Diaschismic family: Difference between revisions

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The 5-limit parent comma for the '''diaschismic family''' is 2048/2025, the [[diaschisma]]. Its monzo is {{monzo| 11 -4 -2 }}, and flipping that yields {{multival| 2 -4 -11 }} for the wedgie for 5-limit '''diaschismic''', or '''srutal''', temperament. This tells us the period is half an octave, the [[Wikipedia: Greatest common divisor|GCD]] of 2 and -4, and that the generator is a fifth. Three periods gives 1800 cents, and decreasing this by two fifths gives the major third. [[34edo]] is a good tuning choice, with [[46edo]], [[56edo]], [[58edo]] or [[80edo]] being other possibilities. Both [[12edo]] and [[22edo]] support it, and retuning them to a MOS of diaschismic gives two scale possibilities.
{{Technical data page}}
The '''diaschismic family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the diaschisma, [[2048/2025]].  


= Srutal (12&34, aka diaschismic) =
== Diaschismic ==
{{Main| Diaschismic }}
 
The [[period]] of diaschismic is half an [[2/1|octave]], and the [[generator]] is a fifth; the [[ploidacot]] is diploid monocot. Three periods gives 1800 cents, and decreasing this by two fifths gives the major third. [[34edo]] is a good tuning choice, with [[46edo]], [[56edo]], [[58edo]], or [[80edo]] being other possibilities. Both [[12edo]] and [[22edo]] support it, and retuning them to a [[mos]] of diaschismic gives two scale possibilities.
 
This temperament is also known as '''srutal''' in the 5-limit, but that name more strictly speaking refers to the [[#Srutal|34d & 46 extension]] to the [[7-limit]] that adds [[4375/4374]] to the comma list.
 
[[Subgroup]]: 2.3.5


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


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


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


[[Tuning ranges]]:  
[[Tuning ranges]]:  
* valid range: ~3/2 = [600.000 to 720.000] (1\2 to 6\10)
* [[5-odd-limit]] [[diamond monotone]]: ~3/2 = [600.000 to 720.000] (1\2 to 6\10)
* nice range: ~3/2 = [701.955, 706.843]
* 5-odd-limit [[diamond tradeoff]]: ~3/2 = [701.955, 706.843]
* strict range: ~3/2 = [701.955, 706.843]
 
{{Optimal ET sequence|legend=1| 10, 12, 22, 34, 46, 80, 206c, 286bc }}


{{Val list|legend=1| 10, 12, 22, 34, 46, 80, 206c, 286bc }}
[[Badness]] (Sintel): 0.467


[[Badness]]: 0.019915
=== 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]].  


== Seven limit extensions ==
Those all keep the same half-octave period and fifth generator.  
The second comma of the [[Normal lists #Normal interval list|normal comma list]] defines which 7-limit family member we are looking at.  


* Pajara derives from [[64/63]] and is a popular and well-known choice.  
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.
* Diaschismic adds 2097152/2066715 to obtain 7-limit harmony by more complex methods, but with greater accuracy.  
* Srutal adds {{monzo| 21 -15 0 1 }}. It does no significant tuning damage, so for that we keep the 5-limit label srutal.  
* Keen adds 2240/2187.
* Echidna [[1728/1715]], the orwellisma.  
* Shrutar [[245/243]], the sensamagic comma.  


Pajara, diaschismic, srutal 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]], the septimal quarter-tone) and echidna has a generator of 9/7.
==== 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]]. The [[S-expression]]-based comma list of this temperament is {[[256/255|S16]], [[289/288|S17]]}.


= Srutal =
== Septimal diaschismic ==
[[Comma list]]: 2048/2025, 4375/4374
{{Main| Diaschismic }}
{{See also| Srutal vs diaschismic }}
 
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.
 
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.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 126/125, 2048/2025
 
{{Mapping|legend=1| 2 0 11 31 | 0 1 -2 -8 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~45/32 = 599.4449{{c}}, ~3/2 = 703.0299{{c}}
: [[error map]]: {{val| -1.110 -0.035 +3.740 -1.391 }}
* [[CWE]]: ~45/32 = 600.0000{{c}}, ~3/2 = 703.7739{{c}}
: error map: {{val| 0.000 +1.819 +6.138 +0.983 }}


[[Tuning ranges]]:  
[[Tuning ranges]]:  
* valid range: ~3/2 = [703.448, 705.882] (34\58 to 20\34)
* 7- and 9-odd-limit [[diamond monotone]]: ~3/2 = [700.000, 705.882] (7\12 to 20\34)
* nice range: ~3/2 = [701.955, 706.843]
* 7- and 9-odd-limit [[diamond tradeoff]]: ~3/2 = [701.955, 706.843]
* strict range: ~3/2 = [703.448, 705.882]
 
{{Optimal ET sequence|legend=1| 12, 34, 46, 58, 104c, 162c }}
 
[[Badness]] (Sintel): 0.959
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 126/125, 176/175, 896/891


POTE generator: ~3/2 = 704.814
Mapping: {{mapping| 2 0 11 31 45 | 0 1 -2 -8 -12 }}


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


Wedgie: &lt;&lt;2 -4 30 -11 42 81||
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]


{{Val list|legend=1| 34d, 46, 80, 126, 206cd, 332bcd }}
{{Optimal ET sequence|legend=0| 12, 34e, 46, 58, 104c, 162ce }}


[[Badness]]: 0.091504
Badness (Sintel): 0.828


== 11-limit ==
=== 13-limit ===
Comma list: 176/175, 896/891, 1331/1323
Subgroup: 2.3.5.7.11.13
 
Comma list: 126/125, 176/175, 196/195, 364/363
 
Mapping: {{mapping| 2 0 11 31 45 55 | 0 1 -2 -8 -12 -15 }}
 
Optimal tunings:
* WE: ~45/32 = 599.4451{{c}}, ~3/2 = 703.0528{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 703.7813{{c}}


Tuning ranges:  
Tuning ranges:  
* valid range: ~3/2 = [704.348, 705.882] (27\46 to 20\34)
* 13- and 15-odd-limit diamond monotone: ~3/2 = [703.448, 704.348] (34\58 to 27\46)
* nice range: ~3/2 = [701.955, 706.843]
* 13-odd-limit diamond tradeoff: ~3/2 = [701.955, 706.843]
* strict range: ~3/2 = [704.348, 705.882]
* 15-odd-limit diamond tradeoff: ~3/2 = [701.955, 711.731]


POTE generator: ~3/2 = 704.856
{{Optimal ET sequence|legend=0| 12f, 34ef, 46, 58, 104c, 162cef }}


Map: [&lt;2 0 11 -42 -28|, &lt;0 1 -2 15 11|]
Badness (Sintel): 0.782


Vals: {{Val list| 34d, 46, 80, 126, 206cd }}
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Badness: 0.035315
Comma list: 126/125, 136/135, 176/175, 196/195, 256/255


== 13-limit ==
Mapping: {{mapping| 2 0 11 31 45 55 5 | 0 1 -2 -8 -12 -15 1 }}
Comma list: 169/168, 176/175, 325/324, 364/363
 
Optimal tunings:
* WE: ~17/12 = 599.6253{{c}}, ~3/2 = 703.3726{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 703.8520{{c}}


Tuning ranges:  
Tuning ranges:  
* valid range: ~3/2 = [704.348, 705.882] (27\46 to 20\34)
* 17-odd-limit diamond monotone: ~3/2 = [703.448, 704.348] (34\58 to 27\46)
* nice range: ~3/2 = [701.955, 706.843]
* 17-odd-limit diamond tradeoff: ~3/2 = [698.955, 711.731]
* strict range: ~3/2 = [704.348, 705.882]
 
{{Optimal ET sequence|legend=0| 12f, 34ef, 46, 58, 104c }}
 
Badness (Sintel): 0.837
 
=== 2.3.5.7.11.13.17.23 subgroup (Na"Naa') ===
<b>Na"Naa'</b> is a remarkable subgroup temperament of {{nowrap| 46 & 58 }} with a prime harmonic of 23. It is yet to be found why it got this strange name.
 
Subgroup: 2.3.5.7.11.13.17.23
 
Comma list: 126/125, 136/135, 176/175, 196/195, 231/230, 256/255
 
Subgroup-val mapping: {{mapping| 2 0 11 31 45 55 5 63 | 0 1 -2 -8 -12 -15 1 -17 }}


POTE generator: ~3/2 = 704.881
Optimal tunings:  
* WE: ~17/12 = 599.6272{{c}}, ~3/2 = 703.4326{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 703.9093{{c}}


Map: [&lt;2 0 11 -42 -28 -18|, &lt;0 1 -2 15 11 8|]
{{Optimal ET sequence|legend=0| 12i, 34efi, 46, 58i, 104ci }}


Vals: {{Val list| 34d, 46, 80, 206cd, 286bcde }}
Badness (Sintel): 0.882


Badness: 0.025286
== Pajara ==
{{Main| Pajara }}


= Pajara =
Pajara is closely associated with 22edo (not to mention [[Paul Erlich]]) but other tunings are possible. The 1/2-octave period serves as both a [[10/7]] and a [[7/5]]. Aside from 22edo, 34 with the val {{val| 34 54 79 96 }} (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.
{{main| Pajara }}


Pajara, with wedgie &lt;&lt;2 -4 -4 -11 -12 2|| is closely associated with 22et (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 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.
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.


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.
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 50/49, 64/63
[[Comma list]]: 50/49, 64/63
{{Mapping|legend=1| 2 0 11 12 | 0 1 -2 -2 }}
[[Optimal tuning]]s:
* [[WE]]: ~7/5 = 598.8483{{c}}, ~3/2 = 705.6906{{c}}
: [[error map]]: {{val| -2.303 +1.432 -5.756 +10.580 }}
* [[CWE]]: ~7/5 = 600.0000{{c}}, ~3/2 = 707.3438{{c}}
: error map: {{val| 0.000 +5.389 -1.001 +16.487 }}


[[Tuning ranges]]:
[[Tuning ranges]]:
* valid range: ~3/2 = [700.000, 720.000] (7\12 to 6\10)
* 7- and 9-odd-limit [[diamond monotone]]: ~3/2 = [700.000, 720.000] (7\12 to 6\10)
* nice range: ~3/2 = [701.955, 715.587]
* 7- and 9-odd-limit [[diamond tradeoff]]: ~3/2 = [701.955, 715.587]
* strict range: ~3/2 = [701.955, 715.587]
 
{{Optimal ET sequence|legend=1| 10, 12, 22, 34d, 56d }}
 
[[Badness]] (Sintel): 0.507
 
=== 2.3.5.7.17 subgroup ===
Subgroup: 2.3.5.7.17


[[POTE_tuning|POTE generator]]: ~3/2 = 707.048
Comma list: 50/49, 64/63, 85/84


[[Mapping]]: [&lt;2 0 11 12|, &lt;0 1 -2 -2|]
Mapping: {{mapping| 2 0 11 12 5 | 0 1 -2 -2 1 }}


{{Val list|legend=1| 10, 12, 22, 34d, 56d }}
Optimal tunings:
* WE: ~7/5 = 599.053{{c}}, ~3/2 = 706.355{{c}}
* CWE: ~7/5 = 600.000{{c}}, ~3/2 = 707.607{{c}}


[[Badness]]: 0.020033
{{Optimal ET sequence|legend=0| 10, 12, 22, 56d }}
 
Badness (Sintel): 0.438
 
=== 11-limit ===
Subgroup: 2.3.5.7.11


== 11-limit ==
Comma list: 50/49, 64/63, 99/98
Comma list: 50/49, 64/63, 99/98
Mapping: {{mapping| 2 0 11 12 26 | 0 1 -2 -2 -6 }}
Optimal tunings:
* WE: ~7/5 = 598.8485{{c}}, ~3/2 = 705.5285{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 707.1826{{c}}


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


[[POTE_tuning|POTE generator]]: ~3/2 = 706.885
{{Optimal ET sequence|legend=0| 10e, 12, 22, 34d, 56d }}


Mapping: [&lt;2 0 11 12 26|, &lt;0 1 -2 -2 -6|]
Badness (Sintel): 0.673


Vals: {{Val list| 10e, 12, 22, 34d, 56d }}
==== 2.3.5.7.11.17 subgroup ====
Subgroup: 2.3.5.7.11.17


Badness: 0.020343
Comma list: 50/49, 64/63, 85/84, 99/98
 
Mapping: {{mapping| 2 0 11 12 26 5 | 0 1 -2 -2 -6 1 }}
 
Optimal tunings:
* WE: ~7/5 = 599.062{{c}}, ~3/2 = 706.095{{c}}
* CWE: ~7/5 = 600.000{{c}}, ~3/2 = 707.370{{c}}
 
{{Optimal ET sequence|legend=0| 10e, 12, 22, 34d, 56d }}
 
Badness (Sintel): 0.645
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


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


Tuning ranges:
Mapping: {{mapping| 2 0 11 12 26 1 | 0 1 -2 -2 -6 2 }}
* valid range: ~3/2 = [700.000, 709.091] (7\12 to 13\22)
 
* nice range: ~3/2 = [701.955, 738.573]
Optimal tunings:  
* strict range: ~3/2 = [701.955, 709.091]
* WE: ~7/5 = 599.9732{{c}}, ~3/2 = 708.8873{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 708.9227{{c}}
 
{{Optimal ET sequence|legend=0| 10e, 12, 22 }}
 
Badness (Sintel): 1.14
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 50/49, 52/51, 64/63, 65/63, 99/98
 
Mapping: {{mapping| 2 0 11 12 26 1 5 | 0 1 -2 -2 -6 2 1 }}
 
Optimal tunings:
* WE: ~7/5 = 599.8871{{c}}, ~3/2 = 708.6725{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 708.8176{{c}}
 
{{Optimal ET sequence|legend=0| 10e, 12, 22 }}
 
Badness (Sintel): 1.06
 
==== Pajarina ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 50/49, 64/63, 78/77, 99/98
 
Mapping: {{mapping| 2 0 11 12 26 36 | 0 1 -2 -2 -6 -9 }}
 
Optimal tunings:
* WE: ~7/5 = 598.7732{{c}}, ~3/2 = 704.6889{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 706.3950{{c}}
 
{{Optimal ET sequence|legend=0| 12f, 22, 34d }}
 
Badness (Sintel): 0.923
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 50/49, 64/63, 78/77, 85/84, 99/98
 
Mapping: {{mapping| 2 0 11 12 26 36 5 | 0 1 -2 -2 -6 -9 1 }}
 
Optimal tunings:
* WE: ~7/5 = 599.0204{{c}}, ~3/2 = 705.2572{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 706.5660{{c}}
 
{{Optimal ET sequence|legend=0| 12f, 22, 34d }}
 
Badness (Sintel): 0.936
 
==== Pajarita ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 40/39, 50/49, 64/63, 66/65
 
Mapping: {{mapping| 2 0 11 12 26 17 | 0 1 -2 -2 -6 -3 }}
 
Optimal tunings:
* WE: ~7/5 = 598.3048{{c}}, ~3/2 = 705.4512{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 707.9238{{c}}
 
{{Optimal ET sequence|legend=0| 10e, 12f, 22f, 34dff }}
 
Badness (Sintel): 0.937
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 40/39, 50/49, 64/63, 66/65, 85/84
 
Mapping: {{mapping| 2 0 11 12 26 17 5 | 0 1 -2 -2 -6 -3 1 }}


POTE generator: ~3/2 = 708.919
Optimal tunings:  
* WE: ~7/5 = 598.6103{{c}}, ~3/2 = 706.3076{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 708.2256{{c}}


Mapping: [&lt;2 0 11 12 26 1|, &lt;0 1 -2 -2 -6 2|]
{{Optimal ET sequence|legend=0| 10e, 12f, 22f }}


Vals: {{Val list| 12, 22 }}
Badness (Sintel): 0.968


Badness: 0.027642
=== Pajarous ===
Subgroup: 2.3.5.7.11


== Pajarous ==
Comma list: 50/49, 55/54, 64/63
Comma list: 50/49, 55/54, 64/63
Mapping: {{mapping| 2 0 11 12 -9 | 0 1 -2 -2 5 }}
Optimal tunings:
* WE: ~7/5 = 599.4055{{c}}, ~3/2 = 708.8747{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 709.5508{{c}}


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


POTE generator: ~3/2 = 709.578
{{Optimal ET sequence|legend=0| 10, 12e, 22, 120bce, 142bce }}


Mapping: [&lt;2 0 11 12 -9|, &lt;0 1 -2 -2 5|]
Badness (Sintel): 0.937


Vals: {{Val list| 10, 12e, 22, 120bce, 142bce }}
==== 2.3.5.7.11.17 subgroup ====
Subgroup: 2.3.5.7.11.13.17


Badness: 0.028349
Comma list: 50/49, 52/51, 55/54, 64/63, 65/63
 
Mapping: {{mapping| 2 0 11 12 -9 1 5 | 0 1 -2 -2 5 2 1 }}
 
Optimal tunings:
* WE: ~7/5 = 599.408{{c}}, ~3/2 = 708.878{{c}}
* CWE: ~7/5 = 600.000{{c}}, ~3/2 = 709.544{{c}}
 
{{Optimal ET sequence|legend=0| 10, 12e, 22 }}
 
Badness (Sintel): 0.766
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


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


POTE generator: ~3/2 = 710.240
Mapping: {{mapping| 2 0 11 12 -9 1 | 0 1 -2 -2 5 2 }}
 
Optimal tunings:
* WE: ~7/5 = 599.9064{{c}}, ~3/2 = 710.1289{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 710.2325{{c}}
 
{{Optimal ET sequence|legend=0| 10, 22 }}
 
Badness (Sintel): 1.04
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 50/49, 52/51, 55/54, 64/63, 65/63
 
Mapping: {{mapping| 2 0 11 12 -9 1 5 | 0 1 -2 -2 5 2 1 }}
 
Optimal tunings:
* WE: ~7/5 = 599.8239{{c}}, ~3/2 = 710.0128{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 710.2067{{c}}


Mapping: [&lt;2 0 11 12 -9 1|, &lt;0 1 -2 -2 5 2|]
{{Optimal ET sequence|legend=0| 10, 22, 54f, 76bdff }}


Vals: {{Val list| 10, 22, 54f, 76bdf }}
Badness (Sintel): 0.930


Badness: 0.025176
==== Pajaro ====
Subgroup: 2.3.5.7.11.13


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


POTE generator ~3/2 = 710.818
Mapping: {{mapping| 2 0 11 12 -9 17 | 0 1 -2 -2 5 -3 }}
 
Optimal tunings:
* WE: ~7/5 = 598.8257{{c}}, ~3/2 = 709.4266{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 710.8414{{c}}
 
{{Optimal ET sequence|legend=0| 10, 22f, 32f }}
 
Badness (Sintel): 1.13
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 40/39, 50/49, 55/54, 64/63, 85/84


Mapping: [&lt;2 0 11 12 -9 17|, &lt;0 1 -2 -2 5 -3|]
Mapping: {{mapping| 2 0 11 12 -9 17 5 | 0 1 -2 -2 5 -3 1 }}


Vals: {{Val list| 10, 22f, 32f, 54f }}
Optimal tunings:  
* WE: ~7/5 = 598.8865{{c}}, ~3/2 = 709.5472{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 710.8704{{c}}


Badness: 0.0274
{{Optimal ET sequence|legend=0| 10, 22f, 32f }}
 
Badness (Sintel): 1.01
 
=== Pajaric ===
Subgroup: 2.3.5.7.11


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


POTE generator: ~3/2 = 705.524
Mapping: {{mapping| 2 0 11 12 7 | 0 1 -2 -2 0 }}
 
Optimal tunings:
* WE: ~7/5 = 597.4807{{c}}, ~3/2 = 702.5616{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 706.0542{{c}}
 
{{Optimal ET sequence|legend=0| 10, 12, 22e }}
 
Badness (Sintel): 0.787
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 40/39, 45/44, 50/49, 56/55
 
Mapping: {{mapping| 2 0 11 12 7 17 | 0 1 -2 -2 0 -3 }}
 
Optimal tunings:
* WE: ~7/5 = 597.1952{{c}}, ~3/2 = 704.1350{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 708.1989{{c}}
 
{{Optimal ET sequence|legend=0| 10, 12f, 22ef }}
 
Badness (Sintel): 0.845
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 34/33, 40/39, 45/44, 50/49, 56/55
 
Mapping: {{mapping| 2 0 11 12 7 17 5 | 0 1 -2 -2 0 -3 1 }}
 
Optimal tunings:
* WE: ~7/5 = 597.6509{{c}}, ~3/2 = 705.7702{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 708.9719{{c}}
 
{{Optimal ET sequence|legend=0| 10, 12f, 22ef }}
 
Badness (Sintel): 0.896
 
=== Hemipaj ===
Subgroup: 2.3.5.7.11
 
Comma list: 50/49, 64/63, 121/120
 
Mapping: {{mapping| 2 1 9 10 8 | 0 2 -4 -4 -1 }}
 
: mapping generators: ~2, ~16/11
 
Optimal tunings:
* WE: ~7/5 = 597.6509{{c}}, ~16/11 = 652.7788{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~16/11 = 653.7119{{c}}
 
{{Optimal ET sequence|legend=0| 2, 20, 22 }}
 
Badness (Sintel): 1.29
 
=== Hemifourths ===
Subgroup: 2.3.5.7.11
 
Comma list: 50/49, 64/63, 243/242
 
Mapping: {{mapping| 2 0 11 12 -1 | 0 2 -4 -4 5 }}
: mapping generators: ~2, ~55/32
 
Optimal tunings:
* WE: ~7/5 = 597.6509{{c}}, ~55/32 = 950.8475{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~55/32 = 953.1172{{c}}
 
{{Optimal ET sequence|legend=0| 10, 24d, 34d }}
 
Badness (Sintel): 1.62
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 50/49, 64/63, 78/77, 144/143
 
Mapping: {{mapping| 2 0 11 12 -1 9 | 0 2 -4 -4 5 -1 }}
 
Optimal tunings:
* WE: ~7/5 = 598.6748{{c}}, ~26/15 = 950.9691{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~26/15 = 953.1052{{c}}
 
{{Optimal ET sequence|legend=0| 10, 24d, 34d }}
 
Badness (Sintel): 1.19
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 50/49, 64/63, 78/77, 85/84, 144/143
 
Mapping: {{mapping| 2 0 11 12 -1 9 5 | 0 2 -4 -4 5 -1 2 }}
 
Optimal tunings:
* WE: ~7/5 = 598.8411{{c}}, ~26/15 = 951.3687{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~26/15 = 953.2169{{c}}
 
{{Optimal ET sequence|legend=0| 10, 24d, 34d }}
 
Badness (Sintel): 1.11
 
== Srutal ==
{{See also| Srutal vs diaschismic }}
 
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]].
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2048/2025, 4375/4374
 
{{Mapping|legend=1| 2 0 11 -42 | 0 1 -2 15 }}
 
[[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 }}
 
[[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]
 
{{Optimal ET sequence|legend=1| 34d, 46, 80, 126, 206cd, 332bcd }}
 
[[Badness]] (Sintel): 2.32
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 176/175, 896/891, 1331/1323
 
Mapping: {{mapping| 2 0 11 -42 -28 | 0 1 -2 15 11 }}
 
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 12 7|, &lt;0 1 -2 -2 0|]
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]


Vals: {{Val list| 10, 12, 22e, 34de }}
{{Optimal ET sequence|legend=0| 34d, 46, 80, 126, 206cd }}


Badness: 0.023798
Badness (Sintel): 1.17


=== 13-limit ===
=== 13-limit ===
Comma list: 40/39, 45/44, 50/49, 56/55
Subgroup: 2.3.5.7.11.13
 
Comma list: 169/168, 176/175, 325/324, 364/363
 
Mapping: {{mapping| 2 0 11 -42 -28 -18 | 0 1 -2 15 11 8 }}
 
Optimal tunings:
* WE: ~45/32 = 599.5490{{c}}, ~3/2 = 704.3516{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 704.8347{{c}}
 
Tuning ranges:
* 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]
 
{{Optimal ET sequence|legend=0| 34d, 46, 80 }}
 
Badness (Sintel): 1.04
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 136/135, 169/168, 176/175, 221/220, 256/255
 
Mapping: {{mapping| 2 0 11 -42 -28 -18 5 | 0 1 -2 15 11 8 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}}
 
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]
 
{{Optimal ET sequence|legend=0| 34d, 46, 80, 126 }}
 
Badness (Sintel): 0.947
 
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 136/135, 169/168, 176/175, 190/189, 221/220, 256/255
 
Mapping: {{mapping| 2 0 11 -42 -28 -18 5 -55 | 0 1 -2 15 11 8 1 20 }}
 
Optimal tunings:
* WE: ~17/12 = 599.6371{{c}}, ~3/2 = 704.4790{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 704.8745{{c}}
 
{{Optimal ET sequence|legend=0| 34dh, 46, 80 }}
 
Badness (Sintel): 1.04
 
==== Srutaloo ====
Srutaloo adds 576/575, 736/729 or 208/207, and rhymes with [[skidoo]].
 
Subgroup: 2.3.5.7.11.13.17.19.23
 
Comma list: 136/135, 169/168, 176/175, 190/189, 208/207, 221/220, 256/255
 
Mapping: {{mapping| 2 0 11 -42 -28 -18 5 -55 -10 | 0 1 -2 15 11 8 1 20 6 }}
 
Optimal tunings:
* WE: ~17/12 = 599.6690{{c}}, ~3/2 = 704.5098{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 704.8713{{c}}
 
{{Optimal ET sequence|legend=0| 34dh, 46, 80 }}
 
Badness (Sintel): 0.971
 
===== 29-limit =====
Subgroup: 2.3.5.7.11.13.17.19.23.29
 
Comma list: 136/135, 169/168, 176/175, 190/189, 208/207, 221/220, 232/231, 256/255
 
Mapping: {{mapping| 2 0 11 -42 -28 -18 5 -55 -10 -76 | 0 1 -2 15 11 8 1 20 6 27 }}
 
Optimal tunings:
* WE: ~17/12 = 599.6664{{c}}, ~3/2 = 704.5138{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 704.8807{{c}}
 
{{Optimal ET sequence|legend=0| 34dhj, 46, 80 }}
 
Badness (Sintel): 1.10
 
===== 31-limit =====
Subgroup: 2.3.5.7.11.13.17.19.23.29.31
 
Comma list: 136/135, 169/168, 176/175, 190/189, 208/207, 217/216, 221/220, 232/231, 256/255
 
Mapping: {{mapping| 2 0 11 -42 -28 -18 5 -55 -10 -76 48 | 0 1 -2 15 11 8 1 20 6 27 -12 }}
 
Optimal tunings:
* WE: ~17/12 = 599.8115{{c}}, ~3/2 = 704.5958{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 704.8086{{c}}
 
{{Optimal ET sequence|legend=0| 46, 80, 126 }}
 
Badness (Sintel): 1.44
 
== 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.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 875/864, 2048/2025
 
{{Mapping|legend=1| 2 0 11 -23 | 0 1 -2 9 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~45/32 = 599.6603{{c}}, ~3/2 = 707.1707{{c}}
: [[error map]]: {{val| -0.679 +4.536 -3.033 -2.591 }}
* [[CWE]]: ~45/32 = 600.0000{{c}}, ~3/2 = 707.5294{{c}}
: error map: {{val| 0.000 +5.574 -1.373 -1.061 }}
 
{{Optimal ET sequence|legend=1| 22, 56, 78, 134b }}
 
[[Badness]] (Sintel): 2.13
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 100/99, 385/384, 1232/1215
 
Mapping: {{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}}


POTE generator: ~3/2 = 707.442
{{Optimal ET sequence|legend=0| 22fh, 46, 68, 114 }}


Map: [&lt;2 0 11 12 7 17|, &lt;0 1 -2 -2 0 -3|]
Badness (Sintel): 1.03


Vals: {{Val list| 10, 12f, 22ef, 34def }}
== 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.


Badness: 0.0205
[[Subgroup]]: 2.3.5.7


== Hemipaj ==
[[Comma list]]: 392/375, 1323/1280
Commas: 50/49, 64/63, 121/120


POTE generator: ~11/8 = 546.383
{{Mapping|legend=1| 2 1 9 11 | 0 2 -4 -5 }}
: mapping generators: ~45/32, ~10/7


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


Vals: {{Val list| 20, 22, 68d, 90d }}
{{Optimal ET sequence|legend=1| 2, 22d, 24 }}


Badness: 0.038890
[[Badness]] (Sintel): 3.99


= Diaschismic =
=== 11-limit ===
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.
Subgroup: 2.3.5.7.11


Diaschismic extends naturally to the 17-limit, for which the same tunings may be used, making it one of the most important of the higher limit rank two temperaments. Adding the 11-limit adds the commas 176/175, 896/891 and 441/440. The 13-limit yields 196/195, 351/350, and 364/363. The 17-limit adds 136/135, 221/220, and 442/441. If you want to explore higher limit harmonies, diaschismic is certainly one excellent way to do it; MOS of 34 notes and even more the 46 note MOS will encompass very great deal of it. Of course 46 or 58 equal provide alternatives which in many ways are similar, particularly in the case of 58.
Comma list: 56/55, 77/75, 1323/1280


[[Comma list]]: 126/125, 2048/2025
Mapping: {{mapping| 2 1 9 11 8 | 0 2 -4 -5 -1 }}


[[POTE_tuning|POTE generator]]: ~3/2 = 703.681
Optimal tunings:
* WE: ~17/12 = 600.2356{{c}}, ~10/7 = 650.3856{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~10/7 = 650.1008{{c}}


[[Mapping]]: [&lt;2 0 11 31|, &lt;0 1 -2 -8|]
{{Optimal ET sequence|legend=0| 2, 22d, 24 }}


{{Val list|legend=1| 12, 46, 58, 104c, 162c }}
Badness (Sintel): 2.10


[[Badness]]: 0.037914
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


== 11-limit ==
Comma list: 56/55, 77/75, 105/104, 507/500
Comma list: 126/125, 176/175, 896/891


POTE generator: ~3/2 = 703.714
Mapping: {{mapping| 2 1 9 11 8 15 | 0 2 -4 -5 -1 -7 }}


Mapping: [&lt;2 0 11 31 45|, &lt;0 1 -2 -8 -12|]
Optimal tunings:
* WE: ~45/32 = 599.9067{{c}}, ~10/7 = 649.4907{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~10/7 = 649.5950{{c}}


Vals: {{Val list| 12, 46, 58, 104c, 162ce }}
{{Optimal ET sequence|legend=0| 2, 24 }}


Badness: 0.025034
Badness (Sintel): 2.12


== 13-limit ==
== Sruti ==
Comma list: 126/125, 196/195, 364/363, 2048/2025
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.


POTE generator: ~3/2 = 703.704
[[Subgroup]]: 2.3.5.7


Mapping: [&lt;2 0 11 31 45 55|, &lt;0 1 -2 -8 -12 -15|]
[[Comma list]]: 2048/2025, 19683/19600


Vals: {{Val list| 46, 58, 104c, 162cef }}
{{Mapping|legend=1| 2 0 11 -15 | 0 2 -4 13 }}
: mapping generators: ~45/32, ~140/81


Badness: 0.018926
[[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 }}


== 17-limit ==
{{Optimal ET sequence|legend=1| 24, 34d, 58, 150cd, 208ccdd, 266ccdd }}
Comma list: 126/125, 136/135, 176/175, 196/195, 256/255


POTE generator: ~3/2 = 703.812
[[Badness]] (Sintel): 2.97


Mapping: [&lt;2 0 11 31 45 55 5|, &lt;0 1 -2 -8 -12 -15 1|]
=== 11-limit ===
Subgroup: 2.3.5.7.11


Vals: {{Val list| 46, 58, 104c }}
Comma list: 176/175, 243/242, 896/891


Badness: 0.016425
Mapping: {{mapping| 2 0 11 -15 -1 | 0 2 -4 13 5 }}


= Keen =
Optimal tunings:
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.
* WE: ~45/32 = 599.1951{{c}}, ~121/70 = 950.5864{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~121/70 = 951.7972{{c}}


Commas: 875/864, 2048/2025
{{Optimal ET sequence|legend=0| 24, 34d, 58, 150cdee, 208ccddee, 266ccddeee }}


[[POTE_tuning|POTE generator]]: 707.571
Badness (Sintel): 1.37


Map: [&lt;2 0 11 -23|, &lt;0 1 -2 9|]
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


EDOs: 22, 56, 78, 134b, 212b, 290b
Comma list: 144/143, 176/175, 351/350, 676/675


Badness: 0.083971
Mapping: {{mapping| 2 0 11 -15 -1 9 | 0 2 -4 13 5 -1 }}


== 11-limit ==
Optimal tunings:
Commas: 100/99, 385/384, 1232/1215
* WE: ~45/32 = 599.1479{{c}}, ~26/15 = 950.5337{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~26/15 = 951.8314{{c}}


[[POTE_tuning|POTE generator]]: 707.609
{{Optimal ET sequence|legend=0| 24, 34d, 58, 150cdeef, 208ccddeeff, 266ccddeeefff }}


Map: [&lt;2 0 11 -23 26|, &lt;0 1 -2 9 -6|]
Badness (Sintel): 0.983


EDOs: 22, 56, 78, 212bf, 290bf
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Badness: 0.045270
Comma list: 136/135, 144/143, 170/169, 176/175, 221/220


= Bidia =
Mapping: {{mapping| 2 0 11 -15 -1 9 5 | 0 2 -4 13 5 -1 2 }}
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
Optimal tunings:
* WE: ~17/12 = 599.3003{{c}}, ~26/15 = 950.7465{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~26/15 = 951.8142{{c}}


POTE generator: ~3/2 = 705.364
{{Optimal ET sequence|legend=0| 24, 34d, 58 }}


Map: [&lt;4 0 22 43|,&lt;0 1 -2 -5|]
Badness (Sintel): 1.05


Wedgie: &lt;&lt;4 -8 -20 -22 -43 -24||
== 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.


EDOs: 12, 56, 68, 80, 148d
[[Subgroup]]: 2.3.5.7


Badness: 0.0565
[[Comma list]]: 49/48, 2048/2025


== 11-limit ==
{{Mapping|legend=1| 2 0 11 4 | 0 2 -4 1 }}
Commas: 176/175, 896/891, 1375/1372
: mapping generators: ~45/32, ~7/4


POTE generator: ~3/2 = 705.087
[[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 }}


Map: [&lt;4 0 22 43 71|,&lt;0 1 -2 -5 -9|]
{{Optimal ET sequence|legend=1| 10, 24, 34 }}


EDOs: 12, 68, 80
[[Badness]] (Sintel): 1.97


Badness: 0.0402
=== 11-limit ===
Subgroup: 2.3.5.7.11


== 13-limit ==
Comma list: 49/48, 56/55, 243/242
Commas: 176/175, 325/324, 640/637, 896/891


POTE generator: ~3/2 = 705.301
Mapping: {{mapping| 2 0 11 4 -1 | 0 2 -4 1 5 }}


Map: [&lt;4 0 22 43 71|,&lt;0 1 -2 -5 -9|]
Optimal tunings:
* WE: ~45/32 = 599.9250{{c}}, ~7/4 = 952.0646{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~7/4 = 952.1784{{c}}


EDOs: 12, 68, 80, 148d, 228bcd, 376bcdf
{{Optimal ET sequence|legend=0| 10, 24, 34 }}


Badness: 0.0411
Badness (Sintel): 1.63


= Echidna =
=== 13-limit ===
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|.
Subgroup: 2.3.5.7.11.13


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.
Comma list: 49/48, 56/55, 91/90, 243/242


Commas: 1728/1715, 2048/2025
Mapping: {{mapping| 2 0 11 4 -1 9 | 0 2 -4 1 5 -1 }}


[[POTE_tuning|POTE generator]]: 434.856
Optimal tunings:
* WE: ~45/32 = 599.7575{{c}}, ~7/4 = 951.9241{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~7/4 = 952.2980{{c}}


Map: [&lt;2 1 9 2|, &lt;0 3 -6 5|]
{{Optimal ET sequence|legend=0| 10, 24, 34, 58d, 92ddef }}


EDOs: 22, 58, 80, 138cd, 218cd
Badness (Sintel): 1.27


Badness: 0.0580
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


== 11-limit ==
Comma list: 49/48, 56/55, 91/90, 119/117, 154/153
Commas: 176/175, 540/539, 896/891


11-limit minimax
Mapping: {{mapping| 2 0 11 4 -1 9 5 | 0 2 -4 1 5 -1 2 }}


[|1 0 0 0 0&gt;, |7/4 0 0 1/4 -1/4&gt;, |2 0 0 -1/2 1/2&gt;,
Optimal tunings:
|37/12 0 0 5/12 -5/12&gt;, |37/12 0 0 -7/12 7/12&gt;<nowiki>]</nowiki>
* WE: ~17/12 = 599.7925{{c}}, ~7/4 = 952.0004{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~7/4 = 952.3178{{c}}


Eigenmonzos: 2, 11/7
{{Optimal ET sequence|legend=0| 10, 24, 34 }}


Minimax generator: (224/11)^(1/12) = 434.792
Badness (Sintel): 1.10


[[POTE_tuning|POTE generator]]: 434.852
== 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.  


Map: [&lt;2 1 9 2 12|, &lt;0 3 -6 5 -7|]
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.


EDOs: 22, 58, 80, 138cde, 218cde
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.


Badness: 0.0260
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]].


== 13-limit ==
[[Subgroup]]: 2.3.5.7
Commas: 176/175, 351/350, 364/363, 540/539


[[POTE_tuning|POTE generator]]: 434.756
[[Comma list]]: 1728/1715, 2048/2025


Map: [&lt;2 1 9 2 12 19|, &lt;0 3 -6 5 -7 -16|]
{{Mapping|legend=1| 2 1 9 2 | 0 3 -6 5 }}
: mapping generators: ~45/32, ~9/7


EDOs: 22, 58, 80, 138cde
[[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 }}


Badness: 0.0237
{{Optimal ET sequence|legend=1| 22, 58, 80, 138cd, 218cd }}


== 17-limit ==
[[Badness]] (Sintel): 1.47
Commas: 136/135, 176/175, 221/220, 256/255, 540/539


[[POTE_tuning|POTE generator]]: 434.816
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;2 1 9 2 12 19 6|, &lt;0 3 -6 5 -7 -16 3|]
Comma list: 176/175, 540/539, 896/891


EDOs: 22, 58, 80, 138cde
Mapping: {{mapping| 2 1 9 2 12 | 0 3 -6 5 -7 }}


Badness: 0.0203
Optimal tunings:
* WE: ~45/32 = 599.3085{{c}}, ~9/7 = 434.3511{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~9/7 = 434.8647{{c}}


= Echidnic =
Minimax tuning:
Commas: 686/675, 1029/1024
* 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


[[POTE_tuning|POTE generator]]: 234.492
{{Optimal ET sequence|legend=0| 22, 58, 80, 138cde, 218cde }}


Map: [&lt;2 2 7 6|, &lt;0 3 -6 -1|]
Badness (Sintel): 0.859


EDOs: 10, 36, 46, 194bcd, 240bcd, 286bcd, 332bcd
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Badness: 0.0722
Comma list: 176/175, 351/350, 364/363, 540/539


== 11-limit ==
Mapping: {{mapping| 2 1 9 2 12 19 | 0 3 -6 5 -7 -16 }}
Commas: 385/384, 441/440, 686/675


[[POTE_tuning|POTE generator]]: 235.096
Optimal tunings:
* WE: ~45/32 = 599.3397{{c}}, ~9/7 = 434.2772{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~9/7 = 434.7864{{c}}


Map: [&lt;2 2 7 6 3|, &lt;0 3 -6 -1 10|]
{{Optimal ET sequence|legend=0| 22, 36f, 58, 80, 138cde }}


EDOs: 10, 46, 102, 148, 342bcd
Badness (Sintel): 0.978


Badness: 0.0451
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


== 13-limit ==
Comma list: 136/135, 176/175, 221/220, 256/255, 540/539
Commas: 91/90, 169/168, 385/384, 441/440


[[POTE_tuning|POTE generator]]: 235.088
Mapping: {{mapping| 2 1 9 2 12 19 6 | 0 3 -6 5 -7 -16 3 }}


Map: [&lt;2 2 7 6 3 7|, &lt;0 3 -6 -1 10 1|]
Optimal tunings:
* WE: ~45/32 = 599.4645{{c}}, ~9/7 = 434.4282{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~9/7 = 434.8340{{c}}


EDOs: 10, 46, 102, 148f, 194bcdf
{{Optimal ET sequence|legend=0| 22, 36f, 58, 80, 138cde }}


Badness: 0.0289
Badness (Sintel): 1.03


Compositions:
== 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.


http://untwelve.org/2011competition_audio/Kosmorsky-A_Stiff_Shot_of_Turpentine.mp3
[[Subgroup]]: 2.3.5.7


(the description says "lemba" which has a similar scale structure but different mapping for 5)
[[Comma list]]: 686/675, 1029/1024


= Shrutar =
{{Mapping|legend=1| 2 2 7 6 | 0 3 -6 -1 }}
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]] makes for a good tuning, but another and excellent choice is a generator of 14^(1/7), making 7s just.
: mapping generators: ~45/32, ~8/7


By adding 121/120 or 176/175 to the commas, shrutar can be extended to the 11-limit, which loses a bit of accuracy, but picks up low-complexity 11-limit harmony, making shrutar quite an interesting 11-limit system. 68, 114 or a 14^(1/7) generator can again be used as tunings.
[[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 }}


Commas: 245/243, 2048/2025
{{Optimal ET sequence|legend=1| 10, 26c, 36, 46 }}


[[POTE_tuning|POTE generator]]: 52.811
[[Badness]] (Sintel): 1.83


Map: [&lt;2 1 9 -2|, &lt;0 2 -4 7|]
=== 11-limit ===
Subgroup: 2.3.5.7.11


EDOs: 22, 46, 68, 182b, 250bc
Comma list: 385/384, 441/440, 686/675


== 11-limit ==
Mapping: {{mapping| 2 2 7 6 3 | 0 3 -6 -1 10 }}
Commas: 121/120, 176/175, 245/243


[[POTE_tuning|POTE generator]]: 52.680
Optimal tunings:
* WE: ~45/32 = 599.8022{{c}}, ~8/7 = 235.0185{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~8/7 = 235.0893{{c}}


Map: [&lt;2 1 9 -2 8|, &lt;0 2 -4 7 -1|]
{{Optimal ET sequence|legend=0| 10, 36e, 46, 102, 148 }}


EDOs: 22, 46, 68, 114, 296bce, 410bce
Badness (Sintel): 1.49


== 13-limit ==
=== 13-limit ===
Commas: 121/120, 176/175, 196/195, 245/243
Subgroup: 2.3.5.7.11.13


POTE generator: ~28/27 = 52.654
Comma list: 91/90, 169/168, 385/384, 441/440


Map: [&lt;2 1 9 -2 8 -10|, &lt;0 2 -4 7 -1 16|]
Mapping: {{mapping| 2 2 7 6 3 7 | 0 3 -6 -1 10 1 }}


EDOs: 22, 24, 46, 68, 114
Optimal tunings:
* WE: ~45/32 = 599.9570{{c}}, ~8/7 = 235.0708{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~8/7 = 235.0862{{c}}


Badness: 0.0281
{{Optimal ET sequence|legend=0| 10, 36e, 46, 102, 148f }}


== 17-limit ==
Badness (Sintel): 1.19
Commas: 121/120, 136/135, 154/153, 176/175, 196/195


POTE generator: ~28/27 = 52.647
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Map: [&lt;2 1 9 -2 8 -10 6|, &lt;0 2 -4 7 -1 16 2|]
Comma list: 91/90, 136/135, 154/153, 169/168, 256/255


EDOs: 22, 24, 46, 68, 114
Mapping: {{mapping| 2 2 7 6 3 7 7 | 0 3 -6 -1 10 1 3 }}


Badness: 0.0187
Optimal tunings:
* WE: ~17/12 = 599.9571{{c}}, ~8/7 = 235.0709{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~8/7 = 235.0860{{c}}


== 19-limit ==
{{Optimal ET sequence|legend=0| 10, 36e, 46, 102, 148f }}
Commas: 121/120, 136/135, 154/153, 176/175, 196/195, 343/342


POTE generator: ~28/27 = 52.730
Badness (Sintel): 0.983


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


EDOs: 22, 24, 46, 68, 114, 182bef
== 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.


Badness: 0.0175
[[Subgroup]]: 2.3.5.7


= Sruti =
[[Comma list]]: 2048/2025, 2401/2400
Commas: 2048/2025, 19683/19600


POTE generator: ~175/144 = 351.876
{{Mapping|legend=1| 2 0 11 8 | 0 4 -8 -3 }}
: mapping generators: ~45/32, ~21/16


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


Wedgie: &lt;&lt;4 -8 26 -22 30 83||
{{Optimal ET sequence|legend=1| 10, …, 58, 68, 126, 446bbccd }}


EDOs: 24, 34d, 58, 150cd, 208cd, 266cd
[[Badness]] (Sintel): 1.86


Badness: 0.1174
=== 11-limit ===
Subgroup: 2.3.5.7.11


== 11-limit ==
Comma list: 176/175, 896/891, 2401/2400
Commas: 176/175, 243/242, 896/891


POTE generator: ~11/9 = 351.863
Mapping: {{mapping| 2 0 11 8 22 | 0 4 -8 -3 -19 }}


Map: [&lt;2 0 11 -15 -1|, &lt;0 2 -4 13 5|]
Optimal tunings:
* WE: ~45/32 = 599.4648{{c}}, ~21/16 = 475.6929{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~21/16 = 476.1507{{c}}


EDOs: 24, 34d, 58, 150cde, 208cde
{{Optimal ET sequence|legend=0| 10e, …, 58, 126, 184c, 310bccde }}


Badness: 0.0415
Badness (Sintel): 1.62


== 13-limit ==
==== 13-limit ====
Commas: 144/143, 176/175, 351/350, 676/675
Subgroup: 2.3.5.7.11.13


POTE generator: ~11/9 = 351.886
Comma list: 176/175, 196/195, 512/507, 676/675


Map: [&lt;2 0 11 -15 -1 9|, &lt;0 2 -4 13 5 -1|]
Mapping: {{mapping| 2 0 11 8 22 9 | 0 4 -8 -3 -19 -2 }}


EDOs: 24, 34d, 58, 150cdef, 208cdef
Optimal tunings:
* WE: ~45/32 = 599.3787{{c}}, ~21/16 = 475.6065{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~21/16 = 476.1345{{c}}


Badness: 0.0238
{{Optimal ET sequence|legend=0| 10e, …, 58, 126f, 184cff }}


= Anguirus =
Badness (Sintel): 1.18
Commas: 49/48, 2048/2025


POTE generator: ~8/7 = 246.979
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


Map: [&lt;2 0 11 4|, &lt;0 2 -4 1|]
Comma list: 136/135, 170/169, 176/175, 196/195, 256/255


Wedgie: &lt;&lt;4 -8 2 -22 -8 27||
Mapping: {{mapping| 2 0 11 8 22 9 5 | 0 4 -8 -3 -19 -2 4 }}


EDOs: 10, 24, 34
Optimal tunings:
* WE: ~17/12 = 599.5077{{c}}, ~21/16 = 475.7713{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~21/16 = 476.1814{{c}}


Badness: 0.0780
{{Optimal ET sequence|legend=0| 10e, 58, 126f }}


== 11-limit ==
Badness (Sintel): 1.21
Commas: 49/48, 56/55, 243/242


POTE generator: ~8/7 = 247.816
=== Quadrafourths ===
Subgroup: 2.3.5.7.11


Map: [&lt;2 0 11 4 -1|, &lt;0 2 -4 1 5|]
Comma list: 243/242, 441/440, 2048/2025


EDOs: 10, 24, 34, 58d, 92de
Mapping: {{mapping| 2 0 11 8 -1 | 0 4 -8 -3 10 }}


Badness: 0.0493
Optimal tunings:
* WE: ~45/32 = 599.2593{{c}}, ~21/16 = 475.4292{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~21/16 = 476.0088{{c}}


== 13-limit ==
{{Optimal ET sequence|legend=0| 10, 48c, 58, 184cee, 242ccdeee }}
Commas: 49/48, 56/55, 91/90, 352/351


POTE generator: ~8/7 = 247.691
Badness (Sintel): 1.62


Map: [&lt;2 0 11 4 -1 9|, &lt;0 2 -4 1 5 -1|]
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


EDOs: 10, 24, 34, 58d, 92def
Comma list: 144/143, 196/195, 243/242, 676/675


Badness: 0.0308
Mapping: {{mapping| 2 0 11 8 -1 9 | 0 4 -8 -3 10 -2 }}


= Shru =
Optimal tunings:
Commas: 392/375, 1323/1280
* WE: ~45/32 = 599.2147{{c}}, ~21/16 = 475.4052{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~21/16 = 476.0253{{c}}


POTE generator: ~64/63 = 50.135
{{Optimal ET sequence|legend=0| 10, 48c, 58, 126eef, 184ceeff, 242ccdeeeff }}


Map: [&lt;2 1 9 11|, &lt;0 2 -4 -5|]
Badness (Sintel): 1.11


Wedgie: &lt;&lt;4 -8 -10 -22 -27 -1||
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


EDOs: 22d, 24
Comma list: 136/135, 144/143, 170/169, 196/195, 221/220


Badness: 0.1576
Mapping: {{mapping| 2 0 11 8 -1 9 5 | 0 4 -8 -3 10 -2 4 }}


== 11-limit ==
Optimal tunings:
Commas: 56/55, 77/75, 1323/1280
* WE: ~17/12 = 599.3353{{c}}, ~21/16 = 475.5495{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~21/16 = 476.0691{{c}}


POTE generator: ~64/63 = 50.130
{{Optimal ET sequence|legend=0| 10, 48c, 58 }}


Map: [&lt;2 1 9 11 8|, &lt;0 2 -4 -5 -1|]
Badness (Sintel): 1.13


EDOs: 22d, 24
== Subgroup extensions ==
=== Srutal archagall (2.3.5.17) ===
{{See also | Fiventeen }}


Badness: 0.0635
Subgroup: 2.3.5.17


== 13-limit ==
Comma list: 136/135, 256/255
Commas: 56/55, 77/75, 105/104, 507/500


POTE generator: ~64/63 = 50.535
Subgroup-val mapping: {{mapping| 2 0 11 5 | 0 1 -2 1 }}
: mapping generators: ~17/12, ~3


Map: [&lt;2 1 9 11 8 15|, &lt;0 2 -4 -5 -1 -7|]
Optimal tunings:  
* WE: ~45/32 = 599.5585{{c}}, ~3/2 = 704.6188{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 705.1356{{c}}


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


Badness: 0.0457
Badness (Sintel): 0.212


[[Category:Theory]]
[[Category:Temperament families]]
[[Category:Temperament family]]
[[Category:Diaschismic family| ]] <!-- main article -->
[[Category:Diaschismic]]
[[Category:Rank 2]]
[[Category:Rank 2]]
[[Category:Todo:Add definition]]
[[Category:Todo:Review]]

Latest revision as of 18:39, 2 June 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The diaschismic family of temperaments tempers out the diaschisma, 2048/2025.

Diaschismic

Main article: Diaschismic

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.

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

Subgroup: 2.3.5

Comma list: 2048/2025

Mapping: [⟨2 0 11], ⟨0 1 -2]]

mapping generators: ~45/32, ~3

Optimal tunings:

  • WE: ~45/32 = 599.4107 ¢, ~3/2 = 704.2059 ¢
error map: ⟨-1.179 +1.072 +1.150]
  • CWE: ~45/32 = 600.0000 ¢, ~3/2 = 704.9585 ¢
error map: ⟨0.000 +3.003 +3.769]

Tuning ranges:

Optimal ET sequence: 10, 12, 22, 34, 46, 80, 206c, 286bc

Badness (Sintel): 0.467

Overview to extensions

7-limit extensions

To get the 7-limit extensions, we add another comma:

  • Septimal diaschismic adds 126/125, the starling comma, to obtain 7-limit harmony by more complex methods than pajara, but with greater accuracy.
  • Pajara adds 50/49 or 64/63 and is a popular and well-known choice.
  • Srutal adds 4375/4374, the ragisma, which is about as accurate as septimal diaschismic but has a much more complex mapping of 7.
  • Keen adds 875/864.

Those all keep the same half-octave period and fifth generator.

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.

Subgroup extensions

Since the diaschisma factors into (256/255)2(289/288) in the 17-limit, it extends naturally to the 2.3.5.17 subgroup as srutal archagall, considered in #Subgroup extensions. The S-expression-based comma list of this temperament is {S16, S17}.

Septimal diaschismic

Main article: Diaschismic
See also: Srutal vs diaschismic

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 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.

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.

Subgroup: 2.3.5.7

Comma list: 126/125, 2048/2025

Mapping: [⟨2 0 11 31], ⟨0 1 -2 -8]]

Optimal tunings:

  • WE: ~45/32 = 599.4449 ¢, ~3/2 = 703.0299 ¢
error map: ⟨-1.110 -0.035 +3.740 -1.391]
  • CWE: ~45/32 = 600.0000 ¢, ~3/2 = 703.7739 ¢
error map: ⟨0.000 +1.819 +6.138 +0.983]

Tuning ranges:

Optimal ET sequence: 12, 34, 46, 58, 104c, 162c

Badness (Sintel): 0.959

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨2 0 11 31 45], ⟨0 1 -2 -8 -12]]

Optimal tunings:

  • WE: ~45/32 = 599.4471 ¢, ~3/2 = 703.0657 ¢
  • CWE: ~45/32 = 600.0000 ¢, ~3/2 = 703.7996 ¢

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]

Optimal ET sequence: 12, 34e, 46, 58, 104c, 162ce

Badness (Sintel): 0.828

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨2 0 11 31 45 55], ⟨0 1 -2 -8 -12 -15]]

Optimal tunings:

  • WE: ~45/32 = 599.4451 ¢, ~3/2 = 703.0528 ¢
  • CWE: ~45/32 = 600.0000 ¢, ~3/2 = 703.7813 ¢

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]

Optimal ET sequence: 12f, 34ef, 46, 58, 104c, 162cef

Badness (Sintel): 0.782

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 126/125, 136/135, 176/175, 196/195, 256/255

Mapping: [⟨2 0 11 31 45 55 5], ⟨0 1 -2 -8 -12 -15 1]]

Optimal tunings:

  • WE: ~17/12 = 599.6253 ¢, ~3/2 = 703.3726 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~3/2 = 703.8520 ¢

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]

Optimal ET sequence: 12f, 34ef, 46, 58, 104c

Badness (Sintel): 0.837

2.3.5.7.11.13.17.23 subgroup (Na"Naa')

Na"Naa' is a remarkable subgroup temperament of 46 & 58 with a prime harmonic of 23. It is yet to be found why it got this strange name.

Subgroup: 2.3.5.7.11.13.17.23

Comma list: 126/125, 136/135, 176/175, 196/195, 231/230, 256/255

Subgroup-val mapping: [⟨2 0 11 31 45 55 5 63], ⟨0 1 -2 -8 -12 -15 1 -17]]

Optimal tunings:

  • WE: ~17/12 = 599.6272 ¢, ~3/2 = 703.4326 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~3/2 = 703.9093 ¢

Optimal ET sequence: 12i, 34efi, 46, 58i, 104ci

Badness (Sintel): 0.882

Pajara

Main article: Pajara

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

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

Subgroup: 2.3.5.7

Comma list: 50/49, 64/63

Mapping: [⟨2 0 11 12], ⟨0 1 -2 -2]]

Optimal tunings:

  • WE: ~7/5 = 598.8483 ¢, ~3/2 = 705.6906 ¢
error map: ⟨-2.303 +1.432 -5.756 +10.580]
  • CWE: ~7/5 = 600.0000 ¢, ~3/2 = 707.3438 ¢
error map: ⟨0.000 +5.389 -1.001 +16.487]

Tuning ranges:

Optimal ET sequence: 10, 12, 22, 34d, 56d

Badness (Sintel): 0.507

2.3.5.7.17 subgroup

Subgroup: 2.3.5.7.17

Comma list: 50/49, 64/63, 85/84

Mapping: [⟨2 0 11 12 5], ⟨0 1 -2 -2 1]]

Optimal tunings:

  • WE: ~7/5 = 599.053 ¢, ~3/2 = 706.355 ¢
  • CWE: ~7/5 = 600.000 ¢, ~3/2 = 707.607 ¢

Optimal ET sequence: 10, 12, 22, 56d

Badness (Sintel): 0.438

11-limit

Subgroup: 2.3.5.7.11

Comma list: 50/49, 64/63, 99/98

Mapping: [⟨2 0 11 12 26], ⟨0 1 -2 -2 -6]]

Optimal tunings:

  • WE: ~7/5 = 598.8485 ¢, ~3/2 = 705.5285 ¢
  • CWE: ~7/5 = 600.0000 ¢, ~3/2 = 707.1826 ¢

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]

Optimal ET sequence: 10e, 12, 22, 34d, 56d

Badness (Sintel): 0.673

2.3.5.7.11.17 subgroup

Subgroup: 2.3.5.7.11.17

Comma list: 50/49, 64/63, 85/84, 99/98

Mapping: [⟨2 0 11 12 26 5], ⟨0 1 -2 -2 -6 1]]

Optimal tunings:

  • WE: ~7/5 = 599.062 ¢, ~3/2 = 706.095 ¢
  • CWE: ~7/5 = 600.000 ¢, ~3/2 = 707.370 ¢

Optimal ET sequence: 10e, 12, 22, 34d, 56d

Badness (Sintel): 0.645

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨2 0 11 12 26 1], ⟨0 1 -2 -2 -6 2]]

Optimal tunings:

  • WE: ~7/5 = 599.9732 ¢, ~3/2 = 708.8873 ¢
  • CWE: ~7/5 = 600.0000 ¢, ~3/2 = 708.9227 ¢

Optimal ET sequence: 10e, 12, 22

Badness (Sintel): 1.14

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨2 0 11 12 26 1 5], ⟨0 1 -2 -2 -6 2 1]]

Optimal tunings:

  • WE: ~7/5 = 599.8871 ¢, ~3/2 = 708.6725 ¢
  • CWE: ~7/5 = 600.0000 ¢, ~3/2 = 708.8176 ¢

Optimal ET sequence: 10e, 12, 22

Badness (Sintel): 1.06

Pajarina

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨2 0 11 12 26 36], ⟨0 1 -2 -2 -6 -9]]

Optimal tunings:

  • WE: ~7/5 = 598.7732 ¢, ~3/2 = 704.6889 ¢
  • CWE: ~7/5 = 600.0000 ¢, ~3/2 = 706.3950 ¢

Optimal ET sequence: 12f, 22, 34d

Badness (Sintel): 0.923

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨2 0 11 12 26 36 5], ⟨0 1 -2 -2 -6 -9 1]]

Optimal tunings:

  • WE: ~7/5 = 599.0204 ¢, ~3/2 = 705.2572 ¢
  • CWE: ~7/5 = 600.0000 ¢, ~3/2 = 706.5660 ¢

Optimal ET sequence: 12f, 22, 34d

Badness (Sintel): 0.936

Pajarita

Subgroup: 2.3.5.7.11.13

Comma list: 40/39, 50/49, 64/63, 66/65

Mapping: [⟨2 0 11 12 26 17], ⟨0 1 -2 -2 -6 -3]]

Optimal tunings:

  • WE: ~7/5 = 598.3048 ¢, ~3/2 = 705.4512 ¢
  • CWE: ~7/5 = 600.0000 ¢, ~3/2 = 707.9238 ¢

Optimal ET sequence: 10e, 12f, 22f, 34dff

Badness (Sintel): 0.937

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 40/39, 50/49, 64/63, 66/65, 85/84

Mapping: [⟨2 0 11 12 26 17 5], ⟨0 1 -2 -2 -6 -3 1]]

Optimal tunings:

  • WE: ~7/5 = 598.6103 ¢, ~3/2 = 706.3076 ¢
  • CWE: ~7/5 = 600.0000 ¢, ~3/2 = 708.2256 ¢

Optimal ET sequence: 10e, 12f, 22f

Badness (Sintel): 0.968

Pajarous

Subgroup: 2.3.5.7.11

Comma list: 50/49, 55/54, 64/63

Mapping: [⟨2 0 11 12 -9], ⟨0 1 -2 -2 5]]

Optimal tunings:

  • WE: ~7/5 = 599.4055 ¢, ~3/2 = 708.8747 ¢
  • CWE: ~7/5 = 600.0000 ¢, ~3/2 = 709.5508 ¢

Tuning ranges:

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

Optimal ET sequence: 10, 12e, 22, 120bce, 142bce

Badness (Sintel): 0.937

2.3.5.7.11.17 subgroup

Subgroup: 2.3.5.7.11.13.17

Comma list: 50/49, 52/51, 55/54, 64/63, 65/63

Mapping: [⟨2 0 11 12 -9 1 5], ⟨0 1 -2 -2 5 2 1]]

Optimal tunings:

  • WE: ~7/5 = 599.408 ¢, ~3/2 = 708.878 ¢
  • CWE: ~7/5 = 600.000 ¢, ~3/2 = 709.544 ¢

Optimal ET sequence: 10, 12e, 22

Badness (Sintel): 0.766

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨2 0 11 12 -9 1], ⟨0 1 -2 -2 5 2]]

Optimal tunings:

  • WE: ~7/5 = 599.9064 ¢, ~3/2 = 710.1289 ¢
  • CWE: ~7/5 = 600.0000 ¢, ~3/2 = 710.2325 ¢

Optimal ET sequence: 10, 22

Badness (Sintel): 1.04

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 50/49, 52/51, 55/54, 64/63, 65/63

Mapping: [⟨2 0 11 12 -9 1 5], ⟨0 1 -2 -2 5 2 1]]

Optimal tunings:

  • WE: ~7/5 = 599.8239 ¢, ~3/2 = 710.0128 ¢
  • CWE: ~7/5 = 600.0000 ¢, ~3/2 = 710.2067 ¢

Optimal ET sequence: 10, 22, 54f, 76bdff

Badness (Sintel): 0.930

Pajaro

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨2 0 11 12 -9 17], ⟨0 1 -2 -2 5 -3]]

Optimal tunings:

  • WE: ~7/5 = 598.8257 ¢, ~3/2 = 709.4266 ¢
  • CWE: ~7/5 = 600.0000 ¢, ~3/2 = 710.8414 ¢

Optimal ET sequence: 10, 22f, 32f

Badness (Sintel): 1.13

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 40/39, 50/49, 55/54, 64/63, 85/84

Mapping: [⟨2 0 11 12 -9 17 5], ⟨0 1 -2 -2 5 -3 1]]

Optimal tunings:

  • WE: ~7/5 = 598.8865 ¢, ~3/2 = 709.5472 ¢
  • CWE: ~7/5 = 600.0000 ¢, ~3/2 = 710.8704 ¢

Optimal ET sequence: 10, 22f, 32f

Badness (Sintel): 1.01

Pajaric

Subgroup: 2.3.5.7.11

Comma list: 45/44, 50/49, 56/55

Mapping: [⟨2 0 11 12 7], ⟨0 1 -2 -2 0]]

Optimal tunings:

  • WE: ~7/5 = 597.4807 ¢, ~3/2 = 702.5616 ¢
  • CWE: ~7/5 = 600.0000 ¢, ~3/2 = 706.0542 ¢

Optimal ET sequence: 10, 12, 22e

Badness (Sintel): 0.787

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨2 0 11 12 7 17], ⟨0 1 -2 -2 0 -3]]

Optimal tunings:

  • WE: ~7/5 = 597.1952 ¢, ~3/2 = 704.1350 ¢
  • CWE: ~7/5 = 600.0000 ¢, ~3/2 = 708.1989 ¢

Optimal ET sequence: 10, 12f, 22ef

Badness (Sintel): 0.845

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨2 0 11 12 7 17 5], ⟨0 1 -2 -2 0 -3 1]]

Optimal tunings:

  • WE: ~7/5 = 597.6509 ¢, ~3/2 = 705.7702 ¢
  • CWE: ~7/5 = 600.0000 ¢, ~3/2 = 708.9719 ¢

Optimal ET sequence: 10, 12f, 22ef

Badness (Sintel): 0.896

Hemipaj

Subgroup: 2.3.5.7.11

Comma list: 50/49, 64/63, 121/120

Mapping: [⟨2 1 9 10 8], ⟨0 2 -4 -4 -1]]

mapping generators: ~2, ~16/11

Optimal tunings:

  • WE: ~7/5 = 597.6509 ¢, ~16/11 = 652.7788 ¢
  • CWE: ~7/5 = 600.0000 ¢, ~16/11 = 653.7119 ¢

Optimal ET sequence: 2, 20, 22

Badness (Sintel): 1.29

Hemifourths

Subgroup: 2.3.5.7.11

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

Mapping: [⟨2 0 11 12 -1], ⟨0 2 -4 -4 5]]

mapping generators: ~2, ~55/32

Optimal tunings:

  • WE: ~7/5 = 597.6509 ¢, ~55/32 = 950.8475 ¢
  • CWE: ~7/5 = 600.0000 ¢, ~55/32 = 953.1172 ¢

Optimal ET sequence: 10, 24d, 34d

Badness (Sintel): 1.62

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨2 0 11 12 -1 9], ⟨0 2 -4 -4 5 -1]]

Optimal tunings:

  • WE: ~7/5 = 598.6748 ¢, ~26/15 = 950.9691 ¢
  • CWE: ~7/5 = 600.0000 ¢, ~26/15 = 953.1052 ¢

Optimal ET sequence: 10, 24d, 34d

Badness (Sintel): 1.19

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 50/49, 64/63, 78/77, 85/84, 144/143

Mapping: [⟨2 0 11 12 -1 9 5], ⟨0 2 -4 -4 5 -1 2]]

Optimal tunings:

  • WE: ~7/5 = 598.8411 ¢, ~26/15 = 951.3687 ¢
  • CWE: ~7/5 = 600.0000 ¢, ~26/15 = 953.2169 ¢

Optimal ET sequence: 10, 24d, 34d

Badness (Sintel): 1.11

Srutal

See also: Srutal vs diaschismic

Srutal can be described as the 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.

Subgroup: 2.3.5.7

Comma list: 2048/2025, 4375/4374

Mapping: [⟨2 0 11 -42], ⟨0 1 -2 15]]

Optimal tunings:

  • WE: ~45/32 = 599.4046 ¢, ~3/2 = 704.1150 ¢
error map: ⟨-1.191 +0.969 +1.289 +0.044]
  • CWE: ~45/32 = 600.0000 ¢, ~3/2 = 704.7646 ¢
error map: ⟨0.000 +2.810 +4.157 +2.643]

Tuning ranges:

Optimal ET sequence: 34d, 46, 80, 126, 206cd, 332bcd

Badness (Sintel): 2.32

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨2 0 11 -42 -28], ⟨0 1 -2 15 11]]

Optimal tunings:

  • WE: ~45/32 = 599.4413 ¢, ~3/2 = 704.1999 ¢
  • CWE: ~45/32 = 600.0000 ¢, ~3/2 = 704.8017 ¢

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]

Optimal ET sequence: 34d, 46, 80, 126, 206cd

Badness (Sintel): 1.17

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 176/175, 325/324, 364/363

Mapping: [⟨2 0 11 -42 -28 -18], ⟨0 1 -2 15 11 8]]

Optimal tunings:

  • WE: ~45/32 = 599.5490 ¢, ~3/2 = 704.3516 ¢
  • CWE: ~45/32 = 600.0000 ¢, ~3/2 = 704.8347 ¢

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]

Optimal ET sequence: 34d, 46, 80

Badness (Sintel): 1.04

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨2 0 11 -42 -28 -18 5], ⟨0 1 -2 15 11 8 1]]

Optimal tunings:

  • WE: ~17/12 = 599.6459 ¢, ~3/2 = 704.4237 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~3/2 = 704.8083 ¢

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]

Optimal ET sequence: 34d, 46, 80, 126

Badness (Sintel): 0.947

19-limit

Subgroup: 2.3.5.7.11.13.17.19

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

Mapping: [⟨2 0 11 -42 -28 -18 5 -55], ⟨0 1 -2 15 11 8 1 20]]

Optimal tunings:

  • WE: ~17/12 = 599.6371 ¢, ~3/2 = 704.4790 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~3/2 = 704.8745 ¢

Optimal ET sequence: 34dh, 46, 80

Badness (Sintel): 1.04

Srutaloo

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

Subgroup: 2.3.5.7.11.13.17.19.23

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

Mapping: [⟨2 0 11 -42 -28 -18 5 -55 -10], ⟨0 1 -2 15 11 8 1 20 6]]

Optimal tunings:

  • WE: ~17/12 = 599.6690 ¢, ~3/2 = 704.5098 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~3/2 = 704.8713 ¢

Optimal ET sequence: 34dh, 46, 80

Badness (Sintel): 0.971

29-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29

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

Mapping: [⟨2 0 11 -42 -28 -18 5 -55 -10 -76], ⟨0 1 -2 15 11 8 1 20 6 27]]

Optimal tunings:

  • WE: ~17/12 = 599.6664 ¢, ~3/2 = 704.5138 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~3/2 = 704.8807 ¢

Optimal ET sequence: 34dhj, 46, 80

Badness (Sintel): 1.10

31-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29.31

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

Mapping: [⟨2 0 11 -42 -28 -18 5 -55 -10 -76 48], ⟨0 1 -2 15 11 8 1 20 6 27 -12]]

Optimal tunings:

  • WE: ~17/12 = 599.8115 ¢, ~3/2 = 704.5958 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~3/2 = 704.8086 ¢

Optimal ET sequence: 46, 80, 126

Badness (Sintel): 1.44

Keen

Keen adds 875/864 as well as 2240/2187 to the set of commas. It may also be described as the 22 & 34 temperament. 78edo is a good tuning choice, and remains a good one in the 11-limit, where the temperament is really more interesting, adding 100/99 and 385/384 to the list of commas.

Subgroup: 2.3.5.7

Comma list: 875/864, 2048/2025

Mapping: [⟨2 0 11 -23], ⟨0 1 -2 9]]

Optimal tunings:

  • WE: ~45/32 = 599.6603 ¢, ~3/2 = 707.1707 ¢
error map: ⟨-0.679 +4.536 -3.033 -2.591]
  • CWE: ~45/32 = 600.0000 ¢, ~3/2 = 707.5294 ¢
error map: ⟨0.000 +5.574 -1.373 -1.061]

Optimal ET sequence: 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: [⟨2 0 11 -23 26], ⟨0 1 -2 9 -6]]

Optimal tunings:

  • WE: ~45/32 = 599.6286 ¢, ~3/2 = 707.1712 ¢
  • CWE: ~45/32 = 600.0000 ¢, ~3/2 = 707.5984 ¢

Optimal ET sequence: 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: [⟨2 0 11 -23 26 -18], ⟨0 1 -2 9 -6 8]]

Optimal tunings:

  • WE: ~45/32 = 599.3498 ¢, ~3/2 = 706.4009 ¢
  • CWE: ~45/32 = 600.0000 ¢, ~3/2 = 707.1309 ¢

Optimal ET sequence: 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: [⟨2 0 11 -23 26 -18 5], ⟨0 1 -2 9 -6 8 1]]

Optimal tunings:

  • WE: ~17/12 = 599.4053 ¢, ~3/2 = 706.4544 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~3/2 = 707.1243 ¢

Optimal ET sequence: 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: [⟨2 0 11 -23 26 36], ⟨0 1 -2 9 -6 -9]]

Optimal tunings:

  • WE: ~45/32 = 599.8547 ¢, ~3/2 = 707.0858 ¢
  • CWE: ~45/32 = 600.0000 ¢, ~3/2 = 707.2596 ¢

Optimal ET sequence: 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: [⟨2 0 11 -23 26 36 5], ⟨0 1 -2 9 -6 -9 1]]

Optimal tunings:

  • WE: ~17/12 = 599.8338 ¢, ~3/2 = 707.0558 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~3/2 = 707.2537 ¢

Optimal ET sequence: 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 12 & 68 temperament; its ploidacot is tetraploid monocot. Scales of bidia 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: [⟨4 0 22 43], ⟨0 1 -2 -5]]

mapping generators: ~25/21, ~3

Optimal tunings:

  • WE: ~25/21 = 299.6887 ¢, ~3/2 = 704.6318 ¢
error map: ⟨-1.245 +1.432 +0.064 +0.854]
  • CWE: ~25/21 = 300.0000 ¢, ~3/2 = 705.5070 ¢
error map: ⟨0.000 +3.552 +2.672 +3.639]

Optimal ET sequence: 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: [⟨4 0 22 43 71], ⟨0 1 -2 -5 -9]]

Optimal tunings:

  • WE: ~25/21 = 299.6809 ¢, ~3/2 = 704.3367 ¢
  • CWE: ~25/21 = 300.0000 ¢, ~3/2 = 705.2170 ¢

Optimal ET sequence: 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: [⟨4 0 22 43 71 -36], ⟨0 1 -2 -5 -9 8]]

Optimal tunings:

  • WE: ~25/21 = 299.7538 ¢, ~3/2 = 704.7222 ¢
  • CWE: ~25/21 = 300.0000 ¢, ~3/2 = 705.3241 ¢

Optimal ET sequence: 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: [⟨4 0 22 43 71 -36 10], ⟨0 1 -2 -5 -9 8 1]]

Optimal tunings:

  • WE: ~25/21 = 299.7883 ¢, ~3/2 = 704.8365 ¢
  • CWE: ~25/21 = 300.0000 ¢, ~3/2 = 705.3496 ¢

Optimal ET sequence: 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: [⟨4 0 22 43 71 -36 10 17], ⟨0 1 -2 -5 -9 8 1 0]]

Optimal tunings:

  • WE: ~19/16 = 299.7967 ¢, ~3/2 = 704.8609 ¢
  • CWE: ~19/16 = 300.0000 ¢, ~3/2 = 705.3519 ¢

Optimal ET sequence: 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: [⟨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 ¢, ~3/2 = 704.8577 ¢
  • CWE: ~19/16 = 300.0000 ¢, ~3/2 = 705.3413 ¢

Optimal ET sequence: 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 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(1/7), 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(1/7) 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: [⟨2 1 9 -2], ⟨0 2 -4 7]]

mapping generators: ~45/32, ~35/24

Optimal tunings:

  • WE: ~45/32 = 599.5401 ¢, ~35/24 = 652.3108 ¢
error map: ⟨-0.920 +2.207 +0.304 -1.730]
  • CWE: ~45/32 = 600.0000 ¢, ~35/24 = 652.7736 ¢
error map: ⟨0.000 +3.592 +2.592 +0.589]

Optimal ET sequence: 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: [⟨2 1 9 -2 8], ⟨0 2 -4 7 -1]]

Optimal tunings:

  • WE: ~45/32 = 599.7721 ¢, ~16/11 = 652.4321 ¢
  • CWE: ~45/32 = 600.0000 ¢, ~16/11 = 652.6672 ¢

Optimal ET sequence: 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: [⟨2 1 9 -2 8 -10], ⟨0 2 -4 7 -1 16]]

Optimal tunings:

  • WE: ~45/32 = 599.7699 ¢, ~16/11 = 652.4035 ¢
  • CWE: ~45/32 = 600.0000 ¢, ~16/11 = 652.6374 ¢

Optimal ET sequence: 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: [⟨2 1 9 -2 8 -10 6], ⟨0 2 -4 7 -1 16 2]]

Optimal tunings:

  • WE: ~17/12 = 599.7995 ¢, ~16/11 = 652.4287 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~16/11 = 652.6334 ¢

Optimal ET sequence: 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: [⟨2 1 9 -2 8 -10 6 -10], ⟨0 2 -4 7 -1 16 2 17]]

Optimal tunings:

  • WE: ~17/12 = 599.8060 ¢, ~16/11 = 652.5190 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~16/11 = 652.7164 ¢

Optimal ET sequence: 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: [⟨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 ¢, ~16/11 = 652.4776 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~16/11 = 652.6926 ¢

Optimal ET sequence: 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: [⟨2 1 9 11], ⟨0 2 -4 -5]]

mapping generators: ~45/32, ~10/7

Optimal tunings:

  • WE: ~45/32 = 600.2519 ¢, ~10/7 = 650.4083 ¢
error map: ⟨+0.504 -0.887 +14.321 -18.096]
  • CWE: ~45/32 = 600.0000 ¢, ~10/7 = 650.1017 ¢
error map: ⟨0.000 -1.752 +13.279 -19.334]

Optimal ET sequence: 2, 22d, 24

Badness (Sintel): 3.99

11-limit

Subgroup: 2.3.5.7.11

Comma list: 56/55, 77/75, 1323/1280

Mapping: [⟨2 1 9 11 8], ⟨0 2 -4 -5 -1]]

Optimal tunings:

  • WE: ~17/12 = 600.2356 ¢, ~10/7 = 650.3856 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~10/7 = 650.1008 ¢

Optimal ET sequence: 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: [⟨2 1 9 11 8 15], ⟨0 2 -4 -5 -1 -7]]

Optimal tunings:

  • WE: ~45/32 = 599.9067 ¢, ~10/7 = 649.4907 ¢
  • CWE: ~45/32 = 600.0000 ¢, ~10/7 = 649.5950 ¢

Optimal ET sequence: 2, 24

Badness (Sintel): 2.12

Sruti

Sruti tempers out 19683/19600, setting itself up as a hemipyth temperament. It has the same semi-octave period as diaschismic, but the generator can be taken as a neutral third or a hemitwelfth. The temperament can be described as 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: [⟨2 0 11 -15], ⟨0 2 -4 13]]

mapping generators: ~45/32, ~140/81

Optimal tunings:

  • WE: ~45/32 = 599.2764 ¢, ~140/81 = 950.7284 ¢
error map: ⟨-1.447 -0.498 +2.813 +1.497]
  • CWE: ~45/32 = 600.0000 ¢, ~140/81 = 951.8227 ¢
error map: ⟨0.000 +1.690 +6.395 +4.869]

Optimal ET sequence: 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: [⟨2 0 11 -15 -1], ⟨0 2 -4 13 5]]

Optimal tunings:

  • WE: ~45/32 = 599.1951 ¢, ~121/70 = 950.5864 ¢
  • CWE: ~45/32 = 600.0000 ¢, ~121/70 = 951.7972 ¢

Optimal ET sequence: 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: [⟨2 0 11 -15 -1 9], ⟨0 2 -4 13 5 -1]]

Optimal tunings:

  • WE: ~45/32 = 599.1479 ¢, ~26/15 = 950.5337 ¢
  • CWE: ~45/32 = 600.0000 ¢, ~26/15 = 951.8314 ¢

Optimal ET sequence: 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: [⟨2 0 11 -15 -1 9 5], ⟨0 2 -4 13 5 -1 2]]

Optimal tunings:

  • WE: ~17/12 = 599.3003 ¢, ~26/15 = 950.7465 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~26/15 = 951.8142 ¢

Optimal ET sequence: 24, 34d, 58

Badness (Sintel): 1.05

Anguirus

As another hemipyth temperament, anguirus tempers out 49/48. It can be described as the 10 & 24 temperament; its ploidacot is diploid dicot, the same as sruti.

Subgroup: 2.3.5.7

Comma list: 49/48, 2048/2025

Mapping: [⟨2 0 11 4], ⟨0 2 -4 1]]

mapping generators: ~45/32, ~7/4

Optimal tunings:

  • WE: ~45/32 = 600.2758 ¢, ~7/4 = 953.4593 ¢
error map: ⟨+0.552 +4.964 +2.883 -14.264]
  • CWE: ~45/32 = 600.0000 ¢, ~7/4 = 953.0188 ¢
error map: ⟨0.000 +4.083 +1.611 -15.807]

Optimal ET sequence: 10, 24, 34

Badness (Sintel): 1.97

11-limit

Subgroup: 2.3.5.7.11

Comma list: 49/48, 56/55, 243/242

Mapping: [⟨2 0 11 4 -1], ⟨0 2 -4 1 5]]

Optimal tunings:

  • WE: ~45/32 = 599.9250 ¢, ~7/4 = 952.0646 ¢
  • CWE: ~45/32 = 600.0000 ¢, ~7/4 = 952.1784 ¢

Optimal ET sequence: 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: [⟨2 0 11 4 -1 9], ⟨0 2 -4 1 5 -1]]

Optimal tunings:

  • WE: ~45/32 = 599.7575 ¢, ~7/4 = 951.9241 ¢
  • CWE: ~45/32 = 600.0000 ¢, ~7/4 = 952.2980 ¢

Optimal ET sequence: 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: [⟨2 0 11 4 -1 9 5], ⟨0 2 -4 1 5 -1 2]]

Optimal tunings:

  • WE: ~17/12 = 599.7925 ¢, ~7/4 = 952.0004 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~7/4 = 952.3178 ¢

Optimal ET sequence: 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 22 & 58 temperament; its ploidacot is diploid alpha-tricot. 58edo or 80edo make for good tunings, or their vals can be added to ⟨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 ¢ and is equal to it if we temper out S99. Three 11/10's then make a 4/3 (tempering out 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: [⟨2 1 9 2], ⟨0 3 -6 5]]

mapping generators: ~45/32, ~9/7

Optimal tunings:

  • WE: ~45/32 = 599.3056 ¢, ~9/7 = 434.3524 ¢
error map: ⟨-1.389 +0.408 +1.322 +1.547]
  • CWE: ~45/32 = 600.0000 ¢, ~9/7 = 434.8327 ¢
error map: ⟨0.000 +2.543 +4.690 +5.338]

Optimal ET sequence: 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: [⟨2 1 9 2 12], ⟨0 3 -6 5 -7]]

Optimal tunings:

  • WE: ~45/32 = 599.3085 ¢, ~9/7 = 434.3511 ¢
  • CWE: ~45/32 = 600.0000 ¢, ~9/7 = 434.8647 ¢

Minimax tuning:

  • 11-odd-limit: ~9/7 = [5/12 0 0 1/12 -1/12⟩
[[1 0 0 0 0⟩, [7/4 0 0 1/4 -1/4⟩, [2 0 0 -1/2 1/2⟩, [37/12 0 0 5/12 -5/12⟩, [37/12 0 0 -7/12 7/12⟩]
unchanged-interval (eigenmonzo) basis: 2.11/7

Optimal ET sequence: 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: [⟨2 1 9 2 12 19], ⟨0 3 -6 5 -7 -16]]

Optimal tunings:

  • WE: ~45/32 = 599.3397 ¢, ~9/7 = 434.2772 ¢
  • CWE: ~45/32 = 600.0000 ¢, ~9/7 = 434.7864 ¢

Optimal ET sequence: 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: [⟨2 1 9 2 12 19 6], ⟨0 3 -6 5 -7 -16 3]]

Optimal tunings:

  • WE: ~45/32 = 599.4645 ¢, ~9/7 = 434.4282 ¢
  • CWE: ~45/32 = 600.0000 ¢, ~9/7 = 434.8340 ¢

Optimal ET sequence: 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 10 & 46 temperament; its ploidacot is diploid tricot.

Subgroup: 2.3.5.7

Comma list: 686/675, 1029/1024

Mapping: [⟨2 2 7 6], ⟨0 3 -6 -1]]

mapping generators: ~45/32, ~8/7

Optimal tunings:

  • WE: ~45/32 = 599.7208 ¢, ~8/7 = 234.8330 ¢
error map: ⟨-0.558 +1.986 +2.733 -5.334]
  • CWE: ~45/32 = 600.0000 ¢, ~8/7 = 234.9539 ¢
error map: ⟨0.000 +2.907 +3.963 -3.780]

Optimal ET sequence: 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: [⟨2 2 7 6 3], ⟨0 3 -6 -1 10]]

Optimal tunings:

  • WE: ~45/32 = 599.8022 ¢, ~8/7 = 235.0185 ¢
  • CWE: ~45/32 = 600.0000 ¢, ~8/7 = 235.0893 ¢

Optimal ET sequence: 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: [⟨2 2 7 6 3 7], ⟨0 3 -6 -1 10 1]]

Optimal tunings:

  • WE: ~45/32 = 599.9570 ¢, ~8/7 = 235.0708 ¢
  • CWE: ~45/32 = 600.0000 ¢, ~8/7 = 235.0862 ¢

Optimal ET sequence: 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: [⟨2 2 7 6 3 7 7], ⟨0 3 -6 -1 10 1 3]]

Optimal tunings:

  • WE: ~17/12 = 599.9571 ¢, ~8/7 = 235.0709 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~8/7 = 235.0860 ¢

Optimal ET sequence: 10, 36e, 46, 102, 148f

Badness (Sintel): 0.983

Music

Quadrasruta

Named by Xenllium in 2022, quadrasruta tempers out 2401/2400, the breedsma, and extends buzzard. It may be described as 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: [⟨2 0 11 8], ⟨0 4 -8 -3]]

mapping generators: ~45/32, ~21/16

Optimal tunings:

  • WE: ~45/32 = 599.4443 ¢, ~21/16 = 475.7746 ¢
error map: ⟨-1.111 +1.143 +1.377 -0.595]
  • CWE: ~45/32 = 600.0000 ¢, ~21/16 = 476.2394 ¢
error map: ⟨0.000 +3.003 +3.771 +2.456]

Optimal ET sequence: 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: [⟨2 0 11 8 22], ⟨0 4 -8 -3 -19]]

Optimal tunings:

  • WE: ~45/32 = 599.4648 ¢, ~21/16 = 475.6929 ¢
  • CWE: ~45/32 = 600.0000 ¢, ~21/16 = 476.1507 ¢

Optimal ET sequence: 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: [⟨2 0 11 8 22 9], ⟨0 4 -8 -3 -19 -2]]

Optimal tunings:

  • WE: ~45/32 = 599.3787 ¢, ~21/16 = 475.6065 ¢
  • CWE: ~45/32 = 600.0000 ¢, ~21/16 = 476.1345 ¢

Optimal ET sequence: 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: [⟨2 0 11 8 22 9 5], ⟨0 4 -8 -3 -19 -2 4]]

Optimal tunings:

  • WE: ~17/12 = 599.5077 ¢, ~21/16 = 475.7713 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~21/16 = 476.1814 ¢

Optimal ET sequence: 10e, 58, 126f

Badness (Sintel): 1.21

Quadrafourths

Subgroup: 2.3.5.7.11

Comma list: 243/242, 441/440, 2048/2025

Mapping: [⟨2 0 11 8 -1], ⟨0 4 -8 -3 10]]

Optimal tunings:

  • WE: ~45/32 = 599.2593 ¢, ~21/16 = 475.4292 ¢
  • CWE: ~45/32 = 600.0000 ¢, ~21/16 = 476.0088 ¢

Optimal ET sequence: 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: [⟨2 0 11 8 -1 9], ⟨0 4 -8 -3 10 -2]]

Optimal tunings:

  • WE: ~45/32 = 599.2147 ¢, ~21/16 = 475.4052 ¢
  • CWE: ~45/32 = 600.0000 ¢, ~21/16 = 476.0253 ¢

Optimal ET sequence: 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: [⟨2 0 11 8 -1 9 5], ⟨0 4 -8 -3 10 -2 4]]

Optimal tunings:

  • WE: ~17/12 = 599.3353 ¢, ~21/16 = 475.5495 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~21/16 = 476.0691 ¢

Optimal ET sequence: 10, 48c, 58

Badness (Sintel): 1.13

Subgroup extensions

Srutal archagall (2.3.5.17)

See also: Fiventeen

Subgroup: 2.3.5.17

Comma list: 136/135, 256/255

Subgroup-val mapping: [⟨2 0 11 5], ⟨0 1 -2 1]]

mapping generators: ~17/12, ~3

Optimal tunings:

  • WE: ~45/32 = 599.5585 ¢, ~3/2 = 704.6188 ¢
  • CWE: ~45/32 = 600.0000 ¢, ~3/2 = 705.1356 ¢

Optimal ET sequence: 10, 12, 22, 34, 80, 114, 194bc

Badness (Sintel): 0.212

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