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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 [[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 }}
 
[[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]])<sup>2</sup>([[289/288]]) in the 17-limit, it extends naturally to the 2.3.5.17 subgroup as ''srutal archagall'', considered in [[#Subgroup extensions_2|#Subgroup extensions]]. The [[S-expression]]-based comma list of this temperament is {[[256/255|S16]], [[289/288|S17]]}.
 
== Septimal diaschismic ==
{{Main| Diaschismic }}
{{See also| Srutal vs diaschismic }}


{{Val list|legend=1| 10, 12, 22, 34, 46, 80, 206c, 286bc }}
A simpler characterization than the one given by the normal comma list is that septimal diaschismic adds [[126/125]] or [[5120/5103]] to the set of commas, and it can also be called {{nowrap| 46 & 58 }}. However described, septimal diaschismic has a 1/2-octave period and a sharp fifth generator like the 5-limit version, but not so sharp, giving a more accurate but more complex temperament. [[104edo]] with the 104c [[val]] provides an excellent tuning, which is close to tuning [[7/4]] just by making the fifth 703.897 cents.


[[Badness]]: 0.019915
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.


== Seven limit extensions ==
[[Subgroup]]: 2.3.5.7
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.
[[Comma list]]: 126/125, 2048/2025
* 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.
{{Mapping|legend=1| 2 0 11 31 | 0 1 -2 -8 }}


= Srutal =
[[Optimal tuning]]s:
[[Comma list]]: 2048/2025, 4375/4374
* [[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]]:
* 7- and 9-odd-limit [[diamond monotone]]: ~3/2 = [700.000, 705.882] (7\12 to 20\34)
* 7- and 9-odd-limit [[diamond tradeoff]]: ~3/2 = [701.955, 706.843]
 
{{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


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


{{Multival|legend=1| 2 -4 30 -11 42 81 }}
Optimal tunings:
* WE: ~45/32 = 599.4471{{c}}, ~3/2 = 703.0657{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 703.7996{{c}}


[[POTE generator]]: ~3/2 = 704.814
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]
 
{{Optimal ET sequence|legend=0| 12f, 34ef, 46, 58, 104c, 162cef }}


Mapping: [{{val|2 0 11 -42 -28}}, {{val|0 1 -2 15 11}}]
Badness (Sintel): 0.782


POTE generator: ~3/2 = 704.856
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Vals: {{Val list| 34d, 46, 80, 126, 206cd }}
Comma list: 126/125, 136/135, 176/175, 196/195, 256/255


Badness: 0.035315
Mapping: {{mapping| 2 0 11 31 45 55 5 | 0 1 -2 -8 -12 -15 1 }}


== 13-limit ==
Optimal tunings:
Comma list: 169/168, 176/175, 325/324, 364/363
* 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]


Mapping: [{{val|2 0 11 -42 -28 -18}}, {{val|0 1 -2 15 11 8}}]
{{Optimal ET sequence|legend=0| 12f, 34ef, 46, 58, 104c }}


POTE generator: ~3/2 = 704.881
Badness (Sintel): 0.837


[[Tuning ranges]]:  
=== 2.3.5.7.11.13.17.23 subgroup (Na"Naa') ===
* valid range: ~3/2 = [703.448, 705.882] (34\58 to 20\34)
<b>Na"Naa'</b> is a remarkable subgroup temperament of {{nowrap| 46 & 58 }} with a prime harmonic of 23. It is yet to be found why it got this strange name.
* nice range: ~3/2 = [701.955, 706.843]
 
* strict range: ~3/2 = [703.448, 705.882]
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 }}
 
Optimal tunings:  
* WE: ~17/12 = 599.6272{{c}}, ~3/2 = 703.4326{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 703.9093{{c}}
 
{{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 {{multival|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 {{val|34 54 79 96}} and 56 with the val {{val|56 89 130 158}} are are interesting alternatives, with more accpetable fifths, and a tetrad which is more clearly a dominant seventh. As such, they are closer to the tuning of 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]]: [{{val|2 0 11 12}}, {{val|0 1 -2 -2}}]
{{Mapping|legend=1| 2 0 11 12 | 0 1 -2 -2 }}


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


[[Tuning ranges]]:
[[Tuning ranges]]:
* 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
 
Comma list: 50/49, 64/63, 85/84


{{Val list|legend=1| 10, 12, 22, 34d, 56d }}
Mapping: {{mapping| 2 0 11 12 5 | 0 1 -2 -2 1 }}


[[Badness]]: 0.020033
Optimal tunings:
* WE: ~7/5 = 599.053{{c}}, ~3/2 = 706.355{{c}}
* CWE: ~7/5 = 600.000{{c}}, ~3/2 = 707.607{{c}}
 
{{Optimal ET sequence|legend=0| 10, 12, 22, 56d }}
 
Badness (Sintel): 0.438
 
=== 11-limit ===
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: [{{val|2 0 11 12 26}}, {{val|0 1 -2 -2 -6}}]
Mapping: {{mapping| 2 0 11 12 26 | 0 1 -2 -2 -6 }}


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


Tuning ranges:
Tuning ranges:
* 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]


Vals: {{Val list| 10e, 12, 22, 34d, 56d }}
{{Optimal ET sequence|legend=0| 10e, 12, 22, 34d, 56d }}


Badness: 0.020343
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: {{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


Mapping: [{{val|2 0 11 12 26 1}}, {{val|0 1 -2 -2 -6 2}}]
Mapping: {{mapping| 2 0 11 12 26 1 | 0 1 -2 -2 -6 2 }}
 
Optimal tunings:
* WE: ~7/5 = 599.9732{{c}}, ~3/2 = 708.8873{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 708.9227{{c}}
 
{{Optimal 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}}


Tuning ranges:
{{Optimal ET sequence|legend=0| 10e, 12f, 22f }}
* valid range: ~3/2 = [700.000, 709.091] (7\12 to 13\22)
* nice range: ~3/2 = [701.955, 738.573]
* strict range: ~3/2 = [701.955, 709.091]


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


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


Tuning ranges:
Tuning ranges:
* 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
 
{{Optimal ET sequence|legend=0| 10, 12e, 22, 120bce, 142bce }}
 
Badness (Sintel): 0.937
 
==== 2.3.5.7.11.17 subgroup ====
Subgroup: 2.3.5.7.11.13.17


Vals: {{Val list| 10, 12e, 22, 120bce, 142bce }}
Comma list: 50/49, 52/51, 55/54, 64/63, 65/63


Badness: 0.028349
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


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


POTE generator: ~3/2 = 710.240
Optimal tunings:  
* WE: ~7/5 = 599.8239{{c}}, ~3/2 = 710.0128{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 710.2067{{c}}


Vals: {{Val list| 10, 22, 54f, 76bdf }}
{{Optimal ET sequence|legend=0| 10, 22, 54f, 76bdff }}


Badness: 0.025176
Badness (Sintel): 0.930
 
==== 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


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


POTE generator ~3/2 = 710.818
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


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


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


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


Badness: 0.023798
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}}
 
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|legend=0| 34d, 46, 80, 126, 206cd }}
 
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: [{{val|2 0 11 12 7 17}}, {{val|0 1 -2 -2 0 -3}}]
Mapping: {{mapping| 2 0 11 -42 -28 -18 5 | 0 1 -2 15 11 8 1 }}


POTE generator: ~3/2 = 707.442
Optimal tunings:  
* WE: ~17/12 = 599.6459{{c}}, ~3/2 = 704.4237{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 704.8083{{c}}


Vals: {{Val list| 10, 12f, 22ef, 34def }}
Tuning ranges:  
* 17-odd-limit diamond monotone: ~3/2 = [704.348, 705.882] (27\46 to 20\34)
* 17-odd-limit diamond tradeoff: ~3/2 = [698.955, 711.731]


Badness: 0.0205
{{Optimal ET sequence|legend=0| 34d, 46, 80, 126 }}


== Hemipaj ==
Badness (Sintel): 0.947
Commas: 50/49, 64/63, 121/120


Mapping: [{{val|2 1 9 10 8}}, {{val|0 2 -4 -4 -1}}]
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19


POTE generator: ~11/8 = 546.383
Comma list: 136/135, 169/168, 176/175, 190/189, 221/220, 256/255


Vals: {{Val list| 20, 22, 68d, 90d }}
Mapping: {{mapping| 2 0 11 -42 -28 -18 5 -55 | 0 1 -2 15 11 8 1 20 }}


Badness: 0.038890
Optimal tunings:  
* WE: ~17/12 = 599.6371{{c}}, ~3/2 = 704.4790{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 704.8745{{c}}


= Diaschismic =
{{Optimal ET sequence|legend=0| 34dh, 46, 80 }}
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 {{multival|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|58EDO]] provides an excellent tuning, but an alternative is to make [[7/4]] just by making the fifth 703.897 cents, as opposed to 703.448 cents for 58et.


Diaschismic extends naturally to the 17-limit, for which the same tunings may be used, making it one of the most important of the higher limit rank two temperaments. Adding the 11-limit adds the commas 176/175, 896/891 and 441/440. The 13-limit yields 196/195, 351/350, and 364/363; the 17-limit adds 136/135, 221/220, and 442/441. If you want to explore higher limit harmonies, diaschismic is certainly one excellent way to do it; MOS of 34 notes and even more the 46 note MOS will encompass very great deal of it. Of course 46 or 58 equal provide alternatives which in many ways are similar, particularly in the case of 58.
Badness (Sintel): 1.04


[[Comma list]]: 126/125, 2048/2025
==== Srutaloo ====
Srutaloo adds 576/575, 736/729 or 208/207, and rhymes with [[skidoo]].


[[Mapping]]: [{{val|2 0 11 31}}, {{val|0 1 -2 -8}}]
Subgroup: 2.3.5.7.11.13.17.19.23


[[POTE generator]]: ~3/2 = 703.681
Comma list: 136/135, 169/168, 176/175, 190/189, 208/207, 221/220, 256/255


{{Val list|legend=1| 12, 46, 58, 104c, 162c }}
Mapping: {{mapping| 2 0 11 -42 -28 -18 5 -55 -10 | 0 1 -2 15 11 8 1 20 6 }}


[[Badness]]: 0.037914
Optimal tunings:  
* WE: ~17/12 = 599.6690{{c}}, ~3/2 = 704.5098{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 704.8713{{c}}


== 11-limit ==
{{Optimal ET sequence|legend=0| 34dh, 46, 80 }}
Comma list: 126/125, 176/175, 896/891


Mapping: [{{val|2 0 11 31 45}}, {{val|0 1 -2 -8 -12}}]
Badness (Sintel): 0.971


POTE generator: ~3/2 = 703.714
===== 29-limit =====
Subgroup: 2.3.5.7.11.13.17.19.23.29


Vals: {{Val list| 12, 46, 58, 104c, 162ce }}
Comma list: 136/135, 169/168, 176/175, 190/189, 208/207, 221/220, 232/231, 256/255


Badness: 0.025034
Mapping: {{mapping| 2 0 11 -42 -28 -18 5 -55 -10 -76 | 0 1 -2 15 11 8 1 20 6 27 }}


== 13-limit ==
Optimal tunings:
Comma list: 126/125, 196/195, 364/363, 2048/2025
* WE: ~17/12 = 599.6664{{c}}, ~3/2 = 704.5138{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 704.8807{{c}}


Mapping: [{{val|2 0 11 31 45 55}}, {{val|0 1 -2 -8 -12 -15}}]
{{Optimal ET sequence|legend=0| 34dhj, 46, 80 }}


POTE generator: ~3/2 = 703.704
Badness (Sintel): 1.10


Vals: {{Val list| 46, 58, 104c, 162cef }}
===== 31-limit =====
Subgroup: 2.3.5.7.11.13.17.19.23.29.31


Badness: 0.018926
Comma list: 136/135, 169/168, 176/175, 190/189, 208/207, 217/216, 221/220, 232/231, 256/255


== 17-limit (Na"Naa') ==
Mapping: {{mapping| 2 0 11 -42 -28 -18 5 -55 -10 -76 48 | 0 1 -2 15 11 8 1 20 6 27 -12 }}
Comma list: 126/125, 136/135, 176/175, 196/195, 256/255


Mapping: [{{val|2 0 11 31 45 55 5}}, {{val|0 1 -2 -8 -12 -15 1}}]
Optimal tunings:  
* WE: ~17/12 = 599.8115{{c}}, ~3/2 = 704.5958{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 704.8086{{c}}


POTE generator: ~3/2 = 703.812
{{Optimal ET sequence|legend=0| 46, 80, 126 }}


Vals: {{Val list| 46, 58, 104c }}
Badness (Sintel): 1.44


Badness: 0.016425
== 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.


= Keen =
[[Subgroup]]: 2.3.5.7
Keen adds 875/864 as well as 2240/2187 to the set of commas, and has wedgie {{multival|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, {{multival|2 -4 18 -12 ...}}, is really more interesting, adding 100/99 and 385/384 to the commas.


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


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


[[POTE generator]]: ~3/2 = 707.571
[[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 }}


{{Val list|legend=1| 22, 56, 78, 134b, 212b, 290bb }}
{{Optimal ET sequence|legend=1| 22, 56, 78, 134b }}


[[Badness]]: 0.083971
[[Badness]] (Sintel): 2.13
 
=== 11-limit ===
Subgroup: 2.3.5.7.11


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


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


POTE generator: ~3/2 = 707.609
{{Optimal ET sequence|legend=0| 22f, 34, 56f }}


Vals: {{Val list| 22, 56, 78, 212be, 290bbe }}
Badness (Sintel): 1.54


Badness: 0.045270
==== Keenic ====
Subgroup: 2.3.5.7.11.13


= Bidia =
Comma list: 91/90, 100/99, 352/351, 385/384
Bidia adds 3136/3125 to the commas, splitting the period into 1/4 octave. It may be called the 12&amp;56 temperament.
 
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
[[Comma list]]: 2048/2025, 3136/3125


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


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


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


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


[[Badness]]: 0.056474
=== 11-limit ===
Subgroup: 2.3.5.7.11


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


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


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


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


Badness: 0.040191
Badness (Sintel): 1.33
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


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


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


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


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


Badness: 0.041137
Badness (Sintel): 1.70
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


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


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


POTE generator: ~3/2 = 705.334
Badness (Sintel): 1.46


Vals: {{Val list| 12, 68, 80, 148d, 228bcd, 376bbcddf }}
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19


Badness: 0.028631
Comma list: 136/135, 176/175, 190/189, 256/255, 325/324, 640/637


= Echidna =
Mapping: {{mapping| 4 0 22 43 71 -36 10 17 | 0 1 -2 -5 -9 8 1 0 }}
Echidna adds 1728/1715 to the commas and takes 9/7 as a generator. It has a wedgie {{multival|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 {{val|138 219 321 388}}.


Echidna becomes more interesting when extended to be an 11-limit temperament by adding 176/175, 896/891 or 540/539 to the commas, where the same tunings can be used as before. It then is able to represent the entire 11-limit diamond to within about six cents of error, within a compass of 24 notes. The 28 note 2MOS gives scope for this, and the 36 note MOS much more.
Optimal tunings:
* WE: ~19/16 = 299.7967{{c}}, ~3/2 = 704.8609{{c}}
* CWE: ~19/16 = 300.0000{{c}}, ~3/2 = 705.3519{{c}}


[[Comma list]]: 1728/1715, 2048/2025
{{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


[[Mapping]]: [{{val|2 1 9 2}}, {{val|0 3 -6 5}}]
Comma list: 136/135, 176/175, 190/189, 253/252, 256/255, 325/324, 640/637


[[POTE generator]]: ~9/7 = 434.856
Mapping: {{mapping| 4 0 22 43 71 -36 10 17 -20 | 0 1 -2 -5 -9 8 1 0 6 }}


{{Val list|legend=1| 22, 58, 80, 138cd, 218cd }}
Optimal tunings:
* WE: ~19/16 = 299.7961{{c}}, ~3/2 = 704.8577{{c}}
* CWE: ~19/16 = 300.0000{{c}}, ~3/2 = 705.3413{{c}}


[[Badness]]: 0.058033
{{Optimal ET sequence|legend=0| 12, 68, 80, 148di }}


== 11-limit ==
Badness (Sintel): 1.24
Comma list: 176/175, 540/539, 896/891


Mapping: [{{val|2 1 9 2 12}}, {{val|0 3 -6 5 -7}}]
== 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.


POTE generator: ~9/7 = 434.852
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.


Minimax tuning:
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.
* 11-odd-limit
: [{{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}}]
: Eigenmonzos: 2, 11/7
: Minimax generator: (224/11)<sup>(1/12)</sup> = 434.792


Vals: {{Val list| 22, 58, 80, 138cde, 218cde }}
[[Subgroup]]: 2.3.5.7


Badness: 0.025987
[[Comma list]]: 245/243, 2048/2025


== 13-limit ==
{{Mapping|legend=1| 2 1 9 -2 | 0 2 -4 7 }}
Comma list: 176/175, 351/350, 364/363, 540/539
: mapping generators: ~45/32, ~35/24


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


POTE generator: ~9/7 = 434.756
{{Optimal ET sequence|legend=1| 22, 46, 68, 182b, 250bc }}


Vals: {{Val list| 22, 58, 80, 138cde }}
[[Badness]] (Sintel): 1.20


Badness: 0.023679
=== 11-limit ===
Subgroup: 2.3.5.7.11


== 17-limit ==
Comma list: 121/120, 176/175, 245/243
Comma list: 136/135, 176/175, 221/220, 256/255, 540/539


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


POTE generator: ~9/7 = 434.816
Optimal tunings:
* WE: ~45/32 = 599.7721{{c}}, ~16/11 = 652.4321{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~16/11 = 652.6672{{c}}


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


Badness: 0.020273
Badness (Sintel): 0.876


= Echidnic =
=== 13-limit ===
[[Comma list]]: 686/675, 1029/1024
Subgroup: 2.3.5.7.11.13


[[Mapping]]: [{{val|2 2 7 6}}, {{val|0 3 -6 -1}}]
Comma list: 121/120, 176/175, 196/195, 245/243


[[POTE generator]]: ~8/7 = 234.492
Mapping: {{mapping| 2 1 9 -2 8 -10 | 0 2 -4 7 -1 16 }}


{{Val list|legend=1| 10, 36, 46, 194bcd, 240bcd, 286bcd, 332bccdd }}
Optimal tunings:
* WE: ~45/32 = 599.7699{{c}}, ~16/11 = 652.4035{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~16/11 = 652.6374{{c}}


[[Badness]]: 0.072246
{{Optimal ET sequence|legend=0| 22f, 46, 68, 114 }}


== 11-limit ==
Badness (Sintel): 1.16
Comma list: 385/384, 441/440, 686/675


Mapping: [{{val|2 2 7 6 3}}, {{val|0 3 -6 -1 10}}]
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


POTE generator: ~8/7 = 235.096
Comma list: 121/120, 136/135, 154/153, 176/175, 196/195


Vals: {{Val list| 10, 36e, 46, 102, 148, 342bcdd }}
Mapping: {{mapping| 2 1 9 -2 8 -10 6 | 0 2 -4 7 -1 16 2 }}


Badness: 0.045127
Optimal tunings:
* WE: ~17/12 = 599.7995{{c}}, ~16/11 = 652.4287{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~16/11 = 652.6334{{c}}


== 13-limit ==
{{Optimal ET sequence|legend=0| 22f, 46, 68, 114 }}
Comma list: 91/90, 169/168, 385/384, 441/440


Mapping: [{{val|2 2 7 6 3 7}}, {{val|0 3 -6 -1 10 1}}]
Badness (Sintel): 0.953


POTE generator: ~8/7 = 235.088
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


Vals: {{Val list| 10, 46, 102, 148f, 194bcdf }}
Comma list: 121/120, 136/135, 154/153, 176/175, 196/195, 343/342


Badness: 0.028874
Mapping: {{mapping| 2 1 9 -2 8 -10 6 -10 | 0 2 -4 7 -1 16 2 17 }}


== 17-limit ==
Optimal tunings:
Comma list: 91/90, 136/135, 154/153, 169/168, 256/255
* WE: ~17/12 = 599.8060{{c}}, ~16/11 = 652.5190{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~16/11 = 652.7164{{c}}


Mapping: [{{val|2 2 7 6 3 7 7}}, {{val|0 3 -6 -1 10 1 3}}]
{{Optimal ET sequence|legend=0| 22fh, 46, 68, 114, 182bef }}


POTE generator: ~8/7 = 235.088
Badness (Sintel): 1.07


Vals: {{Val list| 10, 46, 102, 148f, 194bcdf }}
==== 23-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23


Badness: 0.019304
Comma list: 121/120, 136/135, 154/153, 176/175, 196/195, 253/252, 343/342


== Compositions ==
Mapping: {{mapping| 2 1 9 -2 8 -10 6 -10 -4 | 0 2 -4 7 -1 16 2 17 12 }}
* [http://untwelve.org/2011competition_audio/Kosmorsky-A_Stiff_Shot_of_Turpentine.mp3 A_Stiff_Shot_of_Turpentine] (the description says "lemba" which has a similar scale structure but different mapping for 5)


= Shrutar =
Optimal tunings:
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<sup>(1/7)</sup>, making 7s just.
* WE: ~17/12 = 599.7879{{c}}, ~16/11 = 652.4776{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~16/11 = 652.6926{{c}}


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.
{{Optimal ET sequence|legend=0| 22fh, 46, 68, 114 }}


[[Comma list]]: 245/243, 2048/2025
Badness (Sintel): 1.03


[[Mapping]]: [{{val| 2 1 9 -2 }}, {{val| 0 2 -4 7 }}]
== 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.


[[POTE generator]]: ~36/35 = 52.811
[[Subgroup]]: 2.3.5.7


{{Val list|legend=1| 22, 46, 68, 182b, 250bc }}
[[Comma list]]: 392/375, 1323/1280


== 11-limit ==
{{Mapping|legend=1| 2 1 9 11 | 0 2 -4 -5 }}
Comma list: 121/120, 176/175, 245/243
: mapping generators: ~45/32, ~10/7


Mapping: [{{val| 2 1 9 -2 8 }}, {{val| 0 2 -4 7 -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 }}


POTE generator: ~33/32 = 52.680
{{Optimal ET sequence|legend=1| 2, 22d, 24 }}


Vals: {{Val list| 22, 46, 68, 114, 296bce, 410bce }}
[[Badness]] (Sintel): 3.99


== 13-limit ==
=== 11-limit ===
Comma list: 121/120, 176/175, 196/195, 245/243
Subgroup: 2.3.5.7.11


Mapping: [{{val| 2 1 9 -2 8 -10 }}, {{val| 0 2 -4 7 -1 16 }}]
Comma list: 56/55, 77/75, 1323/1280


POTE generator: ~33/32 = 52.654
Mapping: {{mapping| 2 1 9 11 8 | 0 2 -4 -5 -1 }}


Vals: {{Val list| 22, 24, 46, 68, 114 }}
Optimal tunings:
* WE: ~17/12 = 600.2356{{c}}, ~10/7 = 650.3856{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~10/7 = 650.1008{{c}}


Badness: 0.0281
{{Optimal ET sequence|legend=0| 2, 22d, 24 }}


== 17-limit ==
Badness (Sintel): 2.10
Comma list: 121/120, 136/135, 154/153, 176/175, 196/195


Mapping: [{{val| 2 1 9 -2 8 -10 6 }}, {{val| 0 2 -4 7 -1 16 2 }}]
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


POTE generator: ~33/32 = 52.647
Comma list: 56/55, 77/75, 105/104, 507/500


Vals: {{Val list| 22, 24, 46, 68, 114 }}
Mapping: {{mapping| 2 1 9 11 8 15 | 0 2 -4 -5 -1 -7 }}


Badness: 0.0187
Optimal tunings:
* WE: ~45/32 = 599.9067{{c}}, ~10/7 = 649.4907{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~10/7 = 649.5950{{c}}


== 19-limit ==
{{Optimal ET sequence|legend=0| 2, 24 }}
Comma list: 121/120, 136/135, 154/153, 176/175, 196/195, 343/342


Mapping: [{{val| 2 1 9 -2 8 -10 6 -10 }}, {{val| 0 2 -4 7 -1 16 2 17 }}]
Badness (Sintel): 2.12


POTE generator: ~33/32 = 52.730
== Sruti ==
{{Redirect|Sruti|the tuning concept in Indian music|Shruti}}


Vals: {{Val list| 22, 24, 46, 68, 114, 182bef }}
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.


Badness: 0.0175
[[Subgroup]]: 2.3.5.7


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


[[Mapping]]: [{{val| 2 0 11 -15 }}, {{val| 0 2 -4 13 }}]
{{Mapping|legend=1| 2 0 11 -15 | 0 2 -4 13 }}
: mapping generators: ~45/32, ~140/81


{{Multival|legend=1| 4 -8 26 -22 30 83 }}
[[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 }}


POTE generator: ~175/144 = 351.876
{{Optimal ET sequence|legend=1| 24, 34d, 58, 150cd, 208ccdd, 266ccdd }}


{{Val list|legend=1| 24, 34d, 58, 150cd, 208cd, 266cd }}
[[Badness]] (Sintel): 2.97


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


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


Mapping: [{{val| 2 0 11 -15 -1 }}, {{val| 0 2 -4 13 5 }}]
Mapping: {{mapping| 2 0 11 -15 -1 | 0 2 -4 13 5 }}
 
Optimal tunings:
* WE: ~45/32 = 599.1951{{c}}, ~121/70 = 950.5864{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~121/70 = 951.7972{{c}}


POTE generator: ~11/9 = 351.863
{{Optimal ET sequence|legend=0| 24, 34d, 58, 150cdee, 208ccddee, 266ccddeee }}


Vals: {{Val list| 24, 34d, 58, 150cde, 208cde }}
Badness (Sintel): 1.37


Badness: 0.0415
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


== 13-limit ==
Comma list: 144/143, 176/175, 351/350, 676/675
Comma list: 144/143, 176/175, 351/350, 676/675


Mapping: [{{val| 2 0 11 -15 -1 9 }}, {{val| 0 2 -4 13 5 -1 }}]
Mapping: {{mapping| 2 0 11 -15 -1 9 | 0 2 -4 13 5 -1 }}
 
Optimal tunings:
* WE: ~45/32 = 599.1479{{c}}, ~26/15 = 950.5337{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~26/15 = 951.8314{{c}}
 
{{Optimal ET sequence|legend=0| 24, 34d, 58, 150cdeef, 208ccddeeff, 266ccddeeefff }}
 
Badness (Sintel): 0.983


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


Vals: {{Val list| 24, 34d, 58, 150cdef, 208cdef }}
Comma list: 136/135, 144/143, 170/169, 176/175, 221/220


Badness: 0.0238
Mapping: {{mapping| 2 0 11 -15 -1 9 5 | 0 2 -4 13 5 -1 2 }}
 
Optimal tunings:
* WE: ~17/12 = 599.3003{{c}}, ~26/15 = 950.7465{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~26/15 = 951.8142{{c}}
 
{{Optimal ET sequence|legend=0| 24, 34d, 58 }}
 
Badness (Sintel): 1.05
 
== Anguirus ==
As another hemipyth temperament, anguirus tempers out 49/48. It can be described as the {{nowrap| 10 & 24 }} temperament; its ploidacot is diploid dicot, the same as sruti.
 
[[Subgroup]]: 2.3.5.7


= Anguirus =
[[Comma list]]: 49/48, 2048/2025
[[Comma list]]: 49/48, 2048/2025


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


{{Multival|legend=1| 4 -8 2 -22 -8 27 }}
[[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 }}


[[POTE generator]]: ~8/7 = 246.979
{{Optimal ET sequence|legend=1| 10, 24, 34 }}


{{Val list|legend=1| 10, 24, 34 }}
[[Badness]] (Sintel): 1.97


[[Badness]]: 0.0780
=== 11-limit ===
Subgroup: 2.3.5.7.11


== 11-limit ==
Comma list: 49/48, 56/55, 243/242
Comma list: 49/48, 56/55, 243/242


Mapping: [{{val| 2 0 11 4 -1 }}, {{val| 0 2 -4 1 5 }}]
Mapping: {{mapping| 2 0 11 4 -1 | 0 2 -4 1 5 }}
 
Optimal tunings:
* WE: ~45/32 = 599.9250{{c}}, ~7/4 = 952.0646{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~7/4 = 952.1784{{c}}
 
{{Optimal ET sequence|legend=0| 10, 24, 34 }}
 
Badness (Sintel): 1.63
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 49/48, 56/55, 91/90, 243/242
 
Mapping: {{mapping| 2 0 11 4 -1 9 | 0 2 -4 1 5 -1 }}
 
Optimal tunings:
* WE: ~45/32 = 599.7575{{c}}, ~7/4 = 951.9241{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~7/4 = 952.2980{{c}}
 
{{Optimal ET sequence|legend=0| 10, 24, 34, 58d, 92ddef }}
 
Badness (Sintel): 1.27
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 49/48, 56/55, 91/90, 119/117, 154/153
 
Mapping: {{mapping| 2 0 11 4 -1 9 5 | 0 2 -4 1 5 -1 2 }}
 
Optimal tunings:
* WE: ~17/12 = 599.7925{{c}}, ~7/4 = 952.0004{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~7/4 = 952.3178{{c}}
 
{{Optimal ET sequence|legend=0| 10, 24, 34 }}
 
Badness (Sintel): 1.10
 
== Echidna ==
Echidna adds 1728/1715 to the commas and takes 9/7 as a generator. It may be called the {{nowrap| 22 & 58 }} temperament; its ploidacot is diploid alpha-tricot. [[58edo]] or [[80edo]] make for good tunings, or their vals can be added to {{val| 138 219 321 388 }} (138cde). In most of the tunings it has a significantly sharp 7/4 which some prefer.
 
Echidna becomes more interesting when extended to be an 11-limit temperament by adding 176/175, 540/539 or 896/891 to the commas, where the same tunings can be used as before. It then is able to represent the entire 11-odd-limit diamond to within about six cents of error, within a compass of 24 notes. The 22-note 2mos gives scope for this, and the 36-note mos much more. Better yet, it is related to three important 11-limit edos: 22edo, a trivial tuning, is the smallest consistent in the 11-odd-limit, corresponding to the merge of this temperament with [[hedgehog]]; [[58edo]] is the smallest tuning that is distinctly consistent in the 11-odd-limit and [[80edo]] is the third smallest distinctly consistent in the 11-odd-limit.
 
The generator can be interpreted as 11/10, the period complement of 9/7, as a stack of 11/10 and 9/7 makes [[99/70]] which is extremely close to 600{{cent}} and is equal to it if we temper out [[9801/9800|S99]]. Three 11/10's then make a 4/3 (tempering out [[4000/3993|S10/S11]] thus making 10/9 and 12/11 equidistant from 11/10), implying a flat tuning of 4/3.
 
Like most srutal extensions, the 13- and 17-limit interpretations are possible by observing that since we have tempered out [[176/175]], tempering out [[351/350]] and [[352/351]] which sum to 176/175 is very elegant. In the 17-limit we can equate the half-octave with 17/12 and 24/17 and we can take advantage of the sharp fifth by combining echidna with [[srutal archagall]], leading to a particularly beautiful temperament (one that prefers a very slightly less sharp fifth than srutal archagall). This mapping of 13 and 17 is supported by the patent vals of the three main echidna edos of 22, 58 and 80, of which all except 22 are consistent in the [[17-odd-limit]].
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 1728/1715, 2048/2025
 
{{Mapping|legend=1| 2 1 9 2 | 0 3 -6 5 }}
: mapping generators: ~45/32, ~9/7
 
[[Optimal tuning]]s:
* [[WE]]: ~45/32 = 599.3056{{c}}, ~9/7 = 434.3524{{c}}
: [[error map]]: {{val| -1.389 +0.408 +1.322 +1.547 }}
* [[CWE]]: ~45/32 = 600.0000{{c}}, ~9/7 = 434.8327{{c}}
: error map: {{val| 0.000 +2.543 +4.690 +5.338 }}
 
{{Optimal ET sequence|legend=1| 22, 58, 80, 138cd, 218cd }}
 
[[Badness]] (Sintel): 1.47
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 176/175, 540/539, 896/891
 
Mapping: {{mapping| 2 1 9 2 12 | 0 3 -6 5 -7 }}
 
Optimal tunings:
* WE: ~45/32 = 599.3085{{c}}, ~9/7 = 434.3511{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~9/7 = 434.8647{{c}}
 
Minimax tuning:
* 11-odd-limit: ~9/7 = {{monzo| 5/12 0 0 1/12 -1/12 }}
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 7/4 0 0 1/4 -1/4 }}, {{monzo| 2 0 0 -1/2 1/2 }}, {{monzo| 37/12 0 0 5/12 -5/12 }}, {{monzo| 37/12 0 0 -7/12 7/12 }}]
: unchanged-interval (eigenmonzo) basis: 2.11/7


POTE generator: ~8/7 = 247.816
{{Optimal ET sequence|legend=0| 22, 58, 80, 138cde, 218cde }}


Vals: {{Val list| 10, 24, 34, 58d, 92de }}
Badness (Sintel): 0.859


Badness: 0.0493
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


== 13-limit ==
Comma list: 176/175, 351/350, 364/363, 540/539
Comma list: 49/48, 56/55, 91/90, 352/351


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


POTE generator: ~8/7 = 247.691
Optimal tunings:
* WE: ~45/32 = 599.3397{{c}}, ~9/7 = 434.2772{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~9/7 = 434.7864{{c}}


Vals: {{Val list| 10, 24, 34, 58d, 92def }}
{{Optimal ET sequence|legend=0| 22, 36f, 58, 80, 138cde }}


Badness: 0.0308
Badness (Sintel): 0.978


= Shru =
=== 17-limit ===
[[Comma list]]: 392/375, 1323/1280
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 136/135, 176/175, 221/220, 256/255, 540/539
 
Mapping: {{mapping| 2 1 9 2 12 19 6 | 0 3 -6 5 -7 -16 3 }}
 
Optimal tunings:
* WE: ~45/32 = 599.4645{{c}}, ~9/7 = 434.4282{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~9/7 = 434.8340{{c}}
 
{{Optimal ET sequence|legend=0| 22, 36f, 58, 80, 138cde }}
 
Badness (Sintel): 1.03
 
== Echidnic ==
Echidnic tempers out 686/675 and [[1029/1024]]. It has the same semi-octave period as diaschismic, but slices the generator of a fifth into three ~8/7's. It can be described as the {{nowrap| 10 & 46 }} temperament; its ploidacot is diploid tricot.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 686/675, 1029/1024
 
{{Mapping|legend=1| 2 2 7 6 | 0 3 -6 -1 }}
: mapping generators: ~45/32, ~8/7
 
[[Optimal tuning]]s:
* [[WE]]: ~45/32 = 599.7208{{c}}, ~8/7 = 234.8330{{c}}
: [[error map]]: {{val| -0.558 +1.986 +2.733 -5.334 }}
* [[CWE]]: ~45/32 = 600.0000{{c}}, ~8/7 = 234.9539{{c}}
: error map: {{val| 0.000 +2.907 +3.963 -3.780 }}
 
{{Optimal ET sequence|legend=1| 10, 26c, 36, 46 }}
 
[[Badness]] (Sintel): 1.83
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 385/384, 441/440, 686/675
 
Mapping: {{mapping| 2 2 7 6 3 | 0 3 -6 -1 10 }}
 
Optimal tunings:
* WE: ~45/32 = 599.8022{{c}}, ~8/7 = 235.0185{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~8/7 = 235.0893{{c}}
 
{{Optimal ET sequence|legend=0| 10, 36e, 46, 102, 148 }}
 
Badness (Sintel): 1.49
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 91/90, 169/168, 385/384, 441/440
 
Mapping: {{mapping| 2 2 7 6 3 7 | 0 3 -6 -1 10 1 }}
 
Optimal tunings:
* WE: ~45/32 = 599.9570{{c}}, ~8/7 = 235.0708{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~8/7 = 235.0862{{c}}
 
{{Optimal ET sequence|legend=0| 10, 36e, 46, 102, 148f }}
 
Badness (Sintel): 1.19
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 91/90, 136/135, 154/153, 169/168, 256/255
 
Mapping: {{mapping| 2 2 7 6 3 7 7 | 0 3 -6 -1 10 1 3 }}
 
Optimal tunings:
* WE: ~17/12 = 599.9571{{c}}, ~8/7 = 235.0709{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~8/7 = 235.0860{{c}}
 
{{Optimal ET sequence|legend=0| 10, 36e, 46, 102, 148f }}
 
Badness (Sintel): 0.983
 
; Music
* [https://untwelve.org/competition/2011 ''A Stiff Shot of Turpentine''] [https://untwelve.org/static/audio/competition/2011/Kosmorsky-A_Stiff_Shot_of_Turpentine.mp3 play] by [[Peter Kosmorsky]]
* [https://www.youtube.com/watch?v=VsBXIvBZY6A ''56edo Track (Echidnic16 Scale)''] by [[Budjarn Lambeth]] (2025)
 
== Quadrasruta ==
Named by [[Xenllium]] in 2022, quadrasruta tempers out 2401/2400, the breedsma, and extends [[buzzard]]. It may be described as {{nowrap| 58 & 68 }}; its ploidacot is diploid alpha-tetracot. 126edo may be recommended as a tuning.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2048/2025, 2401/2400
 
{{Mapping|legend=1| 2 0 11 8 | 0 4 -8 -3 }}
: mapping generators: ~45/32, ~21/16
 
[[Optimal tuning]]s:
* [[WE]]: ~45/32 = 599.4443{{c}}, ~21/16 = 475.7746{{c}}
: [[error map]]: {{val| -1.111 +1.143 +1.377 -0.595 }}
* [[CWE]]: ~45/32 = 600.0000{{c}}, ~21/16 = 476.2394{{c}}
: error map: {{val| 0.000 +3.003 +3.771 +2.456 }}
 
{{Optimal ET sequence|legend=1| 10, …, 58, 68, 126, 446bbccd }}
 
[[Badness]] (Sintel): 1.86
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 176/175, 896/891, 2401/2400
 
Mapping: {{mapping| 2 0 11 8 22 | 0 4 -8 -3 -19 }}
 
Optimal tunings:
* WE: ~45/32 = 599.4648{{c}}, ~21/16 = 475.6929{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~21/16 = 476.1507{{c}}
 
{{Optimal ET sequence|legend=0| 10e, …, 58, 126, 184c, 310bccde }}
 
Badness (Sintel): 1.62
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 176/175, 196/195, 512/507, 676/675
 
Mapping: {{mapping| 2 0 11 8 22 9 | 0 4 -8 -3 -19 -2 }}
 
Optimal tunings:
* WE: ~45/32 = 599.3787{{c}}, ~21/16 = 475.6065{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~21/16 = 476.1345{{c}}
 
{{Optimal ET sequence|legend=0| 10e, …, 58, 126f, 184cff }}
 
Badness (Sintel): 1.18
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 136/135, 170/169, 176/175, 196/195, 256/255
 
Mapping: {{mapping| 2 0 11 8 22 9 5 | 0 4 -8 -3 -19 -2 4 }}
 
Optimal tunings:
* WE: ~17/12 = 599.5077{{c}}, ~21/16 = 475.7713{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~21/16 = 476.1814{{c}}
 
{{Optimal ET sequence|legend=0| 10e, 58, 126f }}
 
Badness (Sintel): 1.21
 
=== Quadrafourths ===
Subgroup: 2.3.5.7.11
 
Comma list: 243/242, 441/440, 2048/2025
 
Mapping: {{mapping| 2 0 11 8 -1 | 0 4 -8 -3 10 }}
 
Optimal tunings:
* WE: ~45/32 = 599.2593{{c}}, ~21/16 = 475.4292{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~21/16 = 476.0088{{c}}
 
{{Optimal ET sequence|legend=0| 10, 48c, 58, 184cee, 242ccdeee }}
 
Badness (Sintel): 1.62
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 144/143, 196/195, 243/242, 676/675
 
Mapping: {{mapping| 2 0 11 8 -1 9 | 0 4 -8 -3 10 -2 }}
 
Optimal tunings:
* WE: ~45/32 = 599.2147{{c}}, ~21/16 = 475.4052{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~21/16 = 476.0253{{c}}
 
{{Optimal ET sequence|legend=0| 10, 48c, 58, 126eef, 184ceeff, 242ccdeeeff }}


[[Mapping]]: [{{val| 2 1 9 11 }}, {{val| 0 2 -4 -5 }}]
Badness (Sintel): 1.11


{{Multival|legend=1| 4 -8 -10 -22 -27 -1 }}
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


[[POTE generator]]: ~64/63 =  50.135
Comma list: 136/135, 144/143, 170/169, 196/195, 221/220


{{Val list|legend=1| 22d, 24 }}
Mapping: {{mapping| 2 0 11 8 -1 9 5 | 0 4 -8 -3 10 -2 4 }}


[[Badness]]: 0.1576
Optimal tunings:
* WE: ~17/12 = 599.3353{{c}}, ~21/16 = 475.5495{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~21/16 = 476.0691{{c}}


== 11-limit ==
{{Optimal ET sequence|legend=0| 10, 48c, 58 }}
Comma list: 56/55, 77/75, 1323/1280


Mapping: [{{val| 2 1 9 11 8 }}, {{val| 0 2 -4 -5 -1 }}]
Badness (Sintel): 1.13


POTE generator: ~33/32 = 50.130
== Subgroup extensions ==
=== Srutal archagall (2.3.5.17) ===
{{See also | Fiventeen }}


Vals: {{Val list| 22d, 24 }}
This extension of 5-limit diaschismic adds prime 17 and which with respect to [[MVP archagall]] is able to express the harmonics [[75/1|75]] and [[85/1|85]] in their appropriate prime subgroup. It achieves this by equating [[85/64]] with [[4/3]] by tempering out their difference of [[256/255]] (S16). Therefore it also tempers out [[289/288]] (S17) and thus equates [[17/15]] with [[9/8]] due to tempering out [[136/135]] (S16⋅S17). It could be described as the 10 & 12 temperament with strong emphasis on 12edo being the better tuning on the 2.3.5.17 subgroup, implying ideal tunings of 34edo, 46edo or 80edo.


Badness: 0.0635
Subgroup: 2.3.5.17


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


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


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


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


Badness: 0.0457
Badness (Sintel): 0.212


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