@@ Line 1: / Line 1: @@
-The [[2.3.7 subgroup|2.3.7-subgroup]] [[comma]] for the '''gamelismic clan''' is the gamelisma, [[1029/1024]], with [[monzo]] {{monzo| -10 1 0 3 }}. For any member of the clan, for the rank-3 [[gamelismic family #Gamelismic|gamelismic temperament]] itself, and for the rank-2 2.3.7 temperament [[slendric]] (a.k.a. gamelic), this means three [[~]][[8/7]] intervals give a fifth, [[3/2]]. In fact, we find that 3/2 = (8/7)<sup>3</sup> × 1029/1024. From this it follows that gamelismic temperaments tend to flatten both the fifth and the harmonic seventh, or if they do not, the other of the pair must be flattened even more. [[36edo]] is a good tuning for slendric, though if the full 7-limit is desired, [[72edo]], [[77edo]] or [[118edo]] might be preferred.
+{{Technical data page}}
+The [[2.3.7 subgroup|2.3.7-subgroup]] [[comma]] for the '''gamelismic clan''' is the gamelisma, [[1029/1024]], with [[monzo]] {{monzo| -10 1 0 3 }}. For any member of the clan, for the rank-3 [[gamelismic family #Gamelismic|gamelismic temperament]] itself, and for the rank-2 2.3.7 temperament [[slendric]] (a.k.a. gamelic), this means three [[~]][[8/7]] intervals give a fifth, [[3/2]]. In fact, we find that {{nowrap| 3/2 {{=}} (8/7)<sup>3</sup>⋅(1029/1024) }}. From this it follows that gamelismic temperaments tend to flatten both the fifth and the harmonic seventh, or if they do not, the other of the pair must be flattened even more. [[36edo]] is a good tuning for slendric, though if the full 7-limit is desired, [[72edo]], [[77edo]], or [[118edo]] might be preferred.
+== Slendric ==
+{{Main| Slendric }}
+[[Subgroup]]: 2.3.7
+[[Comma list]]: 1029/1024
+{{Mapping|legend=2| 1 1 3 | 0 3 -1 }}
+{{Mapping|legend=3| 1 1 0 3 | 0 3 0 -1 }}
+: mapping generators: ~2, ~8/7
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.4859{{c}}, ~8/7 = 233.7822{{c}}
+: [[error map]]: {{val| +0.486 -0.123 -1.151 }}
+* [[CWE]]: ~2 = 1200.000{{c}}, ~8/7 = 233.7474{{c}}
+: error map: {{val| 0.000 -0.713 -2.573 }}
+{{Optimal ET sequence|legend=1| 5, 21, 26, 31, 36, 77, 113, 190 }}
+[[Badness]] (Sintel): 0.158
+=== Overview to extensions ===
+==== Full 7-limit extensions ====
 To the gamelisma itself we need to add the comma which appears next on the modified [[Normal lists #Normal interval list|normal comma list]] for the full 7-limit. The second comma on the list for mothra is [[81/80]], for rodan [[245/243]], for guiron [[32805/32768]], for gorgo [[36/35]], and for gidorah [[256/245]]. These all use ~8/7 as a generator, though in the case of gidorah that is the same as ~6/5.
@@ Line 6: / Line 31: @@
 Full 7-limit temperaments discussed elsewhere are:
+* [[Blackwood]] (+28/27) → [[Limmic temperaments #Blackwood|Limmic temperaments]]
 * [[Lemba]] (+50/49) → [[Jubilismic clan #Lemba|Jubilismic clan]]
-* ''[[Echidnic]]'' (+686/675} → [[Diaschismic family #Echidnic|Diaschismic family]]
+* [[Trisected]] (+128/125) → [[Augmented family #Trisected|Augmented family]]
-* [[Valentine]] (+126/125) → [[Starling temperaments #Valentine|Starling temperaments]]
+* ''[[Echidnic]]'' (+686/675) → [[Diaschismic family #Echidnic|Diaschismic family]]
-* [[Superkleismic]] (+875/864) → [[Shibboleth family #Superkleismic|Shibboleth family]]
+* [[Trismegistus]] (+3125/3072) → [[Magic family #Trismegistus|Magic family]]
-* ''[[Blacksmith]]'' (+28/27) → [[Limmic temperaments #Blacksmith|Limmic temperaments]]
-* ''[[Trismegistus]]'' (+3125/3072) → [[Magic family #Trismegistus|Magic family]]
 * [[Hemithirds]] (+3136/3125) → [[Hemimean clan #Hemithirds|Hemimean clan]]
 * ''[[Gamity]]'' (+1071875/1062882) → [[Amity family #Gamity|Amity family]]
@@ Line 17: / Line 41: @@
 * ''[[Heinz]]'' (+78732/78125) → [[Sensipent family #Heinz|Sensipent family]]
 * ''[[Triwell]]'' (+235298/234375) → [[Semicomma family #Triwell|Semicomma family]]
-* ''[[Decades]]'' (+118098/117649) → [[Compton family #Decades|Compton family]]
+* ''[[Gamelstearn]]'' (+118098/117649) → [[Compton family #Gamelstearn|Compton family]]
 The rest are considered below.
-No-five subgroup extensions of slendric include radon, a 2.3.7.11 extension that may be viewed as no-five rodan, and baladic, a 2.3.7.13.17 extension, considered below. Dicussed elsewhere is [[No-fives subgroup temperaments #Gigapyth|gigapyth]] in the 2.3.7.85 subgroup.
+==== Subgroup extensions ====
+No-five subgroup extensions of slendric include radon, a 2.3.7.11-subgroup extension that may be viewed as no-five rodan, considered below, euslendric, a 2.3.7.13-subgroup extension, baladic, a weak 2.3.7.13.17-subgroup extension, and gigapyth, a 2.3.7.85-subgroup extension, considered in [[#Other subgroup extensions]]. Dicussed elsewhere is [[Subgroup temperaments #Trisect|trisect]] in the 2.3.7.11/5 subgroup.
+=== Radon ===
+{{See also|Chromatic pairs #Radon}}
+Radon is the no-fives version of [[rodan]], equating the diatonic major third to [[14/11]].
-== Slendric ==
+Subgroup: 2.3.7.11
-{{Main| Slendric }}
+Comma list: 896/891, 1029/1024
+Subgroup-val mapping: {{mapping| 1 1 3 6 | 0 3 -1 -13 }}
+Gencom mapping: {{mapping| 1 1 0 3 6 | 0 3 0 -1 -13 }}
+Optimal tunings:
+* WE: ~2 = 1199.9708{{c}}, ~8/7 = 234.3748{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.3813{{c}}
+{{Optimal ET sequence|legend=0| 5, …, 36, 41, 87, 128 }}
+Badness (Sintel): 0.619
+== Mothra ==
+{{Main| Mothra }}
-[[Subgroup]]: 2.3.7
+Mothra tempers out [[81/80]] and finds the prime 5 at a stack of four fifths as does any temperament in the [[meantone family]]. It also tempers out [[1728/1715]], the orwellisma. It can be described as the {{nowrap| 26 & 31 }}. Using [[31edo]] with a generator of 6/31 is an excellent tuning choice. However, a pure mos mothra scale is often described as directionless and has limited chord-building potential<ref>[https://www.youtube.com/watch?v=uH3ahBzDSrs 31-EDO Music Theory: Supermajor Hexatonic Scale] by [[Zhea Erose]]</ref>, so something other than a mos may be used as a scale to get the most out of mothra. There are examples of non-mos mothra scales in 31edo [[Strictly proper 7-tone 31edo scales|in the article on strictly proper 7-tone 31edo scales]].
-[[Comma list]]: 1029/1024
+Note that mothra is also called '''cynder''' in the 7-limit, which can be a little confusing sometimes.
-{{Mapping|legend=2| 1 1 3 | 0 3 -1 }}
+Its [[S-expression]]-based comma list is {[[1728/1715|S6/S7]], [[1029/1024|S7/S8]], ([[81/80|S6/S8 = S9]])}, taking advantage of the fact that [[81/80]] is a [[semiparticular]].
-: sval mapping generators: ~2, ~8/7
+[[Subgroup]]: 2.3.5.7
-{{Mapping|legend=3| 1 1 0 3 | 0 3 0 -1 }}
+[[Comma list]]: 81/80, 1029/1024
-: [[gencom]]: [2 8/7; 1029/1024]
+{{Mapping|legend=1| 1 1 0 3 | 0 3 12 -1 }}
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.000, ~8/7 = 233.889
+* [[WE]]: ~2 = 1200.9303{{c}}, ~8/7 = 232.3733{{c}}
-* [[POTE]]: ~2 = 1200.000, ~8/7 = 233.688
+: [[error map]]: {{val| +0.930 -3.905 +2.165 +1.592 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 232.2514{{c}}
+: error map: {{val| 0.000 -5.520 +0.703 -1.077 }}
+[[Algebraic generator]]: Rabrindanath, largest real root of ''x''<sup>8</sup> - 3''x''<sup>2</sup> + 1, or 232.0774 cents.
+[[Minimax tuning]]:
+* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~8/7 = {{monzo| 0 0 1/12 }}
+: {{monzo list| 1 0 0 0 | 1 0 1/4 0 | 0 0 1 0 | 3 0 -1/12 0 }}
+: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5
+{{Optimal ET sequence|legend=1| 5, 21c, 26, 31 }}
+[[Badness]] (Sintel): 0.940
+=== Undecimal mothra ===
+Undecimal mothra is the extension of 7-limit cynder which tempers out 385/384 as is natural in slendric temperaments. It is the simplest extension, supported within a reasonable tuning range (between [[26edo]] and 31edo), and is supported by the patent val of [[5edo]], which implies that it is better behaved as a cluster temperament. It is also notable for being supported by the just tuning of 8/7, and has a restriction to the 2.7.11 subgroup, namely [[amaranthine]], that is a microtemperament.
+Subgroup: 2.3.5.7.11
+Comma list: 81/80, 99/98, 385/384
+Mapping: {{mapping| 1 1 0 3 5 | 0 3 12 -1 -8 }}
+Optimal tunings:
+* WE: ~2 = 1201.3979{{c}}, ~8/7 = 232.3010{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 232.0621{{c}}
+{{Optimal ET sequence|legend=0| 5, 26, 31, 88, 119be, 150be }}
+Badness (Sintel): 0.848
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
+Comma list: 81/80, 99/98, 105/104, 144/143
+Mapping: {{mapping| 1 1 0 3 5 1 | 0 3 12 -1 -8 14 }}
+Optimal tunings:
+* WE: ~2 = 1201.0985{{c}}, ~8/7 = 232.0231{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 231.8425{{c}}
+{{Optimal ET sequence|legend=0| 5, 26, 31, 57, 88 }}
+Badness (Sintel): 0.990
+==== 17-limit ====
+Subgroup: 2.3.5.7.11.13.17
+Comma list: 81/80, 99/98, 105/104, 120/119, 144/143
+Mapping: {{mapping| 1 1 0 3 5 1 | 0 3 12 -1 -8 14 16 }}
+Optimal tunings:
+* WE: ~2 = 1200.9734{{c}}, ~8/7 = 231.8960{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 231.7392{{c}}
+{{Optimal ET sequence|legend=0| 5g, 26, 31, 57, 88 }}
+Badness (Sintel): 1.00
+==== 19-limit ====
+Subgroup: 2.3.5.7.11.13.17.19
+Comma list: 81/80, 99/98, 105/104, 120/119, 144/143, 153/152
+Mapping: {{mapping| 1 1 0 3 5 1 | 0 3 12 -1 -8 14 16 22 }}
+Optimal tunings:
+* WE: ~2 = 1200.9663{{c}}, ~8/7 = 231.8393{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 231.6842{{c}}
+{{Optimal ET sequence|legend=0| 26, 31, 57 }}
+Badness (Sintel): 1.05
+=== Mosura ===
+The [[S-expression]]-based comma list of mosura suggests it might be the most natural extension of 7-limit cynder to the 11-limit: {[[1728/1715|S6/S7]], [[1029/1024|S7/S8]], ([[81/80|S6/S8 = S9]]), [[176/175|S8/S10]]}.
+Subgroup: 2.3.5.7.11
+Comma list: 81/80, 176/175, 540/539
+Mapping: {{mapping| 1 1 0 3 -1 | 0 3 12 -1 23 }}
+Optimal tunings:
+* WE: ~2 = 1200.7675{{c}}, ~8/7 = 232.5673{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 232.4567{{c}}
+{{Optimal ET sequence|legend=0| 5e, 26e, 31, 129 }}
+Badness (Sintel): 1.04
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
+Comma list: 81/80, 144/143, 176/175, 196/195
+Mapping: {{mapping| 1 1 0 3 -1 7 | 0 3 12 -1 23 -17 }}
+Optimal tunings:
+* WE: ~2 = 1199.9347{{c}}, ~8/7 = 232.6275{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 232.6392{{c}}
+{{Optimal ET sequence|legend=0| 31, 67, 98 }}
+Badness (Sintel): 1.52
+==== 17-limit ====
+Subgroup: 2.3.5.7.11.13.17
+Comma list: 81/80, 144/143, 176/175, 189/187, 196/195
+Mapping: {{mapping| 1 1 0 3 -1 7 | 0 3 12 -1 23 -17 -15 }}
+Optimal tunings:
+* WE: ~2 = 1199.7124{{c}}, ~8/7 = 232.6376{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 232.6917{{c}}
+{{Optimal ET sequence|legend=0| 31, 67, 98 }}
+Badness (Sintel): 1.53
+==== 19-limit ====
+Subgroup: 2.3.5.7.11.13.17.19
-{{Optimal ET sequence|legend=1| 36, 77, 113, 190 }}
+Comma list: 81/80, 96/95, 144/143, 153/152, 176/175, 196/195
-=== Radon ===
+Mapping: {{mapping| 1 1 0 3 -1 7 | 0 3 12 -1 23 -17 -15 -9 }}
-[[Subgroup]]: 2.3.7.11
-[[Comma list]]: 896/891, 1029/1024
+Optimal tunings:
+* WE: ~2 = 1199.4885{{c}}, ~8/7 = 232.6310{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 232.7287{{c}}
-{{Mapping|legend=2| 1 1 3 6 | 0 3 -1 -13 }}
+{{Optimal ET sequence|legend=0| 31, 67, 98h }}
-{{Mapping|legend=3| 1 1 0 3 6 | 0 3 0 -1 -13 }}
+Badness (Sintel): 1.50
-: [[gencom]]: [2 8/7; 896/891 1029/1024]
+=== Cyndra ===
+Subgroup: 2.3.5.7.11
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~8/7 = 234.381
+Comma list: 45/44, 81/80, 1029/1024
-{{Optimal ET sequence|legend=1| 36, 41, 87, 128 }}
+Mapping: {{mapping| 1 1 0 3 0 | 0 3 12 -1 18 }}
-=== Baladic ===
+Optimal tunings:
-Baladic is a 2.3.7.13.17 subgroup temperament that attempts to approximate the Maqam Sikah Baladi scale. 36edo is an excellent baladic tuning.
+* WE: ~2 = 1201.1585{{c}}, ~8/7 = 231.5404{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 231.3850{{c}}
-[[Subgroup]]: 2.3.7.13.17
+{{Optimal ET sequence|legend=0| 5e, 21ce, 26 }}
+Badness (Sintel): 1.84
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
-[[Comma list]]: 169/168, 273/272, 289/288
+Comma list: 45/44, 78/77, 81/80, 640/637
-{{Mapping|legend=2| 2 2 6 7 7 | 0 3 -1 1 3 }}
+Mapping: {{mapping| 1 1 0 3 0 1 | 0 3 12 -1 18 14 }}
-: sval mapping generators: ~17/12, ~8/7
+Optimal tunings:
+* WE: ~2 = 1201.1152{{c}}, ~8/7 = 231.5079{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 231.3612{{c}}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~8/7 = 233.616
+{{Optimal ET sequence|legend=0| 5e, 21cef, 26 }}
-{{Optimal ET sequence|legend=1| 10, 26, 36, 154f, 190ffg, 226ffg }}
+Badness (Sintel): 1.41
 == Rodan ==
@@ Line 78: / Line 259: @@
 : ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum #Rodan (5-limit)]].''
-Rodan tempers out 245/243 and can be described as the 41 & 46 temperament. This temperament extends neatly to the 13-limit, though the perfect fifth is sharper than ideal for slendric.
+Rodan tempers out 245/243 and can be described as the {{nowrap| 41 & 46 }} temperament. This temperament is more accurate than mothra and extends neatly to the 13-limit, though the perfect fifth is sharper than ideal for slendric. [[87edo]] is excellent for this, with the 17\87 generator missing the 13-limit CWE tuning by less than a millicent.
 [[Subgroup]]: 2.3.5.7
@@ Line 86: / Line 267: @@
 {{Mapping|legend=1| 1 1 -1 3 | 0 3 17 -1 }}
-{{Multival|legend=1| 3 17 -1 20 -10 -50 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.2146{{c}}, ~8/7 = 234.4587{{c}}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~8/7 = 234.417
+: [[error map]]: {{val| +0.215 +1.636 -0.731 -2.641 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 234.4259{{c}}
+: error map: {{val| 0.000 +1.323 -1.073 -3.252 }}
 [[Minimax tuning]]:
 * [[7-odd-limit|7-]] and [[9-odd-limit]]: ~8/7 = {{monzo| 2/9 0 1/18 -1/18 }}
 : {{monzo list| 1 0 0 0 | 5/3 0 1/6 -1/6 | 25/9 0 17/18 -17/18 | 25/9 0 -1/18 1/18 }}
-: [[eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.7/5
+: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5
 [[Algebraic generator]]: larger root of 20''x''<sup>2</sup> - 36''x'' + 15, or (9 + √6)/10.
@@ Line 99: / Line 282: @@
 {{Optimal ET sequence|legend=1| 41, 87, 128, 215d }}
-[[Badness]] (Smith): 0.037112
+[[Badness]] (Sintel): 0.939
 === 11-limit ===
@@ Line 108: / Line 291: @@
 Mapping: {{mapping| 1 1 -1 3 6 | 0 3 17 -1 -13 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~8/7 = 234.459
+Optimal tunings:
+* WE: ~2 = 1200.0553{{c}}, ~8/7 = 234.4695{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.4594{{c}}
 Minimax tuning:
 * 11-odd-limit: ~8/7 = {{monzo| 4/19 2/19 0 0 -1/19 }}
 : [{{monzo| 1 0 0 0 0 }}, {{monzo| 31/19 6/19 0 0 -3/19 }}, {{monzo| 49/19 34/19 0 0 -17/19 }}, {{monzo| 53/19 -2/19 0 0 1/19 }}, {{monzo| 62/19 -26/19 0 0 13/19 }}]
-: eigenmonzo (unchanged-interval) basis: 2.11/9
+: unchanged-interval (eigenmonzo) basis: 2.11/9
 Algebraic generator: positive root of ''x''<sup>2</sup> + 16''x'' - 31, or √95 - 8.
-{{Optimal ET sequence|legend=0| 41, 46, 87 }}
+{{Optimal ET sequence|legend=0| 41, 87 }}
-Badness (Smith): 0.023093
+Badness (Sintel): 0.763
 ==== 13-limit ====
@@ Line 128: / Line 313: @@
 Mapping: {{mapping| 1 1 -1 3 6 8 | 0 3 17 -1 -13 -22 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~8/7 = 234.482
+Optimal tunings:
+* WE: ~2 = 1199.9868{{c}}, ~8/7 = 234.4796{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.4822{{c}}
 Minimax tuning:
 * 13- and 15-odd-limit: ~8/7 = {{monzo| 3/14 1/14 0 0 0 -1/28 }}
-: eigenmonzo (unchanged-interval) basis: 2.13/9
+: unchanged-interval (eigenmonzo) basis: 2.13/9
 Algebraic generator: Gatetone, positive root of 4''x''<sup>6</sup> - 7''x'' - 1. Recurrence converges slowly.
@@ Line 138: / Line 325: @@
 {{Optimal ET sequence|legend=0| 41, 46, 87 }}
-Badness (Smith): 0.018448
+Badness (Sintel): 0.762
 ===== 17-limit =====
@@ Line 147: / Line 334: @@
 Mapping: {{mapping| 1 1 -1 3 6 8 8 | 0 3 17 -1 -13 -22 -20 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~8/7 = 234.524
+Optimal tunings:
+* WE: ~2 = 1199.8331{{c}}, ~8/7 = 234.4919{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.5254{{c}}
 Minimax tuning:
 * 17-odd-limit: ~8/7 = {{monzo| 3/13 1/13 0 0 0 0 -1/26 }}
-: eigenmonzo (unchanged-interval) basis: 2.17/9
+: unchanged-interval (eigenmonzo) basis: 2.17/9
-{{Optimal ET sequence|legend=0| 41, 46, 87, 220dg, 307dgg }}
+{{Optimal ET sequence|legend=0| 41, 46, 87 }}
-Badness (Smith): 0.016743
+Badness (Sintel): 0.853
 ==== Aerodactyl ====
@@ Line 164: / Line 353: @@
 Mapping: {{mapping| 1 1 -1 3 6 -1 | 0 3 17 -1 -13 24 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~8/7 = 234.639
+Optimal tunings:
+* WE: ~2 = 1200.2997{{c}}, ~8/7 = 234.6972{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.6439{{c}}
-{{Optimal ET sequence|legend=0| 5, 41f, 46, 133ff }}
+{{Optimal ET sequence|legend=0| 5, 41f, 46 }}
-Badness (Smith): 0.033986
+Badness (Sintel): 1.40
 === Aerodino ===
@@ Line 177: / Line 368: @@
 Mapping: {{mapping| 1 1 -1 3 -3 | 0 3 17 -1 33 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~8/7 = 234.728
+Optimal tunings:
+* WE: ~2 = 1199.9179{{c}}, ~8/7 = 234.7123{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.7256{{c}}
-{{Optimal ET sequence|legend=0| 41e, 46 }}
+{{Optimal ET sequence|legend=0| 5e, 41e, 46 }}
-Badness (Smith): 0.054294
+Badness (Sintel): 1.79
 ==== 13-limit ====
@@ Line 190: / Line 383: @@
 Mapping: {{mapping| 1 1 -1 3 -3 -1 | 0 3 17 -1 33 24 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~8/7 = 234.782
+Optimal tunings:
+* WE: ~2 = 1200.0242{{c}}, ~8/7 = 234.7863{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.7824{{c}}
-{{Optimal ET sequence|legend=0| 41ef, 46 }}
+{{Optimal ET sequence|legend=0| 5e, 41ef, 46 }}
-Badness (Smith): 0.035836
+Badness (Sintel): 1.48
 === Varan ===
@@ Line 203: / Line 398: @@
 Mapping: {{mapping| 1 1 -1 3 -2 | 0 3 17 -1 28 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~8/7 = 234.145
+Optimal tunings:
+* WE: ~2 = 1200.3738{{c}}, ~8/7 = 234.2174{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.1586{{c}}
-{{Optimal ET sequence|legend=0| 36ce, 41 }}
+{{Optimal ET sequence|legend=0| 5e, 36ce, 41 }}
-Badness (Smith): 0.044937
+Badness (Sintel): 1.49
 ==== 13-limit ====
@@ Line 216: / Line 413: @@
 Mapping: {{mapping| 1 1 -1 3 -2 0 | 0 3 17 -1 28 19 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~8/7 = 234.089
+Optimal tunings:
+* WE: ~2 = 1200.1389{{c}}, ~8/7 = 234.1162{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.0946{{c}}
-{{Optimal ET sequence|legend=0| 36ce, 41 }}
+{{Optimal ET sequence|legend=0| 5e, 36ce, 41 }}
-Badness (Smith): 0.032284
+Badness (Sintel): 1.33
 == Guiron ==
-Guiron tempers out the [[schisma]], and finds the prime 5 at the diminished fourth as does any temperament in the [[schismatic family]]. It can be described as 36 & 41. It is more complex than rodan, but the optimal tuning is closer to optimal slendric.
+Guiron tempers out the [[schisma]], and finds the prime 5 at the diminished fourth as does any temperament in the [[schismatic family]]. It can be described as the {{nowrap| 36 & 41 }} temperament. It is more complex than rodan, but the optimal tuning is closer to optimal slendric.
 [[Subgroup]]: 2.3.5.7
@@ Line 231: / Line 430: @@
 {{Mapping|legend=1| 1 1 7 3 | 0 3 -24 -1 }}
-: mapping generators: ~2, ~8/7
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.3395{{c}}, ~8/7 = 233.9963{{c}}
-{{Multival|legend=1| 3 -24 -1 -45 -10 65 }}
+: [[error map]]: {{val| +0.340 +0.374 +0.151 -1.804 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 233.9239{{c}}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~8/7 = 233.930
+: error map: {{val| 0.000 -0.183 -0.487 -2.750 }}
 [[Minimax tuning]]:
 * [[7-odd-limit|7-]] and [[9-odd-limit]]: ~8/7 = {{monzo| 7/24 0 -1/24 }}
 : {{monzo list| 1 0 0 0 | 15/8 0 -1/8 0 | 0 0 1 0 | 65/24 0 1/24 0 }}
-: [[eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5
+: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5
 {{Optimal ET sequence|legend=1| 36, 41, 77, 118, 277d }}
-[[Badness]] (Smith): 0.047544
+[[Badness]] (Sintel): 1.20
 === 11-limit ===
@@ Line 253: / Line 452: @@
 Mapping: {{mapping| 1 1 7 3 -2 | 0 3 -24 -1 28 }}
-: mapping generators: ~2, ~8/7
+Optimal tunings:
+* WE: ~2 = 1200.3453{{c}}, ~8/7 = 233.9988{{c}}
-Optimal tuning (POTE): ~2 = 1200.000, ~8/7 = 233.931
+* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.9312{{c}}
 Minimax tuning:
 * 11-odd-limit: ~8/7 = {{monzo| 7/24 0 -1/24 }}
 : [{{monzo| 1 0 0 0 0 }}, {{monzo| 15/8 0 -1/8 0 0 }}, {{monzo| 0 0 1 0 0 }}, {{monzo| 65/24 0 1/24 0 0 }}, {{monzo| 37/6 0 -7/6 0 0 }}]
-: eigenmonzo (unchanged-interval) basis: 2.5
+: unchanged-interval (eigenmonzo) basis: 2.5
 {{Optimal ET sequence|legend=0| 36e, 41, 77, 118, 159, 277d }}
-Badness (Smith): 0.026648
+Badness (Sintel): 0.881
 === 13-limit ===
@@ Line 273: / Line 472: @@
 Mapping: {{mapping| 1 1 7 3 -2 0 | 0 3 -24 -1 28 19 }}
-: mapping generators: ~2, ~8/7
+Optimal tunings:
+* WE: ~2 = 1200.1222{{c}}, ~8/7 = 233.9228{{c}}
-Optimal tuning (POTE): ~2 = 1200.000, ~8/7 = 233.890
+* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.8994{{c}}
 {{Optimal ET sequence|legend=0| 36e, 41, 77, 118 }}
-Badness (Smith): 0.028444
+Badness (Sintel): 1.18
-== Mothra ==
-Mothra tempers out [[81/80]] and finds the prime 5 at a stack of four fifths as does any temperament in the [[meantone family]]. It also tempers out [[1728/1715]], the orwellisma. It can be described as 26 & 31. Using [[31edo]] with a generator of 6/31 is an excellent tuning choice. However, a pure mos mothra scale is often described as directionless and has limited chord-building potential<ref>https://www.youtube.com/watch?v=uH3ahBzDSrs</ref>, so something other than a mos may be used as a scale to get the most out of mothra. There are examples of non-mos mothra scales in 31edo [[Strictly proper 7-tone 31edo scales|in the article on strictly proper 7-tone 31edo scales]].
-Note that mothra is also called '''cynder''' in the 7-limit, which can be a little confusing sometimes.
-Its [[S-expression]]-based comma list is {[[1728/1715|S6/S7]], [[1029/1024|S7/S8]](, [[81/80|S6/S8 = S9]])}, taking advantage of the fact that [[81/80]] is a [[semiparticular]].
-[[Subgroup]]: 2.3.5.7
-[[Comma list]]: 81/80, 1029/1024
-{{Mapping|legend=1| 1 1 0 3 | 0 3 12 -1 }}
-: mapping generators: ~2, ~8/7
-{{Multival|legend=1| 3 12 -1 12 -10 -36 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~8/7 = 232.193
-[[Algebraic generator]]: Rabrindanath, largest real root of ''x''<sup>8</sup> - 3''x''<sup>2</sup> + 1, or 232.0774 cents.
-[[Minimax tuning]]:
-* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~8/7 = {{monzo| 0 0 1/12 }}
-: [{{monzo| 1 0 0 0 }}, {{monzo| 1 0 1/4 0 }}, {{monzo| 0 0 1 0 }}, {{monzo| 3 0 -1/12 0 }}]
-: [[eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5
-{{Optimal ET sequence|legend=1| 5, 26, 31 }}
-[[Badness]] (Smith): 0.037146
-=== 11-limit ===
-Subgroup: 2.3.5.7.11
-Comma list: 81/80, 99/98, 385/384
-Mapping: {{mapping| 1 1 0 3 5 | 0 3 12 -1 -8 }}
-{{Multival|legend=1| 3 12 -1 -8 12 -10 -23 -36 -60 -19 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~8/7 = 232.031
-{{Optimal ET sequence|legend=0| 5, 26, 31, 88, 150be, 181bee }}
-Badness (Smith): 0.025642
-==== 13-limit ====
-Subgroup: 2.3.5.7.11.13
-Comma list: 81/80, 99/98, 105/104, 144/143
-Mapping: {{mapping| 1 1 0 3 5 1 | 0 3 12 -1 -8 14 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~8/7 = 231.811
-{{Optimal ET sequence|legend=0| 5, 26, 31, 57, 88 }}
-Badness (Smith): 0.023954
-; Music
-* ''Prelude for solo piano'' (2014) by [[Chris Vaisvil]] – [https://web.archive.org/web/20201127013310/http://micro.soonlabel.com/16-ET/mothra/20141028_mothra16br4.mp3 play] | [https://www.chrisvaisvil.com/prelude-for-solo-piano-in-mothra16-brat-4-tuning/ blog] – in Mothra[16], brat 4 tuning
-=== Cynder ===
-Subgroup: 2.3.5.7.11
-Comma list: 45/44, 81/80, 1029/1024
-Mapping: {{mapping| 1 1 0 3 0 | 0 3 12 -1 18 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~8/7 = 231.317
-{{Optimal ET sequence|legend=0| 5e, 26, 57e, 83bce }}
-Badness (Smith): 0.055706
-==== 13-limit ====
-Subgroup: 2.3.5.7.11.13
-Comma list: 45/44, 78/77, 81/80, 640/637
-Mapping: {{mapping| 1 1 0 3 0 1 | 0 3 12 -1 18 14 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~8/7 = 231.293
-{{Optimal ET sequence|legend=0| 5e, 26, 57e, 83bce }}
-Badness (Smith): 0.034124
-=== Mosura ===
-The S-expression-based comma list of mosura suggests it might be the most natural extension of 7-limit cynder to the 11-limit: {[[1728/1715|S6/S7]], [[1029/1024|S7/S8]], ([[81/80|S6/S8 = S9]],) [[176/175|S8/S10]]}.
-Subgroup: 2.3.5.7.11
-Comma list: 81/80, 176/175, 540/539
-Mapping: {{mapping| 1 1 0 3 -1 | 0 3 12 -1 23 }}
-Wedgie: {{Multival| 3 12 -1 23 12 -10 26 -36 12 68 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~8/7 = 232.419
-{{Optimal ET sequence|legend=0| 31, 129, 160be, 191bce, 222bce, 253bcee }}
-Badness (Smith): 0.031334
-==== 13-limit ====
-Subgroup: 2.3.5.7.11.13
-Comma list: 81/80, 144/143, 176/175, 196/195
-Mapping: {{mapping| 1 1 0 3 -1 7 | 0 3 12 -1 23 -17 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~8/7 = 232.640
-{{Optimal ET sequence|legend=0| 31, 36, 67, 98 }}
-Badness (Smith): 0.036857
 == Gorgo ==
-In the 5-limit, gorgo tempers out the laconic comma, [[2187/2000]], which is the difference between three [[10/9]]'s and a [[3/2]]. Although a higher-error temperament, it does pop up enough in the low-numbered edos to be useful, most notably in [[16edo]] and [[21edo]]. The only 7-limit extension that makes any sense to use is to add the gamelisma to the comma list.
+: ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum #Laconic]].''
+{{See also| Llywelynsmic clan }}
-=== 5-limit (laconic) ===
+Gorgo tempers the generator of ~8/7 together with ~10/9. It can be described as the {{nowrap| 16 & 21 }} temperament.
-[[Subgroup]]: 2.3.5
-[[Comma list]]: 2187/2000
+If we discard the inaccurate mapping of prime 3, we get [[shoe]], so that the large commas of gorgo are explained practically entirely by the inaccurate 3.
-{{Mapping|legend=1| 1 1 1 | 0 3 7 }}
-{{Multival|legend=1| 3 7 4 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~10/9 = 227.426
-{{Optimal ET sequence|legend=1| 5, 16, 21, 37b }}
-[[Badness]] (Smith): 0.161799
-=== 7-limit ===
 [[Subgroup]]: 2.3.5.7
@@ Line 423: / Line 494: @@
 {{Mapping|legend=1| 1 1 1 3 | 0 3 7 -1 }}
-{{Multival|legend=1| 3 7 -1 4 -10 -22 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.9847{{c}}, ~8/7 = 228.5210{{c}}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~8/7 = 228.334
+: [[error map]]: {{val| +0.985 -15.407 +14.318 +5.607 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 228.4371{{c}}
+: error map: {{val| 0.000 -16.644 +12.746 +2.737 }}
 {{Optimal ET sequence|legend=1| 5, 11c, 16, 21 }}
-[[Badness]] (Smith): 0.060663
+[[Badness]] (Sintel): 1.54
 === 11-limit ===
@@ Line 438: / Line 511: @@
 Mapping: {{mapping| 1 1 1 3 1 | 0 3 7 -1 13 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~8/7 = 227.373
+Optimal tunings:
+* WE: ~2 = 1201.3609{{c}}, ~8/7 = 227.6312{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 227.4955{{c}}
-{{Optimal ET sequence|legend=0| 16, 21, 37b }}
+{{Optimal ET sequence|legend=0| 5e, 16, 21, 37b }}
-Badness (Smith): 0.049500
+Badness (Sintel): 1.64
 ==== 13-limit ====
@@ Line 451: / Line 526: @@
 Mapping: {{mapping| 1 1 1 3 1 2 | 0 3 7 -1 13 9 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~8/7 = 227.230
+Optimal tunings:
+* WE: ~2 = 1201.0996{{c}}, ~8/7 = 227.4378{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 227.3327{{c}}
-{{Optimal ET sequence|legend=0| 16, 21, 37b }}
+{{Optimal ET sequence|legend=0| 5e, 16, 21, 37b }}
-Badness (Smith): 0.032664
+Badness (Sintel): 1.35
 === Spartan ===
@@ Line 464: / Line 541: @@
 Mapping: {{mapping| 1 1 1 3 5 | 0 3 7 -1 -8 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~8/7 = 229.535
+Optimal tunings:
+* WE: ~2 = 1198.9344{{c}}, ~8/7 = 229.3316{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 229.5124{{c}}
-{{Optimal ET sequence|legend=0| 5, 16e, 21, 47c, 68bcce }}
+{{Optimal ET sequence|legend=0| 5, 16e, 21 }}
-Badness (Smith): 0.062683
+Badness (Sintel): 2.07
 ==== 13-limit ====
@@ Line 477: / Line 556: @@
 Mapping: {{mapping| 1 1 1 3 5 2 | 0 3 7 -1 -8 9 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~8/7 = 229.059
+Optimal tunings:
+* WE: ~2 = 1198.3002{{c}}, ~8/7 = 228.7341{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 229.0044{{c}}
-{{Optimal ET sequence|legend=0| 5, 16e, 21, 68bccef }}
+{{Optimal ET sequence|legend=0| 5, 16e, 21 }}
-Badness (Smith): 0.047071
+Badness (Sintel): 1.95
 ; Music
@@ Line 488: / Line 569: @@
 == Gidorah ==
 : ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum #University]].''
+Gidorah is a very low-accuracy temperament where the generator of ~8/7 is lumped together with ~6/5. 16c-, 21cc-, and 26ccc-edo are among the possible tunings.
 [[Subgroup]]: 2.3.5.7
@@ Line 495: / Line 578: @@
 {{Mapping|legend=1| 1 1 2 3 | 0 3 2 -1 }}
-{{Multival|legend=1| 3 2 -1 -4 -10 -8 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1192.4932{{c}}, ~8/7 = 229.3187{{c}}
+: [[error map]]: {{val| -7.507 -21.506 +57.310 -20.665 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 229.6649{{c}}
+: error map: {{val| 0.000 -12.960 +73.016 +1.509 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~8/7 = 230.762
+{{Optimal ET sequence|legend=1| 1b, 5 }}
-{{Optimal ET sequence|legend=1| 5, 16c, 21cc, 26ccc }}
+[[Badness]] (Sintel): 1.58
-[[Badness]] (Smith): 0.062262
 == Oncle ==
 : ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Oncle]].''
+Oncle can be described as the {{nowrap| 31 & 36c }} temperament.
 [[Subgroup]]: 2.3.5.7
@@ Line 512: / Line 599: @@
 {{Mapping|legend=1| 1 1 6 3 | 0 3 -19 -1 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~8/7 = 232.498
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1201.2246{{c}}, ~8/7 = 232.7354{{c}}
+: [[error map]]: {{val| +1.225 -2.524 -0.939 +2.112 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 232.4718{{c}}
+: error map: {{val| 0.000 -4.539 -3.279 -1.298 }}
 {{Optimal ET sequence|legend=1| 31, 98c, 129c, 160bc }}
-[[Badness]] (Smith): 0.088384
+[[Badness]] (Sintel): 2.24
 == Archaeotherium ==
 : ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Archaeotherium]].''
+Archaeotherium can be described as the {{nowrap| 21 & 26 }} temperament.
 [[Subgroup]]: 2.3.5.7
@@ Line 527: / Line 620: @@
 {{Mapping|legend=1| 1 1 5 3 | 0 3 -14 -1 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~8/7 = 230.258
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1202.7179{{c}}, ~8/7 = 230.7800{{c}}
+: [[error map]]: {{val| +2.718 -6.897 -3.644 +8.548 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 230.1909{{c}}
+: error map: {{val| 0.000 -11.382 -8.986 +0.983 }}
-{{Optimal ET sequence|legend=1| 21, 26, 47, 73bc, 99bc }}
+{{Optimal ET sequence|legend=1| 21, 26, 47, 73bc }}
-[[Badness]] (Smith): 0.146306
+[[Badness]] (Sintel): 3.70
 == Clyndro ==
-{{See also| Mavila family }}
+Clyndro tempers out [[135/128]] and finds the interval class of 5 at a stack of -3 fifths as does any temperament in the [[mavila family]]. It can be described as the {{nowrap| 11 & 16 }} temperament.
 [[Subgroup]]: 2.3.5.7
@@ Line 542: / Line 639: @@
 {{Mapping|legend=1| 1 1 4 3 | 0 3 -9 -1 }}
-{{Multival|legend=1| 3 -9 -1 -21 -10 23 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1205.6135{{c}}, ~8/7 = 227.5283{{c}}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~8/7 = 226.469
+: [[error map]]: {{val| +5.613 -13.757 -11.614 +20.486 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 226.3207{{c}}
+: error map: {{val| 0.000 -22.993 -23.200 +4.853 }}
 {{Optimal ET sequence|legend=1| 5c, 11, 16 }}
-[[Badness]] (Smith): 0.159179
+[[Badness]] (Sintel): 4.03
 === 11-limit ===
@@ Line 557: / Line 656: @@
 Mapping: {{mapping| 1 1 4 3 4 | 0 3 -9 -1 -3 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~8/7 = 226.428
+Optimal tunings:
+* WE: ~2 = 1206.2134{{c}}, ~8/7 = 227.6004{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 226.2421{{c}}
 {{Optimal ET sequence|legend=0| 5c, 11, 16 }}
-Badness (Smith): 0.069703
+Badness (Sintel): 2.30
 == Miracle ==
 {{Main| Miracle }}
+: ''For the 5-limit version, see [[Syntonic–31 equivalence continuum #Ampersand]].''
-: ''For the 5-limit version, see [[Ampersand comma #Ampersand]].''
+Miracle is one of the most important entries of this temperament clan. It tempers out [[225/224]], splitting the ~8/7 generator of slendric into 15/14~16/15, and can be described as the {{nowrap| 31 & 41 }} temperament. Its ploidacot is hexacot. It is then extremely natural to equate the neutral third, three generators up, to [[11/9]] and thereby extend miracle to the full [[11-limit]] with essentially no further damage. [[72edo]] makes for an excellent tuning.
 [[Subgroup]]: 2.3.5.7
@@ Line 573: / Line 675: @@
 {{Mapping|legend=1| 1 1 3 3 | 0 6 -7 -2 }}
 : mapping generator: ~2, ~15/14
-{{Multival|legend=1| 6 -7 -2 -25 -20 15 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.8209{{c}}, ~15/14 = 116.7550{{c}}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~15/14 = 116.675
+: [[error map]]: {{val| +0.821 -0.604 -1.136 +0.127 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~15/14 = 116.6756{{c}}
+: error map: {{val| 0.000 -1.901 -3.043 -2.177 }}
 [[Minimax tuning]]:
 * [[7-odd-limit]]: ~15/14 = {{monzo| 2/13 1/13 -1/13 }}
 : {{monzo list| 1 0 0 0 | 25/13 6/13 -6/13 0 | 25/13 -7/13 7/13 0 | 35/13 -2/13 2/13 0 }}
-: [[eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5/3
+: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5/3
 * [[9-odd-limit]]: ~15/14 = {{monzo| 1/19 2/19 -1/19 }}
 : {{monzo list| 1 0 0 0 | 25/19 12/19 -6/19 0 | 50/19 -14/19 7/19 0 | 55/19 -4/19 2/19 0 }}
-: [[eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.9/5
+: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5
 [[Tuning ranges]]:
@@ Line 597: / Line 700: @@
 {{Optimal ET sequence|legend=1| 10, 21, 31, 41, 72 }}
-[[Badness]] (Smith): 0.016742
+[[Badness]] (Sintel): 0.424
 === 11-limit ===
@@ Line 606: / Line 709: @@
 Mapping: {{mapping| 1 1 3 3 2 | 0 6 -7 -2 15 }}
-{{Multival|legend=1| 6 -7 -2 15 -25 -20 3 15 59 49 }}
+Optimal tunings:
+* WE: ~2 = 1200.7626{{c}}, ~15/14 = 116.7069{{c}}
-Optimal tuning (POTE): ~2 = 1200.000, ~15/14 = 116.633
+* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.6469{{c}}
 Minimax tuning:
 * 11-odd-limit: ~15/14 = {{monzo| 1/19 2/19 -1/19 }}
 : [{{monzo| 1 0 0 0 0 }}, {{monzo| 25/19 12/19 -6/19 0 0 }}, {{monzo| 50/19 -14/19 7/19 0 0 }}, {{monzo| 55/19 -4/19 2/19 0 0 }}, {{monzo| 53/19 30/19 -15/19 0 0 }}]
-: eigenmonzo (unchanged-interval) basis: 2.9/5
+: unchanged-interval (eigenmonzo) basis: 2.9/5
 Tuning ranges:
@@ Line 623: / Line 726: @@
 {{Optimal ET sequence|legend=0| 10, 21e, 31, 41, 72, 247c, 319bcde, 391bcde, 463bccde }}
-Badness (Smith): 0.010684
+Badness (Sintel): 0.353
 ==== Miraculous ====
@@ Line 632: / Line 735: @@
 Mapping: {{mapping| 1 1 3 3 2 4 | 0 6 -7 -2 15 -3 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~15/14 = 116.747
+Optimal tunings:
+* WE: ~2 = 1200.1267{{c}}, ~15/14 = 116.7596{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.7488{{c}}
-{{Optimal ET sequence|legend=0| 10, 21e, 31, 41, 72f, 113f, 185cff }}
+{{Optimal ET sequence|legend=0| 10, 21e, 31, 41, 72f }}
-Badness (Smith): 0.018669
+Badness (Sintel): 0.771
 ===== 17-limit =====
@@ Line 645: / Line 750: @@
 Mapping: {{mapping| 1 1 3 3 2 4 4 | 0 6 -7 -2 15 -3 1 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~15/14 = 116.769
+Optimal tunings:
+* WE: ~2 = 1199.6759{{c}}, ~15/14 = 116.7378{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.7657{{c}}
+{{Optimal ET sequence|legend=0| 10, 21e, 31, 41, 72fg }}
+Badness (Sintel): 0.870
+===== 19-limit =====
+Subgroup: 2.3.5.7.11.13.17.19
+Comma list: 105/104, 120/119, 144/143, 154/153, 170/169, 210/209
+{{Todo|complete temperament data|inline=1}}
-{{Optimal ET sequence|legend=0| 10, 21e, 31, 41, 72fg, 113fgg }}
+===== 23-limit =====
+Subgroup: 2.3.5.7.11.13.17.19.23
-Badness (Smith): 0.017084
+Comma list: 105/104, 120/119, 144/143, 154/153, 161/160, 170/169, 210/209
+{{Todo|complete temperament data|inline=1}}
 ==== Benediction ====
@@ Line 658: / Line 779: @@
 Mapping: {{mapping| 1 1 3 3 2 7 | 0 6 -7 -2 15 -34 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~15/14 = 116.574
+Optimal tunings:
+* WE: ~2 = 1199.8601{{c}}, ~15/14 = 116.6572{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.5688{{c}}
 {{Optimal ET sequence|legend=0| 31, 72, 103, 175f }}
-Badness (Smith): 0.015715
+Badness (Sintel): 0.649
 ===== 17-limit =====
@@ Line 671: / Line 794: @@
 Mapping: {{mapping| 1 1 3 3 2 7 7 | 0 6 -7 -2 15 -34 -30 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~15/14 = 116.585
+Optimal tunings:
+* WE: ~2 = 1200.8328{{c}}, ~15/14 = 116.6661{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.5774{{c}}
-{{Optimal ET sequence|legend=0| 31, 72, 103, 175f }}
+{{Optimal ET sequence|legend=0| 31, 72, 103, 175f, 422bcdefffg }}
+Badness (Sintel): 0.639
+===== 19-limit =====
+Subgroup: 2.3.5.7.11.13.17.19
+Comma list: 210/209, 225/224, 243/242, 273/272, 286/285, 375/374
+{{Todo|complete temperament data|inline=1}}
+===== 23-limit =====
+Subgroup: 2.3.5.7.11.13.17.19.23
+Comma list: 162/161, 210/209, 225/224, 231/230, 243/242, 273/272, 286/285
-Badness (Smith): 0.012537
+{{Todo|complete temperament data|inline=1}}
 ==== Manna ====
@@ Line 684: / Line 823: @@
 Mapping: {{mapping| 1 1 3 3 2 0 | 0 6 -7 -2 15 38 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~15/14 = 116.739
+Optimal tunings:
+* WE: ~2 = 1200.7564{{c}}, ~15/14 = 116.8129{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.7528{{c}}
 {{Optimal ET sequence|legend=0| 31f, 41, 72, 185cf, 257cff }}
-Badness (Smith): 0.017012
+Badness (Sintel): 0.703
 ===== 17-limit =====
@@ Line 697: / Line 838: @@
 Mapping: {{mapping| 1 1 3 3 2 0 0 | 0 6 -7 -2 15 38 42 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~15/14 = 116.727
+Optimal tunings:
+* WE: ~2 = 1200.7570{{c}}, ~15/14 = 116.8011{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.7408{{c}}
 {{Optimal ET sequence|legend=0| 31fg, 41, 72, 185cf, 257cff }}
-Badness (Smith): 0.014680
+Badness (Sintel): 0.748
+===== 19-limit =====
+Subgroup: 2.3.5.7.11.13.17.19
+Comma list: 210/209, 225/224, 243/242, 273/272, 325/324, 343/342
+{{Todo|complete temperament data|inline=1}}
+===== 23-limit =====
+Subgroup: 2.3.5.7.11.13.17.19.23
+Comma list: 210/209, 225/224, 243/242, 273/272, 300/299, 325/324, 343/342
+{{Todo|complete temperament data|inline=1}}
 ==== Semimiracle ====
@@ Line 709: / Line 866: @@
 Mapping: {{mapping| 2 2 6 6 4 7 | 0 6 -7 -2 15 2 }}
+: mapping generators: ~55/39, ~15/14
-Optimal tuning (POTE): ~99/70 = 600.000, ~15/14 = 116.624
+Optimal tunings:
+* WE: ~55/39 = 600.4844{{c}}, ~15/14 = 116.7182{{c}}
+* CWE: ~55/39 = 600.0000{{c}}, ~15/14 = 116.6413{{c}}
 {{Optimal ET sequence|legend=0| 10, 62, 72 }}
-Badness (Smith): 0.024622
+Badness (Sintel): 1.02
 ===== 17-limit =====
@@ Line 723: / Line 883: @@
 Mapping: {{mapping| 2 2 6 6 4 7 7 | 0 6 -7 -2 15 2 6 }}
-Optimal tuning (POTE): ~17\12 = 600.000, ~15/14 = 116.628
+Optimal tunings:
+* WE: ~17/12 = 600.5042{{c}}, ~15/14 = 116.7264{{c}}
+* CWE: ~17/12 = 600.0000{{c}}, ~15/14 = 116.6485{{c}}
 {{Optimal ET sequence|legend=0| 10, 62, 72 }}
-Badness (Smith): 0.016130
+Badness (Sintel): 0.822
+===== 19-limit =====
+Subgroup: 2.3.5.7.11.13.17.19
+Comma list: 169/168, 210/209, 221/220, 225/224, 243/242, 273/272
+{{Todo|complete temperament data|inline=1}}
+===== 23-limit =====
+Subgroup: 2.3.5.7.11.13.17.19.23
+Comma list: 169/168, 208/207, 210/209, 221/220, 225/224, 243/242, 273/272
+{{Todo|complete temperament data|inline=1}}
 ==== Hemisecordite ====
@@ Line 735: / Line 911: @@
 Mapping: {{mapping| 1 1 3 3 2 2 | 0 12 -14 -4 30 35 }}
+: mapping generators: ~2, ~27/26
-Optimal tuning (POTE): ~2 = 1200.000, ~27/26 = 58.288
+Optimal tunings:
+* WE: ~2 = 1200.6969{{c}}, ~27/26 = 58.3217{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~27/26 = 58.2964{{c}}
 {{Optimal ET sequence|legend=0| 41, 62, 103, 247c, 350bcde }}
-Badness (Smith): 0.025589
+Badness (Sintel): 1.06
 ===== 17-limit =====
@@ Line 749: / Line 928: @@
 Mapping: {{mapping| 1 1 3 3 2 2 2 | 0 12 -14 -4 30 35 43 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~27/26 = 58.261
+Optimal tunings:
+* WE: ~2 = 1200.6557{{c}}, ~27/26 = 58.2932{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~27/26 = 58.2702{{c}}
 {{Optimal ET sequence|legend=0| 41, 62, 103 }}
-Badness (Smith): 0.022535
+Badness (Sintel): 1.15
+===== 19-limit =====
+Subgroup: 2.3.5.7.11.13.17.19
+Comma list:
+{{Todo|complete temperament data|inline=1}}
+===== 23-limit =====
+Subgroup: 2.3.5.7.11.13.17.19.23
+Comma list:
+{{Todo|complete temperament data|inline=1}}
 ===== Semihemisecordite =====
@@ Line 761: / Line 956: @@
 Mapping: {{mapping| 2 2 6 6 4 4 7 | 0 12 -14 -4 30 35 12 }}
+: mapping generators: ~17/12, ~27/26
-Optimal tuning (POTE): ~17/12 = 600.000, ~27/26 = 58.288
+Optimal tunings:
+* WE: ~17/12 = 600.3951{{c}}, ~27/26 = 58.3260{{c}}
+* CWE: ~17/12 = 600.0000{{c}}, ~27/26 = 58.2974{{c}}
-{{Optimal ET sequence|legend=0| 62, 144g, 206begg, 350bcdeggg }}
+{{Optimal ET sequence|legend=0| 62, 144g, 206begg }}
-Badness (Smith): 0.046958
+Badness (Sintel): 2.39
 ====== 19-limit ======
@@ Line 775: / Line 973: @@
 Mapping: {{mapping| 2 2 6 6 4 4 7 8 | 0 12 -14 -4 30 35 12 5 }}
-Optimal tuning (POTE): ~17/12 = 600.000, ~27/26 = 58.283
+Optimal tunings:
+* WE: ~17/12 = 600.4418{{c}}, ~27/26 = 58.3255{{c}}
+* CWE: ~17/12 = 600.0000{{c}}, ~27/26 = 58.2928{{c}}
 {{Optimal ET sequence|legend=0| 62, 144gh, 206begghh }}
-Badness (Smith): 0.035057
+Badness (Sintel): 2.13
 ====== 23-limit ======
@@ Line 788: / Line 988: @@
 Mapping: {{mapping| 2 2 6 6 4 4 7 8 7 | 0 12 -14 -4 30 35 12 5 21 }}
-Optimal tuning (POTE): ~17/12 = 600.000, ~27/26 = 58.283
+Optimal tunings:
+* WE: ~17/12 = 600.4451{{c}}, ~27/26 = 58.3264{{c}}
+* CWE: ~17/12 = 600.0000{{c}}, ~27/26 = 58.2942{{c}}
 {{Optimal ET sequence|legend=0| 62, 144gh, 206begghhi }}
-Badness (Smith): 0.026421
+Badness (Sintel): 1.89
 ==== Phicordial ====
@@ Line 799: / Line 1,001: @@
 Comma list: 225/224, 243/242, 385/384, 2200/2197
-Mapping: {{mapping| 1 7 -4 1 17 4 | 0 -18 21 6 -45 -1 }}
+Mapping: {{mapping| 1 -11 17 7 -28 3 | 0 18 -21 -6 45 1 }}
+: mapping generators: ~2, ~13/8
-Optimal tuning (POTE): ~2 = 1200.000, ~16/13 = 361.121
+Optimal tunings:
+* WE: ~2 = 1200.7056{{c}}, ~13/8 = 839.3726{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~13/8 = 838.8831{{c}}
-{{Optimal ET sequence|legend=0| 103, 113, 216c }}
+{{Optimal ET sequence|legend=0| 103, 216c, 319bcde, 535bccdef }}
-Badness (Smith): 0.033198
+Badness (Sintel): 1.37
 ===== 17-limit =====
 Subgroup: 2.3.5.7.11.13.17
-Comma list: 225/224, 243/242, 273/272, 441/440, 2200/2197
+Comma list: 225/224, 243/242, 273/272, 385/384, 2200/2197
-Mapping: {{mapping| 1 7 -4 1 17 4 8 | 0 -18 21 6 -45 -1 -13 }}
+Mapping: {{mapping| 1 -11 17 7 -28 3 -5 | 0 18 -21 -6 45 1 13 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~16/13 = 361.123
+Optimal tunings:
+* WE: ~2 = 1200.5918{{c}}, ~13/8 = 839.2912{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~13/8 = 838.8809{{c}}
-{{Optimal ET sequence|legend=0| 103, 113, 216c }}
+{{Optimal ET sequence|legend=0| 103, 216c, 319bcde }}
-Badness (Smith): 0.024705
+Badness (Sintel): 1.26
+===== 19-limit =====
+Subgroup: 2.3.5.7.11.13.17.19
+Comma list: 210/209, 225/224, 243/242, 273/272, 385/384, 2200/2197
+{{Todo|complete temperament data|inline=1}}
+===== 23-limit =====
+Subgroup: 2.3.5.7.11.13.17.19.23
+Comma list: 210/209, 225/224, 243/242, 273/272, 300/299, 385/384, 1105/1104
+{{Todo|complete temperament data|inline=1}}
 === Revelation ===
@@ Line 827: / Line 1,048: @@
 Mapping: {{mapping| 1 1 3 3 5 | 0 6 -7 -2 -16 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~15/14 = 116.277
+Optimal tunings:
+* WE: ~2 = 1201.3320{{c}}, ~15/14 = 116.4057{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.2524{{c}}
 {{Optimal ET sequence|legend=0| 10e, 21, 31 }}
-Badness (Smith): 0.032946
+Badness (Sintel): 1.09
 ==== 13-limit ====
@@ Line 840: / Line 1,063: @@
 Mapping: {{mapping| 1 1 3 3 5 4 | 0 6 -7 -2 -16 -3 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~15/14 = 116.268
+Optimal tunings:
+* WE: ~2 = 1200.6059{{c}}, ~15/14 = 116.3263{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.2564{{c}}
 {{Optimal ET sequence|legend=0| 10e, 21, 31 }}
-Badness (Smith): 0.029452
+Badness (Sintel): 1.22
 === Hemimiracle ===
@@ Line 852: / Line 1,077: @@
 Mapping: {{mapping| 1 1 3 3 4 | 0 12 -14 -4 -11 }}
+: mapping generators: ~2, ~33/32
-Optimal tuning (POTE): ~2 = 1200.000, ~33/32 = 58.408
+Optimal tunings:
+* WE: ~2 = 1200.2902{{c}}, ~33/32 = 58.4217{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~33/32 = 58.4062{{c}}
-{{Optimal ET sequence|legend=0| 20, 21, 41, 144e, 185cee, 226cee }}
+{{Optimal ET sequence|legend=0| 20, 21, 41 }}
-Badness (Smith): 0.059232
+Badness (Sintel): 1.96
 ==== 13-limit ====
@@ Line 866: / Line 1,094: @@
 Mapping: {{mapping| 1 1 3 3 4 4 | 0 12 -14 -4 -11 -6 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~33/32 = 58.430
+Optimal tunings:
+* WE: ~2 = 1199.8454{{c}}, ~33/32 = 58.4220{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~33/32 = 58.4305{{c}}
-{{Optimal ET sequence|legend=0| 20, 21, 41, 144eff, 185ceeff }}
+{{Optimal ET sequence|legend=0| 20, 21, 41 }}
-Badness (Smith): 0.043151
+Badness (Sintel): 1.78
 === Oracle ===
@@ Line 877: / Line 1,107: @@
 Comma list: 121/120, 225/224, 1029/1024
-Mapping: {{mapping| 1 7 -4 1 3 | 0 -12 14 4 1 }}
+Mapping: {{mapping| 1 -5 10 5 4 | 0 12 -14 -4 -1 }}
+: mapping generators: ~2, ~16/11
-Optimal tuning (POTE): ~2 = 1200.000, ~11/8 = 541.668
+Optimal tunings:
+* WE: ~2 = 1201.2122{{c}}, ~16/11 = 658.9974{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~16/11 = 658.3320{{c}}
 {{Optimal ET sequence|legend=0| 11, 20, 31, 82e, 113e, 144ee }}
-Badness (Smith): 0.042687
+Badness (Sintel): 1.41
 == Hemiseven ==
+Unlike miracle which splits 8/7, hemiseven splits ~16/7, an octave above. It can be described as the {{nowrap| 72 & 77 }} temperament; its ploidacot is gamma-hexacot. [[149edo]] is an obvious tuning.
 [[Subgroup]]: 2.3.5.7
 [[Comma list]]: 1029/1024, 19683/19600
-{{Mapping|legend=1| 1 4 14 2 | 0 -6 -29 2 }}
+{{Mapping|legend=1| 1 -2 -15 4 | 0 6 29 -2 }}
+: mapping generators: ~2, ~243/160
-{{Multival|legend=1| 6 29 -2 32 -20 -86 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.5612{{c}}, ~243/160 = 717.0687{{c}}
+: [[error map]]: {{val| +0.561 -0.665 +0.260 -0.718 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~243/160 = 716.7478{{c}}
+: error map: {{val| 0.000 -1.468 -0.629 -2.321 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~320/243 = 483.267
+{{Optimal ET sequence|legend=1| 72, 149, 221, 514bd, 735bcdd }}
-{{Optimal ET sequence|legend=1| 72, 77, 149, 221, 514bd, 735bcdd }}
+[[Badness]] (Sintel): 1.43
-[[Badness]] (Smith): 0.056557
 === 11-limit ===
@@ Line 905: / Line 1,143: @@
 Comma list: 385/384, 441/440, 19683/19600
-Mapping: {{mapping| 1 4 14 2 -5 | 0 -6 -29 2 21 }}
+Mapping: {{mapping| 1 -2 -15 4 16 | 0 6 29 -2 -21 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~320/243 = 483.276
+Optimal tunings:
+* WE: ~2 = 1200.6243{{c}}, ~243/160 = 717.0969{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~243/160 = 716.7292{{c}}
-{{Optimal ET sequence|legend=0| 72, 77, 149, 221e, 293de }}
+{{Optimal ET sequence|legend=0| 72, 149, 221e, 293de }}
-Badness (Smith): 0.028467
+Badness (Sintel): 0.941
 === 13-limit ===
@@ Line 918: / Line 1,158: @@
 Comma list: 351/350, 385/384, 441/440, 676/675
-Mapping: {{mapping| 1 4 14 2 -5 19 | 0 -6 -29 2 21 -38 }}
+Mapping: {{mapping| 1 -2 -15 4 16 -19 | 0 6 29 -2 -21 38 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~120/91 = 483.256
+Optimal tunings:
+* WE: ~2 = 1200.6781{{c}}, ~91/60 = 717.1496{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~91/60 = 716.7520{{c}}
-{{Optimal ET sequence|legend=0| 72, 77, 149, 221ef }}
+{{Optimal ET sequence|legend=0| 72, 149, 221ef }}
-Badness (Smith): 0.021900
+Badness (Sintel): 0.905
 === 17-limit ===
@@ Line 931: / Line 1,173: @@
 Comma list: 273/272, 351/350, 385/384, 441/440, 676/675
-Mapping: {{mapping| 1 4 14 2 -5 19 21 | 0 -6 -29 2 21 -38 -42 }}
+Mapping: {{mapping| 1 -2 -15 4 16 -19 -21 | 0 6 29 -2 -21 38 42 }}
+Optimal tunings:
+* WE: ~2 = 1200.6635{{c}}, ~68/45 = 717.1354{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~68/45 = 716.7472{{c}}
+{{Optimal ET sequence|legend=0| 72, 149, 221ef }}
+Badness (Sintel): 0.800
+== Valentine ==
+{{Main| Valentine }}
+: ''For the 5-limit version, see [[Syntonic–31 equivalence continuum #Valentine (5-limit)]].''
+Valentine tempers out [[126/125]] and [[6144/6125]] as well as 1029/1024. It has a generator of [[~]][[21/20]], three of which make the slendric generator ~8/7. 21/20 can be stripped of its 2 and taken as 3 × 7/5. In this respect it resembles miracle, with a generator of 3 × 5/7, and casablanca, with a generator of 5 × 7/3. These three generators are the simplest in terms of the relationship of tetrads in the [[7-limit symmetrical lattices|lattice of 7-limit tetrads]]. Valentine can be described as the {{nowrap| 31 & 46 }} temperament; its ploidacot is enneacot. [[77edo]], [[108edo]], or [[185edo]] make for excellent tunings, which also happen to be excellent tunings for [[starling]], the rank-3 temperament tempering out 126/125. Hence 7-limit valentine can be used whenever starling is wanted, with the extra tempering out of 1029/1024 having no discernible effect on tuning accuracy. Another tuning for valentine uses (3/2)<sup>1/9</sup> as a generator, giving pure 3/2 fifths. Valentine extends naturally to the 11-limit, tempering out 121/120 and 441/440; 46edo has a valentine generator 3\46 which is only 0.0117 cents sharp of the minimax generator, (11/7)<sup>1/10</sup>.
+Valentine has a very straighforward [[S-expression]]-based comma list in the [[11-limit]] add-23 (i.e. the 2.3.5.7.11.23 subgroup) of {([[176/175|S8/S10 = S22 × S23 × S24]], [[121/120|S11]]), [[441/440|S21]], [[484/483|S22]], [[529/528|S23]], [[576/575|S24]]}, so it is the temperament that equalizes the 20::25 segment of the harmonic series.
+[[Subgroup]]: 2.3.5.7
+[[Comma list]]: 126/125, 1029/1024
+{{Mapping|legend=1| 1 1 2 3 | 0 9 5 -3 }}
+: mapping generators: ~2, ~21/20
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.0749{{c}}, ~21/20 = 77.8687{{c}}
+: [[error map]]: {{val| +0.075 -1.062 +3.179 -2.207 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 77.8673{{c}}
+: error map: {{val| 0.000 -1.149 +3.023 -2.428 }}
+[[Minimax tuning]]:
+* [[7-odd-limit]]: ~21/20 = {{monzo| 1/6 1/12 0 -1/12 }}
+: {{monzo list| 1 0 0 0 | 5/2 3/4 0 -3/4 | 17/6 5/12 0 -5/12 | 5/2 -1/4 0 1/4 }}
+: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/3
+* [[9-odd-limit]]: ~21/20 = {{monzo| 1/21 2/21 0 -1/21}}
+: {{monzo list| 1 0 0 0 | 10/7 6/7 0 -3/7 | 47/21 10/21 0 -5/21 | 20/7 -2/7 0 1/7 }}
+: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/7
+[[Algebraic generator]]: smaller root of ''x''<sup>2</sup> - 89''x'' + 92, or (89 - sqrt (7553))/2, at 77.8616 cents.
+{{Optimal ET sequence|legend=1| 15, 31, 46, 77, 185 }}
+[[Badness]] (Sintel): 0.786
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
+Comma list: 121/120, 126/125, 176/175
+Mapping: {{mapping| 1 1 2 3 3 | 0 9 5 -3 7 }}
+Optimal tunings:
+* WE: ~2 = 1200.3890{{c}}, ~22/21 = 77.9065{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~22/21 = 77.9007{{c}}
+Minimax tuning:
+* 11-odd-limit: ~21/20 = {{monzo| 0 0 0 -1/10 1/10 }}
+: [{{monzo| 1 0 0 0 0 }}, {{monzo| 1 0 0 -9/10 9/10 }}, {{monzo| 2 0 0 -1/2 1/2 }}, {{monzo| 3 0 0 3/10 -3/10 }}, {{monzo| 3 0 0 -7/10 7/10 }}]
+: unchanged-interval (eigenmonzo) basis: 2.11/7
+Algebraic generator: positive root of 4''x''<sup>3</sup> + 15''x''<sup>2</sup> - 21, or else Gontrand2, the smallest positive root of 4''x''<sup>7</sup> - 8''x''<sup>6</sup> + 5.
+{{Optimal ET sequence|legend=0| 15, 31, 46, 77 }}
+Badness (Sintel): 0.552
+==== Valentino ====
+Subgroup: 2.3.5.7.11.13
+Comma list: 121/120, 126/125, 176/175, 196/195
+Mapping: {{mapping| 1 1 2 3 3 5 | 0 9 5 -3 7 -20 }}
+Optimal tunings:
+* WE: ~2 = 1200.1967{{c}}, ~22/21 = 77.9708{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~22/21 = 77.9594{{c}}
+{{Optimal ET sequence|legend=0| 15f, 31, 46, 77 }}
+Badness (Sintel): 0.854
+===== 17-limit =====
+Subgroup: 2.3.5.7.11.13.17
+Comma list: 121/120, 126/125, 154/153, 176/175, 196/195
+Mapping: {{mapping| 1 1 2 3 3 5 5 | 0 9 5 -3 7 -20 -14 }}
+Optimal tunings:
+* WE: ~2 = 1200.0404{{c}}, ~22/21 = 78.0055{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~22/21 = 78.0029{{c}}
+{{Optimal ET sequence|legend=0| 15f, 31, 46, 77, 123e }}
+Badness (Sintel): 0.854
+==== Lupercalia ====
+Subgroup: 2.3.5.7.11.13
+Comma list: 66/65, 105/104, 121/120, 126/125
+Mapping: {{mapping| 1 1 2 3 3 3 | 0 9 5 -3 7 11 }}
+Optimal tunings:
+* WE: ~2 = 1199.9143{{c}}, ~22/21 = 77.7039{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~22/21 = 77.7049{{c}}
+{{Optimal ET sequence|legend=0| 15, 31 }}
+Badness (Sintel): 0.881
+==== Dwynwen ====
+Subgroup: 2.3.5.7.11.13
+Comma list: 91/90, 121/120, 126/125, 176/175
+Mapping: {{mapping| 1 1 2 3 3 2 | 0 9 5 -3 7 26 }}
+Optimal tunings:
+* WE: ~2 = 1200.1306{{c}}, ~22/21 = 78.2273{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~22/21 = 78.2241{{c}}
+{{Optimal ET sequence|legend=0| 15, 31f, 46 }}
+Badness (Sintel): 0.969
+==== Semivalentine ====
+Subgroup: 2.3.5.7.11.13
+Comma list: 121/120, 126/125, 169/168, 176/175
+Mapping: {{mapping| 2 2 4 6 6 7 | 0 9 5 -3 7 3 }}
+: mapping generators: ~55/39, ~22/21
+Optimal tunings:
+* WE: ~55/39 = 600.3497{{c}}, ~22/21 = 77.8845{{c}}
+* CWE: ~55/39 = 600.0000{{c}}, ~22/21 = 77.8715{{c}}
+{{Optimal ET sequence|legend=0| 16, 30, 46, 62, 108ef }}
+Badness (Sintel): 1.35
+==== Hemivalentine ====
+Subgroup: 2.3.5.7.11.13
+Comma list: 121/120, 126/125, 176/175, 343/338
+Mapping: {{mapping| 1 1 2 3 3 4 | 0 18 10 -6 14 -9 }}
+: mapping generators: ~2, ~40/39
+Optimal tunings:
+* WE: ~2 = 1199.6529{{c}}, ~40/39 = 39.0323{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~40/39 = 39.0383{{c}}
+{{Optimal ET sequence|legend=0| 30, 31, 61, 92f }}
+Badness (Sintel): 1.94
+==== Demivalentine ====
+Subgroup: 2.3.5.7.11.13
+Comma list: 121/120, 126/125, 176/175, 676/675
+Mapping: {{mapping| 1 -8 -3 6 -4 -16 | 0 18 10 -6 14 37 }}
+: mapping generators: ~2, ~13/9
+Optimal tunings:
+* WE: ~2 = 1200.3929{{c}}, ~13/9 = 639.1320{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~13/9 = 638.9325{{c}}
+{{Optimal ET sequence|legend=0| 15, 47ef, 62, 77 }}
+Badness (Sintel): 1.44
+=== Hemivalentino ===
+Subgroup: 2.3.5.7.11
+Comma list: 126/125, 243/242, 1029/1024
+Mapping: {{mapping| 1 1 2 3 2 | 0 18 10 -6 45 }}
+Optimal tunings:
+* WE: ~2 = 1200.0816{{c}}, ~45/44 = 38.9236{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~45/44 = 38.9228{{c}}
+{{Optimal ET sequence|legend=0| 31, 92e, 123, 154, 185 }}
+Badness (Sintel): 2.03
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
+Comma list: 126/125, 196/195, 243/242, 1029/1024
+Mapping: {{mapping| 1 1 2 3 2 5 | 0 18 10 -6 45 -40 }}
+Optimal tunings:
+* WE: ~2 = 1199.8782{{c}}, ~45/44 = 38.9440{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~45/44 = 38.9472{{c}}
+{{Optimal ET sequence|legend=0| 31, 123, 154 }}
+Badness (Sintel): 2.39
+==== Hemivalentoid ====
+Subgroup: 2.3.5.7.11.13
+Comma list: 126/125, 144/143, 243/242, 343/338
+Mapping: {{mapping| 1 1 2 3 2 4 | 0 18 10 -6 45 -9 }}
+Optimal tunings:
+* WE: ~2 = 1199.3614{{c}}, ~45/44 = 38.9721{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~45/44 = 38.9839{{c}}
+{{Optimal ET sequence|legend=0| 31, 92ef }}
+Badness (Sintel): 2.39
+== Superkleismic ==
+{{Main| Superkleismic }}
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Shibboleth]].''
+Superkleismic tempers out the keema, [[875/864]], and can be described as the {{nowrap| 15 & 26 }} temperament. It splits the ~7/4 into three ~6/5 generators of around 322 cents. This is noticeably sharper than the [[kleismic]] generator, hence the name.
+In the 11-limit, two generator steps can be identified with ~16/11, and in the 13-limit, the same step can be treated as ~13/9. The [[S-expression]]-based comma list of 13-limit superkleismic is {[[875/864|S5/S6]], [[1029/1024|S7/S8]], [[100/99|S10]], [[144/143|S12]], ([[441/440|S21]])}. Through careful observation of the equivalences therein one can derive the mapping of the full 13-limit.
+Note that the generator is given as 6/5's octave complement, [[5/3]], in the data that follow, since a stack of 9 such generators octave-reduced is the perfect fifth; the [[ploidacot]] of superkleismic is wau-enneacot.
+Superkleismic also sets two intervals of [[21/20]] equal to [[10/9]]; as {{nowrap| 10/9 {{=}} ([[20/19]])⋅([[19/18]]) }}, we can identify 21/20, 20/19, and 19/18 together to add prime 19, tempering out [[361/360]] ({{S|19}}) and [[400/399]] ({{S|20}}). This structure is preserved within the entire superkleismic tuning range between 15edo and 26edo, while extensions for primes 13 and 17 bifurcate and are of higher complexity and lower accuracy.
+edo gives an obvious tuning in all the subgroups.
+[[Subgroup]]: 2.3.5.7
+[[Comma list]]: 875/864, 1029/1024
+{{Mapping|legend=1| 1 -5 -5 5 | 0 9 10 -3 }}
+: mapping generators: ~2, ~5/3
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.7640{{c}}, ~5/3 = 878.6289{{c}}
+: [[error map]]: {{val| +0.764 +1.885 +3.844 -0.893 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/3 = 878.1077{{c}}
+: error map: {{val| 0.000 +1.014 -5.237 -3.149 }}
+{{Optimal ET sequence|legend=1| 11c, 15, 26, 41 }}
+[[Badness]] (Sintel): 1.21
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
+Comma list: 100/99, 245/242, 385/384
+Mapping: {{mapping| 1 -5 -5 5 2 | 0 9 10 -3 2 }}
+Optimal tunings:
+* WE: ~2 = 1200.1691{{c}}, ~5/3 = 878.2772{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 878.1606{{c}}
+{{Optimal ET sequence|legend=0| 11c, 15, 26, 41, 179cde, 220cde, 261ccdee }}
+Badness (Sintel): 0.848
+==== 2.3.5.7.11.19 subgroup ====
+Subgroup: 2.3.5.7.11.19
+Comma list: 100/99, 133/132, 190/189, 385/384
+Mapping: {{mapping| 1 -5 -5 5 2 -6 | 0 9 10 -3 2 14 }}
+Optimal tunings:
+* WE: ~2 = 1200.2289{{c}}, ~5/3 = 878.3409{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 878.1840{{c}}
+{{Optimal ET sequence|legend=0| 11c, 15, 26, 41, 138e }}
+Badness (Sintel): 0.692
+=== 13-limit ===
+Superkleismic in the 13-limit does considerably more damage than in the 11-limit, as indicated by being supported by much fewer [[patent val]]s and having higher Dirichlet badness than its 11-limit counterpart. However, this remains an obvious canonical mapping for prime 13.
+Subgroup: 2.3.5.7.11.13
+Comma list: 100/99, 105/104, 144/143, 245/242
+Mapping: {{mapping| 1 -5 -5 5 2 -8 | 0 9 10 -3 2 16 }}
+Optimal tunings:
+* WE: ~2 = 1200.0261{{c}}, ~5/3 = 878.0252{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 878.0073{{c}}
+{{Optimal ET sequence|legend=0| 11cf, 15, 26, 41 }}
+Badness (Sintel): 0.887
+==== 17-limit ====
+Subgroup: 2.3.5.7.11.13.17
+Comma list: 100/99, 105/104, 120/119, 144/143, 245/242
+Mapping: {{mapping| 1 -5 -5 5 2 -8 -12 | 0 9 10 -3 2 16 22 }}
+Optimal tunings:
+* WE: ~2 = 1200.0488{{c}}, ~5/3 = 877.8872{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 877.8537{{c}}
+{{Optimal ET sequence|legend=0| 11cfg, 15g, 26, 41 }}
+Badness (Sintel): 1.01
+==== 19-limit ====
+Subgroup: 2.3.5.7.11.13.17.19
+Comma list: 100/99, 105/104, 120/119, 144/143, 133/132, 190/189
+Mapping: {{mapping| 1 -5 -5 5 2 -8 -12 -6 | 0 9 10 -3 2 16 22 14 }}
+Optimal tunings:
+* WE: ~2 = 1200.2120{{c}}, ~5/3 = 878.0243{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 877.8789{{c}}
+{{Optimal ET sequence|legend=0| 11cfgh, 15g, 26, 41 }}
+Badness (Sintel): 0.964
+=== Superana ===
+This extension ({{nowrap| 41 & 56 }}) is the counterpart of canonical superkleismic on the other side of 41edo.
+Subgroup: 2.3.5.7.11.13
+Comma list: 100/99, 196/195, 245/242, 385/384
+Mapping: {{mapping| 1 -5 -5 5 2 22 | 0 9 10 -3 2 -25 }}
+Optimal tunings:
+* WE: ~2 = 1199.8272{{c}}, ~5/3 = 878.1538{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 878.2795{{c}}
+{{Optimal ET sequence|legend=0| 15f, 41, 97, 138e }}
+Badness (Sintel): 1.40
+==== 17-limit ====
+Subgroup: 2.3.5.7.11.13.17
+Comma list: 100/99, 154/153, 196/195, 245/242, 256/255
+Mapping: {{mapping| 1 -5 -5 5 2 22 18 | 0 9 10 -3 2 -25 -19 }}
+Optimal tunings:
+* WE: ~2 = 1199.5964{{c}}, ~5/3 = 878.0482{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 878.3444{{c}}
+{{Optimal ET sequence|legend=0| 15f, 41, 56, 97g }}
+Badness (Sintel): 1.45
+==== 19-limit ====
+Subgroup: 2.3.5.7.11.13.17.19
+Comma list: 100/99, 133/132, 154/153, 190/189, 196/195, 256/255
+Mapping: {{mapping| 1 -5 -5 5 2 22 18 -6 | 0 9 10 -3 2 -25 -19 14 }}
+Optimal tunings:
+* WE: ~2 = 1199.6638{{c}}, ~5/3 = 878.1109{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 878.3566{{c}}
+{{Optimal ET sequence|legend=0| 15f, 41, 56, 97g }}
+Badness (Sintel): 1.36
+== Dee leap week ==
+{{Main| Dee leap week }}
+[[Subgroup]]: 2.3.5.7
+[[Comma list]]: 1029/1024, 2460375/2458624
-Optimal tuning (POTE): ~2 = 1200.000, ~45/34 = 483.261
+{{Mapping|legend=1| 1 -5 25 5 | 0 9 -31 -3 }}
-{{Optimal ET sequence|legend=0| 72, 77, 149, 221ef }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.4835{{c}}, ~224/135 = 878.2507{{c}}
+: [[error map]]: {{val| +0.484 -0.117 +0.004 -1.160 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~224/135 = 877.8926{{c}}
+: error map: {{val| 0.000 -0.921 -0.985 -2.504 }}
-Badness (Smith): 0.015701
+{{Optimal ET sequence|legend=1| 41, 108, 149, 190 }}
-== Unidec ==
+[[Badness]] (Sintel): 2.12
-{{Main| Unidec }}
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-=== 5-limit (unidecmic) ===
+Comma list: 385/384, 441/440, 2460375/2458624
-[[Subgroup]]: 2.3.5
-[[Comma list]]: 31381059609/31250000000
+Mapping: {{mapping| 1 -5 25 5 -28 | 0 9 -31 -3 43 }}
-{{Mapping|legend=1| 2 5 8 | 0 -6 -11 }}
+Optimal tunings:
+* WE: ~2 = 1200.4874{{c}}, ~224/135 = 878.2543{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~224/135 = 877.8987{{c}}
-: mapping generators: ~177147/125000, ~10/9
+{{Optimal ET sequence|legend=0| 41, 108e, 149, 190 }}
-[[Optimal tuning]] ([[POTE]]): ~177147/125000 = 600.000, ~10/9 = 183.047
+Badness (Sintel): 1.35
-{{Optimal ET sequence|legend=1| 26, 46, 72, 118, 2524, 2642, 2760, 2878b, …, 5002bc }}
+== Unidec ==
+{{Main| Unidec }}
-[[Badness]] (Smith): 0.082423
+Unidec tempers out the ragisma, [[4375/4374]], and may be described as the {{nowrap| 26 & 46 }} temperament. It has a [[semi-octave]] [[period]] and a generator of ~80/63, two of which minus a period make slendric's generator; its [[ploidacot]] is therefore diploid gamma-hexacot. In the 11-limit, the generator represents [[14/11]]. [[190edo]] makes for an excellent tuning in both the 7-limit and 11-limit.
-=== 7-limit ===
 [[Subgroup]]: 2.3.5.7
 [[Comma list]]: 1029/1024, 4375/4374
-{{Mapping|legend=1| 2 5 8 5 | 0 -6 -11 2 }}
+{{Mapping|legend=1| 2 -1 -3 7 | 0 6 11 -2 }}
-{{Multival|legend=1| 12 22 -4 7 -40 -71 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~1225/864 = 600.2429{{c}}, ~80/63 = 417.0073{{c}}
-[[Optimal tuning]] ([[POTE]]): ~1225/864 = 600.000, ~10/9 = 183.161
+: [[error map]]: {{val| +0.486 -0.154 +0.038 -1.140 }}
+* [[CWE]]: ~1225/864 = 600.0000{{c}}, ~80/63 = 416.8688{{c}}
+: error map: {{val| 0.000 -0.924 -1.090 -2.503 }}
 [[Minimax tuning]]:
 * [[7-odd-limit]]: ~10/9 = {{monzo| 3/26 0 -1/13 1/13 }}
 : {{monzo list| 1 0 0 0 | 47/26 0 6/13 -6/13 | 71/26 0 11/13 -11/13 | 71/26 0 -2/13 2/13 }}
-: [[eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.7/5
+: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5
 * [[9-odd-limit]]: ~10/9 = {{monzo| 5/28 -1/7 0 1/14 }}
 : {{Monzo list| 1 0 0 0 | 10/7 6/7 0 -3/7 | 57/28 11/7 0 -11/14 | 20/7 -2/7 0 1/7 }}
-: [[eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.9/7
+: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/7
 {{Optimal ET sequence|legend=1| 26, 46, 72, 118, 190 }}
-[[Badness]] (Smith): 0.038393
+[[Badness]] (Sintel): 0.972
 === 11-limit ===
@@ Line 985: / Line 1,616: @@
 Comma list: 385/384, 441/440, 4375/4374
-Mapping: {{mapping| 2 5 8 5 6 | 0 -6 -11 2 3 }}
+Mapping: {{mapping| 2 -1 -3 7 9 | 0 6 11 -2 -3 }}
+Optimal tunings:
+* WE: ~99/70 = 600.2497{{c}}, ~14/11 = 417.0085{{c}}
+* CWE: ~99/70 = 600.0000{{c}}, ~14/11 = 416.8543{{c}}
 Minimax tuning:
 * [[11-odd-limit]]: ~10/9 = {{monzo| 5/28 -1/7 0 1/14 }}
 : [{{monzo| 1 0 0 0 0 }}, {{monzo| 10/7 6/7 0 -3/7 0 }}, {{monzo| 57/28 11/7 0 -11/14 0 }}, {{monzo| 20/7 -2/7 0 1/7 0 }}, {{monzo| 99/28 -3/7 0 3/14 0 }}]
-: eigenmonzo (unchanged-interval) basis: 2.9/7
+: unchanged-interval (eigenmonzo) basis: 2.9/7
 {{Optimal ET sequence|legend=0| 26, 46, 72, 118, 190 }}
-Badness (Smith): 0.015479
+Badness (Sintel): 0.512
 ==== Ekadash ====
@@ Line 1,001: / Line 1,636: @@
 Comma list: 385/384, 441/440, 625/624, 729/728
-Mapping: {{mapping| 2 5 8 5 6 19 | 0 -6 -11 2 3 -38 }}
+Mapping: {{mapping| 2 -1 -3 7 9 -19 | 0 6 11 -2 -3 38 }}
-Optimal tuning (POTE): ~99/70 = 600.000, ~10/9 = 183.187
+Optimal tunings:
+* WE: ~99/70 = 600.2497{{c}}, ~14/11 = 417.0085{{c}}
+* CWE: ~99/70 = 600.0000{{c}}, ~14/11 = 416.8543{{c}}
-{{Optimal ET sequence|legend=0| 26f, 46f, 72, 118, 190, 262df, 452cdef }}
+{{Optimal ET sequence|legend=0| 46f, 72, 118, 190, 262df, 452cdef }}
-Badness (Smith): 0.020381
+Badness (Sintel): 0.842
 ==== Hendec ====
@@ Line 1,014: / Line 1,651: @@
 Comma list: 169/168, 325/324, 364/363, 385/384
-Mapping: {{mapping| 2 5 8 5 6 8 | 0 -6 -11 2 3 -2 }}
+Mapping: {{mapping| 2 -1 -3 7 9 6 | 0 6 11 -2 -3 2 }}
-Optimal tuning (POTE): ~91/64 = 600.000, ~10/9 = 183.198
+Optimal tunings:
+* WE: ~91/64 = 600.3825{{c}}, ~14/11 = 417.0678{{c}}
+* CWE: ~91/64 = 600.0000{{c}}, ~14/11 = 416.8290{{c}}
 {{Optimal ET sequence|legend=0| 26, 46, 72, 190ff }}
-Badness (Smith): 0.017707
+Badness (Sintel): 0.732
 ===== 17-limit =====
@@ Line 1,027: / Line 1,666: @@
 Comma list: 169/168, 221/220, 273/272, 325/324, 364/363
-Mapping: {{mapping| 2 5 8 5 6 8 10 | 0 -6 -11 2 3 -2 -6 }}
+Mapping: {{mapping| 2 -1 -3 7 9 6 4 | 0 6 11 -2 -3 2 6 }}
-Optimal tuning (POTE): ~17/12 = 600.000, ~10/9 = 183.196
+Optimal tunings:
+* WE: ~17/12 = 600.3991{{c}}, ~14/11 = 417.0809{{c}}
+* CWE: ~17/12 = 600.0000{{c}}, ~14/11 = 416.8330{{c}}
 {{Optimal ET sequence|legend=0| 26, 46, 72, 190ffg }}
-Badness (Smith): 0.011676
+Badness (Sintel): 0.595
-== Lagaca ==
+== Necromanteion ==
-[[Subgroup]]: 2.3.5.7
+Necromanteion, named by [[Johannes Werpup]] in 2014<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_106371.html Yahoo! Tuning Group | ''Temperament ideas: A cuckoo, and two oracles'']</ref> may be described as the {{nowrap| 31 & 51c }} temperament. The generator is a subfifth representing 35/24, four of which minus two octaves make slendric's generator, so its [[ploidacot]] is beta-dodecacot.
-[[Comma list]]: 1029/1024, 11529602/11390625
-{{Mapping|legend=1| 2 5 2 5 | 0 -9 13 3 }}
-{{Multival|legend=1| 18 -26 -6 -83 -60 59 }}
-[[Optimal tuning]] ([[POTE]]): ~3375/2401 = 600.000, ~15/14 = 122.027
-{{Optimal ET sequence|legend=1| 10, 98, 108, 118 }}
-[[Badness]] (Smith): 0.144345
-== Necromanteion ==
 [[Subgroup]]: 2.3.5.7
 [[Comma list]]: 1029/1024, 5103/5000
-{{Mapping|legend=1| 1 7 10 1 | 0 -12 -17 4 }}
+{{Mapping|legend=1| 1 -5 -7 5 | 0 12 17 -4 }}
+: mapping generators: ~2, ~35/24
-{{Multival|legend=1| 12 17 -4 -1 -40 -57 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.2959{{c}}, ~35/24 = 658.3833{{c}}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~48/35 = 541.779
+: [[error map]]: {{val| +0.296 -2.835 +4.130 -0.879 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~35/24 = 658.2313{{c}}
+: error map: {{val| 0.000 -3.179 +3.619 -1.751 }}
-{{Optimal ET sequence|legend=1| 11c, 20c, 31, 144c, 175c, 206bc, 237bc, 505bbccd }}
+{{Optimal ET sequence|legend=1| 11c, 20c, 31, 144c, 175c }}
-[[Badness]] (Smith): 0.117680
+[[Badness]] (Sintel): 2.98
 === 11-limit ===
@@ Line 1,070: / Line 1,701: @@
 Comma list: 176/175, 243/242, 1029/1024
-Mapping: {{mapping| 1 7 10 1 17 | 0 -12 -17 4 -30 }}
+Mapping: {{mapping| 1 -5 -7 5 -13 | 0 12 17 -4 30 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~15/11 = 541.729
+Optimal tunings:
+* WE: ~2 = 1200.2862{{c}}, ~22/15 = 658.4276{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~22/15 = 658.2805{{c}}
-{{Optimal ET sequence|legend=0| 20ce, 31, 113c, 144c, 175c, 381bccdee }}
+{{Optimal ET sequence|legend=0| 20ce, 31, 113c, 144c }}
-Badness (Smith): 0.053459
+Badness (Sintel): 1.77
 === 13-limit ===
@@ Line 1,083: / Line 1,716: @@
 Comma list: 144/143, 176/175, 243/242, 343/338
-Mapping: {{mapping| 1 7 10 1 17 1 | 0 -12 -17 4 -30 6 }}
+Mapping: {{mapping| 1 -5 -7 5 -13 7 | 0 12 17 -4 30 -6 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~15/11 = 541.606
+Optimal tunings:
+* WE: ~2 = 1199.3663{{c}}, ~22/15 = 658.0465{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~22/15 = 658.3800{{c}}
-{{Optimal ET sequence|legend=0| 20ce, 31, 51ce, 82cf, 113cf, 144cf }}
+{{Optimal ET sequence|legend=0| 20ce, 31, 82cf, 113cf }}
-Badness (Smith): 0.047015
+Badness (Sintel): 1.94
 == Restles ==
 {{See also| Lesser tendoneutralic }}
+Restles may be described as the {{nowrap| 77 & 87 }} temperament, and has a [[ploidacot]] signature of wau-dodecacot. It was named by [[Petr Pařízek]] in 2011 for it is some sort of opposite to [[beatles]]<ref name="petr's long post">[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101780.html Yahoo! Tuning Group | ''Suggested names for the unclasified temperaments'']</ref>.
 [[Subgroup]]: 2.3.5.7
@@ Line 1,099: / Line 1,736: @@
 {{Mapping|legend=1| 1 -2 8 4 | 0 12 -19 -4 }}
+: mapping generators: ~2. ~315/256
-{{Multival|legend=1| 12 -19 -4 -58 -40 44 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.0322{{c}}, ~315/256 = 358.5581{{c}}
+: [[error map]]: {{val| +0.032 +0.678 +1.340 -2.930 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~315/256 = 358.5484{{c}}
+: error map: {{val| 0.000 +0.626 +1.267 -3.019 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~315/256 = 358.548
+{{Optimal ET sequence|legend=1| 77, 87, 164 }}
-{{Optimal ET sequence|legend=1| 10, 77, 87, 164 }}
+[[Badness]] (Sintel): 2.73
-[[Badness]] (Smith): 0.108011
 === 11-limit ===
@@ Line 1,115: / Line 1,755: @@
 Mapping: {{mapping| 1 -2 8 4 -7 | 0 12 -19 -4 35 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~27/22 = 358.571
+Optimal tunings:
+* WE: ~2 = 1200.1110{{c}}, ~27/22 = 358.6045{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~27/22 = 358.5720{{c}}
-{{Optimal ET sequence|legend=0| 10, 77, 87, 164 }}
+{{Optimal ET sequence|legend=0| 77, 87, 164, 251d }}
-Badness (Smith): 0.054655
+Badness (Sintel): 1.81
 === 13-limit ===
@@ Line 1,128: / Line 1,770: @@
 Mapping: {{mapping| 1 -2 8 4 -7 4 | 0 12 -19 -4 35 -1 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~16/13 = 358.574
+Optimal tunings:
+* WE: ~2 = 1200.0482{{c}}, ~~16/13 = 358.5883{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~16/13 = 358.5741{{c}}
-{{Optimal ET sequence|legend=0| 10, 77, 87, 164 }}
+{{Optimal ET sequence|legend=0| 77, 87, 164, 251d }}
-Badness (Smith): 0.028187
+Badness (Sintel): 1.16
+== Lagaca ==
+Cryptically named by [[Petr Pařízek]] in 2011<ref name="petr's long post"/>, lagaca may be described as the {{nowrap| 10 & 118 }} temperament with a [[ploidacot]] signature of diploid wau-enneacot. The name actually refers to the fact that 12 generator steps in this temperament make ~7/3, where "l", "g", "c" are integers alphabetically converted to letters.
+[[Subgroup]]: 2.3.5.7
+[[Comma list]]: 1029/1024, 11529602/11390625
+{{Mapping|legend=1| 2 -4 15 8 | 0 9 -13 -3 }}
+: mapping generators: ~3375/2401, ~450/343
+[[Optimal tuning]]s:
+* [[WE]]: ~3375/2401 = 600.1355{{c}}, ~450/343 = 478.0813{{c}}
+: [[error map]]: {{val| +0.271 +0.235 +0.662 -1.986 }}
+* [[CWE]]: ~3375/2401 = 600.000{{c}}, ~450/343 = 477.9725{{c}}
+: error map: {{val| 0.000 -0.202 +0.043 -2.743 }}
+{{Optimal ET sequence|legend=1| 10, 98, 108, 118 }}
+[[Badness]] (Sintel): 3.65
 == Quartemka ==
 : ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Quartemka]].''
+Quartemka may be described as the {{nowrap| 26 & 61 }} temperament. Its [[ploidacot]] is 18-sheared 21-cot. It was named by [[Petr Pařízek]] in 2011 for its generator is close to 1/4 of the generator for [[emka]]<ref name="petr's long post"/>.
 [[Subgroup]]: 2.3.5.7
@@ Line 1,141: / Line 1,807: @@
 [[Comma list]]: 1029/1024, 1250000/1240029
-{{Mapping|legend=1| 1 4 6 2 | 0 -21 -32 7 }}
+{{Mapping|legend=1| 1 -17 -26 9 | 0 21 32 -7 }}
+: mapping generators: ~2, ~50/27
-{{Multival|legend=1| 21 32 -7 2 -70 -106 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.5278{{c}}, ~50/27 = 1062.4614{{c}}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~27/25 = 138.006
+: [[error map]]: {{val| +0.528 +0.762 -1.272 -1.305 }}
+* [[CWE]]: ~21 = 1200.0000{{c}}, ~50/27 = 1062.0046{{c}}
+: error map: {{val| 0.000 +0.142 -2.167 -2.858 }}
 {{Optimal ET sequence|legend=1| 26, 61, 87, 113, 200 }}
-[[Badness]] (Smith): 0.152287
+[[Badness]] (Sintel): 3.85
 === 11-limit ===
@@ Line 1,156: / Line 1,825: @@
 Comma list: 385/384, 441/440, 800000/793881
-Mapping: {{mapping| 1 4 6 2 3 | 0 -21 -32 7 4 }}
+Mapping: {{mapping| 1 -17 -26 9 7 | 0 21 32 -7 -4 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~27/25 = 137.990
+Optimal tunings:
+* WE: ~2 = 1200.3051{{c}}, ~50/27 = 1062.2805{{c}}
+* CWE: ~21 = 1200.0000{{c}}, ~50/27 = 1062.0147{{c}}
-{{Optimal ET sequence|legend=0| 26, 61, 87, 200, 287d, 487cdd }}
+{{Optimal ET sequence|legend=0| 26, 61, 87, 200, 287d }}
-Badness (Smith): 0.057307
+Badness (Sintel): 1.89
 === 13-limit ===
@@ Line 1,169: / Line 1,840: @@
 Comma list: 325/324, 364/363, 385/384, 2200/2197
-Mapping: {{mapping| 1 4 6 2 3 6 | 0 -21 -32 7 4 -20 }}
+Mapping: {{mapping| 1 -17 -26 9 7 -14 | 0 21 32 -7 -4 20 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~13/12 = 137.990
+Optimal tunings:
+* WE: ~2 = 1200.2708{{c}}, ~24/13 = 1062.2496{{c}}
+* CWE: ~21 = 1200.0000{{c}}, ~24/13 = 1062.0139{{c}}
-{{Optimal ET sequence|legend=0| 26, 61, 87, 200, 487cdd }}
+{{Optimal ET sequence|legend=0| 26, 61, 87, 200 }}
-Badness (Smith): 0.028393
+Badness (Sintel): 1.17
 == Tritriple ==
 : ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tritriple]].''
+Tritriple may be described as the {{nowrap| 103 & 118 }} temperament. Its [[ploidacot]] is iota-beta-27-cot. It was named by [[Petr Pařízek]] in 2011 for its generator is 1/9 of the generator for [[slendric]], so that 3×3 generators [[octave reduction|octave reduced]] give slendric's generator, and another ×3 give the [[3/2|perfect fifth]]<ref name="petr's long post"/>.
 [[Subgroup]]: 2.3.5.7
@@ Line 1,185: / Line 1,860: @@
 {{Mapping|legend=1| 1 -11 -7 7 | 0 27 20 -9 }}
+: mapping generators: ~2, ~864/625
-{{Multival|legend=1| 27 20 -9 -31 -90 -77 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.4239{{c}}, ~864/625 = 559.4921{{c}}
+: [[error map]]: {{val| +0.424 -0.331 +0.561 -1.287 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~864/625 = 559.3015{{c}}
+: error map: {{val| 0.000 -0.815 -0.284 -2.539 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~864/625 = 559.295
+{{Optimal ET sequence|legend=1| 15, …, 88, 103, 118, 221, 339d }}
-{{Optimal ET sequence|legend=1| 15, 88, 103, 118, 339d }}
+[[Badness]] (Sintel): 3.00
-[[Badness]] (Smith): 0.118640
 === 11-limit ===
@@ Line 1,201: / Line 1,879: @@
 Mapping: {{mapping| 1 -11 -7 7 -4 | 0 27 20 -9 16 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~242/175 = 559.293
+Optimal tunings:
+* WE: ~2 = 1200.4953{{c}}, ~242/175 = 559.5243{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~242/175 = 559.3016{{c}}
-{{Optimal ET sequence|legend=0| 15, 88, 103, 118, 339de }}
+{{Optimal ET sequence|legend=0| 15, …, 88, 103, 118, 221e, 339de }}
-Badness (Smith): 0.035350
+Badness (Sintel): 1.17
 == Widefourth ==
@@ Line 1,212: / Line 1,892: @@
 [[Comma list]]: 1029/1024, 48828125/48771072
-{{Mapping|legend=1| 1 16 8 -2 | 0 -33 -13 11 }}
+{{Mapping|legend=1| 1 -17 -5 9 | 0 33 13 -11 }}
-{{Multival|legend=1| 33 13 -11 -56 -110 -62 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.4770{{c}}, ~4608/3125 = 676.0584{{c}}
+: [[error map]]: {{val| +0.477 -0.137 +0.061 -1.175 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~4608/3125 = 675.7954{{c}}
+: error map: {{val| 0.000 -0.705 -0.973 -2.576 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~3125/2304 = 524.210
+{{Optimal ET sequence|legend=1| 16, 71, 87, 103, 190 }}
-{{Optimal ET sequence|legend=1| 16, 55b, 71, 87, 103, 190 }}
+[[Badness]] (Sintel): 3.90
-[[Badness]] (Smith): 0.154117
 === 11-limit ===
@@ Line 1,229: / Line 1,911: @@
 Mapping: {{mapping| 1 16 8 -2 17 | 0 -33 -13 11 -31 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~847/625 = 524.210
+Optimal tunings:
+* WE: ~2 = 1200.4852{{c}}, ~1250/847 = 676.0634{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~1250/847 = 675.7966{{c}}
-{{Optimal ET sequence|legend=0| 16, 55be, 71, 87, 103, 190 }}
+{{Optimal ET sequence|legend=0| 16, 71, 87, 103, 190 }}
-Badness (Smith): 0.040785
+Badness (Sintel): 1.35
 === 13-limit ===
@@ Line 1,242: / Line 1,926: @@
 Mapping: {{mapping| 1 16 8 -2 17 12 | 0 -33 -13 11 -31 -19 }}
-Optimal tuning (POTE): ~2 = 1200.000, ~65/48 = 524.209
+Optimal tunings:
+* WE: ~2 = 1200.4217{{c}}, ~77/52 = 676.0286{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~77/52 = 675.7967{{c}}
+{{Optimal ET sequence|legend=0| 16, 71, 87, 103, 190 }}
+Badness (Sintel): 0.894
+== Other subgroup extensions ==
+=== Euslendric (2.3.7.13) ===
+Forms of slendric in the most optimal range for the 2.3.7 temperament ({{nowrap| 36 & 77 }}) lack an obvious strong mapping of prime 5 or prime 11. However, slendric can extend well to the no-fives no-elevens [[29-limit]] by tempering out [[273/272]], [[343/342]], [[378/377]], [[392/391]], [[513/512]], and [[729/728]], or a comma basis defined in terms of [[S-expression]]s as {S7/S8, S14/S16, S15/S20, S24/S26, S27, S28}. [[113edo]] is an obvious tuning.
+Subgroup: 2.3.7.13
+Comma list: 729/728, 1029/1024
+Subgroup-val mapping: {{mapping| 1 1 3 0 | 0 3 -1 19 }}
+Gencom mapping: {{mapping| 1 1 0 3 0 0 | 0 3 0 -1 0 19 }}
+Optimal tunings:
+* WE: ~2 = 1200.5057{{c}}, ~8/7 = 233.7200{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.6534{{c}}
+{{Optimal ET sequence|legend=0| 5, 31f, 36, 77, 113, 827bdddff }}
+Badness (Sintel): 0.339
+==== 2.3.7.13.17 subgroup ====
+Subgroup: 2.3.7.13.17
+Comma list: 273/272, 729/728, 833/832
+Subgroup-val mapping: {{mapping| 1 1 3 0 0 | 0 3 -1 19 21 }}
+Gencom mapping: {{mapping| 1 1 0 3 0 0 0 | 0 3 0 -1 0 19 21 }}
+Optimal tunings:
+* WE: ~2 = 1200.5282{{c}}, ~8/7 = 233.6492{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.5776{{c}}
+{{Optimal ET sequence|legend=0| 5g, 31fg, 36, 113, 149 }}
+Badness (Sintel): 0.332
+==== 2.3.7.13.17.19 subgroup ====
+Subgroup: 2.3.7.13.17.19
+Comma list: 273/272, 343/342, 513/512, 729/728
+Subgroup-val mapping: {{mapping| 1 1 3 0 0 6 | 0 3 -1 19 21 -9 }}
+Gencom mapping: {{mapping| 1 1 0 3 0 0 0 6 | 0 3 0 -1 0 19 21 -9 }}
+Optimal tunings:
+* WE: ~2 = 1200.3292{{c}}, ~8/7 = 233.6651{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.6106{{c}}
+{{Optimal ET sequence|legend=0| 5g, 36, 77, 113, 262df }}
+Badness (Sintel): 0.380
+==== 2.3.7.13.17.19.23 subgroup ====
+Subgroup: 2.3.7.13.17.19.23
+Comma list: 273/272, 343/342, 392/391, 513/512, 729/728
+Subgroup-val mapping: {{mapping| 1 1 3 0 0 6 9 | 0 3 -1 19 21 -9 -23 }}
+Gencom mapping: {{mapping| 1 1 0 3 0 0 0 6 9 | 0 3 0 -1 0 19 21 -9 -23 }}
+Optimal tunings:
+* WE: ~2 = 1200.3127{{c}}, ~8/7 = 233.6679{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.6091{{c}}
+{{Optimal ET sequence|legend=0| 36, 77, 113, 262df }}
+Badness (Sintel): 0.474
+==== 2.3.7.13.17.19.23.29 subgroup ====
+Subgroup: 2.3.7.13.17.19.23.29
+Comma list: 273/272, 343/342, 378/377, 392/391, 513/512, 609/608
+Subgroup-val mapping: {{mapping| 1 1 3 0 0 6 9 7 | 0 3 -1 19 21 -9 -23 -11 }}
+Gencom mapping: {{mapping| 1 1 0 3 0 0 0 6 9 7 | 0 3 0 -1 0 19 21 -9 -23 -11 }}
+Optimal tunings:
+* WE: ~2 = 1200.2503{{c}}, ~8/7 = 233.6688{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.6208{{c}}
+{{Optimal ET sequence|legend=0| 36, 77, 113 }}
+Badness (Sintel): 0.473
+=== Baladic (2.3.7.13) ===
+Baladic is a 2.3.7.13.17-subgroup temperament that attempts to approximate the Maqam Sikah Baladi scale. It tempers out [[169/168]] ({{S|13}}), which splits [[7/6]] in half ([[13/12]]~[[14/13]]) and one finds that the octave is therefore split in half via the interval [[91/64]], which is then equated to [[17/12]]. 36edo is an excellent baladic tuning.
+Subgroup: 2.3.7.13
+Comma list: 169/168, 1029/1024
+Subgroup-val mapping: {{mapping| 2 2 6 7 | 0 3 -1 1 }}
+Gencom mapping: {{mapping| 2 2 0 6 0 7 | 0 3 0 -1 0 1 }}
+: mapping generators: ~91/64, ~8/7
+Optimal tunings:
+* WE: ~91/64 = 600.4315{{c}}, ~8/7 = 233.7724{{c}}
+* CWE: ~91/64 = 600.0000{{c}}, ~8/7 = 233.7039{{c}}
+{{Optimal ET sequence|legend=0| 10, 26, 36, 154f, 190ff, 226ff, 262dfff }}
+Badness (Sintel): 0.434
+==== 2.3.7.13.17 subgroup ====
+Subgroup: 2.3.7.13.17
+Comma list: 169/168, 273/272, 289/288
+Subgroup-val mapping: {{mapping| 2 2 6 7 7 | 0 3 -1 1 3 }}
+Gencom mapping: {{mapping| 2 2 0 6 0 7 7 | 0 3 0 -1 0 1 3 }}
+Optimal tunings:
+* WE: ~17/12 = 600.4436{{c}}, ~8/7 = 233.7883{{c}}
+* CWE: ~17/12 = 600.0000{{c}}, ~8/7 = 233.7312{{c}}
+{{Optimal ET sequence|legend=0| 10, 26, 36, 154f, 190ffg, 226ffg }}
+Badness (Sintel): 0.253
+=== Gigapyth (2.3.7.85) ===
+Subgroup: 2.3.7.85
+Comma list: 1029/1024, 7225/7203
+Subgroup-val mapping: {{mapping| 1 -2 4 7 | 0 6 -2 -1 }}
+Optimal tunings:
+* WE: ~2 = 1200.8295{{c}}, ~128/85 = 717.2597{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~128/85 = 716.7933{{c}}
+{{Optimal ET sequence|legend=0| 5, 42*, 47, 52, 57, 62, 67, 72, 149*, 370d***, 519bdd***** }}
-{{Optimal ET sequence|legend=0| 16, 55be, 71, 87, 103, 190 }}
+<nowiki/>* Wart for 85
-Badness (Smith): 0.021636
+== References ==
 [[Category:Temperament clans]]

Gamelismic clan: Difference between revisions