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The 2.3.7 [[Just_intonation_subgroups|subgroup]] comma for the gamelismic clan is the gamelisma, 1029/1024, with monzo |-10 1 0 3>. For any member of the clan, for the rank three [[Gamelismic_family#Gamelan|gamelan temperament]] itself, and for the rank two 2.3.7 temperament [[Slendric|slendric]], this means three 8/7 intervals give a fifth, 3/2. In fact, we find that 3/2 = (8/7)^3 * 1029/1024. From this it follows that gamelismic temperaments tend to flatten the both fifth and the 7/4, or if they do not, the other of the pair must be flattened even more. [[36edo]] is a good tuning for gamelismic itself, though if the full 7-limit is desired, [[72edo]], [[77edo]] or [[118edo]] might be preferred.
{{Technical data page}}
The [[2.3.7 subgroup|2.3.7-subgroup]] [[comma]] for the '''gamelismic clan''' is the gamelisma, [[1029/1024]], with [[monzo]] {{monzo| -10 1 0 3 }}. For any member of the clan, for the rank-3 [[gamelismic family #Gamelismic|gamelismic temperament]] itself, and for the rank-2 2.3.7 temperament [[slendric]] (a.k.a. gamelic), this means three [[~]][[8/7]] intervals give a fifth, [[3/2]]. In fact, we find that {{nowrap| 3/2 {{=}} (8/7)<sup>3</sup>⋅(1029/1024) }}. From this it follows that gamelismic temperaments tend to flatten both the fifth and the harmonic seventh, or if they do not, the other of the pair must be flattened even more. [[36edo]] is a good tuning for slendric, though if the full 7-limit is desired, [[72edo]], [[77edo]], or [[118edo]] might be preferred.


7-limit minimax
== Slendric ==
{{Main| Slendric }}


[|1 0 0 0&gt;, |5/2 3/4 0 -3/4&gt;,
[[Subgroup]]: 2.3.7
|5/2 -1/4 1 -3/4&gt;, |5/2 -1/4 0 1/4&gt;]


Eigenmonzos: 2, 7/6, 6/5
[[Comma list]]: 1029/1024


9-limit minimax
{{Mapping|legend=2| 1 1 3 | 0 3 -1 }}


[|1 0 0 0&gt;, |10/7 6/7 0 -3/7&gt;,
{{Mapping|legend=3| 1 1 0 3 | 0 3 0 -1 }}
|10/7 -1/7 1 -3/7&gt;, |20/7 -2/7 0 1/7&gt;]
: mapping generators: ~2, ~8/7


Eigenmonzos: 2, 6/5, 9/7
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.4859{{c}}, ~8/7 = 233.7822{{c}}
: [[error map]]: {{val| +0.486 -0.123 -1.151 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~8/7 = 233.7474{{c}}
: error map: {{val| 0.000 -0.713 -2.573 }}


Lattice basis: 8/7 0.5192 5/4 2.3219
{{Optimal ET sequence|legend=1| 5, 21, 26, 31, 36, 77, 113, 190 }}


Angle(8/7, 5/4) = 90 degrees
[[Badness]] (Sintel): 0.158


Map to lattice: [&lt;0 3 0 -1|, &lt;0 0 1 0|]
=== 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.


Map: [&lt;1 1 0 3|, &lt;0 3 0 -1|, &lt; 0 0 1 0|]
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.


Generators: 2, 5, 7
Full 7-limit temperaments discussed elsewhere are:
* [[Blackwood]] (+28/27) → [[Limmic temperaments #Blackwood|Limmic temperaments]]
* [[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]]


No-fives [[POTE_tuning|POTE generator]]: 233.688
The rest are considered below.


EDOs: 5, 36, 77, 190
==== Subgroup extensions ====
No-five subgroup extensions of slendric include radon, a 2.3.7.11-subgroup extension that may be viewed as no-five rodan, considered below, euslendric, a 2.3.7.13-subgroup extension, baladic, a weak 2.3.7.13.17-subgroup extension, and gigapyth, a 2.3.7.85-subgroup extension, considered in [[#Other subgroup extensions]]. Dicussed elsewhere is [[Subgroup temperaments #Trisect|trisect]] in the 2.3.7.11/5 subgroup.


==Full seven limit children==
=== Radon ===
To the gamelisma itself we need to add the comma which appears next on the modified [[Normal_lists|normal comma list]], which is often a 5-limit comma. 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's 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 major third. Finally, tritikleismic adds 15625/15536 and has a generator of 6/5 with a 1/3 octave period.
{{See also|Chromatic pairs #Radon}}


=Miracle=
Radon is the no-fives version of [[rodan]], equating the diatonic major third to [[14/11]].
[[Comma|Commas]]: 225/224, 1029/1024


[[Minimax_tuning|Minimax tuning]]:
Subgroup: 2.3.7.11


7-limit: [|1 0 0 0&gt;, |25/13 6/13 -6/13 0&gt;,
Comma list: 896/891, 1029/1024
|25/13 -7/13 7/13 0&gt;, |35/13 -2/13 2/13 0&gt;]


[[Eigenmonzo|Eigenmonzos]]: 2, 6/5
Subgroup-val mapping: {{mapping| 1 1 3 6 | 0 3 -1 -13 }}


9-limit: [|1 0 0 0&gt;, |25/19 12/19 -6/19 0&gt;,
Gencom mapping: {{mapping| 1 1 0 3 6 | 0 3 0 -1 -13 }}
|50/19 -14/19 7/19 0&gt;, |55/19 -4/19 2/19 0&gt;]


[[Eigenmonzo|Eigenmonzos]]: 2, 10/9
Optimal tunings:  
* WE: ~2 = 1199.9708{{c}}, ~8/7 = 234.3748{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.3813{{c}}


[[POTE_tuning|POTE generator]]: ~15/14 = 116.675
{{Optimal ET sequence|legend=0| 5, …, 36, 41, 87, 128 }}


Algebraic generator: Secor59, [[Algebraic_number|positive root]] of 15x^6-8x^4-12
Badness (Sintel): 0.619


Map: [&lt;1 1 3 3|, &lt;0 6 -7 -2|]
== Mothra ==
{{Main| Mothra }}


Wedgie: &lt;&lt;6 -7 -2 -25 -20 15||
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]].


EDOs: 10, 21, 31, 41, 72, [[175edo|175]]
Note that mothra is also called '''cynder''' in the 7-limit, which can be a little confusing sometimes.


Badness: 0.0167
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]].


==11-limit==
[[Subgroup]]: 2.3.5.7
[[Comma|Commas]]: 225/224, 243/242, 385/384


[[Minimax_tuning|Minimax tuning]]:
[[Comma list]]: 81/80, 1029/1024


[|1 0 0 0 0&gt;, |25/19 12/19 -6/19 0 0&gt;,
{{Mapping|legend=1| 1 1 0 3 | 0 3 12 -1 }}
|50/19 -14/19 7/19 0 0&gt;, |55/19 -4/19 2/19 0 0&gt;,
|53/19 30/19 -15/19 0 0&gt;]


[[Eigenmonzo|Eigenmonzos]]: 2, 10/9
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.9303{{c}}, ~8/7 = 232.3733{{c}}
: [[error map]]: {{val| +0.930 -3.905 +2.165 +1.592 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 232.2514{{c}}
: error map: {{val| 0.000 -5.520 +0.703 -1.077 }}


[[POTE_tuning|POTE generator]]: ~15/14 = 116.633
[[Algebraic generator]]: Rabrindanath, largest real root of ''x''<sup>8</sup> - 3''x''<sup>2</sup> + 1, or 232.0774 cents.
 
[[Minimax tuning]]:
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~8/7 = {{monzo| 0 0 1/12 }}
: {{monzo list| 1 0 0 0 | 1 0 1/4 0 | 0 0 1 0 | 3 0 -1/12 0 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5
 
{{Optimal ET sequence|legend=1| 5, 21c, 26, 31 }}
 
[[Badness]] (Sintel): 0.940
 
=== Undecimal mothra ===
Undecimal mothra is the extension of 7-limit cynder which tempers out 385/384 as is natural in slendric temperaments. It is the simplest extension, supported within a reasonable tuning range (between [[26edo]] and 31edo), and is supported by the patent val of [[5edo]], which implies that it is better behaved as a cluster temperament. It is also notable for being supported by the just tuning of 8/7, and has a restriction to the 2.7.11 subgroup, namely [[amaranthine]], that is a microtemperament.
 
Subgroup: 2.3.5.7.11
 
Comma list: 81/80, 99/98, 385/384
 
Mapping: {{mapping| 1 1 0 3 5 | 0 3 12 -1 -8 }}
 
Optimal tunings:
* WE: ~2 = 1201.3979{{c}}, ~8/7 = 232.3010{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 232.0621{{c}}
 
{{Optimal ET sequence|legend=0| 5, 26, 31, 88, 119be, 150be }}
 
Badness (Sintel): 0.848
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 81/80, 99/98, 105/104, 144/143
 
Mapping: {{mapping| 1 1 0 3 5 1 | 0 3 12 -1 -8 14 }}
 
Optimal tunings:
* WE: ~2 = 1201.0985{{c}}, ~8/7 = 232.0231{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 231.8425{{c}}
 
{{Optimal ET sequence|legend=0| 5, 26, 31, 57, 88 }}
 
Badness (Sintel): 0.990
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 81/80, 99/98, 105/104, 120/119, 144/143
 
Mapping: {{mapping| 1 1 0 3 5 1 | 0 3 12 -1 -8 14 16 }}
 
Optimal tunings:
* WE: ~2 = 1200.9734{{c}}, ~8/7 = 231.8960{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 231.7392{{c}}
 
{{Optimal ET sequence|legend=0| 5g, 26, 31, 57, 88 }}
 
Badness (Sintel): 1.00
 
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 81/80, 99/98, 105/104, 120/119, 144/143, 153/152
 
Mapping: {{mapping| 1 1 0 3 5 1 | 0 3 12 -1 -8 14 16 22 }}
 
Optimal tunings:
* WE: ~2 = 1200.9663{{c}}, ~8/7 = 231.8393{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 231.6842{{c}}
 
{{Optimal ET sequence|legend=0| 26, 31, 57 }}
 
Badness (Sintel): 1.05
 
=== Mosura ===
The [[S-expression]]-based comma list of mosura suggests it might be the most natural extension of 7-limit cynder to the 11-limit: {[[1728/1715|S6/S7]], [[1029/1024|S7/S8]], ([[81/80|S6/S8 = S9]]), [[176/175|S8/S10]]}.
 
Subgroup: 2.3.5.7.11
 
Comma list: 81/80, 176/175, 540/539
 
Mapping: {{mapping| 1 1 0 3 -1 | 0 3 12 -1 23 }}
 
Optimal tunings:
* WE: ~2 = 1200.7675{{c}}, ~8/7 = 232.5673{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 232.4567{{c}}
 
{{Optimal ET sequence|legend=0| 5e, 26e, 31, 129 }}
 
Badness (Sintel): 1.04
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 81/80, 144/143, 176/175, 196/195
 
Mapping: {{mapping| 1 1 0 3 -1 7 | 0 3 12 -1 23 -17 }}
 
Optimal tunings:
* WE: ~2 = 1199.9347{{c}}, ~8/7 = 232.6275{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 232.6392{{c}}
 
{{Optimal ET sequence|legend=0| 31, 67, 98 }}
 
Badness (Sintel): 1.52
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 81/80, 144/143, 176/175, 189/187, 196/195
 
Mapping: {{mapping| 1 1 0 3 -1 7 | 0 3 12 -1 23 -17 -15 }}
 
Optimal tunings:
* WE: ~2 = 1199.7124{{c}}, ~8/7 = 232.6376{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 232.6917{{c}}
 
{{Optimal ET sequence|legend=0| 31, 67, 98 }}
 
Badness (Sintel): 1.53
 
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 81/80, 96/95, 144/143, 153/152, 176/175, 196/195
 
Mapping: {{mapping| 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 }}
 
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:
* WE: ~2 = 1201.1585{{c}}, ~8/7 = 231.5404{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 231.3850{{c}}
 
{{Optimal ET sequence|legend=0| 5e, 21ce, 26 }}
 
Badness (Sintel): 1.84
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 45/44, 78/77, 81/80, 640/637
 
Mapping: {{mapping| 1 1 0 3 0 1 | 0 3 12 -1 18 14 }}
 
Optimal tunings:
* WE: ~2 = 1201.1152{{c}}, ~8/7 = 231.5079{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 231.3612{{c}}
 
{{Optimal ET sequence|legend=0| 5e, 21cef, 26 }}
 
Badness (Sintel): 1.41
 
== Rodan ==
{{Main| Rodan }}
: ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum #Rodan (5-limit)]].''
 
Rodan tempers out 245/243 and can be described as the {{nowrap| 41 & 46 }} temperament. This temperament 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
 
[[Comma list]]: 245/243, 1029/1024
 
{{Mapping|legend=1| 1 1 -1 3 | 0 3 17 -1 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.2146{{c}}, ~8/7 = 234.4587{{c}}
: [[error map]]: {{val| +0.215 +1.636 -0.731 -2.641 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 234.4259{{c}}
: error map: {{val| 0.000 +1.323 -1.073 -3.252 }}
 
[[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|unchanged-interval (eigenmonzo) basis]]: 2.7/5
 
[[Algebraic generator]]: larger root of 20''x''<sup>2</sup> - 36''x'' + 15, or (9 + √6)/10.
 
{{Optimal ET sequence|legend=1| 41, 87, 128, 215d }}
 
[[Badness]] (Sintel): 0.939
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 245/243, 385/384, 441/440
 
Mapping: {{mapping| 1 1 -1 3 6 | 0 3 17 -1 -13 }}
 
Optimal tunings:
* WE: ~2 = 1200.0553{{c}}, ~8/7 = 234.4695{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.4594{{c}}
 
Minimax tuning:
* 11-odd-limit: ~8/7 = {{monzo| 4/19 2/19 0 0 -1/19 }}
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 31/19 6/19 0 0 -3/19 }}, {{monzo| 49/19 34/19 0 0 -17/19 }}, {{monzo| 53/19 -2/19 0 0 1/19 }}, {{monzo| 62/19 -26/19 0 0 13/19 }}]
: unchanged-interval (eigenmonzo) basis: 2.11/9
 
Algebraic generator: positive root of ''x''<sup>2</sup> + 16''x'' - 31, or √95 - 8.
 
{{Optimal ET sequence|legend=0| 41, 87 }}
 
Badness (Sintel): 0.763
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
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:
* WE: ~2 = 1199.9868{{c}}, ~8/7 = 234.4796{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.4822{{c}}
 
Minimax tuning:
* 13- and 15-odd-limit: ~8/7 = {{monzo| 3/14 1/14 0 0 0 -1/28 }}
: unchanged-interval (eigenmonzo) basis: 2.13/9
 
Algebraic generator: Gatetone, positive root of 4''x''<sup>6</sup> - 7''x'' - 1. Recurrence converges slowly.
 
{{Optimal ET sequence|legend=0| 41, 46, 87 }}
 
Badness (Sintel): 0.762
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 154/153, 196/195, 245/243, 256/255, 273/272
 
Mapping: {{mapping| 1 1 -1 3 6 8 8 | 0 3 17 -1 -13 -22 -20 }}
 
Optimal tunings:
* WE: ~2 = 1199.8331{{c}}, ~8/7 = 234.4919{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.5254{{c}}
 
Minimax tuning:
* 17-odd-limit: ~8/7 = {{monzo| 3/13 1/13 0 0 0 0 -1/26 }}
: unchanged-interval (eigenmonzo) basis: 2.17/9
 
{{Optimal ET sequence|legend=0| 41, 46, 87 }}
 
Badness (Sintel): 0.853
 
==== Aerodactyl ====
Subgroup: 2.3.5.7.11.13
 
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:
* WE: ~2 = 1200.2997{{c}}, ~8/7 = 234.6972{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.6439{{c}}
 
{{Optimal ET sequence|legend=0| 5, 41f, 46 }}
 
Badness (Sintel): 1.40
 
=== Aerodino ===
Subgroup: 2.3.5.7.11
 
Comma list: 176/175, 245/243, 1029/1024
 
Mapping: {{mapping| 1 1 -1 3 -3 | 0 3 17 -1 33 }}
 
Optimal tunings:
* WE: ~2 = 1199.9179{{c}}, ~8/7 = 234.7123{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.7256{{c}}
 
{{Optimal ET sequence|legend=0| 5e, 41e, 46 }}
 
Badness (Sintel): 1.79
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
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:
* WE: ~2 = 1200.0242{{c}}, ~8/7 = 234.7863{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.7824{{c}}
 
{{Optimal ET sequence|legend=0| 5e, 41ef, 46 }}
 
Badness (Sintel): 1.48
 
=== Varan ===
Subgroup: 2.3.5.7.11
 
Comma list: 100/99, 245/243, 1029/1024
 
Mapping: {{mapping| 1 1 -1 3 -2 | 0 3 17 -1 28 }}
 
Optimal tunings:
* WE: ~2 = 1200.3738{{c}}, ~8/7 = 234.2174{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.1586{{c}}
 
{{Optimal ET sequence|legend=0| 5e, 36ce, 41 }}
 
Badness (Sintel): 1.49
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
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:
* WE: ~2 = 1200.1389{{c}}, ~8/7 = 234.1162{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.0946{{c}}
 
{{Optimal ET sequence|legend=0| 5e, 36ce, 41 }}
 
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 & 41 }} temperament. It is more complex than rodan, but the optimal tuning is closer to optimal slendric.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 1029/1024, 10976/10935
 
{{Mapping|legend=1| 1 1 7 3 | 0 3 -24 -1 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.3395{{c}}, ~8/7 = 233.9963{{c}}
: [[error map]]: {{val| +0.340 +0.374 +0.151 -1.804 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 233.9239{{c}}
: error map: {{val| 0.000 -0.183 -0.487 -2.750 }}
 
[[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|unchanged-interval (eigenmonzo) basis]]: 2.5
 
{{Optimal ET sequence|legend=1| 36, 41, 77, 118, 277d }}
 
[[Badness]] (Sintel): 1.20
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 385/384, 441/440, 10976/10935
 
Mapping: {{mapping| 1 1 7 3 -2 | 0 3 -24 -1 28 }}
 
Optimal tunings:
* WE: ~2 = 1200.3453{{c}}, ~8/7 = 233.9988{{c}}
* 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 }}]
: unchanged-interval (eigenmonzo) basis: 2.5
 
{{Optimal ET sequence|legend=0| 36e, 41, 77, 118, 159, 277d }}
 
Badness (Sintel): 0.881
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 196/195, 352/351, 385/384, 729/728
 
Mapping: {{mapping| 1 1 7 3 -2 0 | 0 3 -24 -1 28 19 }}
 
Optimal tunings:
* WE: ~2 = 1200.1222{{c}}, ~8/7 = 233.9228{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.8994{{c}}
 
{{Optimal ET sequence|legend=0| 36e, 41, 77, 118 }}
 
Badness (Sintel): 1.18
 
== Gorgo ==
: ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum #Laconic]].''
{{See also| Llywelynsmic clan }}
 
Gorgo tempers the generator of ~8/7 together with ~10/9. It can be described as the {{nowrap| 16 & 21 }} temperament.
 
If we discard the inaccurate mapping of prime 3, we get [[shoe]], so that the large commas of gorgo are explained practically entirely by the inaccurate 3.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 36/35, 1029/1024
 
{{Mapping|legend=1| 1 1 1 3 | 0 3 7 -1 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.9847{{c}}, ~8/7 = 228.5210{{c}}
: [[error map]]: {{val| +0.985 -15.407 +14.318 +5.607 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 228.4371{{c}}
: error map: {{val| 0.000 -16.644 +12.746 +2.737 }}
 
{{Optimal ET sequence|legend=1| 5, 11c, 16, 21 }}
 
[[Badness]] (Sintel): 1.54
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 36/35, 45/44, 1029/1024
 
Mapping: {{mapping| 1 1 1 3 1 | 0 3 7 -1 13 }}
 
Optimal tunings:
* WE: ~2 = 1201.3609{{c}}, ~8/7 = 227.6312{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 227.4955{{c}}
 
{{Optimal ET sequence|legend=0| 5e, 16, 21, 37b }}
 
Badness (Sintel): 1.64
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
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:
* WE: ~2 = 1201.0996{{c}}, ~8/7 = 227.4378{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 227.3327{{c}}
 
{{Optimal ET sequence|legend=0| 5e, 16, 21, 37b }}
 
Badness (Sintel): 1.35
 
=== Spartan ===
Subgroup: 2.3.5.7.11
 
Comma list: 36/35, 56/55, 1029/1024
 
Mapping: {{mapping| 1 1 1 3 5 | 0 3 7 -1 -8 }}
 
Optimal tunings:
* WE: ~2 = 1198.9344{{c}}, ~8/7 = 229.3316{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 229.5124{{c}}
 
{{Optimal ET sequence|legend=0| 5, 16e, 21 }}
 
Badness (Sintel): 2.07
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
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:
* WE: ~2 = 1198.3002{{c}}, ~8/7 = 228.7341{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 229.0044{{c}}
 
{{Optimal ET sequence|legend=0| 5, 16e, 21 }}
 
Badness (Sintel): 1.95
 
; Music
* [https://web.archive.org/web/20201127012514/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Herman/gorgo-example.mp3 ''Gorgo Example''] by [[Herman Miller]]
 
== Gidorah ==
: ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum #University]].''
 
Gidorah is a very low-accuracy temperament where the generator of ~8/7 is lumped together with ~6/5. 16c-, 21cc-, and 26ccc-edo are among the possible tunings.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 21/20, 144/125
 
{{Mapping|legend=1| 1 1 2 3 | 0 3 2 -1 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1192.4932{{c}}, ~8/7 = 229.3187{{c}}
: [[error map]]: {{val| -7.507 -21.506 +57.310 -20.665 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 229.6649{{c}}
: error map: {{val| 0.000 -12.960 +73.016 +1.509 }}
 
{{Optimal ET sequence|legend=1| 1b, 5 }}
 
[[Badness]] (Sintel): 1.58
 
== Oncle ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Oncle]].''
 
Oncle can be described as the {{nowrap| 31 & 36c }} temperament.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 1029/1024, 2430/2401
 
{{Mapping|legend=1| 1 1 6 3 | 0 3 -19 -1 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.2246{{c}}, ~8/7 = 232.7354{{c}}
: [[error map]]: {{val| +1.225 -2.524 -0.939 +2.112 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 232.4718{{c}}
: error map: {{val| 0.000 -4.539 -3.279 -1.298 }}
 
{{Optimal ET sequence|legend=1| 31, 98c, 129c, 160bc }}
 
[[Badness]] (Sintel): 2.24
 
== Archaeotherium ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Archaeotherium]].''
 
Archaeotherium can be described as the {{nowrap| 21 & 26 }} temperament.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 405/392, 1029/1024
 
{{Mapping|legend=1| 1 1 5 3 | 0 3 -14 -1 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1202.7179{{c}}, ~8/7 = 230.7800{{c}}
: [[error map]]: {{val| +2.718 -6.897 -3.644 +8.548 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 230.1909{{c}}
: error map: {{val| 0.000 -11.382 -8.986 +0.983 }}
 
{{Optimal ET sequence|legend=1| 21, 26, 47, 73bc }}
 
[[Badness]] (Sintel): 3.70
 
== Clyndro ==
Clyndro tempers out [[135/128]] and finds the interval class of 5 at a stack of -3 fifths as does any temperament in the [[mavila family]]. It can be described as the {{nowrap| 11 & 16 }} temperament.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 135/128, 360/343
 
{{Mapping|legend=1| 1 1 4 3 | 0 3 -9 -1 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1205.6135{{c}}, ~8/7 = 227.5283{{c}}
: [[error map]]: {{val| +5.613 -13.757 -11.614 +20.486 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 226.3207{{c}}
: error map: {{val| 0.000 -22.993 -23.200 +4.853 }}
 
{{Optimal ET sequence|legend=1| 5c, 11, 16 }}
 
[[Badness]] (Sintel): 4.03
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 33/32, 45/44, 352/343
 
Mapping: {{mapping| 1 1 4 3 4 | 0 3 -9 -1 -3 }}
 
Optimal tunings:
* WE: ~2 = 1206.2134{{c}}, ~8/7 = 227.6004{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 226.2421{{c}}
 
{{Optimal ET sequence|legend=0| 5c, 11, 16 }}
 
Badness (Sintel): 2.30
 
== Miracle ==
{{Main| Miracle }}
: ''For the 5-limit version, see [[Syntonic–31 equivalence continuum #Ampersand]].''
 
Miracle is one of the most important entries of this temperament clan. It tempers out [[225/224]], splitting the ~8/7 generator of slendric into 15/14~16/15, and can be described as the {{nowrap| 31 & 41 }} temperament. 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
 
[[Comma list]]: 225/224, 1029/1024
 
{{Mapping|legend=1| 1 1 3 3 | 0 6 -7 -2 }}
: mapping generator: ~2, ~15/14
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.8209{{c}}, ~15/14 = 116.7550{{c}}
: [[error map]]: {{val| +0.821 -0.604 -1.136 +0.127 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~15/14 = 116.6756{{c}}
: error map: {{val| 0.000 -1.901 -3.043 -2.177 }}
 
[[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|unchanged-interval (eigenmonzo) basis]]: 2.5/3
* [[9-odd-limit]]: ~15/14 = {{monzo| 1/19 2/19 -1/19 }}
: {{monzo list| 1 0 0 0 | 25/19 12/19 -6/19 0 | 50/19 -14/19 7/19 0 | 55/19 -4/19 2/19 0 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5
 
[[Tuning ranges]]:
* 7-odd-limit [[diamond monotone]]: ~15/14 = [114.286, 120.000] (2\21 to 1\10)
* 9-odd-limit diamond monotone: ~15/14 = [116.129, 120.000] (3\31 to 1\10)
* 7- and 9-odd-limit [[diamond tradeoff]]: ~15/14 = [115.587, 116.993]
 
[[Algebraic generator]]: Secor59, positive root of 15''x''<sup>6</sup> - 8''x''<sup>4</sup> - 12
 
{{Optimal ET sequence|legend=1| 10, 21, 31, 41, 72 }}
 
[[Badness]] (Sintel): 0.424
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 243/242, 385/384
 
Mapping: {{mapping| 1 1 3 3 2 | 0 6 -7 -2 15 }}
 
Optimal tunings:
* WE: ~2 = 1200.7626{{c}}, ~15/14 = 116.7069{{c}}
* 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 }}]
: unchanged-interval (eigenmonzo) basis: 2.9/5
 
Tuning ranges:
* 11-odd-limit diamond monotone: ~15/14 = [116.129, 117.073] (3\31 to 4\41)
* 11-odd-limit diamond tradeoff: ~15/14 = [115.587, 116.993]


Algebraic generator: Secor59
Algebraic generator: Secor59


Map: [&lt;1 1 3 3 2|, &lt;0 6 -7 -2 15|]
{{Optimal ET sequence|legend=0| 10, 21e, 31, 41, 72, 247c, 319bcde, 391bcde, 463bccde }}
 
Badness (Sintel): 0.353
 
==== Miraculous ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 105/104, 144/143, 196/195, 243/242
 
Mapping: {{mapping| 1 1 3 3 2 4 | 0 6 -7 -2 15 -3 }}
 
Optimal tunings:
* WE: ~2 = 1200.1267{{c}}, ~15/14 = 116.7596{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.7488{{c}}
 
{{Optimal ET sequence|legend=0| 10, 21e, 31, 41, 72f }}
 
Badness (Sintel): 0.771
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
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:
* WE: ~2 = 1199.6759{{c}}, ~15/14 = 116.7378{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.7657{{c}}
 
{{Optimal ET sequence|legend=0| 10, 21e, 31, 41, 72fg }}
 
Badness (Sintel): 0.870
 
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 105/104, 120/119, 144/143, 154/153, 170/169, 210/209
 
{{Todo|complete temperament data|inline=1}}
 
===== 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 ====
Subgroup: 2.3.5.7.11.13
 
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:
* WE: ~2 = 1199.8601{{c}}, ~15/14 = 116.6572{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.5688{{c}}
 
{{Optimal ET sequence|legend=0| 31, 72, 103, 175f }}
 
Badness (Sintel): 0.649
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
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:
* WE: ~2 = 1200.8328{{c}}, ~15/14 = 116.6661{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.5774{{c}}
 
{{Optimal ET sequence|legend=0| 31, 72, 103, 175f, 422bcdefffg }}
 
Badness (Sintel): 0.639
 
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 210/209, 225/224, 243/242, 273/272, 286/285, 375/374
 
{{Todo|complete temperament data|inline=1}}
 
===== 23-limit =====
Subgroup: 2.3.5.7.11.13.17.19.23
 
Comma list: 162/161, 210/209, 225/224, 231/230, 243/242, 273/272, 286/285
 
{{Todo|complete temperament data|inline=1}}
 
==== Manna ====
Subgroup: 2.3.5.7.11.13
 
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:
* WE: ~2 = 1200.7564{{c}}, ~15/14 = 116.8129{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.7528{{c}}
 
{{Optimal ET sequence|legend=0| 31f, 41, 72, 185cf, 257cff }}
 
Badness (Sintel): 0.703
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
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:
* WE: ~2 = 1200.7570{{c}}, ~15/14 = 116.8011{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.7408{{c}}
 
{{Optimal ET sequence|legend=0| 31fg, 41, 72, 185cf, 257cff }}
 
Badness (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 ====
Subgroup: 2.3.5.7.11.13
 
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:
* WE: ~55/39 = 600.4844{{c}}, ~15/14 = 116.7182{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~15/14 = 116.6413{{c}}
 
{{Optimal ET sequence|legend=0| 10, 62, 72 }}
 
Badness (Sintel): 1.02
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
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:
* WE: ~17/12 = 600.5042{{c}}, ~15/14 = 116.7264{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~15/14 = 116.6485{{c}}
 
{{Optimal ET sequence|legend=0| 10, 62, 72 }}
 
Badness (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 ====
Subgroup: 2.3.5.7.11.13
 
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:
* WE: ~2 = 1200.6969{{c}}, ~27/26 = 58.3217{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~27/26 = 58.2964{{c}}
 
{{Optimal ET sequence|legend=0| 41, 62, 103, 247c, 350bcde }}
 
Badness (Sintel): 1.06
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 225/224, 243/242, 273/272, 385/384, 847/845
 
Mapping: {{mapping| 1 1 3 3 2 2 2 | 0 12 -14 -4 30 35 43 }}
 
Optimal tunings:
* WE: ~2 = 1200.6557{{c}}, ~27/26 = 58.2932{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~27/26 = 58.2702{{c}}
 
{{Optimal ET sequence|legend=0| 41, 62, 103 }}
 
Badness (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 =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 225/224, 243/242, 289/288, 385/384, 847/845
 
Mapping: {{mapping| 2 2 6 6 4 4 7 | 0 12 -14 -4 30 35 12 }}
: mapping generators: ~17/12, ~27/26
 
Optimal tunings:
* WE: ~17/12 = 600.3951{{c}}, ~27/26 = 58.3260{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~27/26 = 58.2974{{c}}
 
{{Optimal ET sequence|legend=0| 62, 144g, 206begg }}
 
Badness (Sintel): 2.39
 
====== 19-limit ======
Subgroup: 2.3.5.7.11.13.17.19
 
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:
* WE: ~17/12 = 600.4418{{c}}, ~27/26 = 58.3255{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~27/26 = 58.2928{{c}}
 
{{Optimal ET sequence|legend=0| 62, 144gh, 206begghh }}
 
Badness (Sintel): 2.13
 
====== 23-limit ======
Subgroup: 2.3.5.7.11.13.17.19.23
 
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 }}
 
Optimal tunings:
* WE: ~17/12 = 600.4451{{c}}, ~27/26 = 58.3264{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~27/26 = 58.2942{{c}}
 
{{Optimal ET sequence|legend=0| 62, 144gh, 206begghhi }}
 
Badness (Sintel): 1.89
 
==== Phicordial ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 225/224, 243/242, 385/384, 2200/2197
 
Mapping: {{mapping| 1 -11 17 7 -28 3 | 0 18 -21 -6 45 1 }}
: mapping generators: ~2, ~13/8
 
Optimal tunings:
* WE: ~2 = 1200.7056{{c}}, ~13/8 = 839.3726{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/8 = 838.8831{{c}}
 
{{Optimal ET sequence|legend=0| 103, 216c, 319bcde, 535bccdef }}
 
Badness (Sintel): 1.37
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 225/224, 243/242, 273/272, 385/384, 2200/2197
 
Mapping: {{mapping| 1 -11 17 7 -28 3 -5 | 0 18 -21 -6 45 1 13 }}
 
Optimal tunings:
* WE: ~2 = 1200.5918{{c}}, ~13/8 = 839.2912{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/8 = 838.8809{{c}}
 
{{Optimal ET sequence|legend=0| 103, 216c, 319bcde }}
 
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 ===
Subgroup: 2.3.5.7.11
 
Comma list: 99/98, 176/175, 1029/1024
 
Mapping: {{mapping| 1 1 3 3 5 | 0 6 -7 -2 -16 }}
 
Optimal tunings:
* WE: ~2 = 1201.3320{{c}}, ~15/14 = 116.4057{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.2524{{c}}
 
{{Optimal ET sequence|legend=0| 10e, 21, 31 }}
 
Badness (Sintel): 1.09
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
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:
* WE: ~2 = 1200.6059{{c}}, ~15/14 = 116.3263{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.2564{{c}}
 
{{Optimal ET sequence|legend=0| 10e, 21, 31 }}
 
Badness (Sintel): 1.22
 
=== Hemimiracle ===
Subgroup: 2.3.5.7.11
 
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:
* WE: ~2 = 1200.2902{{c}}, ~33/32 = 58.4217{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~33/32 = 58.4062{{c}}
 
{{Optimal ET sequence|legend=0| 20, 21, 41 }}


[[EDO|EDOs]]: 10, 31, 41, [[72edo|72]], 247c, 319bcde, [[391edo|391bcde]], 463bccde
Badness (Sintel): 1.96


Badness: 0.0107
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


[[Chords_of_miracle|Chords of miracle]]
Comma list: 105/104, 196/195, 245/242, 512/507


==Revelation==
Mapping: {{mapping| 1 1 3 3 4 4 | 0 12 -14 -4 -11 -6 }}
Commas: 99/98, 176/175, 1029/1024


POTE generator: ~15/14 = 116.277
Optimal tunings:  
* WE: ~2 = 1199.8454{{c}}, ~33/32 = 58.4220{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~33/32 = 58.4305{{c}}


Map: [&lt;1 1 3 3 5|, &lt;0 6 -7 -2 -16|]
{{Optimal ET sequence|legend=0| 20, 21, 41 }}


EDOs: 10e, 21, 31
Badness (Sintel): 1.78


Badness: 0.0329
=== Oracle ===
Subgroup: 2.3.5.7.11


===13-limit===
Comma list: 121/120, 225/224, 1029/1024
Commas: 66/65, 99/98, 105/104, 512/507


POTE generator: ~15/14 = 116.268
Mapping: {{mapping| 1 -5 10 5 4 | 0 12 -14 -4 -1 }}
: mapping generators: ~2, ~16/11


Map: [&lt;1 1 3 3 5 4|, &lt;0 6 -7 -2 -16 -3|]
Optimal tunings:  
* WE: ~2 = 1201.2122{{c}}, ~16/11 = 658.9974{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/11 = 658.3320{{c}}


EDOs: 10e, 21, 31
{{Optimal ET sequence|legend=0| 11, 20, 31, 82e, 113e, 144ee }}


Badness: 0.0295
Badness (Sintel): 1.41


==Miraculous==
== Hemiseven ==
Commas: 105/104, 144/143, 196/195, 243/242
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.


POTE generator: ~15/14 = 116.747
[[Subgroup]]: 2.3.5.7


Map: [&lt;1 1 3 3 2 4|, &lt;0 6 -7 -2 15 -3|]
[[Comma list]]: 1029/1024, 19683/19600


EDOs: 10, 31, 41, 72f, 113f, 185cff
{{Mapping|legend=1| 1 -2 -15 4 | 0 6 29 -2 }}
: mapping generators: ~2, ~243/160


Badness: 0.0187
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.5612{{c}}, ~243/160 = 717.0687{{c}}
: [[error map]]: {{val| +0.561 -0.665 +0.260 -0.718 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~243/160 = 716.7478{{c}}
: error map: {{val| 0.000 -1.468 -0.629 -2.321 }}


==Benediction==
{{Optimal ET sequence|legend=1| 72, 149, 221, 514bd, 735bcdd }}
Commas: 225/224, 243/242, 351/350, 385 /384


POTE generator: ~15/14 = 116.574
[[Badness]] (Sintel): 1.43


Map: [&lt;1 1 3 3 2 7|, &lt;0 6 -7 -2 15 -34|]
=== 11-limit ===
Subgroup: 2.3.5.7.11


EDOs: 31, 72, 103, [[175edo|175f]]
Comma list: 385/384, 441/440, 19683/19600


Badness: 0.0157
Mapping: {{mapping| 1 -2 -15 4 16 | 0 6 29 -2 -21 }}


===17-limit===
Optimal tunings:
Commas: 225/224, 243/242, 273/272, 351/350, 375/374
* WE: ~2 = 1200.6243{{c}}, ~243/160 = 717.0969{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~243/160 = 716.7292{{c}}


POTE generator: ~15/14 = 116.585
{{Optimal ET sequence|legend=0| 72, 149, 221e, 293de }}


Map: [&lt;1 1 3 3 2 7 7|, &lt;0 6 -7 -2 15 -34 -30|]
Badness (Sintel): 0.941


EDOs: 31, 72, 103, 175f
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


==Manna==
Comma list: 351/350, 385/384, 441/440, 676/675
Commas: 225/224, 243/242, 325/324, 385/384


POTE generator: ~15/14 = 116.739
Mapping: {{mapping| 1 -2 -15 4 16 -19 | 0 6 29 -2 -21 38 }}


Map: [&lt;1 1 3 3 2 0|, &lt;0 6 -7 -2 15 38|]
Optimal tunings:  
* WE: ~2 = 1200.6781{{c}}, ~91/60 = 717.1496{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~91/60 = 716.7520{{c}}


EDOs: 10f, 31f, 41, 72, 113, 185cf, 257cff
{{Optimal ET sequence|legend=0| 72, 149, 221ef }}


Badness: 0.0170
Badness (Sintel): 0.905


==Semimiracle==
=== 17-limit ===
Commas: 169/168, 225/224, 243/242, 385/384
Subgroup: 2.3.5.7.11.13.17


POTE generator: ~15/14 = 116.624
Comma list: 273/272, 351/350, 385/384, 441/440, 676/675


Map: [&lt;2 2 6 6 4 7|, &lt;0 6 -7 -2 15 2|]
Mapping: {{mapping| 1 -2 -15 4 16 -19 -21 | 0 6 29 -2 -21 38 42 }}


EDOs: 10, 62, 72
Optimal tunings:  
* WE: ~2 = 1200.6635{{c}}, ~68/45 = 717.1354{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~68/45 = 716.7472{{c}}


Badness: 0.0246
{{Optimal ET sequence|legend=0| 72, 149, 221ef }}


===17-limit===
Badness (Sintel): 0.800
Commas: 169/168, 221/220, 225/224, 243/242, 273/272


POTE generator: ~15/14 = 116.628
== Valentine ==
{{Main| Valentine }}
: ''For the 5-limit version, see [[Syntonic–31 equivalence continuum #Valentine (5-limit)]].''


Map: [&lt;2 2 6 6 4 7 7|, &lt;0 6 -7 -2 15 2 6|]
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>.


EDOs: 10, 62, 72
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.


Badness: 0.0161
[[Subgroup]]: 2.3.5.7


==Hemimiracle==
[[Comma list]]: 126/125, 1029/1024
Commas: 225/224, 245/242, 1029/1024


POTE generator: ~33/32 = 58.408
{{Mapping|legend=1| 1 1 2 3 | 0 9 5 -3 }}
: mapping generators: ~2, ~21/20


Map: [&lt;1 1 3 3 4|, &lt;0 12 -14 -4 -11|]
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0749{{c}}, ~21/20 = 77.8687{{c}}
: [[error map]]: {{val| +0.075 -1.062 +3.179 -2.207 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 77.8673{{c}}
: error map: {{val| 0.000 -1.149 +3.023 -2.428 }}


EDOs: 20, 21, 41, 144e, 185cee, 226cee
[[Minimax tuning]]:
* [[7-odd-limit]]: ~21/20 = {{monzo| 1/6 1/12 0 -1/12 }}
: {{monzo list| 1 0 0 0 | 5/2 3/4 0 -3/4 | 17/6 5/12 0 -5/12 | 5/2 -1/4 0 1/4 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/3
* [[9-odd-limit]]: ~21/20 = {{monzo| 1/21 2/21 0 -1/21}}
: {{monzo list| 1 0 0 0 | 10/7 6/7 0 -3/7 | 47/21 10/21 0 -5/21 | 20/7 -2/7 0 1/7 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/7


Badness: 0.0592
[[Algebraic generator]]: smaller root of ''x''<sup>2</sup> - 89''x'' + 92, or (89 - sqrt (7553))/2, at 77.8616 cents.  


===13-limit===
{{Optimal ET sequence|legend=1| 15, 31, 46, 77, 185 }}
Commas: 105/104, 196/195, 245/242, 512/507


POTE generator: ~33/32 = 58.430
[[Badness]] (Sintel): 0.786


Map: [&lt;1 1 3 3 4 4|, &lt;0 12 -14 -4 -11 -6|]
=== 11-limit ===
Subgroup: 2.3.5.7.11


EDOs: 20, 21, 41, 144eff, 185ceeff
Comma list: 121/120, 126/125, 176/175


Badness: 0.0432
Mapping: {{mapping| 1 1 2 3 3 | 0 9 5 -3 7 }}


==Oracle==
Optimal tunings:
Commas: 121/120, 225/224, 1029/1024
* WE: ~2 = 1200.3890{{c}}, ~22/21 = 77.9065{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/21 = 77.9007{{c}}


POTE generator: ~11/8 = 541.668
Minimax tuning:
* 11-odd-limit: ~21/20 = {{monzo| 0 0 0 -1/10 1/10 }}
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 1 0 0 -9/10 9/10 }}, {{monzo| 2 0 0 -1/2 1/2 }}, {{monzo| 3 0 0 3/10 -3/10 }}, {{monzo| 3 0 0 -7/10 7/10 }}]
: unchanged-interval (eigenmonzo) basis: 2.11/7


Map: [&lt;1 7 -4 1 3|, &lt;0 -12 14 4 1|]
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.


EDOs: 11, 20, 31, 51, 82e, 113e, 144ee
{{Optimal ET sequence|legend=0| 15, 31, 46, 77 }}


Badness: 0.0427
Badness (Sintel): 0.552


==Hemisecordite==
==== Valentino ====
Commas: 225/224, 243/242, 385/384, 847/845
Subgroup: 2.3.5.7.11.13


POTE generator: ~27/26 = 58.288
Comma list: 121/120, 126/125, 176/175, 196/195


Map: [&lt;1 1 3 3 2 2|, &lt;0 12 -14 -4 30 35|]
Mapping: {{mapping| 1 1 2 3 3 5 | 0 9 5 -3 7 -20 }}


EDOs: 41, 62, 103, 247c, 350bcde
Optimal tunings:  
* WE: ~2 = 1200.1967{{c}}, ~22/21 = 77.9708{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/21 = 77.9594{{c}}


Badness: 0.0256
{{Optimal ET sequence|legend=0| 15f, 31, 46, 77 }}


===17-limit===
Badness (Sintel): 0.854
Commas: 225/224, 243/242, 273/272, 385/384, 847/845


POTE generator: ~27/26 = 58.261
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


Map: [&lt;1 1 3 3 2 2 2|, &lt;0 12 -14 -4 30 35 43|]
Comma list: 121/120, 126/125, 154/153, 176/175, 196/195


EDOs: 41, 62, 103
Mapping: {{mapping| 1 1 2 3 3 5 5 | 0 9 5 -3 7 -20 -14 }}


Badness: 0.0225
Optimal tunings:  
* WE: ~2 = 1200.0404{{c}}, ~22/21 = 78.0055{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/21 = 78.0029{{c}}


==Phicordial==
{{Optimal ET sequence|legend=0| 15f, 31, 46, 77, 123e }}
Commas: 225/224, 243/242, 385/384, 2200/2197


POTE generator: ~16/13 = 361.121
Badness (Sintel): 0.854


Map: [&lt;1 7 -4 1 17 4|, &lt;0 -18 21 6 -45 -1|]
==== Lupercalia ====
Subgroup: 2.3.5.7.11.13


EDOs: 10, 103, 113, 216c
Comma list: 66/65, 105/104, 121/120, 126/125


Badness: 0.0332
Mapping: {{mapping| 1 1 2 3 3 3 | 0 9 5 -3 7 11 }}


===17-limit===
Optimal tunings:
Commas: 225/224, 243/242, 273/272, 441/440, 2200/2197
* WE: ~2 = 1199.9143{{c}}, ~22/21 = 77.7039{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/21 = 77.7049{{c}}


POTE generator: ~16/13 = 361.123
{{Optimal ET sequence|legend=0| 15, 31 }}


Map: [&lt;1 7 -4 1 17 4 8|, &lt;0 -18 21 6 -45 -1 -13|]
Badness (Sintel): 0.881


EDOs: 10, 103, 113, 216c
==== Dwynwen ====
Subgroup: 2.3.5.7.11.13


Badness: 0.0247
Comma list: 91/90, 121/120, 126/125, 176/175


==Music==
Mapping: {{mapping| 1 1 2 3 3 2 | 0 9 5 -3 7 26 }}
By [[Gene_Ward_Smith|Gene Ward Smith]]


[http://www.archive.org/details/RachmaninoffPlaysBlackjack Rachmaninoff Plays Blackjack] [http://www.archive.org/download/RachmaninoffPlaysBlackjack/rachman.mp3 play]
Optimal tunings:  
* WE: ~2 = 1200.1306{{c}}, ~22/21 = 78.2273{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/21 = 78.2241{{c}}


By [[Joseph_Pehrson|Joseph Pehrson]]
{{Optimal ET sequence|legend=0| 15, 31f, 46 }}


[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Pehrson/blackandjill.mp3 Black and Jill]
Badness (Sintel): 0.969


[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Pehrson/josephpehrson+blacklight.mp3 Blacklight]
==== Semivalentine ====
Subgroup: 2.3.5.7.11.13


[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Pehrson/josephpehrson+blackjack.mp3 Blackjack]
Comma list: 121/120, 126/125, 169/168, 176/175


=Rodan=
Mapping: {{mapping| 2 2 4 6 6 7 | 0 9 5 -3 7 3 }}
[[Comma|Commas]]: 245/243, 1029/1024
: mapping generators: ~55/39, ~22/21


7+9 limit minimax tuning: [|1 0 0 0&gt;, |5/3 0 1/6 -1/6&gt;,
Optimal tunings:  
|25/9 0 17/18 -17/18&gt;, |25/9 0 -1/18 1/18&gt;]
* WE: ~55/39 = 600.3497{{c}}, ~22/21 = 77.8845{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~22/21 = 77.8715{{c}}


[[Eigenmonzo|Eigenmonzos]]: 2, 7/5
{{Optimal ET sequence|legend=0| 16, 30, 46, 62, 108ef }}


[[POTE_tuning|POTE generator]]: ~8/7 = 234.417
Badness (Sintel): 1.35


Algebraic generator: [[Algebraic_number|larger root]] of 20x^2-36x+15, or (9+√6)/10.
==== Hemivalentine ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 1 -1 3|, &lt;0 3 17 -1|]
Comma list: 121/120, 126/125, 176/175, 343/338


EDOs: 5, 41, 87, 128, 215d
Mapping: {{mapping| 1 1 2 3 3 4 | 0 18 10 -6 14 -9 }}
: mapping generators: ~2, ~40/39


Badness: 0.0371
Optimal tunings:  
* WE: ~2 = 1199.6529{{c}}, ~40/39 = 39.0323{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~40/39 = 39.0383{{c}}


==11-limit==
{{Optimal ET sequence|legend=0| 30, 31, 61, 92f }}
[[Comma|Commas]]: 245/243, 385/384, 441/440


[[Minimax_tuning|Minimax tuning]]: [|1 0 0 0 0&gt;, |31/19 6/19 0 0 -3/19&gt;,
Badness (Sintel): 1.94
|49/19 34/19 0 0 -17/19&gt;, |53/19 -2/19 0 0 1/19&gt;,
|62/19 -26/19 0 0 13/19&gt;]


[[Eigenmonzo|Eigenmonzos]]: 2, 11/9
==== Demivalentine ====
Subgroup: 2.3.5.7.11.13


[[POTE_tuning|POTE generator]]: ~8/7 = 234.459
Comma list: 121/120, 126/125, 176/175, 676/675


Algebraic generator: [[Algebraic_number|positive root]] of x^2+16x-31, or √95-8.
Mapping: {{mapping| 1 -8 -3 6 -4 -16 | 0 18 10 -6 14 37 }}
: mapping generators: ~2, ~13/9


Map: [&lt;1 1 -1 3 6|, &lt;0 3 17 -1 -13|]
Optimal tunings:  
* WE: ~2 = 1200.3929{{c}}, ~13/9 = 639.1320{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/9 = 638.9325{{c}}


EDOs: 5, 41, 46, 87
{{Optimal ET sequence|legend=0| 15, 47ef, 62, 77 }}


Badness: 0.0231
Badness (Sintel): 1.44


[[Chords_of_rodan|Chords of rodan]]
=== Hemivalentino ===
Subgroup: 2.3.5.7.11


==13-limit==
Comma list: 126/125, 243/242, 1029/1024
Commas: 196/195, 245/243, 352/351, 364/363


13 and 15 limit minimax
Mapping: {{mapping| 1 1 2 3 2 | 0 18 10 -6 45 }}


[|1 0 0 0 0 0&gt;, |23/14 3/14 0 0 0 -3/28&gt;,
Optimal tunings:
|37/14 17/14 0 0 0 -17/28&gt;, |39/14 -1/14 0 0 0 1/28&gt;,
* WE: ~2 = 1200.0816{{c}}, ~45/44 = 38.9236{{c}}
|45/14 -13/14 0 0 0 13/28&gt;, |23/7 -11/7 0 0 0 11/14&gt;]
* CWE: ~2 = 1200.0000{{c}}, ~45/44 = 38.9228{{c}}


Eigenmonzos: 2, 13/9
{{Optimal ET sequence|legend=0| 31, 92e, 123, 154, 185 }}


[[POTE_tuning|POTE generator]]: ~8/7 = 234.482
Badness (Sintel): 2.03


Algebraic generator: Gatetone, positive root of 4x^6-7x-1. Recurrence converges slowly.
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 1 -1 3 6 8|, &lt;0 3 17 -1 -13 -22|]
Comma list: 126/125, 196/195, 243/242, 1029/1024


Generators: 2, 8/7
Mapping: {{mapping| 1 1 2 3 2 5 | 0 18 10 -6 45 -40 }}


EDOs: 41, 46, [[87edo|87]]
Optimal tunings:  
* WE: ~2 = 1199.8782{{c}}, ~45/44 = 38.9440{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~45/44 = 38.9472{{c}}


Badness: 0.0184
{{Optimal ET sequence|legend=0| 31, 123, 154 }}


==17-limit==
Badness (Sintel): 2.39
Commas: 154/153, 196/195, 245/243, 256/255, 273/272


Eigenmonzos: 2, 18/17
==== Hemivalentoid ====
Subgroup: 2.3.5.7.11.13


[[POTE_tuning|POTE generator]]: ~8/7 = 234.524
Comma list: 126/125, 144/143, 243/242, 343/338


Map: [&lt;1 1 -1 3 6 8 8|, &lt;0 3 17 -1 -13 -22 -20|]
Mapping: {{mapping| 1 1 2 3 2 4 | 0 18 10 -6 45 -9 }}


Edos: 41, 46, 87, 220dg, 307dgg
Optimal tunings:  
* WE: ~2 = 1199.3614{{c}}, ~45/44 = 38.9721{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~45/44 = 38.9839{{c}}


Badness: 0.0167
{{Optimal ET sequence|legend=0| 31, 92ef }}


===Music===
Badness (Sintel): 2.39
By [[Gene_Ward_Smith|Gene Ward Smith]]


[http://www.archive.org/details/Pianodactyl Pianodactyl] [http://www.archive.org/download/Pianodactyl/pianodactyl.mp3 play]
== Superkleismic ==
{{Main| Superkleismic }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Shibboleth]].''


==Aerodactyl==
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.
Commas: 91/90, 245/243, 385/384, 441/440


[[POTE_tuning|POTE generator]]: ~8/7 = 234.639
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.  


Map: [&lt;1 1 -1 3 6 -1|, &lt;0 3 17 -1 -13 24|
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.


EDOs: 5, 41f, 46, 51c
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.


Badness: 0.0340
41edo gives an obvious tuning in all the subgroups.  


==Aerodino==
[[Subgroup]]: 2.3.5.7
Commas: 176/175, 245/243, 1029/1024


POTE generator: ~8/7 = 234.728
[[Comma list]]: 875/864, 1029/1024


Map: [&lt;1 1 -1 3 -3|, &lt;0 3 17 -1 33|
{{Mapping|legend=1| 1 -5 -5 5 | 0 9 10 -3 }}
: mapping generators: ~2, ~5/3


EDOS: 5e, 41e, 46
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.7640{{c}}, ~5/3 = 878.6289{{c}}
: [[error map]]: {{val| +0.764 +1.885 +3.844 -0.893 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/3 = 878.1077{{c}}
: error map: {{val| 0.000 +1.014 -5.237 -3.149 }}


Badness: 0.0543
{{Optimal ET sequence|legend=1| 11c, 15, 26, 41 }}


===13-limit===
[[Badness]] (Sintel): 1.21
Commas: 91/90, 176/175, 245/243, 847/845


POTE generator: ~8/7 = 234.782
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 1 -1 3 -3 -1|, &lt;0 3 17 -1 33 24|
Comma list: 100/99, 245/242, 385/384


EDOs: 5e, 46
Mapping: {{mapping| 1 -5 -5 5 2 | 0 9 10 -3 2 }}


Badness: 0.0358
Optimal tunings:  
* WE: ~2 = 1200.1691{{c}}, ~5/3 = 878.2772{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 878.1606{{c}}


==Varan==
{{Optimal ET sequence|legend=0| 11c, 15, 26, 41, 179cde, 220cde, 261ccdee }}
Commas: 100/99, 245/243, 1029/1024


POTE generator: ~8/7 = 234.145
Badness (Sintel): 0.848


Map: [&lt;1 1 -1 3 -2|, &lt;0 3 17 -1 28|
==== 2.3.5.7.11.19 subgroup ====
Subgroup: 2.3.5.7.11.19


EDOs: 5e, 41, 46e
Comma list: 100/99, 133/132, 190/189, 385/384


Badness: 0.0449
Mapping: {{mapping| 1 -5 -5 5 2 -6 | 0 9 10 -3 2 14 }}


===13-limit===
Optimal tunings:
Commas: 100/99, 105/104, 245/243, 352/351
* WE: ~2 = 1200.2289{{c}}, ~5/3 = 878.3409{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 878.1840{{c}}


POTE generator: ~8/7 = 234.089
{{Optimal ET sequence|legend=0| 11c, 15, 26, 41, 138e }}


Map: [&lt;1 1 -1 3 -2 0|, &lt;0 3 17 -1 28 19|
Badness (Sintel): 0.692


EDOs: 5e, 41
=== 13-limit ===
Superkleismic in the 13-limit does considerably more damage than in the 11-limit, as indicated by being supported by much fewer [[patent val]]s and having higher Dirichlet badness than its 11-limit counterpart. However, this remains an obvious canonical mapping for prime 13.


Badness: 0.0323
Subgroup: 2.3.5.7.11.13


=[[Starling temperaments|Valentine]]=
Comma list: 100/99, 105/104, 144/143, 245/242
Comma: 1990656/1953125


POTE generator: ~25/24 = 78.039
Mapping: {{mapping| 1 -5 -5 5 2 -8 | 0 9 10 -3 2 16 }}


Map: [&lt;1 1 2|, &lt;0 9 5|]
Optimal tunings:  
* WE: ~2 = 1200.0261{{c}}, ~5/3 = 878.0252{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 878.0073{{c}}


EDOs: 15, 31, 46, 77, 123
{{Optimal ET sequence|legend=0| 11cf, 15, 26, 41 }}


Badness: 0.1228
Badness (Sintel): 0.887


==7-limit==
==== 17-limit ====
[[Comma|Commas]]: 126/125, 1029/1024
Subgroup: 2.3.5.7.11.13.17


[[Minimax_tuning|Minimax tuning]]:
Comma list: 100/99, 105/104, 120/119, 144/143, 245/242


7-limit: [|1 0 0 0&gt;, |5/2 3/4 0 -3/4&gt;,
Mapping: {{mapping| 1 -5 -5 5 2 -8 -12 | 0 9 10 -3 2 16 22 }}
|17/6 5/12 0 -5/12&gt;, [5/2 -1/4 0 1/4&gt;]


Eigenmonzos: 2, 7/6
Optimal tunings:  
* WE: ~2 = 1200.0488{{c}}, ~5/3 = 877.8872{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 877.8537{{c}}


9-limit: [|1 0 0 0&gt;, |10/7 6/7 0 -3/7&gt;,
{{Optimal ET sequence|legend=0| 11cfg, 15g, 26, 41 }}
|47/21 10/21 0 -5/21&gt;, |20/7 -2/7 0 1/7&gt;]


Eigenmonzos: 2, 9/7
Badness (Sintel): 1.01


[[POTE_tuning|POTE generator]]: ~21/20 = 77.864
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


Algebraic generator: [[Algebraic_number|smaller root]] of x^2-89x+92, or (89-√7553)/2, at 77.8616 cents.
Comma list: 100/99, 105/104, 120/119, 144/143, 133/132, 190/189


Map: [&lt;1 1 2 3|, &lt;0 9 5 -3|]
Mapping: {{mapping| 1 -5 -5 5 2 -8 -12 -6 | 0 9 10 -3 2 16 22 14 }}


Wedgie: &lt;&lt;9 5 -3 -13 -30 -21||
Optimal tunings:  
* WE: ~2 = 1200.2120{{c}}, ~5/3 = 878.0243{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 877.8789{{c}}


EDOs: 15, 31, 46, 77, 185, 262cd
{{Optimal ET sequence|legend=0| 11cfgh, 15g, 26, 41 }}


Badness: 0.0311
Badness (Sintel): 0.964


==11-limit==
=== Superana ===
[[Comma|Commas]]: 121/120, 126/125, 176/175
This extension ({{nowrap| 41 & 56 }}) is the counterpart of canonical superkleismic on the other side of 41edo.


[[Minimax_tuning|Minimax tuning]]:
Subgroup: 2.3.5.7.11.13


[|1 0 0 0 0&gt;, |1 0 0 -9/10 9/10&gt;,
Comma list: 100/99, 196/195, 245/242, 385/384
|2 0 0 -1/2 1/2&gt;, |3 0 0 3/10 -3/10&gt;, |3 0 0 -7/10 7/10&gt;]


Eigenmonzos: 2, 11/7
Mapping: {{mapping| 1 -5 -5 5 2 22 | 0 9 10 -3 2 -25 }}


[[POTE_tuning|POTE generator]]: ~21/20 = 77.881
Optimal tunings:  
* WE: ~2 = 1199.8272{{c}}, ~5/3 = 878.1538{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 878.2795{{c}}


Algebraic generator: [[Algebraic_number|positive root]] of 4x^3+15x^2-21, or else Gontrand2, the smallest positive root of 4x^7-8x^6+5.
{{Optimal ET sequence|legend=0| 15f, 41, 97, 138e }}


Map: [&lt;1 1 2 3 3|, &lt;0 9 5 -3 7|]
Badness (Sintel): 1.40


Generators: 2, 21/20
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


[[EDO|EDOs]]: 15, 31, 46, 77, 262cdee, 339cdeee
Comma list: 100/99, 154/153, 196/195, 245/242, 256/255


Badness: 0.0167
Mapping: {{mapping| 1 -5 -5 5 2 22 18 | 0 9 10 -3 2 -25 -19 }}


[[Chords_of_valentine|Chords of valentine]]
Optimal tunings:
* WE: ~2 = 1199.5964{{c}}, ~5/3 = 878.0482{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 878.3444{{c}}


=Unidec=
{{Optimal ET sequence|legend=0| 15f, 41, 56, 97g }}
Comma: 31381059609/31250000000


POTE generator: ~10/9 = 183.047
Badness (Sintel): 1.45


Map: [&lt;2 5 8|, &lt;0 -6 -11|]
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


EDOs: 26, 46, 72, 118, 2524, 2642, 2760, 5002bc
Comma list: 100/99, 133/132, 154/153, 190/189, 196/195, 256/255


Badness: 0.0824
Mapping: {{mapping| 1 -5 -5 5 2 22 18 -6 | 0 9 10 -3 2 -25 -19 14 }}


==7-limit==
Optimal tunings:
[[Comma|Commas]]: 1029/1024, 4375/4374
* WE: ~2 = 1199.6638{{c}}, ~5/3 = 878.1109{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 878.3566{{c}}


[[Minimax_tuning|Minimax tuning]]:
{{Optimal ET sequence|legend=0| 15f, 41, 56, 97g }}


7-limit: [|1 0 0 0&gt;, |47/26 0 6/13 -6/13&gt;,
Badness (Sintel): 1.36
|71/26 0 11/13 -11/13&gt;, |71/26 0 -2/13 2/13&gt;]


[[Eigenmonzo|Eigenmonzos]]: 2, 7/5
== Dee leap week ==
{{Main| Dee leap week }}


9-limit: [|1 0 0 0&gt;, |10/7 6/7 0 -3/7&gt;,
[[Subgroup]]: 2.3.5.7
|57/28 11/7 0 -11/14&gt;, |20/7 -2/7 0 1/7&gt;]


[[Eigenmonzo|Eigenmonzos]]: 2, 9/7
[[Comma list]]: 1029/1024, 2460375/2458624


[[POTE_tuning|POTE generator]]: ~10/9 = 183.161
{{Mapping|legend=1| 1 -5 25 5 | 0 9 -31 -3 }}


Map: [&lt;2 5 8 5|, &lt;0 -6 -11 2|]
[[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 }}


Wedgie: &lt;&lt;12 22 -4 7 -40 -71||
{{Optimal ET sequence|legend=1| 41, 108, 149, 190 }}


EDOs: 26, 46, 72, 118, 190
[[Badness]] (Sintel): 2.12


Badness: 0.0384
=== 11-limit ===
Subgroup: 2.3.5.7.11


==11-limit==
Comma list: 385/384, 441/440, 2460375/2458624
[[Comma|Commas]]: 385/384, 441/440, 4375/4374


[[Minimax_tuning|Minimax tuning]]:
Mapping: {{mapping| 1 -5 25 5 -28 | 0 9 -31 -3 43 }}


[|1 0 0 0 0&gt;, |10/7 6/7 0 -3/7 0&gt;, |57/28 11/7 0 -11/14 0&gt;,
Optimal tunings:
|20/7 -2/7 0 1/7 0&gt;, |99/28 -3/7 0 3/14 0&gt;]
* WE: ~2 = 1200.4874{{c}}, ~224/135 = 878.2543{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~224/135 = 877.8987{{c}}


[[Eigenmonzo|Eigenmonzos]]: 2, 9/7
{{Optimal ET sequence|legend=0| 41, 108e, 149, 190 }}


[[POTE_tuning|POTE generator]]: 183.165
Badness (Sintel): 1.35


Map: [&lt;2 5 8 5 6|, &lt;0 -6 -11 2 3|]
== Unidec ==
{{Main| Unidec }}


EDOs: 26, 46, 72, 118, 190
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.


Badness: 0.0155
[[Subgroup]]: 2.3.5.7


[[Chords_of_unidec|Chords of unidec]]
[[Comma list]]: 1029/1024, 4375/4374


==Ekadash==
{{Mapping|legend=1| 2 -1 -3 7 | 0 6 11 -2 }}
Commas: 385/384, 441/440, 625/624, 729/728


[[POTE_tuning|POTE generator]]: ~10/9 = 183.187
[[Optimal tuning]]s:
* [[WE]]: ~1225/864 = 600.2429{{c}}, ~80/63 = 417.0073{{c}}
: [[error map]]: {{val| +0.486 -0.154 +0.038 -1.140 }}
* [[CWE]]: ~1225/864 = 600.0000{{c}}, ~80/63 = 416.8688{{c}}
: error map: {{val| 0.000 -0.924 -1.090 -2.503 }}


Map: [&lt;2 5 8 5 6 19|, &lt;0 -6 -11 2 3 -38|]
[[Minimax tuning]]:
* [[7-odd-limit]]: ~10/9 = {{monzo| 3/26 0 -1/13 1/13 }}
: {{monzo list| 1 0 0 0 | 47/26 0 6/13 -6/13 | 71/26 0 11/13 -11/13 | 71/26 0 -2/13 2/13 }}
: [[eigenmonzo basis|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|unchanged-interval (eigenmonzo) basis]]: 2.9/7


EDOs: 20cf, 26f, 46f, 72, 118, 190, 262df, 452cdef
{{Optimal ET sequence|legend=1| 26, 46, 72, 118, 190 }}


Badness: 0.0204
[[Badness]] (Sintel): 0.972


==Hendec==
=== 11-limit ===
Commas: 169/168, 325/324, 364/363, 1716/1715
Subgroup: 2.3.5.7.11


[[POTE_tuning|POTE generator]]: ~10/9 = 183.187
Comma list: 385/384, 441/440, 4375/4374


Map: [&lt;2 5 8 5 6 8|, &lt;0 -6 -11 2 3 -2|]
Mapping: {{mapping| 2 -1 -3 7 9 | 0 6 11 -2 -3 }}


EDOs: 26, 46, 72
Optimal tunings:  
* WE: ~99/70 = 600.2497{{c}}, ~14/11 = 417.0085{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~14/11 = 416.8543{{c}}


Badness: 0.0177
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 }}]
: unchanged-interval (eigenmonzo) basis: 2.9/7


===17-limit===
{{Optimal ET sequence|legend=0| 26, 46, 72, 118, 190 }}
Commas: 169/168, 221/220, 273/272, 325/324, 364/363


[[POTE_tuning|POTE generator]]: ~10/9 = 183.196
Badness (Sintel): 0.512


Map: [&lt;2 5 8 5 6 8 10|, &lt;0 -6 -11 2 3 -2 -6|]
==== Ekadash ====
Subgroup: 2.3.5.7.11.13


EDOs: 26, 46, 72
Comma list: 385/384, 441/440, 625/624, 729/728


===Music===
Mapping: {{mapping| 2 -1 -3 7 9 -19 | 0 6 11 -2 -3 38 }}
[[Technical_Notes_for_Newbeams#Track notes:-Hypnocloudsmack 2|Hypnocloudsmack 2]] by [[Andrew_Heathwaite|Andrew Heathwaite]]


=Hemithirds=
Optimal tunings:
* WE: ~99/70 = 600.2497{{c}}, ~14/11 = 417.0085{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~14/11 = 416.8543{{c}}


==7-limit==
{{Optimal ET sequence|legend=0| 46f, 72, 118, 190, 262df, 452cdef }}
[[Comma|Commas]]: 1029/1024, 3136/3125


[[Minimax_tuning|Minimax tuning]]:
Badness (Sintel): 0.842


7-limit: [|1 0 0 0&gt;, |5/2 3/4 0 -3/4&gt;,
==== Hendec ====
|11/5 -1/10 0 1/10&gt;, |5/2 -1/4 0 1/4&gt;]
Subgroup: 2.3.5.7.11.13


[[Eigenmonzo|Eigenmonzos]]: 2, 7/6
Comma list: 169/168, 325/324, 364/363, 385/384


9-limit: [|1 0 0 0&gt;, |10/7 6/7 0 -3/7&gt;,
Mapping: {{mapping| 2 -1 -3 7 9 6 | 0 6 11 -2 -3 2 }}
|82/35 -4/35 0 2/35&gt;, |20/7 -2/7 0 1/7&gt;]


[[Eigenmonzo|Eigenmonzos]]: 2, 7/6
Optimal tunings:  
* WE: ~91/64 = 600.3825{{c}}, ~14/11 = 417.0678{{c}}
* CWE: ~91/64 = 600.0000{{c}}, ~14/11 = 416.8290{{c}}


POTE generator: ~28/25 = 193.244
{{Optimal ET sequence|legend=0| 26, 46, 72, 190ff }}


Map: [&lt;1 4 2 2|, &lt;0 -15 2 5|]
Badness (Sintel): 0.732


[[EDO|EDOs]]: [[31edo|31]], [[56edo|56]], [[87edo|87]], [[118edo|118]], [[149edo|149]], [[180edo|180]]
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


Badness: 0.0443
Comma list: 169/168, 221/220, 273/272, 325/324, 364/363
==11-limit==
[[Comma|Commas]]: 385/384, 441/440, 3136/3125


[[Minimax_tuning|Minimax tuning]]:
Mapping: {{mapping| 2 -1 -3 7 9 6 4 | 0 6 11 -2 -3 2 6 }}


[|1 0 0 0 0&gt;, |11/9 0 0 -5/9 5/9&gt;, |64/27 0 0 2/27 -2/27&gt;,
Optimal tunings:
|79/27 0 0 5/27 -5/27&gt;, |79/27 0 0 -22/27 22/27&gt;]
* WE: ~17/12 = 600.3991{{c}}, ~14/11 = 417.0809{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~14/11 = 416.8330{{c}}


[[Eigenmonzo|Eigenmonzos]]: 2, 11/7
{{Optimal ET sequence|legend=0| 26, 46, 72, 190ffg }}


POTE generator: ~28/25 = 193.227
Badness (Sintel): 0.595


Map: [&lt;1 4 2 2 7|, &lt;0 -15 2 5 -22|]
== 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, so its [[ploidacot]] is beta-dodecacot.


EDOs: 31, 87, 118
[[Subgroup]]: 2.3.5.7


Badness: 0.0190
[[Comma list]]: 1029/1024, 5103/5000


[[Chords_of_hemithirds|Chords of hemithirds]]
{{Mapping|legend=1| 1 -5 -7 5 | 0 12 17 -4 }}
: mapping generators: ~2, ~35/24


==13-limit==
[[Optimal tuning]]s:
Commas: 196/195 352/351 1001/1000 1029/1024
* [[WE]]: ~2 = 1200.2959{{c}}, ~35/24 = 658.3833{{c}}
: [[error map]]: {{val| +0.296 -2.835 +4.130 -0.879 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~35/24 = 658.2313{{c}}
: error map: {{val| 0.000 -3.179 +3.619 -1.751 }}


POTE generator: ~28/25 = 193.166
{{Optimal ET sequence|legend=1| 11c, 20c, 31, 144c, 175c }}


Map: [&lt;1 4 2 2 7 0|, &lt;0 -15 2 5 -22 23|]
[[Badness]] (Sintel): 2.98


EDOs: 31, 56, 87, 118, 205d
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.0217
Comma list: 176/175, 243/242, 1029/1024


=Hemiseven=
Mapping: {{mapping| 1 -5 -7 5 -13 | 0 12 17 -4 30 }}


==7-limit==
Optimal tunings:
Commas: 1029/1024, 19683/19600
* WE: ~2 = 1200.2862{{c}}, ~22/15 = 658.4276{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/15 = 658.2805{{c}}


POTE generator: ~320/243 = 483.267
{{Optimal ET sequence|legend=0| 20ce, 31, 113c, 144c }}


Map: [&lt;1 4 14 2|, &lt;0 -6 -29 2|]
Badness (Sintel): 1.77


Wedgie: &lt;&lt;6 29 -2 32 -20 -86||
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


EDOs: 5, 72, 77, 149, 221, 514bd, 735bcd
Comma list: 144/143, 176/175, 243/242, 343/338


Badness: 0.0566
Mapping: {{mapping| 1 -5 -7 5 -13 7 | 0 12 17 -4 30 -6 }}


==11-limit==
Optimal tunings:
Commas: 385/384, 441/440, 19683/19600
* WE: ~2 = 1199.3663{{c}}, ~22/15 = 658.0465{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/15 = 658.3800{{c}}


POTE generator: ~320/243 = 483.276
{{Optimal ET sequence|legend=0| 20ce, 31, 82cf, 113cf }}


Map: [&lt;1 4 14 2 -5|, &lt;0 -6 -29 2 21|]
Badness (Sintel): 1.94


EDOs: 72, 77, 149, 221e, 293de
== Restles ==
{{See also| Lesser tendoneutralic }}


Badness: 0.0285
Restles may be described as the {{nowrap| 77 & 87 }} temperament, and has a [[ploidacot]] signature of wau-dodecacot. It was named by [[Petr Pařízek]] in 2011 for it is some sort of opposite to [[beatles]]<ref name="petr's long post">[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101780.html Yahoo! Tuning Group | ''Suggested names for the unclasified temperaments'']</ref>.  


==13-limit==
[[Subgroup]]: 2.3.5.7
Commas: 351/350, 385/384, 441/440, 676/675


POTE generator: ~320/243 = 483.256
[[Comma list]]: 1029/1024, 153664/151875


Map: [&lt;1 4 14 2 -5 19|, &lt;0 -6 -29 2 21 -38|]
{{Mapping|legend=1| 1 -2 8 4 | 0 12 -19 -4 }}
: mapping generators: ~2. ~315/256


EDOs: 72, 77, 149, 221ef
[[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 }}


==17-limit==
{{Optimal ET sequence|legend=1| 77, 87, 164 }}
Commas: 273/272, 351/350, 385/384, 441/440, 676/675


POTE generator: ~320/243 = 483.261
[[Badness]] (Sintel): 2.73


Map: [&lt;1 4 14 2 -5 19 21|, &lt;0 -6 -29 2 21 -38 -42|]
=== 11-limit ===
Subgroup: 2.3.5.7.11


EDOs: 72, 77, 149, 221ef
Comma list: 385/384, 441/440, 153664/151875


=Tritikleismic=
Mapping: {{mapping| 1 -2 8 4 -7 | 0 12 -19 -4 35 }}


==7-limit==
Optimal tunings:
[[Comma|Commas]]: 1029/1024, 15625/15552
* WE: ~2 = 1200.1110{{c}}, ~27/22 = 358.6045{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~27/22 = 358.5720{{c}}


[[Minimax_tunings|Minimax tunings]]:
{{Optimal ET sequence|legend=0| 77, 87, 164, 251d }}


7-limit: [|1 0 0 0&gt;, |2 0 6/7 -6/7&gt;,
Badness (Sintel): 1.81
|8/3 0 5/7 -5/7&gt;, |8/3 0 -2/7 2/7&gt;]


[[Eigenmonzo|Eigenmonzos]]: 2, 7/5
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


9-limit: [|1 0 0 0&gt;, |10/7 6/7 0 -3/7&gt;,
Comma list: 196/195, 352/351, 385/384, 676/675
|46/21 5/7 0 -5/14&gt;, |20/7 -2/7 0 1/7&gt;]


[[Eigenmonzo|Eigenmonzos]]: 2, 9/7
Mapping: {{mapping| 1 -2 8 4 -7 4 | 0 12 -19 -4 35 -1 }}


POTE generator: ~6/5 = 316.872
Optimal tunings:  
* WE: ~2 = 1200.0482{{c}}, ~~16/13 = 358.5883{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/13 = 358.5741{{c}}


Map: [&lt;3 0 3 10|, &lt;0 6 5 -2|]
{{Optimal ET sequence|legend=0| 77, 87, 164, 251d }}


[[EDO|EDOs]]: [[15edo|15]], [[72edo|72]], [[87edo|87]], [[159edo|159]], [[231edo|231]]
Badness (Sintel): 1.16


Badness: 0.0563
== 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.  


==11-limit==
[[Subgroup]]: 2.3.5.7
[[Comma|Commas]]: 385/384, 441/440, 4000/3993


[[Minimax_tuning|Minimax tuning]]:
[[Comma list]]: 1029/1024, 11529602/11390625


[|1 0 0 0 0&gt;, |10/7 6/7 0 -3/7 0&gt;, |46/21 5/7 0 -5/14 0&gt;,
{{Mapping|legend=1| 2 -4 15 8 | 0 9 -13 -3 }}
|20/7 -2/7 0 1/7 0&gt;, |71/21 3/7 0 -3/14 0&gt;]
: mapping generators: ~3375/2401, ~450/343


[[Eigenmonzo|Eigenmonzos]]: 2, 9/7
[[Optimal tuning]]s:
* [[WE]]: ~3375/2401 = 600.1355{{c}}, ~450/343 = 478.0813{{c}}
: [[error map]]: {{val| +0.271 +0.235 +0.662 -1.986 }}
* [[CWE]]: ~3375/2401 = 600.000{{c}}, ~450/343 = 477.9725{{c}}
: error map: {{val| 0.000 -0.202 +0.043 -2.743 }}


POTE generator: ~6/5 = 316.881
{{Optimal ET sequence|legend=1| 10, 98, 108, 118 }}


Map: [&lt;3 0 3 10 8|, &lt;0 6 5 -2 3|]
[[Badness]] (Sintel): 3.65


EDOs: 72, 159, 231
== Quartemka ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Quartemka]].''


Badness: 0.0193
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"/>.  


==13-limit==
[[Subgroup]]: 2.3.5.7
Commas: 325/324, 364/363, 441/440, 625/624


Map: [&lt;3 0 3 10 8 0|, &lt;0 6 5 -2 3 14|]
[[Comma list]]: 1029/1024, 1250000/1240029


EDOs: 15, 72, 87, 159
{{Mapping|legend=1| 1 -17 -26 9 | 0 21 32 -7 }}
: mapping generators: ~2, ~50/27


==17-limit==
[[Optimal tuning]]s:
Commas: 273/272, 325/324, 364/363, 375/374, 385/384
* [[WE]]: ~2 = 1200.5278{{c}}, ~50/27 = 1062.4614{{c}}
: [[error map]]: {{val| +0.528 +0.762 -1.272 -1.305 }}
* [[CWE]]: ~21 = 1200.0000{{c}}, ~50/27 = 1062.0046{{c}}
: error map: {{val| 0.000 +0.142 -2.167 -2.858 }}


Map: [&lt;3 0 3 10 8 0 -2|, &lt;0 6 5 -2 3 14 18|]
{{Optimal ET sequence|legend=1| 26, 61, 87, 113, 200 }}


EDOs: 15g, 72, 87, 159
[[Badness]] (Sintel): 3.85


=Superkleismic=
=== 11-limit ===
Commas: 1029/1024, 875/864
Subgroup: 2.3.5.7.11


POTE generator: ~6/5 = 321.930
Comma list: 385/384, 441/440, 800000/793881


Map: [&lt;1 4 5 2|, &lt;0 -9 -10 3|]
Mapping: {{mapping| 1 -17 -26 9 7 | 0 21 32 -7 -4 }}


EDOs: 11, 15, 26, 41
Optimal tunings:  
* WE: ~2 = 1200.3051{{c}}, ~50/27 = 1062.2805{{c}}
* CWE: ~21 = 1200.0000{{c}}, ~50/27 = 1062.0147{{c}}


Badness: 0.0479
{{Optimal ET sequence|legend=0| 26, 61, 87, 200, 287d }}


==11-limit==
Badness (Sintel): 1.89
Commas: 100/99, 245/242, 385/384


POTE generator: ~6/5 = 321.847
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 4 5 2 4|, &lt;0 -9 -10 3 -2|]
Comma list: 325/324, 364/363, 385/384, 2200/2197


EDOs: 15, 26, 41, 261
Mapping: {{mapping| 1 -17 -26 9 7 -14 | 0 21 32 -7 -4 20 }}


Badness: 0.0257
Optimal tunings:  
* WE: ~2 = 1200.2708{{c}}, ~24/13 = 1062.2496{{c}}
* CWE: ~21 = 1200.0000{{c}}, ~24/13 = 1062.0139{{c}}


==13-limit==
{{Optimal ET sequence|legend=0| 26, 61, 87, 200 }}
Commas: 100/99, 105/104, 245/243, 1188/1183


POTE generator: ~6/5 = 321.994
Badness (Sintel): 1.17


Map: [&lt;1 4 5 2 4 8|, &lt;0 -9 -10 3 -2 -16|]
== Tritriple ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tritriple]].''


EDOs: 11, 15, 26, 41
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"/>.


Badness: 0.0215
[[Subgroup]]: 2.3.5.7


=Gorgo=
[[Comma list]]: 1029/1024, 1959552/1953125
Commas: 36/35, 1029/1024


[[POTE_tuning|POTE generator]]: ~8/7 = 228.334
{{Mapping|legend=1| 1 -11 -7 7 | 0 27 20 -9 }}
: mapping generators: ~2, ~864/625


Map: [&lt;1 1 1 3|, &lt;0 3 7 -1|]
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.4239{{c}}, ~864/625 = 559.4921{{c}}
: [[error map]]: {{val| +0.424 -0.331 +0.561 -1.287 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~864/625 = 559.3015{{c}}
: error map: {{val| 0.000 -0.815 -0.284 -2.539 }}


Wedgie: &lt;&lt;3 7 -1 4 -10 -22||
{{Optimal ET sequence|legend=1| 15, …, 88, 103, 118, 221, 339d }}


EDOs: 5, 16, 21
[[Badness]] (Sintel): 3.00


Badness: 0.0607
=== 11-limit ===
Subgroup: 2.3.5.7.11


==11-limit==
Comma list: 385/384, 441/440, 43923/43750
Commas: 36/35, 56/55, 1029/1024


POTE generator: ~8/7 = 229.535
Mapping: {{mapping| 1 -11 -7 7 -4 | 0 27 20 -9 16 }}


Map: [&lt;1 1 1 3 5|, &lt;0 3 7 -1 -8|]
Optimal tunings:  
* WE: ~2 = 1200.4953{{c}}, ~242/175 = 559.5243{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~242/175 = 559.3016{{c}}


EDOs: 5, 16e, 21, 47c, 68bce
{{Optimal ET sequence|legend=0| 15, …, 88, 103, 118, 221e, 339de }}


Badness: 0.0627
Badness (Sintel): 1.17


==13-limit==
== Widefourth ==
Commas: 27/26, 36/35, 56/55, 507/500
[[Subgroup]]: 2.3.5.7


POTE generator: ~8/7 = 229.059
[[Comma list]]: 1029/1024, 48828125/48771072


Map: [&lt;1 1 1 3 5 2|, &lt;0 3 7 -1 -8 9|]
{{Mapping|legend=1| 1 -17 -5 9 | 0 33 13 -11 }}


EDOs 5, 21, 68bcef
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.4770{{c}}, ~4608/3125 = 676.0584{{c}}
: [[error map]]: {{val| +0.477 -0.137 +0.061 -1.175 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~4608/3125 = 675.7954{{c}}
: error map: {{val| 0.000 -0.705 -0.973 -2.576 }}


Badness: 0.0471
{{Optimal ET sequence|legend=1| 16, 71, 87, 103, 190 }}


==Music==
[[Badness]] (Sintel): 3.90
[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Herman/gorgo-example.mp3 Gorgo Example] by [[Herman_Miller|Herman Miller]]


=Lemba=
=== 11-limit ===
Commas: 50/49, 525/512
Subgroup: 2.3.5.7.11


[[POTE_tuning|POTE generator]]: ~8/7 = 232.089
Comma list: 385/384, 441/440, 234375/234256


Map: [&lt;2 2 5 6|, &lt;0 3 -1 -1|]
Mapping: {{mapping| 1 16 8 -2 17 | 0 -33 -13 11 -31 }}


Wedgie: &lt;&lt;6 -2 -2 -17 -20 1||
Optimal tunings:  
* WE: ~2 = 1200.4852{{c}}, ~1250/847 = 676.0634{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~1250/847 = 675.7966{{c}}


EDOs: 10, 16, 26
{{Optimal ET sequence|legend=0| 16, 71, 87, 103, 190 }}


Badness: 0.0622
Badness (Sintel): 1.35


==Music==
=== 13-limit ===
By [[Herman_Miller|Herman Miller]]
Subgroup: 2.3.5.7.11.13


[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Herman/LembaGalatsia.mp3 Lemba Galatsia]
Comma list: 385/384, 441/440, 625/624, 847/845


[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Herman/lemba-gpo-test.mp3 GPO Lemba]
Mapping: {{mapping| 1 16 8 -2 17 12 | 0 -33 -13 11 -31 -19 }}


=Gidorah=
Optimal tunings:
Commas: 21/20, 144/125
* WE: ~2 = 1200.4217{{c}}, ~77/52 = 676.0286{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~77/52 = 675.7967{{c}}


POTE generator: ~8/7 = 230.762
{{Optimal ET sequence|legend=0| 16, 71, 87, 103, 190 }}


Map: [&lt;1 1 2 3|, &lt;0 3 2 -1|]
Badness (Sintel): 0.894


EDOs: 5, 11, 16c, 21cc, 26ccc
== 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.


Badness: 0.0623
Subgroup: 2.3.7.13


=Clyndro=
Comma list: 729/728, 1029/1024
Commas: 135/128, 360/343


POTE generator: ~8/7 = 226.469
Subgroup-val mapping: {{mapping| 1 1 3 0 | 0 3 -1 19 }}


Map: [&lt;1 1 4 3|, &lt;0 3 -9 -1|]
Gencom mapping: {{mapping| 1 1 0 3 0 0 | 0 3 0 -1 0 19 }}


EDOs: 5c, 11, 16
Optimal tunings:  
* WE: ~2 = 1200.5057{{c}}, ~8/7 = 233.7200{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.6534{{c}}


Badness: 0.1592
{{Optimal ET sequence|legend=0| 5, 31f, 36, 77, 113, 827bdddff }}


==11-limit==
Badness (Sintel): 0.339
Commas: 33/32, 45/44, 352/343


POTE generator: ~8/7 = 226.428
==== 2.3.7.13.17 subgroup ====
Subgroup: 2.3.7.13.17


Map: [&lt;1 1 4 3 4|, &lt;0 3 -9 -1 -3|]
Comma list: 273/272, 729/728, 833/832


EDOs: 5c, 11, 16
Subgroup-val mapping: {{mapping| 1 1 3 0 0 | 0 3 -1 19 21 }}


Badness: 0.0697
Gencom mapping: {{mapping| 1 1 0 3 0 0 0 | 0 3 0 -1 0 19 21 }}


=Necromanteion=
Optimal tunings:
Commas: 1029/1024, 5103/5000
* WE: ~2 = 1200.5282{{c}}, ~8/7 = 233.6492{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.5776{{c}}


POTE generator: ~48/35 = 541.779
{{Optimal ET sequence|legend=0| 5g, 31fg, 36, 113, 149 }}


Map: [&lt;1 7 10 1|, &lt;0 -12 -17 4|]
Badness (Sintel): 0.332


EDOs: 11c, 20c, 31, 51c, 82c, 113c, 144c, 175c, 206bc, 237bc, 505bcd
==== 2.3.7.13.17.19 subgroup ====
Subgroup: 2.3.7.13.17.19


Badness: 0.1177
Comma list: 273/272, 343/342, 513/512, 729/728


==11-limit==
Subgroup-val mapping: {{mapping| 1 1 3 0 0 6 | 0 3 -1 19 21 -9 }}
Commas: 176/175, 243/242, 1029/1024


POTE generator: ~15/11 = 541.729
Gencom mapping: {{mapping| 1 1 0 3 0 0 0 6 | 0 3 0 -1 0 19 21 -9 }}


Map: [&lt;1 7 10 1 17|, &lt;0 -12 -17 4 -30|]
Optimal tunings:  
* WE: ~2 = 1200.3292{{c}}, ~8/7 = 233.6651{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.6106{{c}}


EDOs: 31, 82c, 113c, 144c, 175c, 350bcde, 381bcde
{{Optimal ET sequence|legend=0| 5g, 36, 77, 113, 262df }}


Badness: 0.0535
Badness (Sintel): 0.380


==13-limit==
==== 2.3.7.13.17.19.23 subgroup ====
Commas: 144/143, 176/175, 243/242, 343/338
Subgroup: 2.3.7.13.17.19.23


POTE generator: ~15/11 = 541.606
Comma list: 273/272, 343/342, 392/391, 513/512, 729/728


Map: [&lt;1 7 10 1 17 1|, &lt;0 -12 -17 4 -30 6|]
Subgroup-val mapping: {{mapping| 1 1 3 0 0 6 9 | 0 3 -1 19 21 -9 -23 }}


EDOs: 31, 51ce, 82cf, 113cf, 144cf
Gencom mapping: {{mapping| 1 1 0 3 0 0 0 6 9 | 0 3 0 -1 0 19 21 -9 -23 }}


Badness: 0.0470
Optimal tunings:  
* WE: ~2 = 1200.3127{{c}}, ~8/7 = 233.6679{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.6091{{c}}


=Widefourth=
{{Optimal ET sequence|legend=0| 36, 77, 113, 262df }}
Commas: 1029/1024, 48828125/48771072


POTE generator: ~3125/2304 = 524.210
Badness (Sintel): 0.474


Map: [&lt;1 16 8 -2|, &lt;0 -33 -13 11|]
==== 2.3.7.13.17.19.23.29 subgroup ====
Subgroup: 2.3.7.13.17.19.23.29


Wedgie: &lt;&lt;33 13 -11 -56 -110 -62||
Comma list: 273/272, 343/342, 378/377, 392/391, 513/512, 609/608


EDOs: 16, 71, 87, 103, 190
Subgroup-val mapping: {{mapping| 1 1 3 0 0 6 9 7 | 0 3 -1 19 21 -9 -23 -11 }}


Badness: 0.1541
Gencom mapping: {{mapping| 1 1 0 3 0 0 0 6 9 7 | 0 3 0 -1 0 19 21 -9 -23 -11 }}


==11-limit==
Optimal tunings:
Commas: 385/384, 441/440, 234375/234256
* WE: ~2 = 1200.2503{{c}}, ~8/7 = 233.6688{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.6208{{c}}


POTE generator: ~3125/2304 = 524.210
{{Optimal ET sequence|legend=0| 36, 77, 113 }}


Map: [&lt;1 16 8 -2 17|, &lt;0 -33 -13 11 -31|]
Badness (Sintel): 0.473


EDOs: 16, 71, 87, 103, 190
=== 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.


Badness: 0.0408
Subgroup: 2.3.7.13


==13-limit==
Comma list: 169/168, 1029/1024
Commas: 385/384, 441/440, 625/624, 847/845


POTE generator: ~65/48 = 524.209
Subgroup-val mapping: {{mapping| 2 2 6 7 | 0 3 -1 1 }}


Map: [&lt;1 16 8 -2 17 12|, &lt;0 -33 -13 11 -31 -19|]
Gencom mapping: {{mapping| 2 2 0 6 0 7 | 0 3 0 -1 0 1 }}
: mapping generators: ~91/64, ~8/7


EDOs: 16, 71, 87, 103, 190
Optimal tunings:  
* WE: ~91/64 = 600.4315{{c}}, ~8/7 = 233.7724{{c}}
* CWE: ~91/64 = 600.0000{{c}}, ~8/7 = 233.7039{{c}}


Badness: 0.0216
{{Optimal ET sequence|legend=0| 10, 26, 36, 154f, 190ff, 226ff, 262dfff }}


=Tritriple=
Badness (Sintel): 0.434
Comma: |31 20 -27&gt;


POTE generator: ~864/625 = 559.332
==== 2.3.7.13.17 subgroup ====
Subgroup: 2.3.7.13.17


Map: [&lt;1 -11 -7|, &lt;0 27 20|]
Comma list: 169/168, 273/272, 289/288


EDOs: 15, 103, 118, 133, 959, 1077
Subgroup-val mapping: {{mapping| 2 2 6 7 7 | 0 3 -1 1 3 }}


Badness: 0.2836
Gencom mapping: {{mapping| 2 2 0 6 0 7 7 | 0 3 0 -1 0 1 3 }}


==7-limit==
Optimal tunings:
Commas: 1029/1024, 1959552/1953125
* WE: ~17/12 = 600.4436{{c}}, ~8/7 = 233.7883{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~8/7 = 233.7312{{c}}


POTE generator: ~864/625 = 559.295
{{Optimal ET sequence|legend=0| 10, 26, 36, 154f, 190ffg, 226ffg }}


Map: [&lt;1 -11 -7 7|, &lt;0 27 20 -9|]
Badness (Sintel): 0.253


EDOs: 15, 103, 118, 133, 339d
=== Gigapyth (2.3.7.85) ===
Subgroup: 2.3.7.85


Badness: 0.1186
Comma list: 1029/1024, 7225/7203


==11-limit==
Subgroup-val mapping: {{mapping| 1 -2 4 7 | 0 6 -2 -1 }}
Commas: 385/384, 441/440, 43923/43750


POTE generator: ~242/175 = 559.293
Optimal tunings:  
* WE: ~2 = 1200.8295{{c}}, ~128/85 = 717.2597{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~128/85 = 716.7933{{c}}


Map: [&lt;1 -11 -7 7 -4|, &lt;0 27 20 -9 16|]
{{Optimal ET sequence|legend=0| 5, 42*, 47, 52, 57, 62, 67, 72, 149*, 370d***, 519bdd***** }}


EDOs: 15, 103, 118, 133, 339de
<nowiki/>* Wart for 85


Badness: 0.0353
== References ==


[[Category:clan]]
[[Category:Temperament clans]]
[[Category:family]]
[[Category:Gamelismic clan| ]] <!-- main article -->
[[Category:gamelismic]]
[[Category:Rank 2]]
[[Category:listen]]
[[Category:Listen]]
[[Category:miracle]]
[[Category:theory]]
Retrieved from "https://en.xen.wiki/w/Gamelismic_clan"