.573]
== Pajara ==
{{Main| Pajara }}


strict range: []
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.


POTE generator: ~3/2 = 708.919
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.


Map: [&lt;2 0 11 12 26 1|, &lt;0 1 -2 -2 -6 2|]
[[Subgroup]]: 2.3.5.7


EDOs: 12, 22
[[Comma list]]: 50/49, 64/63


Badness: 0.0276
{{Mapping|legend=1| 2 0 11 12 | 0 1 -2 -2 }}


==Pajarous==
[[Optimal tuning]]s:
Commas: 50/49, 55/54, 64/63
* [[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 }}


valid range: 709.091 (22)
[[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]


nice range: [701.955, 715.803]
{{Optimal ET sequence|legend=1| 10, 12, 22, 34d, 56d }}


strict range: 709.091
[[Badness]] (Sintel): 0.507


POTE generator: ~3/2 = 709.578
=== 2.3.5.7.17 subgroup ===
Subgroup: 2.3.5.7.17


Map: [&lt;2 0 11 12 -9|, &lt;0 1 -2 -2 5|]
Comma list: 50/49, 64/63, 85/84


EDOs: 10, 12e, 22, 120bce, 142bce
{{Mapping|legend=0| 2 0 11 12 5 | 0 1 -2 -2 1 }}


Badness: 0.0283
Optimal tunings:  
* WE: ~7/5 = 599.053{{c}}, ~3/2 = 706.355{{c}}
* CWE: ~7/5 = 600.000{{c}}, ~3/2 = 707.607{{c}}


===13-limit===
{{Optimal ET sequence|legend=0| 10, 12, 22, 56d }}
Commas: 50/49, 55/54, 64/63, 65/63


valid range: []
Badness (Sintel): 0.438


nice range: [701.955, 738.573]
=== 11-limit ===
Subgroup: 2.3.5.7.11


strict range: []
Comma list: 50/49, 64/63, 99/98


POTE generator: ~3/2 = 710.240
{{Mapping|legend=0| 2 0 11 12 26 | 0 1 -2 -2 -6 }}


Map: [&lt;2 0 11 12 -9 1|, &lt;0 1 -2 -2 5 2|]
Optimal tunings:  
* WE: ~7/5 = 598.8485{{c}}, ~3/2 = 705.5285{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 707.1826{{c}}


EDOs: 10, 22, 54f, 76bdf
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]


Badness: 0.0252
{{Optimal ET sequence|legend=0| 10e, 12, 22, 34d, 56d }}


==Pajaric==
Badness (Sintel): 0.673
Commas: 45/44, 50/49, 56/55


POTE generator: ~3/2 = 705.524
==== 2.3.5.7.11.17 subgroup ====
Subgroup: 2.3.5.7.11.17


Map: [&lt;2 0 11 12 7|, &lt;0 1 -2 -2 0|]
Comma list: 50/49, 64/63, 85/84, 99/98


EDOs: 10, 12, 22e, 34de
{{Mapping|legend=0| 2 0 11 12 26 5 | 0 1 -2 -2 -6 1 }}


Badness: 0.0238
Optimal tunings:  
* WE: ~7/5 = 599.062{{c}}, ~3/2 = 706.095{{c}}
* CWE: ~7/5 = 600.000{{c}}, ~3/2 = 707.370{{c}}


===13-limit===
{{Optimal ET sequence|legend=0| 10e, 12, 22, 34d, 56d }}
Commas: 40/39, 45/44, 50/49, 56/55


POTE generator: ~3/2 = 707.442
Badness (Sintel): 0.645


Map: [&lt;2 0 11 12 7 17|, &lt;0 1 -2 -2 0 -3|]
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


EDOs: 10, 12f, 22ef, 34def
Comma list: 50/49, 64/63, 65/63, 99/98


Badness: 0.0205
{{Mapping|legend=0| 2 0 11 12 26 1 | 0 1 -2 -2 -6 2 }}


==Pajaro==
Optimal tunings:
Commas: 40/39, 50/49, 55/54, 64/63
* WE: ~7/5 = 599.9732{{c}}, ~3/2 = 708.8873{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 708.9227{{c}}


POTE generator ~3/2 = 710.818
{{Optimal ET sequence|legend=0| 10e, 12, 22 }}


Map: [&lt;2 0 11 12 -9 17|, &lt;0 1 -2 -2 5 -3|]
Badness (Sintel): 1.14


EDOs: 10, 22f, 32f, 54f
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


Badness: 0.0274
Comma list: 50/49, 52/51, 64/63, 65/63, 99/98


==Hemipaj==
{{Mapping|legend=0| 2 0 11 12 26 1 5 | 0 1 -2 -2 -6 2 1 }}
Commas: 50/49, 64/63, 121/120


POTE generator: ~11/8 = 546.383
Optimal tunings:  
* WE: ~7/5 = 599.8871{{c}}, ~3/2 = 708.6725{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 708.8176{{c}}


Map: [&lt;2 1 9 10 8|, &lt;0 2 -4 -4 -1|]
{{Optimal ET sequence|legend=0| 10e, 12, 22 }}


EDOs: 20, 22, 68d, 90d
Badness (Sintel): 1.06


Badness: 0.0389
==== Pajarina ====
Subgroup: 2.3.5.7.11.13


=Diaschismic=
Comma list: 50/49, 64/63, 78/77, 99/98
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]] 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 58et.


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.
{{Mapping|legend=0| 2 0 11 12 26 36 | 0 1 -2 -2 -6 -9 }}


Commas: 126/125, 2048/2025
Optimal tunings:  
* WE: ~7/5 = 598.7732{{c}}, ~3/2 = 704.6889{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 706.3950{{c}}


[[POTE_tuning|POTE generator]]: 703.681
{{Optimal ET sequence|legend=0| 12f, 22, 34d }}


Map: [&lt;2 0 11 31|, &lt;0 1 -2 -8|]
Badness (Sintel): 0.923


EDOs: 46, 58, 104c, 162c
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


Badness: 0.0379
Comma list: 50/49, 64/63, 78/77, 85/84, 99/98


==11-limit==
{{Mapping|legend=0| 2 0 11 12 26 36 5 | 0 1 -2 -2 -6 -9 1 }}
Commas: 126/125, 176/175, 896/891


[[POTE_tuning|POTE generator]]: 703.714
Optimal tunings:  
* WE: ~7/5 = 599.0204{{c}}, ~3/2 = 705.2572{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 706.5660{{c}}


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


EDOs: 46, 58, 104c, 162ce
Badness (Sintel): 0.936


==13-limit==
==== Pajarita ====
Commas: 126/125, 196/195, 364/363, 2048/2025
Subgroup: 2.3.5.7.11.13


[[POTE_tuning|POTE generator]]: 703.704
Comma list: 40/39, 50/49, 64/63, 66/65


Map: [&lt;2 0 11 31 45 55|, &lt;0 1 -2 -8 -12 -15|]
{{Mapping|legend=0| 2 0 11 12 26 17 | 0 1 -2 -2 -6 -3 }}


EDOs: [[46edo|46]], [[58edo|58]], [[104edo|104c]], [[162edo|162cef]]
Optimal tunings:  
* WE: ~7/5 = 598.3048{{c}}, ~3/2 = 705.4512{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 707.9238{{c}}


==17-limit==
{{Optimal ET sequence|legend=0| 10e, 12f, 22f, 34dff }}
Commas: 126/125, 136/135, 176/175, 196/195, 256/255


[[POTE_tuning|POTE generator]]: 703.812
Badness (Sintel): 0.937


Map: [&lt;2 0 11 31 45 55 5|, &lt;0 1 -2 -8 -12 -15 1|]
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


EDOs: 46, 58, 104c
Comma list: 40/39, 50/49, 64/63, 66/65, 85/84


=Keen=
{{Mapping|legend=0| 2 0 11 12 26 17 5 | 0 1 -2 -2 -6 -3 1 }}
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.


Commas: 2048/2025, 875/864
Optimal tunings:  
* WE: ~7/5 = 598.6103{{c}}, ~3/2 = 706.3076{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 708.2256{{c}}


[[POTE_tuning|POTE generator]]: 707.571
{{Optimal ET sequence|legend=0| 10e, 12f, 22f }}


Map: [&lt;2 0 11 -23|, &lt;0 1 -2 9|]
Badness (Sintel): 0.968


EDOs: 22, 56, 78, 134b, 212b, 290b
=== Pajarous ===
Subgroup: 2.3.5.7.11


==11-limit==
Comma list: 50/49, 55/54, 64/63
Commas: 100/99, 385/384, 1232/1215


[[POTE_tuning|POTE generator]]: 707.609
{{Mapping|legend=0| 2 0 11 12 -9 | 0 1 -2 -2 5 }}


Map: [&lt;2 0 11 -23 26|, &lt;0 1 -2 9 -6|]
Optimal tunings:  
* WE: ~7/5 = 599.4055{{c}}, ~3/2 = 708.8747{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 709.5508{{c}}


EDOs: 22, 56, 78, 212bf, 290bf
Tuning ranges:
* 11-odd-limit diamond monotone: ~3/2 = 709.091 (13\22)
* 11-odd-limit diamond tradeoff: ~3/2 = [701.955, 715.803]


=Bidia=
{{Optimal ET sequence|legend=0| 10, 12e, 22, 120bce, 142bce }}
Bidia adds 3136/3125 to the commas, splitting the period into 1/4 octave. It may be called the 12&amp;56 temperament.


Commas: 2048/2025, 3136/3125
Badness (Sintel): 0.937


POTE generator: ~3/2 = 705.364
==== 2.3.5.7.11.17 subgroup ====
Subgroup: 2.3.5.7.11.13.17


Map: [&lt;4 0 22 43|,&lt;0 1 -2 -5|]
Comma list: 50/49, 52/51, 55/54, 64/63, 65/63


Wedgie: &lt;&lt;4 -8 -20 -22 -43 -24||
{{Mapping|legend=0| 2 0 11 12 -9 1 5 | 0 1 -2 -2 5 2 1 }}


EDOs: 12, 56, 68, 80, 148d
Optimal tunings:  
* WE: ~7/5 = 599.408{{c}}, ~3/2 = 708.878{{c}}
* CWE: ~7/5 = 600.000{{c}}, ~3/2 = 709.544{{c}}


Badness: 0.0565
{{Optimal ET sequence|legend=0| 10, 12e, 22 }}


==11-limit==
Badness (Sintel): 0.766
Commas: 176/175, 896/891, 1375/1372


POTE generator: ~3/2 = 705.087
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;4 0 22 43 71|,&lt;0 1 -2 -5 -9|]
Comma list: 50/49, 55/54, 64/63, 65/63


EDOs: 12, 68, 80
{{Mapping|legend=0| 2 0 11 12 -9 1 | 0 1 -2 -2 5 2 }}


Badness: 0.0402
Optimal tunings:  
* WE: ~7/5 = 599.9064{{c}}, ~3/2 = 710.1289{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 710.2325{{c}}


==13-limit==
{{Optimal ET sequence|legend=0| 10, 22 }}
Commas: 176/175, 325/324, 640/637, 896/891


POTE generator: ~3/2 = 705.301
Badness (Sintel): 1.04


Map: [&lt;4 0 22 43 71|,&lt;0 1 -2 -5 -9|]
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


EDOs: 12, 68, 80, 148d, 228bcd, 376bcdf
Comma list: 50/49, 52/51, 55/54, 64/63, 65/63


Badness: 0.0411
{{Mapping|legend=0| 2 0 11 12 -9 1 5 | 0 1 -2 -2 5 2 1 }}


=Echidna=
Optimal tunings:
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|.
* WE: ~7/5 = 599.8239{{c}}, ~3/2 = 710.0128{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 710.2067{{c}}


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


Commas: 2048/2025, 1728/1715
Badness (Sintel): 0.930


[[POTE_tuning|POTE generator]]: 434.856
==== Pajaro ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;2 1 9 2|, &lt;0 3 -6 5|]
Comma list: 40/39, 50/49, 55/54, 64/63


EDOs: 22, 58, 80, 138cd, 218cd
{{Mapping|legend=0| 2 0 11 12 -9 17 | 0 1 -2 -2 5 -3 }}


Badness: 0.0580
Optimal tunings:  
* WE: ~7/5 = 598.8257{{c}}, ~3/2 = 709.4266{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 710.8414{{c}}


==11-limit==
{{Optimal ET sequence|legend=0| 10, 22f, 32f }}
Commas: 176/175, 896/891, 540/539


11-limit minimax
Badness (Sintel): 1.13


[|1 0 0 0 0&gt;, |7/4 0 0 1/4 -1/4&gt;, |2 0 0 -1/2 1/2&gt;,
===== 17-limit =====
|37/12 0 0 5/12 -5/12&gt;, |37/12 0 0 -7/12 7/12&gt;<nowiki>]</nowiki>
Subgroup: 2.3.5.7.11.13.17


Eigenmonzos: 2, 11/7
Comma list: 40/39, 50/49, 55/54, 64/63, 85/84


Minimax generator: (224/11)^(1/12) = 434.792
{{Mapping|legend=0| 2 0 11 12 -9 17 5 | 0 1 -2 -2 5 -3 1 }}


[[POTE_tuning|POTE generator]]: 434.852
Optimal tunings:  
* WE: ~7/5 = 598.8865{{c}}, ~3/2 = 709.5472{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 710.8704{{c}}


Map: [&lt;2 1 9 2 12|, &lt;0 3 -6 5 -7|]
{{Optimal ET sequence|legend=0| 10, 22f, 32f }}


EDOs: 22, 58, 80, 138cde, 218cde
Badness (Sintel): 1.01


Badness: 0.0260
=== Pajaric ===
Subgroup: 2.3.5.7.11


==13-limit==
Comma list: 45/44, 50/49, 56/55
Commas: 176/175, 351/350, 364/363, 540/539


[[POTE_tuning|POTE generator]]: 434.756
{{Mapping|legend=0| 2 0 11 12 7 | 0 1 -2 -2 0 }}


Map: [&lt;2 1 9 2 12 19|, &lt;0 3 -6 5 -7 -16|]
Optimal tunings:  
* WE: ~7/5 = 597.4807{{c}}, ~3/2 = 702.5616{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 706.0542{{c}}


EDOs: 22, 58, 80, 138cde
{{Optimal ET sequence|legend=0| 10, 12, 22e }}


Badness: 0.0237
Badness (Sintel): 0.787


==17-limit==
==== 13-limit ====
Commas: 136/135, 176/175, 221/220, 256/255, 540/539
Subgroup: 2.3.5.7.11.13


[[POTE_tuning|POTE generator]]: 434.816
Comma list: 40/39, 45/44, 50/49, 56/55


Map: [&lt;2 1 9 2 12 19 6|, &lt;0 3 -6 5 -7 -16 3|]
{{Mapping|legend=0| 2 0 11 12 7 17 | 0 1 -2 -2 0 -3 }}


EDOs: 22, 58, 80, 138cde
Optimal tunings:  
* WE: ~7/5 = 597.1952{{c}}, ~3/2 = 704.1350{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 708.1989{{c}}


Badness: 0.0203
{{Optimal ET sequence|legend=0| 10, 12f, 22ef }}


=Echidnic=
Badness (Sintel): 0.845
Commas: 686/675, 1029/1024


[[POTE_tuning|POTE generator]]: 234.492
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


Map: [&lt;2 2 7 6|, &lt;0 3 -6 -1|]
Comma list: 34/33, 40/39, 45/44, 50/49, 56/55


EDOs: 10, 36, 46, 194bcd, 240bcd, 286bcd, 332bcd
{{Mapping|legend=0| 2 0 11 12 7 17 5 | 0 1 -2 -2 0 -3 1 }}


Badness: 0.0722
Optimal tunings:  
* WE: ~7/5 = 597.6509{{c}}, ~3/2 = 705.7702{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 708.9719{{c}}


==11-limit==
{{Optimal ET sequence|legend=0| 10, 12f, 22ef }}
Commas: 385/384, 441/440, 686/675


[[POTE_tuning|POTE generator]]: 235.096
Badness (Sintel): 0.896


Map: [&lt;2 2 7 6 3|, &lt;0 3 -6 -1 10|]
=== Hemipaj ===
Subgroup: 2.3.5.7.11


EDOs: 10, 46, 102, 148, 342bcd
Comma list: 50/49, 64/63, 121/120


Badness: 0.0451
{{Mapping|legend=0| 2 1 9 10 8 | 0 2 -4 -4 -1 }}


==13-limit==
: mapping generators: ~2, ~16/11
Commas: 91/90, 169/168, 385/384, 441/440


[[POTE_tuning|POTE generator]]: 235.088
Optimal tunings:  
* WE: ~7/5 = 597.6509{{c}}, ~16/11 = 652.7788{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~16/11 = 653.7119{{c}}


Map: [&lt;2 2 7 6 3 7|, &lt;0 3 -6 -1 10 1|]
{{Optimal ET sequence|legend=0| 2, 20, 22 }}


EDOs: 10, 46, 102, 148f, 194bcdf
Badness (Sintel): 1.29


Badness: 0.0289
=== Hemifourths ===
Subgroup: 2.3.5.7.11


Compositions:
Comma list: 50/49, 64/63, 243/242


http://untwelve.org/2011competition_audio/Kosmorsky-A_Stiff_Shot_of_Turpentine.mp3
{{Mapping|legend=0| 2 0 11 12 -1 | 0 2 -4 -4 5 }}
: mapping generators: ~2, ~55/32


(the description says "lemba" which has a similar scale structure but different mapping for 5)
Optimal tunings:
* WE: ~7/5 = 597.6509{{c}}, ~55/32 = 950.8475{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~55/32 = 953.1172{{c}}


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


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.
Badness (Sintel): 1.62


Commas: 2048/2025, 245/243
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


[[POTE_tuning|POTE generator]]: 52.811
Comma list: 50/49, 64/63, 78/77, 144/143


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


EDOs: 22, 46, 68, 182b, 250bc
Optimal tunings:  
* WE: ~7/5 = 598.6748{{c}}, ~26/15 = 950.9691{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~26/15 = 953.1052{{c}}


==11-limit==
{{Optimal ET sequence|legend=0| 10, 24d, 34d }}
Commas: 2048/2025, 245/243, 121/120


[[POTE_tuning|POTE generator]]: 52.680
Badness (Sintel): 1.19


Map: [&lt;2 1 9 -2 8|, &lt;0 2 -4 7 -1|]
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


EDOs: 22, 46, 68, 114, 296bce, 410bce
Comma list: 50/49, 64/63, 78/77, 85/84, 144/143


==13-limit==
{{Mapping|legend=0| 2 0 11 12 -1 9 5 | 0 2 -4 -4 5 -1 2 }}
Commas: 121/120, 176/175, 196/195, 245/243


POTE generator: ~28/27 = 52.654
Optimal tunings:  
* WE: ~7/5 = 598.8411{{c}}, ~26/15 = 951.3687{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~26/15 = 953.2169{{c}}


Map: [&lt;2 1 9 -2 8 -10|, &lt;0 2 -4 7 -1 16|]
{{Optimal ET sequence|legend=0| 10, 24d, 34d }}


EDOs: 22, 24, 46, 68, 114
Badness (Sintel): 1.11


Badness: 0.0281
== Srutal ==
{{See also| Srutal vs diaschismic }}


==17-limit==
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]].
Commas: 121/120, 136/135, 154/153, 176/175, 196/195


POTE generator: ~28/27 = 52.647
[[Subgroup]]: 2.3.5.7


Map: [&lt;2 1 9 -2 8 -10 6|, &lt;0 2 -4 7 -1 16 2|]
[[Comma list]]: 2048/2025, 4375/4374


EDOs: 22, 24, 46, 68, 114
{{Mapping|legend=1| 2 0 11 -42 | 0 1 -2 15 }}


Badness: 0.0187
[[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 }}


==19-limit==
[[Tuning ranges]]:
Commas: 121/120, 136/135, 154/153, 176/175, 196/195, 343/342
* 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]


POTE generator: ~28/27 = 52.730
{{Optimal ET sequence|legend=1| 34d, 46, 80, 126, 206cd, 332bcd }}


Map: [&lt;2 1 9 -2 8 -10 6 -10|, &lt;0 2 -4 7 -1 16 2 17|]
[[Badness]] (Sintel): 2.32


EDOs: 22, 24, 46, 68, 114, 182bef
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.0175
Comma list: 176/175, 896/891, 1331/1323


=Sruti=
{{Mapping|legend=0| 2 0 11 -42 -28 | 0 1 -2 15 11 }}
Commas: 2048/2025, 19683/19600


POTE generator: ~175/144 = 351.876
Optimal tunings:  
* WE: ~45/32 = 599.4413{{c}}, ~3/2 = 704.1999{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 704.8017{{c}}


Map: [&lt;2 0 11 -15|, &lt;0 2 -4 13|]
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]


Wedgie: &lt;&lt;4 -8 26 -22 30 83||
{{Optimal ET sequence|legend=0| 34d, 46, 80, 126, 206cd }}


EDOs: 24, 34d, 58, 150cd, 208cd, 266cd
Badness (Sintel): 1.17


Badness: 0.1174
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


==11-limit==
Comma list: 169/168, 176/175, 325/324, 364/363
Commas: 176/175, 243/242, 896/891


POTE generator: ~11/9 = 351.863
{{Mapping|legend=0| 2 0 11 -42 -28 -18 | 0 1 -2 15 11 8 }}


Map: [&lt;2 0 11 -15 -1|, &lt;0 2 -4 13 5|]
Optimal tunings:  
* WE: ~45/32 = 599.5490{{c}}, ~3/2 = 704.3516{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 704.8347{{c}}


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


Badness: 0.0415
{{Optimal ET sequence|legend=0| 34d, 46, 80 }}


==13-limit==
Badness (Sintel): 1.04
Commas: 144/143, 176/175, 351/350, 676/675


POTE generator: ~11/9 = 351.886
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Map: [&lt;2 0 11 -15 -1 9|, &lt;0 2 -4 13 5 -1|]
Comma list: 136/135, 169/168, 176/175, 221/220, 256/255


EDOs: 24, 34d, 58, 150cdef, 208cdef
{{Mapping|legend=0| 2 0 11 -42 -28 -18 5 | 0 1 -2 15 11 8 1 }}


Badness: 0.0238
Optimal tunings:  
* WE: ~17/12 = 599.6459{{c}}, ~3/2 = 704.4237{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 704.8083{{c}}


=Anguirus=
Tuning ranges:
Commas: 49/48, 2048/2025
* 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]


POTE generator: ~8/7 = 246.979
{{Optimal ET sequence|legend=0| 34d, 46, 80, 126 }}


Map: [&lt;2 0 11 4|, &lt;0 2 -4 1|]
Badness (Sintel): 0.947


Wedgie: &lt;&lt;4 -8 2 -22 -8 27||
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19


EDOs: 10, 24, 34
Comma list: 136/135, 169/168, 176/175, 190/189, 221/220, 256/255


Badness: 0.0780
{{Mapping|legend=0| 2 0 11 -42 -28 -18 5 -55 | 0 1 -2 15 11 8 1 20 }}


==11-limit==
Optimal tunings:
Commas: 49/48, 56/55, 243/242
* WE: ~17/12 = 599.6371{{c}}, ~3/2 = 704.4790{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 704.8745{{c}}


POTE generator: ~8/7 = 247.816
{{Optimal ET sequence|legend=0| 34dh, 46, 80 }}


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


EDOs: 10, 24, 34, 58d, 92de
==== Srutaloo ====
Srutaloo adds 576/575, 736/729 or 208/207, and rhymes with [[skidoo]].


Badness: 0.0493
Subgroup: 2.3.5.7.11.13.17.19.23


==13-limit==
Comma list: 136/135, 169/168, 176/175, 190/189, 208/207, 221/220, 256/255
Commas: 49/48 56/55 91/90 352/351


POTE generator: ~8/7 = 247.691
{{Mapping|legend=0| 2 0 11 -42 -28 -18 5 -55 -10 | 0 1 -2 15 11 8 1 20 6 }}


Map: [&lt;2 0 11 4 -1 9|, &lt;0 2 -4 1 5 -1|]
Optimal tunings:  
* WE: ~17/12 = 599.6690{{c}}, ~3/2 = 704.5098{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 704.8713{{c}}


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


Badness: 0.0308
Badness (Sintel): 0.971


=Shru=
===== 29-limit =====
Commas: 392/375, 1323/1280
Subgroup: 2.3.5.7.11.13.17.19.23.29


POTE generator: ~64/63 =  50.135
Comma list: 136/135, 169/168, 176/175, 190/189, 208/207, 221/220, 232/231, 256/255


Map: [&lt;2 1 9 11|, &lt;0 2 -4 -5|]
{{Mapping|legend=0| 2 0 11 -42 -28 -18 5 -55 -10 -76 | 0 1 -2 15 11 8 1 20 6 27 }}


Wedgie: &lt;&lt;4 -8 -10 -22 -27 -1||
Optimal tunings:  
* WE: ~17/12 = 599.6664{{c}}, ~3/2 = 704.5138{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 704.8807{{c}}


EDOs: 22d, 24
{{Optimal ET sequence|legend=0| 34dhj, 46, 80 }}


Badness: 0.1576
Badness (Sintel): 1.10


==11-limit==
===== 31-limit =====
Commas: 56/55, 77/75, 1323/1280
Subgroup: 2.3.5.7.11.13.17.19.23.29.31


POTE generator: ~64/63 = 50.130
Comma list: 136/135, 169/168, 176/175, 190/189, 208/207, 217/216, 221/220, 232/231, 256/255


Map: [&lt;2 1 9 11 8|, &lt;0 2 -4 -5 -1|]
{{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 }}


EDOs: 22d, 24
Optimal tunings:  
* WE: ~17/12 = 599.8115{{c}}, ~3/2 = 704.5958{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 704.8086{{c}}


Badness: 0.0635
{{Optimal ET sequence|legend=0| 46, 80, 126 }}


==13-limit==
Badness (Sintel): 1.44
Commas: 56/55, 77/75, 105/104, 507/500


POTE generator: ~64/63 = 50.535
== 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.


Map: [&lt;2 1 9 11 8 15|, &lt;0 2 -4 -5 -1 -7|]
[[Subgroup]]: 2.3.5.7


EDOs: 24
[[Comma list]]: 875/864, 2048/2025


Badness: 0.0457
{{Mapping|legend=1| 2 0 11 -23 | 0 1 -2 9 }}
[[Category:diaschismic]]
 
[[Category:family]]
[[Optimal tuning]]s:
[[Category:theory]]
* [[WE]]: ~45/32 = 599.6603{{c}}, ~3/2 = 707.1707{{c}}
[[Category:todo:add_definition]]
: [[error map]]: {{val| -0.679 +4.536 -3.033 -2.591 }}
[[Category:todo:intro]]
* [[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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