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{{Technical data page}}
{{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.
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.
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.
Miracle adds [[33075/32768]] and uses the [[secor]], half an ~8/7, as generator. Lemba adds [[525/512]] to the list, and has a half-octave [[period]]. Valentine adds [[6144/6125]] with a generator of ~21/20 and superkleismic adds [[875/864]] with a generator of ~6/5. Unidec adds [[4375/4374]], and has a generator of ~10/9 with a half-octave period. Hemithirds adds [[65625/65536]] with a generator half of a classical major third. Finally, tritikleismic adds [[15625/15552]] and has a generator of 6/5 with a 1/3-octave period.
Full 7-limit temperaments discussed elsewhere are:
* [[Lemba]] (+50/49) → [[Jubilismic clan #Lemba|Jubilismic clan]]
* ''[[Echidnic]]'' (+686/675} → [[Diaschismic family #Echidnic|Diaschismic family]]
* [[Blackwood]] (+28/27) → [[Limmic temperaments #Blackwood|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]]
* ''[[Tritikleismic]]'' (+15625/15552) → [[Kleismic family #Tritikleismic|Kleismic family]]
* ''[[Heinz]]'' (+78732/78125) → [[Sensipent family #Heinz|Sensipent family]]
* ''[[Triwell]]'' (+235298/234375) → [[Semicomma family #Triwell|Semicomma 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.


== Slendric ==
== Slendric ==
Line 39: Line 19:
* [[CWE]]: ~2 = 1200.000{{c}}, ~8/7 = 233.7474{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~8/7 = 233.7474{{c}}
: error map: {{val| 0.000 -0.713 -2.573 }}
: error map: {{val| 0.000 -0.713 -2.573 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~8/7 = 233.889{{c}}
: [[error map]]: {{val| 0.000 -0.288 -2.715 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~8/7 = 233.688{{c}}
: error map: {{val| 0.000 -0.892 -2.513 }} -->


{{Optimal ET sequence|legend=1| 36, 77, 113, 190 }}
{{Optimal ET sequence|legend=1| 5, 21, 26, 31, 36, 77, 113, 190 }}


[[Badness]] (Sintel): 0.158
[[Badness]] (Sintel): 0.158


=== Euslendric ===
=== Overview to extensions ===
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.
==== 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]].  


Subgroup: 2.3.7.13
Miracle adds [[33075/32768]] and uses the [[secor]], half an ~8/7, as generator. Lemba adds [[525/512]] to the list, and has a half-octave [[period]]. Valentine adds [[6144/6125]] with a generator of ~[[21/20]] and superkleismic adds [[875/864]] with a generator of ~6/5. Unidec adds [[4375/4374]], and has a generator of ~[[10/9]] with a half-octave period. Hemithirds adds [[65625/65536]] with a generator half of a classical major third. Finally, tritikleismic adds [[15625/15552]] and has a generator of 6/5 with a 1/3-octave period.


Comma list: 729/728, 1029/1024
Full 7-limit temperaments discussed elsewhere are:
* [[Blackwood]] (+28/27) → [[Blackwood family #Blackwood|Blackwood family]]
* [[Lemba]] (+50/49) → [[Jubilismic clan #Lemba|Jubilismic clan]]
* [[Trisected]] (+128/125) → [[Augmented family #Trisected|Augmented family]]
* ''[[Echidnic]]'' (+686/675) → [[Diaschismic family #Echidnic|Diaschismic family]]
* [[Trismegistus]] (+3125/3072) → [[Magic family #Trismegistus|Magic family]]
* [[Hemithirds]] (+3136/3125) → [[Hemimean clan #Hemithirds|Hemimean clan]]
* ''[[Gamity]]'' (+1071875/1062882) → [[Amity family #Gamity|Amity family]]
* ''[[Tritikleismic]]'' (+15625/15552) → [[Kleismic family #Tritikleismic|Kleismic family]]
* ''[[Heinz]]'' (+78732/78125) → [[Sensipent family #Heinz|Sensipent family]]
* ''[[Triwell]]'' (+235298/234375) → [[Semicomma family #Triwell|Semicomma family]]
* ''[[Gamelstearn]]'' (+118098/117649) → [[Compton family #Gamelstearn|Compton family]]


Sval mapping: {{mapping| 1 1 3 0 | 0 3 -1 19 }}
The rest are considered below.


Gencom mapping: {{mapping| 1 1 0 3 0 0 | 0 3 0 -1 0 19 }}
==== 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.


Optimal tunings:
=== Radon ===
* WE: ~2 = 1200.5057{{c}}, ~8/7 = 233.7200{{c}}
{{See also|Chromatic pairs #Radon}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.6534{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~8/7 = 233.734{{c}}
* POTE: ~2 = 1200.000{{c}}, ~8/7 = 233.622{{c}} -->


{{Optimal ET sequence|legend=0| 5, 31f, 36, 77, 113, 827bdddff }}
Radon is the no-fives version of [[rodan]], equating the diatonic major third to [[14/11]].


Badness (Sintel): 0.339
Subgroup: 2.3.7.11


==== 2.3.7.13.17 subgroup ====
Comma list: 896/891, 1029/1024
Subgroup: 2.3.7.13.17


Comma list: 273/272, 729/728, 833/832
Subgroup-val mapping: {{mapping| 1 1 3 6 | 0 3 -1 -13 }}


Sval mapping: {{mapping| 1 1 3 0 0 | 0 3 -1 19 21 }}
Gencom mapping: {{mapping| 1 1 0 3 6 | 0 3 0 -1 -13 }}
 
Gencom mapping: {{mapping| 1 1 0 3 0 0 0 | 0 3 0 -1 0 19 21 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.5282{{c}}, ~8/7 = 233.6492{{c}}
* WE: ~2 = 1199.9708{{c}}, ~8/7 = 234.3748{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.5776{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.3813{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~8/7 = 233.657{{c}}
* POTE: ~2 = 1200.000{{c}}, ~8/7 = 233.546{{c}} -->


{{Optimal ET sequence|legend=0| 5g, 31fg, 36, 113, 149 }}
{{Optimal ET sequence|legend=0| 5, …, 36, 41, 87, 128 }}


Badness (Sintel): 0.332
Badness (Sintel): 0.619


==== 2.3.7.13.17.19 subgroup ====
== Mothra ==
Subgroup: 2.3.7.13.17.19
{{Main| Mothra }}


Comma list: 273/272, 343/342, 513/512, 729/728
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]].


Sval mapping: {{mapping| 1 1 3 0 0 6 | 0 3 -1 19 21 -9 }}
Note that mothra is also called '''cynder''' in the 7-limit, which can be a little confusing sometimes.


Gencom mapping: {{mapping| 1 1 0 3 0 0 0 6 | 0 3 0 -1 0 19 21 -9 }}
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


Optimal tunings:  
[[Comma list]]: 81/80, 1029/1024
* WE: ~2 = 1200.3292{{c}}, ~8/7 = 233.6651{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.6106{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~8/7 = 233.657{{c}}
* POTE: ~2 = 1200.000{{c}}, ~8/7 = 233.601{{c}} -->


{{Optimal ET sequence|legend=0| 5g, 36, 77, 113, 262df }}
{{Mapping|legend=1| 1 1 0 3 | 0 3 12 -1 }}


Badness (Sintel): 0.380
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.9303{{c}}, ~8/7 = 232.3733{{c}}
: [[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 }}


==== 2.3.7.13.17.19.23 subgroup ====
[[Algebraic generator]]: Rabrindanath, largest real root of ''x''<sup>8</sup> - 3''x''<sup>2</sup> + 1, or 232.0774 cents.
Subgroup: 2.3.7.13.17.19.23


Comma list: 273/272, 343/342, 392/391, 513/512, 729/728
[[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


Sval mapping: {{mapping| 1 1 3 0 0 6 9 | 0 3 -1 19 21 -9 -23 }}
{{Optimal ET sequence|legend=1| 5, 21c, 26, 31 }}


Gencom mapping: {{mapping| 1 1 0 3 0 0 0 6 9 | 0 3 0 -1 0 19 21 -9 -23 }}
[[Badness]] (Sintel): 0.940


Optimal tunings:
=== Undecimal mothra ===
* WE: ~2 = 1200.3127{{c}}, ~8/7 = 233.6679{{c}}
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.
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.6091{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~8/7 = 233.624{{c}}
* POTE: ~2 = 1200.000{{c}}, ~8/7 = 233.607{{c}} -->


{{Optimal ET sequence|legend=0| 36, 77, 113, 262df }}
Subgroup: 2.3.5.7.11


Badness (Sintel): 0.474
Comma list: 81/80, 99/98, 385/384


==== 2.3.7.13.17.19.23.29 subgroup ====
Mapping: {{mapping| 1 1 0 3 5 | 0 3 12 -1 -8 }}
Subgroup: 2.3.7.13.17.19.23.29


Comma list: 273/272, 343/342, 378/377, 392/391, 513/512, 609/608
Optimal tunings:  
* WE: ~2 = 1201.3979{{c}}, ~8/7 = 232.3010{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 232.0621{{c}}


Sval mapping: {{mapping| 1 1 3 0 0 6 9 7 | 0 3 -1 19 21 -9 -23 -11 }}
{{Optimal ET sequence|legend=0| 5, 26, 31, 88, 119be, 150be }}


Gencom mapping: {{mapping| 1 1 0 3 0 0 0 6 9 7 | 0 3 0 -1 0 19 21 -9 -23 -11 }}
Badness (Sintel): 0.848


Optimal tunings:
==== 13-limit ====
* WE: ~2 = 1200.2503{{c}}, ~8/7 = 233.6688{{c}}
Subgroup: 2.3.5.7.11.13
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.6208{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~8/7 = 233.626{{c}}
* POTE: ~2 = 1200.000{{c}}, ~8/7 = 233.620{{c}} -->


{{Optimal ET sequence|legend=0| 36, 77, 113 }}
Comma list: 81/80, 99/98, 105/104, 144/143


Badness (Sintel): 0.473
Mapping: {{mapping| 1 1 0 3 5 1 | 0 3 12 -1 -8 14 }}


=== Radon ===
Optimal tunings:
Radon is the no-fives version of [[rodan]], equating the diatonic major third to [[14/11]].
* 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 }}


Subgroup: 2.3.7.11
Badness (Sintel): 0.990


Comma list: 896/891, 1029/1024
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


Sval mapping: {{mapping| 1 1 3 6 | 0 3 -1 -13 }}
Comma list: 81/80, 99/98, 105/104, 120/119, 144/143


Gencom mapping: {{mapping| 1 1 0 3 6 | 0 3 0 -1 -13 }}
Mapping: {{mapping| 1 1 0 3 5 1 | 0 3 12 -1 -8 14 16 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.9708{{c}}, ~8/7 = 234.3748{{c}}
* WE: ~2 = 1200.9734{{c}}, ~8/7 = 231.8960{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.3813{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 231.7392{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~8/7 = 234.384{{c}}
: error map: {{val| 0.000 +1.197 -3.210 +1.691 }}
* POTE: ~2 = 1200.000{{c}}, ~8/7 = 234.381{{c}}
: error map: {{val| 0.000 +1.187 -3.206 +1.735 }} -->


{{Optimal ET sequence|legend=0| 5, …, 36, 41, 87, 128 }}
{{Optimal ET sequence|legend=0| 5g, 26, 31, 57, 88 }}


Badness (Sintel): 0.619
Badness (Sintel): 1.00


=== Baladic ===
==== 19-limit ====
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: 2.3.5.7.11.13.17.19


Subgroup: 2.3.7.13
Comma list: 81/80, 99/98, 105/104, 120/119, 144/143, 153/152


Comma list: 169/168, 1029/1024
Mapping: {{mapping| 1 1 0 3 5 1 | 0 3 12 -1 -8 14 16 22 }}
 
Sval 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:  
Optimal tunings:  
* WE: ~91/64 = 600.4315{{c}}, ~8/7 = 233.7724{{c}}
* WE: ~2 = 1200.9663{{c}}, ~8/7 = 231.8393{{c}}
* CWE: ~91/64 = 600.0000{{c}}, ~8/7 = 233.7039{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 231.6842{{c}}
<!-- * POTE: ~91/64 = 600.0000{{c}}, ~8/7 = 233.6044{{c}} -->


{{Optimal ET sequence|legend=0| 10, 26, 36, 154f, 190ff, 226ff, 262dfff }}
{{Optimal ET sequence|legend=0| 26, 31, 57 }}


Badness (Sintel): 0.434
Badness (Sintel): 1.05


==== 2.3.7.13.17 subgroup ====
=== Mosura ===
Subgroup: 2.3.7.13.17
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]]}.


Comma list: 169/168, 273/272, 289/288
Subgroup: 2.3.5.7.11


Sval mapping: {{mapping| 2 2 6 7 7 | 0 3 -1 1 3 }}
Comma list: 81/80, 176/175, 540/539


Gencom mapping: {{mapping| 2 2 0 6 0 7 7 | 0 3 0 -1 0 1 3 }}
Mapping: {{mapping| 1 1 0 3 -1 | 0 3 12 -1 23 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~17/12 = 600.4436{{c}}, ~8/7 = 233.7883{{c}}
* WE: ~2 = 1200.7675{{c}}, ~8/7 = 232.5673{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~8/7 = 233.7312{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 232.4567{{c}}
<!-- * CTE: ~17/12 = 600.000{{c}}, ~8/7 = 234.138{{c}}
* POTE: ~17/12 = 600.000{{c}}, ~8/7 = 233.616{{c}} -->


{{Optimal ET sequence|legend=0| 10, 26, 36, 154f, 190ffg, 226ffg }}
{{Optimal ET sequence|legend=0| 5e, 26e, 31, 129 }}


Badness (Sintel): 0.253
Badness (Sintel): 1.04


== Mothra ==
==== 13-limit ====
{{Main| Mothra }}
Subgroup: 2.3.5.7.11.13


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: 81/80, 144/143, 176/175, 196/195


Note that mothra is also called '''cynder''' in the 7-limit, which can be a little confusing sometimes.
Mapping: {{mapping| 1 1 0 3 -1 7 | 0 3 12 -1 23 -17 }}


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]].
Optimal tunings:
* WE: ~2 = 1199.9347{{c}}, ~8/7 = 232.6275{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 232.6392{{c}}


[[Subgroup]]: 2.3.5.7
{{Optimal ET sequence|legend=0| 31, 67, 98 }}


[[Comma list]]: 81/80, 1029/1024
Badness (Sintel): 1.52


{{Mapping|legend=1| 1 1 0 3 | 0 3 12 -1 }}
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


[[Optimal tuning]]s:  
Comma list: 81/80, 144/143, 176/175, 189/187, 196/195
* [[WE]]: ~2 = 1200.9303{{c}}, ~8/7 = 232.3733{{c}}
: [[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 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~8/7 = 232.400{{c}}
: [[error map]]: {{val| 0.000 -4.756 +2.482 -1.226 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~8/7 = 232.193{{c}}
: error map: {{val| 0.000 -5.375 +0.005 -1.019 }} -->


[[Algebraic generator]]: Rabrindanath, largest real root of ''x''<sup>8</sup> - 3''x''<sup>2</sup> + 1, or 232.0774 cents.
Mapping: {{mapping| 1 1 0 3 -1 7 | 0 3 12 -1 23 -17 -15 }}


[[Minimax tuning]]:  
Optimal tunings:  
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~8/7 = {{monzo| 0 0 1/12 }}
* WE: ~2 = 1199.7124{{c}}, ~8/7 = 232.6376{{c}}
: {{monzo list| 1 0 0 0 | 1 0 1/4 0 | 0 0 1 0 | 3 0 -1/12 0 }}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 232.6917{{c}}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5


{{Optimal ET sequence|legend=1| 5, 21c, 26, 31 }}
{{Optimal ET sequence|legend=0| 31, 67, 98 }}


[[Badness]] (Sintel): 0.940
Badness (Sintel): 1.53


=== Undecimal mothra ===
==== 19-limit ====
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: 2.3.5.7.11.13.17.19


Subgroup: 2.3.5.7.11
Comma list: 81/80, 96/95, 144/143, 153/152, 176/175, 196/195


Comma list: 81/80, 99/98, 385/384
Mapping: {{mapping| 1 1 0 3 -1 7 | 0 3 12 -1 23 -17 -15 -9 }}
 
Mapping: {{mapping| 1 1 0 3 5 | 0 3 12 -1 -8 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1201.3979{{c}}, ~8/7 = 232.3010{{c}}
* WE: ~2 = 1199.4885{{c}}, ~8/7 = 232.6310{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 232.0621{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 232.7287{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~8/7 = 232.203{{c}}
* POTE: ~2 = 1200.000{{c}}, ~8/7 = 232.031{{c}} -->


{{Optimal ET sequence|legend=0| 5, 26, 31, 88, 119be, 150be }}
{{Optimal ET sequence|legend=0| 31, 67, 98h }}


Badness (Sintel): 0.848
Badness (Sintel): 1.50


==== 13-limit ====
=== Cyndra ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11


Comma list: 81/80, 99/98, 105/104, 144/143
Comma list: 45/44, 81/80, 1029/1024


Mapping: {{mapping| 1 1 0 3 5 1 | 0 3 12 -1 -8 14 }}
Mapping: {{mapping| 1 1 0 3 0 | 0 3 12 -1 18 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1201.0985{{c}}, ~8/7 = 232.0231{{c}}
* WE: ~2 = 1201.1585{{c}}, ~8/7 = 231.5404{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 231.8425{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 231.3850{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~8/7 = 231.993{{c}}
* POTE: ~2 = 1200.000{{c}}, ~8/7 = 231.811{{c}} -->


{{Optimal ET sequence|legend=0| 5, 26, 31, 57, 88 }}
{{Optimal ET sequence|legend=0| 5e, 21ce, 26 }}


Badness (Sintel): 0.990
Badness (Sintel): 1.84


; Music
==== 13-limit ====
* ''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
Subgroup: 2.3.5.7.11.13


==== 17-limit ====
Comma list: 45/44, 78/77, 81/80, 640/637
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 0 1 | 0 3 12 -1 18 14 }}
 
Mapping: {{mapping| 1 1 0 3 5 1 | 0 3 12 -1 -8 14 16 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.9734{{c}}, ~8/7 = 231.8960{{c}}
* WE: ~2 = 1201.1152{{c}}, ~8/7 = 231.5079{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 231.7392{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 231.3612{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~8/7 = 231.891{{c}}
* POTE: ~2 = 1200.000{{c}}, ~8/7 = 231.708{{c}} -->


{{Optimal ET sequence|legend=0| 5g, 26, 31, 57, 88 }}
{{Optimal ET sequence|legend=0| 5e, 21cef, 26 }}


Badness (Sintel): 1.00
Badness (Sintel): 1.41


==== 19-limit ====
== Rodan ==
Subgroup: 2.3.5.7.11.13.17.19
{{Main| Rodan }}
: ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum #Rodan (5-limit)]].''


Comma list: 81/80, 99/98, 105/104, 120/119, 144/143, 153/152
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.


Mapping: {{mapping| 1 1 0 3 5 1 | 0 3 12 -1 -8 14 16 22 }}
[[Subgroup]]: 2.3.5.7


Optimal tunings:  
[[Comma list]]: 245/243, 1029/1024
* WE: ~2 = 1200.9663{{c}}, ~8/7 = 231.8393{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 231.6842{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~8/7 = 231.837{{c}}
* POTE: ~2 = 1200.000{{c}}, ~8/7 = 231.653{{c}} -->


{{Optimal ET sequence|legend=0| 26, 31, 57 }}
{{Mapping|legend=1| 1 1 -1 3 | 0 3 17 -1 }}


Badness (Sintel): 1.05
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.2146{{c}}, ~8/7 = 234.4587{{c}}
: [[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 }}


=== Mosura ===
[[Minimax tuning]]:
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]]}.
* [[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|unchanged-interval (eigenmonzo) basis]]: 2.7/5
 
[[Algebraic generator]]: larger root of 20''x''<sup>2</sup> - 36''x'' + 15, or (9 + √6)/10.
 
{{Optimal ET sequence|legend=1| 41, 87, 128, 215d }}
 
[[Badness]] (Sintel): 0.939


=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 81/80, 176/175, 540/539
Comma list: 245/243, 385/384, 441/440


Mapping: {{mapping| 1 1 0 3 -1 | 0 3 12 -1 23 }}
Mapping: {{mapping| 1 1 -1 3 6 | 0 3 17 -1 -13 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.7675{{c}}, ~8/7 = 232.5673{{c}}
* WE: ~2 = 1200.0553{{c}}, ~8/7 = 234.4695{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 232.4567{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.4594{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~8/7 = 232.557{{c}}
* POTE: ~2 = 1200.000{{c}}, ~8/7 = 232.419{{c}} -->


{{Optimal ET sequence|legend=0| 5e, 26e, 31, 129 }}
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 }}]
: unchanged-interval (eigenmonzo) basis: 2.11/9


Badness (Sintel): 1.04
Algebraic generator: positive root of ''x''<sup>2</sup> + 16''x'' - 31, or √95 - 8.


==== 13-limit ====
{{Optimal ET sequence|legend=0| 41, 87 }}
 
Badness (Sintel): 0.763
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 81/80, 144/143, 176/175, 196/195
Comma list: 196/195, 245/243, 352/351, 364/363


Mapping: {{mapping| 1 1 0 3 -1 7 | 0 3 12 -1 23 -17 }}
Mapping: {{mapping| 1 1 -1 3 6 8 | 0 3 17 -1 -13 -22 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.9347{{c}}, ~8/7 = 232.6275{{c}}
* WE: ~2 = 1199.9868{{c}}, ~8/7 = 234.4796{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 232.6392{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.4822{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~8/7 = 232.635{{c}}
 
* POTE: ~2 = 1200.000{{c}}, ~8/7 = 232.640{{c}} -->
Minimax tuning:
* 13- and 15-odd-limit: ~8/7 = {{monzo| 3/14 1/14 0 0 0 -1/28 }}
: unchanged-interval (eigenmonzo) basis: 2.13/9
 
Algebraic generator: Gatetone, positive root of 4''x''<sup>6</sup> - 7''x'' - 1. Recurrence converges slowly.


{{Optimal ET sequence|legend=0| 31, 67, 98 }}
{{Optimal ET sequence|legend=0| 41, 46, 87 }}


Badness (Sintel): 1.52
Badness (Sintel): 0.762


==== 17-limit ====
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 81/80, 144/143, 176/175, 189/187, 196/195
Comma list: 154/153, 196/195, 245/243, 256/255, 273/272


Mapping: {{mapping| 1 1 0 3 -1 7 | 0 3 12 -1 23 -17 -15 }}
Mapping: {{mapping| 1 1 -1 3 6 8 8 | 0 3 17 -1 -13 -22 -20 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.7124{{c}}, ~8/7 = 232.6376{{c}}
* WE: ~2 = 1199.8331{{c}}, ~8/7 = 234.4919{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 232.6917{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.5254{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~8/7 = 232.681{{c}}
 
* POTE: ~2 = 1200.000{{c}}, ~8/7 = 232.693{{c}} -->
Minimax tuning:
* 17-odd-limit: ~8/7 = {{monzo| 3/13 1/13 0 0 0 0 -1/26 }}
: unchanged-interval (eigenmonzo) basis: 2.17/9


{{Optimal ET sequence|legend=0| 31, 67, 98 }}
{{Optimal ET sequence|legend=0| 41, 46, 87 }}


Badness (Sintel): 1.53
Badness (Sintel): 0.853


==== 19-limit ====
==== Aerodactyl ====
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13


Comma list: 81/80, 96/95, 144/143, 153/152, 176/175, 196/195
Comma list: 91/90, 245/243, 385/384, 441/440


Mapping: {{mapping| 1 1 0 3 -1 7 | 0 3 12 -1 23 -17 -15 -9 }}
Mapping: {{mapping| 1 1 -1 3 6 -1 | 0 3 17 -1 -13 24 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.4885{{c}}, ~8/7 = 232.6310{{c}}
* WE: ~2 = 1200.2997{{c}}, ~8/7 = 234.6972{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 232.7287{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.6439{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~8/7 = 232.717{{c}}
* POTE: ~2 = 1200.000{{c}}, ~8/7 = 232.730{{c}} -->


{{Optimal ET sequence|legend=0| 31, 67, 98h }}
{{Optimal ET sequence|legend=0| 5, 41f, 46 }}


Badness (Sintel): 1.50
Badness (Sintel): 1.40


=== Cyndra ===
=== Aerodino ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 45/44, 81/80, 1029/1024
Comma list: 176/175, 245/243, 1029/1024


Mapping: {{mapping| 1 1 0 3 0 | 0 3 12 -1 18 }}
Mapping: {{mapping| 1 1 -1 3 -3 | 0 3 17 -1 33 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1201.1585{{c}}, ~8/7 = 231.5404{{c}}
* WE: ~2 = 1199.9179{{c}}, ~8/7 = 234.7123{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 231.3850{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.7256{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~8/7 = 231.566{{c}}
* POTE: ~2 = 1200.000{{c}}, ~8/7 = 231.317{{c}} -->


{{Optimal ET sequence|legend=0| 5e, 21ce, 26 }}
{{Optimal ET sequence|legend=0| 5e, 41e, 46 }}


Badness (Sintel): 1.84
Badness (Sintel): 1.79


==== 13-limit ====
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 45/44, 78/77, 81/80, 640/637
Comma list: 91/90, 176/175, 245/243, 847/845


Mapping: {{mapping| 1 1 0 3 0 1 | 0 3 12 -1 18 14 }}
Mapping: {{mapping| 1 1 -1 3 -3 -1 | 0 3 17 -1 33 24 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1201.1152{{c}}, ~8/7 = 231.5079{{c}}
* WE: ~2 = 1200.0242{{c}}, ~8/7 = 234.7863{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 231.3612{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.7824{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~8/7 = 231.546{{c}}
* POTE: ~2 = 1200.000{{c}}, ~8/7 = 231.293{{c}} -->


{{Optimal ET sequence|legend=0| 5e, 21cef, 26 }}
{{Optimal ET sequence|legend=0| 5e, 41ef, 46 }}


Badness (Sintel): 1.41
Badness (Sintel): 1.48


== Rodan ==
=== Varan ===
{{Main| Rodan }}
Subgroup: 2.3.5.7.11
: ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum #Rodan (5-limit)]].''


Rodan tempers out 245/243 and can be described as the {{nowrap| 41 & 46 }} temperament. This temperament extends neatly to the 13-limit, though the perfect fifth is sharper than ideal for slendric.
Comma list: 100/99, 245/243, 1029/1024


[[Subgroup]]: 2.3.5.7
Mapping: {{mapping| 1 1 -1 3 -2 | 0 3 17 -1 28 }}


[[Comma list]]: 245/243, 1029/1024
Optimal tunings:  
* WE: ~2 = 1200.3738{{c}}, ~8/7 = 234.2174{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.1586{{c}}


{{Mapping|legend=1| 1 1 -1 3 | 0 3 17 -1 }}
{{Optimal ET sequence|legend=0| 5e, 36ce, 41 }}


[[Optimal tuning]]s:  
Badness (Sintel): 1.49
* [[WE]]: ~2 = 1200.2146{{c}}, ~8/7 = 234.4587{{c}}
: [[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 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~8/7 = 234.450{{c}}
: [[error map]]: {{val| 0.000 +1.396 -0.660 -3.276 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~8/7 = 234.417{{c}}
: error map: {{val| 0.000 +1.295 -1.229 -3.243 }} -->


[[Minimax tuning]]:
==== 13-limit ====
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~8/7 = {{monzo| 2/9 0 1/18 -1/18 }}
Subgroup: 2.3.5.7.11.13
: {{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|unchanged-interval (eigenmonzo) basis]]: 2.7/5


[[Algebraic generator]]: larger root of 20''x''<sup>2</sup> - 36''x'' + 15, or (9 + √6)/10.
Comma list: 100/99, 105/104, 245/243, 352/351


{{Optimal ET sequence|legend=1| 41, 87, 128, 215d }}
Mapping: {{mapping| 1 1 -1 3 -2 0 | 0 3 17 -1 28 19 }}


[[Badness]] (Sintel): 0.939
Optimal tunings:  
* WE: ~2 = 1200.1389{{c}}, ~8/7 = 234.1162{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.0946{{c}}


=== 11-limit ===
{{Optimal ET sequence|legend=0| 5e, 36ce, 41 }}
Subgroup: 2.3.5.7.11


Comma list: 245/243, 385/384, 441/440
Badness (Sintel): 1.33


Mapping: {{mapping| 1 1 -1 3 6 | 0 3 17 -1 -13 }}
== 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 the {{nowrap| 36 & 41 }} temperament. It is more complex than rodan, but the optimal tuning is closer to optimal slendric.


Optimal tunings:  
[[Subgroup]]: 2.3.5.7
* WE: ~2 = 1200.0553{{c}}, ~8/7 = 234.4695{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.4594{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~8/7 = 234.463{{c}}
* POTE: ~2 = 1200.000{{c}}, ~8/7 = 234.459{{c}} -->


Minimax tuning:
[[Comma list]]: 1029/1024, 10976/10935
* 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 }}]
: unchanged-interval (eigenmonzo) basis: 2.11/9


Algebraic generator: positive root of ''x''<sup>2</sup> + 16''x'' - 31, or √95 - 8.
{{Mapping|legend=1| 1 1 7 3 | 0 3 -24 -1 }}


{{Optimal ET sequence|legend=0| 41, 87 }}
[[Optimal tuning]]s:
 
* [[WE]]: ~2 = 1200.3395{{c}}, ~8/7 = 233.9963{{c}}
Badness (Sintel): 0.763
: [[error map]]: {{val| +0.340 +0.374 +0.151 -1.804 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 233.9239{{c}}
: error map: {{val| 0.000 -0.183 -0.487 -2.750 }}


==== 13-limit ====
[[Minimax tuning]]:
Subgroup: 2.3.5.7.11.13
* [[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|unchanged-interval (eigenmonzo) basis]]: 2.5


Comma list: 196/195, 245/243, 352/351, 364/363
{{Optimal ET sequence|legend=1| 36, 41, 77, 118, 277d }}


Mapping: {{mapping| 1 1 -1 3 6 8 | 0 3 17 -1 -13 -22 }}
[[Badness]] (Sintel): 1.20


Optimal tunings:
=== 11-limit ===
* WE: ~2 = 1199.9868{{c}}, ~8/7 = 234.4796{{c}}
Subgroup: 2.3.5.7.11
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.4822{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~8/7 = 234.482{{c}}
* POTE: ~2 = 1200.000{{c}}, ~8/7 = 234.482{{c}} -->


Minimax tuning:  
Comma list: 385/384, 441/440, 10976/10935
* 13- and 15-odd-limit: ~8/7 = {{monzo| 3/14 1/14 0 0 0 -1/28 }}
: unchanged-interval (eigenmonzo) basis: 2.13/9


Algebraic generator: Gatetone, positive root of 4''x''<sup>6</sup> - 7''x'' - 1. Recurrence converges slowly.
Mapping: {{mapping| 1 1 7 3 -2 | 0 3 -24 -1 28 }}


{{Optimal ET sequence|legend=0| 41, 46, 87 }}
Optimal tunings:  
 
* WE: ~2 = 1200.3453{{c}}, ~8/7 = 233.9988{{c}}
Badness (Sintel): 0.762
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.9312{{c}}
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 154/153, 196/195, 245/243, 256/255, 273/272
 
Mapping: {{mapping| 1 1 -1 3 6 8 8 | 0 3 17 -1 -13 -22 -20 }}
 
Optimal tunings:  
* WE: ~2 = 1199.8331{{c}}, ~8/7 = 234.4919{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.5254{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~8/7 = 234.532{{c}}
* POTE: ~2 = 1200.000{{c}}, ~8/7 = 234.524{{c}} -->


Minimax tuning:
Minimax tuning:
* 17-odd-limit: ~8/7 = {{monzo| 3/13 1/13 0 0 0 0 -1/26 }}
* 11-odd-limit: ~8/7 = {{monzo| 7/24 0 -1/24 }}
: unchanged-interval (eigenmonzo) basis: 2.17/9
: [{{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 }}]
: unchanged-interval (eigenmonzo) basis: 2.5


{{Optimal ET sequence|legend=0| 41, 46, 87 }}
{{Optimal ET sequence|legend=0| 36e, 41, 77, 118, 159, 277d }}


Badness (Sintel): 0.853
Badness (Sintel): 0.881


==== Aerodactyl ====
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 91/90, 245/243, 385/384, 441/440
Comma list: 196/195, 352/351, 385/384, 729/728


Mapping: {{mapping| 1 1 -1 3 6 -1 | 0 3 17 -1 -13 24 }}
Mapping: {{mapping| 1 1 7 3 -2 0 | 0 3 -24 -1 28 19 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.2997{{c}}, ~8/7 = 234.6972{{c}}
* WE: ~2 = 1200.1222{{c}}, ~8/7 = 233.9228{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.6439{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.8994{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~8/7 = 234.670{{c}}
* POTE: ~2 = 1200.000{{c}}, ~8/7 = 234.639{{c}} -->


{{Optimal ET sequence|legend=0| 5, 41f, 46 }}
{{Optimal ET sequence|legend=0| 36e, 41, 77, 118 }}


Badness (Sintel): 1.40
Badness (Sintel): 1.18


=== Aerodino ===
== Gorgo ==
Subgroup: 2.3.5.7.11
: ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum #Laconic]].''
{{See also| Llywelynsmic clan }}


Comma list: 176/175, 245/243, 1029/1024
Gorgo tempers the generator of ~8/7 together with ~10/9. It can be described as the {{nowrap| 16 & 21 }} temperament.


Mapping: {{mapping| 1 1 -1 3 -3 | 0 3 17 -1 33 }}
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.


Optimal tunings:  
[[Subgroup]]: 2.3.5.7
* WE: ~2 = 1199.9179{{c}}, ~8/7 = 234.7123{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.7256{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~8/7 = 234.719{{c}}
* POTE: ~2 = 1200.000{{c}}, ~8/7 = 234.728{{c}} -->


{{Optimal ET sequence|legend=0| 5e, 41e, 46 }}
[[Comma list]]: 36/35, 1029/1024


Badness (Sintel): 1.79
{{Mapping|legend=1| 1 1 1 3 | 0 3 7 -1 }}


==== 13-limit ====
[[Optimal tuning]]s:
Subgroup: 2.3.5.7.11.13
* [[WE]]: ~2 = 1200.9847{{c}}, ~8/7 = 228.5210{{c}}
: [[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 }}


Comma list: 91/90, 176/175, 245/243, 847/845
{{Optimal ET sequence|legend=1| 5, 11c, 16, 21 }}


Mapping: {{mapping| 1 1 -1 3 -3 -1 | 0 3 17 -1 33 24 }}
[[Badness]] (Sintel): 1.54


Optimal tunings:
=== 11-limit ===
* WE: ~2 = 1200.0242{{c}}, ~8/7 = 234.7863{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.7824{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~8/7 = 234.786{{c}}
* POTE: ~2 = 1200.000{{c}}, ~8/7 = 234.782{{c}} -->
 
{{Optimal ET sequence|legend=0| 5e, 41ef, 46 }}
 
Badness (Sintel): 1.48
 
=== Varan ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 100/99, 245/243, 1029/1024
Comma list: 36/35, 45/44, 1029/1024


Mapping: {{mapping| 1 1 -1 3 -2 | 0 3 17 -1 28 }}
Mapping: {{mapping| 1 1 1 3 1 | 0 3 7 -1 13 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.3738{{c}}, ~8/7 = 234.2174{{c}}
* WE: ~2 = 1201.3609{{c}}, ~8/7 = 227.6312{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.1586{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 227.4955{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~8/7 = 234.197{{c}}
* POTE: ~2 = 1200.000{{c}}, ~8/7 = 234.145{{c}} -->


{{Optimal ET sequence|legend=0| 5e, 36ce, 41 }}
{{Optimal ET sequence|legend=0| 5e, 16, 21, 37b }}


Badness (Sintel): 1.49
Badness (Sintel): 1.64


==== 13-limit ====
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 100/99, 105/104, 245/243, 352/351
Comma list: 27/26, 36/35, 45/44, 507/500


Mapping: {{mapping| 1 1 -1 3 -2 0 | 0 3 17 -1 28 19 }}
Mapping: {{mapping| 1 1 1 3 1 2 | 0 3 7 -1 13 9 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.1389{{c}}, ~8/7 = 234.1162{{c}}
* WE: ~2 = 1201.0996{{c}}, ~8/7 = 227.4378{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.0946{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 227.3327{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~8/7 = 234.111{{c}}
 
* POTE: ~2 = 1200.000{{c}}, ~8/7 = 234.089{{c}} -->
{{Optimal ET sequence|legend=0| 5e, 16, 21, 37b }}


{{Optimal ET sequence|legend=0| 5e, 36ce, 41 }}
Badness (Sintel): 1.35


Badness (Sintel): 1.33
=== Spartan ===
Subgroup: 2.3.5.7.11


== Guiron ==
Comma list: 36/35, 56/55, 1029/1024
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
Mapping: {{mapping| 1 1 1 3 5 | 0 3 7 -1 -8 }}


[[Comma list]]: 1029/1024, 10976/10935
Optimal tunings:  
* WE: ~2 = 1198.9344{{c}}, ~8/7 = 229.3316{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 229.5124{{c}}


{{Mapping|legend=1| 1 1 7 3 | 0 3 -24 -1 }}
{{Optimal ET sequence|legend=0| 5, 16e, 21 }}


[[Optimal tuning]]s:  
Badness (Sintel): 2.07
* [[WE]]: ~2 = 1200.3395{{c}}, ~8/7 = 233.9963{{c}}
: [[error map]]: {{val| +0.340 +0.374 +0.151 -1.804 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 233.9239{{c}}
: error map: {{val| 0.000 -0.183 -0.487 -2.750 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~8/7 = 233.903{{c}}
: [[error map]]: {{val| 0.000 -0.246 +0.012 -2.729 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~8/7 = 233.930{{c}}
: error map: {{val| 0.000 -0.165 -0.637 -2.756 }} -->


[[Minimax tuning]]:
==== 13-limit ====
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~8/7 = {{monzo| 7/24 0 -1/24 }}
Subgroup: 2.3.5.7.11.13
: {{monzo list| 1 0 0 0 | 15/8 0 -1/8 0 | 0 0 1 0 | 65/24 0 1/24 0 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5


{{Optimal ET sequence|legend=1| 36, 41, 77, 118, 277d }}
Comma list: 27/26, 36/35, 56/55, 507/500


[[Badness]] (Sintel): 1.20
Mapping: {{mapping| 1 1 1 3 5 2 | 0 3 7 -1 -8 9 }}


=== 11-limit ===
Optimal tunings:
Subgroup: 2.3.5.7.11
* WE: ~2 = 1198.3002{{c}}, ~8/7 = 228.7341{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 229.0044{{c}}


Comma list: 385/384, 441/440, 10976/10935
{{Optimal ET sequence|legend=0| 5, 16e, 21 }}


Mapping: {{mapping| 1 1 7 3 -2 | 0 3 -24 -1 28 }}
Badness (Sintel): 1.95


Optimal tunings:
; Music
* WE: ~2 = 1200.3453{{c}}, ~8/7 = 233.9988{{c}}
* [https://web.archive.org/web/20201127012514/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Herman/gorgo-example.mp3 ''Gorgo Example''] by [[Herman Miller]]
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.9312{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~8/7 = 233.930{{c}}
* POTE: ~2 = 1200.000{{c}}, ~8/7 = 233.931{{c}} -->


Minimax tuning:
== Gidorah ==
* 11-odd-limit: ~8/7 = {{monzo| 7/24 0 -1/24 }}
: ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum #University]].''
: [{{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 }}]
: unchanged-interval (eigenmonzo) basis: 2.5


{{Optimal ET sequence|legend=0| 36e, 41, 77, 118, 159, 277d }}
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.


Badness (Sintel): 0.881
[[Subgroup]]: 2.3.5.7


=== 13-limit ===
[[Comma list]]: 21/20, 144/125
Subgroup: 2.3.5.7.11.13


Comma list: 196/195, 352/351, 385/384, 729/728
{{Mapping|legend=1| 1 1 2 3 | 0 3 2 -1 }}


Mapping: {{mapping| 1 1 7 3 -2 0 | 0 3 -24 -1 28 19 }}
[[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 tunings:
{{Optimal ET sequence|legend=1| 1b, 5 }}
* WE: ~2 = 1200.1222{{c}}, ~8/7 = 233.9228{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.8994{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~8/7 = 233.902{{c}}
* POTE: ~2 = 1200.000{{c}}, ~8/7 = 233.899{{c}} -->


{{Optimal ET sequence|legend=0| 36e, 41, 77, 118 }}
[[Badness]] (Sintel): 1.58


Badness (Sintel): 1.18
== Oncle ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Oncle]].''


== Gorgo ==
Oncle can be described as the {{nowrap| 31 & 36c }} temperament.  
: ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum #Laconic]].''
{{See also| Llywelynsmic clan }}
 
Gorgo tempers the generator of ~8/7 together with ~10/9. It can be described as the {{nowrap| 16 & 21 }} temperament.
 
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, meaning that this temperament is much more accurate than its comma list suggests.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 36/35, 1029/1024
[[Comma list]]: 1029/1024, 2430/2401


{{Mapping|legend=1| 1 1 1 3 | 0 3 7 -1 }}
{{Mapping|legend=1| 1 1 6 3 | 0 3 -19 -1 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.9847{{c}}, ~8/7 = 228.5210{{c}}
* [[WE]]: ~2 = 1201.2246{{c}}, ~8/7 = 232.7354{{c}}
: [[error map]]: {{val| +0.985 -15.407 +14.318 +5.607 }}
: [[error map]]: {{val| +1.225 -2.524 -0.939 +2.112 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 228.4371{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 232.4718{{c}}
: error map: {{val| 0.000 -16.644 +12.746 +2.737 }}
: error map: {{val| 0.000 -4.539 -3.279 -1.298 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~8/7 = 228.724{{c}}
: [[error map]]: {{val| 0.000 -15.782 +14.756 +2.450 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~8/7 = 228.334{{c}}
: error map: {{val| 0.000 -16.954 +12.022 +2.840 }} -->


{{Optimal ET sequence|legend=1| 5, 11c, 16, 21 }}
{{Optimal ET sequence|legend=1| 31, 98c, 129c, 160bc }}


[[Badness]] (Sintel): 1.54
[[Badness]] (Sintel): 2.24


=== 11-limit ===
== Archaeotherium ==
Subgroup: 2.3.5.7.11
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Archaeotherium]].''


Comma list: 36/35, 45/44, 1029/1024
Archaeotherium can be described as the {{nowrap| 21 & 26 }} temperament.


Mapping: {{mapping| 1 1 1 3 1 | 0 3 7 -1 13 }}
[[Subgroup]]: 2.3.5.7


Optimal tunings:  
[[Comma list]]: 405/392, 1029/1024
* WE: ~2 = 1201.3609{{c}}, ~8/7 = 227.6312{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 227.4955{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~8/7 = 227.833{{c}}
* POTE: ~2 = 1200.000{{c}}, ~8/7 = 227.373{{c}} -->


{{Optimal ET sequence|legend=0| 5e, 16, 21, 37b }}
{{Mapping|legend=1| 1 1 5 3 | 0 3 -14 -1 }}


Badness (Sintel): 1.64
[[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 }}


==== 13-limit ====
{{Optimal ET sequence|legend=1| 21, 26, 47, 73bc }}
Subgroup: 2.3.5.7.11.13


Comma list: 27/26, 36/35, 45/44, 507/500
[[Badness]] (Sintel): 3.70


Mapping: {{mapping| 1 1 1 3 1 2 | 0 3 7 -1 13 9 }}
== Clyndro ==
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.


Optimal tunings:  
[[Subgroup]]: 2.3.5.7
* WE: ~2 = 1201.0996{{c}}, ~8/7 = 227.4378{{c}}
 
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 227.3327{{c}}
[[Comma list]]: 135/128, 360/343
<!-- * CTE: ~2 = 1200.000{{c}}, ~8/7 = 227.633{{c}}
 
* POTE: ~2 = 1200.000{{c}}, ~8/7 = 227.230{{c}} -->
{{Mapping|legend=1| 1 1 4 3 | 0 3 -9 -1 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1205.6135{{c}}, ~8/7 = 227.5283{{c}}
: [[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=0| 5e, 16, 21, 37b }}
{{Optimal ET sequence|legend=1| 5c, 11, 16 }}


Badness (Sintel): 1.35
[[Badness]] (Sintel): 4.03


=== Spartan ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 36/35, 56/55, 1029/1024
Comma list: 33/32, 45/44, 352/343


Mapping: {{mapping| 1 1 1 3 5 | 0 3 7 -1 -8 }}
Mapping: {{mapping| 1 1 4 3 4 | 0 3 -9 -1 -3 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1198.9344{{c}}, ~8/7 = 229.3316{{c}}
* WE: ~2 = 1206.2134{{c}}, ~8/7 = 227.6004{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 229.5124{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 226.2421{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~8/7 = 229.420{{c}}
* POTE: ~2 = 1200.000{{c}}, ~8/7 = 229.535{{c}} -->


{{Optimal ET sequence|legend=0| 5, 16e, 21 }}
{{Optimal ET sequence|legend=0| 5c, 11, 16 }}


Badness (Sintel): 2.07
Badness (Sintel): 2.30


==== 13-limit ====
== Miracle ==
Subgroup: 2.3.5.7.11.13
{{Main| Miracle }}
: ''For the 5-limit version, see [[Syntonic–31 equivalence continuum #Ampersand]].''


Comma list: 27/26, 36/35, 56/55, 507/500
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.


Mapping: {{mapping| 1 1 1 3 5 2 | 0 3 7 -1 -8 9 }}
[[Subgroup]]: 2.3.5.7


Optimal tunings:  
[[Comma list]]: 225/224, 1029/1024
* WE: ~2 = 1198.3002{{c}}, ~8/7 = 228.7341{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 229.0044{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~8/7 = 228.758{{c}}
* POTE: ~2 = 1200.000{{c}}, ~8/7 = 229.059{{c}} -->


{{Optimal ET sequence|legend=0| 5, 16e, 21 }}
{{Mapping|legend=1| 1 1 3 3 | 0 6 -7 -2 }}
: mapping generator: ~2, ~15/14


Badness (Sintel): 1.95
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.8209{{c}}, ~15/14 = 116.7550{{c}}
: [[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 }}


; Music
[[Minimax tuning]]:
* [https://web.archive.org/web/20201127012514/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Herman/gorgo-example.mp3 ''Gorgo Example''] by [[Herman Miller]]
* [[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|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|unchanged-interval (eigenmonzo) basis]]: 2.9/5


== Gidorah ==
[[Tuning ranges]]:
: ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum #University]].''
* 7-odd-limit [[diamond monotone]]: ~15/14 = [114.286, 120.000] (2\21 to 1\10)
* 9-odd-limit diamond monotone: ~15/14 = [116.129, 120.000] (3\31 to 1\10)
* 7- and 9-odd-limit [[diamond tradeoff]]: ~15/14 = [115.587, 116.993]


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.
[[Algebraic generator]]: Secor59, positive root of 15''x''<sup>6</sup> - 8''x''<sup>4</sup> - 12


[[Subgroup]]: 2.3.5.7
{{Optimal ET sequence|legend=1| 10, 21, 31, 41, 72 }}


[[Comma list]]: 21/20, 144/125
[[Badness]] (Sintel): 0.424


{{Mapping|legend=1| 1 1 2 3 | 0 3 2 -1 }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


[[Optimal tuning]]s:  
Comma list: 225/224, 243/242, 385/384
* [[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 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~8/7 = 227.100{{c}}
: [[error map]]: {{val| 0.000 -20.655 +67.886 +4.074 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~8/7 = 230.762{{c}}
: error map: {{val| 0.000 -9.668 +75.211 +0.412 }} -->


{{Optimal ET sequence|legend=1| 1b, 5 }}
Mapping: {{mapping| 1 1 3 3 2 | 0 6 -7 -2 15 }}


[[Badness]] (Sintel): 1.58
Optimal tunings:  
* WE: ~2 = 1200.7626{{c}}, ~15/14 = 116.7069{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.6469{{c}}


== Oncle ==
Minimax tuning:
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Oncle]].''
* 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 }}]
: unchanged-interval (eigenmonzo) basis: 2.9/5


Oncle can be described as the {{nowrap| 31 & 36c }} temperament.  
Tuning ranges:
* 11-odd-limit diamond monotone: ~15/14 = [116.129, 117.073] (3\31 to 4\41)
* 11-odd-limit diamond tradeoff: ~15/14 = [115.587, 116.993]


[[Subgroup]]: 2.3.5.7
Algebraic generator: Secor59


[[Comma list]]: 1029/1024, 2430/2401
{{Optimal ET sequence|legend=0| 10, 21e, 31, 41, 72, 247c, 319bcde, 391bcde, 463bccde }}


{{Mapping|legend=1| 1 1 6 3 | 0 3 -19 -1 }}
Badness (Sintel): 0.353


[[Optimal tuning]]s:
==== Miraculous ====
* [[WE]]: ~2 = 1201.2246{{c}}, ~8/7 = 232.7354{{c}}
Subgroup: 2.3.5.7.11.13
: [[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 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~8/7 = 232.383{{c}}
: [[error map]]: {{val| 0.000 -4.807 -1.585 -1.209 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~8/7 = 232.498{{c}}
: error map: {{val| 0.000 -4.461 -3.778 -1.324 }} -->


{{Optimal ET sequence|legend=1| 31, 98c, 129c, 160bc }}
Comma list: 105/104, 144/143, 196/195, 243/242


[[Badness]] (Sintel): 2.24
Mapping: {{mapping| 1 1 3 3 2 4 | 0 6 -7 -2 15 -3 }}


== Archaeotherium ==
Optimal tunings:
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Archaeotherium]].''
* WE: ~2 = 1200.1267{{c}}, ~15/14 = 116.7596{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.7488{{c}}


Archaeotherium can be described as the {{nowrap| 21 & 26 }} temperament.
{{Optimal ET sequence|legend=0| 10, 21e, 31, 41, 72f }}


[[Subgroup]]: 2.3.5.7
Badness (Sintel): 0.771


[[Comma list]]: 405/392, 1029/1024
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


{{Mapping|legend=1| 1 1 5 3 | 0 3 -14 -1 }}
Comma list: 105/104, 120/119, 144/143, 154/153, 170/169


[[Optimal tuning]]s:
Mapping: {{mapping| 1 1 3 3 2 4 4 | 0 6 -7 -2 15 -3 1 }}
* [[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 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~8/7 = 229.951{{c}}
: [[error map]]: {{val| 0.000 -12.102 -5.626 +1.223 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~8/7 = 230.258{{c}}
: error map: {{val| 0.000 -11.180 -9.933 +0.916 }} -->


{{Optimal ET sequence|legend=1| 21, 26, 47, 73bc }}
Optimal tunings:
* WE: ~2 = 1199.6759{{c}}, ~15/14 = 116.7378{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.7657{{c}}


[[Badness]] (Sintel): 3.70
{{Optimal ET sequence|legend=0| 10, 21e, 31, 41, 72fg }}


== Clyndro ==
Badness (Sintel): 0.870
{{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.  
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


[[Subgroup]]: 2.3.5.7
Comma list: 105/104, 120/119, 144/143, 154/153, 170/169, 210/209


[[Comma list]]: 135/128, 360/343
{{Todo|complete temperament data|inline=1}}


{{Mapping|legend=1| 1 1 4 3 | 0 3 -9 -1 }}
===== 23-limit =====
Subgroup: 2.3.5.7.11.13.17.19.23


[[Optimal tuning]]s:  
Comma list: 105/104, 120/119, 144/143, 154/153, 161/160, 170/169, 210/209
* [[WE]]: ~2 = 1205.6135{{c}}, ~8/7 = 227.5283{{c}}
: [[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 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~8/7 = 225.752{{c}}
: [[error map]]: {{val| 0.000 -24.699 -18.081 +5.422 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~8/7 = 226.469{{c}}
: error map: {{val| 0.000 -22.548 -24.534 +4.705 }} -->


{{Optimal ET sequence|legend=1| 5c, 11, 16 }}
{{Todo|complete temperament data|inline=1}}


[[Badness]] (Sintel): 4.03
==== Benediction ====
Subgroup: 2.3.5.7.11.13


=== 11-limit ===
Comma list: 225/224, 243/242, 351/350, 385/384
Subgroup: 2.3.5.7.11


Comma list: 33/32, 45/44, 352/343
Mapping: {{mapping| 1 1 3 3 2 7 | 0 6 -7 -2 15 -34 }}
 
Mapping: {{mapping| 1 1 4 3 4 | 0 3 -9 -1 -3 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1206.2134{{c}}, ~8/7 = 227.6004{{c}}
* WE: ~2 = 1199.8601{{c}}, ~15/14 = 116.6572{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 226.2421{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.5688{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~8/7 = 225.384{{c}}
* POTE: ~2 = 1200.000{{c}}, ~8/7 = 226.428{{c}} -->


{{Optimal ET sequence|legend=0| 5c, 11, 16 }}
{{Optimal ET sequence|legend=0| 31, 72, 103, 175f }}


Badness (Sintel): 2.30
Badness (Sintel): 0.649


== Miracle ==
===== 17-limit =====
{{Main| Miracle }}
Subgroup: 2.3.5.7.11.13.17
: ''For the 5-limit version, see [[Syntonic–31 equivalence continuum #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. 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.
Comma list: 225/224, 243/242, 273/272, 351/350, 375/374


[[Subgroup]]: 2.3.5.7
Mapping: {{mapping| 1 1 3 3 2 7 7 | 0 6 -7 -2 15 -34 -30 }}


[[Comma list]]: 225/224, 1029/1024
Optimal tunings:  
* WE: ~2 = 1200.8328{{c}}, ~15/14 = 116.6661{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.5774{{c}}


{{Mapping|legend=1| 1 1 3 3 | 0 6 -7 -2 }}
{{Optimal ET sequence|legend=0| 31, 72, 103, 175f, 422bcdefffg }}


: mapping generator: ~2, ~15/14
Badness (Sintel): 0.639


[[Optimal tuning]]s:
===== 19-limit =====
* [[CTE]]: ~2 = 1200.000{{c}}, ~15/14 = 116.677{{c}}
Subgroup: 2.3.5.7.11.13.17.19
: [[error map]]: {{val| 0.000 -1.892 -3.054 -2.180 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~15/14 = 116.675{{c}}
: error map: {{val| 0.000 -1.904 -3.040 -2.176 }}


[[Minimax tuning]]:
Comma list: 210/209, 225/224, 243/242, 273/272, 286/285, 375/374
* [[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|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|unchanged-interval (eigenmonzo) basis]]: 2.9/5


[[Tuning ranges]]:
{{Todo|complete temperament data|inline=1}}
* 7-odd-limit [[diamond monotone]]: ~15/14 = [114.286, 120.000] (2\21 to 1\10)
* 9-odd-limit diamond monotone: ~15/14 = [116.129, 120.000] (3\31 to 1\10)
* 7- and 9-odd-limit [[diamond tradeoff]]: ~15/14 = [115.587, 116.993]


[[Algebraic generator]]: Secor59, positive root of 15''x''<sup>6</sup> - 8''x''<sup>4</sup> - 12
===== 23-limit =====
Subgroup: 2.3.5.7.11.13.17.19.23


{{Optimal ET sequence|legend=1| 10, 21, 31, 41, 72 }}
Comma list: 162/161, 210/209, 225/224, 231/230, 243/242, 273/272, 286/285


[[Badness]] (Smith): 0.016742
{{Todo|complete temperament data|inline=1}}


=== 11-limit ===
==== Manna ====
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11.13


Comma list: 225/224, 243/242, 385/384
Comma list: 225/224, 243/242, 325/324, 385/384


Mapping: {{mapping| 1 1 3 3 2 | 0 6 -7 -2 15 }}
Mapping: {{mapping| 1 1 3 3 2 0 | 0 6 -7 -2 15 38 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~15/14 = 116.711{{c}}
* WE: ~2 = 1200.7564{{c}}, ~15/14 = 116.8129{{c}}
* POTE: ~2 = 1200.000{{c}}, ~15/14 = 116.633{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.7528{{c}}


Minimax tuning:
{{Optimal ET sequence|legend=0| 31f, 41, 72, 185cf, 257cff }}
* 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 }}]
: unchanged-interval (eigenmonzo) basis: 2.9/5


Tuning ranges:
Badness (Sintel): 0.703
* 11-odd-limit diamond monotone: ~15/14 = [116.129, 117.073] (3\31 to 4\41)
* 11-odd-limit diamond tradeoff: ~15/14 = [115.587, 116.993]


Algebraic generator: Secor59
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


{{Optimal ET sequence|legend=0| 10, 21e, 31, 41, 72, 247c, 319bcde, 391bcde, 463bccde }}
Comma list: 225/224, 243/242, 273/272, 325/324, 385/384


Badness (Smith): 0.010684
Mapping: {{mapping| 1 1 3 3 2 0 0 | 0 6 -7 -2 15 38 42 }}
 
==== Miraculous ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 105/104, 144/143, 196/195, 243/242
 
Mapping: {{mapping| 1 1 3 3 2 4 | 0 6 -7 -2 15 -3 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~15/14 = 116.758{{c}}
* WE: ~2 = 1200.7570{{c}}, ~15/14 = 116.8011{{c}}
* POTE: ~2 = 1200.000{{c}}, ~15/14 = 116.747{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.7408{{c}}


{{Optimal ET sequence|legend=0| 10, 21e, 31, 41, 72f, 113f, 185cff }}
{{Optimal ET sequence|legend=0| 31fg, 41, 72, 185cf, 257cff }}


Badness (Smith): 0.018669
Badness (Sintel): 0.748


===== 17-limit =====
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 105/104, 120/119, 144/143, 154/153, 170/169
Comma list: 210/209, 225/224, 243/242, 273/272, 325/324, 343/342


Mapping: {{mapping| 1 1 3 3 2 4 4 | 0 6 -7 -2 15 -3 1 }}
{{Todo|complete temperament data|inline=1}}


Optimal tunings:
===== 23-limit =====
* CTE: ~2 = 1200.000{{c}}, ~15/14 = 116.742{{c}}
Subgroup: 2.3.5.7.11.13.17.19.23
* POTE: ~2 = 1200.000{{c}}, ~15/14 = 116.769{{c}}


{{Optimal ET sequence|legend=0| 10, 21e, 31, 41, 72fg }}
Comma list: 210/209, 225/224, 243/242, 273/272, 300/299, 325/324, 343/342


Badness (Smith): 0.017084
{{Todo|complete temperament data|inline=1}}


==== Benediction ====
==== Semimiracle ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 225/224, 243/242, 351/350, 385/384
Comma list: 169/168, 225/224, 243/242, 385/384


Mapping: {{mapping| 1 1 3 3 2 7 | 0 6 -7 -2 15 -34 }}
Mapping: {{mapping| 2 2 6 6 4 7 | 0 6 -7 -2 15 2 }}
: mapping generators: ~55/39, ~15/14


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~15/14 = 116.541{{c}}
* WE: ~55/39 = 600.4844{{c}}, ~15/14 = 116.7182{{c}}
* POTE: ~2 = 1200.000{{c}}, ~15/14 = 116.574{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~15/14 = 116.6413{{c}}


{{Optimal ET sequence|legend=0| 31, 72, 103, 175f }}
{{Optimal ET sequence|legend=0| 10, 62, 72 }}


Badness (Smith): 0.015715
Badness (Sintel): 1.02


===== 17-limit =====
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 225/224, 243/242, 273/272, 351/350, 375/374
Comma list: 169/168, 221/220, 225/224, 243/242, 273/272


Mapping: {{mapping| 1 1 3 3 2 7 7 | 0 6 -7 -2 15 -34 -30 }}
Mapping: {{mapping| 2 2 6 6 4 7 7 | 0 6 -7 -2 15 2 6 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~15/14 = 116.529{{c}}
* WE: ~17/12 = 600.5042{{c}}, ~15/14 = 116.7264{{c}}
* POTE: ~2 = 1200.000{{c}}, ~15/14 = 116.585{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~15/14 = 116.6485{{c}}


{{Optimal ET sequence|legend=0| 31, 72, 103, 175f, 422bcdefffg }}
{{Optimal ET sequence|legend=0| 10, 62, 72 }}


Badness (Smith): 0.012537
Badness (Sintel): 0.822


==== Manna ====
===== 19-limit =====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 225/224, 243/242, 325/324, 385/384
Comma list: 169/168, 210/209, 221/220, 225/224, 243/242, 273/272


Mapping: {{mapping| 1 1 3 3 2 0 | 0 6 -7 -2 15 38 }}
{{Todo|complete temperament data|inline=1}}


Optimal tunings:
===== 23-limit =====
* CTE: ~2 = 1200.000{{c}}, ~15/14 = 116.814{{c}}
Subgroup: 2.3.5.7.11.13.17.19.23
* POTE: ~2 = 1200.000{{c}}, ~15/14 = 116.739{{c}}


{{Optimal ET sequence|legend=0| 31f, 41, 72, 185cf, 257cff }}
Comma list: 169/168, 208/207, 210/209, 221/220, 225/224, 243/242, 273/272


Badness (Smith): 0.017012
{{Todo|complete temperament data|inline=1}}


===== 17-limit =====
==== Hemisecordite ====
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13


Comma list: 225/224, 243/242, 273/272, 325/324, 385/384
Comma list: 225/224, 243/242, 385/384, 847/845


Mapping: {{mapping| 1 1 3 3 2 0 0 | 0 6 -7 -2 15 38 42 }}
Mapping: {{mapping| 1 1 3 3 2 2 | 0 12 -14 -4 30 35 }}
: mapping generators: ~2, ~27/26


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~15/14 = 116.802{{c}}
* WE: ~2 = 1200.6969{{c}}, ~27/26 = 58.3217{{c}}
* POTE: ~2 = 1200.000{{c}}, ~15/14 = 116.727{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~27/26 = 58.2964{{c}}


{{Optimal ET sequence|legend=0| 31fg, 41, 72, 185cf, 257cff }}
{{Optimal ET sequence|legend=0| 41, 62, 103, 247c, 350bcde }}


Badness (Smith): 0.014680
Badness (Sintel): 1.06


==== Semimiracle ====
===== 17-limit =====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13.17


Comma list: 169/168, 225/224, 243/242, 385/384
Comma list: 225/224, 243/242, 273/272, 385/384, 847/845


Mapping: {{mapping| 2 2 6 6 4 7 | 0 6 -7 -2 15 2 }}
Mapping: {{mapping| 1 1 3 3 2 2 2 | 0 12 -14 -4 30 35 43 }}
 
: mapping generators: ~55/39, ~15/14


Optimal tunings:  
Optimal tunings:  
* CTE: ~55/39 = 600.000{{c}}, ~15/14 = 116.735{{c}}
* WE: ~2 = 1200.6557{{c}}, ~27/26 = 58.2932{{c}}
* POTE: ~55/39 = 600.000{{c}}, ~15/14 = 116.624{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~27/26 = 58.2702{{c}}


{{Optimal ET sequence|legend=0| 10, 62, 72 }}
{{Optimal ET sequence|legend=0| 41, 62, 103 }}


Badness (Smith): 0.024622
Badness (Sintel): 1.15


===== 17-limit =====
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 169/168, 221/220, 225/224, 243/242, 273/272
Comma list:  


Mapping: {{mapping| 2 2 6 6 4 7 7 | 0 6 -7 -2 15 2 6 }}
{{Todo|complete temperament data|inline=1}}


Optimal tunings:
===== 23-limit =====
* CTE: ~17/12 = 600.000{{c}}, ~15/14 = 116.771{{c}}
Subgroup: 2.3.5.7.11.13.17.19.23
* POTE: ~17/12 = 600.000{{c}}, ~15/14 = 116.628{{c}}


{{Optimal ET sequence|legend=0| 10, 62, 72 }}
Comma list:


Badness (Smith): 0.016130
{{Todo|complete temperament data|inline=1}}


==== Hemisecordite ====
===== Semihemisecordite =====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13.17


Comma list: 225/224, 243/242, 385/384, 847/845
Comma list: 225/224, 243/242, 289/288, 385/384, 847/845
 
Mapping: {{mapping| 1 1 3 3 2 2 | 0 12 -14 -4 30 35 }}
 
: mapping generators: ~2, ~27/26
 
Optimal tunings:
* CTE: ~2 = 1200.000{{c}}, ~27/26 = 58.337{{c}}
* POTE: ~2 = 1200.000{{c}}, ~27/26 = 58.288{{c}}
 
{{Optimal ET sequence|legend=0| 41, 62, 103, 247c, 350bcde }}
 
Badness (Smith): 0.025589
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 225/224, 243/242, 273/272, 385/384, 847/845
 
Mapping: {{mapping| 1 1 3 3 2 2 2 | 0 12 -14 -4 30 35 43 }}
 
Optimal tunings:
* CTE: ~2 = 1200.000{{c}}, ~27/26 = 58.312{{c}}
* POTE: ~2 = 1200.000{{c}}, ~27/26 = 58.261{{c}}
 
{{Optimal ET sequence|legend=0| 41, 62, 103 }}
 
Badness (Smith): 0.022535
 
===== Semihemisecordite =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 225/224, 243/242, 289/288, 385/384, 847/845


Mapping: {{mapping| 2 2 6 6 4 4 7 | 0 12 -14 -4 30 35 12 }}
Mapping: {{mapping| 2 2 6 6 4 4 7 | 0 12 -14 -4 30 35 12 }}
: mapping generators: ~17/12, ~27/26
: mapping generators: ~17/12, ~27/26


Optimal tunings:  
Optimal tunings:  
* CTE: ~17/12 = 600.000{{c}}, ~27/26 = 58.350{{c}}
* WE: ~17/12 = 600.3951{{c}}, ~27/26 = 58.3260{{c}}
* POTE: ~17/12 = 600.000{{c}}, ~27/26 = 58.288{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~27/26 = 58.2974{{c}}


{{Optimal ET sequence|legend=0| 62, 144g, 206begg }}
{{Optimal ET sequence|legend=0| 62, 144g, 206begg }}


Badness (Smith): 0.046958
Badness (Sintel): 2.39


====== 19-limit ======
====== 19-limit ======
Line 1,134: Line 974:


Optimal tunings:  
Optimal tunings:  
* CTE: ~17/12 = 600.000{{c}}, ~27/26 = 58.356{{c}}
* WE: ~17/12 = 600.4418{{c}}, ~27/26 = 58.3255{{c}}
* POTE: ~17/12 = 600.000{{c}}, ~27/26 = 58.283{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~27/26 = 58.2928{{c}}


{{Optimal ET sequence|legend=0| 62, 144gh, 206begghh }}
{{Optimal ET sequence|legend=0| 62, 144gh, 206begghh }}


Badness (Smith): 0.035057
Badness (Sintel): 2.13


====== 23-limit ======
====== 23-limit ======
Line 1,149: Line 989:


Optimal tunings:  
Optimal tunings:  
* CTE: ~17/12 = 600.000{{c}}, ~27/26 = 58.366{{c}}
* WE: ~17/12 = 600.4451{{c}}, ~27/26 = 58.3264{{c}}
* POTE: ~17/12 = 600.000{{c}}, ~27/26 = 58.283{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~27/26 = 58.2942{{c}}


{{Optimal ET sequence|legend=0| 62, 144gh, 206begghhi }}
{{Optimal ET sequence|legend=0| 62, 144gh, 206begghhi }}


Badness (Smith): 0.026421
Badness (Sintel): 1.89


==== Phicordial ====
==== Phicordial ====
Line 1,161: Line 1,001:
Comma list: 225/224, 243/242, 385/384, 2200/2197
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
: mapping generators: ~2, ~16/13


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~16/13 = 361.096{{c}}
* WE: ~2 = 1200.7056{{c}}, ~13/8 = 839.3726{{c}}
* POTE: ~2 = 1200.000{{c}}, ~16/13 = 361.121{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/8 = 838.8831{{c}}


{{Optimal ET sequence|legend=0| 103, 216c, 319bcde, 535bccdef }}
{{Optimal ET sequence|legend=0| 103, 216c, 319bcde, 535bccdef }}


Badness (Smith): 0.033198
Badness (Sintel): 1.37


===== 17-limit =====
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
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 tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~16/13 = 361.098{{c}}
* WE: ~2 = 1200.5918{{c}}, ~13/8 = 839.2912{{c}}
* POTE: ~2 = 1200.000{{c}}, ~16/13 = 361.123{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/8 = 838.8809{{c}}


{{Optimal ET sequence|legend=0| 103, 216c, 319bcde }}
{{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 ===
=== Revelation ===
Line 1,196: Line 1,049:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~15/14 = 116.142{{c}}
* WE: ~2 = 1201.3320{{c}}, ~15/14 = 116.4057{{c}}
* POTE: ~2 = 1200.000{{c}}, ~15/14 = 116.277{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.2524{{c}}


{{Optimal ET sequence|legend=0| 10e, 21, 31 }}
{{Optimal ET sequence|legend=0| 10e, 21, 31 }}


Badness (Smith): 0.032946
Badness (Sintel): 1.09


==== 13-limit ====
==== 13-limit ====
Line 1,211: Line 1,064:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~15/14 = 116.194{{c}}
* WE: ~2 = 1200.6059{{c}}, ~15/14 = 116.3263{{c}}
* POTE: ~2 = 1200.000{{c}}, ~15/14 = 116.268{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.2564{{c}}


{{Optimal ET sequence|legend=0| 10e, 21, 31 }}
{{Optimal ET sequence|legend=0| 10e, 21, 31 }}


Badness (Smith): 0.029452
Badness (Sintel): 1.22


=== Hemimiracle ===
=== Hemimiracle ===
Line 1,224: Line 1,077:


Mapping: {{mapping| 1 1 3 3 4 | 0 12 -14 -4 -11 }}
Mapping: {{mapping| 1 1 3 3 4 | 0 12 -14 -4 -11 }}
: mapping generators: ~2, ~33/32
: mapping generators: ~2, ~33/32


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~33/32 = 58.399{{c}}
* WE: ~2 = 1200.2902{{c}}, ~33/32 = 58.4217{{c}}
* POTE: ~2 = 1200.000{{c}}, ~33/32 = 58.408{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~33/32 = 58.4062{{c}}


{{Optimal ET sequence|legend=0| 20, 21, 41 }}
{{Optimal ET sequence|legend=0| 20, 21, 41 }}


Badness (Smith): 0.059232
Badness (Sintel): 1.96


==== 13-limit ====
==== 13-limit ====
Line 1,243: Line 1,095:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~33/32 = 58.436{{c}}
* WE: ~2 = 1199.8454{{c}}, ~33/32 = 58.4220{{c}}
* POTE: ~2 = 1200.000{{c}}, ~33/32 = 58.430{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~33/32 = 58.4305{{c}}


{{Optimal ET sequence|legend=0| 20, 21, 41 }}
{{Optimal ET sequence|legend=0| 20, 21, 41 }}


Badness (Smith): 0.043151
Badness (Sintel): 1.78


=== Oracle ===
=== Oracle ===
The name is a portmanteau of [[orwell]] and [[miracle]]: Oracle is a weak extension of 7-limit miracle, splitting its ~[[16/15]] generator and an octave into two ~[[16/11]] generators. Additionally, when [[restriction|restricted]] to the 2.15.7.11 subgroup, oracle's generator corresponds to 2 stacked orwell generators.
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 121/120, 225/224, 1029/1024
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
: mapping generators: ~2, ~11/8


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~11/8 = 541.670{{c}}
* WE: ~2 = 1201.2122{{c}}, ~16/11 = 658.9974{{c}}
* POTE: ~2 = 1200.000{{c}}, ~11/8 = 541.668{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/11 = 658.3320{{c}}


{{Optimal ET sequence|legend=0| 11, 20, 31, 82e, 113e, 144ee }}
{{Optimal ET sequence|legend=0| 11, 20, 31, 82e, 113e, 144ee }}


Badness (Smith): 0.042687
Badness (Sintel): 1.41


== Hemiseven ==
== Hemiseven ==
Unlike miracle which splits ~8/7, hemiseven splits ~7/4. It can be described as the {{nowrap| 72 & 77 }} temperament. [[149edo]] is an obvious tuning.  
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
[[Subgroup]]: 2.3.5.7
Line 1,274: Line 1,127:
[[Comma list]]: 1029/1024, 19683/19600
[[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, ~320/243
: mapping generators: ~2, ~243/160


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~320/243 = 483.215{{c}}
* [[WE]]: ~2 = 1200.5612{{c}}, ~243/160 = 717.0687{{c}}
: [[error map]]: {{val| 0.000 -1.247 +0.441 -2.395 }}
: [[error map]]: {{val| +0.561 -0.665 +0.260 -0.718 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~320/243 = 483.267{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~243/160 = 716.7478{{c}}
: error map: {{val| 0.000 -1.554 -1.043 -2.293 }}
: error map: {{val| 0.000 -1.468 -0.629 -2.321 }}


{{Optimal ET sequence|legend=1| 72, 149, 221, 514bd, 735bcdd }}
{{Optimal ET sequence|legend=1| 72, 149, 221, 514bd, 735bcdd }}


[[Badness]] (Smith): 0.056557
[[Badness]] (Sintel): 1.43


=== 11-limit ===
=== 11-limit ===
Line 1,292: Line 1,145:
Comma list: 385/384, 441/440, 19683/19600
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 tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~320/243 = 483.247{{c}}
* WE: ~2 = 1200.6243{{c}}, ~243/160 = 717.0969{{c}}
* POTE: ~2 = 1200.000{{c}}, ~320/243 = 483.276{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~243/160 = 716.7292{{c}}


{{Optimal ET sequence|legend=0| 72, 149, 221e, 293de }}
{{Optimal ET sequence|legend=0| 72, 149, 221e, 293de }}


Badness (Smith): 0.028467
Badness (Sintel): 0.941


=== 13-limit ===
=== 13-limit ===
Line 1,307: Line 1,160:
Comma list: 351/350, 385/384, 441/440, 676/675
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 tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~120/91 = 483.213{{c}}
* WE: ~2 = 1200.6781{{c}}, ~91/60 = 717.1496{{c}}
* POTE: ~2 = 1200.000{{c}}, ~120/91 = 483.255{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~91/60 = 716.7520{{c}}


{{Optimal ET sequence|legend=0| 72, 149, 221ef }}
{{Optimal ET sequence|legend=0| 72, 149, 221ef }}


Badness (Smith): 0.021900
Badness (Sintel): 0.905


=== 17-limit ===
=== 17-limit ===
Line 1,322: Line 1,175:
Comma list: 273/272, 351/350, 385/384, 441/440, 676/675
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:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~45/34 = 483.213{{c}}
* WE: ~2 = 1200.6635{{c}}, ~68/45 = 717.1354{{c}}
* POTE: ~2 = 1200.000{{c}}, ~45/34 = 483.261{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~68/45 = 716.7472{{c}}


{{Optimal ET sequence|legend=0| 72, 149, 221ef }}
{{Optimal ET sequence|legend=0| 72, 149, 221ef }}


Badness (Smith): 0.015701
Badness (Sintel): 0.800


== Valentine ==
== Valentine ==
Line 1,336: Line 1,189:
: ''For the 5-limit version, see [[Syntonic–31 equivalence continuum #Valentine (5-limit)]].''
: ''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 also be described as the {{nowrap| 31 & 46 }} temperament, and [[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 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.
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.
Line 1,348: Line 1,201:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~21/20 = 77.878{{c}}
* [[WE]]: ~2 = 1200.0749{{c}}, ~21/20 = 77.8687{{c}}
: [[error map]]: {{val| 0.000 -1.057 +3.074 -2.459 }}
: [[error map]]: {{val| +0.075 -1.062 +3.179 -2.207 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~21/20 = 77.864{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 77.8673{{c}}
: error map: {{val| 0.000 -1.181 +3.005 -2.417 }}
: error map: {{val| 0.000 -1.149 +3.023 -2.428 }}


[[Minimax tuning]]:
[[Minimax tuning]]:
Line 1,363: Line 1,216:
[[Algebraic generator]]: smaller root of ''x''<sup>2</sup> - 89''x'' + 92, or (89 - sqrt (7553))/2, at 77.8616 cents.  
[[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, 262cd }}
{{Optimal ET sequence|legend=1| 15, 31, 46, 77, 185 }}


[[Badness]] (Smith): 0.031056
[[Badness]] (Sintel): 0.786


=== 11-limit ===
=== 11-limit ===
Line 1,373: Line 1,226:


Mapping: {{mapping| 1 1 2 3 3 | 0 9 5 -3 7 }}
Mapping: {{mapping| 1 1 2 3 3 | 0 9 5 -3 7 }}
: mapping generators: ~2, ~21/20


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~22/21 = 77.963{{c}}
* WE: ~2 = 1200.3890{{c}}, ~22/21 = 77.9065{{c}}
* POTE: ~2 = 1200.000{{c}}, ~22/21 = 77.881{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/21 = 77.9007{{c}}


Minimax tuning:
Minimax tuning:
Line 1,387: Line 1,238:
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.
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, 262cdee, 339cdeee }}
{{Optimal ET sequence|legend=0| 15, 31, 46, 77 }}


Badness (Smith): 0.016687
Badness (Sintel): 0.552


==== Valentino ====
==== Valentino ====
Line 1,399: Line 1,250:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~22/21 = 77.968{{c}}
* WE: ~2 = 1200.1967{{c}}, ~22/21 = 77.9708{{c}}
* POTE: ~2 = 1200.000{{c}}, ~22/21 = 77.958{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/21 = 77.9594{{c}}


{{Optimal ET sequence|legend=0| 15f, 31, 46, 77 }}
{{Optimal ET sequence|legend=0| 15f, 31, 46, 77 }}


Badness (Smith): 0.020665
Badness (Sintel): 0.854


===== 17-limit =====
===== 17-limit =====
Line 1,414: Line 1,265:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~22/21 = 78.003{{c}}
* WE: ~2 = 1200.0404{{c}}, ~22/21 = 78.0055{{c}}
* POTE: ~2 = 1200.000{{c}}, ~22/21 = 78.003{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/21 = 78.0029{{c}}


{{Optimal ET sequence|legend=0| 15f, 31, 46, 77, 123e }}
{{Optimal ET sequence|legend=0| 15f, 31, 46, 77, 123e }}


Badness (Smith): 0.016768
Badness (Sintel): 0.854


==== Lupercalia ====
==== Lupercalia ====
Line 1,429: Line 1,280:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~22/21 = 77.694{{c}}
* WE: ~2 = 1199.9143{{c}}, ~22/21 = 77.7039{{c}}
* POTE: ~2 = 1200.000{{c}}, ~22/21 = 77.709{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/21 = 77.7049{{c}}


{{Optimal ET sequence|legend=0| 15, 31 }}
{{Optimal ET sequence|legend=0| 15, 31 }}


Badness (Smith): 0.021328
Badness (Sintel): 0.881


==== Dwynwen ====
==== Dwynwen ====
Line 1,444: Line 1,295:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~22/21 = 78.243{{c}}
* WE: ~2 = 1200.1306{{c}}, ~22/21 = 78.2273{{c}}
* POTE: ~2 = 1200.000{{c}}, ~22/21 = 78.219{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/21 = 78.2241{{c}}


{{Optimal ET sequence|legend=0| 15, 31f, 46 }}
{{Optimal ET sequence|legend=0| 15, 31f, 46 }}


Badness (Smith): 0.023461
Badness (Sintel): 0.969


==== Semivalentine ====
==== Semivalentine ====
Line 1,457: Line 1,308:


Mapping: {{mapping| 2 2 4 6 6 7 | 0 9 5 -3 7 3 }}
Mapping: {{mapping| 2 2 4 6 6 7 | 0 9 5 -3 7 3 }}
: mapping generators: ~55/39, ~22/21
: mapping generators: ~55/39, ~22/21


Optimal tunings:  
Optimal tunings:  
* CTE: ~55/39 = 600.000{{c}}, ~22/21 = 77.997{{c}}
* WE: ~55/39 = 600.3497{{c}}, ~22/21 = 77.8845{{c}}
* POTE: ~55/39 = 600.000{{c}}, ~22/21 = 77.839{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~22/21 = 77.8715{{c}}


{{Optimal ET sequence|legend=0| 16, 30, 46, 62, 108ef }}
{{Optimal ET sequence|legend=0| 16, 30, 46, 62, 108ef }}


Badness (Smith): 0.032749
Badness (Sintel): 1.35


==== Hemivalentine ====
==== Hemivalentine ====
Line 1,474: Line 1,324:


Mapping: {{mapping| 1 1 2 3 3 4 | 0 18 10 -6 14 -9 }}
Mapping: {{mapping| 1 1 2 3 3 4 | 0 18 10 -6 14 -9 }}
: mapping generators: ~2, ~40/39
: mapping generators: ~2, ~40/39


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~40/39 = 39.014{{c}}
* WE: ~2 = 1199.6529{{c}}, ~40/39 = 39.0323{{c}}
* POTE: ~2 = 1200.000{{c}}, ~40/39 = 39.044{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~40/39 = 39.0383{{c}}


{{Optimal ET sequence|legend=0| 30, 31, 61, 92f }}
{{Optimal ET sequence|legend=0| 30, 31, 61, 92f }}


Badness (Smith): 0.047059
Badness (Sintel): 1.94


==== Demivalentine ====
==== Demivalentine ====
Line 1,491: Line 1,340:


Mapping: {{mapping| 1 -8 -3 6 -4 -16 | 0 18 10 -6 14 37 }}
Mapping: {{mapping| 1 -8 -3 6 -4 -16 | 0 18 10 -6 14 37 }}
: mapping generators: ~2, ~13/9
: mapping generators: ~2, ~13/9


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~13/9 = 638.964{{c}}
* WE: ~2 = 1200.3929{{c}}, ~13/9 = 639.1320{{c}}
* CWE: ~2 = 1200.000{{c}}, ~13/9 = 638.932{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/9 = 638.9325{{c}}


{{Optimal ET sequence|legend=0| 15, 47ef, 62, 77 }}
{{Optimal ET sequence|legend=0| 15, 47ef, 62, 77 }}


Badness (Smith): 0.0349
Badness (Sintel): 1.44


=== Hemivalentino ===
=== Hemivalentino ===
Line 1,510: Line 1,358:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~45/44 = 38.928{{c}}
* WE: ~2 = 1200.0816{{c}}, ~45/44 = 38.9236{{c}}
* POTE: ~2 = 1200.000{{c}}, ~45/44 = 38.921{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~45/44 = 38.9228{{c}}


{{Optimal ET sequence|legend=0| 31, 92e, 123, 154, 185 }}
{{Optimal ET sequence|legend=0| 31, 92e, 123, 154, 185 }}


Badness (Smith): 0.061275
Badness (Sintel): 2.03


==== 13-limit ====
==== 13-limit ====
Line 1,525: Line 1,373:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~45/44 = 38.944{{c}}
* WE: ~2 = 1199.8782{{c}}, ~45/44 = 38.9440{{c}}
* POTE: ~2 = 1200.000{{c}}, ~45/44 = 38.948{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~45/44 = 38.9472{{c}}


{{Optimal ET sequence|legend=0| 31, 123, 154 }}
{{Optimal ET sequence|legend=0| 31, 123, 154 }}


Badness (Smith): 0.057919
Badness (Sintel): 2.39


==== Hemivalentoid ====
==== Hemivalentoid ====
Line 1,540: Line 1,388:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~40/39 = 38.946{{c}}
* WE: ~2 = 1199.3614{{c}}, ~45/44 = 38.9721{{c}}
* POTE: ~2 = 1200.000{{c}}, ~40/39 = 38.993{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~45/44 = 38.9839{{c}}


{{Optimal ET sequence|legend=0| 31, 92ef }}
{{Optimal ET sequence|legend=0| 31, 92ef }}


Badness (Smith): 0.057931
Badness (Sintel): 2.39


== Superkleismic ==
== Superkleismic ==
Line 1,551: Line 1,399:
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Shibboleth]].''
: ''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. Its [[ploidacot]] is wau-enneacot. 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.
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 {{nowrap| S19 {{=}} [[361/360]] }} and {{nowrap| S20 {{=}} [[400/399]] }}.  
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.  


41edo gives an obvious tuning in all the subgroups.  
41edo gives an obvious tuning in all the subgroups.  
Line 1,561: Line 1,413:
[[Comma list]]: 875/864, 1029/1024
[[Comma list]]: 875/864, 1029/1024


{{Mapping|legend=1| 1 4 5 2 | 0 -9 -10 3 }}
{{Mapping|legend=1| 1 -5 -5 5 | 0 9 10 -3 }}
 
: mapping generators: ~2, ~5/3
: mapping generators: ~2, ~6/5


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~6/5 = 321.798{{c}}
* [[WE]]: ~2 = 1200.7640{{c}}, ~5/3 = 878.6289{{c}}
* [[POTE]]: ~2 = 1200.000{{c}}, ~6/5 = 321.930{{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 }}
{{Optimal ET sequence|legend=1| 11c, 15, 26, 41 }}


[[Badness]]:
[[Badness]] (Sintel): 1.21
* Smith: 0.0479
* Sintel: 1.21


=== 11-limit ===
=== 11-limit ===
Line 1,580: Line 1,431:
Comma list: 100/99, 245/242, 385/384
Comma list: 100/99, 245/242, 385/384


Mapping: {{mapping| 1 4 5 2 4 | 0 -9 -10 3 -2 }}
Mapping: {{mapping| 1 -5 -5 5 2 | 0 9 10 -3 2 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~6/5 = 321.815{{c}}
* WE: ~2 = 1200.1691{{c}}, ~5/3 = 878.2772{{c}}
* POTE: ~2 = 1200.000{{c}}, ~6/5 = 321.847{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 878.1606{{c}}


{{Optimal ET sequence|legend=0| 11c, 15, 26, 41, 179cde, 220cde, 261ccdee }}
{{Optimal ET sequence|legend=0| 11c, 15, 26, 41, 179cde, 220cde, 261ccdee }}


Badness:
Badness (Sintel): 0.848
* Smith: 0.0257
* Sintel: 0.848


==== 2.3.5.7.11.19 subgroup ====
==== 2.3.5.7.11.19 subgroup ====
The following structure is preserved within the entire superkleismic tuning range between 15edo and 26edo, while extensions to 13 and 17 bifurcate and are of higher badness. These primes are of lower complexity than 13 and 17 are in the below extensions.
Subgroup: 2.3.5.7.11.19
Subgroup: 2.3.5.7.11.19


Comma list: 100/99, 133/132, 190/189, 385/384
Comma list: 100/99, 133/132, 190/189, 385/384


Mapping: {{mapping| 1 4 5 2 4 8 | 0 -9 -10 3 -2 -14}}
Mapping: {{mapping| 1 -5 -5 5 2 -6 | 0 9 10 -3 2 14 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~6/5 = 321.779{{c}}
* WE: ~2 = 1200.2289{{c}}, ~5/3 = 878.3409{{c}}
* POTE: ~2 = 1200.000{{c}}, ~6/5 = 321.827{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 878.1840{{c}}


Optimal ET sequence: {{Optimal ET sequence| 11c, 15, 26, 41, 138e, 179cde, 220cdeh }}
{{Optimal ET sequence|legend=0| 11c, 15, 26, 41, 138e }}


Badness (Sintel): 0.692
Badness (Sintel): 0.692
Line 1,616: Line 1,463:
Comma list: 100/99, 105/104, 144/143, 245/242
Comma list: 100/99, 105/104, 144/143, 245/242


Mapping: {{mapping| 1 4 5 2 4 8 | 0 -9 -10 3 -2 -16 }}
Mapping: {{mapping| 1 -5 -5 5 2 -8 | 0 9 10 -3 2 16 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~6/5 = 321.986{{c}}
* WE: ~2 = 1200.0261{{c}}, ~5/3 = 878.0252{{c}}
* POTE: ~2 = 1200.000{{c}}, ~6/5 = 321.994{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 878.0073{{c}}


{{Optimal ET sequence|legend=0| 11cf, 15, 26, 41 }}
{{Optimal ET sequence|legend=0| 11cf, 15, 26, 41 }}


Badness:
Badness (Sintel): 0.887
* Smith: 0.0215
* Sintel: 0.887


==== 17-limit ====
==== 17-limit ====
Line 1,633: Line 1,478:
Comma list: 100/99, 105/104, 120/119, 144/143, 245/242
Comma list: 100/99, 105/104, 120/119, 144/143, 245/242


Mapping: {{mapping| 1 4 5 2 4 8 10 | 0 -9 -10 3 -2 -16 -22 }}
Mapping: {{mapping| 1 -5 -5 5 2 -8 -12 | 0 9 10 -3 2 16 22 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~6/5 = 322.136{{c}}
* WE: ~2 = 1200.0488{{c}}, ~5/3 = 877.8872{{c}}
* POTE: ~2 = 1200.000{{c}}, ~6/5 = 322.149{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 877.8537{{c}}


{{Optimal ET sequence|legend=0| 11cfg, 15g, 26, 41 }}
{{Optimal ET sequence|legend=0| 11cfg, 15g, 26, 41 }}


Badness (Sintel): 1.007
Badness (Sintel): 1.01


==== 19-limit ====
==== 19-limit ====
Line 1,648: Line 1,493:
Comma list: 100/99, 105/104, 120/119, 144/143, 133/132, 190/189
Comma list: 100/99, 105/104, 120/119, 144/143, 133/132, 190/189


Mapping: {{mapping| 1 4 5 2 4 8 10 8 | 0 -9 -10 3 -2 -16 -22 -14 }}
Mapping: {{mapping| 1 -5 -5 5 2 -8 -12 -6 | 0 9 10 -3 2 16 22 14 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~6/5 = 322.084{{c}}
* WE: ~2 = 1200.2120{{c}}, ~5/3 = 878.0243{{c}}
* CWE: ~2 = 1200.000{{c}}, ~6/5 = 322.121{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 877.8789{{c}}


{{Optimal ET sequence|legend=0| 11cfgh, 15g, 26, 41 }}
{{Optimal ET sequence|legend=0| 11cfgh, 15g, 26, 41 }}
Line 1,663: Line 1,508:
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 100/99, 196/195, 385/384, 441/440
Comma list: 100/99, 196/195, 245/242, 385/384


Mapping: {{mapping| 1 4 5 2 4 -3 | 0 -9 -10 3 -2 25 }}
Mapping: {{mapping| 1 -5 -5 5 2 22 | 0 9 10 -3 2 -25 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~6/5 = 321.724{{c}}
* WE: ~2 = 1199.8272{{c}}, ~5/3 = 878.1538{{c}}
* POTE: ~2 = 1200.000{{c}}, ~6/5 = 321.719{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 878.2795{{c}}


{{Optimal ET sequence|legend=0| 15f, 26f, 41, 97, 138e }}
{{Optimal ET sequence|legend=0| 15f, 41, 97, 138e }}


Badness (Sintel): 1.402
Badness (Sintel): 1.40


==== 17-limit ====
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 100/99, 154/153, 196/195, 256/255, 273/272
Comma list: 100/99, 154/153, 196/195, 245/242, 256/255


Mapping: {{mapping| 1 4 5 2 4 -3 -1 | 0 -9 -10 3 -2 25 19 }}
Mapping: {{mapping| 1 -5 -5 5 2 22 18 | 0 9 10 -3 2 -25 -19 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~6/5 = 321.650{{c}}
* WE: ~2 = 1199.5964{{c}}, ~5/3 = 878.0482{{c}}
* POTE: ~2 = 1200.000{{c}}, ~6/5 = 321.657{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 878.3444{{c}}


{{Optimal ET sequence|legend=0| 15f, 26fg, 41, 56, 97g }}
{{Optimal ET sequence|legend=0| 15f, 41, 56, 97g }}


Badness (Sintel): 1.452
Badness (Sintel): 1.45


==== 19-limit ====
==== 19-limit ====
Line 1,695: Line 1,540:
Comma list: 100/99, 133/132, 154/153, 190/189, 196/195, 256/255
Comma list: 100/99, 133/132, 154/153, 190/189, 196/195, 256/255


Mapping: {{mapping| 1 4 5 2 4 -3 -1 8 | 0 -9 -10 3 -2 25 19 -14 }}
Mapping: {{mapping| 1 -5 -5 5 2 22 18 -6 | 0 9 10 -3 2 -25 -19 14 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~6/5 = 321.646{{c}}
* WE: ~2 = 1199.6638{{c}}, ~5/3 = 878.1109{{c}}
* POTE: ~2 = 1200.000{{c}}, ~6/5 = 321.643{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 878.3566{{c}}


{{Optimal ET sequence|legend=0| 15f, 26fg, 41, 56, 97g }}
{{Optimal ET sequence|legend=0| 15f, 41, 56, 97g }}


Badness (Sintel): 1.355
Badness (Sintel): 1.36


== Unidec ==
== Dee leap week ==
{{Main| Unidec }}
{{Main| Dee leap week }}


=== 5-limit (unidecmic) ===
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5


[[Comma list]]: 31381059609/31250000000
[[Comma list]]: 1029/1024, 2460375/2458624


{{Mapping|legend=1| 2 5 8 | 0 -6 -11 }}
{{Mapping|legend=1| 1 -5 25 5 | 0 9 -31 -3 }}
: mapping generators: ~177147/125000, ~10/9


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~177147/125000 = 600.000{{c}}, ~10/9 = 183.041{{c}}
* [[WE]]: ~2 = 1200.4835{{c}}, ~224/135 = 878.2507{{c}}
: [[error map]]: {{val| 0.000 -0.201 +0.235 }}
: [[error map]]: {{val| +0.484 -0.117 +0.004 -1.160 }}
* [[POTE]]: ~177147/125000 = 600.000{{c}}, ~10/9 = 183.047{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~224/135 = 877.8926{{c}}
: error map: {{val| 0.000 -0.236 +0.172 }}
: error map: {{val| 0.000 -0.921 -0.985 -2.504 }}
 
{{Optimal ET sequence|legend=1| 41, 108, 149, 190 }}
 
[[Badness]] (Sintel): 2.12
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 385/384, 441/440, 2460375/2458624
 
Mapping: {{mapping| 1 -5 25 5 -28 | 0 9 -31 -3 43 }}
 
Optimal tunings:
* WE: ~2 = 1200.4874{{c}}, ~224/135 = 878.2543{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~224/135 = 877.8987{{c}}
 
{{Optimal ET sequence|legend=0| 41, 108e, 149, 190 }}
 
Badness (Sintel): 1.35


{{Optimal ET sequence|legend=1| 26, 46, 72, 118, 2524, 2642, 2760 }}
== 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
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 1029/1024, 4375/4374
[[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 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~1225/864 = 600.000{{c}}, ~10/9 = 183.060{{c}}
* [[WE]]: ~1225/864 = 600.2429{{c}}, ~80/63 = 417.0073{{c}}
: [[error map]]: {{val| 0.000 -0.313 +0.030 -2.707 }}
: [[error map]]: {{val| +0.486 -0.154 +0.038 -1.140 }}
* [[POTE]]: ~1225/864 = 600.000{{c}}, ~10/9 = 183.161{{c}}
* [[CWE]]: ~1225/864 = 600.0000{{c}}, ~80/63 = 416.8688{{c}}
: error map: {{val| 0.000 -0.924 -1.090 -2.503 }}
: error map: {{val| 0.000 -0.924 -1.090 -2.503 }}


Line 1,749: Line 1,611:
{{Optimal ET sequence|legend=1| 26, 46, 72, 118, 190 }}
{{Optimal ET sequence|legend=1| 26, 46, 72, 118, 190 }}


[[Badness]] (Smith): 0.038393
[[Badness]] (Sintel): 0.972


=== 11-limit ===
=== 11-limit ===
Line 1,756: Line 1,618:
Comma list: 385/384, 441/440, 4375/4374
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:  
Optimal tunings:  
* CTE: ~99/70 = 600.000{{c}}, ~10/9 = 183.074{{c}}
* WE: ~99/70 = 600.2497{{c}}, ~14/11 = 417.0085{{c}}
* CWE: ~99/70 = 600.000{{c}}, ~10/9 = 183.146{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~14/11 = 416.8543{{c}}


Minimax tuning:
Minimax tuning:
Line 1,769: Line 1,631:
{{Optimal ET sequence|legend=0| 26, 46, 72, 118, 190 }}
{{Optimal ET sequence|legend=0| 26, 46, 72, 118, 190 }}


Badness (Smith): 0.015479
Badness (Sintel): 0.512


==== Ekadash ====
==== Ekadash ====
Line 1,776: Line 1,638:
Comma list: 385/384, 441/440, 625/624, 729/728
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 tunings:  
Optimal tunings:  
* CTE: ~99/70 = 600.000{{c}}, ~10/9 = 183.125{{c}}
* WE: ~99/70 = 600.2497{{c}}, ~14/11 = 417.0085{{c}}
* POTE: ~99/70 = 600.000{{c}}, ~10/9 = 183.187{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~14/11 = 416.8543{{c}}


{{Optimal ET sequence|legend=0| 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 ====
==== Hendec ====
Line 1,791: Line 1,653:
Comma list: 169/168, 325/324, 364/363, 385/384
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 tunings:  
Optimal tunings:  
* CTE: ~91/64 = 600.000{{c}}, ~10/9 = 183.048{{c}}
* WE: ~91/64 = 600.3825{{c}}, ~14/11 = 417.0678{{c}}
* POTE: ~91/64 = 600.000{{c}}, ~10/9 = 183.198{{c}}
* CWE: ~91/64 = 600.0000{{c}}, ~14/11 = 416.8290{{c}}


{{Optimal ET sequence|legend=0| 26, 46, 72, 190ff }}
{{Optimal ET sequence|legend=0| 26, 46, 72, 190ff }}


Badness (Smith): 0.017707
Badness (Sintel): 0.732


===== 17-limit =====
===== 17-limit =====
Line 1,806: Line 1,668:
Comma list: 169/168, 221/220, 273/272, 325/324, 364/363
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 tunings:  
Optimal tunings:  
* CTE: ~17/12 = 600.000{{c}}, ~10/9 = 183.020{{c}}
* WE: ~17/12 = 600.3991{{c}}, ~14/11 = 417.0809{{c}}
* POTE: ~17/12 = 600.000{{c}}, ~10/9 = 183.196{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~14/11 = 416.8330{{c}}


{{Optimal ET sequence|legend=0| 26, 46, 72, 190ffg }}
{{Optimal ET sequence|legend=0| 26, 46, 72, 190ffg }}


Badness (Smith): 0.011676
Badness (Sintel): 0.595
 
== Restles ==
{{See also| Lesser tendoneutralic }}
 
Restles may be described as the {{nowrap| 77 & 87 }} temperament, and has a [[ploidacot]] signature of gamma-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>.


== Lagaca ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 1029/1024, 11529602/11390625
[[Comma list]]: 1029/1024, 153664/151875


{{Mapping|legend=1| 2 5 2 5 | 0 -9 13 3 }}
{{Mapping|legend=1| 1 -2 8 4 | 0 12 -19 -4 }}
 
: mapping generators: ~2. ~315/256
: mapping generators: ~3375/2401, ~15/14


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~3375/2401 = 600.000{{c}}, ~15/14 = 122.031{{c}}
* [[WE]]: ~2 = 1200.0322{{c}}, ~315/256 = 358.5581{{c}}
: [[error map]]: {{val| 0.000 -0.232 +0.087 -2.734 }}
: [[error map]]: {{val| +0.032 +0.678 +1.340 -2.930 }}
* [[POTE]]: ~3375/2401 = 600.000{{c}}, ~15/14 = 122.027{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~315/256 = 358.5484{{c}}
: error map: {{val| 0.000 -0.195 +0.033 -2.746 }}
: error map: {{val| 0.000 +0.626 +1.267 -3.019 }}


{{Optimal ET sequence|legend=1| 10, 98, 108, 118 }}
{{Optimal ET sequence|legend=1| 77, 87, 164 }}
 
[[Badness]] (Sintel): 2.73
 
=== 11-limit ===
Subgroup: 2.3.5.7.11


[[Badness]] (Smith): 0.144345
Comma list: 385/384, 441/440, 153664/151875


== Dee leap week ==
Mapping: {{mapping| 1 -2 8 4 -7 | 0 12 -19 -4 35 }}
{{Main|Dee leap week}}


[[Subgroup]]: 2.3.5.7
Optimal tunings:  
* WE: ~2 = 1200.1110{{c}}, ~27/22 = 358.6045{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~27/22 = 358.5720{{c}}


[[Comma list]]: 1029/1024, 2460375/2458624
{{Optimal ET sequence|legend=0| 77, 87, 164, 251d }}


{{Mapping|legend=1| 1 4 -6 2 | 0 -9 31 3 }}
Badness (Sintel): 1.81


[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~135/112 = 322.123{{c}}
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


=== 11-limit ===
Comma list: 196/195, 352/351, 385/384, 676/675
Subgroup: 2.3.5.7.11


Comma list: 385/384, 441/440, 2460375/2458624
Mapping: {{mapping| 1 -2 8 4 -7 4 | 0 12 -19 -4 35 -1 }}


Mapping: {{mapping| 1 4 -6 2 15 | 0 -9 31 3 43 }}
Optimal tunings:  
* WE: ~2 = 1200.0482{{c}}, ~~16/13 = 358.5883{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/13 = 358.5741{{c}}


Optimal tuning (CTE): ~2 = 1200.000{{c}}, ~135/112 = 322.097
{{Optimal ET sequence|legend=0| 77, 87, 164, 251d }}


{{Optimal ET sequence|legend=0| 41, 149, 190, 231 }}
Badness (Sintel): 1.16


== Necromanteion ==
== Necromanteion ==
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. Therefore, its [[ploidacot]] is wau-dodecacot.
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 1029/1024, 5103/5000
[[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, ~48/35
: mapping generators: ~2, ~35/24


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~48/35 = 541.743{{c}}
* [[WE]]: ~2 = 1200.2959{{c}}, ~35/24 = 658.3833{{c}}
: [[error map]]: {{val| 0.000 -2.872 +4.053 -1.853 }}
: [[error map]]: {{val| +0.296 -2.835 +4.130 -0.879 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~48/35 = 541.779{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~35/24 = 658.2313{{c}}
: error map: {{val| 0.000 -3.304 +3.442 -1.710 }}
: error map: {{val| 0.000 -3.179 +3.619 -1.751 }}


{{Optimal ET sequence|legend=1| 11c, 20c, 31, 144c, 175c }}
{{Optimal ET sequence|legend=1| 11c, 20c, 31, 144c, 175c }}


[[Badness]] (Smith): 0.117680
[[Badness]] (Sintel): 2.98


=== 11-limit ===
=== 11-limit ===
Line 1,880: Line 1,755:
Comma list: 176/175, 243/242, 1029/1024
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 tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~15/11 = 541.695{{c}}
* WE: ~2 = 1200.2862{{c}}, ~22/15 = 658.4276{{c}}
* POTE: ~2 = 1200.000{{c}}, ~15/11 = 541.729{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/15 = 658.2805{{c}}


{{Optimal ET sequence|legend=0| 20ce, 31, 113c, 144c }}
{{Optimal ET sequence|legend=0| 20ce, 31, 113c, 144c }}


Badness (Smith): 0.053459
Badness (Sintel): 1.77


=== 13-limit ===
=== 13-limit ===
Line 1,895: Line 1,770:
Comma list: 144/143, 176/175, 243/242, 343/338
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 tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~15/11 = 541.673{{c}}
* WE: ~2 = 1199.3663{{c}}, ~22/15 = 658.0465{{c}}
* POTE: ~2 = 1200.000{{c}}, ~15/11 = 541.606{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/15 = 658.3800{{c}}


{{Optimal ET sequence|legend=0| 20ce, 31, 82cf, 113cf }}
{{Optimal ET sequence|legend=0| 20ce, 31, 82cf, 113cf }}


Badness (Smith): 0.047015
Badness (Sintel): 1.94


== Restles ==
== Lagaca ==
{{See also| Lesser tendoneutralic }}
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
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 1029/1024, 153664/151875
[[Comma list]]: 1029/1024, 11529602/11390625


{{Mapping|legend=1| 1 -2 8 4 | 0 12 -19 -4 }}
{{Mapping|legend=1| 2 -4 15 8 | 0 9 -13 -3 }}
: mapping generators: ~2. ~315/256
: 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
 
[[Comma list]]: 1029/1024, 1250000/1240029
 
{{Mapping|legend=1| 1 -17 -26 9 | 0 21 32 -7 }}
: mapping generators: ~2, ~50/27


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~315/256 = 358.548{{c}}
* [[WE]]: ~2 = 1200.5278{{c}}, ~50/27 = 1062.4614{{c}}
: [[error map]]: {{val| 0.000 +0.620 +1.275 -3.018 }}
: [[error map]]: {{val| +0.528 +0.762 -1.272 -1.305 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~315/256 = 358.548{{c}}
* [[CWE]]: ~21 = 1200.0000{{c}}, ~50/27 = 1062.0046{{c}}
: error map: {{val| 0.000 +0.627 +1.265 -3.020 }}
: error map: {{val| 0.000 +0.142 -2.167 -2.858 }}


{{Optimal ET sequence|legend=1| 77, 87, 164 }}
{{Optimal ET sequence|legend=1| 26, 61, 87, 113, 200 }}


[[Badness]] (Smith): 0.108011
[[Badness]] (Sintel): 3.85


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 385/384, 441/440, 153664/151875
Comma list: 385/384, 441/440, 800000/793881


Mapping: {{mapping| 1 -2 8 4 -7 | 0 12 -19 -4 35 }}
Mapping: {{mapping| 1 -17 -26 9 7 | 0 21 32 -7 -4 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~27/22 = 358.575{{c}}
* WE: ~2 = 1200.3051{{c}}, ~50/27 = 1062.2805{{c}}
* POTE: ~2 = 1200.000{{c}}, ~27/22 = 358.571{{c}}
* CWE: ~21 = 1200.0000{{c}}, ~50/27 = 1062.0147{{c}}


{{Optimal ET sequence|legend=0| 77, 87, 164, 251d }}
{{Optimal ET sequence|legend=0| 26, 61, 87, 200, 287d }}


Badness (Smith): 0.054655
Badness (Sintel): 1.89


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 196/195, 352/351, 385/384, 676/675
Comma list: 325/324, 364/363, 385/384, 2200/2197


Mapping: {{mapping| 1 -2 8 4 -7 4 | 0 12 -19 -4 35 -1 }}
Mapping: {{mapping| 1 -17 -26 9 7 -14 | 0 21 32 -7 -4 20 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~16/13 = 358.576{{c}}
* WE: ~2 = 1200.2708{{c}}, ~24/13 = 1062.2496{{c}}
* POTE: ~2 = 1200.000{{c}}, ~16/13 = 358.574{{c}}
* CWE: ~21 = 1200.0000{{c}}, ~24/13 = 1062.0139{{c}}


{{Optimal ET sequence|legend=0| 77, 87, 164, 251d }}
{{Optimal ET sequence|legend=0| 26, 61, 87, 200 }}


Badness (Smith): 0.028187
Badness (Sintel): 1.17


== Quartemka ==
== Tritriple ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Quartemka]].''
: ''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
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 1029/1024, 1250000/1240029
[[Comma list]]: 1029/1024, 1959552/1953125


{{Mapping|legend=1| 1 4 6 2 | 0 -21 -32 7 }}
{{Mapping|legend=1| 1 -11 -7 7 | 0 27 20 -9 }}
: mapping generators: ~2, ~27/25
: mapping generators: ~2, ~864/625


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~27/25 = 137.971{{c}}
* [[WE]]: ~2 = 1200.4239{{c}}, ~864/625 = 559.4921{{c}}
: [[error map]]: {{val| 0.000 +0.658 -1.380 -3.030 }}
: [[error map]]: {{val| +0.424 -0.331 +0.561 -1.287 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~27/25 = 138.006{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~864/625 = 559.3015{{c}}
: error map: {{val| 0.000 -0.075 -2.496 -2.786 }}
: error map: {{val| 0.000 -0.815 -0.284 -2.539 }}


{{Optimal ET sequence|legend=1| 26, 61, 87, 113, 200 }}
{{Optimal ET sequence|legend=1| 15, …, 88, 103, 118, 221, 339d }}


[[Badness]] (Smith): 0.152287
[[Badness]] (Sintel): 3.00


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 385/384, 441/440, 800000/793881
Comma list: 385/384, 441/440, 43923/43750


Mapping: {{mapping| 1 4 6 2 3 | 0 -21 -32 7 4 }}
Mapping: {{mapping| 1 -11 -7 7 -4 | 0 27 20 -9 16 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~27/25 = 137.970{{c}}
* WE: ~2 = 1200.4953{{c}}, ~242/175 = 559.5243{{c}}
* POTE: ~2 = 1200.000{{c}}, ~27/25 = 137.990{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~242/175 = 559.3016{{c}}


{{Optimal ET sequence|legend=0| 26, 61, 87, 200, 287d }}
{{Optimal ET sequence|legend=0| 15, …, 88, 103, 118, 221e, 339de }}


Badness (Smith): 0.057307
Badness (Sintel): 1.17


=== 13-limit ===
== Widefourth ==
Subgroup: 2.3.5.7.11.13
[[Subgroup]]: 2.3.5.7


Comma list: 325/324, 364/363, 385/384, 2200/2197
[[Comma list]]: 1029/1024, 48828125/48771072


Mapping: {{mapping| 1 4 6 2 3 6 | 0 -21 -32 7 4 -20 }}
{{Mapping|legend=1| 1 -17 -5 9 | 0 33 13 -11 }}


Optimal tunings:  
[[Optimal tuning]]s:  
* CTE: ~2 = 1200.000{{c}}, ~13/12 = 137.971{{c}}
* [[WE]]: ~2 = 1200.4770{{c}}, ~4608/3125 = 676.0584{{c}}
* POTE: ~2 = 1200.000{{c}}, ~13/12 = 137.990{{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 ET sequence|legend=0| 26, 61, 87, 200 }}
{{Optimal ET sequence|legend=1| 16, 71, 87, 103, 190 }}


Badness (Smith): 0.028393
[[Badness]] (Sintel): 3.90


== Tritriple ==
=== 11-limit ===
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tritriple]].''
Subgroup: 2.3.5.7.11


[[Subgroup]]: 2.3.5.7
Comma list: 385/384, 441/440, 234375/234256


[[Comma list]]: 1029/1024, 1959552/1953125
Mapping: {{mapping| 1 16 8 -2 17 | 0 -33 -13 11 -31 }}


{{Mapping|legend=1| 1 -11 -7 7 | 0 27 20 -9 }}
Optimal tunings:
: mapping generators: ~2, ~864/625
* WE: ~2 = 1200.4852{{c}}, ~1250/847 = 676.0634{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~1250/847 = 675.7966{{c}}


[[Optimal tuning]]s:
{{Optimal ET sequence|legend=0| 16, 71, 87, 103, 190 }}
* [[CTE]]: ~2 = 1200.000{{c}}, ~864/625 = 559.320{{c}}
: [[error map]]: {{val| 0.000 -0.317 +0.085 -2.705 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~864/625 = 559.295{{c}}
: error map: {{val| 0.000 -1.003 -0.423 -2.477 }}


{{Optimal ET sequence|legend=1| 15, …, 88, 103, 118, 221, 339d }}
Badness (Sintel): 1.35


[[Badness]] (Smith): 0.118640
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


=== 11-limit ===
Comma list: 385/384, 441/440, 625/624, 847/845
Subgroup: 2.3.5.7.11


Comma list: 385/384, 441/440, 43923/43750
Mapping: {{mapping| 1 16 8 -2 17 12 | 0 -33 -13 11 -31 -19 }}
 
Mapping: {{mapping| 1 -11 -7 7 -4 | 0 27 20 -9 16 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~242/175 = 559.327{{c}}
* WE: ~2 = 1200.4217{{c}}, ~77/52 = 676.0286{{c}}
* POTE: ~2 = 1200.000{{c}}, ~242/175 = 559.293{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~77/52 = 675.7967{{c}}


{{Optimal ET sequence|legend=0| 15, …, 88, 103, 118, 221e, 339de }}
{{Optimal ET sequence|legend=0| 16, 71, 87, 103, 190 }}


Badness (Smith): 0.035350
Badness (Sintel): 0.894


== Widefourth ==
== Other subgroup extensions ==
[[Subgroup]]: 2.3.5.7
=== 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.


[[Comma list]]: 1029/1024, 48828125/48771072
Subgroup: 2.3.7.13


{{Mapping|legend=1| 1 16 8 -2 | 0 -33 -13 11 }}
Comma list: 729/728, 1029/1024


[[Optimal tuning]]s:
Subgroup-val mapping: {{mapping| 1 1 3 0 | 0 3 -1 19 }}
* [[CTE]]: ~2 = 1200.000{{c}}, ~3125/2304 = 524.188{{c}}
: [[error map]]: {{val| 0.000 -0.154 -0.756 -2.759 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~3125/2304 = 524.210{{c}}
: error map: {{val| 0.000 -0.892 -1.047 -2.513 }}


{{Optimal ET sequence|legend=1| 16, 71, 87, 103, 190 }}
Gencom mapping: {{mapping| 1 1 0 3 0 0 | 0 3 0 -1 0 19 }}


[[Badness]] (Smith): 0.154117
Optimal tunings:  
* WE: ~2 = 1200.5057{{c}}, ~8/7 = 233.7200{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.6534{{c}}


=== 11-limit ===
{{Optimal ET sequence|legend=0| 5, 31f, 36, 77, 113, 827bdddff }}
Subgroup: 2.3.5.7.11


Comma list: 385/384, 441/440, 234375/234256
Badness (Sintel): 0.339


Mapping: {{mapping| 1 16 8 -2 17 | 0 -33 -13 11 -31 }}
==== 2.3.7.13.17 subgroup ====
Subgroup: 2.3.7.13.17


Optimal tunings:  
Comma list: 273/272, 729/728, 833/832
* CTE: ~2 = 1200.000{{c}}, ~847/625 = 524.183{{c}}
* POTE: ~2 = 1200.000{{c}}, ~847/625 = 524.210{{c}}


{{Optimal ET sequence|legend=0| 16, 71, 87, 103, 190 }}
Subgroup-val mapping: {{mapping| 1 1 3 0 0 | 0 3 -1 19 21 }}


Badness (Smith): 0.040785
Gencom mapping: {{mapping| 1 1 0 3 0 0 0 | 0 3 0 -1 0 19 21 }}


=== 13-limit ===
Optimal tunings:
Subgroup: 2.3.5.7.11.13
* WE: ~2 = 1200.5282{{c}}, ~8/7 = 233.6492{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.5776{{c}}


Comma list: 385/384, 441/440, 625/624, 847/845
{{Optimal ET sequence|legend=0| 5g, 31fg, 36, 113, 149 }}


Mapping: {{mapping| 1 16 8 -2 17 12 | 0 -33 -13 11 -31 -19 }}
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:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~65/48 = 524.183{{c}}
* WE: ~2 = 1200.3127{{c}}, ~8/7 = 233.6679{{c}}
* POTE: ~2 = 1200.000{{c}}, ~65/48 = 524.209{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.6091{{c}}


{{Optimal ET sequence|legend=0| 16, 71, 87, 103, 190 }}
{{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***** }}


Badness (Smith): 0.021636
<nowiki/>* Wart for 85


== References ==
== References ==


[[Category:Temperament clans]]
[[Category:Temperament clans]]
[[Category:Pages with mostly numerical content]]
[[Category:Gamelismic clan| ]] <!-- main article -->
[[Category:Gamelismic clan| ]] <!-- main article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
[[Category:Listen]]
[[Category:Listen]]
Retrieved from "https://en.xen.wiki/w/Gamelismic_clan"