Gamelismic clan: Difference between revisions

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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)3 × 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)3⋅(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]]

* ''[[Blacksmith]]'' (+28/27) → [[Limmic temperaments #Blacksmith|Limmic temperaments]]

* ''[[Trismegistus]]'' (+3125/3072) → [[Magic family #Trismegistus|Magic family]]

* [[Hemithirds]] (+3136/3125) → [[Hemimean clan #Hemithirds|Hemimean clan]]

* ''[[Gamity]]'' (+1071875/1062882) → [[Amity family #Gamity|Amity family]]

* ''[[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 ==

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~~: sval mapping generators: ~2, ~8/7~~

: mapping generators: ~2, ~8/7

: ~~[[gencom]]~~: [2 8/7~~; 1029/1024]~~

[[Optimal tuning]]s:

* [[~~CTE~~]]: ~2 = 1200.~~000~~, ~8/7 = 233.~~889~~

* [[WE]]: ~2 = 1200.4859{{c}}, ~8/7 = 233.7822{{c}}

: [[error map]]: {{val| 0.~~000~~ -0.~~288~~ -2.~~715~~ }}

: [[error map]]: {{val| +0.486 -0.123 -1.151 }}

* [[~~POTE~~]]: ~2 = 1200.000, ~8/7 = 233.~~688~~

* [[CWE]]: ~2 = 1200.000{{c}}, ~8/7 = 233.7474{{c}}

: error map: {{val| 0.000 -0.~~892~~ -2.~~513~~ }}

: error map: {{val| 0.000 -0.713 -2.573 }}

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

~~=== Radon ===~~

[[Badness]] (Sintel): 0.158

[[~~Subgroup~~]]: 2.~~3.7.11~~

[[~~Comma~~ list]]~~: 896~~/~~891~~, ~~1029~~/~~1024~~

=== Overview to extensions ===

==== Full 7-limit extensions ====

To the gamelisma itself we need to add the comma which appears next on the modified [[Normal lists #Normal interval list|normal comma list]] for the full 7-limit. The second comma on the list for mothra is [[81/80]], for rodan [[245/243]], for guiron [[32805/32768]], for gorgo [[36/35]], and for gidorah [[256/245]]. These all use ~8/7 as a generator, though in the case of gidorah that is the same as ~[[6/5]].

~~{{Mapping|legend=2| 1 1 3~~ 6 ~~| 0 3~~ -1 -~~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.

~~{{Mapping~~|~~legend=3~~| ~~1 1 0 3 6~~ | ~~0 3 0 -1 -13 }}~~

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]]

~~: [[gencom]]: [2 8/7; 896/891 1029/1024]~~

The rest are considered below.

~~[[Optimal tuning]]s:~~

==== Subgroup extensions ====

* [[CTE]]: ~2 ~~= 1200~~.~~000~~, ~~~8/~~7 ~~= 234~~.~~384~~

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.

~~: [[error map]]: {{val| 0~~.~~000 +1~~.~~197~~ -3.~~210 +1~~.~~691 }}~~

* [[~~POTE~~]]~~: ~2 = 1200~~.~~000, ~8/7 = 234.381~~

~~: error map: {{val~~| 0.~~000 +1~~.~~187 -3~~.~~206 +1~~.~~735 }}~~

=== Radon ===

~~=== Baladic ===~~

Radon is the no-fives version of [[rodan]], equating the diatonic major third to [[14/11]].

~~Baladic~~ is ~~a 2.3.7.13.17 subgroup temperament that attempts~~ to ~~approximate the Maqam Sikah Baladi scale. 36edo is an excellent baladic tuning~~.

[[Subgroup]]: 2.3.7.13

Subgroup: 2.3.7.11

[[Comma list]]: ~~169~~/~~168~~, 1029/1024

Comma list: 896/891, 1029/1024

~~[[Gencom]]~~: ~~[91/64 8/7; 169/168 1029/1024]~~

Subgroup-val mapping: {{mapping| 1 1 3 6 | 0 3 -1 -13 }}

~~[[Mapping|Sval~~ mapping]]: [{{~~val~~|~~2 2~~ 6 ~~7}}, {{val~~|0 3 -1 1}}]

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

~~[[Tp tuning|POL2 generator]]~~: ~8/7 = ~~233~~.~~6044~~

Optimal tunings:

* WE: ~2 = 1199.9708{{c}}, ~8/7 = 234.3748{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.3813{{c}}

{{Optimal ET sequence|legend=1| 10, 26, 36, ~~154…~~, ~~190…~~, ~~226…, 262…~~ }}

~~[[Tp tuning #T2 tuning|RMS error]]~~: 0.~~5452 cents~~

Badness (Sintel): 0.619

~~==== 2.3.7.13.17 ====~~

~~Subgroup: 2.3.7.13.17~~

~~[[Comma list]]: 169/168, 273/272, 289/288~~

~~{{Mapping|legend=2| 2 2 6 7 7 | 0 3 -1 1 3 }}~~

~~: sval mapping generators: ~17/12, ~8/7~~

~~[[Optimal tuning]]s:~~

* [[CTE]]: ~2 = 1200.000, ~8/7 = 234.138

* [[POTE]]: ~2 = 1200.000, ~8/7 = 233.616

~~{{Optimal ET sequence|legend=1| 10, 26, 36, 154f, 190ffg, 226ffg }}~~

== Mothra ==

Mothra tempers out [[81/80]] and finds the prime 5 at a stack of four fifths as does any temperament in the [[meantone family]]. It also tempers out [[1728/1715]], the orwellisma. It can be described as the {{nowrap|26 &~~amp;~~ 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]].

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]].

Note that mothra is also called '''cynder''' in the 7-limit, which can be a little confusing sometimes.

Its [[S-expression]]-based comma list is {[[1728/1715|S6/S7]], [[1029/1024|S7/S8]](, [[81/80|S6/S8 = S9]])}, taking advantage of the fact that [[81/80]] is a [[semiparticular]].

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

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~~: mapping generators: ~2, ~8/7~~

[[Optimal tuning]]s:

* [[~~CTE~~]]: ~2 = 1200.~~000~~, ~8/7 = 232.~~400~~

* [[WE]]: ~2 = 1200.9303{{c}}, ~8/7 = 232.3733{{c}}

: [[error map]]: {{val| 0.~~000~~ -4.~~756~~ +2.~~482 -~~1.~~226~~ }}

: [[error map]]: {{val| +0.930 -3.905 +2.165 +1.592 }}

* [[~~POTE~~]]: ~2 = 1200.~~000~~, ~8/7 = 232.~~193~~

* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 232.2514{{c}}

: error map: {{val| 0.000 -5.~~375~~ +0.~~005~~ -1.~~019~~ }}

: error map: {{val| 0.000 -5.520 +0.703 -1.077 }}

[[Algebraic generator]]: Rabrindanath, largest real root of ''x''8 - 3''x''2 + 1, or 232.0774 cents.

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[[Minimax tuning]]:

* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~8/7 = {{monzo| 0 0 1/12 }}

: [{{monzo| 1 0 0 0 ~~}}, {{monzo~~| 1 0 1/4 0 ~~}}, {{monzo~~| 0 0 1 0 ~~}}, {{monzo~~| 3 0 -1/12 0 }}]

: {{monzo list| 1 0 0 0 | 1 0 1/4 0 | 0 0 1 0 | 3 0 -1/12 0 }}

: [[eigenmonzo basis|~~eigenmonzo (~~unchanged-interval) basis]]: 2.5

: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5

[[Badness]] (~~Smith~~): 0.~~037146~~

[[Badness]] (Sintel): 0.940

=== Undecimal mothra ===

Undecimal mothra is the extension of 7-limit cynder which tempers out 385/384 as is natural in slendric temperaments. It is the simplest extension, supported within a reasonable tuning range (between [[26edo]] and 31edo), and is supported by the patent val of [[5edo]], which implies that it is better behaved as a cluster temperament. It is also notable for being supported by the just tuning of 8/7, and has a restriction to the 2.7.11 subgroup, namely [[amaranthine]], that is a microtemperament.

~~=== 11-limit ===~~

Subgroup: 2.3.5.7.11

Comma list: 81/80, 99/98, 385/384

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

Optimal tunings:

* ~~CTE~~: ~2 = ~~1200~~.~~000~~, ~8/7 = 232.~~203~~

* WE: ~2 = 1201.3979{{c}}, ~8/7 = 232.3010{{c}}

* ~~POTE~~: ~2 = 1200.~~000~~, ~8/7 = 232.~~031~~

* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 232.0621{{c}}

Badness (~~Smith~~): 0.~~025642~~

Badness (Sintel): 0.848

==== 13-limit ====

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Comma list: 81/80, 99/98, 105/104, 144/143

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

Optimal tunings:

* ~~CTE~~: ~2 = ~~1200~~.~~000~~, ~8/7 = ~~231~~.~~993~~

* WE: ~2 = 1201.0985{{c}}, ~8/7 = 232.0231{{c}}

* ~~POTE~~: ~2 = 1200.~~000~~, ~8/7 = 231.~~811~~

* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 231.8425{{c}}

Badness (~~Smith~~): 0.~~023954~~

Badness (Sintel): 0.990

==== 17-limit ====

Subgroup: 2.3.5.7.11.13.17

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

Optimal tunings:

* WE: ~2 = 1200.9734{{c}}, ~8/7 = 231.8960{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 231.7392{{c}}

Badness (Sintel): 1.00

==== 19-limit ====

Subgroup: 2.3.5.7.11.13.17.19

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

Optimal tunings:

* WE: ~2 = 1200.9663{{c}}, ~8/7 = 231.8393{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 231.6842{{c}}

~~; Music~~

Badness (Sintel): 1.05

* ''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

=== Mosura ===

The [[S-expression]]-based comma list of mosura suggests it might be the most natural extension of 7-limit cynder to the 11-limit: {[[1728/1715|S6/S7]], [[1029/1024|S7/S8]], ([[81/80|S6/S8 = S9]],) [[176/175|S8/S10]]}.

The [[S-expression]]-based comma list of mosura suggests it might be the most natural extension of 7-limit cynder to the 11-limit: {[[1728/1715|S6/S7]], [[1029/1024|S7/S8]], ([[81/80|S6/S8 = S9]]), [[176/175|S8/S10]]}.

Subgroup: 2.3.5.7.11

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Comma list: 81/80, 176/175, 540/539

~~Mapping:~~ {{~~mapping~~| 1 1 0 3 -1 | 0 3 12 -1 23 }}

Optimal tunings:

* ~~CTE~~: ~2 = 1200.~~000~~, ~8/7 = 232.~~557~~

* WE: ~2 = 1200.7675{{c}}, ~8/7 = 232.5673{{c}}

* ~~POTE~~: ~2 = 1200.~~000~~, ~8/7 = 232.~~419~~

* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 232.4567{{c}}

Badness (~~Smith~~): 0.~~031334~~

Badness (Sintel): 1.04

==== 13-limit ====

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

~~Mapping:~~ {{~~mapping~~| 1 1 0 3 -1 7 | 0 3 12 -1 23 -17 }}

Optimal tunings:

* ~~CTE~~: ~2 = ~~1200~~.~~000~~, ~8/7 = 232.~~635~~

* WE: ~2 = 1199.9347{{c}}, ~8/7 = 232.6275{{c}}

* ~~POTE~~: ~2 = 1200.~~000~~, ~8/7 = 232.~~640~~

* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 232.6392{{c}}

Badness (~~Smith~~): 0.~~036857~~

Badness (Sintel): 1.52

==== 17-limit ====

Subgroup: 2.3.5.7.11.13.17

Comma list: 81/80, 144/143, 176/175, 189/187, 196/195

Optimal tunings:

* WE: ~2 = 1199.7124{{c}}, ~8/7 = 232.6376{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 232.6917{{c}}

Badness (Sintel): 1.53

==== 19-limit ====

Subgroup: 2.3.5.7.11.13.17.19

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

Optimal tunings:

* WE: ~2 = 1199.4885{{c}}, ~8/7 = 232.6310{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 232.7287{{c}}

=== ~~Cynder~~ ===

Badness (Sintel): 1.50

=== Cyndra ===

Subgroup: 2.3.5.7.11

Comma list: 45/44, 81/80, 1029/1024

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

Optimal tunings:

* ~~CTE~~: ~2 = ~~1200~~.~~000~~, ~8/7 = 231.~~566~~

* WE: ~2 = 1201.1585{{c}}, ~8/7 = 231.5404{{c}}

* ~~POTE~~: ~2 = 1200.~~000~~, ~8/7 = 231.~~317~~

* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 231.3850{{c}}

Badness (~~Smith~~): 0.~~055706~~

Badness (Sintel): 1.84

==== 13-limit ====

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Comma list: 45/44, 78/77, 81/80, 640/637

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

Optimal tunings:

* ~~CTE~~: ~2 = ~~1200~~.~~000~~, ~8/7 = 231.~~546~~

* WE: ~2 = 1201.1152{{c}}, ~8/7 = 231.5079{{c}}

* ~~POTE~~: ~2 = 1200.~~000~~, ~8/7 = 231.~~293~~

* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 231.3612{{c}}

Badness (~~Smith~~): 0.~~034124~~

Badness (Sintel): 1.41

== Rodan ==

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: ''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 &~~amp;~~ 46}} temperament. This temperament extends neatly to the 13-limit, though the perfect fifth is sharper than ideal for slendric.

Rodan tempers out 245/243 and can be described as the {{nowrap| 41 & 46 }} temperament. This temperament is more accurate than mothra and extends neatly to the 13-limit, though the perfect fifth is sharper than ideal for slendric. [[87edo]] is excellent for this, with the 17\87 generator missing the 13-limit CWE tuning by less than a millicent.

[[Subgroup]]: 2.3.5.7

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[[Optimal tuning]]s:

* [[~~CTE~~]]: ~2 = 1200.~~000~~, ~8/7 = 234.~~450~~

* [[WE]]: ~2 = 1200.2146{{c}}, ~8/7 = 234.4587{{c}}

: [[error map]]: {{val| 0.~~000~~ +1.~~396~~ -0.~~660~~ -3.~~276~~ }}

: [[error map]]: {{val| +0.215 +1.636 -0.731 -2.641 }}

* [[~~POTE~~]]: ~2 = 1200.~~000~~, ~8/7 = 234.~~417~~

* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 234.4259{{c}}

: error map: {{val| 0.000 +1.~~295~~ -1.~~229~~ -3.~~243~~ }}

: error map: {{val| 0.000 +1.323 -1.073 -3.252 }}

[[Minimax tuning]]:

* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~8/7 = {{monzo| 2/9 0 1/18 -1/18 }}

: {{monzo list| 1 0 0 0 | 5/3 0 1/6 -1/6 | 25/9 0 17/18 -17/18 | 25/9 0 -1/18 1/18 }}

: [[eigenmonzo basis|~~eigenmonzo (~~unchanged-interval) basis]]: 2.7/5

: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5

[[Algebraic generator]]: larger root of 20''x''2 - 36''x'' + 15, or (9 + √6)/10.

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[[Badness]] (~~Smith~~): 0.~~037112~~

[[Badness]] (Sintel): 0.939

=== 11-limit ===

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Comma list: 245/243, 385/384, 441/440

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

Optimal tunings:

* ~~CTE~~: ~2 = 1200.~~000~~, ~8/7 = 234.~~463~~

* WE: ~2 = 1200.0553{{c}}, ~8/7 = 234.4695{{c}}

* ~~POTE~~: ~2 = 1200.~~000~~, ~8/7 = 234.~~459~~

* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.4594{{c}}

Minimax tuning:

* 11-odd-limit: ~8/7 = {{monzo| 4/19 2/19 0 0 -1/19 }}

: [{{monzo| 1 0 0 0 0 }}, {{monzo| 31/19 6/19 0 0 -3/19 }}, {{monzo| 49/19 34/19 0 0 -17/19 }}, {{monzo| 53/19 -2/19 0 0 1/19 }}, {{monzo| 62/19 -26/19 0 0 13/19 }}]

: ~~eigenmonzo (~~unchanged-interval) basis: 2.11/9

: unchanged-interval (eigenmonzo) basis: 2.11/9

Algebraic generator: positive root of ''x''2 + 16''x'' - 31, or √95 - 8.

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Badness (~~Smith~~): 0.~~023093~~

Badness (Sintel): 0.763

==== 13-limit ====

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Comma list: 196/195, 245/243, 352/351, 364/363

~~Mapping:~~ {{~~mapping~~| 1 1 -1 3 6 8 | 0 3 17 -1 -13 -22 }}

Optimal tunings:

* ~~CTE~~: ~2 = ~~1200~~.~~000~~, ~8/7 = 234.~~482~~

* WE: ~2 = 1199.9868{{c}}, ~8/7 = 234.4796{{c}}

* ~~POTE~~: ~2 = 1200.~~000~~, ~8/7 = 234.~~482~~

* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.4822{{c}}

Minimax tuning:

* 13- and 15-odd-limit: ~8/7 = {{monzo| 3/14 1/14 0 0 0 -1/28 }}

: ~~eigenmonzo (~~unchanged-interval) basis: 2.13/9

: unchanged-interval (eigenmonzo) basis: 2.13/9

Algebraic generator: Gatetone, positive root of 4''x''6 - 7''x'' - 1. Recurrence converges slowly.

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Line 325:

Badness (~~Smith~~): 0.~~018448~~

Badness (Sintel): 0.762

===== 17-limit =====

Line 302:

Line 332:

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:

* ~~CTE~~: ~2 = ~~1200~~.~~000~~, ~8/7 = 234.~~532~~

* WE: ~2 = 1199.8331{{c}}, ~8/7 = 234.4919{{c}}

* ~~POTE~~: ~2 = 1200.~~000~~, ~8/7 = 234.~~524~~

* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.5254{{c}}

Minimax tuning:

* 17-odd-limit: ~8/7 = {{monzo| 3/13 1/13 0 0 0 0 -1/26 }}

: ~~eigenmonzo (~~unchanged-interval) basis: 2.17/9

: unchanged-interval (eigenmonzo) basis: 2.17/9

Badness (~~Smith~~): 0.~~016743~~

Badness (Sintel): 0.853

==== Aerodactyl ====

Line 321:

Line 351:

Comma list: 91/90, 245/243, 385/384, 441/440

~~Mapping:~~ {{~~mapping~~| 1 1 -1 3 6 -1 | 0 3 17 -1 -13 24 }}

Optimal tunings:

* ~~CTE~~: ~2 = 1200.~~000~~, ~8/7 = 234.~~670~~

* WE: ~2 = 1200.2997{{c}}, ~8/7 = 234.6972{{c}}

* ~~POTE~~: ~2 = 1200.~~000~~, ~8/7 = 234.~~639~~

* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.6439{{c}}

Badness (~~Smith~~): 0.~~033986~~

Badness (Sintel): 1.40

=== Aerodino ===

Line 336:

Line 366:

Comma list: 176/175, 245/243, 1029/1024

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

Optimal tunings:

* ~~CTE~~: ~2 = ~~1200~~.~~000~~, ~8/7 = 234.~~719~~

* WE: ~2 = 1199.9179{{c}}, ~8/7 = 234.7123{{c}}

* ~~POTE~~: ~2 = 1200.~~000~~, ~8/7 = 234.~~728~~

* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.7256{{c}}

Badness (~~Smith~~): 0.~~054294~~

Badness (Sintel): 1.79

==== 13-limit ====

Line 351:

Line 381:

Comma list: 91/90, 176/175, 245/243, 847/845

~~Mapping:~~ {{~~mapping~~| 1 1 -1 3 -3 -1 | 0 3 17 -1 33 24 }}

Optimal tunings:

* ~~CTE~~: ~2 = 1200.~~000~~, ~8/7 = 234.~~786~~

* WE: ~2 = 1200.0242{{c}}, ~8/7 = 234.7863{{c}}

* ~~POTE~~: ~2 = 1200.~~000~~, ~8/7 = 234.~~782~~

* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.7824{{c}}

Badness (~~Smith~~): 0.~~035836~~

Badness (Sintel): 1.48

=== Varan ===

Line 366:

Line 396:

Comma list: 100/99, 245/243, 1029/1024

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

Optimal tunings:

* ~~CTE~~: ~2 = 1200.~~000~~, ~8/7 = 234.~~197~~

* WE: ~2 = 1200.3738{{c}}, ~8/7 = 234.2174{{c}}

* ~~POTE~~: ~2 = 1200.~~000~~, ~8/7 = 234.~~145~~

* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.1586{{c}}

Badness (~~Smith~~): 0.~~044937~~

Badness (Sintel): 1.49

==== 13-limit ====

Line 381:

Line 411:

Comma list: 100/99, 105/104, 245/243, 352/351

~~Mapping:~~ {{~~mapping~~| 1 1 -1 3 -2 0 | 0 3 17 -1 28 19 }}

Optimal tunings:

* ~~CTE~~: ~2 = 1200.~~000~~, ~8/7 = 234.~~111~~

* WE: ~2 = 1200.1389{{c}}, ~8/7 = 234.1162{{c}}

* ~~POTE~~: ~2 = 1200.~~000~~, ~8/7 = 234.~~089~~

* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.0946{{c}}

Badness (~~Smith~~): 0.~~032284~~

Badness (Sintel): 1.33

== Guiron ==

Guiron tempers out the [[schisma]], and finds the prime 5 at the diminished fourth as does any temperament in the [[schismatic family]]. It can be described as the {{nowrap|36 &~~amp;~~ 41}} temperament. It is more complex than rodan, but the optimal tuning is closer to optimal slendric.

Guiron tempers out the [[schisma]], and finds the prime 5 at the diminished fourth as does any temperament in the [[schismatic family]]. It can be described as the {{nowrap| 36 & 41 }} temperament. It is more complex than rodan, but the optimal tuning is closer to optimal slendric.

[[Subgroup]]: 2.3.5.7

Line 399:

Line 429:

~~: mapping generators: ~2, ~8/7~~

[[Optimal tuning]]s:

* [[~~CTE~~]]: ~2 = 1200.~~000~~, ~8/7 = 233.~~903~~

* [[WE]]: ~2 = 1200.3395{{c}}, ~8/7 = 233.9963{{c}}

: [[error map]]: {{val| 0.~~000 -~~0.~~246~~ +0.~~012~~ -2.~~729~~ }}

: [[error map]]: {{val| +0.340 +0.374 +0.151 -1.804 }}

* [[~~POTE~~]]: ~2 = 1200.~~000~~, ~8/7 = 233.~~930~~

* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 233.9239{{c}}

: error map: {{val| 0.000 -0.~~165~~ -0.~~637~~ -2.~~756~~ }}

: error map: {{val| 0.000 -0.183 -0.487 -2.750 }}

[[Minimax tuning]]:

* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~8/7 = {{monzo| 7/24 0 -1/24 }}

: {{monzo list| 1 0 0 0 | 15/8 0 -1/8 0 | 0 0 1 0 | 65/24 0 1/24 0 }}

: [[eigenmonzo basis|~~eigenmonzo (~~unchanged-interval) basis]]: 2.5

: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5

[[Badness]] (~~Smith~~): 0.~~047544~~

[[Badness]] (Sintel): 1.20

=== 11-limit ===

Line 422:

Line 450:

Comma list: 385/384, 441/440, 10976/10935

~~Mapping:~~ {{~~mapping~~| 1 1 7 3 -2 | 0 3 -24 -1 28 }}

~~: mapping generators: ~2, ~8/7~~

Optimal tunings:

* ~~CTE~~: ~2 = 1200.~~000~~, ~8/7 = 233.~~930~~

* WE: ~2 = 1200.3453{{c}}, ~8/7 = 233.9988{{c}}

* ~~POTE~~: ~2 = 1200.~~000~~, ~8/7 = 233.~~931~~

* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.9312{{c}}

Minimax tuning:

* 11-odd-limit: ~8/7 = {{monzo| 7/24 0 -1/24 }}

: [{{monzo| 1 0 0 0 0 }}, {{monzo| 15/8 0 -1/8 0 0 }}, {{monzo| 0 0 1 0 0 }}, {{monzo| 65/24 0 1/24 0 0 }}, {{monzo| 37/6 0 -7/6 0 0 }}]

: ~~eigenmonzo (~~unchanged-interval) basis: 2.5

: unchanged-interval (eigenmonzo) basis: 2.5

Badness (~~Smith~~): 0.~~026648~~

Badness (Sintel): 0.881

=== 13-limit ===

Line 444:

Line 470:

Comma list: 196/195, 352/351, 385/384, 729/728

~~Mapping:~~ {{~~mapping~~| 1 1 7 3 -2 0 | 0 3 -24 -1 28 19 }}

~~: mapping generators: ~2, ~8/7~~

Optimal tunings:

* ~~CTE~~: ~2 = 1200.~~000~~, ~8/7 = 233.~~902~~

* WE: ~2 = 1200.1222{{c}}, ~8/7 = 233.9228{{c}}

* ~~POTE~~: ~2 = 1200.~~000~~, ~8/7 = 233.~~899~~

* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.8994{{c}}

Badness (~~Smith~~): 0.~~028444~~

Badness (Sintel): 1.18

== Gorgo ==

: ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum #Laconic]].''

Gorgo tempers the generator of ~8/7 together with ~10/9. It can be described as the {{nowrap| 16 & 21 }} temperament.

~~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.

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

Line 472:

Line 495:

[[Optimal tuning]]s:

* [[~~CTE~~]]: ~2 = 1200.~~000~~, ~8/7 = 228.~~724~~

* [[WE]]: ~2 = 1200.9847{{c}}, ~8/7 = 228.5210{{c}}

: [[error map]]: {{val| 0.~~000~~ -15.~~782~~ +14.~~756~~ +2.~~450~~ }}

: [[error map]]: {{val| +0.985 -15.407 +14.318 +5.607 }}

* [[~~POTE~~]]: ~2 = 1200.~~000~~, ~8/7 = 228.~~334~~

* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 228.4371{{c}}

: error map: {{val| 0.000 -16.~~954~~ +12.~~022~~ +2.~~840~~ }}

: error map: {{val| 0.000 -16.644 +12.746 +2.737 }}

[[Badness]] (~~Smith~~): 0.~~060663~~

[[Badness]] (Sintel): 1.54

=== 11-limit ===

Line 486:

Line 509:

Comma list: 36/35, 45/44, 1029/1024

~~Mapping:~~ {{~~mapping~~| 1 1 1 3 1 | 0 3 7 -1 13 }}

Optimal tunings:

* ~~CTE~~: ~2 = ~~1200~~.~~000~~, ~8/7 = 227.~~833~~

* WE: ~2 = 1201.3609{{c}}, ~8/7 = 227.6312{{c}}

* ~~POTE~~: ~2 = 1200.~~000~~, ~8/7 = 227.~~373~~

* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 227.4955{{c}}

Badness (~~Smith~~): 0.~~049500~~

Badness (Sintel): 1.64

==== 13-limit ====

Line 501:

Line 524:

Comma list: 27/26, 36/35, 45/44, 507/500

~~Mapping:~~ {{~~mapping~~| 1 1 1 3 1 2 | 0 3 7 -1 13 9 }}

Optimal tunings:

* ~~CTE~~: ~2 = ~~1200~~.~~000~~, ~8/7 = 227.~~633~~

* WE: ~2 = 1201.0996{{c}}, ~8/7 = 227.4378{{c}}

* ~~POTE~~: ~2 = 1200.~~000~~, ~8/7 = 227.~~230~~

* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 227.3327{{c}}

Badness (~~Smith~~): 0.~~032664~~

Badness (Sintel): 1.35

=== Spartan ===

Line 516:

Line 539:

Comma list: 36/35, 56/55, 1029/1024

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

Optimal tunings:

* ~~CTE~~: ~2 = ~~1200~~.~~000~~, ~8/7 = 229.~~420~~

* WE: ~2 = 1198.9344{{c}}, ~8/7 = 229.3316{{c}}

* ~~POTE~~: ~2 = 1200.~~000~~, ~8/7 = 229.~~535~~

* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 229.5124{{c}}

Badness (~~Smith~~): 0.~~062683~~

Badness (Sintel): 2.07

==== 13-limit ====

Line 531:

Line 554:

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

~~Mapping:~~ {{~~mapping~~| 1 1 1 3 5 2 | 0 3 7 -1 -8 9 }}

Optimal tunings:

* ~~CTE~~: ~2 = ~~1200~~.~~000~~, ~8/7 = 228.~~758~~

* WE: ~2 = 1198.3002{{c}}, ~8/7 = 228.7341{{c}}

* ~~POTE~~: ~2 = 1200.~~000~~, ~8/7 = 229.~~059~~

* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 229.0044{{c}}

Badness (~~Smith~~): 0.~~047071~~

Badness (Sintel): 1.95

; Music

Line 556:

Line 579:

[[Optimal tuning]]s:

* [[~~CTE~~]]: ~2 = ~~1200~~.~~000~~, ~8/7 = ~~227~~.~~100~~

* [[WE]]: ~2 = 1192.4932{{c}}, ~8/7 = 229.3187{{c}}

: [[error map]]: {{val| 0.~~000~~ -20.~~655~~ +67.~~886 +4~~.~~074~~ }}

: [[error map]]: {{val| -7.507 -21.506 +57.310 -20.665 }}

* [[~~POTE~~]]: ~2 = 1200.~~000~~, ~8/7 = ~~230~~.~~762~~

* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 229.6649{{c}}

: error map: {{val| 0.000 -9.~~668~~ +75.~~211~~ +0.~~412~~ }}

: error map: {{val| 0.000 -12.960 +73.016 +1.509 }}

[[Badness]] (~~Smith~~): 0.~~062262~~

[[Badness]] (Sintel): 1.58

== Oncle ==

: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Oncle]].''

Oncle can be described as the {{nowrap|31 &~~amp;~~ 36c}} ~~temperamnet~~.

Oncle can be described as the {{nowrap| 31 & 36c }} temperament.

[[Subgroup]]: 2.3.5.7

Line 577:

Line 600:

[[Optimal tuning]]s:

* [[~~CTE~~]]: ~2 = ~~1200~~.~~000~~, ~8/7 = 232.~~383~~

* [[WE]]: ~2 = 1201.2246{{c}}, ~8/7 = 232.7354{{c}}

: [[error map]]: {{val| 0.~~000~~ -4.~~807~~ -1.~~585 -1~~.~~209~~ }}

: [[error map]]: {{val| +1.225 -2.524 -0.939 +2.112 }}

* [[~~POTE~~]]: ~2 = 1200.~~000~~, ~8/7 = 232.~~498~~

* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 232.4718{{c}}

: error map: {{val| 0.000 -4.~~461~~ -3.~~778~~ -1.~~324~~ }}

: error map: {{val| 0.000 -4.539 -3.279 -1.298 }}

[[Badness]] (~~Smith~~): 0.~~088384~~

[[Badness]] (Sintel): 2.24

== Archaeotherium ==

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

Archaeotherium can be described as the {{nowrap|21 &~~amp;~~ 26}} ~~temperamnet~~.

Archaeotherium can be described as the {{nowrap| 21 & 26 }} temperament.

[[Subgroup]]: 2.3.5.7

Line 598:

Line 621:

[[Optimal tuning]]s:

* [[~~CTE~~]]: ~2 = ~~1200~~.~~000~~, ~8/7 = ~~229~~.~~951~~

* [[WE]]: ~2 = 1202.7179{{c}}, ~8/7 = 230.7800{{c}}

: [[error map]]: {{val| 0.~~000~~ -12.~~102~~ -5.~~626~~ +1.~~223~~ }}

: [[error map]]: {{val| +2.718 -6.897 -3.644 +8.548 }}

* [[~~POTE~~]]: ~2 = 1200.~~000~~, ~8/7 = 230.~~258~~

* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 230.1909{{c}}

: error map: {{val| 0.000 -11.~~180~~ -9.~~933~~ +0.~~916~~ }}

: error map: {{val| 0.000 -11.382 -8.986 +0.983 }}

[[Badness]] (~~Smith~~): 0.~~146306~~

[[Badness]] (Sintel): 3.70

== Clyndro ==

~~{{See also| Mavila family }}~~

Clyndro tempers out [[135/128]] and finds the interval class of 5 at a stack of -3 fifths as does any temperament in the [[mavila family]]. It can be described as the {{nowrap| 11 & 16 }} temperament.

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 &~~amp;~~ 16}} ~~temperamnet~~.

[[Subgroup]]: 2.3.5.7

Line 619:

Line 640:

[[Optimal tuning]]s:

* [[~~CTE~~]]: ~2 = ~~1200~~.~~000~~, ~8/7 = ~~225~~.~~752~~

* [[WE]]: ~2 = 1205.6135{{c}}, ~8/7 = 227.5283{{c}}

: [[error map]]: {{val| 0.~~000~~ -24.~~699~~ -18.~~081~~ +5.~~422~~ }}

: [[error map]]: {{val| +5.613 -13.757 -11.614 +20.486 }}

* [[~~POTE~~]]: ~2 = 1200.~~000~~, ~8/7 = 226.~~469~~

* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 226.3207{{c}}

: error map: {{val| 0.000 -22.~~548~~ -24.~~534~~ +4.~~705~~ }}

: error map: {{val| 0.000 -22.993 -23.200 +4.853 }}

[[Badness]] (~~Smith~~): 0.~~159179~~

[[Badness]] (Sintel): 4.03

=== 11-limit ===

Line 633:

Line 654:

Comma list: 33/32, 45/44, 352/343

~~Mapping:~~ {{~~mapping~~| 1 1 4 3 4 | 0 3 -9 -1 -3 }}

Optimal tunings:

* ~~CTE~~: ~2 = ~~1200~~.~~000~~, ~8/7 = ~~225~~.~~384~~

* WE: ~2 = 1206.2134{{c}}, ~8/7 = 227.6004{{c}}

* ~~POTE~~: ~2 = 1200.~~000~~, ~8/7 = 226.~~428~~

* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 226.2421{{c}}

Badness (~~Smith~~): 0.~~069703~~

Badness (Sintel): 2.30

== Miracle ==

Line 647:

Line 668:

: ''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 &~~amp;~~ 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.

Miracle is one of the most important entries of this temperament clan. It tempers out [[225/224]], splitting the ~8/7 generator of slendric into [[15/14]]~[[16/15]], and can be described as the {{nowrap| 31 & 41 }} temperament. Its ploidacot is hexacot. It is then extremely natural to equate the neutral third, three generators up, to [[11/9]] and thereby extend miracle to the full [[11-limit]] with essentially no further damage. [[72edo]] makes for an excellent tuning.

[[Subgroup]]: 2.3.5.7

Line 654:

Line 675:

: mapping generator: ~2, ~15/14

[[Optimal tuning]]s:

* [[~~CTE~~]]: ~2 = 1200.~~000~~, ~15/14 = 116.~~677~~

* [[WE]]: ~2 = 1200.8209{{c}}, ~15/14 = 116.7550{{c}}

: [[error map]]: {{val| 0.~~000~~ -1.~~892~~ -3.~~054 -2~~.~~180~~ }}

: [[error map]]: {{val| +0.821 -0.604 -1.136 +0.127 }}

* [[~~POTE~~]]: ~2 = 1200.~~000~~, ~15/14 = 116.~~675~~

* [[CWE]]: ~2 = 1200.0000{{c}}, ~15/14 = 116.6756{{c}}

: error map: {{val| 0.000 -1.~~904~~ -3.~~040~~ -2.~~176~~ }}

: error map: {{val| 0.000 -1.901 -3.043 -2.177 }}

[[Minimax tuning]]:

* [[7-odd-limit]]: ~15/14 = {{monzo| 2/13 1/13 -1/13 }}

: {{monzo list| 1 0 0 0 | 25/13 6/13 -6/13 0 | 25/13 -7/13 7/13 0 | 35/13 -2/13 2/13 0 }}

: [[eigenmonzo basis|~~eigenmonzo (~~unchanged-interval) basis]]: 2.5/3

: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5/3

* [[9-odd-limit]]: ~15/14 = {{monzo| 1/19 2/19 -1/19 }}

: {{monzo list| 1 0 0 0 | 25/19 12/19 -6/19 0 | 50/19 -14/19 7/19 0 | 55/19 -4/19 2/19 0 }}

: [[eigenmonzo basis|~~eigenmonzo (~~unchanged-interval) basis]]: 2.9/5

: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5

[[Tuning ranges]]:

Line 680:

Line 700:

[[Badness]] (~~Smith~~): 0.~~016742~~

[[Badness]] (Sintel): 0.424

=== 11-limit ===

Line 687:

Line 707:

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

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

Optimal tunings:

* ~~CTE~~: ~2 = 1200.~~000~~, ~15/14 = 116.~~711~~

* WE: ~2 = 1200.7626{{c}}, ~15/14 = 116.7069{{c}}

* ~~POTE~~: ~2 = 1200.~~000~~, ~15/14 = 116.~~633~~

* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.6469{{c}}

Minimax tuning:

* 11-odd-limit: ~15/14 = {{monzo| 1/19 2/19 -1/19 }}

: [{{monzo| 1 0 0 0 0 }}, {{monzo| 25/19 12/19 -6/19 0 0 }}, {{monzo| 50/19 -14/19 7/19 0 0 }}, {{monzo| 55/19 -4/19 2/19 0 0 }}, {{monzo| 53/19 30/19 -15/19 0 0 }}]

: ~~eigenmonzo (~~unchanged-interval) basis: 2.9/5

: unchanged-interval (eigenmonzo) basis: 2.9/5

Tuning ranges:

Line 706:

Line 726:

{{Optimal ET sequence|legend=0| 10, 21e, 31, 41, 72, 247c, 319bcde, 391bcde, 463bccde }}

Badness (~~Smith~~): 0.~~010684~~

Badness (Sintel): 0.353

==== Miraculous ====

Line 713:

Line 733:

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:

* ~~CTE~~: ~2 = 1200.~~000~~, ~15/14 = 116.~~758~~

* WE: ~2 = 1200.1267{{c}}, ~15/14 = 116.7596{{c}}

* ~~POTE~~: ~2 = 1200.~~000~~, ~15/14 = 116.~~747~~

* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.7488{{c}}

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

Badness (~~Smith~~): 0.~~018669~~

Badness (Sintel): 0.771

===== 17-limit =====

Line 728:

Line 748:

Comma list: 105/104, 120/119, 144/143, 154/153, 170/169

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

Optimal tunings:

* ~~CTE~~: ~2 = ~~1200~~.~~000~~, ~15/14 = 116.~~742~~

* WE: ~2 = 1199.6759{{c}}, ~15/14 = 116.7378{{c}}

* ~~POTE~~: ~2 = 1200.~~000~~, ~15/14 = 116.~~769~~

* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.7657{{c}}

Badness (~~Smith~~): 0.~~017084~~

Badness (Sintel): 0.870

===== 19-limit =====

Subgroup: 2.3.5.7.11.13.17.19

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

===== 23-limit =====

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 105/104, 120/119, 144/143, 154/153, 161/160, 170/169, 210/209

==== Benediction ====

Line 743:

Line 777:

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

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

Optimal tunings:

* ~~CTE~~: ~2 = ~~1200~~.~~000~~, ~15/14 = 116.~~541~~

* WE: ~2 = 1199.8601{{c}}, ~15/14 = 116.6572{{c}}

* ~~POTE~~: ~2 = 1200.~~000~~, ~15/14 = 116.~~574~~

* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.5688{{c}}

Badness (~~Smith~~): 0.~~015715~~

Badness (Sintel): 0.649

===== 17-limit =====

Line 758:

Line 792:

Comma list: 225/224, 243/242, 273/272, 351/350, 375/374

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

Optimal tunings:

* ~~CTE~~: ~2 = 1200.~~000~~, ~15/14 = 116.~~529~~

* WE: ~2 = 1200.8328{{c}}, ~15/14 = 116.6661{{c}}

* ~~POTE~~: ~2 = 1200.~~000~~, ~15/14 = 116.~~585~~

* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.5774{{c}}

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

Badness (~~Smith~~): 0.~~012537~~

Badness (Sintel): 0.639

===== 19-limit =====

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 210/209, 225/224, 243/242, 273/272, 286/285, 375/374

===== 23-limit =====

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 162/161, 210/209, 225/224, 231/230, 243/242, 273/272, 286/285

==== Manna ====

Line 773:

Line 821:

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

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

Optimal tunings:

* ~~CTE~~: ~2 = 1200.~~000~~, ~15/14 = 116.~~814~~

* WE: ~2 = 1200.7564{{c}}, ~15/14 = 116.8129{{c}}

* ~~POTE~~: ~2 = 1200.~~000~~, ~15/14 = 116.~~739~~

* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.7528{{c}}

Badness (~~Smith~~): 0.~~017012~~

Badness (Sintel): 0.703

===== 17-limit =====

Line 788:

Line 836:

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

~~Mapping:~~ {{~~mapping~~| 1 1 3 3 2 0 0 | 0 6 -7 -2 15 38 42 }}

Optimal tunings:

* ~~CTE~~: ~2 = 1200.~~000~~, ~15/14 = 116.~~802~~

* WE: ~2 = 1200.7570{{c}}, ~15/14 = 116.8011{{c}}

* ~~POTE~~: ~2 = 1200.~~000~~, ~15/14 = 116.~~727~~

* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.7408{{c}}

Badness (~~Smith~~): 0.~~014680~~

Badness (Sintel): 0.748

===== 19-limit =====

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 210/209, 225/224, 243/242, 273/272, 325/324, 343/342

===== 23-limit =====

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 210/209, 225/224, 243/242, 273/272, 300/299, 325/324, 343/342

==== Semimiracle ====

Line 803:

Line 865:

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

~~Mapping:~~ {{~~mapping~~| 2 2 6 6 4 7 | 0 6 -7 -2 15 2 }}

: mapping generators: ~55/39, ~15/14

Optimal tunings:

* ~~CTE~~: ~55/39 = 600.~~000~~, ~15/14 = 116.~~735~~

* WE: ~55/39 = 600.4844{{c}}, ~15/14 = 116.7182{{c}}

* ~~POTE~~: ~55/39 = 600.~~000~~, ~15/14 = 116.~~624~~

* CWE: ~55/39 = 600.0000{{c}}, ~15/14 = 116.6413{{c}}

Badness (~~Smith~~): 0.~~024622~~

Badness (Sintel): 1.02

===== 17-limit =====

Line 820:

Line 881:

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

~~Mapping:~~ {{~~mapping~~| 2 2 6 6 4 7 7 | 0 6 -7 -2 15 2 6 }}

Optimal tunings:

* ~~CTE~~: ~17/12 = 600.~~000~~, ~15/14 = 116.~~771~~

* WE: ~17/12 = 600.5042{{c}}, ~15/14 = 116.7264{{c}}

* ~~POTE~~: ~17/12 = 600.~~000~~, ~15/14 = 116.~~628~~

* CWE: ~17/12 = 600.0000{{c}}, ~15/14 = 116.6485{{c}}

Badness (~~Smith~~): 0.~~016130~~

Badness (Sintel): 0.822

===== 19-limit =====

Subgroup: 2.3.5.7.11.13.17.19

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

===== 23-limit =====

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 169/168, 208/207, 210/209, 221/220, 225/224, 243/242, 273/272

==== Hemisecordite ====

Line 835:

Line 910:

Comma list: 225/224, 243/242, 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~~, ~27/26 = 58.~~337~~

* WE: ~2 = 1200.6969{{c}}, ~27/26 = 58.3217{{c}}

* ~~POTE~~: ~2 = 1200.~~000~~, ~27/26 = 58.~~288~~

* CWE: ~2 = 1200.0000{{c}}, ~27/26 = 58.2964{{c}}

Badness (~~Smith~~): 0.~~025589~~

Badness (Sintel): 1.06

===== 17-limit =====

Line 852:

Line 926:

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~~, ~27/26 = 58.~~312~~

* WE: ~2 = 1200.6557{{c}}, ~27/26 = 58.2932{{c}}

* ~~POTE~~: ~2 = 1200.~~000~~, ~27/26 = 58.~~261~~

* CWE: ~2 = 1200.0000{{c}}, ~27/26 = 58.2702{{c}}

Badness (~~Smith~~): 0.~~022535~~

Badness (Sintel): 1.15

===== 19-limit =====

Subgroup: 2.3.5.7.11.13.17.19

Comma list:

===== 23-limit =====

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list:

===== Semihemisecordite =====

Line 867:

Line 955:

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 generators: ~17/12, ~27/26

Optimal tunings:

* ~~CTE~~: ~17/12 = 600.~~000~~, ~27/26 = 58.~~350~~

* WE: ~17/12 = 600.3951{{c}}, ~27/26 = 58.3260{{c}}

* ~~POTE~~: ~17/12 = 600.~~000~~, ~27/26 = 58.~~288~~

* CWE: ~17/12 = 600.0000{{c}}, ~27/26 = 58.2974{{c}}

Badness (~~Smith~~): 0.~~046958~~

Badness (Sintel): 2.39

====== 19-limit ======

Line 884:

Line 971:

Comma list: 209/208, 225/224, 243/242, 289/288, 361/360, 385/384

~~Mapping:~~ {{~~mapping~~| 2 2 6 6 4 4 7 8 | 0 12 -14 -4 30 35 12 5 }}

Optimal tunings:

* ~~CTE~~: ~17/12 = 600.~~000~~, ~27/26 = 58.~~356~~

* WE: ~17/12 = 600.4418{{c}}, ~27/26 = 58.3255{{c}}

* ~~POTE~~: ~17/12 = 600.~~000~~, ~27/26 = 58.~~283~~

* CWE: ~17/12 = 600.0000{{c}}, ~27/26 = 58.2928{{c}}

Badness (~~Smith~~): 0.~~035057~~

Badness (Sintel): 2.13

====== 23-limit ======

Line 899:

Line 986:

Comma list: 209/208, 225/224, 243/242, 289/288, 323/322, 361/360, 385/384

~~Mapping:~~ {{~~mapping~~| 2 2 6 6 4 4 7 8 7 | 0 12 -14 -4 30 35 12 5 21 }}

{{Mapping|legend=0| 2 2 6 6 4 4 7 8 7 | 0 12 -14 -4 30 35 12 5 21 }}

Optimal tunings:

* ~~CTE~~: ~17/12 = 600.~~000~~, ~27/26 = 58.~~366~~

* WE: ~17/12 = 600.4451{{c}}, ~27/26 = 58.3264{{c}}

* ~~POTE~~: ~17/12 = 600.~~000~~, ~27/26 = 58.~~283~~

* CWE: ~17/12 = 600.0000{{c}}, ~27/26 = 58.2942{{c}}

Badness (~~Smith~~): 0.~~026421~~

Badness (Sintel): 1.89

==== Phicordial ====

Line 914:

Line 1,001:

Comma list: 225/224, 243/242, 385/384, 2200/2197

~~Mapping:~~ {{~~mapping~~| 1 7 -~~4 1 17 4~~ | 0 -18 21 6 -45 -1 }}

: mapping generators: ~2, ~13/8

: mapping generators: ~2, ~16/13

Optimal tunings:

* ~~CTE~~: ~2 = 1200.~~000~~, ~16/13 = ~~361~~.~~096~~

* WE: ~2 = 1200.7056{{c}}, ~13/8 = 839.3726{{c}}

* ~~POTE~~: ~2 = 1200.~~000~~, ~16/13 = ~~361~~.~~121~~

* CWE: ~2 = 1200.0000{{c}}, ~13/8 = 838.8831{{c}}

Badness (~~Smith~~): 0.~~033198~~

Badness (Sintel): 1.37

===== 17-limit =====

Subgroup: 2.3.5.7.11.13.17

Comma list: 225/224, 243/242, 273/272, ~~441~~/~~440~~, 2200/2197

Comma list: 225/224, 243/242, 273/272, 385/384, 2200/2197

~~Mapping:~~ {{~~mapping~~| 1 7 -~~4 1 17 4 8~~ | 0 -18 21 6 -45 -1 -13 }}

Optimal tunings:

* ~~CTE~~: ~2 = 1200.~~000~~, ~16/13 = ~~361~~.~~098~~

* WE: ~2 = 1200.5918{{c}}, ~13/8 = 839.2912{{c}}

* ~~POTE~~: ~2 = 1200.~~000~~, ~16/13 = ~~361~~.~~123~~

* CWE: ~2 = 1200.0000{{c}}, ~13/8 = 838.8809{{c}}

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

===== 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

=== Revelation ===

Line 946:

Line 1,046:

Comma list: 99/98, 176/175, 1029/1024

~~Mapping:~~ {{~~mapping~~| 1 1 3 3 5 | 0 6 -7 -2 -16 }}

Optimal tunings:

* ~~CTE~~: ~2 = ~~1200~~.~~000~~, ~15/14 = 116.~~142~~

* WE: ~2 = 1201.3320{{c}}, ~15/14 = 116.4057{{c}}

* ~~POTE~~: ~2 = 1200.~~000~~, ~15/14 = 116.~~277~~

* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.2524{{c}}

Badness (~~Smith~~): 0.~~032946~~

Badness (Sintel): 1.09

==== 13-limit ====

Line 961:

Line 1,061:

Comma list: 66/65, 99/98, 105/104, 512/507

~~Mapping:~~ {{~~mapping~~| 1 1 3 3 5 4 | 0 6 -7 -2 -16 -3 }}

Optimal tunings:

* ~~CTE~~: ~2 = 1200.~~000~~, ~15/14 = 116.~~194~~

* WE: ~2 = 1200.6059{{c}}, ~15/14 = 116.3263{{c}}

* ~~POTE~~: ~2 = 1200.~~000~~, ~15/14 = 116.~~268~~

* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.2564{{c}}

Badness (~~Smith~~): 0.~~029452~~

Badness (Sintel): 1.22

=== Hemimiracle ===

Line 976:

Line 1,076:

Comma list: 225/224, 245/242, 1029/1024

~~Mapping:~~ {{~~mapping~~| 1 1 3 3 4 | 0 12 -14 -4 -11 }}

: mapping generators: ~2, ~33/32

Optimal tunings:

* ~~CTE~~: ~2 = 1200.~~000~~, ~33/32 = 58.~~399~~

* WE: ~2 = 1200.2902{{c}}, ~33/32 = 58.4217{{c}}

* ~~POTE~~: ~2 = 1200.~~000~~, ~33/32 = 58.~~408~~

* CWE: ~2 = 1200.0000{{c}}, ~33/32 = 58.4062{{c}}

Badness (~~Smith~~): 0.~~059232~~

Badness (Sintel): 1.96

==== 13-limit ====

Line 993:

Line 1,092:

Comma list: 105/104, 196/195, 245/242, 512/507

~~Mapping:~~ {{~~mapping~~| 1 1 3 3 4 4 | 0 12 -14 -4 -11 -6 }}

Optimal tunings:

* ~~CTE~~: ~2 = ~~1200~~.~~000~~, ~33/32 = 58.~~436~~

* WE: ~2 = 1199.8454{{c}}, ~33/32 = 58.4220{{c}}

* ~~POTE~~: ~2 = 1200.~~000~~, ~33/32 = 58.~~430~~

* CWE: ~2 = 1200.0000{{c}}, ~33/32 = 58.4305{{c}}

Badness (~~Smith~~): 0.~~043151~~

Badness (Sintel): 1.78

=== 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

Comma list: 121/120, 225/224, 1029/1024

~~Mapping:~~ {{~~mapping~~| 1 7 -4 ~~1 3~~ | 0 -12 14 4 1 }}

: mapping generators: ~2, ~16/11

: mapping generators: ~2, ~11/8

Optimal tunings:

* ~~CTE~~: ~2 = ~~1200~~.~~000~~, ~11/8 = ~~541~~.~~670~~

* WE: ~2 = 1201.2122{{c}}, ~16/11 = 658.9974{{c}}

* ~~POTE~~: ~2 = 1200.~~000~~, ~11/8 = ~~541~~.~~668~~

* CWE: ~2 = 1200.0000{{c}}, ~16/11 = 658.3320{{c}}

Badness (~~Smith~~): 0.~~042687~~

Badness (Sintel): 1.41

== Hemiseven ==

Unlike miracle which splits ~8/7, hemiseven splits ~7/4. It can be described as the {{nowrap|72 &~~amp;~~ 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

Line 1,027:

Line 1,127:

[[Comma list]]: 1029/1024, 19683/19600

: mapping generators: ~2, ~243/160

: mapping generators: ~2, ~~~320~~/~~243~~

[[Optimal tuning]]s:

* [[~~CTE~~]]: ~2 = 1200.~~000~~, ~~~320~~/~~243~~ = ~~483~~.~~215~~

* [[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~~, ~~~320~~/~~243~~ = ~~483~~.~~267~~

* [[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 }}

[[Badness]] (~~Smith~~): 0.~~056557~~

[[Badness]] (Sintel): 1.43

=== 11-limit ===

Line 1,046:

Line 1,145:

Comma list: 385/384, 441/440, 19683/19600

~~Mapping:~~ {{~~mapping~~| 1 ~~4 14~~ 2 -5 | 0 -6 -29 2 21 }}

Optimal tunings:

* ~~CTE~~: ~2 = 1200.~~000~~, ~~~320~~/~~243~~ = ~~483~~.~~247~~

* WE: ~2 = 1200.6243{{c}}, ~243/160 = 717.0969{{c}}

* ~~POTE~~: ~2 = 1200.~~000~~, ~~~320~~/~~243~~ = ~~483~~.~~276~~

* CWE: ~2 = 1200.0000{{c}}, ~243/160 = 716.7292{{c}}

Badness (~~Smith~~): 0.~~028467~~

Badness (Sintel): 0.941

=== 13-limit ===

Line 1,061:

Line 1,160:

Comma list: 351/350, 385/384, 441/440, 676/675

~~Mapping:~~ {{~~mapping~~| 1 4 ~~14 2~~ -5 19 | 0 -6 -29 2 21 -38 }}

Optimal tunings:

* ~~CTE~~: ~2 = 1200.~~000~~, ~~~120~~/91 = ~~483~~.~~213~~

* WE: ~2 = 1200.6781{{c}}, ~91/60 = 717.1496{{c}}

* ~~POTE~~: ~2 = 1200.~~000~~, ~~~120~~/91 = ~~483~~.~~255~~

* CWE: ~2 = 1200.0000{{c}}, ~91/60 = 716.7520{{c}}

Badness (~~Smith~~): 0.~~021900~~

Badness (Sintel): 0.905

=== 17-limit ===

Line 1,076:

Line 1,175:

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|legend=0| 1 -2 -15 4 16 -19 -21 | 0 6 29 -2 -21 38 42 }}

Optimal tunings:

* ~~CTE~~: ~2 = 1200.~~000~~, ~45~~/34~~ = ~~483~~.~~213~~

* WE: ~2 = 1200.6635{{c}}, ~68/45 = 717.1354{{c}}

* ~~POTE~~: ~2 = 1200.~~000~~, ~45~~/34~~ = ~~483~~.~~261~~

* CWE: ~2 = 1200.0000{{c}}, ~68/45 = 716.7472{{c}}

Badness (~~Smith~~): 0.~~015701~~

Badness (Sintel): 0.800

== Valentine ==

Line 1,090:

Line 1,189:

: ''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 [[~~The Seven Limit Symmetrical Lattices~~|lattice of 7-limit tetrads]]. Valentine can ~~also~~ be described as the {{nowrap|31 &~~amp;~~ 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)1/9 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)1/10.

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)1/9 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]])1/10.

Valentine has a very straighforward [[S-expression]]-based comma list in the [[11-limit]] add-23 (~~aka~~ 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 ~~that~~ it's 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.

[[Subgroup]]: 2.3.5.7

Line 1,099:

Line 1,198:

: mapping generators: ~2, ~21/20

[[Optimal tuning]]s:

* [[~~CTE~~]]: ~2 = 1200.~~000~~, ~21/20 = 77.~~878~~

* [[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~~, ~21/20 = 77.~~864~~

* [[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]]:

* [[7-odd-limit]]: ~21/20 = {{monzo| 1/6 1/12 0 -1/12 }}

: {{monzo list| 1 0 0 0 | 5/2 3/4 0 -3/4 | 17/6 5/12 0 -5/12 | 5/2 -1/4 0 1/4 }}

: [[eigenmonzo basis|~~eigenmonzo (~~unchanged-interval) basis]]: 2.7/3

: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/3

* [[9-odd-limit]]: ~21/20 = {{monzo| 1/21 2/21 0 -1/21}}

: {{monzo list| 1 0 0 0 | 10/7 6/7 0 -3/7 | 47/21 10/21 0 -5/21 | 20/7 -2/7 0 1/7 }}

: [[eigenmonzo basis|~~eigenmonzo (~~unchanged-interval) basis]]: 2.9/7

: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/7

[[Algebraic generator]]: smaller root of ''x''2 - 89''x'' + 92, or (89 - sqrt (7553))/2, at 77.8616 cents.

{{Optimal ET sequence|legend=1| 15, 31, 46, 77, 185~~, 262cd~~ }}

[[Badness]] (~~Smith~~): 0.~~031056~~

[[Badness]] (Sintel): 0.786

=== 11-limit ===

Line 1,127:

Line 1,225:

Comma list: 121/120, 126/125, 176/175

~~Mapping:~~ {{~~mapping~~| 1 1 2 3 3 | 0 9 5 -3 7 }}

~~: mapping generators: ~2, ~21/20~~

Optimal tunings:

* ~~CTE~~: ~2 = 1200.~~000~~, ~22/21 = 77.~~963~~

* WE: ~2 = 1200.3890{{c}}, ~22/21 = 77.9065{{c}}

* ~~POTE~~: ~2 = 1200.~~000~~, ~22/21 = 77.~~881~~

* CWE: ~2 = 1200.0000{{c}}, ~22/21 = 77.9007{{c}}

Minimax tuning:

* 11-odd-limit: ~21/20 = {{monzo| 0 0 0 -1/10 1/10 }}

: [{{monzo| 1 0 0 0 0 }}, {{monzo| 1 0 0 -9/10 9/10 }}, {{monzo| 2 0 0 -1/2 1/2 }}, {{monzo| 3 0 0 3/10 -3/10 }}, {{monzo| 3 0 0 -7/10 7/10 }}]

: ~~eigenmonzo (~~unchanged-interval) basis: 2.11/7

: unchanged-interval (eigenmonzo) basis: 2.11/7

Algebraic generator: positive root of 4''x''3 + 15''x''2 - 21, or else Gontrand2, the smallest positive root of 4''x''7 - 8''x''6 + 5.

{{Optimal ET sequence|legend=0| 15, 31, 46, 77~~, 262cdee, 339cdeee~~ }}

Badness (~~Smith~~): 0.~~016687~~

Badness (Sintel): 0.552

==== Valentino ====

Line 1,151:

Line 1,247:

Comma list: 121/120, 126/125, 176/175, 196/195

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

Optimal tunings:

* ~~CTE~~: ~2 = 1200.~~000~~, ~22/21 = 77.~~968~~

* WE: ~2 = 1200.1967{{c}}, ~22/21 = 77.9708{{c}}

* ~~POTE~~: ~2 = 1200.~~000~~, ~22/21 = 77.~~958~~

* CWE: ~2 = 1200.0000{{c}}, ~22/21 = 77.9594{{c}}

Badness (~~Smith~~): 0.~~020665~~

Badness (Sintel): 0.854

===== 17-limit =====

Line 1,166:

Line 1,262:

Comma list: 121/120, 126/125, 154/153, 176/175, 196/195

~~Mapping:~~ {{~~mapping~~| 1 1 2 3 3 5 5 | 0 9 5 -3 7 -20 -14 }}

Optimal tunings:

* ~~CTE~~: ~2 = 1200.~~000~~, ~22/21 = 78.~~003~~

* WE: ~2 = 1200.0404{{c}}, ~22/21 = 78.0055{{c}}

* ~~POTE~~: ~2 = 1200.~~000~~, ~22/21 = 78.~~003~~

* CWE: ~2 = 1200.0000{{c}}, ~22/21 = 78.0029{{c}}

Badness (~~Smith~~): 0.~~016768~~

Badness (Sintel): 0.854

==== Lupercalia ====

Line 1,181:

Line 1,277:

Comma list: 66/65, 105/104, 121/120, 126/125

~~Mapping:~~ {{~~mapping~~| 1 1 2 3 3 3 | 0 9 5 -3 7 11 }}

Optimal tunings:

* ~~CTE~~: ~2 = ~~1200~~.~~000~~, ~22/21 = 77.~~694~~

* WE: ~2 = 1199.9143{{c}}, ~22/21 = 77.7039{{c}}

* ~~POTE~~: ~2 = 1200.~~000~~, ~22/21 = 77.~~709~~

* CWE: ~2 = 1200.0000{{c}}, ~22/21 = 77.7049{{c}}

Badness (~~Smith~~): 0.~~021328~~

Badness (Sintel): 0.881

==== Dwynwen ====

Line 1,196:

Line 1,292:

Comma list: 91/90, 121/120, 126/125, 176/175

~~Mapping:~~ {{~~mapping~~| 1 1 2 3 3 2 | 0 9 5 -3 7 26 }}

Optimal tunings:

* ~~CTE~~: ~2 = 1200.~~000~~, ~22/21 = 78.~~243~~

* WE: ~2 = 1200.1306{{c}}, ~22/21 = 78.2273{{c}}

* ~~POTE~~: ~2 = 1200.~~000~~, ~22/21 = 78.~~219~~

* CWE: ~2 = 1200.0000{{c}}, ~22/21 = 78.2241{{c}}

Badness (~~Smith~~): 0.~~023461~~

Badness (Sintel): 0.969

==== Semivalentine ====

Line 1,211:

Line 1,307:

Comma list: 121/120, 126/125, 169/168, 176/175

~~Mapping:~~ {{~~mapping~~| 2 2 4 6 6 7 | 0 9 5 -3 7 3 }}

: mapping generators: ~55/39, ~22/21

Optimal tunings:

* ~~CTE~~: ~55/39 = 600.~~000~~, ~22/21 = 77.~~997~~

* WE: ~55/39 = 600.3497{{c}}, ~22/21 = 77.8845{{c}}

* ~~POTE~~: ~55/39 = 600.~~000~~, ~22/21 = 77.~~839~~

* CWE: ~55/39 = 600.0000{{c}}, ~22/21 = 77.8715{{c}}

Badness (~~Smith~~): 0.~~032749~~

Badness (Sintel): 1.35

==== Hemivalentine ====

Line 1,228:

Line 1,323:

Comma list: 121/120, 126/125, 176/175, 343/338

~~Mapping:~~ {{~~mapping~~| 1 1 2 3 3 4 | 0 18 10 -6 14 -9 }}

: mapping generators: ~2, ~40/39

Optimal tunings:

* ~~CTE~~: ~2 = ~~1200~~.~~000~~, ~40/39 = 39.~~014~~

* WE: ~2 = 1199.6529{{c}}, ~40/39 = 39.0323{{c}}

* ~~POTE~~: ~2 = 1200.~~000~~, ~40/39 = 39.~~044~~

* CWE: ~2 = 1200.0000{{c}}, ~40/39 = 39.0383{{c}}

Badness (~~Smith~~): 0.~~047059~~

Badness (Sintel): 1.94

==== Demivalentine ====

Line 1,245:

Line 1,339:

Comma list: 121/120, 126/125, 176/175, 676/675

~~Mapping:~~ {{~~mapping~~| 1 -8 -3 6 -4 -16 | 0 18 10 -6 14 37 }}

: mapping generators: ~2, ~13/9

Optimal tunings:

* ~~CTE~~: ~2 = 1200.~~000~~, ~13/9 = ~~638~~.~~964~~

* WE: ~2 = 1200.3929{{c}}, ~13/9 = 639.1320{{c}}

* CWE: ~2 = 1200.~~000~~, ~13/9 = 638.~~932~~

* CWE: ~2 = 1200.0000{{c}}, ~13/9 = 638.9325{{c}}

Badness (~~Smith~~): 0.~~0349~~

Badness (Sintel): 1.44

=== Hemivalentino ===

Line 1,262:

Line 1,355:

Comma list: 126/125, 243/242, 1029/1024

~~Mapping:~~ {{~~mapping~~| 1 1 2 3 2 | 0 18 10 -6 45 }}

Optimal tunings:

* ~~CTE~~: ~2 = 1200.~~000~~, ~45/44 = 38.~~928~~

* WE: ~2 = 1200.0816{{c}}, ~45/44 = 38.9236{{c}}

* ~~POTE~~: ~2 = 1200.~~000~~, ~45/44 = 38.~~921~~

* CWE: ~2 = 1200.0000{{c}}, ~45/44 = 38.9228{{c}}

Badness (~~Smith~~): 0.~~061275~~

Badness (Sintel): 2.03

==== 13-limit ====

Line 1,277:

Line 1,370:

Comma list: 126/125, 196/195, 243/242, 1029/1024

~~Mapping:~~ {{~~mapping~~| 1 1 2 3 2 5 | 0 18 10 -6 45 -40 }}

Optimal tunings:

* ~~CTE~~: ~2 = ~~1200~~.~~000~~, ~45/44 = 38.~~944~~

* WE: ~2 = 1199.8782{{c}}, ~45/44 = 38.9440{{c}}

* ~~POTE~~: ~2 = 1200.~~000~~, ~45/44 = 38.~~948~~

* CWE: ~2 = 1200.0000{{c}}, ~45/44 = 38.9472{{c}}

Badness (~~Smith~~): 0.~~057919~~

Badness (Sintel): 2.39

==== Hemivalentoid ====

Line 1,292:

Line 1,385:

Comma list: 126/125, 144/143, 243/242, 343/338

~~Mapping:~~ {{~~mapping~~| 1 1 2 3 2 4 | 0 18 10 -6 45 -9 }}

Optimal tunings:

* ~~CTE~~: ~2 = ~~1200~~.~~000~~, ~40/39 = 38.~~946~~

* WE: ~2 = 1199.3614{{c}}, ~45/44 = 38.9721{{c}}

* ~~POTE~~: ~2 = 1200.~~000~~, ~40/39 = 38.~~993~~

* CWE: ~2 = 1200.0000{{c}}, ~45/44 = 38.9839{{c}}

Badness (~~Smith~~): 0.~~057931~~

Badness (Sintel): 2.39

== Superkleismic ==

Line 1,306:

Line 1,399:

: ''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.

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]] }}.

In the 11-limit, two generator steps can be identified with ~16/11, and in the 13-limit, the same step can be treated as ~13/9. The [[S-expression]]-based comma list of 13-limit superkleismic is {[[875/864|S5/S6]], [[1029/1024|S7/S8]], [[100/99|S10]], [[144/143|S12]], ([[441/440|S21]])}. Through careful observation of the equivalences therein one can derive the mapping of the full 13-limit.

Note that the generator is given as 6/5's octave complement, [[5/3]], in the data that follow, since a stack of 9 such generators octave-reduced is the perfect fifth; the [[ploidacot]] of superkleismic is wau-enneacot.

Superkleismic also sets two intervals of [[21/20]] equal to [[10/9]]; as {{nowrap| 10/9 {{=}} ([[20/19]])⋅([[19/18]]) }}, we can identify 21/20, 20/19, and 19/18 together to add prime 19, tempering out [[361/360]] ({{S|19}}) and [[400/399]] ({{S|20}}). This structure is preserved within the entire superkleismic tuning range between 15edo and 26edo, while extensions for primes 13 and 17 bifurcate and are of higher complexity and lower accuracy.

41edo gives an obvious tuning in all the subgroups.

Line 1,316:

Line 1,413:

[[Comma list]]: 875/864, 1029/1024

: mapping generators: ~2, ~5/3

: mapping generators: ~2, ~6/5

[[Optimal tuning]]s:

* [[~~CTE~~]]: ~2 = 1200.~~000~~, ~6/5 = ~~321~~.~~798~~

* [[WE]]: ~2 = 1200.7640{{c}}, ~5/3 = 878.6289{{c}}

* [[~~POTE~~]]: ~2 = 1200.~~000~~, ~6/5 ~~= 321~~.~~930~~

: [[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 }}

[[Badness]]:

[[Badness]] (Sintel): 1.21

* Smith: 0.0479

* Dirichlet: 1.21

=== 11-limit ===

Line 1,335:

Line 1,431:

Comma list: 100/99, 245/242, 385/384

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

Optimal tunings:

* ~~CTE~~: ~2 = 1200.~~000~~, ~6/5 = ~~321~~.~~815~~

* WE: ~2 = 1200.1691{{c}}, ~5/3 = 878.2772{{c}}

* ~~POTE~~: ~2 = 1200.~~000~~, ~6/5 = ~~321~~.~~847~~

* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 878.1606{{c}}

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

Badness:

Badness (Sintel): 0.848

* Smith: 0.0257

* Dirichlet: 0.848

==== 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

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

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

Optimal tunings:

* ~~CTE~~: ~2 = ~~1\1~~, ~6/5 = ~~321~~.~~779~~

* WE: ~2 = 1200.2289{{c}}, ~5/3 = 878.3409{{c}}

* ~~POTE~~: ~2 = ~~1\1~~, ~6/5 = ~~321~~.~~827~~

* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 878.1840{{c}}

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

Badness (Sintel): 0.692

Line 1,371:

Line 1,463:

Comma list: 100/99, 105/104, 144/143, 245/242

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

Optimal tunings:

* ~~CTE~~: ~2 = 1200.~~000~~, ~6/5 = ~~321~~.~~986~~

* WE: ~2 = 1200.0261{{c}}, ~5/3 = 878.0252{{c}}

* ~~POTE~~: ~2 = 1200.~~000~~, ~6/5 = ~~321~~.~~994~~

* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 878.0073{{c}}

Badness:

Badness (Sintel): 0.887

* Smith: 0.0215

* Dirichlet: 0.887

==== 17-limit ====

Line 1,388:

Line 1,478:

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 }}

Optimal tunings:

* ~~CTE~~: ~2 = 1200.~~000~~, ~6/5 = ~~322~~.~~136~~

* WE: ~2 = 1200.0488{{c}}, ~5/3 = 877.8872{{c}}

* ~~POTE~~: ~2 = 1200.~~000~~, ~6/5 = ~~322~~.~~149~~

* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 877.8537{{c}}

Badness (Sintel): 1.~~007~~

Badness (Sintel): 1.01

==== 19-limit ====

Line 1,403:

Line 1,493:

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|legend=0| 1 -5 -5 5 2 -8 -12 -6 | 0 9 10 -3 2 16 22 14 }}

Optimal tunings:

* ~~CTE~~: ~2 = 1200.~~000~~, ~6/5 = ~~322~~.~~084~~

* WE: ~2 = 1200.2120{{c}}, ~5/3 = 878.0243{{c}}

* CWE: ~2 = 1200.~~000~~, ~6/5 = ~~322~~.~~121~~

* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 877.8789{{c}}

Line 1,414:

Line 1,504:

=== Superana ===

This extension (41 & 56) is the counterpart of canonical superkleismic on the other side of 41edo.

This extension ({{nowrap| 41 & 56 }}) is the counterpart of canonical superkleismic on the other side of 41edo.

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 196/195, 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 }}

Optimal tunings:

* ~~CTE~~: ~2 = ~~1200~~.~~000~~, ~6/5 = ~~321~~.~~724~~

* WE: ~2 = 1199.8272{{c}}, ~5/3 = 878.1538{{c}}

* ~~POTE~~: ~2 = 1200.~~000~~, ~6/5 = ~~321~~.~~719~~

* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 878.2795{{c}}

Badness (Sintel): 1.~~402~~

Badness (Sintel): 1.40

==== 17-limit ====

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 }}

Optimal tunings:

* ~~CTE~~: ~2 = ~~1200~~.~~000~~, ~6/5 = ~~321~~.~~650~~

* WE: ~2 = 1199.5964{{c}}, ~5/3 = 878.0482{{c}}

* ~~POTE~~: ~2 = 1200.~~000~~, ~6/5 = ~~321~~.~~657~~

* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 878.3444{{c}}

Badness (Sintel): 1.~~452~~

Badness (Sintel): 1.45

==== 19-limit ====

Line 1,450:

Line 1,540:

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|legend=0| 1 -5 -5 5 2 22 18 -6 | 0 9 10 -3 2 -25 -19 14 }}

Optimal tunings:

* ~~CTE~~: ~2 = 1200.~~000~~, ~6/5 = ~~321~~.~~646~~

* WE: ~2 = 1199.6638{{c}}, ~5/3 = 878.1109{{c}}

* ~~POTE~~: ~2 = 1200.~~000~~, ~6/5 = ~~321~~.~~643~~

* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 878.3566{{c}}

Badness (Sintel): 1.36

== Dee leap week ==

[[Subgroup]]: 2.3.5.7

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

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1200.4835{{c}}, ~224/135 = 878.2507{{c}}

: [[error map]]: {{val| +0.484 -0.117 +0.004 -1.160 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~224/135 = 877.8926{{c}}

: error map: {{val| 0.000 -0.921 -0.985 -2.504 }}

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

Badness (Sintel): 1.~~355~~

[[Badness]] (Sintel): 2.12

== ~~Unidec~~ ==

=== 11-limit ===

~~{{Main| Unidec }}~~

Subgroup: 2.3.5.7.11

~~=== 5-limit (unidecmic) ===~~

Comma list: 385/384, 441/440, 2460375/2458624

~~[[Subgroup]]~~: ~~2.3.5~~

~~[[Comma list]]: 31381059609/31250000000~~

Optimal tunings:

* WE: ~2 = 1200.4874{{c}}, ~224/135 = 878.2543{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~224/135 = 877.8987{{c}}

~~: mapping generators: ~177147/125000~~, ~~~10/9~~

~~[[Optimal tuning]]s~~:

Badness (Sintel): 1.35

* [[CTE]]: ~177147/125000 = 600.000, ~10/9 = 183.041

~~: [[error map]]: {{val| 0.000 -0.201 +0.235 }}~~

* [[POTE]]: ~177147/125000 = 600.000, ~10/9 = 183.047

~~: error map: {{val| 0.000 -0.236 +0~~.~~172 }}~~

{{~~Optimal ET sequence|legend=1~~| ~~26, 46, 72, 118, 2524, 2642, 2760~~ }}

== Unidec ==

[[~~Badness~~]] ~~(Smith): 0~~.~~082423~~

Unidec tempers out the ragisma, [[4375/4374]], and may be described as the {{nowrap| 26 & 46 }} temperament. It has a [[semi-octave]] [[period]] and a generator of ~[[80/63]], two of which minus a period make slendric's generator; its [[ploidacot]] is therefore diploid gamma-hexacot. In the 11-limit, the generator represents [[14/11]]. [[190edo]] makes for an excellent tuning in both the 7-limit and 11-limit.

~~=== 7-limit ===~~

[[Subgroup]]: 2.3.5.7

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

[[Optimal tuning]]s:

* [[~~CTE~~]]: ~1225/864 = 600.~~000~~, ~10/9 = ~~183~~.~~060~~

* [[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~~, ~10/9 = ~~183~~.~~161~~

* [[CWE]]: ~1225/864 = 600.0000{{c}}, ~80/63 = 416.8688{{c}}

: error map: {{val| 0.000 -0.924 -1.090 -2.503 }}

Line 1,498:

Line 1,604:

* [[7-odd-limit]]: ~10/9 = {{monzo| 3/26 0 -1/13 1/13 }}

: {{monzo list| 1 0 0 0 | 47/26 0 6/13 -6/13 | 71/26 0 11/13 -11/13 | 71/26 0 -2/13 2/13 }}

: [[eigenmonzo basis|~~eigenmonzo (~~unchanged-interval) basis]]: 2.7/5

: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5

* [[9-odd-limit]]: ~10/9 = {{monzo| 5/28 -1/7 0 1/14 }}

: {{Monzo list| 1 0 0 0 | 10/7 6/7 0 -3/7 | 57/28 11/7 0 -11/14 | 20/7 -2/7 0 1/7 }}

: [[eigenmonzo basis|~~eigenmonzo (~~unchanged-interval) basis]]: 2.9/7

: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/7

[[Badness]] (~~Smith~~): 0.~~038393~~

[[Badness]] (Sintel): 0.972

=== 11-limit ===

Line 1,512:

Line 1,618:

Comma list: 385/384, 441/440, 4375/4374

~~Mapping:~~ {{~~mapping~~| 2 ~~5 8 5 6~~ | 0 -6 -11 2 3 }}

Optimal tunings:

* ~~CTE~~: ~99/70 = 600.~~000~~, ~10/9 = ~~183~~.~~074~~

* WE: ~99/70 = 600.2497{{c}}, ~14/11 = 417.0085{{c}}

* CWE: ~99/70 = 600.~~000~~, ~10/9 = ~~183~~.~~146~~

* CWE: ~99/70 = 600.0000{{c}}, ~14/11 = 416.8543{{c}}

Minimax tuning:

* [[11-odd-limit]]: ~10/9 = {{monzo| 5/28 -1/7 0 1/14 }}

: [{{monzo| 1 0 0 0 0 }}, {{monzo| 10/7 6/7 0 -3/7 0 }}, {{monzo| 57/28 11/7 0 -11/14 0 }}, {{monzo| 20/7 -2/7 0 1/7 0 }}, {{monzo| 99/28 -3/7 0 3/14 0 }}]

: ~~eigenmonzo (~~unchanged-interval) basis: 2.9/7

: unchanged-interval (eigenmonzo) basis: 2.9/7

Badness (~~Smith~~): 0.~~015479~~

Badness (Sintel): 0.512

==== Ekadash ====

Line 1,532:

Line 1,638:

Comma list: 385/384, 441/440, 625/624, 729/728

~~Mapping:~~ {{~~mapping~~| 2 ~~5 8 5 6~~ 19 | 0 -6 -11 2 3 -38 }}

Optimal tunings:

* ~~CTE~~: ~99/70 = 600.~~000~~, ~10/9 = ~~183~~.~~125~~

* WE: ~99/70 = 600.2497{{c}}, ~14/11 = 417.0085{{c}}

* ~~POTE~~: ~99/70 = 600.~~000~~, ~10/9 = ~~183~~.~~187~~

* CWE: ~99/70 = 600.0000{{c}}, ~14/11 = 416.8543{{c}}

{{Optimal ET sequence|legend=0| 46f, 72, 118, 190, 262df, 452cdef }}

Badness (~~Smith~~): 0.~~020381~~

Badness (Sintel): 0.842

==== Hendec ====

Line 1,547:

Line 1,653:

Comma list: 169/168, 325/324, 364/363, 385/384

~~Mapping:~~ {{~~mapping~~| 2 ~~5 8 5~~ 6 8 | 0 -6 -11 2 3 -2 }}

Optimal tunings:

* ~~CTE~~: ~91/64 = 600.~~000~~, ~10/9 = ~~183~~.~~048~~

* WE: ~91/64 = 600.3825{{c}}, ~14/11 = 417.0678{{c}}

* ~~POTE~~: ~91/64 = 600.~~000~~, ~10/9 = ~~183~~.~~198~~

* CWE: ~91/64 = 600.0000{{c}}, ~14/11 = 416.8290{{c}}

Badness (~~Smith~~): 0.~~017707~~

Badness (Sintel): 0.732

===== 17-limit =====

Line 1,562:

Line 1,668:

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 }}

Optimal tunings:

* ~~CTE~~: ~17/12 = 600.~~000~~, ~10/9 = ~~183~~.~~020~~

* WE: ~17/12 = 600.3991{{c}}, ~14/11 = 417.0809{{c}}

* ~~POTE~~: ~17/12 = 600.~~000~~, ~10/9 = ~~183~~.~~196~~

* CWE: ~17/12 = 600.0000{{c}}, ~14/11 = 416.8330{{c}}

Badness (~~Smith~~): 0.~~011676~~

Badness (Sintel): 0.595

== Restles ==

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

[[Comma list]]: 1029/1024, ~~11529602~~/~~11390625~~

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

: mapping generators: ~2. ~315/256

: ~~mapping generators~~: ~~~3375~~/~~2401~~, ~15/14

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1200.0322{{c}}, ~315/256 = 358.5581{{c}}

: [[error map]]: {{val| +0.032 +0.678 +1.340 -2.930 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~315/256 = 358.5484{{c}}

: error map: {{val| 0.000 +0.626 +1.267 -3.019 }}

~~[[Optimal tuning]]s:~~

* [[CTE]]: ~3375/2401 = 600.000, ~15/14 = 122.031

~~: [[error map]]:~~ {{~~val~~| ~~0.000 -0.232 +0.087 -2.734 }}~~

* [[POTE]]: ~3375/2401 = ~~600.000~~, ~~~15/14 = 122.027~~

~~: error map: {{val| 0.000 -0.195 +0.033 -2.746~~ }}

~~{{Optimal ET sequence|legend=1| 10, 98, 108, 118 }}~~

[[Badness]] (Sintel): 2.73

~~[[Badness]] (Smith)~~: 0.~~144345~~

=== 11-limit ===

Subgroup: 2.3.5.7.11

~~== Dee leap week ==~~

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

~~{{Main|Dee leap week}}~~

~~Subgroup:~~ 2~~.3.5.~~7

~~Comma list~~: ~~1029~~/~~1024~~, ~~2460375~~/~~2458624~~

Optimal tunings:

* WE: ~2 = 1200.1110{{c}}, ~27/22 = 358.6045{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~27/22 = 358.5720{{c}}

~~Mapping: [~~{{~~val~~|~~1 4 -6 2}}~~, ~~{{val|0 -9 31 3~~}}]

~~Optimal tuning~~ (~~CTE~~): ~~~135/112 = 322~~.~~123~~

Badness (Sintel): 1.81

=== 11-limit ===

=== 13-limit ===

Subgroup: 2.3.5.7.11

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 352/351, 385/384, 676/675

~~Comma list: 385/384, 441/440, 2460375/2458624~~

~~Mapping~~: [{{~~val|1 4 -6~~ 2 15}}, {{~~val|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): ~135/112~~ = ~~322.097~~

~~{{Optimal ET sequence|legend=~~1~~|41, 149, 190, 231}}~~

Badness (Sintel): 1.16

== 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

[[Comma list]]: 1029/1024, 5103/5000

: mapping generators: ~2, ~35/24

: mapping generators: ~2, ~48/35

[[Optimal tuning]]s:

* [[~~CTE~~]]: ~2 = 1200.~~000~~, ~48/35 = ~~541~~.~~743~~

* [[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~~, ~48/35 = ~~541~~.~~779~~

* [[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 }}

[[Badness]] (~~Smith~~): 0.~~117680~~

[[Badness]] (Sintel): 2.98

=== 11-limit ===

Line 1,637:

Line 1,755:

Comma list: 176/175, 243/242, 1029/1024

~~Mapping:~~ {{~~mapping~~| 1 7 ~~10 1 17~~ | 0 -12 -17 4 -30 }}

Optimal tunings:

* ~~CTE~~: ~2 = 1200.~~000~~, ~15~~/11~~ = ~~541~~.~~695~~

* WE: ~2 = 1200.2862{{c}}, ~22/15 = 658.4276{{c}}

* ~~POTE~~: ~2 = 1200.~~000~~, ~15~~/11~~ = ~~541~~.~~729~~

* CWE: ~2 = 1200.0000{{c}}, ~22/15 = 658.2805{{c}}

Badness (~~Smith~~): 0.~~053459~~

Badness (Sintel): 1.77

=== 13-limit ===

Line 1,652:

Line 1,770:

Comma list: 144/143, 176/175, 243/242, 343/338

~~Mapping:~~ {{~~mapping~~| 1 7 ~~10 1 17 1~~ | 0 -12 -17 4 -30 6 }}

Optimal tunings:

* ~~CTE~~: ~2 = ~~1200~~.~~000~~, ~15~~/11~~ = ~~541~~.~~673~~

* WE: ~2 = 1199.3663{{c}}, ~22/15 = 658.0465{{c}}

* ~~POTE~~: ~2 = 1200.~~000~~, ~15~~/11~~ = ~~541~~.~~606~~

* CWE: ~2 = 1200.0000{{c}}, ~22/15 = 658.3800{{c}}

Badness (~~Smith~~): 0.~~047015~~

Badness (Sintel): 1.94

== ~~Restles~~ ==

== Lagaca ==

Cryptically named by [[Petr Pařízek]] in 2011<ref name="petr's long post"/>, lagaca may be described as the {{nowrap| 10 & 118 }} temperament with a [[ploidacot]] signature of diploid wau-enneacot. The name actually refers to the fact that 12 generator steps in this temperament make ~[[7/3]], where "l", "g", "c" are integers alphabetically converted to letters.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 1029/1024, ~~153664~~/~~151875~~

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

: mapping generators: ~3375/2401, ~450/343

: mapping generators: ~2. ~~~315~~/~~256~~

[[Optimal tuning]]s:

* [[~~CTE~~]]: ~2 = ~~1200~~.~~000~~, ~~~315~~/~~256~~ = ~~358~~.~~548~~

* [[WE]]: ~3375/2401 = 600.1355{{c}}, ~450/343 = 478.0813{{c}}

: [[error map]]: {{val| 0.~~000~~ +0.~~620~~ +1.~~275~~ -3.~~018~~ }}

: [[error map]]: {{val| +0.271 +0.235 +0.662 -1.986 }}

* [[~~POTE~~]]: ~2 = ~~1200~~.000, ~~~315~~/~~256~~ = ~~358~~.~~548~~

* [[CWE]]: ~3375/2401 = 600.000{{c}}, ~450/343 = 477.9725{{c}}

: error map: {{val| 0.000 +0.~~627~~ +1.~~265~~ -3.~~020~~ }}

: error map: {{val| 0.000 -0.202 +0.043 -2.743 }}

{{Optimal ET sequence|legend=1| 77, 87, ~~164 }}~~

~~[[Badness]] (Smith): 0.108011~~

~~=== 11-limit ===~~

~~Subgroup: 2.3.5.7.11~~

~~Comma list: 385/384~~, ~~441/440, 153664/151875~~

~~Mapping: {{mapping| 1 -2 8 4 -7 | 0 12 -19 -4 35 }}~~

~~Optimal tunings:~~

* CTE: ~2 = 1200.000, ~27/22 = 358.575

* POTE: ~2 = 1200.000, ~27/22 = 358.571

~~{{Optimal ET sequence|legend=0| 77, 87, 164, 251d }}~~

~~Badness (Smith): 0.054655~~

~~=== 13-limit ===~~

~~Subgroup: 2.3.5.7.11.13~~

~~Comma list: 196/195, 352/351, 385/384, 676/675~~

~~Mapping: {{mapping| 1 -2 8 4 -7 4 | 0 12 -19 -4 35 -1~~ }}

~~Optimal tunings:~~

[[Badness]] (Sintel): 3.65

* CTE: ~2 = 1200.000, ~16/13 = 358.576

* POTE: ~2 = 1200.000, ~16/13 = 358.574

~~{{Optimal ET sequence|legend=0| 77, 87, 164, 251d }}~~

Badness (~~Smith~~): 0.~~028187~~

== Quartemka ==

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

Quartemka may be described as the {{nowrap| 26 & 61 }} temperament. Its [[ploidacot]] is 18-sheared 21-cot. It was named by [[Petr Pařízek]] in 2011 for its generator is close to 1/4 of the generator for [[emka]]<ref name="petr's long post"/>.

[[Subgroup]]: 2.3.5.7

Line 1,720:

Line 1,809:

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

: mapping generators: ~2, ~50/27

: mapping generators: ~2, ~27~~/25~~

[[Optimal tuning]]s:

* [[~~CTE~~]]: ~2 = 1200.~~000~~, ~27~~/25~~ = ~~137~~.~~971~~

* [[WE]]: ~2 = 1200.5278{{c}}, ~50/27 = 1062.4614{{c}}

: [[error map]]: {{val| 0.~~000~~ +0.~~658~~ -1.~~380~~ -3.~~030~~ }}

: [[error map]]: {{val| +0.528 +0.762 -1.272 -1.305 }}

* [[~~POTE~~]]: ~2 = 1200.~~000~~, ~27~~/25~~ = ~~138~~.~~006~~

* [[CWE]]: ~21 = 1200.0000{{c}}, ~50/27 = 1062.0046{{c}}

: error map: {{val| 0.000 -0.~~075~~ -2.~~496~~ -2.~~786~~ }}

: error map: {{val| 0.000 +0.142 -2.167 -2.858 }}

[[Badness]] (~~Smith~~): 0.~~152287~~

[[Badness]] (Sintel): 3.85

=== 11-limit ===

Line 1,739:

Line 1,827:

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

~~Mapping:~~ {{~~mapping~~| 1 ~~4 6 2 3~~ | 0 -21 -32 7 4 }}

Optimal tunings:

* ~~CTE~~: ~2 = 1200.~~000~~, ~27~~/25~~ = ~~137~~.~~970~~

* WE: ~2 = 1200.3051{{c}}, ~50/27 = 1062.2805{{c}}

* ~~POTE~~: ~2 = 1200.~~000~~, ~27~~/25~~ = ~~137~~.~~990~~

* CWE: ~21 = 1200.0000{{c}}, ~50/27 = 1062.0147{{c}}

Badness (~~Smith~~): 0.~~057307~~

Badness (Sintel): 1.89

=== 13-limit ===

Line 1,754:

Line 1,842:

Comma list: 325/324, 364/363, 385/384, 2200/2197

~~Mapping:~~ {{~~mapping~~| 1 ~~4 6 2 3 6~~ | 0 -21 -32 7 4 -20 }}

Optimal tunings:

* ~~CTE~~: ~2 = 1200.~~000~~, ~13~~/12~~ = ~~137~~.~~971~~

* WE: ~2 = 1200.2708{{c}}, ~24/13 = 1062.2496{{c}}

* ~~POTE~~: ~2 = 1200.~~000~~, ~13~~/12~~ = ~~137~~.~~990~~

* CWE: ~21 = 1200.0000{{c}}, ~24/13 = 1062.0139{{c}}

Badness (~~Smith~~): 0.~~028393~~

Badness (Sintel): 1.17

== Tritriple ==

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

Tritriple may be described as the {{nowrap| 103 & 118 }} temperament. Its [[ploidacot]] is iota-beta-27-cot. It was named by [[Petr Pařízek]] in 2011 for its generator is 1/9 of the generator for [[slendric]], so that 3×3 generators [[octave reduction|octave reduced]] give slendric's generator, and another ×3 give the [[3/2|perfect fifth]]<ref name="petr's long post"/>.

[[Subgroup]]: 2.3.5.7

Line 1,772:

Line 1,862:

: mapping generators: ~2, ~864/625

[[Optimal tuning]]s:

* [[~~CTE~~]]: ~2 = 1200.~~000~~, ~864/625 = 559.~~320~~

* [[WE]]: ~2 = 1200.4239{{c}}, ~864/625 = 559.4921{{c}}

: [[error map]]: {{val| 0.~~000~~ -0.~~317~~ +0.~~085~~ -2.~~705~~ }}

: [[error map]]: {{val| +0.424 -0.331 +0.561 -1.287 }}

* [[~~POTE~~]]: ~2 = 1200.~~000~~, ~864/625 = 559.~~295~~

* [[CWE]]: ~2 = 1200.0000{{c}}, ~864/625 = 559.3015{{c}}

: error map: {{val| 0.000 -1.~~003~~ -0.~~423~~ -2.~~477~~ }}

: error map: {{val| 0.000 -0.815 -0.284 -2.539 }}

{{Optimal ET sequence|legend=1| 15, …, 88, 103, 118, 221, 339d }}

[[Badness]] (~~Smith~~): 0.~~118640~~

[[Badness]] (Sintel): 3.00

=== 11-limit ===

Line 1,790:

Line 1,879:

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

~~Mapping:~~ {{~~mapping~~| 1 -11 -7 7 -4 | 0 27 20 -9 16 }}

Optimal tunings:

* ~~CTE~~: ~2 = 1200.~~000~~, ~242/175 = 559.~~327~~

* WE: ~2 = 1200.4953{{c}}, ~242/175 = 559.5243{{c}}

* ~~POTE~~: ~2 = 1200.~~000~~, ~242/175 = 559.~~293~~

* CWE: ~2 = 1200.0000{{c}}, ~242/175 = 559.3016{{c}}

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

Badness (~~Smith~~): 0.~~035350~~

Badness (Sintel): 1.17

== Widefourth ==

Line 1,805:

Line 1,894:

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

[[Optimal tuning]]s:

* [[~~CTE~~]]: ~2 = 1200.~~000~~, ~3125~~/2304~~ = ~~524~~.~~188~~

* [[WE]]: ~2 = 1200.4770{{c}}, ~4608/3125 = 676.0584{{c}}

: [[error map]]: {{val| 0.~~000~~ -0.~~154 -~~0.~~756~~ -2.~~759~~ }}

: [[error map]]: {{val| +0.477 -0.137 +0.061 -1.175 }}

* [[~~POTE~~]]: ~2 = 1200.~~000~~, ~3125~~/2304~~ = ~~524~~.~~210~~

* [[CWE]]: ~2 = 1200.0000{{c}}, ~4608/3125 = 675.7954{{c}}

: error map: {{val| 0.000 -0.~~892~~ -1.~~047~~ -2.~~513~~ }}

: error map: {{val| 0.000 -0.705 -0.973 -2.576 }}

[[Badness]] (~~Smith~~): 0.~~154117~~

[[Badness]] (Sintel): 3.90

=== 11-limit ===

Line 1,822:

Line 1,911:

Comma list: 385/384, 441/440, 234375/234256

~~Mapping:~~ {{~~mapping~~| 1 16 8 -2 17 | 0 -33 -13 11 -31 }}

Optimal tunings:

* ~~CTE~~: ~2 = 1200.~~000~~, ~847~~/625~~ = ~~524~~.~~183~~

* WE: ~2 = 1200.4852{{c}}, ~1250/847 = 676.0634{{c}}

* ~~POTE~~: ~2 = 1200.~~000~~, ~847~~/625~~ = ~~524~~.~~210~~

* CWE: ~2 = 1200.0000{{c}}, ~1250/847 = 675.7966{{c}}

Badness (~~Smith~~): 0.~~040785~~

Badness (Sintel): 1.35

=== 13-limit ===

Line 1,837:

Line 1,926:

Comma list: 385/384, 441/440, 625/624, 847/845

~~Mapping:~~ {{~~mapping~~| 1 16 8 -2 17 12 | 0 -33 -13 11 -31 -19 }}

Optimal tunings:

* ~~CTE~~: ~2 = 1200.~~000~~, ~65/48 = ~~524~~.~~183~~

* WE: ~2 = 1200.4217{{c}}, ~77/52 = 676.0286{{c}}

* ~~POTE~~: ~2 = 1200.~~000~~, ~65/48 = ~~524~~.~~209~~

* CWE: ~2 = 1200.0000{{c}}, ~77/52 = 675.7967{{c}}

Badness (~~Smith~~): 0.~~021636~~

Badness (Sintel): 0.894

== Other subgroup extensions ==

=== Euslendric (2.3.7.13) ===

Forms of slendric in the most optimal range for the 2.3.7 temperament ({{nowrap| 36 & 77 }}) lack an obvious strong mapping of prime 5 or prime 11. However, slendric can extend well to the no-fives no-elevens [[29-limit]] by tempering out [[273/272]], [[343/342]], [[378/377]], [[392/391]], [[513/512]], and [[729/728]], or a comma basis defined in terms of [[S-expression]]s as {S7/S8, S14/S16, S15/S20, S24/S26, S27, S28}. [[113edo]] is an obvious tuning.

Subgroup: 2.3.7.13

Comma list: 729/728, 1029/1024

Subgroup-val mapping: {{mapping| 1 1 3 0 | 0 3 -1 19 }}

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

Optimal tunings:

* WE: ~2 = 1200.5057{{c}}, ~8/7 = 233.7200{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.6534{{c}}

{{Optimal ET sequence|legend=0| 5, 31f, 36, 77, 113, 827bdddff }}

~~== Notes ==~~

Badness (Sintel): 0.339

==== 2.3.7.13.17 subgroup ====

Subgroup: 2.3.7.13.17

Comma list: 273/272, 729/728, 833/832

Subgroup-val mapping: {{mapping| 1 1 3 0 0 | 0 3 -1 19 21 }}

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

Optimal tunings:

* WE: ~2 = 1200.5282{{c}}, ~8/7 = 233.6492{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.5776{{c}}

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}}

Badness (Sintel): 0.380

==== 2.3.7.13.17.19.23 subgroup ====

Subgroup: 2.3.7.13.17.19.23

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

Subgroup-val mapping: {{mapping| 1 1 3 0 0 6 9 | 0 3 -1 19 21 -9 -23 }}

Gencom mapping: {{mapping| 1 1 0 3 0 0 0 6 9 | 0 3 0 -1 0 19 21 -9 -23 }}

Optimal tunings:

* WE: ~2 = 1200.3127{{c}}, ~8/7 = 233.6679{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.6091{{c}}

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}}

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***** }}

<nowiki/>* Wart for 85

== References ==

[[Category:Gamelismic clan| ]]

[[Category:Temperament clans]]

[[Category:~~Gamelismic clan| ]] <!~~-~~- main article -->~~

[[Category:Catalogs of rank-2 temperaments]]

~~[[Category:Rank~~ 2]]

[[Category:Listen]]

@@ Line 1: / Line 1: @@
 {{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]]
-* ''[[Blacksmith]]'' (+28/27) → [[Limmic temperaments #Blacksmith|Limmic temperaments]]
-* ''[[Trismegistus]]'' (+3125/3072) → [[Magic family #Trismegistus|Magic family]]
-* [[Hemithirds]] (+3136/3125) → [[Hemimean clan #Hemithirds|Hemimean clan]]
-* ''[[Gamity]]'' (+1071875/1062882) → [[Amity family #Gamity|Amity family]]
-* ''[[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 ==
@@ Line 30: / Line 10: @@
 {{Mapping|legend=2| 1 1 3 | 0 3 -1 }}
-: sval mapping generators: ~2, ~8/7
 {{Mapping|legend=3| 1 1 0 3 | 0 3 0 -1 }}
+: mapping generators: ~2, ~8/7
-: [[gencom]]: [2 8/7; 1029/1024]
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.000, ~8/7 = 233.889
+* [[WE]]: ~2 = 1200.4859{{c}}, ~8/7 = 233.7822{{c}}
-: [[error map]]: {{val| 0.000 -0.288 -2.715 }}
+: [[error map]]: {{val| +0.486 -0.123 -1.151 }}
-* [[POTE]]: ~2 = 1200.000, ~8/7 = 233.688
+* [[CWE]]: ~2 = 1200.000{{c}}, ~8/7 = 233.7474{{c}}
-: error map: {{val| 0.000 -0.892 -2.513 }}
+: error map: {{val| 0.000 -0.713 -2.573 }}
-{{Optimal ET sequence|legend=1| 36, 77, 113, 190 }}
+{{Optimal ET sequence|legend=1| 5, 21, 26, 31, 36, 77, 113, 190 }}
-=== Radon ===
+[[Badness]] (Sintel): 0.158
-[[Subgroup]]: 2.3.7.11
-[[Comma list]]: 896/891, 1029/1024
+=== Overview to extensions ===
+==== Full 7-limit extensions ====
+To the gamelisma itself we need to add the comma which appears next on the modified [[Normal lists #Normal interval list|normal comma list]] for the full 7-limit. The second comma on the list for mothra is [[81/80]], for rodan [[245/243]], for guiron [[32805/32768]], for gorgo [[36/35]], and for gidorah [[256/245]]. These all use ~8/7 as a generator, though in the case of gidorah that is the same as ~[[6/5]].
-{{Mapping|legend=2| 1 1 3 6 | 0 3 -1 -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.
-{{Mapping|legend=3| 1 1 0 3 6 | 0 3 0 -1 -13 }}
+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]]
-: [[gencom]]: [2 8/7; 896/891 1029/1024]
+The rest are considered below.
-[[Optimal tuning]]s:
+==== Subgroup extensions ====
-* [[CTE]]: ~2 = 1200.000, ~8/7 = 234.384
+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.
-: [[error map]]: {{val| 0.000 +1.197 -3.210 +1.691 }}
-* [[POTE]]: ~2 = 1200.000, ~8/7 = 234.381
-: error map: {{val| 0.000 +1.187 -3.206 +1.735 }}
-{{Optimal ET sequence|legend=1| 36, 41, 87, 128 }}
+=== Radon ===
+{{See also|Chromatic pairs #Radon}}
-=== Baladic ===
+Radon is the no-fives version of [[rodan]], equating the diatonic major third to [[14/11]].
-Baladic is a 2.3.7.13.17 subgroup temperament that attempts to approximate the Maqam Sikah Baladi scale. 36edo is an excellent baladic tuning.
-[[Subgroup]]: 2.3.7.13
+Subgroup: 2.3.7.11
-[[Comma list]]: 169/168, 1029/1024
+Comma list: 896/891, 1029/1024
-[[Gencom]]: [91/64 8/7; 169/168 1029/1024]
+Subgroup-val mapping: {{mapping| 1 1 3 6 | 0 3 -1 -13 }}
-[[Mapping|Sval mapping]]: [{{val|2 2 6 7}}, {{val|0 3 -1 1}}]
+Gencom mapping: {{mapping| 1 1 0 3 6 | 0 3 0 -1 -13 }}
-[[Tp tuning|POL2 generator]]: ~8/7 = 233.6044
+Optimal tunings:
+* WE: ~2 = 1199.9708{{c}}, ~8/7 = 234.3748{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.3813{{c}}
-{{Optimal ET sequence|legend=1| 10, 26, 36, 154…, 190…, 226…, 262… }}
+{{Optimal ET sequence|legend=0| 5, …, 36, 41, 87, 128 }}
-[[Tp tuning #T2 tuning|RMS error]]: 0.5452 cents
+Badness (Sintel): 0.619
-==== 2.3.7.13.17 ====
-Subgroup: 2.3.7.13.17
-[[Comma list]]: 169/168, 273/272, 289/288
-{{Mapping|legend=2| 2 2 6 7 7 | 0 3 -1 1 3 }}
-: sval mapping generators: ~17/12, ~8/7
-[[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.000, ~8/7 = 234.138
-* [[POTE]]: ~2 = 1200.000, ~8/7 = 233.616
-{{Optimal ET sequence|legend=1| 10, 26, 36, 154f, 190ffg, 226ffg }}
 == Mothra ==
-{{main|Mothra}}
+{{Main| Mothra }}
-Mothra tempers out [[81/80]] and finds the prime 5 at a stack of four fifths as does any temperament in the [[meantone family]]. It also tempers out [[1728/1715]], the orwellisma. It can be described as the {{nowrap|26 &amp; 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]].
+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]].
 Note that mothra is also called '''cynder''' in the 7-limit, which can be a little confusing sometimes.
-Its [[S-expression]]-based comma list is {[[1728/1715|S6/S7]], [[1029/1024|S7/S8]](, [[81/80|S6/S8 = S9]])}, taking advantage of the fact that [[81/80]] is a [[semiparticular]].
+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
@@ Line 110: / Line 83: @@
 {{Mapping|legend=1| 1 1 0 3 | 0 3 12 -1 }}
-: mapping generators: ~2, ~8/7
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.000, ~8/7 = 232.400
+* [[WE]]: ~2 = 1200.9303{{c}}, ~8/7 = 232.3733{{c}}
-: [[error map]]: {{val| 0.000 -4.756 +2.482 -1.226 }}
+: [[error map]]: {{val| +0.930 -3.905 +2.165 +1.592 }}
-* [[POTE]]: ~2 = 1200.000, ~8/7 = 232.193
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 232.2514{{c}}
-: error map: {{val| 0.000 -5.375 +0.005 -1.019 }}
+: error map: {{val| 0.000 -5.520 +0.703 -1.077 }}
 [[Algebraic generator]]: Rabrindanath, largest real root of ''x''<sup>8</sup> - 3''x''<sup>2</sup> + 1, or 232.0774 cents.
@@ Line 123: / Line 94: @@
 [[Minimax tuning]]:
 * [[7-odd-limit|7-]] and [[9-odd-limit]]: ~8/7 = {{monzo| 0 0 1/12 }}
-: [{{monzo| 1 0 0 0 }}, {{monzo| 1 0 1/4 0 }}, {{monzo| 0 0 1 0 }}, {{monzo| 3 0 -1/12 0 }}]
+: {{monzo list| 1 0 0 0 | 1 0 1/4 0 | 0 0 1 0 | 3 0 -1/12 0 }}
-: [[eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5
+: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5
 {{Optimal ET sequence|legend=1| 5, 21c, 26, 31 }}
-[[Badness]] (Smith): 0.037146
+[[Badness]] (Sintel): 0.940
+=== Undecimal mothra ===
+Undecimal mothra is the extension of 7-limit cynder which tempers out 385/384 as is natural in slendric temperaments. It is the simplest extension, supported within a reasonable tuning range (between [[26edo]] and 31edo), and is supported by the patent val of [[5edo]], which implies that it is better behaved as a cluster temperament. It is also notable for being supported by the just tuning of 8/7, and has a restriction to the 2.7.11 subgroup, namely [[amaranthine]], that is a microtemperament.
-=== 11-limit ===
 Subgroup: 2.3.5.7.11
 Comma list: 81/80, 99/98, 385/384
-Mapping: {{mapping| 1 1 0 3 5 | 0 3 12 -1 -8 }}
+{{Mapping|legend=0| 1 1 0 3 5 | 0 3 12 -1 -8 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~8/7 = 232.203
+* WE: ~2 = 1201.3979{{c}}, ~8/7 = 232.3010{{c}}
-* POTE: ~2 = 1200.000, ~8/7 = 232.031
+* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 232.0621{{c}}
 {{Optimal ET sequence|legend=0| 5, 26, 31, 88, 119be, 150be }}
-Badness (Smith): 0.025642
+Badness (Sintel): 0.848
 ==== 13-limit ====
@@ Line 150: / Line 123: @@
 Comma list: 81/80, 99/98, 105/104, 144/143
-Mapping: {{mapping| 1 1 0 3 5 1 | 0 3 12 -1 -8 14 }}
+{{Mapping|legend=0| 1 1 0 3 5 1 | 0 3 12 -1 -8 14 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~8/7 = 231.993
+* WE: ~2 = 1201.0985{{c}}, ~8/7 = 232.0231{{c}}
-* POTE: ~2 = 1200.000, ~8/7 = 231.811
+* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 231.8425{{c}}
 {{Optimal ET sequence|legend=0| 5, 26, 31, 57, 88 }}
-Badness (Smith): 0.023954
+Badness (Sintel): 0.990
+==== 17-limit ====
+Subgroup: 2.3.5.7.11.13.17
+Comma list: 81/80, 99/98, 105/104, 120/119, 144/143
+{{Mapping|legend=0| 1 1 0 3 5 1 | 0 3 12 -1 -8 14 16 }}
+Optimal tunings:
+* WE: ~2 = 1200.9734{{c}}, ~8/7 = 231.8960{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 231.7392{{c}}
+{{Optimal ET sequence|legend=0| 5g, 26, 31, 57, 88 }}
+Badness (Sintel): 1.00
+==== 19-limit ====
+Subgroup: 2.3.5.7.11.13.17.19
+Comma list: 81/80, 99/98, 105/104, 120/119, 144/143, 153/152
+{{Mapping|legend=0| 1 1 0 3 5 1 | 0 3 12 -1 -8 14 16 22 }}
+Optimal tunings:
+* WE: ~2 = 1200.9663{{c}}, ~8/7 = 231.8393{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 231.6842{{c}}
+{{Optimal ET sequence|legend=0| 26, 31, 57 }}
-; Music
+Badness (Sintel): 1.05
-* ''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
 === Mosura ===
-The [[S-expression]]-based comma list of mosura suggests it might be the most natural extension of 7-limit cynder to the 11-limit: {[[1728/1715|S6/S7]], [[1029/1024|S7/S8]], ([[81/80|S6/S8 = S9]],) [[176/175|S8/S10]]}.
+The [[S-expression]]-based comma list of mosura suggests it might be the most natural extension of 7-limit cynder to the 11-limit: {[[1728/1715|S6/S7]], [[1029/1024|S7/S8]], ([[81/80|S6/S8 = S9]]), [[176/175|S8/S10]]}.
 Subgroup: 2.3.5.7.11
@@ Line 170: / Line 170: @@
 Comma list: 81/80, 176/175, 540/539
-Mapping: {{mapping| 1 1 0 3 -1 | 0 3 12 -1 23 }}
+{{Mapping|legend=0| 1 1 0 3 -1 | 0 3 12 -1 23 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~8/7 = 232.557
+* WE: ~2 = 1200.7675{{c}}, ~8/7 = 232.5673{{c}}
-* POTE: ~2 = 1200.000, ~8/7 = 232.419
+* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 232.4567{{c}}
 {{Optimal ET sequence|legend=0| 5e, 26e, 31, 129 }}
-Badness (Smith): 0.031334
+Badness (Sintel): 1.04
 ==== 13-limit ====
@@ Line 185: / Line 185: @@
 Comma list: 81/80, 144/143, 176/175, 196/195
-Mapping: {{mapping| 1 1 0 3 -1 7 | 0 3 12 -1 23 -17 }}
+{{Mapping|legend=0| 1 1 0 3 -1 7 | 0 3 12 -1 23 -17 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~8/7 = 232.635
+* WE: ~2 = 1199.9347{{c}}, ~8/7 = 232.6275{{c}}
-* POTE: ~2 = 1200.000, ~8/7 = 232.640
+* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 232.6392{{c}}
 {{Optimal ET sequence|legend=0| 31, 67, 98 }}
-Badness (Smith): 0.036857
+Badness (Sintel): 1.52
+==== 17-limit ====
+Subgroup: 2.3.5.7.11.13.17
+Comma list: 81/80, 144/143, 176/175, 189/187, 196/195
+{{Mapping|legend=0| 1 1 0 3 -1 7 | 0 3 12 -1 23 -17 -15 }}
+Optimal tunings:
+* WE: ~2 = 1199.7124{{c}}, ~8/7 = 232.6376{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 232.6917{{c}}
+{{Optimal ET sequence|legend=0| 31, 67, 98 }}
+Badness (Sintel): 1.53
+==== 19-limit ====
+Subgroup: 2.3.5.7.11.13.17.19
+Comma list: 81/80, 96/95, 144/143, 153/152, 176/175, 196/195
+{{Mapping|legend=0| 1 1 0 3 -1 7 | 0 3 12 -1 23 -17 -15 -9 }}
+Optimal tunings:
+* WE: ~2 = 1199.4885{{c}}, ~8/7 = 232.6310{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 232.7287{{c}}
+{{Optimal ET sequence|legend=0| 31, 67, 98h }}
-=== Cynder ===
+Badness (Sintel): 1.50
+=== Cyndra ===
 Subgroup: 2.3.5.7.11
 Comma list: 45/44, 81/80, 1029/1024
-Mapping: {{mapping| 1 1 0 3 0 | 0 3 12 -1 18 }}
+{{Mapping|legend=0| 1 1 0 3 0 | 0 3 12 -1 18 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~8/7 = 231.566
+* WE: ~2 = 1201.1585{{c}}, ~8/7 = 231.5404{{c}}
-* POTE: ~2 = 1200.000, ~8/7 = 231.317
+* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 231.3850{{c}}
 {{Optimal ET sequence|legend=0| 5e, 21ce, 26 }}
-Badness (Smith): 0.055706
+Badness (Sintel): 1.84
 ==== 13-limit ====
@@ Line 215: / Line 245: @@
 Comma list: 45/44, 78/77, 81/80, 640/637
-Mapping: {{mapping| 1 1 0 3 0 1 | 0 3 12 -1 18 14 }}
+{{Mapping|legend=0| 1 1 0 3 0 1 | 0 3 12 -1 18 14 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~8/7 = 231.546
+* WE: ~2 = 1201.1152{{c}}, ~8/7 = 231.5079{{c}}
-* POTE: ~2 = 1200.000, ~8/7 = 231.293
+* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 231.3612{{c}}
 {{Optimal ET sequence|legend=0| 5e, 21cef, 26 }}
-Badness (Smith): 0.034124
+Badness (Sintel): 1.41
 == Rodan ==
@@ Line 229: / Line 259: @@
 : ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum #Rodan (5-limit)]].''
-Rodan tempers out 245/243 and can be described as the {{nowrap|41 &amp; 46}} temperament. This temperament extends neatly to the 13-limit, though the perfect fifth is sharper than ideal for slendric.
+Rodan tempers out 245/243 and can be described as the {{nowrap| 41 & 46 }} temperament. This temperament is more accurate than mothra and extends neatly to the 13-limit, though the perfect fifth is sharper than ideal for slendric. [[87edo]] is excellent for this, with the 17\87 generator missing the 13-limit CWE tuning by less than a millicent.
 [[Subgroup]]: 2.3.5.7
@@ Line 238: / Line 268: @@
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.000, ~8/7 = 234.450
+* [[WE]]: ~2 = 1200.2146{{c}}, ~8/7 = 234.4587{{c}}
-: [[error map]]: {{val| 0.000 +1.396 -0.660 -3.276 }}
+: [[error map]]: {{val| +0.215 +1.636 -0.731 -2.641 }}
-* [[POTE]]: ~2 = 1200.000, ~8/7 = 234.417
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 234.4259{{c}}
-: error map: {{val| 0.000 +1.295 -1.229 -3.243 }}
+: error map: {{val| 0.000 +1.323 -1.073 -3.252 }}
 [[Minimax tuning]]:
 * [[7-odd-limit|7-]] and [[9-odd-limit]]: ~8/7 = {{monzo| 2/9 0 1/18 -1/18 }}
 : {{monzo list| 1 0 0 0 | 5/3 0 1/6 -1/6 | 25/9 0 17/18 -17/18 | 25/9 0 -1/18 1/18 }}
-: [[eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.7/5
+: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5
 [[Algebraic generator]]: larger root of 20''x''<sup>2</sup> - 36''x'' + 15, or (9 + √6)/10.
@@ Line 252: / Line 282: @@
 {{Optimal ET sequence|legend=1| 41, 87, 128, 215d }}
-[[Badness]] (Smith): 0.037112
+[[Badness]] (Sintel): 0.939
 === 11-limit ===
@@ Line 259: / Line 289: @@
 Comma list: 245/243, 385/384, 441/440
-Mapping: {{mapping| 1 1 -1 3 6 | 0 3 17 -1 -13 }}
+{{Mapping|legend=0| 1 1 -1 3 6 | 0 3 17 -1 -13 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~8/7 = 234.463
+* WE: ~2 = 1200.0553{{c}}, ~8/7 = 234.4695{{c}}
-* POTE: ~2 = 1200.000, ~8/7 = 234.459
+* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.4594{{c}}
 Minimax tuning:
 * 11-odd-limit: ~8/7 = {{monzo| 4/19 2/19 0 0 -1/19 }}
 : [{{monzo| 1 0 0 0 0 }}, {{monzo| 31/19 6/19 0 0 -3/19 }}, {{monzo| 49/19 34/19 0 0 -17/19 }}, {{monzo| 53/19 -2/19 0 0 1/19 }}, {{monzo| 62/19 -26/19 0 0 13/19 }}]
-: eigenmonzo (unchanged-interval) basis: 2.11/9
+: unchanged-interval (eigenmonzo) basis: 2.11/9
 Algebraic generator: positive root of ''x''<sup>2</sup> + 16''x'' - 31, or √95 - 8.
@@ Line 274: / Line 304: @@
 {{Optimal ET sequence|legend=0| 41, 87 }}
-Badness (Smith): 0.023093
+Badness (Sintel): 0.763
 ==== 13-limit ====
@@ Line 281: / Line 311: @@
 Comma list: 196/195, 245/243, 352/351, 364/363
-Mapping: {{mapping| 1 1 -1 3 6 8 | 0 3 17 -1 -13 -22 }}
+{{Mapping|legend=0| 1 1 -1 3 6 8 | 0 3 17 -1 -13 -22 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~8/7 = 234.482
+* WE: ~2 = 1199.9868{{c}}, ~8/7 = 234.4796{{c}}
-* POTE: ~2 = 1200.000, ~8/7 = 234.482
+* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.4822{{c}}
 Minimax tuning:
 * 13- and 15-odd-limit: ~8/7 = {{monzo| 3/14 1/14 0 0 0 -1/28 }}
-: eigenmonzo (unchanged-interval) basis: 2.13/9
+: unchanged-interval (eigenmonzo) basis: 2.13/9
 Algebraic generator: Gatetone, positive root of 4''x''<sup>6</sup> - 7''x'' - 1. Recurrence converges slowly.
@@ Line 295: / Line 325: @@
 {{Optimal ET sequence|legend=0| 41, 46, 87 }}
-Badness (Smith): 0.018448
+Badness (Sintel): 0.762
 ===== 17-limit =====
@@ Line 302: / Line 332: @@
 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 }}
+{{Mapping|legend=0| 1 1 -1 3 6 8 8 | 0 3 17 -1 -13 -22 -20 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~8/7 = 234.532
+* WE: ~2 = 1199.8331{{c}}, ~8/7 = 234.4919{{c}}
-* POTE: ~2 = 1200.000, ~8/7 = 234.524
+* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.5254{{c}}
 Minimax tuning:
 * 17-odd-limit: ~8/7 = {{monzo| 3/13 1/13 0 0 0 0 -1/26 }}
-: eigenmonzo (unchanged-interval) basis: 2.17/9
+: unchanged-interval (eigenmonzo) basis: 2.17/9
 {{Optimal ET sequence|legend=0| 41, 46, 87 }}
-Badness (Smith): 0.016743
+Badness (Sintel): 0.853
 ==== Aerodactyl ====
@@ Line 321: / Line 351: @@
 Comma list: 91/90, 245/243, 385/384, 441/440
-Mapping: {{mapping| 1 1 -1 3 6 -1 | 0 3 17 -1 -13 24 }}
+{{Mapping|legend=0| 1 1 -1 3 6 -1 | 0 3 17 -1 -13 24 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~8/7 = 234.670
+* WE: ~2 = 1200.2997{{c}}, ~8/7 = 234.6972{{c}}
-* POTE: ~2 = 1200.000, ~8/7 = 234.639
+* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.6439{{c}}
 {{Optimal ET sequence|legend=0| 5, 41f, 46 }}
-Badness (Smith): 0.033986
+Badness (Sintel): 1.40
 === Aerodino ===
@@ Line 336: / Line 366: @@
 Comma list: 176/175, 245/243, 1029/1024
-Mapping: {{mapping| 1 1 -1 3 -3 | 0 3 17 -1 33 }}
+{{Mapping|legend=0| 1 1 -1 3 -3 | 0 3 17 -1 33 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~8/7 = 234.719
+* WE: ~2 = 1199.9179{{c}}, ~8/7 = 234.7123{{c}}
-* POTE: ~2 = 1200.000, ~8/7 = 234.728
+* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.7256{{c}}
 {{Optimal ET sequence|legend=0| 5e, 41e, 46 }}
-Badness (Smith): 0.054294
+Badness (Sintel): 1.79
 ==== 13-limit ====
@@ Line 351: / Line 381: @@
 Comma list: 91/90, 176/175, 245/243, 847/845
-Mapping: {{mapping| 1 1 -1 3 -3 -1 | 0 3 17 -1 33 24 }}
+{{Mapping|legend=0| 1 1 -1 3 -3 -1 | 0 3 17 -1 33 24 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~8/7 = 234.786
+* WE: ~2 = 1200.0242{{c}}, ~8/7 = 234.7863{{c}}
-* POTE: ~2 = 1200.000, ~8/7 = 234.782
+* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.7824{{c}}
 {{Optimal ET sequence|legend=0| 5e, 41ef, 46 }}
-Badness (Smith): 0.035836
+Badness (Sintel): 1.48
 === Varan ===
@@ Line 366: / Line 396: @@
 Comma list: 100/99, 245/243, 1029/1024
-Mapping: {{mapping| 1 1 -1 3 -2 | 0 3 17 -1 28 }}
+{{Mapping|legend=0| 1 1 -1 3 -2 | 0 3 17 -1 28 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~8/7 = 234.197
+* WE: ~2 = 1200.3738{{c}}, ~8/7 = 234.2174{{c}}
-* POTE: ~2 = 1200.000, ~8/7 = 234.145
+* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.1586{{c}}
 {{Optimal ET sequence|legend=0| 5e, 36ce, 41 }}
-Badness (Smith): 0.044937
+Badness (Sintel): 1.49
 ==== 13-limit ====
@@ Line 381: / Line 411: @@
 Comma list: 100/99, 105/104, 245/243, 352/351
-Mapping: {{mapping| 1 1 -1 3 -2 0 | 0 3 17 -1 28 19 }}
+{{Mapping|legend=0| 1 1 -1 3 -2 0 | 0 3 17 -1 28 19 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~8/7 = 234.111
+* WE: ~2 = 1200.1389{{c}}, ~8/7 = 234.1162{{c}}
-* POTE: ~2 = 1200.000, ~8/7 = 234.089
+* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.0946{{c}}
 {{Optimal ET sequence|legend=0| 5e, 36ce, 41 }}
-Badness (Smith): 0.032284
+Badness (Sintel): 1.33
 == Guiron ==
-Guiron tempers out the [[schisma]], and finds the prime 5 at the diminished fourth as does any temperament in the [[schismatic family]]. It can be described as the {{nowrap|36 &amp; 41}} temperament. It is more complex than rodan, but the optimal tuning is closer to optimal slendric.
+Guiron tempers out the [[schisma]], and finds the prime 5 at the diminished fourth as does any temperament in the [[schismatic family]]. It can be described as the {{nowrap| 36 & 41 }} temperament. It is more complex than rodan, but the optimal tuning is closer to optimal slendric.
 [[Subgroup]]: 2.3.5.7
@@ Line 399: / Line 429: @@
 {{Mapping|legend=1| 1 1 7 3 | 0 3 -24 -1 }}
-: mapping generators: ~2, ~8/7
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.000, ~8/7 = 233.903
+* [[WE]]: ~2 = 1200.3395{{c}}, ~8/7 = 233.9963{{c}}
-: [[error map]]: {{val| 0.000 -0.246 +0.012 -2.729 }}
+: [[error map]]: {{val| +0.340 +0.374 +0.151 -1.804 }}
-* [[POTE]]: ~2 = 1200.000, ~8/7 = 233.930
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 233.9239{{c}}
-: error map: {{val| 0.000 -0.165 -0.637 -2.756 }}
+: error map: {{val| 0.000 -0.183 -0.487 -2.750 }}
 [[Minimax tuning]]:
 * [[7-odd-limit|7-]] and [[9-odd-limit]]: ~8/7 = {{monzo| 7/24 0 -1/24 }}
 : {{monzo list| 1 0 0 0 | 15/8 0 -1/8 0 | 0 0 1 0 | 65/24 0 1/24 0 }}
-: [[eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5
+: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5
 {{Optimal ET sequence|legend=1| 36, 41, 77, 118, 277d }}
-[[Badness]] (Smith): 0.047544
+[[Badness]] (Sintel): 1.20
 === 11-limit ===
@@ Line 422: / Line 450: @@
 Comma list: 385/384, 441/440, 10976/10935
-Mapping: {{mapping| 1 1 7 3 -2 | 0 3 -24 -1 28 }}
+{{Mapping|legend=0| 1 1 7 3 -2 | 0 3 -24 -1 28 }}
-: mapping generators: ~2, ~8/7
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~8/7 = 233.930
+* WE: ~2 = 1200.3453{{c}}, ~8/7 = 233.9988{{c}}
-* POTE: ~2 = 1200.000, ~8/7 = 233.931
+* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.9312{{c}}
 Minimax tuning:
 * 11-odd-limit: ~8/7 = {{monzo| 7/24 0 -1/24 }}
 : [{{monzo| 1 0 0 0 0 }}, {{monzo| 15/8 0 -1/8 0 0 }}, {{monzo| 0 0 1 0 0 }}, {{monzo| 65/24 0 1/24 0 0 }}, {{monzo| 37/6 0 -7/6 0 0 }}]
-: eigenmonzo (unchanged-interval) basis: 2.5
+: unchanged-interval (eigenmonzo) basis: 2.5
 {{Optimal ET sequence|legend=0| 36e, 41, 77, 118, 159, 277d }}
-Badness (Smith): 0.026648
+Badness (Sintel): 0.881
 === 13-limit ===
@@ Line 444: / Line 470: @@
 Comma list: 196/195, 352/351, 385/384, 729/728
-Mapping: {{mapping| 1 1 7 3 -2 0 | 0 3 -24 -1 28 19 }}
+{{Mapping|legend=0| 1 1 7 3 -2 0 | 0 3 -24 -1 28 19 }}
-: mapping generators: ~2, ~8/7
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~8/7 = 233.902
+* WE: ~2 = 1200.1222{{c}}, ~8/7 = 233.9228{{c}}
-* POTE: ~2 = 1200.000, ~8/7 = 233.899
+* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.8994{{c}}
 {{Optimal ET sequence|legend=0| 36e, 41, 77, 118 }}
-Badness (Smith): 0.028444
+Badness (Sintel): 1.18
 == Gorgo ==
 : ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum #Laconic]].''
+{{See also| Llywelynsmic clan }}
-{{ See also | Shoe }}
+Gorgo tempers the generator of ~8/7 together with ~10/9. It can be described as the {{nowrap| 16 & 21 }} temperament.
-Gorgo tempers the generator of ~8/7 together with ~10/9. It can be described as the {{nowrap|16 &amp; 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.
-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
@@ Line 472: / Line 495: @@
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.000, ~8/7 = 228.724
+* [[WE]]: ~2 = 1200.9847{{c}}, ~8/7 = 228.5210{{c}}
-: [[error map]]: {{val| 0.000 -15.782 +14.756 +2.450 }}
+: [[error map]]: {{val| +0.985 -15.407 +14.318 +5.607 }}
-* [[POTE]]: ~2 = 1200.000, ~8/7 = 228.334
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 228.4371{{c}}
-: error map: {{val| 0.000 -16.954 +12.022 +2.840 }}
+: error map: {{val| 0.000 -16.644 +12.746 +2.737 }}
 {{Optimal ET sequence|legend=1| 5, 11c, 16, 21 }}
-[[Badness]] (Smith): 0.060663
+[[Badness]] (Sintel): 1.54
 === 11-limit ===
@@ Line 486: / Line 509: @@
 Comma list: 36/35, 45/44, 1029/1024
-Mapping: {{mapping| 1 1 1 3 1 | 0 3 7 -1 13 }}
+{{Mapping|legend=0| 1 1 1 3 1 | 0 3 7 -1 13 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~8/7 = 227.833
+* WE: ~2 = 1201.3609{{c}}, ~8/7 = 227.6312{{c}}
-* POTE: ~2 = 1200.000, ~8/7 = 227.373
+* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 227.4955{{c}}
 {{Optimal ET sequence|legend=0| 5e, 16, 21, 37b }}
-Badness (Smith): 0.049500
+Badness (Sintel): 1.64
 ==== 13-limit ====
@@ Line 501: / Line 524: @@
 Comma list: 27/26, 36/35, 45/44, 507/500
-Mapping: {{mapping| 1 1 1 3 1 2 | 0 3 7 -1 13 9 }}
+{{Mapping|legend=0| 1 1 1 3 1 2 | 0 3 7 -1 13 9 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~8/7 = 227.633
+* WE: ~2 = 1201.0996{{c}}, ~8/7 = 227.4378{{c}}
-* POTE: ~2 = 1200.000, ~8/7 = 227.230
+* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 227.3327{{c}}
 {{Optimal ET sequence|legend=0| 5e, 16, 21, 37b }}
-Badness (Smith): 0.032664
+Badness (Sintel): 1.35
 === Spartan ===
@@ Line 516: / Line 539: @@
 Comma list: 36/35, 56/55, 1029/1024
-Mapping: {{mapping| 1 1 1 3 5 | 0 3 7 -1 -8 }}
+{{Mapping|legend=0| 1 1 1 3 5 | 0 3 7 -1 -8 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~8/7 = 229.420
+* WE: ~2 = 1198.9344{{c}}, ~8/7 = 229.3316{{c}}
-* POTE: ~2 = 1200.000, ~8/7 = 229.535
+* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 229.5124{{c}}
 {{Optimal ET sequence|legend=0| 5, 16e, 21 }}
-Badness (Smith): 0.062683
+Badness (Sintel): 2.07
 ==== 13-limit ====
@@ Line 531: / Line 554: @@
 Comma list: 27/26, 36/35, 56/55, 507/500
-Mapping: {{mapping| 1 1 1 3 5 2 | 0 3 7 -1 -8 9 }}
+{{Mapping|legend=0| 1 1 1 3 5 2 | 0 3 7 -1 -8 9 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~8/7 = 228.758
+* WE: ~2 = 1198.3002{{c}}, ~8/7 = 228.7341{{c}}
-* POTE: ~2 = 1200.000, ~8/7 = 229.059
+* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 229.0044{{c}}
 {{Optimal ET sequence|legend=0| 5, 16e, 21 }}
-Badness (Smith): 0.047071
+Badness (Sintel): 1.95
 ; Music
@@ Line 556: / Line 579: @@
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.000, ~8/7 = 227.100
+* [[WE]]: ~2 = 1192.4932{{c}}, ~8/7 = 229.3187{{c}}
-: [[error map]]: {{val| 0.000 -20.655 +67.886 +4.074 }}
+: [[error map]]: {{val| -7.507 -21.506 +57.310 -20.665 }}
-* [[POTE]]: ~2 = 1200.000, ~8/7 = 230.762
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 229.6649{{c}}
-: error map: {{val| 0.000 -9.668 +75.211 +0.412 }}
+: error map: {{val| 0.000 -12.960 +73.016 +1.509 }}
 {{Optimal ET sequence|legend=1| 1b, 5 }}
-[[Badness]] (Smith): 0.062262
+[[Badness]] (Sintel): 1.58
 == Oncle ==
 : ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Oncle]].''
-Oncle can be described as the {{nowrap|31 &amp; 36c}} temperamnet.
+Oncle can be described as the {{nowrap| 31 & 36c }} temperament.
 [[Subgroup]]: 2.3.5.7
@@ Line 577: / Line 600: @@
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.000, ~8/7 = 232.383
+* [[WE]]: ~2 = 1201.2246{{c}}, ~8/7 = 232.7354{{c}}
-: [[error map]]: {{val| 0.000 -4.807 -1.585 -1.209 }}
+: [[error map]]: {{val| +1.225 -2.524 -0.939 +2.112 }}
-* [[POTE]]: ~2 = 1200.000, ~8/7 = 232.498
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 232.4718{{c}}
-: error map: {{val| 0.000 -4.461 -3.778 -1.324 }}
+: error map: {{val| 0.000 -4.539 -3.279 -1.298 }}
 {{Optimal ET sequence|legend=1| 31, 98c, 129c, 160bc }}
-[[Badness]] (Smith): 0.088384
+[[Badness]] (Sintel): 2.24
 == Archaeotherium ==
 : ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Archaeotherium]].''
-Archaeotherium can be described as the {{nowrap|21 &amp; 26}} temperamnet.
+Archaeotherium can be described as the {{nowrap| 21 & 26 }} temperament.
 [[Subgroup]]: 2.3.5.7
@@ Line 598: / Line 621: @@
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.000, ~8/7 = 229.951
+* [[WE]]: ~2 = 1202.7179{{c}}, ~8/7 = 230.7800{{c}}
-: [[error map]]: {{val| 0.000 -12.102 -5.626 +1.223 }}
+: [[error map]]: {{val| +2.718 -6.897 -3.644 +8.548 }}
-* [[POTE]]: ~2 = 1200.000, ~8/7 = 230.258
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 230.1909{{c}}
-: error map: {{val| 0.000 -11.180 -9.933 +0.916 }}
+: error map: {{val| 0.000 -11.382 -8.986 +0.983 }}
-{{Optimal ET sequence|legend=1| 21, 26, 47, 73bc, 99bc }}
+{{Optimal ET sequence|legend=1| 21, 26, 47, 73bc }}
-[[Badness]] (Smith): 0.146306
+[[Badness]] (Sintel): 3.70
 == Clyndro ==
-{{See also| Mavila family }}
+Clyndro tempers out [[135/128]] and finds the interval class of 5 at a stack of -3 fifths as does any temperament in the [[mavila family]]. It can be described as the {{nowrap| 11 & 16 }} temperament.
-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 &amp; 16}} temperamnet.
 [[Subgroup]]: 2.3.5.7
@@ Line 619: / Line 640: @@
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.000, ~8/7 = 225.752
+* [[WE]]: ~2 = 1205.6135{{c}}, ~8/7 = 227.5283{{c}}
-: [[error map]]: {{val| 0.000 -24.699 -18.081 +5.422 }}
+: [[error map]]: {{val| +5.613 -13.757 -11.614 +20.486 }}
-* [[POTE]]: ~2 = 1200.000, ~8/7 = 226.469
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 226.3207{{c}}
-: error map: {{val| 0.000 -22.548 -24.534 +4.705 }}
+: error map: {{val| 0.000 -22.993 -23.200 +4.853 }}
 {{Optimal ET sequence|legend=1| 5c, 11, 16 }}
-[[Badness]] (Smith): 0.159179
+[[Badness]] (Sintel): 4.03
 === 11-limit ===
@@ Line 633: / Line 654: @@
 Comma list: 33/32, 45/44, 352/343
-Mapping: {{mapping| 1 1 4 3 4 | 0 3 -9 -1 -3 }}
+{{Mapping|legend=0| 1 1 4 3 4 | 0 3 -9 -1 -3 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~8/7 = 225.384
+* WE: ~2 = 1206.2134{{c}}, ~8/7 = 227.6004{{c}}
-* POTE: ~2 = 1200.000, ~8/7 = 226.428
+* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 226.2421{{c}}
 {{Optimal ET sequence|legend=0| 5c, 11, 16 }}
-Badness (Smith): 0.069703
+Badness (Sintel): 2.30
 == Miracle ==
@@ Line 647: / Line 668: @@
 : ''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 &amp; 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.
+Miracle is one of the most important entries of this temperament clan. It tempers out [[225/224]], splitting the ~8/7 generator of slendric into [[15/14]]~[[16/15]], and can be described as the {{nowrap| 31 & 41 }} temperament. Its ploidacot is hexacot. It is then extremely natural to equate the neutral third, three generators up, to [[11/9]] and thereby extend miracle to the full [[11-limit]] with essentially no further damage. [[72edo]] makes for an excellent tuning.
 [[Subgroup]]: 2.3.5.7
@@ Line 654: / Line 675: @@
 {{Mapping|legend=1| 1 1 3 3 | 0 6 -7 -2 }}
 : mapping generator: ~2, ~15/14
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.000, ~15/14 = 116.677
+* [[WE]]: ~2 = 1200.8209{{c}}, ~15/14 = 116.7550{{c}}
-: [[error map]]: {{val| 0.000 -1.892 -3.054 -2.180 }}
+: [[error map]]: {{val| +0.821 -0.604 -1.136 +0.127 }}
-* [[POTE]]: ~2 = 1200.000, ~15/14 = 116.675
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~15/14 = 116.6756{{c}}
-: error map: {{val| 0.000 -1.904 -3.040 -2.176 }}
+: error map: {{val| 0.000 -1.901 -3.043 -2.177 }}
 [[Minimax tuning]]:
 * [[7-odd-limit]]: ~15/14 = {{monzo| 2/13 1/13 -1/13 }}
 : {{monzo list| 1 0 0 0 | 25/13 6/13 -6/13 0 | 25/13 -7/13 7/13 0 | 35/13 -2/13 2/13 0 }}
-: [[eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5/3
+: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5/3
 * [[9-odd-limit]]: ~15/14 = {{monzo| 1/19 2/19 -1/19 }}
 : {{monzo list| 1 0 0 0 | 25/19 12/19 -6/19 0 | 50/19 -14/19 7/19 0 | 55/19 -4/19 2/19 0 }}
-: [[eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.9/5
+: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5
 [[Tuning ranges]]:
@@ Line 680: / Line 700: @@
 {{Optimal ET sequence|legend=1| 10, 21, 31, 41, 72 }}
-[[Badness]] (Smith): 0.016742
+[[Badness]] (Sintel): 0.424
 === 11-limit ===
@@ Line 687: / Line 707: @@
 Comma list: 225/224, 243/242, 385/384
-Mapping: {{mapping| 1 1 3 3 2 | 0 6 -7 -2 15 }}
+{{Mapping|legend=0| 1 1 3 3 2 | 0 6 -7 -2 15 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~15/14 = 116.711
+* WE: ~2 = 1200.7626{{c}}, ~15/14 = 116.7069{{c}}
-* POTE: ~2 = 1200.000, ~15/14 = 116.633
+* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.6469{{c}}
 Minimax tuning:
 * 11-odd-limit: ~15/14 = {{monzo| 1/19 2/19 -1/19 }}
 : [{{monzo| 1 0 0 0 0 }}, {{monzo| 25/19 12/19 -6/19 0 0 }}, {{monzo| 50/19 -14/19 7/19 0 0 }}, {{monzo| 55/19 -4/19 2/19 0 0 }}, {{monzo| 53/19 30/19 -15/19 0 0 }}]
-: eigenmonzo (unchanged-interval) basis: 2.9/5
+: unchanged-interval (eigenmonzo) basis: 2.9/5
 Tuning ranges:
@@ Line 706: / Line 726: @@
 {{Optimal ET sequence|legend=0| 10, 21e, 31, 41, 72, 247c, 319bcde, 391bcde, 463bccde }}
-Badness (Smith): 0.010684
+Badness (Sintel): 0.353
 ==== Miraculous ====
@@ Line 713: / Line 733: @@
 Comma list: 105/104, 144/143, 196/195, 243/242
-Mapping: {{mapping| 1 1 3 3 2 4 | 0 6 -7 -2 15 -3 }}
+{{Mapping|legend=0| 1 1 3 3 2 4 | 0 6 -7 -2 15 -3 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~15/14 = 116.758
+* WE: ~2 = 1200.1267{{c}}, ~15/14 = 116.7596{{c}}
-* POTE: ~2 = 1200.000, ~15/14 = 116.747
+* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.7488{{c}}
-{{Optimal ET sequence|legend=0| 10, 21e, 31, 41, 72f, 113f, 185cff }}
+{{Optimal ET sequence|legend=0| 10, 21e, 31, 41, 72f }}
-Badness (Smith): 0.018669
+Badness (Sintel): 0.771
 ===== 17-limit =====
@@ Line 728: / Line 748: @@
 Comma list: 105/104, 120/119, 144/143, 154/153, 170/169
-Mapping: {{mapping| 1 1 3 3 2 4 4 | 0 6 -7 -2 15 -3 1 }}
+{{Mapping|legend=0| 1 1 3 3 2 4 4 | 0 6 -7 -2 15 -3 1 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~15/14 = 116.742
+* WE: ~2 = 1199.6759{{c}}, ~15/14 = 116.7378{{c}}
-* POTE: ~2 = 1200.000, ~15/14 = 116.769
+* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.7657{{c}}
 {{Optimal ET sequence|legend=0| 10, 21e, 31, 41, 72fg }}
-Badness (Smith): 0.017084
+Badness (Sintel): 0.870
+===== 19-limit =====
+Subgroup: 2.3.5.7.11.13.17.19
+Comma list: 105/104, 120/119, 144/143, 154/153, 170/169, 210/209
+{{Todo|complete temperament data|inline=1}}
+===== 23-limit =====
+Subgroup: 2.3.5.7.11.13.17.19.23
+Comma list: 105/104, 120/119, 144/143, 154/153, 161/160, 170/169, 210/209
+{{Todo|complete temperament data|inline=1}}
 ==== Benediction ====
@@ Line 743: / Line 777: @@
 Comma list: 225/224, 243/242, 351/350, 385/384
-Mapping: {{mapping| 1 1 3 3 2 7 | 0 6 -7 -2 15 -34 }}
+{{Mapping|legend=0| 1 1 3 3 2 7 | 0 6 -7 -2 15 -34 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~15/14 = 116.541
+* WE: ~2 = 1199.8601{{c}}, ~15/14 = 116.6572{{c}}
-* POTE: ~2 = 1200.000, ~15/14 = 116.574
+* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.5688{{c}}
 {{Optimal ET sequence|legend=0| 31, 72, 103, 175f }}
-Badness (Smith): 0.015715
+Badness (Sintel): 0.649
 ===== 17-limit =====
@@ Line 758: / Line 792: @@
 Comma list: 225/224, 243/242, 273/272, 351/350, 375/374
-Mapping: {{mapping| 1 1 3 3 2 7 7 | 0 6 -7 -2 15 -34 -30 }}
+{{Mapping|legend=0| 1 1 3 3 2 7 7 | 0 6 -7 -2 15 -34 -30 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~15/14 = 116.529
+* WE: ~2 = 1200.8328{{c}}, ~15/14 = 116.6661{{c}}
-* POTE: ~2 = 1200.000, ~15/14 = 116.585
+* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.5774{{c}}
 {{Optimal ET sequence|legend=0| 31, 72, 103, 175f, 422bcdefffg }}
-Badness (Smith): 0.012537
+Badness (Sintel): 0.639
+===== 19-limit =====
+Subgroup: 2.3.5.7.11.13.17.19
+Comma list: 210/209, 225/224, 243/242, 273/272, 286/285, 375/374
+{{Todo|complete temperament data|inline=1}}
+===== 23-limit =====
+Subgroup: 2.3.5.7.11.13.17.19.23
+Comma list: 162/161, 210/209, 225/224, 231/230, 243/242, 273/272, 286/285
+{{Todo|complete temperament data|inline=1}}
 ==== Manna ====
@@ Line 773: / Line 821: @@
 Comma list: 225/224, 243/242, 325/324, 385/384
-Mapping: {{mapping| 1 1 3 3 2 0 | 0 6 -7 -2 15 38 }}
+{{Mapping|legend=0| 1 1 3 3 2 0 | 0 6 -7 -2 15 38 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~15/14 = 116.814
+* WE: ~2 = 1200.7564{{c}}, ~15/14 = 116.8129{{c}}
-* POTE: ~2 = 1200.000, ~15/14 = 116.739
+* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.7528{{c}}
 {{Optimal ET sequence|legend=0| 31f, 41, 72, 185cf, 257cff }}
-Badness (Smith): 0.017012
+Badness (Sintel): 0.703
 ===== 17-limit =====
@@ Line 788: / Line 836: @@
 Comma list: 225/224, 243/242, 273/272, 325/324, 385/384
-Mapping: {{mapping| 1 1 3 3 2 0 0 | 0 6 -7 -2 15 38 42 }}
+{{Mapping|legend=0| 1 1 3 3 2 0 0 | 0 6 -7 -2 15 38 42 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~15/14 = 116.802
+* WE: ~2 = 1200.7570{{c}}, ~15/14 = 116.8011{{c}}
-* POTE: ~2 = 1200.000, ~15/14 = 116.727
+* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.7408{{c}}
 {{Optimal ET sequence|legend=0| 31fg, 41, 72, 185cf, 257cff }}
-Badness (Smith): 0.014680
+Badness (Sintel): 0.748
+===== 19-limit =====
+Subgroup: 2.3.5.7.11.13.17.19
+Comma list: 210/209, 225/224, 243/242, 273/272, 325/324, 343/342
+{{Todo|complete temperament data|inline=1}}
+===== 23-limit =====
+Subgroup: 2.3.5.7.11.13.17.19.23
+Comma list: 210/209, 225/224, 243/242, 273/272, 300/299, 325/324, 343/342
+{{Todo|complete temperament data|inline=1}}
 ==== Semimiracle ====
@@ Line 803: / Line 865: @@
 Comma list: 169/168, 225/224, 243/242, 385/384
-Mapping: {{mapping| 2 2 6 6 4 7 | 0 6 -7 -2 15 2 }}
+{{Mapping|legend=0| 2 2 6 6 4 7 | 0 6 -7 -2 15 2 }}
 : mapping generators: ~55/39, ~15/14
 Optimal tunings:
-* CTE: ~55/39 = 600.000, ~15/14 = 116.735
+* WE: ~55/39 = 600.4844{{c}}, ~15/14 = 116.7182{{c}}
-* POTE: ~55/39 = 600.000, ~15/14 = 116.624
+* CWE: ~55/39 = 600.0000{{c}}, ~15/14 = 116.6413{{c}}
 {{Optimal ET sequence|legend=0| 10, 62, 72 }}
-Badness (Smith): 0.024622
+Badness (Sintel): 1.02
 ===== 17-limit =====
@@ Line 820: / Line 881: @@
 Comma list: 169/168, 221/220, 225/224, 243/242, 273/272
-Mapping: {{mapping| 2 2 6 6 4 7 7 | 0 6 -7 -2 15 2 6 }}
+{{Mapping|legend=0| 2 2 6 6 4 7 7 | 0 6 -7 -2 15 2 6 }}
 Optimal tunings:
-* CTE: ~17/12 = 600.000, ~15/14 = 116.771
+* WE: ~17/12 = 600.5042{{c}}, ~15/14 = 116.7264{{c}}
-* POTE: ~17/12 = 600.000, ~15/14 = 116.628
+* CWE: ~17/12 = 600.0000{{c}}, ~15/14 = 116.6485{{c}}
 {{Optimal ET sequence|legend=0| 10, 62, 72 }}
-Badness (Smith): 0.016130
+Badness (Sintel): 0.822
+===== 19-limit =====
+Subgroup: 2.3.5.7.11.13.17.19
+Comma list: 169/168, 210/209, 221/220, 225/224, 243/242, 273/272
+{{Todo|complete temperament data|inline=1}}
+===== 23-limit =====
+Subgroup: 2.3.5.7.11.13.17.19.23
+Comma list: 169/168, 208/207, 210/209, 221/220, 225/224, 243/242, 273/272
+{{Todo|complete temperament data|inline=1}}
 ==== Hemisecordite ====
@@ Line 835: / Line 910: @@
 Comma list: 225/224, 243/242, 385/384, 847/845
-Mapping: {{mapping| 1 1 3 3 2 2 | 0 12 -14 -4 30 35 }}
+{{Mapping|legend=0| 1 1 3 3 2 2 | 0 12 -14 -4 30 35 }}
 : mapping generators: ~2, ~27/26
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~27/26 = 58.337
+* WE: ~2 = 1200.6969{{c}}, ~27/26 = 58.3217{{c}}
-* POTE: ~2 = 1200.000, ~27/26 = 58.288
+* CWE: ~2 = 1200.0000{{c}}, ~27/26 = 58.2964{{c}}
 {{Optimal ET sequence|legend=0| 41, 62, 103, 247c, 350bcde }}
-Badness (Smith): 0.025589
+Badness (Sintel): 1.06
 ===== 17-limit =====
@@ Line 852: / Line 926: @@
 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 }}
+{{Mapping|legend=0| 1 1 3 3 2 2 2 | 0 12 -14 -4 30 35 43 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~27/26 = 58.312
+* WE: ~2 = 1200.6557{{c}}, ~27/26 = 58.2932{{c}}
-* POTE: ~2 = 1200.000, ~27/26 = 58.261
+* CWE: ~2 = 1200.0000{{c}}, ~27/26 = 58.2702{{c}}
 {{Optimal ET sequence|legend=0| 41, 62, 103 }}
-Badness (Smith): 0.022535
+Badness (Sintel): 1.15
+===== 19-limit =====
+Subgroup: 2.3.5.7.11.13.17.19
+Comma list:
+{{Todo|complete temperament data|inline=1}}
+===== 23-limit =====
+Subgroup: 2.3.5.7.11.13.17.19.23
+Comma list:
+{{Todo|complete temperament data|inline=1}}
 ===== Semihemisecordite =====
@@ Line 867: / Line 955: @@
 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|legend=0| 2 2 6 6 4 4 7 | 0 12 -14 -4 30 35 12 }}
 : mapping generators: ~17/12, ~27/26
 Optimal tunings:
-* CTE: ~17/12 = 600.000, ~27/26 = 58.350
+* WE: ~17/12 = 600.3951{{c}}, ~27/26 = 58.3260{{c}}
-* POTE: ~17/12 = 600.000, ~27/26 = 58.288
+* CWE: ~17/12 = 600.0000{{c}}, ~27/26 = 58.2974{{c}}
 {{Optimal ET sequence|legend=0| 62, 144g, 206begg }}
-Badness (Smith): 0.046958
+Badness (Sintel): 2.39
 ====== 19-limit ======
@@ Line 884: / Line 971: @@
 Comma list: 209/208, 225/224, 243/242, 289/288, 361/360, 385/384
-Mapping: {{mapping| 2 2 6 6 4 4 7 8 | 0 12 -14 -4 30 35 12 5 }}
+{{Mapping|legend=0| 2 2 6 6 4 4 7 8 | 0 12 -14 -4 30 35 12 5 }}
 Optimal tunings:
-* CTE: ~17/12 = 600.000, ~27/26 = 58.356
+* WE: ~17/12 = 600.4418{{c}}, ~27/26 = 58.3255{{c}}
-* POTE: ~17/12 = 600.000, ~27/26 = 58.283
+* CWE: ~17/12 = 600.0000{{c}}, ~27/26 = 58.2928{{c}}
 {{Optimal ET sequence|legend=0| 62, 144gh, 206begghh }}
-Badness (Smith): 0.035057
+Badness (Sintel): 2.13
 ====== 23-limit ======
@@ Line 899: / Line 986: @@
 Comma list: 209/208, 225/224, 243/242, 289/288, 323/322, 361/360, 385/384
-Mapping: {{mapping| 2 2 6 6 4 4 7 8 7 | 0 12 -14 -4 30 35 12 5 21 }}
+{{Mapping|legend=0| 2 2 6 6 4 4 7 8 7 | 0 12 -14 -4 30 35 12 5 21 }}
 Optimal tunings:
-* CTE: ~17/12 = 600.000, ~27/26 = 58.366
+* WE: ~17/12 = 600.4451{{c}}, ~27/26 = 58.3264{{c}}
-* POTE: ~17/12 = 600.000, ~27/26 = 58.283
+* CWE: ~17/12 = 600.0000{{c}}, ~27/26 = 58.2942{{c}}
 {{Optimal ET sequence|legend=0| 62, 144gh, 206begghhi }}
-Badness (Smith): 0.026421
+Badness (Sintel): 1.89
 ==== Phicordial ====
@@ Line 914: / Line 1,001: @@
 Comma list: 225/224, 243/242, 385/384, 2200/2197
-Mapping: {{mapping| 1 7 -4 1 17 4 | 0 -18 21 6 -45 -1 }}
+{{Mapping|legend=0| 1 -11 17 7 -28 3 | 0 18 -21 -6 45 1 }}
+: mapping generators: ~2, ~13/8
-: mapping generators: ~2, ~16/13
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~16/13 = 361.096
+* WE: ~2 = 1200.7056{{c}}, ~13/8 = 839.3726{{c}}
-* POTE: ~2 = 1200.000, ~16/13 = 361.121
+* CWE: ~2 = 1200.0000{{c}}, ~13/8 = 838.8831{{c}}
 {{Optimal ET sequence|legend=0| 103, 216c, 319bcde, 535bccdef }}
-Badness (Smith): 0.033198
+Badness (Sintel): 1.37
 ===== 17-limit =====
 Subgroup: 2.3.5.7.11.13.17
-Comma list: 225/224, 243/242, 273/272, 441/440, 2200/2197
+Comma list: 225/224, 243/242, 273/272, 385/384, 2200/2197
-Mapping: {{mapping| 1 7 -4 1 17 4 8 | 0 -18 21 6 -45 -1 -13 }}
+{{Mapping|legend=0| 1 -11 17 7 -28 3 -5 | 0 18 -21 -6 45 1 13 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~16/13 = 361.098
+* WE: ~2 = 1200.5918{{c}}, ~13/8 = 839.2912{{c}}
-* POTE: ~2 = 1200.000, ~16/13 = 361.123
+* CWE: ~2 = 1200.0000{{c}}, ~13/8 = 838.8809{{c}}
 {{Optimal ET sequence|legend=0| 103, 216c, 319bcde }}
-Badness (Smith): 0.024705
+Badness (Sintel): 1.26
+===== 19-limit =====
+Subgroup: 2.3.5.7.11.13.17.19
+Comma list: 210/209, 225/224, 243/242, 273/272, 385/384, 2200/2197
+{{Todo|complete temperament data|inline=1}}
+===== 23-limit =====
+Subgroup: 2.3.5.7.11.13.17.19.23
+Comma list: 210/209, 225/224, 243/242, 273/272, 300/299, 385/384, 1105/1104
+{{Todo|complete temperament data|inline=1}}
 === Revelation ===
@@ Line 946: / Line 1,046: @@
 Comma list: 99/98, 176/175, 1029/1024
-Mapping: {{mapping| 1 1 3 3 5 | 0 6 -7 -2 -16 }}
+{{Mapping|legend=0| 1 1 3 3 5 | 0 6 -7 -2 -16 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~15/14 = 116.142
+* WE: ~2 = 1201.3320{{c}}, ~15/14 = 116.4057{{c}}
-* POTE: ~2 = 1200.000, ~15/14 = 116.277
+* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.2524{{c}}
 {{Optimal ET sequence|legend=0| 10e, 21, 31 }}
-Badness (Smith): 0.032946
+Badness (Sintel): 1.09
 ==== 13-limit ====
@@ Line 961: / Line 1,061: @@
 Comma list: 66/65, 99/98, 105/104, 512/507
-Mapping: {{mapping| 1 1 3 3 5 4 | 0 6 -7 -2 -16 -3 }}
+{{Mapping|legend=0| 1 1 3 3 5 4 | 0 6 -7 -2 -16 -3 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~15/14 = 116.194
+* WE: ~2 = 1200.6059{{c}}, ~15/14 = 116.3263{{c}}
-* POTE: ~2 = 1200.000, ~15/14 = 116.268
+* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.2564{{c}}
 {{Optimal ET sequence|legend=0| 10e, 21, 31 }}
-Badness (Smith): 0.029452
+Badness (Sintel): 1.22
 === Hemimiracle ===
@@ Line 976: / Line 1,076: @@
 Comma list: 225/224, 245/242, 1029/1024
-Mapping: {{mapping| 1 1 3 3 4 | 0 12 -14 -4 -11 }}
+{{Mapping|legend=0| 1 1 3 3 4 | 0 12 -14 -4 -11 }}
 : mapping generators: ~2, ~33/32
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~33/32 = 58.399
+* WE: ~2 = 1200.2902{{c}}, ~33/32 = 58.4217{{c}}
-* POTE: ~2 = 1200.000, ~33/32 = 58.408
+* CWE: ~2 = 1200.0000{{c}}, ~33/32 = 58.4062{{c}}
 {{Optimal ET sequence|legend=0| 20, 21, 41 }}
-Badness (Smith): 0.059232
+Badness (Sintel): 1.96
 ==== 13-limit ====
@@ Line 993: / Line 1,092: @@
 Comma list: 105/104, 196/195, 245/242, 512/507
-Mapping: {{mapping| 1 1 3 3 4 4 | 0 12 -14 -4 -11 -6 }}
+{{Mapping|legend=0| 1 1 3 3 4 4 | 0 12 -14 -4 -11 -6 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~33/32 = 58.436
+* WE: ~2 = 1199.8454{{c}}, ~33/32 = 58.4220{{c}}
-* POTE: ~2 = 1200.000, ~33/32 = 58.430
+* CWE: ~2 = 1200.0000{{c}}, ~33/32 = 58.4305{{c}}
 {{Optimal ET sequence|legend=0| 20, 21, 41 }}
-Badness (Smith): 0.043151
+Badness (Sintel): 1.78
 === 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
 Comma list: 121/120, 225/224, 1029/1024
-Mapping: {{mapping| 1 7 -4 1 3 | 0 -12 14 4 1 }}
+{{Mapping|legend=0| 1 -5 10 5 4 | 0 12 -14 -4 -1 }}
+: mapping generators: ~2, ~16/11
-: mapping generators: ~2, ~11/8
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~11/8 = 541.670
+* WE: ~2 = 1201.2122{{c}}, ~16/11 = 658.9974{{c}}
-* POTE: ~2 = 1200.000, ~11/8 = 541.668
+* CWE: ~2 = 1200.0000{{c}}, ~16/11 = 658.3320{{c}}
 {{Optimal ET sequence|legend=0| 11, 20, 31, 82e, 113e, 144ee }}
-Badness (Smith): 0.042687
+Badness (Sintel): 1.41
 == Hemiseven ==
-Unlike miracle which splits ~8/7, hemiseven splits ~7/4. It can be described as the {{nowrap|72 &amp; 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
@@ Line 1,027: / Line 1,127: @@
 [[Comma list]]: 1029/1024, 19683/19600
-{{Mapping|legend=1| 1 4 14 2 | 0 -6 -29 2 }}
+{{Mapping|legend=1| 1 -2 -15 4 | 0 6 29 -2 }}
+: mapping generators: ~2, ~243/160
-: mapping generators: ~2, ~320/243
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.000, ~320/243 = 483.215
+* [[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, ~320/243 = 483.267
+* [[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 }}
-[[Badness]] (Smith): 0.056557
+[[Badness]] (Sintel): 1.43
 === 11-limit ===
@@ Line 1,046: / Line 1,145: @@
 Comma list: 385/384, 441/440, 19683/19600
-Mapping: {{mapping| 1 4 14 2 -5 | 0 -6 -29 2 21 }}
+{{Mapping|legend=0| 1 -2 -15 4 16 | 0 6 29 -2 -21 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~320/243 = 483.247
+* WE: ~2 = 1200.6243{{c}}, ~243/160 = 717.0969{{c}}
-* POTE: ~2 = 1200.000, ~320/243 = 483.276
+* CWE: ~2 = 1200.0000{{c}}, ~243/160 = 716.7292{{c}}
 {{Optimal ET sequence|legend=0| 72, 149, 221e, 293de }}
-Badness (Smith): 0.028467
+Badness (Sintel): 0.941
 === 13-limit ===
@@ Line 1,061: / Line 1,160: @@
 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|legend=0| 1 -2 -15 4 16 -19 | 0 6 29 -2 -21 38 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~120/91 = 483.213
+* WE: ~2 = 1200.6781{{c}}, ~91/60 = 717.1496{{c}}
-* POTE: ~2 = 1200.000, ~120/91 = 483.255
+* CWE: ~2 = 1200.0000{{c}}, ~91/60 = 716.7520{{c}}
 {{Optimal ET sequence|legend=0| 72, 149, 221ef }}
-Badness (Smith): 0.021900
+Badness (Sintel): 0.905
 === 17-limit ===
@@ Line 1,076: / Line 1,175: @@
 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|legend=0| 1 -2 -15 4 16 -19 -21 | 0 6 29 -2 -21 38 42 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~45/34 = 483.213
+* WE: ~2 = 1200.6635{{c}}, ~68/45 = 717.1354{{c}}
-* POTE: ~2 = 1200.000, ~45/34 = 483.261
+* CWE: ~2 = 1200.0000{{c}}, ~68/45 = 716.7472{{c}}
 {{Optimal ET sequence|legend=0| 72, 149, 221ef }}
-Badness (Smith): 0.015701
+Badness (Sintel): 0.800
 == Valentine ==
@@ Line 1,090: / Line 1,189: @@
 : ''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 [[The Seven Limit Symmetrical Lattices|lattice of 7-limit tetrads]]. Valentine can also be described as the {{nowrap|31 &amp; 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 (aka 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 that it's 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.
 [[Subgroup]]: 2.3.5.7
@@ Line 1,099: / Line 1,198: @@
 {{Mapping|legend=1| 1 1 2 3 | 0 9 5 -3 }}
 : mapping generators: ~2, ~21/20
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.000, ~21/20 = 77.878
+* [[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, ~21/20 = 77.864
+* [[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]]:
 * [[7-odd-limit]]: ~21/20 = {{monzo| 1/6 1/12 0 -1/12 }}
 : {{monzo list| 1 0 0 0 | 5/2 3/4 0 -3/4 | 17/6 5/12 0 -5/12 | 5/2 -1/4 0 1/4 }}
-: [[eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.7/3
+: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/3
 * [[9-odd-limit]]: ~21/20 = {{monzo| 1/21 2/21 0 -1/21}}
 : {{monzo list| 1 0 0 0 | 10/7 6/7 0 -3/7 | 47/21 10/21 0 -5/21 | 20/7 -2/7 0 1/7 }}
-: [[eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.9/7
+: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/7
 [[Algebraic generator]]: smaller root of ''x''<sup>2</sup> - 89''x'' + 92, or (89 - sqrt (7553))/2, at 77.8616 cents.
-{{Optimal ET sequence|legend=1| 15, 31, 46, 77, 185, 262cd }}
+{{Optimal ET sequence|legend=1| 15, 31, 46, 77, 185 }}
-[[Badness]] (Smith): 0.031056
+[[Badness]] (Sintel): 0.786
 === 11-limit ===
@@ Line 1,127: / Line 1,225: @@
 Comma list: 121/120, 126/125, 176/175
-Mapping: {{mapping| 1 1 2 3 3 | 0 9 5 -3 7 }}
+{{Mapping|legend=0| 1 1 2 3 3 | 0 9 5 -3 7 }}
-: mapping generators: ~2, ~21/20
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~22/21 = 77.963
+* WE: ~2 = 1200.3890{{c}}, ~22/21 = 77.9065{{c}}
-* POTE: ~2 = 1200.000, ~22/21 = 77.881
+* CWE: ~2 = 1200.0000{{c}}, ~22/21 = 77.9007{{c}}
 Minimax tuning:
 * 11-odd-limit: ~21/20 = {{monzo| 0 0 0 -1/10 1/10 }}
 : [{{monzo| 1 0 0 0 0 }}, {{monzo| 1 0 0 -9/10 9/10 }}, {{monzo| 2 0 0 -1/2 1/2 }}, {{monzo| 3 0 0 3/10 -3/10 }}, {{monzo| 3 0 0 -7/10 7/10 }}]
-: eigenmonzo (unchanged-interval) basis: 2.11/7
+: unchanged-interval (eigenmonzo) basis: 2.11/7
 Algebraic generator: positive root of 4''x''<sup>3</sup> + 15''x''<sup>2</sup> - 21, or else Gontrand2, the smallest positive root of 4''x''<sup>7</sup> - 8''x''<sup>6</sup> + 5.
-{{Optimal ET sequence|legend=0| 15, 31, 46, 77, 262cdee, 339cdeee }}
+{{Optimal ET sequence|legend=0| 15, 31, 46, 77 }}
-Badness (Smith): 0.016687
+Badness (Sintel): 0.552
 ==== Valentino ====
@@ Line 1,151: / Line 1,247: @@
 Comma list: 121/120, 126/125, 176/175, 196/195
-Mapping: {{mapping| 1 1 2 3 3 5 | 0 9 5 -3 7 -20 }}
+{{Mapping|legend=0| 1 1 2 3 3 5 | 0 9 5 -3 7 -20 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~22/21 = 77.968
+* WE: ~2 = 1200.1967{{c}}, ~22/21 = 77.9708{{c}}
-* POTE: ~2 = 1200.000, ~22/21 = 77.958
+* CWE: ~2 = 1200.0000{{c}}, ~22/21 = 77.9594{{c}}
 {{Optimal ET sequence|legend=0| 15f, 31, 46, 77 }}
-Badness (Smith): 0.020665
+Badness (Sintel): 0.854
 ===== 17-limit =====
@@ Line 1,166: / Line 1,262: @@
 Comma list: 121/120, 126/125, 154/153, 176/175, 196/195
-Mapping: {{mapping| 1 1 2 3 3 5 5 | 0 9 5 -3 7 -20 -14 }}
+{{Mapping|legend=0| 1 1 2 3 3 5 5 | 0 9 5 -3 7 -20 -14 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~22/21 = 78.003
+* WE: ~2 = 1200.0404{{c}}, ~22/21 = 78.0055{{c}}
-* POTE: ~2 = 1200.000, ~22/21 = 78.003
+* CWE: ~2 = 1200.0000{{c}}, ~22/21 = 78.0029{{c}}
 {{Optimal ET sequence|legend=0| 15f, 31, 46, 77, 123e }}
-Badness (Smith): 0.016768
+Badness (Sintel): 0.854
 ==== Lupercalia ====
@@ Line 1,181: / Line 1,277: @@
 Comma list: 66/65, 105/104, 121/120, 126/125
-Mapping: {{mapping| 1 1 2 3 3 3 | 0 9 5 -3 7 11 }}
+{{Mapping|legend=0| 1 1 2 3 3 3 | 0 9 5 -3 7 11 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~22/21 = 77.694
+* WE: ~2 = 1199.9143{{c}}, ~22/21 = 77.7039{{c}}
-* POTE: ~2 = 1200.000, ~22/21 = 77.709
+* CWE: ~2 = 1200.0000{{c}}, ~22/21 = 77.7049{{c}}
 {{Optimal ET sequence|legend=0| 15, 31 }}
-Badness (Smith): 0.021328
+Badness (Sintel): 0.881
 ==== Dwynwen ====
@@ Line 1,196: / Line 1,292: @@
 Comma list: 91/90, 121/120, 126/125, 176/175
-Mapping: {{mapping| 1 1 2 3 3 2 | 0 9 5 -3 7 26 }}
+{{Mapping|legend=0| 1 1 2 3 3 2 | 0 9 5 -3 7 26 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~22/21 = 78.243
+* WE: ~2 = 1200.1306{{c}}, ~22/21 = 78.2273{{c}}
-* POTE: ~2 = 1200.000, ~22/21 = 78.219
+* CWE: ~2 = 1200.0000{{c}}, ~22/21 = 78.2241{{c}}
 {{Optimal ET sequence|legend=0| 15, 31f, 46 }}
-Badness (Smith): 0.023461
+Badness (Sintel): 0.969
 ==== Semivalentine ====
@@ Line 1,211: / Line 1,307: @@
 Comma list: 121/120, 126/125, 169/168, 176/175
-Mapping: {{mapping| 2 2 4 6 6 7 | 0 9 5 -3 7 3 }}
+{{Mapping|legend=0| 2 2 4 6 6 7 | 0 9 5 -3 7 3 }}
 : mapping generators: ~55/39, ~22/21
 Optimal tunings:
-* CTE: ~55/39 = 600.000, ~22/21 = 77.997
+* WE: ~55/39 = 600.3497{{c}}, ~22/21 = 77.8845{{c}}
-* POTE: ~55/39 = 600.000, ~22/21 = 77.839
+* CWE: ~55/39 = 600.0000{{c}}, ~22/21 = 77.8715{{c}}
 {{Optimal ET sequence|legend=0| 16, 30, 46, 62, 108ef }}
-Badness (Smith): 0.032749
+Badness (Sintel): 1.35
 ==== Hemivalentine ====
@@ Line 1,228: / Line 1,323: @@
 Comma list: 121/120, 126/125, 176/175, 343/338
-Mapping: {{mapping| 1 1 2 3 3 4 | 0 18 10 -6 14 -9 }}
+{{Mapping|legend=0| 1 1 2 3 3 4 | 0 18 10 -6 14 -9 }}
 : mapping generators: ~2, ~40/39
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~40/39 = 39.014
+* WE: ~2 = 1199.6529{{c}}, ~40/39 = 39.0323{{c}}
-* POTE: ~2 = 1200.000, ~40/39 = 39.044
+* CWE: ~2 = 1200.0000{{c}}, ~40/39 = 39.0383{{c}}
 {{Optimal ET sequence|legend=0| 30, 31, 61, 92f }}
-Badness (Smith): 0.047059
+Badness (Sintel): 1.94
 ==== Demivalentine ====
@@ Line 1,245: / Line 1,339: @@
 Comma list: 121/120, 126/125, 176/175, 676/675
-Mapping: {{mapping| 1 -8 -3 6 -4 -16 | 0 18 10 -6 14 37 }}
+{{Mapping|legend=0| 1 -8 -3 6 -4 -16 | 0 18 10 -6 14 37 }}
 : mapping generators: ~2, ~13/9
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~13/9 = 638.964
+* WE: ~2 = 1200.3929{{c}}, ~13/9 = 639.1320{{c}}
-* CWE: ~2 = 1200.000, ~13/9 = 638.932
+* CWE: ~2 = 1200.0000{{c}}, ~13/9 = 638.9325{{c}}
 {{Optimal ET sequence|legend=0| 15, 47ef, 62, 77 }}
-Badness (Smith): 0.0349
+Badness (Sintel): 1.44
 === Hemivalentino ===
@@ Line 1,262: / Line 1,355: @@
 Comma list: 126/125, 243/242, 1029/1024
-Mapping: {{mapping| 1 1 2 3 2 | 0 18 10 -6 45 }}
+{{Mapping|legend=0| 1 1 2 3 2 | 0 18 10 -6 45 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~45/44 = 38.928
+* WE: ~2 = 1200.0816{{c}}, ~45/44 = 38.9236{{c}}
-* POTE: ~2 = 1200.000, ~45/44 = 38.921
+* CWE: ~2 = 1200.0000{{c}}, ~45/44 = 38.9228{{c}}
 {{Optimal ET sequence|legend=0| 31, 92e, 123, 154, 185 }}
-Badness (Smith): 0.061275
+Badness (Sintel): 2.03
 ==== 13-limit ====
@@ Line 1,277: / Line 1,370: @@
 Comma list: 126/125, 196/195, 243/242, 1029/1024
-Mapping: {{mapping| 1 1 2 3 2 5 | 0 18 10 -6 45 -40 }}
+{{Mapping|legend=0| 1 1 2 3 2 5 | 0 18 10 -6 45 -40 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~45/44 = 38.944
+* WE: ~2 = 1199.8782{{c}}, ~45/44 = 38.9440{{c}}
-* POTE: ~2 = 1200.000, ~45/44 = 38.948
+* CWE: ~2 = 1200.0000{{c}}, ~45/44 = 38.9472{{c}}
 {{Optimal ET sequence|legend=0| 31, 123, 154 }}
-Badness (Smith): 0.057919
+Badness (Sintel): 2.39
 ==== Hemivalentoid ====
@@ Line 1,292: / Line 1,385: @@
 Comma list: 126/125, 144/143, 243/242, 343/338
-Mapping: {{mapping| 1 1 2 3 2 4 | 0 18 10 -6 45 -9 }}
+{{Mapping|legend=0| 1 1 2 3 2 4 | 0 18 10 -6 45 -9 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~40/39 = 38.946
+* WE: ~2 = 1199.3614{{c}}, ~45/44 = 38.9721{{c}}
-* POTE: ~2 = 1200.000, ~40/39 = 38.993
+* CWE: ~2 = 1200.0000{{c}}, ~45/44 = 38.9839{{c}}
 {{Optimal ET sequence|legend=0| 31, 92ef }}
-Badness (Smith): 0.057931
+Badness (Sintel): 2.39
 == Superkleismic ==
@@ Line 1,306: / Line 1,399: @@
 : ''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.
-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]] }}.
+In the 11-limit, two generator steps can be identified with ~16/11, and in the 13-limit, the same step can be treated as ~13/9. The [[S-expression]]-based comma list of 13-limit superkleismic is {[[875/864|S5/S6]], [[1029/1024|S7/S8]], [[100/99|S10]], [[144/143|S12]], ([[441/440|S21]])}. Through careful observation of the equivalences therein one can derive the mapping of the full 13-limit.
+Note that the generator is given as 6/5's octave complement, [[5/3]], in the data that follow, since a stack of 9 such generators octave-reduced is the perfect fifth; the [[ploidacot]] of superkleismic is wau-enneacot.
+Superkleismic also sets two intervals of [[21/20]] equal to [[10/9]]; as {{nowrap| 10/9 {{=}} ([[20/19]])⋅([[19/18]]) }}, we can identify 21/20, 20/19, and 19/18 together to add prime 19, tempering out [[361/360]] ({{S|19}}) and [[400/399]] ({{S|20}}). This structure is preserved within the entire superkleismic tuning range between 15edo and 26edo, while extensions for primes 13 and 17 bifurcate and are of higher complexity and lower accuracy.
 edo gives an obvious tuning in all the subgroups.
@@ Line 1,316: / Line 1,413: @@
 [[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:
-* [[CTE]]: ~2 = 1200.000, ~6/5 = 321.798
+* [[WE]]: ~2 = 1200.7640{{c}}, ~5/3 = 878.6289{{c}}
-* [[POTE]]: ~2 = 1200.000, ~6/5 = 321.930
+: [[error map]]: {{val| +0.764 +1.885 +3.844 -0.893 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/3 = 878.1077{{c}}
+: error map: {{val| 0.000 +1.014 -5.237 -3.149 }}
 {{Optimal ET sequence|legend=1| 11c, 15, 26, 41 }}
-[[Badness]]:
+[[Badness]] (Sintel): 1.21
-* Smith: 0.0479
-* Dirichlet: 1.21
 === 11-limit ===
@@ Line 1,335: / Line 1,431: @@
 Comma list: 100/99, 245/242, 385/384
-Mapping: {{mapping| 1 4 5 2 4 | 0 -9 -10 3 -2 }}
+{{Mapping|legend=0| 1 -5 -5 5 2 | 0 9 10 -3 2 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~6/5 = 321.815
+* WE: ~2 = 1200.1691{{c}}, ~5/3 = 878.2772{{c}}
-* POTE: ~2 = 1200.000, ~6/5 = 321.847
+* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 878.1606{{c}}
 {{Optimal ET sequence|legend=0| 11c, 15, 26, 41, 179cde, 220cde, 261ccdee }}
-Badness:
+Badness (Sintel): 0.848
-* Smith: 0.0257
-* Dirichlet: 0.848
 ==== 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
 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|legend=0| 1 -5 -5 5 2 -6 | 0 9 10 -3 2 14 }}
 Optimal tunings:
-* CTE: ~2 = 1\1, ~6/5 = 321.779
+* WE: ~2 = 1200.2289{{c}}, ~5/3 = 878.3409{{c}}
-* POTE: ~2 = 1\1, ~6/5 = 321.827
+* 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
@@ Line 1,371: / Line 1,463: @@
 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|legend=0| 1 -5 -5 5 2 -8 | 0 9 10 -3 2 16 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~6/5 = 321.986
+* WE: ~2 = 1200.0261{{c}}, ~5/3 = 878.0252{{c}}
-* POTE: ~2 = 1200.000, ~6/5 = 321.994
+* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 878.0073{{c}}
 {{Optimal ET sequence|legend=0| 11cf, 15, 26, 41 }}
-Badness:
+Badness (Sintel): 0.887
-* Smith: 0.0215
-* Dirichlet: 0.887
 ==== 17-limit ====
@@ Line 1,388: / Line 1,478: @@
 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|legend=0| 1 -5 -5 5 2 -8 -12 | 0 9 10 -3 2 16 22 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~6/5 = 322.136
+* WE: ~2 = 1200.0488{{c}}, ~5/3 = 877.8872{{c}}
-* POTE: ~2 = 1200.000, ~6/5 = 322.149
+* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 877.8537{{c}}
 {{Optimal ET sequence|legend=0| 11cfg, 15g, 26, 41 }}
-Badness (Sintel): 1.007
+Badness (Sintel): 1.01
 ==== 19-limit ====
@@ Line 1,403: / Line 1,493: @@
 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|legend=0| 1 -5 -5 5 2 -8 -12 -6 | 0 9 10 -3 2 16 22 14 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~6/5 = 322.084
+* WE: ~2 = 1200.2120{{c}}, ~5/3 = 878.0243{{c}}
-* CWE: ~2 = 1200.000, ~6/5 = 322.121
+* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 877.8789{{c}}
 {{Optimal ET sequence|legend=0| 11cfgh, 15g, 26, 41 }}
@@ Line 1,414: / Line 1,504: @@
 === Superana ===
-This extension (41 & 56) is the counterpart of canonical superkleismic on the other side of 41edo.
+This extension ({{nowrap| 41 & 56 }}) is the counterpart of canonical superkleismic on the other side of 41edo.
 Subgroup: 2.3.5.7.11.13
-Comma list: 100/99, 196/195, 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|legend=0| 1 -5 -5 5 2 22 | 0 9 10 -3 2 -25 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~6/5 = 321.724
+* WE: ~2 = 1199.8272{{c}}, ~5/3 = 878.1538{{c}}
-* POTE: ~2 = 1200.000, ~6/5 = 321.719
+* 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 ====
 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|legend=0| 1 -5 -5 5 2 22 18 | 0 9 10 -3 2 -25 -19 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~6/5 = 321.650
+* WE: ~2 = 1199.5964{{c}}, ~5/3 = 878.0482{{c}}
-* POTE: ~2 = 1200.000, ~6/5 = 321.657
+* 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 ====
@@ Line 1,450: / Line 1,540: @@
 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|legend=0| 1 -5 -5 5 2 22 18 -6 | 0 9 10 -3 2 -25 -19 14 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~6/5 = 321.646
+* WE: ~2 = 1199.6638{{c}}, ~5/3 = 878.1109{{c}}
-* POTE: ~2 = 1200.000, ~6/5 = 321.643
+* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 878.3566{{c}}
+{{Optimal ET sequence|legend=0| 15f, 41, 56, 97g }}
+Badness (Sintel): 1.36
+== Dee leap week ==
+{{Main| Dee leap week }}
+[[Subgroup]]: 2.3.5.7
+[[Comma list]]: 1029/1024, 2460375/2458624
+{{Mapping|legend=1| 1 -5 25 5 | 0 9 -31 -3 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.4835{{c}}, ~224/135 = 878.2507{{c}}
+: [[error map]]: {{val| +0.484 -0.117 +0.004 -1.160 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~224/135 = 877.8926{{c}}
+: error map: {{val| 0.000 -0.921 -0.985 -2.504 }}
-{{Optimal ET sequence|legend=0| 15f, 26fg, 41, 56, 97g }}
+{{Optimal ET sequence|legend=1| 41, 108, 149, 190 }}
-Badness (Sintel): 1.355
+[[Badness]] (Sintel): 2.12
-== Unidec ==
+=== 11-limit ===
-{{Main| Unidec }}
+Subgroup: 2.3.5.7.11
-=== 5-limit (unidecmic) ===
+Comma list: 385/384, 441/440, 2460375/2458624
-[[Subgroup]]: 2.3.5
-[[Comma list]]: 31381059609/31250000000
+{{Mapping|legend=0| 1 -5 25 5 -28 | 0 9 -31 -3 43 }}
-{{Mapping|legend=1| 2 5 8 | 0 -6 -11 }}
+Optimal tunings:
+* WE: ~2 = 1200.4874{{c}}, ~224/135 = 878.2543{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~224/135 = 877.8987{{c}}
-: mapping generators: ~177147/125000, ~10/9
+{{Optimal ET sequence|legend=0| 41, 108e, 149, 190 }}
-[[Optimal tuning]]s:
+Badness (Sintel): 1.35
-* [[CTE]]: ~177147/125000 = 600.000, ~10/9 = 183.041
-: [[error map]]: {{val| 0.000 -0.201 +0.235 }}
-* [[POTE]]: ~177147/125000 = 600.000, ~10/9 = 183.047
-: error map: {{val| 0.000 -0.236 +0.172 }}
-{{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
 [[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:
-* [[CTE]]: ~1225/864 = 600.000, ~10/9 = 183.060
+* [[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, ~10/9 = 183.161
+* [[CWE]]: ~1225/864 = 600.0000{{c}}, ~80/63 = 416.8688{{c}}
 : error map: {{val| 0.000 -0.924 -1.090 -2.503 }}
@@ Line 1,498: / Line 1,604: @@
 * [[7-odd-limit]]: ~10/9 = {{monzo| 3/26 0 -1/13 1/13 }}
 : {{monzo list| 1 0 0 0 | 47/26 0 6/13 -6/13 | 71/26 0 11/13 -11/13 | 71/26 0 -2/13 2/13 }}
-: [[eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.7/5
+: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5
 * [[9-odd-limit]]: ~10/9 = {{monzo| 5/28 -1/7 0 1/14 }}
 : {{Monzo list| 1 0 0 0 | 10/7 6/7 0 -3/7 | 57/28 11/7 0 -11/14 | 20/7 -2/7 0 1/7 }}
-: [[eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.9/7
+: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/7
 {{Optimal ET sequence|legend=1| 26, 46, 72, 118, 190 }}
-[[Badness]] (Smith): 0.038393
+[[Badness]] (Sintel): 0.972
 === 11-limit ===
@@ Line 1,512: / Line 1,618: @@
 Comma list: 385/384, 441/440, 4375/4374
-Mapping: {{mapping| 2 5 8 5 6 | 0 -6 -11 2 3 }}
+{{Mapping|legend=0| 2 -1 -3 7 9 | 0 6 11 -2 -3 }}
 Optimal tunings:
-* CTE: ~99/70 = 600.000, ~10/9 = 183.074
+* WE: ~99/70 = 600.2497{{c}}, ~14/11 = 417.0085{{c}}
-* CWE: ~99/70 = 600.000, ~10/9 = 183.146
+* CWE: ~99/70 = 600.0000{{c}}, ~14/11 = 416.8543{{c}}
 Minimax tuning:
 * [[11-odd-limit]]: ~10/9 = {{monzo| 5/28 -1/7 0 1/14 }}
 : [{{monzo| 1 0 0 0 0 }}, {{monzo| 10/7 6/7 0 -3/7 0 }}, {{monzo| 57/28 11/7 0 -11/14 0 }}, {{monzo| 20/7 -2/7 0 1/7 0 }}, {{monzo| 99/28 -3/7 0 3/14 0 }}]
-: eigenmonzo (unchanged-interval) basis: 2.9/7
+: unchanged-interval (eigenmonzo) basis: 2.9/7
 {{Optimal ET sequence|legend=0| 26, 46, 72, 118, 190 }}
-Badness (Smith): 0.015479
+Badness (Sintel): 0.512
 ==== Ekadash ====
@@ Line 1,532: / Line 1,638: @@
 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|legend=0| 2 -1 -3 7 9 -19 | 0 6 11 -2 -3 38 }}
 Optimal tunings:
-* CTE: ~99/70 = 600.000, ~10/9 = 183.125
+* WE: ~99/70 = 600.2497{{c}}, ~14/11 = 417.0085{{c}}
-* POTE: ~99/70 = 600.000, ~10/9 = 183.187
+* CWE: ~99/70 = 600.0000{{c}}, ~14/11 = 416.8543{{c}}
 {{Optimal ET sequence|legend=0| 46f, 72, 118, 190, 262df, 452cdef }}
-Badness (Smith): 0.020381
+Badness (Sintel): 0.842
 ==== Hendec ====
@@ Line 1,547: / Line 1,653: @@
 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|legend=0| 2 -1 -3 7 9 6 | 0 6 11 -2 -3 2 }}
 Optimal tunings:
-* CTE: ~91/64 = 600.000, ~10/9 = 183.048
+* WE: ~91/64 = 600.3825{{c}}, ~14/11 = 417.0678{{c}}
-* POTE: ~91/64 = 600.000, ~10/9 = 183.198
+* CWE: ~91/64 = 600.0000{{c}}, ~14/11 = 416.8290{{c}}
 {{Optimal ET sequence|legend=0| 26, 46, 72, 190ff }}
-Badness (Smith): 0.017707
+Badness (Sintel): 0.732
 ===== 17-limit =====
@@ Line 1,562: / Line 1,668: @@
 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|legend=0| 2 -1 -3 7 9 6 4 | 0 6 11 -2 -3 2 6 }}
 Optimal tunings:
-* CTE: ~17/12 = 600.000, ~10/9 = 183.020
+* WE: ~17/12 = 600.3991{{c}}, ~14/11 = 417.0809{{c}}
-* POTE: ~17/12 = 600.000, ~10/9 = 183.196
+* CWE: ~17/12 = 600.0000{{c}}, ~14/11 = 416.8330{{c}}
 {{Optimal ET sequence|legend=0| 26, 46, 72, 190ffg }}
-Badness (Smith): 0.011676
+Badness (Sintel): 0.595
+== 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
-[[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:
+* [[WE]]: ~2 = 1200.0322{{c}}, ~315/256 = 358.5581{{c}}
+: [[error map]]: {{val| +0.032 +0.678 +1.340 -2.930 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~315/256 = 358.5484{{c}}
+: error map: {{val| 0.000 +0.626 +1.267 -3.019 }}
-[[Optimal tuning]]s:
+{{Optimal ET sequence|legend=1| 77, 87, 164 }}
-* [[CTE]]: ~3375/2401 = 600.000, ~15/14 = 122.031
-: [[error map]]: {{val| 0.000 -0.232 +0.087 -2.734 }}
-* [[POTE]]: ~3375/2401 = 600.000, ~15/14 = 122.027
-: error map: {{val| 0.000 -0.195 +0.033 -2.746 }}
-{{Optimal ET sequence|legend=1| 10, 98, 108, 118 }}
+[[Badness]] (Sintel): 2.73
-[[Badness]] (Smith): 0.144345
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-== Dee leap week ==
+Comma list: 385/384, 441/440, 153664/151875
-{{Main|Dee leap week}}
-Subgroup: 2.3.5.7
+{{Mapping|legend=0| 1 -2 8 4 -7 | 0 12 -19 -4 35 }}
-Comma list: 1029/1024, 2460375/2458624
+Optimal tunings:
+* WE: ~2 = 1200.1110{{c}}, ~27/22 = 358.6045{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~27/22 = 358.5720{{c}}
-Mapping: [{{val|1 4 -6 2}}, {{val|0 -9 31 3}}]
+{{Optimal ET sequence|legend=0| 77, 87, 164, 251d }}
-Optimal tuning (CTE): ~135/112 = 322.123
+Badness (Sintel): 1.81
-=== 11-limit ===
+=== 13-limit ===
-Subgroup: 2.3.5.7.11
+Subgroup: 2.3.5.7.11.13
+Comma list: 196/195, 352/351, 385/384, 676/675
-Comma list: 385/384, 441/440, 2460375/2458624
+{{Mapping|legend=0| 1 -2 8 4 -7 4 | 0 12 -19 -4 35 -1 }}
-Mapping: [{{val|1 4 -6 2 15}}, {{val|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): ~135/112 = 322.097
+{{Optimal ET sequence|legend=0| 77, 87, 164, 251d }}
-{{Optimal ET sequence|legend=1|41, 149, 190, 231}}
+Badness (Sintel): 1.16
 == 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
 [[Comma list]]: 1029/1024, 5103/5000
-{{Mapping|legend=1| 1 7 10 1 | 0 -12 -17 4 }}
+{{Mapping|legend=1| 1 -5 -7 5 | 0 12 17 -4 }}
+: mapping generators: ~2, ~35/24
-: mapping generators: ~2, ~48/35
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.000, ~48/35 = 541.743
+* [[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, ~48/35 = 541.779
+* [[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 }}
-[[Badness]] (Smith): 0.117680
+[[Badness]] (Sintel): 2.98
 === 11-limit ===
@@ Line 1,637: / Line 1,755: @@
 Comma list: 176/175, 243/242, 1029/1024
-Mapping: {{mapping| 1 7 10 1 17 | 0 -12 -17 4 -30 }}
+{{Mapping|legend=0| 1 -5 -7 5 -13 | 0 12 17 -4 30 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~15/11 = 541.695
+* WE: ~2 = 1200.2862{{c}}, ~22/15 = 658.4276{{c}}
-* POTE: ~2 = 1200.000, ~15/11 = 541.729
+* CWE: ~2 = 1200.0000{{c}}, ~22/15 = 658.2805{{c}}
 {{Optimal ET sequence|legend=0| 20ce, 31, 113c, 144c }}
-Badness (Smith): 0.053459
+Badness (Sintel): 1.77
 === 13-limit ===
@@ Line 1,652: / Line 1,770: @@
 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|legend=0| 1 -5 -7 5 -13 7 | 0 12 17 -4 30 -6 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~15/11 = 541.673
+* WE: ~2 = 1199.3663{{c}}, ~22/15 = 658.0465{{c}}
-* POTE: ~2 = 1200.000, ~15/11 = 541.606
+* CWE: ~2 = 1200.0000{{c}}, ~22/15 = 658.3800{{c}}
 {{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
-[[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: ~3375/2401, ~450/343
-: mapping generators: ~2. ~315/256
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.000, ~315/256 = 358.548
+* [[WE]]: ~3375/2401 = 600.1355{{c}}, ~450/343 = 478.0813{{c}}
-: [[error map]]: {{val| 0.000 +0.620 +1.275 -3.018 }}
+: [[error map]]: {{val| +0.271 +0.235 +0.662 -1.986 }}
-* [[POTE]]: ~2 = 1200.000, ~315/256 = 358.548
+* [[CWE]]: ~3375/2401 = 600.000{{c}}, ~450/343 = 477.9725{{c}}
-: error map: {{val| 0.000 +0.627 +1.265 -3.020 }}
+: error map: {{val| 0.000 -0.202 +0.043 -2.743 }}
-{{Optimal ET sequence|legend=1| 77, 87, 164 }}
+{{Optimal ET sequence|legend=1| 10, 98, 108, 118 }}
-[[Badness]] (Smith): 0.108011
-=== 11-limit ===
-Subgroup: 2.3.5.7.11
-Comma list: 385/384, 441/440, 153664/151875
-Mapping: {{mapping| 1 -2 8 4 -7 | 0 12 -19 -4 35 }}
-Optimal tunings:
-* CTE: ~2 = 1200.000, ~27/22 = 358.575
-* POTE: ~2 = 1200.000, ~27/22 = 358.571
-{{Optimal ET sequence|legend=0| 77, 87, 164, 251d }}
-Badness (Smith): 0.054655
-=== 13-limit ===
-Subgroup: 2.3.5.7.11.13
-Comma list: 196/195, 352/351, 385/384, 676/675
-Mapping: {{mapping| 1 -2 8 4 -7 4 | 0 12 -19 -4 35 -1 }}
-Optimal tunings:
+[[Badness]] (Sintel): 3.65
-* CTE: ~2 = 1200.000, ~16/13 = 358.576
-* POTE: ~2 = 1200.000, ~16/13 = 358.574
-{{Optimal ET sequence|legend=0| 77, 87, 164, 251d }}
-Badness (Smith): 0.028187
 == Quartemka ==
 : ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Quartemka]].''
+Quartemka may be described as the {{nowrap| 26 & 61 }} temperament. Its [[ploidacot]] is 18-sheared 21-cot. It was named by [[Petr Pařízek]] in 2011 for its generator is close to 1/4 of the generator for [[emka]]<ref name="petr's long post"/>.
 [[Subgroup]]: 2.3.5.7
@@ Line 1,720: / Line 1,809: @@
 [[Comma list]]: 1029/1024, 1250000/1240029
-{{Mapping|legend=1| 1 4 6 2 | 0 -21 -32 7 }}
+{{Mapping|legend=1| 1 -17 -26 9 | 0 21 32 -7 }}
+: mapping generators: ~2, ~50/27
-: mapping generators: ~2, ~27/25
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.000, ~27/25 = 137.971
+* [[WE]]: ~2 = 1200.5278{{c}}, ~50/27 = 1062.4614{{c}}
-: [[error map]]: {{val| 0.000 +0.658 -1.380 -3.030 }}
+: [[error map]]: {{val| +0.528 +0.762 -1.272 -1.305 }}
-* [[POTE]]: ~2 = 1200.000, ~27/25 = 138.006
+* [[CWE]]: ~21 = 1200.0000{{c}}, ~50/27 = 1062.0046{{c}}
-: error map: {{val| 0.000 -0.075 -2.496 -2.786 }}
+: error map: {{val| 0.000 +0.142 -2.167 -2.858 }}
 {{Optimal ET sequence|legend=1| 26, 61, 87, 113, 200 }}
-[[Badness]] (Smith): 0.152287
+[[Badness]] (Sintel): 3.85
 === 11-limit ===
@@ Line 1,739: / Line 1,827: @@
 Comma list: 385/384, 441/440, 800000/793881
-Mapping: {{mapping| 1 4 6 2 3 | 0 -21 -32 7 4 }}
+{{Mapping|legend=0| 1 -17 -26 9 7 | 0 21 32 -7 -4 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~27/25 = 137.970
+* WE: ~2 = 1200.3051{{c}}, ~50/27 = 1062.2805{{c}}
-* POTE: ~2 = 1200.000, ~27/25 = 137.990
+* CWE: ~21 = 1200.0000{{c}}, ~50/27 = 1062.0147{{c}}
 {{Optimal ET sequence|legend=0| 26, 61, 87, 200, 287d }}
-Badness (Smith): 0.057307
+Badness (Sintel): 1.89
 === 13-limit ===
@@ Line 1,754: / Line 1,842: @@
 Comma list: 325/324, 364/363, 385/384, 2200/2197
-Mapping: {{mapping| 1 4 6 2 3 6 | 0 -21 -32 7 4 -20 }}
+{{Mapping|legend=0| 1 -17 -26 9 7 -14 | 0 21 32 -7 -4 20 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~13/12 = 137.971
+* WE: ~2 = 1200.2708{{c}}, ~24/13 = 1062.2496{{c}}
-* POTE: ~2 = 1200.000, ~13/12 = 137.990
+* CWE: ~21 = 1200.0000{{c}}, ~24/13 = 1062.0139{{c}}
 {{Optimal ET sequence|legend=0| 26, 61, 87, 200 }}
-Badness (Smith): 0.028393
+Badness (Sintel): 1.17
 == Tritriple ==
 : ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tritriple]].''
+Tritriple may be described as the {{nowrap| 103 & 118 }} temperament. Its [[ploidacot]] is iota-beta-27-cot. It was named by [[Petr Pařízek]] in 2011 for its generator is 1/9 of the generator for [[slendric]], so that 3×3 generators [[octave reduction|octave reduced]] give slendric's generator, and another ×3 give the [[3/2|perfect fifth]]<ref name="petr's long post"/>.
 [[Subgroup]]: 2.3.5.7
@@ Line 1,772: / Line 1,862: @@
 {{Mapping|legend=1| 1 -11 -7 7 | 0 27 20 -9 }}
 : mapping generators: ~2, ~864/625
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.000, ~864/625 = 559.320
+* [[WE]]: ~2 = 1200.4239{{c}}, ~864/625 = 559.4921{{c}}
-: [[error map]]: {{val| 0.000 -0.317 +0.085 -2.705 }}
+: [[error map]]: {{val| +0.424 -0.331 +0.561 -1.287 }}
-* [[POTE]]: ~2 = 1200.000, ~864/625 = 559.295
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~864/625 = 559.3015{{c}}
-: error map: {{val| 0.000 -1.003 -0.423 -2.477 }}
+: error map: {{val| 0.000 -0.815 -0.284 -2.539 }}
 {{Optimal ET sequence|legend=1| 15, …, 88, 103, 118, 221, 339d }}
-[[Badness]] (Smith): 0.118640
+[[Badness]] (Sintel): 3.00
 === 11-limit ===
@@ Line 1,790: / Line 1,879: @@
 Comma list: 385/384, 441/440, 43923/43750
-Mapping: {{mapping| 1 -11 -7 7 -4 | 0 27 20 -9 16 }}
+{{Mapping|legend=0| 1 -11 -7 7 -4 | 0 27 20 -9 16 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~242/175 = 559.327
+* WE: ~2 = 1200.4953{{c}}, ~242/175 = 559.5243{{c}}
-* POTE: ~2 = 1200.000, ~242/175 = 559.293
+* CWE: ~2 = 1200.0000{{c}}, ~242/175 = 559.3016{{c}}
 {{Optimal ET sequence|legend=0| 15, …, 88, 103, 118, 221e, 339de }}
-Badness (Smith): 0.035350
+Badness (Sintel): 1.17
 == Widefourth ==
@@ Line 1,805: / Line 1,894: @@
 [[Comma list]]: 1029/1024, 48828125/48771072
-{{Mapping|legend=1| 1 16 8 -2 | 0 -33 -13 11 }}
+{{Mapping|legend=1| 1 -17 -5 9 | 0 33 13 -11 }}
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.000, ~3125/2304 = 524.188
+* [[WE]]: ~2 = 1200.4770{{c}}, ~4608/3125 = 676.0584{{c}}
-: [[error map]]: {{val| 0.000 -0.154 -0.756 -2.759 }}
+: [[error map]]: {{val| +0.477 -0.137 +0.061 -1.175 }}
-* [[POTE]]: ~2 = 1200.000, ~3125/2304 = 524.210
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~4608/3125 = 675.7954{{c}}
-: error map: {{val| 0.000 -0.892 -1.047 -2.513 }}
+: error map: {{val| 0.000 -0.705 -0.973 -2.576 }}
 {{Optimal ET sequence|legend=1| 16, 71, 87, 103, 190 }}
-[[Badness]] (Smith): 0.154117
+[[Badness]] (Sintel): 3.90
 === 11-limit ===
@@ Line 1,822: / Line 1,911: @@
 Comma list: 385/384, 441/440, 234375/234256
-Mapping: {{mapping| 1 16 8 -2 17 | 0 -33 -13 11 -31 }}
+{{Mapping|legend=0| 1 16 8 -2 17 | 0 -33 -13 11 -31 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~847/625 = 524.183
+* WE: ~2 = 1200.4852{{c}}, ~1250/847 = 676.0634{{c}}
-* POTE: ~2 = 1200.000, ~847/625 = 524.210
+* CWE: ~2 = 1200.0000{{c}}, ~1250/847 = 675.7966{{c}}
 {{Optimal ET sequence|legend=0| 16, 71, 87, 103, 190 }}
-Badness (Smith): 0.040785
+Badness (Sintel): 1.35
 === 13-limit ===
@@ Line 1,837: / Line 1,926: @@
 Comma list: 385/384, 441/440, 625/624, 847/845
-Mapping: {{mapping| 1 16 8 -2 17 12 | 0 -33 -13 11 -31 -19 }}
+{{Mapping|legend=0| 1 16 8 -2 17 12 | 0 -33 -13 11 -31 -19 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~65/48 = 524.183
+* WE: ~2 = 1200.4217{{c}}, ~77/52 = 676.0286{{c}}
-* POTE: ~2 = 1200.000, ~65/48 = 524.209
+* CWE: ~2 = 1200.0000{{c}}, ~77/52 = 675.7967{{c}}
 {{Optimal ET sequence|legend=0| 16, 71, 87, 103, 190 }}
-Badness (Smith): 0.021636
+Badness (Sintel): 0.894
+== Other subgroup extensions ==
+=== Euslendric (2.3.7.13) ===
+Forms of slendric in the most optimal range for the 2.3.7 temperament ({{nowrap| 36 & 77 }}) lack an obvious strong mapping of prime 5 or prime 11. However, slendric can extend well to the no-fives no-elevens [[29-limit]] by tempering out [[273/272]], [[343/342]], [[378/377]], [[392/391]], [[513/512]], and [[729/728]], or a comma basis defined in terms of [[S-expression]]s as {S7/S8, S14/S16, S15/S20, S24/S26, S27, S28}. [[113edo]] is an obvious tuning.
+Subgroup: 2.3.7.13
+Comma list: 729/728, 1029/1024
+Subgroup-val mapping: {{mapping| 1 1 3 0 | 0 3 -1 19 }}
+Gencom mapping: {{mapping| 1 1 0 3 0 0 | 0 3 0 -1 0 19 }}
+Optimal tunings:
+* WE: ~2 = 1200.5057{{c}}, ~8/7 = 233.7200{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.6534{{c}}
+{{Optimal ET sequence|legend=0| 5, 31f, 36, 77, 113, 827bdddff }}
-== Notes ==
+Badness (Sintel): 0.339
+==== 2.3.7.13.17 subgroup ====
+Subgroup: 2.3.7.13.17
+Comma list: 273/272, 729/728, 833/832
+Subgroup-val mapping: {{mapping| 1 1 3 0 0 | 0 3 -1 19 21 }}
+Gencom mapping: {{mapping| 1 1 0 3 0 0 0 | 0 3 0 -1 0 19 21 }}
+Optimal tunings:
+* WE: ~2 = 1200.5282{{c}}, ~8/7 = 233.6492{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.5776{{c}}
+{{Optimal ET sequence|legend=0| 5g, 31fg, 36, 113, 149 }}
+Badness (Sintel): 0.332
+==== 2.3.7.13.17.19 subgroup ====
+Subgroup: 2.3.7.13.17.19
+Comma list: 273/272, 343/342, 513/512, 729/728
+Subgroup-val mapping: {{mapping| 1 1 3 0 0 6 | 0 3 -1 19 21 -9 }}
+Gencom mapping: {{mapping| 1 1 0 3 0 0 0 6 | 0 3 0 -1 0 19 21 -9 }}
+Optimal tunings:
+* WE: ~2 = 1200.3292{{c}}, ~8/7 = 233.6651{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.6106{{c}}
+{{Optimal ET sequence|legend=0| 5g, 36, 77, 113, 262df }}
+Badness (Sintel): 0.380
+==== 2.3.7.13.17.19.23 subgroup ====
+Subgroup: 2.3.7.13.17.19.23
+Comma list: 273/272, 343/342, 392/391, 513/512, 729/728
+Subgroup-val mapping: {{mapping| 1 1 3 0 0 6 9 | 0 3 -1 19 21 -9 -23 }}
+Gencom mapping: {{mapping| 1 1 0 3 0 0 0 6 9 | 0 3 0 -1 0 19 21 -9 -23 }}
+Optimal tunings:
+* WE: ~2 = 1200.3127{{c}}, ~8/7 = 233.6679{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.6091{{c}}
+{{Optimal ET sequence|legend=0| 36, 77, 113, 262df }}
+Badness (Sintel): 0.474
+==== 2.3.7.13.17.19.23.29 subgroup ====
+Subgroup: 2.3.7.13.17.19.23.29
+Comma list: 273/272, 343/342, 378/377, 392/391, 513/512, 609/608
+Subgroup-val mapping: {{mapping| 1 1 3 0 0 6 9 7 | 0 3 -1 19 21 -9 -23 -11 }}
+Gencom mapping: {{mapping| 1 1 0 3 0 0 0 6 9 7 | 0 3 0 -1 0 19 21 -9 -23 -11 }}
+Optimal tunings:
+* WE: ~2 = 1200.2503{{c}}, ~8/7 = 233.6688{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.6208{{c}}
+{{Optimal ET sequence|legend=0| 36, 77, 113 }}
+Badness (Sintel): 0.473
+=== Baladic (2.3.7.13) ===
+Baladic is a 2.3.7.13.17-subgroup temperament that attempts to approximate the Maqam Sikah Baladi scale. It tempers out [[169/168]] ({{S|13}}), which splits [[7/6]] in half ([[13/12]]~[[14/13]]) and one finds that the octave is therefore split in half via the interval [[91/64]], which is then equated to [[17/12]]. 36edo is an excellent baladic tuning.
+Subgroup: 2.3.7.13
+Comma list: 169/168, 1029/1024
+Subgroup-val mapping: {{mapping| 2 2 6 7 | 0 3 -1 1 }}
+Gencom mapping: {{mapping| 2 2 0 6 0 7 | 0 3 0 -1 0 1 }}
+: mapping generators: ~91/64, ~8/7
+Optimal tunings:
+* WE: ~91/64 = 600.4315{{c}}, ~8/7 = 233.7724{{c}}
+* CWE: ~91/64 = 600.0000{{c}}, ~8/7 = 233.7039{{c}}
+{{Optimal ET sequence|legend=0| 10, 26, 36, 154f, 190ff, 226ff, 262dfff }}
+Badness (Sintel): 0.434
+==== 2.3.7.13.17 subgroup ====
+Subgroup: 2.3.7.13.17
+Comma list: 169/168, 273/272, 289/288
+Subgroup-val mapping: {{mapping| 2 2 6 7 7 | 0 3 -1 1 3 }}
+Gencom mapping: {{mapping| 2 2 0 6 0 7 7 | 0 3 0 -1 0 1 3 }}
+Optimal tunings:
+* WE: ~17/12 = 600.4436{{c}}, ~8/7 = 233.7883{{c}}
+* CWE: ~17/12 = 600.0000{{c}}, ~8/7 = 233.7312{{c}}
+{{Optimal ET sequence|legend=0| 10, 26, 36, 154f, 190ffg, 226ffg }}
+Badness (Sintel): 0.253
+=== Gigapyth (2.3.7.85) ===
+Subgroup: 2.3.7.85
+Comma list: 1029/1024, 7225/7203
+Subgroup-val mapping: {{mapping| 1 -2 4 7 | 0 6 -2 -1 }}
+Optimal tunings:
+* WE: ~2 = 1200.8295{{c}}, ~128/85 = 717.2597{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~128/85 = 716.7933{{c}}
+{{Optimal ET sequence|legend=0| 5, 42*, 47, 52, 57, 62, 67, 72, 149*, 370d***, 519bdd***** }}
+<nowiki/>* Wart for 85
+== References ==
+[[Category:Gamelismic clan| ]] <!-- main article -->
 [[Category:Temperament clans]]
-[[Category:Gamelismic clan| ]] <!-- main article -->
+[[Category:Catalogs of rank-2 temperaments]]
-[[Category:Rank 2]]
 [[Category:Listen]]