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{{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|]
==== 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]]}.


Wedgie: &lt;&lt;2 -4 30 -11 42 81||
== Septimal diaschismic ==
{{Main| Diaschismic }}
{{See also| Srutal vs diaschismic }}


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


Badness: 0.0915
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.


==11-limit==
[[Subgroup]]: 2.3.5.7
Commas: 176/175, 896/891, 1331/1323


valid range: [704.348, 705.882] (46 to 34d)
[[Comma list]]: 126/125, 2048/2025


nice range: [701.955, 706.843]
{{Mapping|legend=1| 2 0 11 31 | 0 1 -2 -8 }}


strict range: [704.348, 705.882]
[[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 }}


POTE generator: ~3/2 = 704.856
[[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]


Map: [&lt;2 0 11 -42 -28|, &lt;0 1 -2 15 11|]
{{Optimal ET sequence|legend=1| 12, 34, 46, 58, 104c, 162c }}


EDOs: 46, 80, 126, 206cd
[[Badness]] (Sintel): 0.959


Badness: 0.0353
=== 11-limit ===
Subgroup: 2.3.5.7.11


==13-limit==
Comma list: 126/125, 176/175, 896/891
Commas: 169/168, 176/175, 325/324, 364/363


valid range: [704.348, 705.882] (46 to 34d)
Mapping: {{mapping| 2 0 11 31 45 | 0 1 -2 -8 -12 }}


nice range: [701.955, 706.843]
Optimal tunings:  
* WE: ~45/32 = 599.4471{{c}}, ~3/2 = 703.0657{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 703.7996{{c}}


strict range: [704.348, 705.882]
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]


POTE generator: ~3/2 = 704.881
{{Optimal ET sequence|legend=0| 12, 34e, 46, 58, 104c, 162ce }}


Map: [&lt;2 0 11 -42 -28 -18|, &lt;0 1 -2 15 11 8|]
Badness (Sintel): 0.828


EDOs: 34d, 46, 80, 206cd, 286bcde
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Badness: 0.0253
Comma list: 126/125, 176/175, 196/195, 364/363


=Pajara=
Mapping: {{mapping| 2 0 11 31 45 55 | 0 1 -2 -8 -12 -15 }}
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.
Optimal tunings:
* WE: ~45/32 = 599.4451{{c}}, ~3/2 = 703.0528{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 703.7813{{c}}


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.
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]


Commas: 50/49, 64/63
{{Optimal ET sequence|legend=0| 12f, 34ef, 46, 58, 104c, 162cef }}


valid range: [700.000, 720.000] (12 to 10)
Badness (Sintel): 0.782


nice range: [701.955, 715.587]
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


strict range: [701.955, 715.587]
Comma list: 126/125, 136/135, 176/175, 196/195, 256/255


[[POTE_tuning|POTE generator]]: 707.048
Mapping: {{mapping| 2 0 11 31 45 55 5 | 0 1 -2 -8 -12 -15 1 }}


Map: [&lt;2 0 11 12|, &lt;0 1 -2 -2|]
Optimal tunings:  
* WE: ~17/12 = 599.6253{{c}}, ~3/2 = 703.3726{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 703.8520{{c}}


EDOs: 22, 34, 56
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]


Badness: 0.0200
{{Optimal ET sequence|legend=0| 12f, 34ef, 46, 58, 104c }}


==11-limit==
Badness (Sintel): 0.837
Commas: 50/49, 64/63, 99/98


valid range: [700.000, 709.091] (12 to 22)
=== 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.


nice range: [701.955, 715.587]
Subgroup: 2.3.5.7.11.13.17.23


strict range: [701.955, 709.091]
Comma list: 126/125, 136/135, 176/175, 196/195, 231/230, 256/255


[[POTE_tuning|POTE generator]]: 706.885
Subgroup-val mapping: {{mapping| 2 0 11 31 45 55 5 63 | 0 1 -2 -8 -12 -15 1 -17 }}


Map: [&lt;2 0 11 12 26|, &lt;0 1 -2 -2 -6|]
Optimal tunings:  
* WE: ~17/12 = 599.6272{{c}}, ~3/2 = 703.4326{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 703.9093{{c}}


EDOs: 22, 34, 56, 146
{{Optimal ET sequence|legend=0| 12i, 34efi, 46, 58i, 104ci }}


Badness: 0.0203
Badness (Sintel): 0.882


==13-limit==
== Pajara ==
Commas: 50/49, 64/63, 65/63, 99/98
{{Main| Pajara }}


valid 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.


nice range: [701.955, 738.573]
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.


strict range: []
[[Subgroup]]: 2.3.5.7


POTE generator: ~3/2 = 708.919
[[Comma list]]: 50/49, 64/63


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


EDOs: 12, 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 }}


Badness: 0.0276
[[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]


==Pajarous==
{{Optimal ET sequence|legend=1| 10, 12, 22, 34d, 56d }}
Commas: 50/49, 55/54, 64/63


valid range: 709.091 (22)
[[Badness]] (Sintel): 0.507


nice range: [701.955, 715.803]
=== 2.3.5.7.17 subgroup ===
Subgroup: 2.3.5.7.17


strict range: 709.091
Comma list: 50/49, 64/63, 85/84


POTE generator: ~3/2 = 709.578
Mapping: {{mapping| 2 0 11 12 5 | 0 1 -2 -2 1 }}


Map: [&lt;2 0 11 12 -9|, &lt;0 1 -2 -2 5|]
Optimal tunings:  
* WE: ~7/5 = 599.053{{c}}, ~3/2 = 706.355{{c}}
* CWE: ~7/5 = 600.000{{c}}, ~3/2 = 707.607{{c}}


EDOs: 10, 12e, 22, 120bce, 142bce
{{Optimal ET sequence|legend=0| 10, 12, 22, 56d }}


Badness: 0.0283
Badness (Sintel): 0.438


===13-limit===
=== 11-limit ===
Commas: 50/49, 55/54, 64/63, 65/63
Subgroup: 2.3.5.7.11


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


nice range: [701.955, 738.573]
Mapping: {{mapping| 2 0 11 12 26 | 0 1 -2 -2 -6 }}


strict range: []
Optimal tunings:  
* WE: ~7/5 = 598.8485{{c}}, ~3/2 = 705.5285{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 707.1826{{c}}


POTE generator: ~3/2 = 710.240
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]


Map: [&lt;2 0 11 12 -9 1|, &lt;0 1 -2 -2 5 2|]
{{Optimal ET sequence|legend=0| 10e, 12, 22, 34d, 56d }}


EDOs: 10, 22, 54f, 76bdf
Badness (Sintel): 0.673


Badness: 0.0252
==== 2.3.5.7.11.17 subgroup ====
Subgroup: 2.3.5.7.11.17


==Pajaric==
Comma list: 50/49, 64/63, 85/84, 99/98
Commas: 45/44, 50/49, 56/55


POTE generator: ~3/2 = 705.524
Mapping: {{mapping| 2 0 11 12 26 5 | 0 1 -2 -2 -6 1 }}


Map: [&lt;2 0 11 12 7|, &lt;0 1 -2 -2 0|]
Optimal tunings:  
* WE: ~7/5 = 599.062{{c}}, ~3/2 = 706.095{{c}}
* CWE: ~7/5 = 600.000{{c}}, ~3/2 = 707.370{{c}}


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


Badness: 0.0238
Badness (Sintel): 0.645


===13-limit===
==== 13-limit ====
Commas: 40/39, 45/44, 50/49, 56/55
Subgroup: 2.3.5.7.11.13


POTE generator: ~3/2 = 707.442
Comma list: 50/49, 64/63, 65/63, 99/98


Map: [&lt;2 0 11 12 7 17|, &lt;0 1 -2 -2 0 -3|]
Mapping: {{mapping| 2 0 11 12 26 1 | 0 1 -2 -2 -6 2 }}


EDOs: 10, 12f, 22ef, 34def
Optimal tunings:  
* WE: ~7/5 = 599.9732{{c}}, ~3/2 = 708.8873{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 708.9227{{c}}


Badness: 0.0205
{{Optimal ET sequence|legend=0| 10e, 12, 22 }}


==Pajaro==
Badness (Sintel): 1.14
Commas: 40/39, 50/49, 55/54, 64/63


POTE generator ~3/2 = 710.818
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


Map: [&lt;2 0 11 12 -9 17|, &lt;0 1 -2 -2 5 -3|]
Comma list: 50/49, 52/51, 64/63, 65/63, 99/98


EDOs: 10, 22f, 32f, 54f
Mapping: {{mapping| 2 0 11 12 26 1 5 | 0 1 -2 -2 -6 2 1 }}


Badness: 0.0274
Optimal tunings:  
* WE: ~7/5 = 599.8871{{c}}, ~3/2 = 708.6725{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 708.8176{{c}}


==Hemipaj==
{{Optimal ET sequence|legend=0| 10e, 12, 22 }}
Commas: 50/49, 64/63, 121/120


POTE generator: ~11/8 = 546.383
Badness (Sintel): 1.06


Map: [&lt;2 1 9 10 8|, &lt;0 2 -4 -4 -1|]
==== Pajarina ====
Subgroup: 2.3.5.7.11.13


EDOs: 20, 22, 68d, 90d
Comma list: 50/49, 64/63, 78/77, 99/98


Badness: 0.0389
Mapping: {{mapping| 2 0 11 12 26 36 | 0 1 -2 -2 -6 -9 }}


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


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


Commas: 126/125, 2048/2025
Badness (Sintel): 0.923


[[POTE_tuning|POTE generator]]: 703.681
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


Map: [&lt;2 0 11 31|, &lt;0 1 -2 -8|]
Comma list: 50/49, 64/63, 78/77, 85/84, 99/98


EDOs: 46, 58, 104c, 162c
Mapping: {{mapping| 2 0 11 12 26 36 5 | 0 1 -2 -2 -6 -9 1 }}


Badness: 0.0379
Optimal tunings:  
* WE: ~7/5 = 599.0204{{c}}, ~3/2 = 705.2572{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 706.5660{{c}}


==11-limit==
{{Optimal ET sequence|legend=0| 12f, 22, 34d }}
Commas: 126/125, 176/175, 896/891


[[POTE_tuning|POTE generator]]: 703.714
Badness (Sintel): 0.936


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


EDOs: 46, 58, 104c, 162ce
Comma list: 40/39, 50/49, 64/63, 66/65


==13-limit==
Mapping: {{mapping| 2 0 11 12 26 17 | 0 1 -2 -2 -6 -3 }}
Commas: 126/125, 196/195, 364/363, 2048/2025


[[POTE_tuning|POTE generator]]: 703.704
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|, &lt;0 1 -2 -8 -12 -15|]
{{Optimal ET sequence|legend=0| 10e, 12f, 22f, 34dff }}


EDOs: [[46edo|46]], [[58edo|58]], [[104edo|104c]], [[162edo|162cef]]
Badness (Sintel): 0.937


==17-limit==
===== 17-limit =====
Commas: 126/125, 136/135, 176/175, 196/195, 256/255
Subgroup: 2.3.5.7.11.13.17


[[POTE_tuning|POTE generator]]: 703.812
Comma list: 40/39, 50/49, 64/63, 66/65, 85/84


Map: [&lt;2 0 11 31 45 55 5|, &lt;0 1 -2 -8 -12 -15 1|]
Mapping: {{mapping| 2 0 11 12 26 17 5 | 0 1 -2 -2 -6 -3 1 }}


EDOs: 46, 58, 104c
Optimal tunings:  
* WE: ~7/5 = 598.6103{{c}}, ~3/2 = 706.3076{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 708.2256{{c}}


=Keen=
{{Optimal ET sequence|legend=0| 10e, 12f, 22f }}
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
Badness (Sintel): 0.968


[[POTE_tuning|POTE generator]]: 707.571
=== Pajarous ===
Subgroup: 2.3.5.7.11


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


EDOs: 22, 56, 78, 134b, 212b, 290b
Mapping: {{mapping| 2 0 11 12 -9 | 0 1 -2 -2 5 }}


==11-limit==
Optimal tunings:
Commas: 100/99, 385/384, 1232/1215
* WE: ~7/5 = 599.4055{{c}}, ~3/2 = 708.8747{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 709.5508{{c}}


[[POTE_tuning|POTE generator]]: 707.609
Tuning ranges:
* 11-odd-limit diamond monotone: ~3/2 = 709.091 (13\22)
* 11-odd-limit diamond tradeoff: ~3/2 = [701.955, 715.803]


Map: [&lt;2 0 11 -23 26|, &lt;0 1 -2 9 -6|]
{{Optimal ET sequence|legend=0| 10, 12e, 22, 120bce, 142bce }}


EDOs: 22, 56, 78, 212bf, 290bf
Badness (Sintel): 0.937


=Bidia=
==== 2.3.5.7.11.17 subgroup ====
Bidia adds 3136/3125 to the commas, splitting the period into 1/4 octave. It may be called the 12&amp;56 temperament.
Subgroup: 2.3.5.7.11.13.17


Commas: 2048/2025, 3136/3125
Comma list: 50/49, 52/51, 55/54, 64/63, 65/63


POTE generator: ~3/2 = 705.364
Mapping: {{mapping| 2 0 11 12 -9 1 5 | 0 1 -2 -2 5 2 1 }}


Map: [&lt;4 0 22 43|,&lt;0 1 -2 -5|]
Optimal tunings:  
* WE: ~7/5 = 599.408{{c}}, ~3/2 = 708.878{{c}}
* CWE: ~7/5 = 600.000{{c}}, ~3/2 = 709.544{{c}}


Wedgie: &lt;&lt;4 -8 -20 -22 -43 -24||
{{Optimal ET sequence|legend=0| 10, 12e, 22 }}


EDOs: 12, 56, 68, 80, 148d
Badness (Sintel): 0.766


Badness: 0.0565
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


==11-limit==
Comma list: 50/49, 55/54, 64/63, 65/63
Commas: 176/175, 896/891, 1375/1372


POTE generator: ~3/2 = 705.087
Mapping: {{mapping| 2 0 11 12 -9 1 | 0 1 -2 -2 5 2 }}


Map: [&lt;4 0 22 43 71|,&lt;0 1 -2 -5 -9|]
Optimal tunings:  
* WE: ~7/5 = 599.9064{{c}}, ~3/2 = 710.1289{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 710.2325{{c}}


EDOs: 12, 68, 80
{{Optimal ET sequence|legend=0| 10, 22 }}


Badness: 0.0402
Badness (Sintel): 1.04


==13-limit==
===== 17-limit =====
Commas: 176/175, 325/324, 640/637, 896/891
Subgroup: 2.3.5.7.11.13.17


POTE generator: ~3/2 = 705.301
Comma list: 50/49, 52/51, 55/54, 64/63, 65/63


Map: [&lt;4 0 22 43 71|,&lt;0 1 -2 -5 -9|]
Mapping: {{mapping| 2 0 11 12 -9 1 5 | 0 1 -2 -2 5 2 1 }}


EDOs: 12, 68, 80, 148d, 228bcd, 376bcdf
Optimal tunings:  
* WE: ~7/5 = 599.8239{{c}}, ~3/2 = 710.0128{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 710.2067{{c}}


Badness: 0.0411
{{Optimal ET sequence|legend=0| 10, 22, 54f, 76bdff }}


=Echidna=
Badness (Sintel): 0.930
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.
==== Pajaro ====
Subgroup: 2.3.5.7.11.13


Commas: 2048/2025, 1728/1715
Comma list: 40/39, 50/49, 55/54, 64/63


[[POTE_tuning|POTE generator]]: 434.856
Mapping: {{mapping| 2 0 11 12 -9 17 | 0 1 -2 -2 5 -3 }}


Map: [&lt;2 1 9 2|, &lt;0 3 -6 5|]
Optimal tunings:  
* WE: ~7/5 = 598.8257{{c}}, ~3/2 = 709.4266{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 710.8414{{c}}


EDOs: 22, 58, 80, 138cd, 218cd
{{Optimal ET sequence|legend=0| 10, 22f, 32f }}


Badness: 0.0580
Badness (Sintel): 1.13


==11-limit==
===== 17-limit =====
Commas: 176/175, 896/891, 540/539
Subgroup: 2.3.5.7.11.13.17


11-limit minimax
Comma list: 40/39, 50/49, 55/54, 64/63, 85/84


[|1 0 0 0 0&gt;, |7/4 0 0 1/4 -1/4&gt;, |2 0 0 -1/2 1/2&gt;,
Mapping: {{mapping| 2 0 11 12 -9 17 5 | 0 1 -2 -2 5 -3 1 }}
|37/12 0 0 5/12 -5/12&gt;, |37/12 0 0 -7/12 7/12&gt;<nowiki>]</nowiki>


Eigenmonzos: 2, 11/7
Optimal tunings:  
* WE: ~7/5 = 598.8865{{c}}, ~3/2 = 709.5472{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 710.8704{{c}}


Minimax generator: (224/11)^(1/12) = 434.792
{{Optimal ET sequence|legend=0| 10, 22f, 32f }}


[[POTE_tuning|POTE generator]]: 434.852
Badness (Sintel): 1.01


Map: [&lt;2 1 9 2 12|, &lt;0 3 -6 5 -7|]
=== Pajaric ===
Subgroup: 2.3.5.7.11


EDOs: 22, 58, 80, 138cde, 218cde
Comma list: 45/44, 50/49, 56/55


Badness: 0.0260
Mapping: {{mapping| 2 0 11 12 7 | 0 1 -2 -2 0 }}


==13-limit==
Optimal tunings:
Commas: 176/175, 351/350, 364/363, 540/539
* WE: ~7/5 = 597.4807{{c}}, ~3/2 = 702.5616{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 706.0542{{c}}


[[POTE_tuning|POTE generator]]: 434.756
{{Optimal ET sequence|legend=0| 10, 12, 22e }}


Map: [&lt;2 1 9 2 12 19|, &lt;0 3 -6 5 -7 -16|]
Badness (Sintel): 0.787


EDOs: 22, 58, 80, 138cde
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Badness: 0.0237
Comma list: 40/39, 45/44, 50/49, 56/55


==17-limit==
Mapping: {{mapping| 2 0 11 12 7 17 | 0 1 -2 -2 0 -3 }}
Commas: 136/135, 176/175, 221/220, 256/255, 540/539


[[POTE_tuning|POTE generator]]: 434.816
Optimal tunings:  
* WE: ~7/5 = 597.1952{{c}}, ~3/2 = 704.1350{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 708.1989{{c}}


Map: [&lt;2 1 9 2 12 19 6|, &lt;0 3 -6 5 -7 -16 3|]
{{Optimal ET sequence|legend=0| 10, 12f, 22ef }}


EDOs: 22, 58, 80, 138cde
Badness (Sintel): 0.845


Badness: 0.0203
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


=Echidnic=
Comma list: 34/33, 40/39, 45/44, 50/49, 56/55
Commas: 686/675, 1029/1024


[[POTE_tuning|POTE generator]]: 234.492
Mapping: {{mapping| 2 0 11 12 7 17 5 | 0 1 -2 -2 0 -3 1 }}


Map: [&lt;2 2 7 6|, &lt;0 3 -6 -1|]
Optimal tunings:  
* WE: ~7/5 = 597.6509{{c}}, ~3/2 = 705.7702{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 708.9719{{c}}


EDOs: 10, 36, 46, 194bcd, 240bcd, 286bcd, 332bcd
{{Optimal ET sequence|legend=0| 10, 12f, 22ef }}


Badness: 0.0722
Badness (Sintel): 0.896


==11-limit==
=== Hemipaj ===
Commas: 385/384, 441/440, 686/675
Subgroup: 2.3.5.7.11


[[POTE_tuning|POTE generator]]: 235.096
Comma list: 50/49, 64/63, 121/120


Map: [&lt;2 2 7 6 3|, &lt;0 3 -6 -1 10|]
Mapping: {{mapping| 2 1 9 10 8 | 0 2 -4 -4 -1 }}


EDOs: 10, 46, 102, 148, 342bcd
: mapping generators: ~2, ~16/11


Badness: 0.0451
Optimal tunings:  
* WE: ~7/5 = 597.6509{{c}}, ~16/11 = 652.7788{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~16/11 = 653.7119{{c}}


==13-limit==
{{Optimal ET sequence|legend=0| 2, 20, 22 }}
Commas: 91/90, 169/168, 385/384, 441/440


[[POTE_tuning|POTE generator]]: 235.088
Badness (Sintel): 1.29


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


EDOs: 10, 46, 102, 148f, 194bcdf
Comma list: 50/49, 64/63, 243/242


Badness: 0.0289
Mapping: {{mapping| 2 0 11 12 -1 | 0 2 -4 -4 5 }}
: mapping generators: ~2, ~55/32


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


http://untwelve.org/2011competition_audio/Kosmorsky-A_Stiff_Shot_of_Turpentine.mp3
{{Optimal ET sequence|legend=0| 10, 24d, 34d }}


(the description says "lemba" which has a similar scale structure but different mapping for 5)
Badness (Sintel): 1.62


=Shrutar=
==== 13-limit ====
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.
Subgroup: 2.3.5.7.11.13


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.
Comma list: 50/49, 64/63, 78/77, 144/143


Commas: 2048/2025, 245/243
Mapping: {{mapping| 2 0 11 12 -1 9 | 0 2 -4 -4 5 -1 }}


[[POTE_tuning|POTE generator]]: 52.811
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|, &lt;0 2 -4 7|]
{{Optimal ET sequence|legend=0| 10, 24d, 34d }}


EDOs: 22, 46, 68, 182b, 250bc
Badness (Sintel): 1.19


==11-limit==
==== 17-limit ====
Commas: 2048/2025, 245/243, 121/120
Subgroup: 2.3.5.7.11.13.17


[[POTE_tuning|POTE generator]]: 52.680
Comma list: 50/49, 64/63, 78/77, 85/84, 144/143


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


EDOs: 22, 46, 68, 114, 296bce, 410bce
Optimal tunings:  
* WE: ~7/5 = 598.8411{{c}}, ~26/15 = 951.3687{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~26/15 = 953.2169{{c}}


==13-limit==
{{Optimal ET sequence|legend=0| 10, 24d, 34d }}
Commas: 121/120, 176/175, 196/195, 245/243


POTE generator: ~28/27 = 52.654
Badness (Sintel): 1.11


Map: [&lt;2 1 9 -2 8 -10|, &lt;0 2 -4 7 -1 16|]
== Srutal ==
{{See also| Srutal vs diaschismic }}


EDOs: 22, 24, 46, 68, 114
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]].


Badness: 0.0281
[[Subgroup]]: 2.3.5.7


==17-limit==
[[Comma list]]: 2048/2025, 4375/4374
Commas: 121/120, 136/135, 154/153, 176/175, 196/195


POTE generator: ~28/27 = 52.647
{{Mapping|legend=1| 2 0 11 -42 | 0 1 -2 15 }}


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


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


Badness: 0.0187
{{Optimal ET sequence|legend=1| 34d, 46, 80, 126, 206cd, 332bcd }}


==19-limit==
[[Badness]] (Sintel): 2.32
Commas: 121/120, 136/135, 154/153, 176/175, 196/195, 343/342


POTE generator: ~28/27 = 52.730
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;2 1 9 -2 8 -10 6 -10|, &lt;0 2 -4 7 -1 16 2 17|]
Comma list: 176/175, 896/891, 1331/1323


EDOs: 22, 24, 46, 68, 114, 182bef
Mapping: {{mapping| 2 0 11 -42 -28 | 0 1 -2 15 11 }}


Badness: 0.0175
Optimal tunings:  
* WE: ~45/32 = 599.4413{{c}}, ~3/2 = 704.1999{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 704.8017{{c}}


=Sruti=
Tuning ranges:
Commas: 2048/2025, 19683/19600
* 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]


POTE generator: ~175/144 = 351.876
{{Optimal ET sequence|legend=0| 34d, 46, 80, 126, 206cd }}


Map: [&lt;2 0 11 -15|, &lt;0 2 -4 13|]
Badness (Sintel): 1.17


Wedgie: &lt;&lt;4 -8 26 -22 30 83||
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


EDOs: 24, 34d, 58, 150cd, 208cd, 266cd
Comma list: 169/168, 176/175, 325/324, 364/363


Badness: 0.1174
Mapping: {{mapping| 2 0 11 -42 -28 -18 | 0 1 -2 15 11 8 }}


==11-limit==
Optimal tunings:
Commas: 176/175, 243/242, 896/891
* WE: ~45/32 = 599.5490{{c}}, ~3/2 = 704.3516{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 704.8347{{c}}


POTE generator: ~11/9 = 351.863
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]


Map: [&lt;2 0 11 -15 -1|, &lt;0 2 -4 13 5|]
{{Optimal ET sequence|legend=0| 34d, 46, 80 }}


EDOs: 24, 34d, 58, 150cde, 208cde
Badness (Sintel): 1.04


Badness: 0.0415
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


==13-limit==
Comma list: 136/135, 169/168, 176/175, 221/220, 256/255
Commas: 144/143, 176/175, 351/350, 676/675


POTE generator: ~11/9 = 351.886
Mapping: {{mapping| 2 0 11 -42 -28 -18 5 | 0 1 -2 15 11 8 1 }}


Map: [&lt;2 0 11 -15 -1 9|, &lt;0 2 -4 13 5 -1|]
Optimal tunings:  
* WE: ~17/12 = 599.6459{{c}}, ~3/2 = 704.4237{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 704.8083{{c}}


EDOs: 24, 34d, 58, 150cdef, 208cdef
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]


Badness: 0.0238
{{Optimal ET sequence|legend=0| 34d, 46, 80, 126 }}


=Anguirus=
Badness (Sintel): 0.947
Commas: 49/48, 2048/2025


POTE generator: ~8/7 = 246.979
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19


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


Wedgie: &lt;&lt;4 -8 2 -22 -8 27||
Mapping: {{mapping| 2 0 11 -42 -28 -18 5 -55 | 0 1 -2 15 11 8 1 20 }}


EDOs: 10, 24, 34
Optimal tunings:  
* WE: ~17/12 = 599.6371{{c}}, ~3/2 = 704.4790{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 704.8745{{c}}


Badness: 0.0780
{{Optimal ET sequence|legend=0| 34dh, 46, 80 }}


==11-limit==
Badness (Sintel): 1.04
Commas: 49/48, 56/55, 243/242


POTE generator: ~8/7 = 247.816
==== Srutaloo ====
Srutaloo adds 576/575, 736/729 or 208/207, and rhymes with [[skidoo]].


Map: [&lt;2 0 11 4 -1|, &lt;0 2 -4 1 5|]
Subgroup: 2.3.5.7.11.13.17.19.23


EDOs: 10, 24, 34, 58d, 92de
Comma list: 136/135, 169/168, 176/175, 190/189, 208/207, 221/220, 256/255


Badness: 0.0493
Mapping: {{mapping| 2 0 11 -42 -28 -18 5 -55 -10 | 0 1 -2 15 11 8 1 20 6 }}


==13-limit==
Optimal tunings:
Commas: 49/48 56/55 91/90 352/351
* WE: ~17/12 = 599.6690{{c}}, ~3/2 = 704.5098{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 704.8713{{c}}


POTE generator: ~8/7 = 247.691
{{Optimal ET sequence|legend=0| 34dh, 46, 80 }}


Map: [&lt;2 0 11 4 -1 9|, &lt;0 2 -4 1 5 -1|]
Badness (Sintel): 0.971


EDOs: 10, 24, 34, 58d, 92def
===== 29-limit =====
Subgroup: 2.3.5.7.11.13.17.19.23.29


Badness: 0.0308
Comma list: 136/135, 169/168, 176/175, 190/189, 208/207, 221/220, 232/231, 256/255


=Shru=
Mapping: {{mapping| 2 0 11 -42 -28 -18 5 -55 -10 -76 | 0 1 -2 15 11 8 1 20 6 27 }}
Commas: 392/375, 1323/1280


POTE generator: ~64/63 = 50.135
Optimal tunings:  
* WE: ~17/12 = 599.6664{{c}}, ~3/2 = 704.5138{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 704.8807{{c}}


Map: [&lt;2 1 9 11|, &lt;0 2 -4 -5|]
{{Optimal ET sequence|legend=0| 34dhj, 46, 80 }}


Wedgie: &lt;&lt;4 -8 -10 -22 -27 -1||
Badness (Sintel): 1.10


EDOs: 22d, 24
===== 31-limit =====
Subgroup: 2.3.5.7.11.13.17.19.23.29.31


Badness: 0.1576
Comma list: 136/135, 169/168, 176/175, 190/189, 208/207, 217/216, 221/220, 232/231, 256/255


==11-limit==
Mapping: {{mapping| 2 0 11 -42 -28 -18 5 -55 -10 -76 48 | 0 1 -2 15 11 8 1 20 6 27 -12 }}
Commas: 56/55, 77/75, 1323/1280


POTE generator: ~64/63 = 50.130
Optimal tunings:  
* WE: ~17/12 = 599.8115{{c}}, ~3/2 = 704.5958{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 704.8086{{c}}


Map: [&lt;2 1 9 11 8|, &lt;0 2 -4 -5 -1|]
{{Optimal ET sequence|legend=0| 46, 80, 126 }}


EDOs: 22d, 24
Badness (Sintel): 1.44


Badness: 0.0635
== 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.


==13-limit==
[[Subgroup]]: 2.3.5.7
Commas: 56/55, 77/75, 105/104, 507/500


POTE generator: ~64/63 = 50.535
[[Comma list]]: 875/864, 2048/2025


Map: [&lt;2 1 9 11 8 15|, &lt;0 2 -4 -5 -1 -7|]
{{Mapping|legend=1| 2 0 11 -23 | 0 1 -2 9 }}


EDOs: 24
[[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 }}


Badness: 0.0457
{{Optimal ET sequence|legend=1| 22, 56, 78, 134b }}


[[Category:Theory]]
[[Badness]] (Sintel): 2.13
[[Category:Temperament family]]
[[Category:Diaschismic]]


[[Category:Todo:add_definition]]
=== 11-limit ===
[[Category:Todo:intro]]
Subgroup: 2.3.5.7.11
 
Comma list: 100/99, 385/384, 1232/1215
 
Mapping: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 4 0 22 43 71 -36 10 17 | 0 1 -2 -5 -9 8 1 0 }}
 
Optimal tunings:
* WE: ~19/16 = 299.7967{{c}}, ~3/2 = 704.8609{{c}}
* CWE: ~19/16 = 300.0000{{c}}, ~3/2 = 705.3519{{c}}
 
{{Optimal ET sequence|legend=0| 12, 68, 80, 148d }}
 
Badness (Sintel): 1.25
 
=== 23-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23
 
Comma list: 136/135, 176/175, 190/189, 253/252, 256/255, 325/324, 640/637
 
Mapping: {{mapping| 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: {{mapping| 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 ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 121/120, 176/175, 196/195, 245/243
 
Mapping: {{mapping| 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: {{mapping| 2 1 9 -2 8 -10 6 | 0 2 -4 7 -1 16 2 }}
 
Optimal tunings:
* WE: ~17/12 = 599.7995{{c}}, ~16/11 = 652.4287{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~16/11 = 652.6334{{c}}
 
{{Optimal ET sequence|legend=0| 22f, 46, 68, 114 }}
 
Badness (Sintel): 0.953
 
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 121/120, 136/135, 154/153, 176/175, 196/195, 343/342
 
Mapping: {{mapping| 2 1 9 -2 8 -10 6 -10 | 0 2 -4 7 -1 16 2 17 }}
 
Optimal tunings:
* WE: ~17/12 = 599.8060{{c}}, ~16/11 = 652.5190{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~16/11 = 652.7164{{c}}
 
{{Optimal ET sequence|legend=0| 22fh, 46, 68, 114, 182bef }}
 
Badness (Sintel): 1.07
 
==== 23-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23
 
Comma list: 121/120, 136/135, 154/153, 176/175, 196/195, 253/252, 343/342
 
Mapping: {{mapping| 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.
 
[[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: {{mapping| 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: {{mapping| 2 1 9 11 8 15 | 0 2 -4 -5 -1 -7 }}
 
Optimal tunings:
* WE: ~45/32 = 599.9067{{c}}, ~10/7 = 649.4907{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~10/7 = 649.5950{{c}}
 
{{Optimal ET sequence|legend=0| 2, 24 }}
 
Badness (Sintel): 2.12
 
== Sruti ==
{{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: {{mapping| 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: {{mapping| 2 0 11 -15 -1 9 | 0 2 -4 13 5 -1 }}
 
Optimal tunings:
* WE: ~45/32 = 599.1479{{c}}, ~26/15 = 950.5337{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~26/15 = 951.8314{{c}}
 
{{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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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
 
== Echidna ==
Echidna adds 1728/1715 to the commas and takes 9/7 as a generator. It may be called the {{nowrap| 22 & 58 }} temperament; its ploidacot is diploid alpha-tricot. [[58edo]] or [[80edo]] make for good tunings, or their vals can be added to {{val| 138 219 321 388 }} (138cde). In most of the tunings it has a significantly sharp 7/4 which some prefer.
 
Echidna becomes more interesting when extended to be an 11-limit temperament by adding 176/175, 540/539 or 896/891 to the commas, where the same tunings can be used as before. It then is able to represent the entire 11-odd-limit diamond to within about six cents of error, within a compass of 24 notes. The 22-note 2mos gives scope for this, and the 36-note mos much more. Better yet, it is related to three important 11-limit edos: 22edo, a trivial tuning, is the smallest consistent in the 11-odd-limit, corresponding to the merge of this temperament with [[hedgehog]]; [[58edo]] is the smallest tuning that is distinctly consistent in the 11-odd-limit and [[80edo]] is the third smallest distinctly consistent in the 11-odd-limit.
 
The generator can be interpreted as 11/10, the period complement of 9/7, as a stack of 11/10 and 9/7 makes [[99/70]] which is extremely close to 600{{cent}} and is equal to it if we temper out [[9801/9800|S99]]. Three 11/10's then make a 4/3 (tempering out [[4000/3993|S10/S11]] thus making 10/9 and 12/11 equidistant from 11/10), implying a flat tuning of 4/3.
 
Like most srutal extensions, the 13- and 17-limit interpretations are possible by observing that since we have tempered out [[176/175]], tempering out [[351/350]] and [[352/351]] which sum to 176/175 is very elegant. In the 17-limit we can equate the half-octave with 17/12 and 24/17 and we can take advantage of the sharp fifth by combining echidna with [[srutal archagall]], leading to a particularly beautiful temperament (one that prefers a very slightly less sharp fifth than srutal archagall). This mapping of 13 and 17 is supported by the patent vals of the three main echidna edos of 22, 58 and 80, of which all except 22 are consistent in the [[17-odd-limit]].
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 1728/1715, 2048/2025
 
{{Mapping|legend=1| 2 1 9 2 | 0 3 -6 5 }}
: mapping generators: ~45/32, ~9/7
 
[[Optimal tuning]]s:
* [[WE]]: ~45/32 = 599.3056{{c}}, ~9/7 = 434.3524{{c}}
: [[error map]]: {{val| -1.389 +0.408 +1.322 +1.547 }}
* [[CWE]]: ~45/32 = 600.0000{{c}}, ~9/7 = 434.8327{{c}}
: error map: {{val| 0.000 +2.543 +4.690 +5.338 }}
 
{{Optimal ET sequence|legend=1| 22, 58, 80, 138cd, 218cd }}
 
[[Badness]] (Sintel): 1.47
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 176/175, 540/539, 896/891
 
Mapping: {{mapping| 2 1 9 2 12 | 0 3 -6 5 -7 }}
 
Optimal tunings:
* WE: ~45/32 = 599.3085{{c}}, ~9/7 = 434.3511{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~9/7 = 434.8647{{c}}
 
Minimax tuning:
* 11-odd-limit: ~9/7 = {{monzo| 5/12 0 0 1/12 -1/12 }}
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 7/4 0 0 1/4 -1/4 }}, {{monzo| 2 0 0 -1/2 1/2 }}, {{monzo| 37/12 0 0 5/12 -5/12 }}, {{monzo| 37/12 0 0 -7/12 7/12 }}]
: unchanged-interval (eigenmonzo) basis: 2.11/7
 
{{Optimal ET sequence|legend=0| 22, 58, 80, 138cde, 218cde }}
 
Badness (Sintel): 0.859
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 176/175, 351/350, 364/363, 540/539
 
Mapping: {{mapping| 2 1 9 2 12 19 | 0 3 -6 5 -7 -16 }}
 
Optimal tunings:
* WE: ~45/32 = 599.3397{{c}}, ~9/7 = 434.2772{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~9/7 = 434.7864{{c}}
 
{{Optimal ET sequence|legend=0| 22, 36f, 58, 80, 138cde }}
 
Badness (Sintel): 0.978
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 136/135, 176/175, 221/220, 256/255, 540/539
 
Mapping: {{mapping| 2 1 9 2 12 19 6 | 0 3 -6 5 -7 -16 3 }}
 
Optimal tunings:
* WE: ~45/32 = 599.4645{{c}}, ~9/7 = 434.4282{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~9/7 = 434.8340{{c}}
 
{{Optimal ET sequence|legend=0| 22, 36f, 58, 80, 138cde }}
 
Badness (Sintel): 1.03
 
== 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 2 0 11 8 22 | 0 4 -8 -3 -19 }}
 
Optimal tunings:
* WE: ~45/32 = 599.4648{{c}}, ~21/16 = 475.6929{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~21/16 = 476.1507{{c}}
 
{{Optimal ET sequence|legend=0| 10e, …, 58, 126, 184c, 310bccde }}
 
Badness (Sintel): 1.62
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 176/175, 196/195, 512/507, 676/675
 
Mapping: {{mapping| 2 0 11 8 22 9 | 0 4 -8 -3 -19 -2 }}
 
Optimal tunings:
* WE: ~45/32 = 599.3787{{c}}, ~21/16 = 475.6065{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~21/16 = 476.1345{{c}}
 
{{Optimal ET sequence|legend=0| 10e, …, 58, 126f, 184cff }}
 
Badness (Sintel): 1.18
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 136/135, 170/169, 176/175, 196/195, 256/255
 
Mapping: {{mapping| 2 0 11 8 22 9 5 | 0 4 -8 -3 -19 -2 4 }}
 
Optimal tunings:
* WE: ~17/12 = 599.5077{{c}}, ~21/16 = 475.7713{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~21/16 = 476.1814{{c}}
 
{{Optimal ET sequence|legend=0| 10e, 58, 126f }}
 
Badness (Sintel): 1.21
 
=== Quadrafourths ===
Subgroup: 2.3.5.7.11
 
Comma list: 243/242, 441/440, 2048/2025
 
Mapping: {{mapping| 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: {{mapping| 2 0 11 8 -1 9 | 0 4 -8 -3 10 -2 }}
 
Optimal tunings:
* WE: ~45/32 = 599.2147{{c}}, ~21/16 = 475.4052{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~21/16 = 476.0253{{c}}
 
{{Optimal ET sequence|legend=0| 10, 48c, 58, 126eef, 184ceeff, 242ccdeeeff }}
 
Badness (Sintel): 1.11
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 136/135, 144/143, 170/169, 196/195, 221/220
 
Mapping: {{mapping| 2 0 11 8 -1 9 5 | 0 4 -8 -3 10 -2 4 }}
 
Optimal tunings:
* WE: ~17/12 = 599.3353{{c}}, ~21/16 = 475.5495{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~21/16 = 476.0691{{c}}
 
{{Optimal ET sequence|legend=0| 10, 48c, 58 }}
 
Badness (Sintel): 1.13
 
== Subgroup extensions ==
=== Srutal archagall (2.3.5.17) ===
{{See also | Fiventeen }}
 
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
 
Subgroup-val mapping: {{mapping| 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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