@@ Line 1: / Line 1: @@
-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]].
+== 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.
-== Srutal aka diaschismic ==
 [[Subgroup]]: 2.3.5
@@ Line 7: / Line 14: @@
 {{Mapping|legend=1| 2 0 11 | 0 1 -2 }}
 : mapping generators: ~45/32, ~3
-[[Optimal tuning]] ([[POTE]]): ~45/32 = 1\2, ~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]]:
-* 5-odd-limit [[diamond monotone]]: ~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)
 * 5-odd-limit [[diamond tradeoff]]: ~3/2 = [701.955, 706.843]
-* 5-odd-limit diamond monotone and tradeoff: ~3/2 = [701.955, 706.843]
 {{Optimal ET sequence|legend=1| 10, 12, 22, 34, 46, 80, 206c, 286bc }}
-[[Badness]]: 0.019915
+[[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 derives from [[64/63]] and is a popular and well-known choice.
+* Pajara adds [[50/49]] or [[64/63]] and is a popular and well-known choice.
-* Diaschismic adds [[126/125]], the starling comma, to obtain 7-limit harmony by more complex methods, but with greater accuracy.
+* Srutal adds [[4375/4374]], the ragisma, which is about as accurate as septimal diaschismic but has a much more complex mapping of 7.
-* Srutal adds [[4375/4374]], the ragisma. It does no significant tuning damage, so we keep the 5-limit label srutal.
 * Keen adds [[875/864]].
-* Bidia adds [[3136/3125]], the hemimean comma.
-* Echidna adds [[1728/1715]], the orwellisma.
-* Shrutar adds [[245/243]], the sensamagic comma.
-Pajara, diaschismic, srutal and keen keep the same half-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. Bidia has a quarter-octave period and a fifth generator.
+Those all keep the same half-octave period and fifth generator.
-==== Subgroup extensions ====
+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.
-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, resulting in ''srutal archagall''.
-=== Srutal archagall ===
+Temperaments discussed elsewhere include [[stearnsmic clan #Echidna|echidna]]. The rest are considered below.
-Subgroup: 2.3.5.17
-Comma list: 136/135, 256/255
+==== 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]]}.
-Sval mapping: {{mapping| 2 0 11 5 | 0 1 -2 1 }}
+More generally, one may note that since the fifth of diaschismic is sharp in any good tuning, it is compatible with the logic of [[parapyth]] in making ~13/11 a minor third and ~14/11 a major third, leading to the rank-3 temperament [[varda]], which multiple temps here are rank-2 tunings of, such as 17-limit diaschismic and srutal (and their obvious extensions to prime 23, na"naa' and srutaloo, both by mapping ~23/16 as an augmented fourth aka. tritone).
-Optimal tuning (CTE): ~17/12 = 1\2, ~3/2 = 705.1272
+== Septimal diaschismic ==
+{{Main| Diaschismic }}
-{{Optimal ET sequence|legend=1| 10, 12, 22, 34, 80, 114, 194bc }}
+{{See also| Srutal vs diaschismic }}
-Badness: 0.00575
+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.
-== Srutal ==
+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.
-{{See also| Srutal vs diaschismic }}
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 2048/2025, 4375/4374
+[[Comma list]]: 126/125, 2048/2025
-{{Mapping|legend=1| 2 0 11 -42 | 0 1 -2 15 }}
+{{Mapping|legend=1| 2 0 11 31 | 0 1 -2 -8 }}
-{{Multival|legend=1| 2 -4 30 -11 42 81 }}
-[[Optimal tuning]] ([[POTE]]): ~45/32 = 1\2, ~3/2 = 704.814
+[[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]]:
-* 7- and 9-odd-limit [[diamond monotone]]: ~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)
 * 7- and 9-odd-limit [[diamond tradeoff]]: ~3/2 = [701.955, 706.843]
-* 7- and 9-odd-limit diamond monotone and tradeoff: ~3/2 = [703.448, 705.882]
-{{Optimal ET sequence|legend=1| 34d, 46, 80, 126, 206cd, 332bcd }}
+{{Optimal ET sequence|legend=1| 12, 34, 46, 58, 104c, 162c }}
-[[Badness]]: 0.091504
+[[Badness]] (Sintel): 0.959
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 176/175, 896/891, 1331/1323
+Comma list: 126/125, 176/175, 896/891
-Mapping: {{mapping| 2 0 11 -42 -28 | 0 1 -2 15 11 }}
+{{Mapping|legend=0| 2 0 11 31 45 | 0 1 -2 -8 -12 }}
-Optimal tuning (POTE): ~45/32 = 1\2, ~3/2 = 704.856
+Optimal tunings:
+* WE: ~45/32 = 599.4471{{c}}, ~3/2 = 703.0657{{c}}
+* CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 703.7996{{c}}
 Tuning ranges:
-* 11-odd-limit diamond monotone: ~3/2 = [704.348, 705.882] (27\46 to 20\34)
+* 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]
-* 11-odd-limit diamond monotone and tradeoff: ~3/2 = [704.348, 705.882]
-{{Optimal ET sequence|legend=1| 34d, 46, 80, 126, 206cd }}
+{{Optimal ET sequence|legend=0| 12, 34e, 46, 58, 104c, 162ce }}
-Badness: 0.035315
+Badness (Sintel): 0.828
 === 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 169/168, 176/175, 325/324, 364/363
+Comma list: 126/125, 176/175, 196/195, 364/363
-Mapping: {{mapping| 2 0 11 -42 -28 -18 | 0 1 -2 15 11 8 }}
+{{Mapping|legend=0| 2 0 11 31 45 55 | 0 1 -2 -8 -12 -15 }}
-Optimal tuning (POTE): ~45/32 = 1\2, ~3/2 = 704.881
+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:
-* 13- and 15-odd-limit diamond monotone: ~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)
 * 13-odd-limit diamond tradeoff: ~3/2 = [701.955, 706.843]
 * 15-odd-limit diamond tradeoff: ~3/2 = [701.955, 711.731]
-* 13- and 15-odd-limit diamond monotone and tradeoff: ~3/2 = [704.348, 705.882]
-{{Optimal ET sequence|legend=1| 34d, 46, 80, 206cd, 286bcde }}
+{{Optimal ET sequence|legend=0| 12f, 34ef, 46, 58, 104c, 162cef }}
-Badness: 0.025286
+Badness (Sintel): 0.782
 === 17-limit ===
 Subgroup: 2.3.5.7.11.13.17
-Comma list: 136/135, 169/168, 176/175, 221/220, 256/255
+Comma list: 126/125, 136/135, 176/175, 196/195, 256/255
-Mapping: {{mapping| 2 0 11 -42 -28 -18 5 | 0 1 -2 15 11 8 1 }}
+{{Mapping|legend=0| 2 0 11 31 45 55 5 | 0 1 -2 -8 -12 -15 1 }}
-Optimal tuning (POTE): ~17/12 = 1\2, ~3/2 = 704.840
+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:
-* 17-odd-limit diamond monotone: ~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)
 * 17-odd-limit diamond tradeoff: ~3/2 = [698.955, 711.731]
-* 17-odd-limit diamond monotone and tradeoff: ~3/2 = [704.348, 705.882]
-{{Optimal ET sequence|legend=1| 34d, 46, 80, 126, 206cd }}
+{{Optimal ET sequence|legend=0| 12f, 34ef, 46, 58, 104c }}
-Badness: 0.018594
+Badness (Sintel): 0.837
-=== 19-limit ===
+=== 2.3.5.7.11.13.17.23 subgroup (Na"Naa') ===
-Srutal, shrutar and bidia have similar 19-limit properties, tempering 190/189, related rank-3 [[julius]].
+<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: 2.3.5.7.11.13.17.19
+Subgroup: 2.3.5.7.11.13.17.23
-Comma list: 136/135, 169/168, 176/175, 190/189, 221/220, 256/255
+Comma list: 126/125, 136/135, 176/175, 196/195, 231/230, 256/255
-Mapping: {{mapping| 2 0 11 -42 -28 -18 5 -55 | 0 1 -2 15 11 8 1 20 }}
-Optimal tuning (POTE): ~17/12 = 1\2, ~3/2 = 704.905
-{{Optimal ET sequence|legend=1| 34dh, 46, 80, 206cd }}
-Badness: 0.017063
-==== Srutaloo ====
-Srutaloo adds 576/575, 736/729 or 208/207, 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 tuning (POTE): ~17/12 = 1\2, ~3/2 = 704.899
-{{Optimal ET sequence|legend=1| 34dh, 46, 80, 206cd }}
-Badness: 0.013555
-===== 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 }}
+{{Mapping|legend=2| 2 0 11 31 45 55 5 63 | 0 1 -2 -8 -12 -15 1 -17 }}
-Optimal tuning (POTE): ~17/12 = 1\2, ~3/2 = 704.906
+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=1| 34dhj, 46, 80, 206cd }}
+{{Optimal ET sequence|legend=0| 12i, 34efi, 46, 58i, 104ci }}
-Badness: 0.013203
+Badness (Sintel): 0.882
 == Pajara ==
 {{Main| 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 }} 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 12edo and of common practice Western music in general, while retaining the distictiveness of a sharp fifth.
+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.
-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.
+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
@@ Line 184: / Line 165: @@
 {{Mapping|legend=1| 2 0 11 12 | 0 1 -2 -2 }}
-{{Multival|legend=1| 2 -4 -4 -11 -12 2 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~7/5 = 598.8483{{c}}, ~3/2 = 705.6906{{c}}
-[[Optimal tuning]] ([[POTE]]): ~7/5 = 1\2, ~3/2 = 707.048
+: [[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]]:
 * 7- and 9-odd-limit [[diamond monotone]]: ~3/2 = [700.000, 720.000] (7\12 to 6\10)
 * 7- and 9-odd-limit [[diamond tradeoff]]: ~3/2 = [701.955, 715.587]
-* 7- and 9-odd-limit diamond monotone and tradeoff: ~3/2 = [701.955, 715.587]
 {{Optimal ET sequence|legend=1| 10, 12, 22, 34d, 56d }}
-[[Badness]]: 0.020033
+[[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|legend=0| 2 0 11 12 5 | 0 1 -2 -2 1 }}
+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 ===
@@ Line 202: / Line 199: @@
 Comma list: 50/49, 64/63, 99/98
-Mapping: {{mapping| 2 0 11 12 26 | 0 1 -2 -2 -6 }}
+{{Mapping|legend=0| 2 0 11 12 26 | 0 1 -2 -2 -6 }}
-Optimal tuning (POTE): ~7/5 = 1\2, ~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:
 * 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]
-* 11-odd-limit diamond monotone and tradeoff: ~3/2 = [701.955, 709.091]
-{{Optimal ET sequence|legend=1| 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|legend=0| 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 ====
@@ Line 220: / Line 233: @@
 Comma list: 50/49, 64/63, 65/63, 99/98
-Mapping: {{mapping| 2 0 11 12 26 1 | 0 1 -2 -2 -6 2 }}
+{{Mapping|legend=0| 2 0 11 12 26 1 | 0 1 -2 -2 -6 2 }}
-Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 708.919
+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=1| 10e, 12, 22 }}
+{{Optimal ET sequence|legend=0| 10e, 12, 22 }}
-Badness: 0.027642
+Badness (Sintel): 1.14
 ===== 17-limit =====
@@ Line 233: / Line 248: @@
 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 }}
+{{Mapping|legend=0| 2 0 11 12 26 1 5 | 0 1 -2 -2 -6 2 1 }}
-Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 708.806
+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=1| 10e, 12, 22 }}
+{{Optimal ET sequence|legend=0| 10e, 12, 22 }}
-Badness: 0.020899
+Badness (Sintel): 1.06
 ==== Pajarina ====
@@ Line 246: / Line 263: @@
 Comma list: 50/49, 64/63, 78/77, 99/98
-Mapping: {{mapping| 2 0 11 12 26 36 | 0 1 -2 -2 -6 -9 }}
+{{Mapping|legend=0| 2 0 11 12 26 36 | 0 1 -2 -2 -6 -9 }}
-Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 706.133
+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=1| 12f, 22, 34d }}
+{{Optimal ET sequence|legend=0| 12f, 22, 34d }}
-Badness: 0.022327
+Badness (Sintel): 0.923
 ===== 17-limit =====
@@ Line 259: / Line 278: @@
 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 }}
+{{Mapping|legend=0| 2 0 11 12 26 36 5 | 0 1 -2 -2 -6 -9 1 }}
-Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 706.410
+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=1| 12f, 22, 34d }}
+{{Optimal ET sequence|legend=0| 12f, 22, 34d }}
-Badness: 0.018375
+Badness (Sintel): 0.936
 ==== Pajarita ====
@@ Line 272: / Line 293: @@
 Comma list: 40/39, 50/49, 64/63, 66/65
-Mapping: {{mapping| 2 0 11 12 26 17 | 0 1 -2 -2 -6 -3 }}
+{{Mapping|legend=0| 2 0 11 12 26 17 | 0 1 -2 -2 -6 -3 }}
-Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 707.450
+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=1| 10e, 12f, 22f }}
+{{Optimal ET sequence|legend=0| 10e, 12f, 22f, 34dff }}
-Badness: 0.022677
+Badness (Sintel): 0.937
 ===== 17-limit =====
@@ Line 285: / Line 308: @@
 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 }}
+{{Mapping|legend=0| 2 0 11 12 26 17 5 | 0 1 -2 -2 -6 -3 1 }}
-Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 707.947
+Optimal tunings:
+* WE: ~7/5 = 598.6103{{c}}, ~3/2 = 706.3076{{c}}
+* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 708.2256{{c}}
-{{Optimal ET sequence|legend=1| 10e, 12f, 22f }}
+{{Optimal ET sequence|legend=0| 10e, 12f, 22f }}
-Badness: 0.019007
+Badness (Sintel): 0.968
 === Pajarous ===
@@ Line 298: / Line 323: @@
 Comma list: 50/49, 55/54, 64/63
-Mapping: {{mapping| 2 0 11 12 -9 | 0 1 -2 -2 5 }}
+{{Mapping|legend=0| 2 0 11 12 -9 | 0 1 -2 -2 5 }}
-Optimal tuning (POTE): ~7/5 = 1\2, ~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:
 * 11-odd-limit diamond monotone: ~3/2 = 709.091 (13\22)
 * 11-odd-limit diamond tradeoff: ~3/2 = [701.955, 715.803]
-* 11-odd-limit diamond monotone and tradeoff: ~3/2 = 709.091
-{{Optimal ET sequence|legend=1| 10, 12e, 22, 120bce, 142bce }}
+{{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
+Comma list: 50/49, 52/51, 55/54, 64/63, 65/63
+{{Mapping|legend=0| 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: 0.028349
+Badness (Sintel): 0.766
 ==== 13-limit ====
@@ Line 316: / Line 357: @@
 Comma list: 50/49, 55/54, 64/63, 65/63
-Mapping: {{mapping| 2 0 11 12 -9 1 | 0 1 -2 -2 5 2 }}
+{{Mapping|legend=0| 2 0 11 12 -9 1 | 0 1 -2 -2 5 2 }}
-Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 710.240
+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=1| 10, 22, 54f, 76bdff }}
+{{Optimal ET sequence|legend=0| 10, 22 }}
-Badness: 0.025176
+Badness (Sintel): 1.04
 ===== 17-limit =====
@@ Line 329: / Line 372: @@
 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 }}
+{{Mapping|legend=0| 2 0 11 12 -9 1 5 | 0 1 -2 -2 5 2 1 }}
-Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 710.221
+Optimal tunings:
+* WE: ~7/5 = 599.8239{{c}}, ~3/2 = 710.0128{{c}}
+* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 710.2067{{c}}
-{{Optimal ET sequence|legend=1| 10, 22, 54f, 76bdff }}
+{{Optimal ET sequence|legend=0| 10, 22, 54f, 76bdff }}
-Badness: 0.018249
+Badness (Sintel): 0.930
 ==== Pajaro ====
@@ Line 342: / Line 387: @@
 Comma list: 40/39, 50/49, 55/54, 64/63
-Mapping: {{mapping| 2 0 11 12 -9 17 | 0 1 -2 -2 5 -3 }}
+{{Mapping|legend=0| 2 0 11 12 -9 17 | 0 1 -2 -2 5 -3 }}
-Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 710.818
+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=1| 10, 22f, 32f, 54ff }}
+{{Optimal ET sequence|legend=0| 10, 22f, 32f }}
-Badness: 0.027355
+Badness (Sintel): 1.13
 ===== 17-limit =====
@@ Line 355: / Line 402: @@
 Comma list: 40/39, 50/49, 55/54, 64/63, 85/84
-Mapping: {{mapping| 2 0 11 12 -9 17 5 | 0 1 -2 -2 5 -3 1 }}
+{{Mapping|legend=0| 2 0 11 12 -9 17 5 | 0 1 -2 -2 5 -3 1 }}
-Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 710.866
+Optimal tunings:
+* WE: ~7/5 = 598.8865{{c}}, ~3/2 = 709.5472{{c}}
+* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 710.8704{{c}}
-{{Optimal ET sequence|legend=1| 10, 22f, 32f, 54ff }}
+{{Optimal ET sequence|legend=0| 10, 22f, 32f }}
-Badness: 0.019844
+Badness (Sintel): 1.01
 === Pajaric ===
@@ Line 368: / Line 417: @@
 Comma list: 45/44, 50/49, 56/55
-Mapping: {{mapping| 2 0 11 12 7 | 0 1 -2 -2 0 }}
+{{Mapping|legend=0| 2 0 11 12 7 | 0 1 -2 -2 0 }}
-Optimal tuning (POTE): ~7/5 = 1\2, ~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}}
-{{Optimal ET sequence|legend=1| 10, 12, 22e, 34dee }}
+{{Optimal ET sequence|legend=0| 10, 12, 22e }}
-Badness: 0.023798
+Badness (Sintel): 0.787
 ==== 13-limit ====
@@ Line 381: / Line 432: @@
 Comma list: 40/39, 45/44, 50/49, 56/55
-Mapping: {{mapping| 2 0 11 12 7 17 | 0 1 -2 -2 0 -3 }}
+{{Mapping|legend=0| 2 0 11 12 7 17 | 0 1 -2 -2 0 -3 }}
-Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 707.442
+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=1| 10, 12f, 22ef }}
+{{Optimal ET sequence|legend=0| 10, 12f, 22ef }}
-Badness: 0.020461
+Badness (Sintel): 0.845
 ==== 17-limit ====
@@ Line 394: / Line 447: @@
 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 }}
+{{Mapping|legend=0| 2 0 11 12 7 17 5 | 0 1 -2 -2 0 -3 1 }}
-Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 708.544
+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=1| 10, 12f, 22ef }}
+{{Optimal ET sequence|legend=0| 10, 12f, 22ef }}
-Badness: 0.017592
+Badness (Sintel): 0.896
 === Hemipaj ===
@@ Line 407: / Line 462: @@
 Comma list: 50/49, 64/63, 121/120
-Mapping: {{mapping| 2 1 9 10 8 | 0 2 -4 -4 -1 }}
+{{Mapping|legend=0| 2 1 9 10 8 | 0 2 -4 -4 -1 }}
+: mapping generators: ~2, ~16/11
-Optimal tuning (POTE): ~7/5 = 1\2, ~11/8 = 546.383
+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=1| 20, 22, 68d, 90d }}
+{{Optimal ET sequence|legend=0| 2, 20, 22 }}
-Badness: 0.038890
+Badness (Sintel): 1.29
 === Hemifourths ===
@@ Line 420: / Line 479: @@
 Comma list: 50/49, 64/63, 243/242
-Mapping: {{mapping| 2 0 11 12 -1 | 0 2 -4 -4 5 }}
+{{Mapping|legend=0| 2 0 11 12 -1 | 0 2 -4 -4 5 }}
+: mapping generators: ~2, ~55/32
-Optimal tuning (POTE): ~7/5 = 1\2, ~55/32 = 953.093
+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=1| 10, 24d, 34d }}
+{{Optimal ET sequence|legend=0| 10, 24d, 34d }}
-Badness: 0.048885
+Badness (Sintel): 1.62
 ==== 13-limit ====
@@ Line 433: / Line 495: @@
 Comma list: 50/49, 64/63, 78/77, 144/143
-Mapping: {{mapping| 2 0 11 12 -1 9 | 0 2 -4 -4 5 -1 }}
+{{Mapping|legend=0| 2 0 11 12 -1 9 | 0 2 -4 -4 5 -1 }}
-Optimal tuning (POTE): ~7/5 = 1\2, ~26/15 = 953.074
+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=1| 10, 24d, 34d }}
+{{Optimal ET sequence|legend=0| 10, 24d, 34d }}
-Badness: 0.028755
+Badness (Sintel): 1.19
 ==== 17-limit ====
@@ Line 446: / Line 510: @@
 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 }}
+{{Mapping|legend=0| 2 0 11 12 -1 9 5 | 0 2 -4 -4 5 -1 2 }}
-Optimal tuning (POTE): ~7/5 = 1\2, ~26/15 = 953.210
+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=1| 10, 24d, 34d }}
+{{Optimal ET sequence|legend=0| 10, 24d, 34d }}
-Badness: 0.021790
+Badness (Sintel): 1.11
-== Diaschismic ==
+== Srutal ==
 {{See also| Srutal vs diaschismic }}
-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 a 1/2-octave 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 58edo.
+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]].
-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. 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.
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 126/125, 2048/2025
+[[Comma list]]: 2048/2025, 4375/4374
-{{Mapping|legend=1| 2 0 11 31 | 0 1 -2 -8 }}
+{{Mapping|legend=1| 2 0 11 -42 | 0 1 -2 15 }}
-{{Multival|legend=1| 2 -4 -16 -11 -31 -26 }}
-[[Optimal tuning]] ([[POTE]]): ~45/32 = 1\2, ~3/2 = 703.681
+[[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 = [700.000, 705.882] (7\12 to 20\34)
+* 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]
-* 7- and 9-odd-limit diamond monotone and tradeoff: ~3/2 = [701.955, 705.882]
-{{Optimal ET sequence|legend=1| 12, 46, 58, 104c, 162c }}
+{{Optimal ET sequence|legend=1| 34d, 46, 80, 126, 206cd, 332bcd }}
-[[Badness]]: 0.037914
+[[Badness]] (Sintel): 2.32
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 126/125, 176/175, 896/891
+Comma list: 176/175, 896/891, 1331/1323
-Mapping: {{mapping| 2 0 11 31 45  0 1 -2 -8 -12 }}
+{{Mapping|legend=0| 2 0 11 -42 -28 | 0 1 -2 15 11 }}
-Optimal tuning (POTE): ~45/32 = 1\2, ~3/2 = 703.714
+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 = [700.000, 704.348] (7\12 to 27\46)
+* 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]
-* 11-odd-limit diamond monotone and tradeoff: ~3/2 = [701.955, 704.348]
-{{Optimal ET sequence|legend=1| 12, 46, 58, 104c, 162ce }}
+{{Optimal ET sequence|legend=0| 34d, 46, 80, 126, 206cd }}
-Badness: 0.025034
+Badness (Sintel): 1.17
 === 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 126/125, 176/175, 196/195, 364/363
+Comma list: 169/168, 176/175, 325/324, 364/363
-Mapping: {{mapping| 2 0 11 31 45 55 | 0 1 -2 -8 -12 -15 }}
+{{Mapping|legend=0| 2 0 11 -42 -28 -18 | 0 1 -2 15 11 8 }}
-Optimal tuning (POTE): ~45/32 = 1\2, ~3/2 = 703.704
+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 = [703.448, 704.348] (34\58 to 27\46)
+* 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]
-* 13- and 15-odd-limit diamond monotone and tradeoff: ~3/2 = [703.448, 704.348]
-{{Optimal ET sequence|legend=1| 46, 58, 104c, 162cef }}
+{{Optimal ET sequence|legend=0| 34d, 46, 80 }}
-Badness: 0.018926
+Badness (Sintel): 1.04
 === 17-limit ===
 Subgroup: 2.3.5.7.11.13.17
-Comma list: 126/125, 136/135, 176/175, 196/195, 256/255
+Comma list: 136/135, 169/168, 176/175, 221/220, 256/255
-Mapping: {{mapping| 2 0 11 31 45 55 5 | 0 1 -2 -8 -12 -15 1 }}
+{{Mapping|legend=0| 2 0 11 -42 -28 -18 5 | 0 1 -2 15 11 8 1 }}
-Optimal tuning (POTE): ~17/12 = 1\2, ~3/2 = 703.812
+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 = [703.448, 704.348] (34\58 to 27\46)
+* 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]
-* 17-odd-limit diamond monotone and tradeoff: ~3/2 = [703.448, 704.348]
-{{Optimal ET sequence|legend=1| 46, 58, 104c }}
+{{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|legend=0| 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|legend=0| 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|legend=0| 2 0 11 -42 -28 -18 5 -55 -10 -76 | 0 1 -2 15 11 8 1 20 6 27 }}
-Badness: 0.016425
+Optimal tunings:
+* WE: ~17/12 = 599.6664{{c}}, ~3/2 = 704.5138{{c}}
+* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 704.8807{{c}}
-==== Na"Naa' ====
+{{Optimal ET sequence|legend=0| 34dhj, 46, 80 }}
-<b>Na"Naa'</b> is a remarkable subgroup temperament of 46&amp;58 with a prime harmonic of 23.
-Subgroup: 2.3.5.7.11.13.17.23
+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
-Comma list: 126/125, 136/135, 176/175, 196/195, 231/230, 256/255
+{{Mapping|legend=0| 2 0 11 -42 -28 -18 5 -55 -10 -76 48 | 0 1 -2 15 11 8 1 20 6 27 -12 }}
-Sval mapping: {{mapping| 2 0 11 31 45 55 5 63 | 0 1 -2 -8 -12 -15 1 -17 }}
+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 tuning (POTE): ~17/12 = 1\2, ~3/2 = 703.870
+{{Optimal ET sequence|legend=0| 46, 80, 126 }}
-{{Optimal ET sequence|legend=1| 46, 58i, 104ci }}
+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 &amp; 56 temperament. [[78edo]] 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.
+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
@@ Line 557: / Line 674: @@
 {{Mapping|legend=1| 2 0 11 -23 | 0 1 -2 9 }}
-{{Multival|legend=1| 2 -4 18 -11 23 53 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~45/32 = 599.6603{{c}}, ~3/2 = 707.1707{{c}}
-[[Optimal tuning]] ([[POTE]]): ~45/32 = 1\2, ~3/2 = 707.571
+: [[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, 212b, 290bb }}
+{{Optimal ET sequence|legend=1| 22, 56, 78, 134b }}
-[[Badness]]: 0.083971
+[[Badness]] (Sintel): 2.13
 === 11-limit ===
@@ Line 570: / Line 689: @@
 Comma list: 100/99, 385/384, 1232/1215
-Mapping: {{mapping| 2 0 11 -23 26 | 0 1 -2 9 -6 }}
+{{Mapping|legend=0| 2 0 11 -23 26 | 0 1 -2 9 -6 }}
-Optimal tuning (POTE): ~45/32 = 1\2, ~3/2 = 707.609
+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=1| 22, 56, 78, 212be, 290bbe }}
+{{Optimal ET sequence|legend=0| 22, 56, 78 }}
-Badness: 0.045270
+Badness (Sintel): 1.50
 ==== 13-limit ====
@@ Line 583: / Line 704: @@
 Comma list: 100/99, 105/104, 144/143, 1078/1053
-Mapping: {{mapping| 2 0 11 -23 26 -18 | 0 1 -2 9 -6 8 }}
+{{Mapping|legend=0| 2 0 11 -23 26 -18 | 0 1 -2 9 -6 8 }}
-Optimal tuning (POTE): ~45/32 = 1\2, ~3/2 = 707.167
+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=1| 22f, 34, 56f }}
+{{Optimal ET sequence|legend=0| 22f, 34, 56f }}
-Badness: 0.044877
+Badness (Sintel): 1.85
 ===== 17-limit =====
@@ Line 596: / Line 719: @@
 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 }}
+{{Mapping|legend=0| 2 0 11 -23 26 -18 5 | 0 1 -2 9 -6 8 1}}
-Optimal tuning (POTE): ~17/12 = 1\2, ~3/2 = 707.155
+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=1| 22f, 34, 56f }}
+{{Optimal ET sequence|legend=0| 22f, 34, 56f }}
-Badness: 0.030297
+Badness (Sintel): 1.54
 ==== Keenic ====
@@ Line 609: / Line 734: @@
 Comma list: 91/90, 100/99, 352/351, 385/384
-Mapping: {{mapping| 2 0 11 -23 26 36 | 0 1 -2 9 -6 -9 }}
+{{Mapping|legend=0| 2 0 11 -23 26 36 | 0 1 -2 9 -6 -9 }}
-Optimal tuning (POTE): ~45/32 = 1\2, ~3/2 = 707.257
+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=1| 22, 34, 56 }}
+{{Optimal ET sequence|legend=0| 22, 34, 56 }}
-Badness: 0.040351
+Badness (Sintel): 1.67
 ===== 17-limit =====
@@ Line 622: / Line 749: @@
 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 }}
+{{Mapping|legend=0| 2 0 11 -23 26 36 5 | 0 1 -2 9 -6 -9 1 }}
-Optimal tuning (POTE): ~17/12 = 1\2, ~3/2 = 707.252
+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=1| 22, 34, 56 }}
+{{Optimal ET sequence|legend=0| 22, 34, 56 }}
-Badness: 0.026917
+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 &amp; 56 temperament.
+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
@@ Line 638: / Line 767: @@
 {{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}}
-[[Optimal tuning]] ([[POTE]]): ~25/21 = 1\4, ~3/2 = 705.364
+: [[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 }}
+{{Optimal ET sequence|legend=1| 12, …, 56, 68, 80, 148d }}
-[[Badness]]: 0.056474
+[[Badness]] (Sintel): 1.43
 === 11-limit ===
@@ Line 652: / Line 784: @@
 Comma list: 176/175, 896/891, 1375/1372
-Mapping: {{mapping| 4 0 22 43 71 | 0 1 -2 -5 -9 }}
+{{Mapping|legend=0| 4 0 22 43 71 | 0 1 -2 -5 -9 }}
-Optimal tuning (POTE): ~25/21 = 1\4, ~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}}
-{{Optimal ET sequence|legend=1| 12, 68, 80 }}
+{{Optimal ET sequence|legend=0| 12, 56e, 68, 80 }}
-Badness: 0.040191
+Badness (Sintel): 1.33
 === 13-limit ===
@@ Line 665: / Line 799: @@
 Comma list: 176/175, 325/324, 640/637, 896/891
-Mapping: {{mapping| 4 0 22 43 71 -36 | 0 1 -2 -5 -9 8 }}
+{{Mapping|legend=0| 4 0 22 43 71 -36 | 0 1 -2 -5 -9 8 }}
-Optimal tuning (POTE): ~25/21 = 1\4, ~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}}
-{{Optimal ET sequence|legend=1| 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 ===
@@ Line 678: / Line 814: @@
 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 }}
+{{Mapping|legend=0| 4 0 22 43 71 -36 10 | 0 1 -2 -5 -9 8 1 }}
-Optimal tuning (POTE): ~25/21 = 1\4, ~3/2 = 705.334
+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=1| 12, 68, 80, 148d, 228bcd, 376bbcddf }}
+{{Optimal ET sequence|legend=0| 12, 68, 80, 148d }}
-Badness: 0.028631
+Badness (Sintel): 1.46
 === 19-limit ===
@@ Line 691: / Line 829: @@
 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 }}
+{{Mapping|legend=0| 4 0 22 43 71 -36 10 17 | 0 1 -2 -5 -9 8 1 0 }}
-Optimal tuning (POTE): ~19/16 = 1\4, ~3/2 = 705.339
+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=1| 12, 68, 80, 148d, 376bbcddfh }}
+{{Optimal ET sequence|legend=0| 12, 68, 80, 148d }}
-Badness: 0.020590
+Badness (Sintel): 1.25
-== Echidna ==
+=== 23-limit ===
-Echidna adds 1728/1715 to the commas and takes 9/7 as a generator. It may be called the 22 &amp; 58 temperament. [[58edo]] or [[80edo]] make for good tunings, or their vals can be added to {{val| 138 219 321 388 }}.
+Subgroup: 2.3.5.7.11.13.17.19.23
-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: 136/135, 176/175, 190/189, 253/252, 256/255, 325/324, 640/637
-[[Subgroup]]: 2.3.5.7
+{{Mapping|legend=0| 4 0 22 43 71 -36 10 17 -20 | 0 1 -2 -5 -9 8 1 0 6 }}
-[[Comma list]]: 1728/1715, 2048/2025
+Optimal tunings:
+* WE: ~19/16 = 299.7961{{c}}, ~3/2 = 704.8577{{c}}
+* CWE: ~19/16 = 300.0000{{c}}, ~3/2 = 705.3413{{c}}
-{{Mapping|legend=1| 2 1 9 2 | 0 3 -6 5 }}
+{{Optimal ET sequence|legend=0| 12, 68, 80, 148di }}
-{{Multival|legend=1| 6 -12 10 -33 -1 57 }}
+Badness (Sintel): 1.24
-[[Optimal tuning]] ([[POTE]]): ~45/32 = 1\2, ~9/7 = 434.856
+== 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.
-{{Optimal ET sequence|legend=1| 22, 58, 80, 138cd, 218cd }}
+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.
-[[Badness]]: 0.058033
+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-limit ===
+[[Subgroup]]: 2.3.5.7
-Subgroup: 2.3.5.7.11
-Comma list: 176/175, 540/539, 896/891
+[[Comma list]]: 245/243, 2048/2025
-Mapping: {{mapping| 2 1 9 2 12 | 0 3 -6 5 -7 }}
+{{Mapping|legend=1| 2 1 9 -2 | 0 2 -4 7 }}
+: mapping generators: ~45/32, ~35/24
-Optimal tuning (POTE): ~45/32 = 1\2, ~9/7 = 434.852
+[[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 }}
-Minimax tuning:
+{{Optimal ET sequence|legend=1| 22, 46, 68, 182b, 250bc }}
-* 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 }}]
-: Eigenmonzo (unchanged-interval) basis: 2.11/7
-{{Optimal ET sequence|legend=1| 22, 58, 80, 138cde, 218cde }}
+[[Badness]] (Sintel): 1.20
-Badness: 0.025987
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-=== 13-limit ===
+Comma list: 121/120, 176/175, 245/243
-Subgroup: 2.3.5.7.11.13
-Comma list: 176/175, 351/350, 364/363, 540/539
+{{Mapping|legend=0| 2 1 9 -2 8 | 0 2 -4 7 -1 }}
-Mapping: {{mapping| 2 1 9 2 12 19 | 0 3 -6 5 -7 -16 }}
+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 tuning (POTE): ~45/32 = 1\2, ~9/7 = 434.756
+{{Optimal ET sequence|legend=0| 22, 46, 68, 114 }}
-{{Optimal ET sequence|legend=1| 22, 58, 80, 138cde }}
+Badness (Sintel): 0.876
-Badness: 0.023679
+=== 13-limit ===
+Note that an alternative, simpler mapping for prime 13 exists at -7 gens, as in [[#Shru]]. This mapping is especially valuable when considering shrutar as a no-7's 17-limit temperament (22 & 24, supported by 46, 68f and 70), as the complexity from mapping 7 at +7 gens is one of the motivations for picking a +16 gen mapping for 13. This temperament isn't as accurate, but it is simple, and makes many [[MOS]]es, specifically 2L 2''n''s for ''n'' up to [[2L 20s|10]], followed by [[22L 2s]] for the 24-note MOS, with a quarter-tone generator equivalent to ~11/8.
-=== 17-limit ===
+Subgroup: 2.3.5.7.11.13
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 136/135, 176/175, 221/220, 256/255, 540/539
+Comma list: 121/120, 176/175, 196/195, 245/243
-Mapping: {{mapping| 2 1 9 2 12 19 6 | 0 3 -6 5 -7 -16 3 }}
+{{Mapping|legend=0| 2 1 9 -2 8 -10 | 0 2 -4 7 -1 16 }}
-Optimal tuning (POTE): ~17/12 = 1\2, ~9/7 = 434.816
+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=1| 22, 58, 80, 138cde }}
+{{Optimal ET sequence|legend=0| 22f, 46, 68, 114 }}
-Badness: 0.020273
+Badness (Sintel): 1.16
-== Echidnic ==
+==== 17-limit ====
-[[Subgroup]]: 2.3.5.7
+Subgroup: 2.3.5.7.11.13.17
-[[Comma list]]: 686/675, 1029/1024
+Comma list: 121/120, 136/135, 154/153, 176/175, 196/195
-{{Mapping|legend=1| 2 2 7 6 | 0 3 -6 -1 }}
+{{Mapping|legend=0| 2 1 9 -2 8 -10 6 | 0 2 -4 7 -1 16 2 }}
-{{Multival|legend=1| 6 -12 -2 -33 -20 29 }}
+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 tuning]] ([[POTE]]): ~45/32 = 1\2, ~8/7 = 234.492
+{{Optimal ET sequence|legend=0| 22f, 46, 68, 114 }}
-{{Optimal ET sequence|legend=1| 10, 36, 46, 194bcd, 240bcd, 286bcd, 332bccdd }}
+Badness (Sintel): 0.953
-[[Badness]]: 0.072246
+==== 19-limit ====
+Subgroup: 2.3.5.7.11.13.17.19
-=== 11-limit ===
+Comma list: 121/120, 136/135, 154/153, 176/175, 196/195, 343/342
-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 }}
+{{Mapping|legend=0| 2 1 9 -2 8 -10 6 -10 | 0 2 -4 7 -1 16 2 17 }}
-Optimal tuning (POTE): ~45/32 = 1\2, ~8/7 = 235.096
+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=1| 10, 36e, 46, 102, 148, 342bcdd }}
+{{Optimal ET sequence|legend=0| 22fh, 46, 68, 114, 182bef }}
-Badness: 0.045127
+Badness (Sintel): 1.07
-=== 13-limit ===
+==== 23-limit ====
-Subgroup: 2.3.5.7.11.13
+Subgroup: 2.3.5.7.11.13.17.19.23
-Comma list: 91/90, 169/168, 385/384, 441/440
+Comma list: 121/120, 136/135, 154/153, 176/175, 196/195, 253/252, 343/342
-Mapping: {{mapping| 2 2 7 6 3 7 | 0 3 -6 -1 10 1 }}
+{{Mapping|legend=0| 2 1 9 -2 8 -10 6 -10 -4 | 0 2 -4 7 -1 16 2 17 12 }}
-Optimal tuning (POTE): ~45/32 = 1\2, ~8/7 = 235.088
+Optimal tunings:
+* WE: ~17/12 = 599.7879{{c}}, ~16/11 = 652.4776{{c}}
+* CWE: ~17/12 = 600.0000{{c}}, ~16/11 = 652.6926{{c}}
-{{Optimal ET sequence|legend=1| 10, 46, 102, 148f, 194bcdf }}
+{{Optimal ET sequence|legend=0| 22fh, 46, 68, 114 }}
-Badness: 0.028874
+Badness (Sintel): 1.03
-=== 17-limit ===
+== Shru ==
-Subgroup: 2.3.5.7.11.13.17
+Shru tempers out 392/375 and slices the compound semitone into two generators of ~10/7. Its ploidacot is diploid alpha-dicot, the same as shrutar. It can be seen as one way of merging [[parapyth]] with [[diaschismic]] (and hence as a lower-accuracy rank 2 tuning of [[varda]]).
-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 tuning (POTE): ~17/12 = 1\2, ~8/7 = 235.088
-{{Optimal ET sequence|legend=1| 10, 46, 102, 148f, 194bcdf }}
-Badness: 0.019304
-; Compositions
-* [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]]
-== Shrutar ==
-Shrutar adds 245/243 to the commas, and also tempers out 6144/6125. 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 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.
+Note that shru is especially efficient as a no-7's 17-limit temperament, as its 7 is the highest damage, and without it, the other primes can be tuned more accurately.
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 245/243, 2048/2025
+[[Comma list]]: 392/375, 1323/1280
-{{Mapping|legend=1| 2 1 9 -2 | 0 2 -4 7 }}
+{{Mapping|legend=1| 2 1 9 11 | 0 2 -4 -5 }}
+: mapping generators: ~45/32, ~10/7
-{{Multival|legend=1| 4 -8 14 -22 11 55 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~45/32 = 600.2519{{c}}, ~10/7 = 650.4083{{c}}
+: [[error map]]: {{val| +0.504 -0.887 +14.321 -18.096 }}
+* [[CWE]]: ~45/32 = 600.0000{{c}}, ~10/7 = 650.1017{{c}}
+: error map: {{val| 0.000 -1.752 +13.279 -19.334 }}
-[[Optimal tuning]] ([[POTE]]): ~45/32 = 1\2, ~35/24 = 652.811
+{{Optimal ET sequence|legend=1| 2, 22d, 24 }}
-{{Optimal ET sequence|legend=1| 22, 46, 68, 182b, 250bc }}
-[[Badness]]: 0.047377
+[[Badness]] (Sintel): 3.99
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 121/120, 176/175, 245/243
+Comma list: 56/55, 77/75, 1323/1280
-Mapping: {{mapping| 2 1 9 -2 8 | 0 2 -4 7 -1 }}
+{{Mapping|legend=0| 2 1 9 11 8 | 0 2 -4 -5 -1 }}
-Optimal tuning (POTE): ~45/32 = 1\2, ~16/11 = 652.680
+Optimal tunings:
+* WE: ~17/12 = 600.2356{{c}}, ~10/7 = 650.3856{{c}}
+* CWE: ~17/12 = 600.0000{{c}}, ~10/7 = 650.1008{{c}}
-{{Optimal ET sequence|legend=1| 22, 46, 68, 114, 296bce, 410bce }}
+{{Optimal ET sequence|legend=0| 2, 22d, 24 }}
-Badness: 0.026489
+Badness (Sintel): 2.10
 === 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 121/120, 176/175, 196/195, 245/243
+Comma list: 56/55, 77/75, 105/104, 507/500
-Mapping: {{mapping| 2 1 9 -2 8 -10 | 0 2 -4 7 -1 16 }}
+{{Mapping|legend=0| 2 1 9 11 8 15 | 0 2 -4 -5 -1 -7 }}
-Optimal tuning (POTE): ~45/32 = 1\2, ~16/11 = 652.654
+Optimal tunings:
+* WE: ~45/32 = 599.9067{{c}}, ~10/7 = 649.4907{{c}}
+* CWE: ~45/32 = 600.0000{{c}}, ~10/7 = 649.5950{{c}}
-{{Optimal ET sequence|legend=1| 22f, 24f, 46, 68, 114 }}
+{{Optimal ET sequence|legend=0| 2, 24 }}
-Badness: 0.028057
+Badness (Sintel): 2.12
-=== 17-limit ===
+== Sruti ==
-Subgroup: 2.3.5.7.11.13.17
+{{Redirect|Sruti|the tuning concept in Indian music|Shruti}}
-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 tuning (POTE): ~17/12 = 1\2, ~16/11 = 652.647
-{{Optimal ET sequence|legend=1| 22f, 24f, 46, 68, 114 }}
-Badness: 0.018716
-=== 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 tuning (POTE): ~17/12 = 1\2, ~16/11 = 652.730
-{{Optimal ET sequence|legend=1| 22fh, 24fh, 46, 68, 114, 182bef }}
-Badness: 0.017540
+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.
-== Sruti ==
 [[Subgroup]]: 2.3.5.7
@@ Line 896: / Line 1,017: @@
 {{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}}
-[[Optimal tuning]] ([[POTE]]): ~45/32 = 1\2, ~140/81 = 951.876
+: [[error map]]: {{val| -1.447 -0.498 +2.813 +1.497 }}
+* [[CWE]]: ~45/32 = 600.0000{{c}}, ~140/81 = 951.8227{{c}}
+: error map: {{val| 0.000 +1.690 +6.395 +4.869 }}
 {{Optimal ET sequence|legend=1| 24, 34d, 58, 150cd, 208ccdd, 266ccdd }}
-[[Badness]]: 0.117358
+[[Badness]] (Sintel): 2.97
 === 11-limit ===
@@ Line 910: / Line 1,034: @@
 Comma list: 176/175, 243/242, 896/891
-Mapping: {{mapping| 2 0 11 -15 -1 | 0 2 -4 13 5 }}
+{{Mapping|legend=0| 2 0 11 -15 -1 | 0 2 -4 13 5 }}
-Optimal tuning (POTE): ~45/32 = 1\2, ~121/70 = 951.863
+Optimal tunings:
+* WE: ~45/32 = 599.1951{{c}}, ~121/70 = 950.5864{{c}}
+* CWE: ~45/32 = 600.0000{{c}}, ~121/70 = 951.7972{{c}}
-{{Optimal ET sequence|legend=1| 24, 34d, 58 }}
+{{Optimal ET sequence|legend=0| 24, 34d, 58, 150cdee, 208ccddee, 266ccddeee }}
-Badness: 0.041459
+Badness (Sintel): 1.37
 === 13-limit ===
@@ Line 923: / Line 1,049: @@
 Comma list: 144/143, 176/175, 351/350, 676/675
-Mapping: {{mapping| 2 0 11 -15 -1 9 | 0 2 -4 13 5 -1 }}
+{{Mapping|legend=0| 2 0 11 -15 -1 9 | 0 2 -4 13 5 -1 }}
-Optimal tuning (POTE): ~45/32 = 1\2, ~26/15 = 951.886
+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=1| 24, 34d, 58, 150cdeef, 208ccddeeff }}
+{{Optimal ET sequence|legend=0| 24, 34d, 58, 150cdeef, 208ccddeeff, 266ccddeeefff }}
-Badness: 0.023791
+Badness (Sintel): 0.983
 === 17-limit ===
@@ Line 936: / Line 1,064: @@
 Comma list: 136/135, 144/143, 170/169, 176/175, 221/220
-Mapping: {{mapping| 2 0 11 -15 -1 9 5 | 0 2 -4 13 5 -1 2 }}
+{{Mapping|legend=0| 2 0 11 -15 -1 9 5 | 0 2 -4 13 5 -1 2 }}
-Optimal tuning (POTE): ~17/12 = 1\2, ~26/15 = 951.857
+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=1| 24, 34d, 58 }}
+{{Optimal ET sequence|legend=0| 24, 34d, 58 }}
-Badness: 0.020536
+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
@@ Line 950: / Line 1,082: @@
 {{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}}
-[[Optimal tuning]] ([[POTE]]): ~45/32 = 1\2, ~7/4 = 953.021
+: [[error map]]: {{val| +0.552 +4.964 +2.883 -14.264 }}
+* [[CWE]]: ~45/32 = 600.0000{{c}}, ~7/4 = 953.0188{{c}}
+: error map: {{val| 0.000 +4.083 +1.611 -15.807 }}
 {{Optimal ET sequence|legend=1| 10, 24, 34 }}
-[[Badness]]: 0.077955
+[[Badness]] (Sintel): 1.97
 === 11-limit ===
@@ Line 964: / Line 1,099: @@
 Comma list: 49/48, 56/55, 243/242
-Mapping: {{mapping| 2 0 11 4 -1 | 0 2 -4 1 5 }}
+{{Mapping|legend=0| 2 0 11 4 -1 | 0 2 -4 1 5 }}
-Optimal tuning (POTE): ~45/32 = 1\2, ~7/4 = 952.184
+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=1| 10, 24, 34, 58d, 92de }}
+{{Optimal ET sequence|legend=0| 10, 24, 34 }}
-Badness: 0.049253
+Badness (Sintel): 1.63
 === 13-limit ===
@@ Line 977: / Line 1,114: @@
 Comma list: 49/48, 56/55, 91/90, 243/242
-Mapping: {{mapping| 2 0 11 4 -1 9 | 0 2 -4 1 5 -1 }}
+{{Mapping|legend=0| 2 0 11 4 -1 9 | 0 2 -4 1 5 -1 }}
-Optimal tuning (POTE): ~45/32 = 1\2, ~7/4 = 952.309
+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=1| 10, 24, 34, 58d, 92ddef }}
+{{Optimal ET sequence|legend=0| 10, 24, 34, 58d, 92ddef }}
-Badness: 0.030829
+Badness (Sintel): 1.27
 === 17-limit ===
@@ Line 990: / Line 1,129: @@
 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 }}
+{{Mapping|legend=0| 2 0 11 4 -1 9 5 | 0 2 -4 1 5 -1 2 }}
+Optimal tunings:
+* WE: ~17/12 = 599.7925{{c}}, ~7/4 = 952.0004{{c}}
+* CWE: ~17/12 = 600.0000{{c}}, ~7/4 = 952.3178{{c}}
-Optimal tuning (POTE): ~17/12 = 1\2, ~7/4 = 952.330
+{{Optimal ET sequence|legend=0| 10, 24, 34 }}
-{{Optimal ET sequence|legend=1| 10, 24, 34, 58d, 92ddef }}
+Badness (Sintel): 1.10
-Badness: 0.021796
+== 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.
-== Shru ==
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 392/375, 1323/1280
+[[Comma list]]: 686/675, 1029/1024
-{{Mapping|legend=1| 2 1 9 11 | 0 2 -4 -5 }}
+{{Mapping|legend=1| 2 2 7 6 | 0 3 -6 -1 }}
+: mapping generators: ~45/32, ~8/7
-{{Multival|legend=1| 4 -8 -10 -22 -27 -1 }}
-[[Optimal tuning]] ([[POTE]]): ~45/32 = 1\2, ~10/7 = 650.135
+[[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| 2, 22d, 24 }}
+{{Optimal ET sequence|legend=1| 10, 26c, 36, 46 }}
-[[Badness]]: 0.157619
+[[Badness]] (Sintel): 1.83
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 56/55, 77/75, 1323/1280
+Comma list: 385/384, 441/440, 686/675
-Mapping: {{mapping| 2 1 9 11 8 | 0 2 -4 -5 -1 }}
+{{Mapping|legend=0| 2 2 7 6 3 | 0 3 -6 -1 10 }}
-Optimal tuning (POTE): ~45/32 = 1\2, ~10/7 = 650.130
+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=1| 2, 22d, 24 }}
+{{Optimal ET sequence|legend=0| 10, 36e, 46, 102, 148 }}
-Badness: 0.063483
+Badness (Sintel): 1.49
 === 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 56/55, 77/75, 105/104, 507/500
+Comma list: 91/90, 169/168, 385/384, 441/440
+{{Mapping|legend=0| 2 2 7 6 3 7 | 0 3 -6 -1 10 1 }}
+Optimal tunings:
+* WE: ~45/32 = 599.9570{{c}}, ~8/7 = 235.0708{{c}}
+* CWE: ~45/32 = 600.0000{{c}}, ~8/7 = 235.0862{{c}}
+{{Optimal ET sequence|legend=0| 10, 36e, 46, 102, 148f }}
+Badness (Sintel): 1.19
+=== 17-limit ===
+Subgroup: 2.3.5.7.11.13.17
+Comma list: 91/90, 136/135, 154/153, 169/168, 256/255
+{{Mapping|legend=0| 2 2 7 6 3 7 7 | 0 3 -6 -1 10 1 3 }}
-Mapping: {{mapping| 2 1 9 11 8 15 | 0 2 -4 -5 -1 -7 }}
+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 tuning (POTE): ~45/32 = 1\2, ~10/7 = 650.535
+{{Optimal ET sequence|legend=0| 10, 36e, 46, 102, 148f }}
-{{Optimal ET sequence|legend=1| 22df, 24 }}
+Badness (Sintel): 0.983
-Badness: 0.045731
+; 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
@@ Line 1,045: / Line 1,216: @@
 {{Mapping|legend=1| 2 0 11 8 | 0 4 -8 -3 }}
+: mapping generators: ~45/32, ~21/16
-{{Multival|legend=1| 8 -16 -6 -44 -32 31 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~45/32 = 599.4443{{c}}, ~21/16 = 475.7746{{c}}
-[[Optimal tuning]] ([[POTE]]): ~45/32 = 1\2, ~21/16 = 476.216
+: [[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, 38c, 48c, 58, 68, 126 }}
+{{Optimal ET sequence|legend=1| 10, …, 58, 68, 126, 446bbccd }}
-[[Badness]]: 0.073569
+[[Badness]] (Sintel): 1.86
 === 11-limit ===
@@ Line 1,059: / Line 1,233: @@
 Comma list: 176/175, 896/891, 2401/2400
-Mapping: {{mapping| 2 0 11 8 22 | 0 4 -8 -3 -19 }}
+{{Mapping|legend=0| 2 0 11 8 22 | 0 4 -8 -3 -19 }}
-Optimal tuning (POTE): ~45/32 = 1\2, ~21/16 = 476.118
+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=1| 58, 126, 184c, 310bccde }}
+{{Optimal ET sequence|legend=0| 10e, …, 58, 126, 184c, 310bccde }}
-Badness: 0.049018
+Badness (Sintel): 1.62
 ==== 13-limit ====
@@ Line 1,072: / Line 1,248: @@
 Comma list: 176/175, 196/195, 512/507, 676/675
-Mapping: {{mapping| 2 0 11 8 22 9 | 0 4 -8 -3 -19 -2 }}
+{{Mapping|legend=0| 2 0 11 8 22 9 | 0 4 -8 -3 -19 -2 }}
-Optimal tuning (POTE): ~45/32 = 1\2, ~21/16 = 476.099
+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=1| 58, 126f, 184cff }}
+{{Optimal ET sequence|legend=0| 10e, …, 58, 126f, 184cff }}
-Badness: 0.028463
+Badness (Sintel): 1.18
 ==== 17-limit ====
@@ Line 1,085: / Line 1,263: @@
 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 }}
+{{Mapping|legend=0| 2 0 11 8 22 9 5 | 0 4 -8 -3 -19 -2 4 }}
-Optimal tuning (POTE): ~17/12 = 1\2, ~21/16 = 476.162
+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=1| 58, 126f }}
+{{Optimal ET sequence|legend=0| 10e, 58, 126f }}
-Badness: 0.023820
+Badness (Sintel): 1.21
 === Quadrafourths ===
@@ Line 1,098: / Line 1,278: @@
 Comma list: 243/242, 441/440, 2048/2025
-Mapping: {{mapping| 2 0 11 8 -1 | 0 4 -8 -3 10 }}
+{{Mapping|legend=0| 2 0 11 8 -1 | 0 4 -8 -3 10 }}
-Optimal tuning (POTE): ~45/32 = 1\2, ~21/16 = 476.017
+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=1| 10, 38c, 48c, 58 }}
+{{Optimal ET sequence|legend=0| 10, 48c, 58, 184cee, 242ccdeee }}
-Badness: 0.049114
+Badness (Sintel): 1.62
 ==== 13-limit ====
@@ Line 1,111: / Line 1,293: @@
 Comma list: 144/143, 196/195, 243/242, 676/675
-Mapping: {{mapping| 2 0 11 8 -1 9 | 0 4 -8 -3 10 -2 }}
+{{Mapping|legend=0| 2 0 11 8 -1 9 | 0 4 -8 -3 10 -2 }}
-Optimal tuning (POTE): ~45/32 = 1\2, ~21/16 = 476.028
+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=1| 10, 38c, 48c, 58 }}
+{{Optimal ET sequence|legend=0| 10, 48c, 58, 126eef, 184ceeff, 242ccdeeeff }}
-Badness: 0.026743
+Badness (Sintel): 1.11
 ==== 17-limit ====
@@ Line 1,124: / Line 1,308: @@
 Comma list: 136/135, 144/143, 170/169, 196/195, 221/220
-Mapping: {{mapping| 2 0 11 8 -1 9 5 | 0 4 -8 -3 10 -2 4 }}
+{{Mapping|legend=0| 2 0 11 8 -1 9 5 | 0 4 -8 -3 10 -2 4 }}
+Optimal tunings:
+* WE: ~17/12 = 599.3353{{c}}, ~21/16 = 475.5495{{c}}
+* CWE: ~17/12 = 600.0000{{c}}, ~21/16 = 476.0691{{c}}
+{{Optimal ET sequence|legend=0| 10, 48c, 58 }}
+Badness (Sintel): 1.13
+== Subgroup extensions ==
+=== Srutal archagall (2.3.5.17) ===
+{{See also | Fiventeen }}
+This extension of 5-limit diaschismic adds prime 17 and which with respect to [[MVP archagall]] is able to express the harmonics [[75/1|75]] and [[85/1|85]] in their appropriate prime subgroup. It achieves this by equating [[85/64]] with [[4/3]] by tempering out their difference of [[256/255]] (S16). Therefore it also tempers out [[289/288]] (S17) and thus equates [[17/15]] with [[9/8]] due to tempering out [[136/135]] (S16⋅S17). It could be described as the 10 & 12 temperament with strong emphasis on 12edo being the better tuning on the 2.3.5.17 subgroup, implying ideal tunings of 34edo, 46edo or 80edo.
+Subgroup: 2.3.5.17
+Comma list: 136/135, 256/255
+{{Mapping|legend=2| 2 0 11 5 | 0 1 -2 1 }}
+: mapping generators: ~17/12, ~3
-Optimal tuning (POTE): ~17/12 = 1\2, ~21/16 = 476.077
+Optimal tunings:
+* WE: ~45/32 = 599.5585{{c}}, ~3/2 = 704.6188{{c}}
+* CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 705.1356{{c}}
-{{Optimal ET sequence|legend=1| 10, 38c, 48c, 58, 126eef, 184ceeff }}
+{{Optimal ET sequence|legend=0| 10, 12, 22, 34, 80, 114, 194bc }}
-Badness: 0.022239
+Badness (Sintel): 0.212
+[[Category:Diaschismic family| ]] <!-- main article -->
 [[Category:Temperament families]]
-[[Category:Diaschismic family| ]] <!-- main article -->
+[[Category:Catalogs of rank-2 temperaments]]
-[[Category:Diaschismic| ]] <!-- key article -->
-[[Category:Rank 2]]

Diaschismic family: Difference between revisions