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The 2.3.7 [[Just_intonation_subgroups|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 three [[Gamelismic family #Gamelan|gamelan temperament]] itself, and for the rank two 2.3.7 temperament [[slendric]], this means three [[8/7]] intervals give a fifth, [[3/2]]. In fact, we find that 3/2 = (8/7)<sup>3</sup> × 1029/1024. From this it follows that gamelismic temperaments tend to flatten both the fifth and the 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.


= Slendric =
== Slendric ==
{{main| Slendric }}
{{Main| Slendric }}


Period: 1\1
[[Subgroup]]: 2.3.7


Optimal ([[POTE]]) generator: ~8/7 = 233.688
[[Comma list]]: 1029/1024


EDO generator: [[36edo|7\36]], [[41edo|8\41]], [[77edo|15\77]]
{{Mapping|legend=2| 1 1 3 | 0 3 -1 }}


Scales (Scala files): [[slendric5]], [[slendric6]], [[slendric11]], [[slendric16]]
{{Mapping|legend=3| 1 1 0 3 | 0 3 0 -1 }}
: mapping generators: ~2, ~8/7


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
[[Optimal tuning]]s:
<div style="line-height:1.6;">Technical data</div>
* [[WE]]: ~2 = 1200.4859{{c}}, ~8/7 = 233.7822{{c}}
<div class="mw-collapsible-content">
: [[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 }}


Subgroup: 2.3.7
{{Optimal ET sequence|legend=1| 5, 21, 26, 31, 36, 77, 113, 190 }}


[[Comma list]]: 1029/1024
[[Badness]] (Sintel): 0.158


[[Sval]] [[mapping]]: [{{val| 1 1 3 }}, {{val| 0 3 -1 }}]  
=== 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 generators: ~2, ~8/7
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.


Gencom mapping: [{{val| 1 1 0 3 }}, {{val| 0 3 0 -1 }}]
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]]


[[Gencom]]: [2 8/7; 1029/1024]
The rest are considered below.


{{Val list|legend=1| 36, 77, 113, 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.


</div></div>
=== Radon ===
{{See also|Chromatic pairs #Radon}}


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


Discussed elsewhere are [[Archytas clan #Blacksmith|blacksmith]], [[Meantone family #Mothra|mothra]], [[Schismatic family #Guiron|guiron]] and [[Sensipent family #Heinz|heinz]]. The rest are considered below.
Subgroup: 2.3.7.11


== Subgroup extensions ==
Comma list: 896/891, 1029/1024
No-five subgroup extensions of slendric include [[Chromatic pairs #Radon|radon]], the 2.3.7.11 extension that may be viewed as no-five rodan, and baladic, the 2.3.7.13.17 extension, considered below.


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


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


Optimal ([[POTE]]) generator: ~15/14 = 116.675
Optimal tunings:  
* WE: ~2 = 1199.9708{{c}}, ~8/7 = 234.3748{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.3813{{c}}


EDO generators: [[31edo|3\31]], [[41edo|4\41]], [[72edo|7\72]]
{{Optimal ET sequence|legend=0| 5, …, 36, 41, 87, 128 }}


Scales (Scala files): [[Miracle 10]], [[Blackjack]]
Badness (Sintel): 0.619


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
== Mothra ==
<div style="line-height:1.6;">Technical data</div>
{{Main| Mothra }}
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7
Mothra tempers out [[81/80]] and finds the prime 5 at a stack of four fifths as does any temperament in the [[meantone family]]. It also tempers out [[1728/1715]], the orwellisma. It can be described as the {{nowrap| 26 & 31 }}. Using [[31edo]] with a generator of 6/31 is an excellent tuning choice. However, a pure mos mothra scale is often described as directionless and has limited chord-building potential<ref>[https://www.youtube.com/watch?v=uH3ahBzDSrs 31-EDO Music Theory: Supermajor Hexatonic Scale] by [[Zhea Erose]]</ref>, so something other than a mos may be used as a scale to get the most out of mothra. There are examples of non-mos mothra scales in 31edo [[Strictly proper 7-tone 31edo scales|in the article on strictly proper 7-tone 31edo scales]].


[[Comma list]]: 225/224, 1029/1024
Note that mothra is also called '''cynder''' in the 7-limit, which can be a little confusing sometimes.


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


{{Multival|legend=1| 6 -7 -2 -25 -20 15 }}
[[Subgroup]]: 2.3.5.7


[[Minimax tuning]]:
[[Comma list]]: 81/80, 1029/1024
* [[7-odd-limit]]:
: [{{monzo| 1 0 0 0 }}, {{monzo| 25/13 6/13 -6/13 0 }}, {{monzo| 25/13 -7/13 7/13 0 }}, {{monzo| 35/13 -2/13 2/13 0 }}]
: [[Eigenmonzo]]s: 2, 6/5
* [[9-odd-limit]]:
: [{{monzo| 1 0 0 0 }}, {{monzo| 25/19 12/19 -6/19 0 }}, {{monzo| 50/19 -14/19 7/19 0 }}, {{monzo| 55/19 -4/19 2/19 0 }}]
: [[Eigenmonzo]]s: 2, 10/9


Algebraic generator: Secor59, [[Algebraic number|positive root]] of 15''x''<sup>6</sup> - 8''x''<sup>4</sup> - 12
{{Mapping|legend=1| 1 1 0 3 | 0 3 12 -1 }}


{{Val list|legend=1| 10, 21, 31, 41, 72 }}
[[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 }}


[[Badness]]: 0.0167
[[Algebraic generator]]: Rabrindanath, largest real root of ''x''<sup>8</sup> - 3''x''<sup>2</sup> + 1, or 232.0774 cents.


</div></div>
[[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


== 11-limit ==
{{Optimal ET sequence|legend=1| 5, 21c, 26, 31 }}


Period: 1\1
[[Badness]] (Sintel): 0.940


Optimal ([[POTE]]) generator: ~15/14 = 116.633
=== 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.


EDO generators: [[31edo|3\31]], [[41edo|4\41]], [[72edo|7\72]]
Subgroup: 2.3.5.7.11
 
Comma list: 81/80, 99/98, 385/384


Scales (Scala files): [[Miracle 10]], [[Blackjack]]
Mapping: {{mapping| 1 1 0 3 5 | 0 3 12 -1 -8 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Optimal tunings:
<div style="line-height:1.6;">Technical data</div>
* WE: ~2 = 1201.3979{{c}}, ~8/7 = 232.3010{{c}}
<div class="mw-collapsible-content">
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 232.0621{{c}}


Subgroup: 2.3.5.7.11
{{Optimal ET sequence|legend=0| 5, 26, 31, 88, 119be, 150be }}


Comma list: 225/224, 243/242, 385/384
Badness (Sintel): 0.848


Mapping: [{{val| 1 1 3 3 2 }}, {{val| 0 6 -7 -2 15 }}]
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Minimax tuning:
Comma list: 81/80, 99/98, 105/104, 144/143
* [[11-odd-limit]]:
: [{{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]]s: 2, 10/9


Algebraic generator: Secor59
Mapping: {{mapping| 1 1 0 3 5 1 | 0 3 12 -1 -8 14 }}


{{Val list|legend=1| 10, 21e, 31, 41, 72, 247c, 319bcde, 391bcde, 463bccde }}
Optimal tunings:
* WE: ~2 = 1201.0985{{c}}, ~8/7 = 232.0231{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 231.8425{{c}}


Badness: 0.0107
{{Optimal ET sequence|legend=0| 5, 26, 31, 57, 88 }}


</div></div>
Badness (Sintel): 0.990


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


Period: 1\1
Comma list: 81/80, 99/98, 105/104, 120/119, 144/143


Optimal ([[POTE]]) generator: ~15/14 = 116.747
Mapping: {{mapping| 1 1 0 3 5 1 | 0 3 12 -1 -8 14 16 }}


EDO generators: [[31edo|3\31]], [[41edo|4\41]]
Optimal tunings:  
* WE: ~2 = 1200.9734{{c}}, ~8/7 = 231.8960{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 231.7392{{c}}


Scales (Scala files): [[Miracle 10]], [[Blackjack]]
{{Optimal ET sequence|legend=0| 5g, 26, 31, 57, 88 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Badness (Sintel): 1.00
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


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


Comma list: 105/104, 144/143, 196/195, 243/242
Comma list: 81/80, 99/98, 105/104, 120/119, 144/143, 153/152


Mapping: [{{val| 1 1 3 3 2 4 }}, {{val| 0 6 -7 -2 15 -3 }}]
Mapping: {{mapping| 1 1 0 3 5 1 | 0 3 12 -1 -8 14 16 22 }}


{{Val list|legend=1| 10, 21e, 31, 41, 72f, 113f, 185cff }}
Optimal tunings:
* WE: ~2 = 1200.9663{{c}}, ~8/7 = 231.8393{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 231.6842{{c}}


Badness: 0.0187
{{Optimal ET sequence|legend=0| 26, 31, 57 }}


</div></div>
Badness (Sintel): 1.05


=== Benediction ===
=== 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]]}.


Period: 1\1
Subgroup: 2.3.5.7.11


Optimal ([[POTE]]) generator: ~15/14 = 116.574
Comma list: 81/80, 176/175, 540/539


EDO generators: [[31edo|3\31]], [[72edo|7\72]], [[103edo|10\103]]
Mapping: {{mapping| 1 1 0 3 -1 | 0 3 12 -1 23 }}


Scales (Scala files):  
Optimal tunings:  
* WE: ~2 = 1200.7675{{c}}, ~8/7 = 232.5673{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 232.4567{{c}}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
{{Optimal ET sequence|legend=0| 5e, 26e, 31, 129 }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Badness (Sintel): 1.04
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 225/224, 243/242, 351/350, 385/384
Comma list: 81/80, 144/143, 176/175, 196/195


Mapping: [{{val| 1 1 3 3 2 7 }}, {{val| 0 6 -7 -2 15 -34 }}]
Mapping: {{mapping| 1 1 0 3 -1 7 | 0 3 12 -1 23 -17 }}


{{Val list|legend=1| 31, 72, 103, 175f }}
Optimal tunings:
* WE: ~2 = 1199.9347{{c}}, ~8/7 = 232.6275{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 232.6392{{c}}


Badness: 0.0157
{{Optimal ET sequence|legend=0| 31, 67, 98 }}


</div></div>
Badness (Sintel): 1.52


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


Period: 1\1
Comma list: 81/80, 144/143, 176/175, 189/187, 196/195


Optimal ([[POTE]]) generator: ~15/14 = 116.585
Mapping: {{mapping| 1 1 0 3 -1 7 | 0 3 12 -1 23 -17 -15 }}


EDO generators: [[31edo|3\31]], [[72edo|7\72]], [[103edo|10\103]]
Optimal tunings:  
* WE: ~2 = 1199.7124{{c}}, ~8/7 = 232.6376{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 232.6917{{c}}


Scales (Scala files):
{{Optimal ET sequence|legend=0| 31, 67, 98 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Badness (Sintel): 1.53
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11.13.17
==== 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


Comma list: 225/224, 243/242, 273/272, 351/350, 375/374
Mapping: {{mapping| 1 1 0 3 -1 7 | 0 3 12 -1 23 -17 -15 -9 }}


Mapping: [{{val| 1 1 3 3 2 7 7 }}, {{val| 0 6 -7 -2 15 -34 -30 }}]
Optimal tunings:  
* WE: ~2 = 1199.4885{{c}}, ~8/7 = 232.6310{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 232.7287{{c}}


{{Val list|legend=1| 31, 72, 103, 175f }}
{{Optimal ET sequence|legend=0| 31, 67, 98h }}


</div></div>
Badness (Sintel): 1.50


=== Manna ===
=== Cyndra ===
Subgroup: 2.3.5.7.11


Period: 1\1
Comma list: 45/44, 81/80, 1029/1024


Optimal ([[POTE]]) generator: ~15/14 = 116.739
Mapping: {{mapping| 1 1 0 3 0 | 0 3 12 -1 18 }}


EDO generators: [[41edo|4\41]], [[72edo|7\72]], [[113edo|11\113]]
Optimal tunings:  
* WE: ~2 = 1201.1585{{c}}, ~8/7 = 231.5404{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 231.3850{{c}}


Scales (Scala files):
{{Optimal ET sequence|legend=0| 5e, 21ce, 26 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Badness (Sintel): 1.84
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


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


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


Mapping: [{{val| 1 1 3 3 2 0 }}, {{val| 0 6 -7 -2 15 38 }}]
{{Optimal ET sequence|legend=0| 5e, 21cef, 26 }}


{{Val list|legend=1| 31f, 41, 72, 185cf, 257cff }}
Badness (Sintel): 1.41


Badness: 0.0170
== Rodan ==
{{Main| Rodan }}
: ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum #Rodan (5-limit)]].''


</div></div>
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.


=== Semimiracle ===
[[Subgroup]]: 2.3.5.7


Period: 1\2
[[Comma list]]: 245/243, 1029/1024


Optimal ([[POTE]]) generator: ~15/14 = 116.624
{{Mapping|legend=1| 1 1 -1 3 | 0 3 17 -1 }}


EDO generators: [[62edo|6\62]], [[72edo|7\72]], [[82edo|8\82]]
[[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 }}


Scales (Scala files):  
[[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


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
[[Algebraic generator]]: larger root of 20''x''<sup>2</sup> - 36''x'' + 15, or (9 + √6)/10.
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11.13
{{Optimal ET sequence|legend=1| 41, 87, 128, 215d }}


Comma list: 169/168, 225/224, 243/242, 385/384
[[Badness]] (Sintel): 0.939


Mapping: [{{val| 2 2 6 6 4 7 }}, {{val| 0 6 -7 -2 15 2 }}]
=== 11-limit ===
Subgroup: 2.3.5.7.11


{{Val list|legend=1| 62, 72 }}
Comma list: 245/243, 385/384, 441/440


Badness: 0.0246
Mapping: {{mapping| 1 1 -1 3 6 | 0 3 17 -1 -13 }}


</div></div>
Optimal tunings:
* WE: ~2 = 1200.0553{{c}}, ~8/7 = 234.4695{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.4594{{c}}


==== 17-limit ====
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


Period: 1\2
Algebraic generator: positive root of ''x''<sup>2</sup> + 16''x'' - 31, or √95 - 8.


Optimal ([[POTE]]) generator: ~15/14 = 116.628
{{Optimal ET sequence|legend=0| 41, 87 }}


EDO generators: [[62edo|6\62]], [[72edo|7\72]], [[82edo|8\82]]
Badness (Sintel): 0.763


Scales (Scala files):  
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Comma list: 196/195, 245/243, 352/351, 364/363
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11.13.17
Mapping: {{mapping| 1 1 -1 3 6 8 | 0 3 17 -1 -13 -22 }}


Comma list: 169/168, 221/220, 225/224, 243/242, 273/272
Optimal tunings:  
* WE: ~2 = 1199.9868{{c}}, ~8/7 = 234.4796{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.4822{{c}}


Mapping: [{{val| 2 2 6 6 4 7 7 }}, {{val| 0 6 -7 -2 15 2 6 }}]
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


{{Val list|legend=1| 62, 72 }}
Algebraic generator: Gatetone, positive root of 4''x''<sup>6</sup> - 7''x'' - 1. Recurrence converges slowly.


Badness: 0.0161
{{Optimal ET sequence|legend=0| 41, 46, 87 }}


</div></div>
Badness (Sintel): 0.762


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


Period: 1\1
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 ([[POTE]]) generator: ~27/26 = 58.288
Optimal tunings:  
* WE: ~2 = 1199.8331{{c}}, ~8/7 = 234.4919{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.5254{{c}}


EDO generators: [[41edo|2\41]], [[62edo|3\62]], [[103edo|5\103]]
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


Scales (Scala files):
{{Optimal ET sequence|legend=0| 41, 46, 87 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Badness (Sintel): 0.853
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


==== Aerodactyl ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 225/224, 243/242, 385/384, 847/845
Comma list: 91/90, 245/243, 385/384, 441/440


Mapping: [{{val| 1 1 3 3 2 2 }}, {{val| 0 12 -14 -4 30 35 }}]
Mapping: {{mapping| 1 1 -1 3 6 -1 | 0 3 17 -1 -13 24 }}


{{Val list|legend=1| 41, 62, 103, 247c, 350bcde }}
Optimal tunings:
* WE: ~2 = 1200.2997{{c}}, ~8/7 = 234.6972{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.6439{{c}}


Badness: 0.0256
{{Optimal ET sequence|legend=0| 5, 41f, 46 }}


</div></div>
Badness (Sintel): 1.40


==== 17-limit ====
=== Aerodino ===
Subgroup: 2.3.5.7.11


Period: 1\1
Comma list: 176/175, 245/243, 1029/1024


Optimal ([[POTE]]) generator: ~27/26 = 58.261
Mapping: {{mapping| 1 1 -1 3 -3 | 0 3 17 -1 33 }}


EDO generators: [[41edo|2\41]], [[62edo|3\62]], [[103edo|5\103]]
Optimal tunings:  
* WE: ~2 = 1199.9179{{c}}, ~8/7 = 234.7123{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.7256{{c}}


Scales (Scala files):
{{Optimal ET sequence|legend=0| 5e, 41e, 46 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Badness (Sintel): 1.79
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11.13.17
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Comma list: 225/224, 243/242, 273/272, 385/384, 847/845
Comma list: 91/90, 176/175, 245/243, 847/845


Mapping: [{{val| 1 1 3 3 2 2 2 }}, {{val| 0 12 -14 -4 30 35 43 }}]
Mapping: {{mapping| 1 1 -1 3 -3 -1 | 0 3 17 -1 33 24 }}


{{Val list|legend=1| 41, 62, 103 }}
Optimal tunings:
* WE: ~2 = 1200.0242{{c}}, ~8/7 = 234.7863{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.7824{{c}}


Badness: 0.0225
{{Optimal ET sequence|legend=0| 5e, 41ef, 46 }}


</div></div>
Badness (Sintel): 1.48


=== Phicordial ===
=== Varan ===
Subgroup: 2.3.5.7.11


Period: 1\1
Comma list: 100/99, 245/243, 1029/1024


Optimal ([[POTE]]) generator: ~16/13 = 361.121
Mapping: {{mapping| 1 1 -1 3 -2 | 0 3 17 -1 28 }}


EDO generators: [[103edo|31\103]], [[113edo|34\113]]
Optimal tunings:  
* WE: ~2 = 1200.3738{{c}}, ~8/7 = 234.2174{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.1586{{c}}


Scales (Scala files):
{{Optimal ET sequence|legend=0| 5e, 36ce, 41 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Badness (Sintel): 1.49
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


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


Comma list: 225/224, 243/242, 385/384, 2200/2197
Comma list: 100/99, 105/104, 245/243, 352/351


Mapping: [{{val| 1 7 -4 1 17 4 }}, {{val| 0 -18 21 6 -45 -1 }}]
Mapping: {{mapping| 1 1 -1 3 -2 0 | 0 3 17 -1 28 19 }}


{{Val list|legend=1| 103, 113, 216c }}
Optimal tunings:
* WE: ~2 = 1200.1389{{c}}, ~8/7 = 234.1162{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.0946{{c}}


Badness: 0.0332
{{Optimal ET sequence|legend=0| 5e, 36ce, 41 }}


</div></div>
Badness (Sintel): 1.33


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


Period: 1\1
[[Subgroup]]: 2.3.5.7


Optimal ([[POTE]]) generator: ~16/13 = 361.123
[[Comma list]]: 1029/1024, 10976/10935


EDO generators: [[103edo|31\103]], [[113edo|34\113]]
{{Mapping|legend=1| 1 1 7 3 | 0 3 -24 -1 }}


Scales (Scala files):  
[[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 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
[[Minimax tuning]]:
<div style="line-height:1.6;">Technical data</div>
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~8/7 = {{monzo| 7/24 0 -1/24 }}
<div class="mw-collapsible-content">
: {{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


Subgroup: 2.3.5.7.11.13.17
{{Optimal ET sequence|legend=1| 36, 41, 77, 118, 277d }}


Comma list: 225/224, 243/242, 273/272, 441/440, 2200/2197
[[Badness]] (Sintel): 1.20


Mapping: [{{val| 1 7 -4 1 17 4 8 }}, {{val| 0 -18 21 6 -45 -1 -13 }}]
=== 11-limit ===
Subgroup: 2.3.5.7.11


{{Val list|legend=1| 103, 113, 216c }}
Comma list: 385/384, 441/440, 10976/10935


Badness: 0.0247
Mapping: {{mapping| 1 1 7 3 -2 | 0 3 -24 -1 28 }}


</div></div>
Optimal tunings:
* WE: ~2 = 1200.3453{{c}}, ~8/7 = 233.9988{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.9312{{c}}


== Revelation ==
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


Period: 1\1
{{Optimal ET sequence|legend=0| 36e, 41, 77, 118, 159, 277d }}


Optimal ([[POTE]]) generator: ~15/14 = 116.277
Badness (Sintel): 0.881


EDO generators: [[21edo|2\21]], [[31edo|3\31]]
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Scales (Scala files):  
Comma list: 196/195, 352/351, 385/384, 729/728


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Mapping: {{mapping| 1 1 7 3 -2 0 | 0 3 -24 -1 28 19 }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11
Optimal tunings:  
* WE: ~2 = 1200.1222{{c}}, ~8/7 = 233.9228{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.8994{{c}}


Comma list: 99/98, 176/175, 1029/1024
{{Optimal ET sequence|legend=0| 36e, 41, 77, 118 }}


Mapping: [{{val| 1 1 3 3 5 }}, {{val| 0 6 -7 -2 -16 }}]
Badness (Sintel): 1.18


{{Val list|legend=1| 10e, 21, 31 }}
== Gorgo ==
: ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum #Laconic]].''
{{See also| Llywelynsmic clan }}


Badness: 0.0329
Gorgo tempers the generator of ~8/7 together with ~10/9. It can be described as the {{nowrap| 16 & 21 }} temperament.  


</div></div>
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.


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


Period: 1\1
[[Comma list]]: 36/35, 1029/1024


Optimal ([[POTE]]) generator: ~15/14 = 116.268
{{Mapping|legend=1| 1 1 1 3 | 0 3 7 -1 }}


EDO generators: [[21edo|2\21]], [[31edo|3\31]]
[[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 }}


Scales (Scala files):
{{Optimal ET sequence|legend=1| 5, 11c, 16, 21 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
[[Badness]] (Sintel): 1.54
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


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


Comma list: 66/65, 99/98, 105/104, 512/507
Comma list: 36/35, 45/44, 1029/1024


Mapping: [{{val| 1 1 3 3 5 4 }}, {{val| 0 6 -7 -2 -16 -3 }}]
Mapping: {{mapping| 1 1 1 3 1 | 0 3 7 -1 13 }}


{{Val list|legend=1| 10e, 21, 31 }}
Optimal tunings:
* WE: ~2 = 1201.3609{{c}}, ~8/7 = 227.6312{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 227.4955{{c}}


Badness: 0.0295
{{Optimal ET sequence|legend=0| 5e, 16, 21, 37b }}


</div></div>
Badness (Sintel): 1.64


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


Period: 1\1
Comma list: 27/26, 36/35, 45/44, 507/500


Optimal ([[POTE]]) generator: ~33/32 = 58.408
Mapping: {{mapping| 1 1 1 3 1 2 | 0 3 7 -1 13 9 }}


EDO generators: [[20edo|1\20]], [[21edo|1\21]], [[41edo|2\41]]
Optimal tunings:  
* WE: ~2 = 1201.0996{{c}}, ~8/7 = 227.4378{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 227.3327{{c}}


Scales (Scala files):
{{Optimal ET sequence|legend=0| 5e, 16, 21, 37b }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Badness (Sintel): 1.35
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


=== Spartan ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 225/224, 245/242, 1029/1024
Comma list: 36/35, 56/55, 1029/1024


Mapping: [{{val| 1 1 3 3 4 }}, {{val| 0 12 -14 -4 -11 }}]
Mapping: {{mapping| 1 1 1 3 5 | 0 3 7 -1 -8 }}


{{Val list|legend=1| 20, 21, 41, 144e, 185cee, 226cee }}
Optimal tunings:
* WE: ~2 = 1198.9344{{c}}, ~8/7 = 229.3316{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 229.5124{{c}}


Badness: 0.0592
{{Optimal ET sequence|legend=0| 5, 16e, 21 }}


</div></div>
Badness (Sintel): 2.07


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


Period: 1\1
Comma list: 27/26, 36/35, 56/55, 507/500


Optimal ([[POTE]]) generator: ~33/32 = 58.430
Mapping: {{mapping| 1 1 1 3 5 2 | 0 3 7 -1 -8 9 }}


EDO generators: [[20edo|1\20]], [[21edo|1\21]], [[41edo|2\41]]
Optimal tunings:  
* WE: ~2 = 1198.3002{{c}}, ~8/7 = 228.7341{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 229.0044{{c}}


Scales (Scala files):
{{Optimal ET sequence|legend=0| 5, 16e, 21 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Badness (Sintel): 1.95
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11.13
; 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]]


Comma list: 105/104, 196/195, 245/242, 512/507
== Gidorah ==
: ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum #University]].''


Mapping: [{{val| 1 1 3 3 4 4 }}, {{val| 0 12 -14 -4 -11 -6 }}]
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.


{{Val list|legend=1| 20, 21, 41, 144eff, 185ceeff }}
[[Subgroup]]: 2.3.5.7


Badness: 0.0432
[[Comma list]]: 21/20, 144/125


</div></div>
{{Mapping|legend=1| 1 1 2 3 | 0 3 2 -1 }}


== Oracle ==
[[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 }}


Period: 1\1
{{Optimal ET sequence|legend=1| 1b, 5 }}


Optimal ([[POTE]]) generator: ~11/8 = 541.668
[[Badness]] (Sintel): 1.58


EDO generators: [[11edo|5\11]], [[20edo|9\20]], [[31edo|14\31]]
== Oncle ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Oncle]].''


Scales (Scala files):
Oncle can be described as the {{nowrap| 31 & 36c }} temperament.


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
[[Subgroup]]: 2.3.5.7
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11
[[Comma list]]: 1029/1024, 2430/2401


Comma list: 121/120, 225/224, 1029/1024
{{Mapping|legend=1| 1 1 6 3 | 0 3 -19 -1 }}


Mapping: [{{val| 1 7 -4 1 3 }}, {{val| 0 -12 14 4 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 }}


{{Val list|legend=1| 11, 20, 31, 82e, 113e, 144ee }}
{{Optimal ET sequence|legend=1| 31, 98c, 129c, 160bc }}


Badness: 0.0427
[[Badness]] (Sintel): 2.24


</div></div>
== Archaeotherium ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Archaeotherium]].''


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


Period: 1\1
[[Subgroup]]: 2.3.5.7


Optimal ([[POTE]]) generator: ~8/7 = 234.417
[[Comma list]]: 405/392, 1029/1024


EDO generators: [[41edo|8\41]], [[46edo|9\46]], [[87edo|17\87]]
{{Mapping|legend=1| 1 1 5 3 | 0 3 -14 -1 }}


Scales (Scala files): [[rodan26opt]], [[rodan31opt]], [[rodan41opt]]
[[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 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
{{Optimal ET sequence|legend=1| 21, 26, 47, 73bc }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7
[[Badness]] (Sintel): 3.70


[[Comma list]]: 245/243, 1029/1024
== 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.


[[Mapping]]: [{{val| 1 1 -1 3 }}, {{val| 0 3 17 -1 }}]
[[Subgroup]]: 2.3.5.7


[[Minimax tuning]]:  
[[Comma list]]: 135/128, 360/343
* 7- and 9-odd-limit:
: [{{monzo| 1 0 0 0 }}, {{monzo| 5/3 0 1/6 -1/6 }}, {{monzo| 25/9 0 17/18 -17/18 }}, {{monzo| 25/9 0 -1/18 1/18 }}]
: [[Eigenmonzo]]s: 2, 7/5


Algebraic generator: [[Algebraic number|larger root]] of 20''x''<sup>2</sup> - 36''x'' + 15, or (9 + √6)/10.
{{Mapping|legend=1| 1 1 4 3 | 0 3 -9 -1 }}


{{Val list|legend=1| 41, 87, 128, 215d }}
[[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 }}


[[Badness]]: 0.0371
{{Optimal ET sequence|legend=1| 5c, 11, 16 }}


</div></div>
[[Badness]] (Sintel): 4.03


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


Period: 1\1
Comma list: 33/32, 45/44, 352/343


Optimal ([[POTE]]) generator: ~8/7 = 234.459
Mapping: {{mapping| 1 1 4 3 4 | 0 3 -9 -1 -3 }}


EDO generators: [[41edo|8\41]], [[46edo|9\46]], [[87edo|17\87]]
Optimal tunings:  
* WE: ~2 = 1206.2134{{c}}, ~8/7 = 227.6004{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 226.2421{{c}}


Scales (Scala files): [[rodan26opt]], [[rodan31opt]], [[rodan41opt]]
{{Optimal ET sequence|legend=0| 5c, 11, 16 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Badness (Sintel): 2.30
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11
== Miracle ==
{{Main| Miracle }}
: ''For the 5-limit version, see [[Syntonic–31 equivalence continuum #Ampersand]].''


Comma list: 245/243, 385/384, 441/440
Miracle is one of the most important entries of this temperament clan. It tempers out [[225/224]], splitting the ~8/7 generator of slendric into 15/14~16/15, and can be described as the {{nowrap| 31 & 41 }} temperament. Its ploidacot is hexacot. It is then extremely natural to equate the neutral third, three generators up, to [[11/9]] and thereby extend miracle to the full [[11-limit]] with essentially no further damage. [[72edo]] makes for an excellent tuning.


Mapping: [{{val| 1 1 -1 3 6 }}, {{val| 0 3 17 -1 -13 }}]
[[Subgroup]]: 2.3.5.7


Minimax tuning:
[[Comma list]]: 225/224, 1029/1024
* 11-odd-limit:
: [{{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]]s: 2, 11/9


Algebraic generator: [[Algebraic number|positive root]] of ''x''<sup>2</sup> + 16''x'' - 31, or √95 - 8.
{{Mapping|legend=1| 1 1 3 3 | 0 6 -7 -2 }}
: mapping generator: ~2, ~15/14


{{Val list|legend=1| 41, 46, 87 }}
[[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 }}


Badness: 0.0231
[[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


</div></div>
[[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]


=== 13-limit ===
[[Algebraic generator]]: Secor59, positive root of 15''x''<sup>6</sup> - 8''x''<sup>4</sup> - 12


Period: 1\1
{{Optimal ET sequence|legend=1| 10, 21, 31, 41, 72 }}


Optimal ([[POTE]]) generator: ~8/7 = 234.482
[[Badness]] (Sintel): 0.424


EDO generators: [[41edo|8\41]], [[46edo|9\46]], [[87edo|17\87]]
=== 11-limit ===
Subgroup: 2.3.5.7.11


Scales (Scala files): [[rodan26opt]], [[rodan31opt]], [[rodan41opt]]
Comma list: 225/224, 243/242, 385/384


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Mapping: {{mapping| 1 1 3 3 2 | 0 6 -7 -2 15 }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11.13
Optimal tunings:  
* WE: ~2 = 1200.7626{{c}}, ~15/14 = 116.7069{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.6469{{c}}


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


Mapping: [{{val| 1 1 -1 3 6 8 }}, {{val| 0 3 17 -1 -13 -22 }}]
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]


Minimax tuning:  
Algebraic generator: Secor59
* 13- and 15-odd-limit:
: [{{monzo| 1 0 0 0 0 0 }}, {{monzo| 23/14 3/14 0 0 0 -3/28 }}, {{monzo| 37/14 17/14 0 0 0 -17/28 }}, {{monzo| 39/14 -1/14 0 0 0 1/28 }}, {{monzo| 45/14 -13/14 0 0 0 13/28 }}, {{monzo| 23/7 -11/7 0 0 0 11/14 }}]
: Eigenmonzos: 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| 10, 21e, 31, 41, 72, 247c, 319bcde, 391bcde, 463bccde }}


{{Val list|legend=1| 41, 46, 87 }}
Badness (Sintel): 0.353


Badness: 0.0184
==== Miraculous ====
Subgroup: 2.3.5.7.11.13


</div></div>
Comma list: 105/104, 144/143, 196/195, 243/242


==== 17-limit ====
Mapping: {{mapping| 1 1 3 3 2 4 | 0 6 -7 -2 15 -3 }}


Period: 1\1
Optimal tunings:  
* WE: ~2 = 1200.1267{{c}}, ~15/14 = 116.7596{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.7488{{c}}


Optimal ([[POTE]]) generator: ~8/7 = 234.524
{{Optimal ET sequence|legend=0| 10, 21e, 31, 41, 72f }}


EDO generators: [[41edo|8\41]], [[46edo|9\46]], [[87edo|17\87]]
Badness (Sintel): 0.771


Scales (Scala files): [[rodan26opt]], [[rodan31opt]], [[rodan41opt]]
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Comma list: 105/104, 120/119, 144/143, 154/153, 170/169
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11.13.17
Mapping: {{mapping| 1 1 3 3 2 4 4 | 0 6 -7 -2 15 -3 1 }}


Comma list: 154/153, 196/195, 245/243, 256/255, 273/272
Optimal tunings:  
* WE: ~2 = 1199.6759{{c}}, ~15/14 = 116.7378{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.7657{{c}}


Mapping: [{{val| 1 1 -1 3 6 8 8 }}, {{val| 0 3 17 -1 -13 -22 -20 }}]
{{Optimal ET sequence|legend=0| 10, 21e, 31, 41, 72fg }}


Minimax tuning:  
Badness (Sintel): 0.870
* 17-odd-limit eigenmonzos: 2, 18/17


{{Val list|legend=1| 41, 46, 87, 220dg, 307dgg }}
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


Badness: 0.0167
Comma list: 105/104, 120/119, 144/143, 154/153, 170/169, 210/209


</div></div>
{{Todo|complete temperament data|inline=1}}


=== Aerodactyl ===
===== 23-limit =====
Subgroup: 2.3.5.7.11.13.17.19.23


Period: 1\1
Comma list: 105/104, 120/119, 144/143, 154/153, 161/160, 170/169, 210/209


Optimal ([[POTE]]) generator: ~8/7 = 234.639
{{Todo|complete temperament data|inline=1}}


EDO generators: [[41edo|8\41]], [[46edo|9\46]], [[51edo|10\51]]
==== Benediction ====
Subgroup: 2.3.5.7.11.13


Scales (Scala files):  
Comma list: 225/224, 243/242, 351/350, 385/384


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Mapping: {{mapping| 1 1 3 3 2 7 | 0 6 -7 -2 15 -34 }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11.13
Optimal tunings:  
* WE: ~2 = 1199.8601{{c}}, ~15/14 = 116.6572{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.5688{{c}}


Comma list: 91/90, 245/243, 385/384, 441/440
{{Optimal ET sequence|legend=0| 31, 72, 103, 175f }}


Mapping: [{{val| 1 1 -1 3 6 -1 }}, {{val| 0 3 17 -1 -13 24 }}]
Badness (Sintel): 0.649


{{Val list|legend=1| 41f, 46, 133ff }}
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


Badness: 0.0340
Comma list: 225/224, 243/242, 273/272, 351/350, 375/374


</div></div>
Mapping: {{mapping| 1 1 3 3 2 7 7 | 0 6 -7 -2 15 -34 -30 }}


== Aerodino ==
Optimal tunings:
* WE: ~2 = 1200.8328{{c}}, ~15/14 = 116.6661{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.5774{{c}}


Period: 1\1
{{Optimal ET sequence|legend=0| 31, 72, 103, 175f, 422bcdefffg }}


Optimal ([[POTE]]) generator: ~8/7 = 234.728
Badness (Sintel): 0.639


EDO generators: [[41edo|8\41]], [[46edo|9\46]], [[51edo|10\51]]
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


Scales (Scala files):  
Comma list: 210/209, 225/224, 243/242, 273/272, 286/285, 375/374


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
{{Todo|complete temperament data|inline=1}}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11
===== 23-limit =====
Subgroup: 2.3.5.7.11.13.17.19.23


Comma list: 176/175, 245/243, 1029/1024
Comma list: 162/161, 210/209, 225/224, 231/230, 243/242, 273/272, 286/285


Mapping: [{{val| 1 1 -1 3 -3 }}, {{val| 0 3 17 -1 33 }}]
{{Todo|complete temperament data|inline=1}}


{{Val list|legend=1| 41e, 46 }}
==== Manna ====
Subgroup: 2.3.5.7.11.13


Badness: 0.0543
Comma list: 225/224, 243/242, 325/324, 385/384


</div></div>
Mapping: {{mapping| 1 1 3 3 2 0 | 0 6 -7 -2 15 38 }}


=== 13-limit ===
Optimal tunings:
* WE: ~2 = 1200.7564{{c}}, ~15/14 = 116.8129{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.7528{{c}}


Period: 1\1
{{Optimal ET sequence|legend=0| 31f, 41, 72, 185cf, 257cff }}


Optimal ([[POTE]]) generator: ~8/7 = 234.782
Badness (Sintel): 0.703


EDO generators: [[41edo|8\41]], [[46edo|9\46]], [[51edo|10\51]]
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


Scales (Scala files):  
Comma list: 225/224, 243/242, 273/272, 325/324, 385/384


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Mapping: {{mapping| 1 1 3 3 2 0 0 | 0 6 -7 -2 15 38 42 }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Comma list: 91/90, 176/175, 245/243, 847/845
Optimal tunings:  
* WE: ~2 = 1200.7570{{c}}, ~15/14 = 116.8011{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.7408{{c}}


Mapping: [{{val| 1 1 -1 3 -3 -1 }}, {{val| 0 3 17 -1 33 24 }}]
{{Optimal ET sequence|legend=0| 31fg, 41, 72, 185cf, 257cff }}


{{Val list|legend=1| 41ef, 46 }}
Badness (Sintel): 0.748


Badness: 0.0358
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


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


== Varan ==
{{Todo|complete temperament data|inline=1}}


Period: 1\1
===== 23-limit =====
Subgroup: 2.3.5.7.11.13.17.19.23


Optimal ([[POTE]]) generator: ~8/7 = 234.145
Comma list: 210/209, 225/224, 243/242, 273/272, 300/299, 325/324, 343/342


EDO generators: [[36edo|7\36]], [[41edo|8\41]], [[46edo|9\46]]
{{Todo|complete temperament data|inline=1}}


Scales (Scala files):  
==== Semimiracle ====
Subgroup: 2.3.5.7.11.13


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Comma list: 169/168, 225/224, 243/242, 385/384
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Comma list: 100/99, 245/243, 1029/1024
Mapping: {{mapping| 2 2 6 6 4 7 | 0 6 -7 -2 15 2 }}
: mapping generators: ~55/39, ~15/14


Mapping: [{{val| 1 1 -1 3 -2 }}, {{val| 0 3 17 -1 28 }}]
Optimal tunings:  
* WE: ~55/39 = 600.4844{{c}}, ~15/14 = 116.7182{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~15/14 = 116.6413{{c}}


{{Val list|legend=1| 36ce, 41 }}
{{Optimal ET sequence|legend=0| 10, 62, 72 }}


Badness: 0.0449
Badness (Sintel): 1.02


</div></div>
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


=== 13-limit ===
Comma list: 169/168, 221/220, 225/224, 243/242, 273/272


Period: 1\1
Mapping: {{mapping| 2 2 6 6 4 7 7 | 0 6 -7 -2 15 2 6 }}


Optimal ([[POTE]]) generator: ~8/7 = 234.089
Optimal tunings:  
* WE: ~17/12 = 600.5042{{c}}, ~15/14 = 116.7264{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~15/14 = 116.6485{{c}}


EDO generators: [[36edo|7\36]], [[41edo|8\41]], [[46edo|9\46]]
{{Optimal ET sequence|legend=0| 10, 62, 72 }}


Scales (Scala files):  
Badness (Sintel): 0.822


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
===== 19-limit =====
<div style="line-height:1.6;">Technical data</div>
Subgroup: 2.3.5.7.11.13.17.19
<div class="mw-collapsible-content">


Comma list: 100/99, 105/104, 245/243, 352/351
Comma list: 169/168, 210/209, 221/220, 225/224, 243/242, 273/272


Mapping: [{{val| 1 1 -1 3 -2 0 }}, {{val| 0 3 17 -1 28 19 }}]
{{Todo|complete temperament data|inline=1}}


{{Val list|legend=1| 36ce, 41 }}
===== 23-limit =====
Subgroup: 2.3.5.7.11.13.17.19.23


Badness: 0.0323
Comma list: 169/168, 208/207, 210/209, 221/220, 225/224, 243/242, 273/272


</div></div>
{{Todo|complete temperament data|inline=1}}


= Valentine =
==== Hemisecordite ====
{{main| Valentine }}
Subgroup: 2.3.5.7.11.13
{{see also| Starling temperaments #Valentine }}


== 5-limit ==
Comma list: 225/224, 243/242, 385/384, 847/845


Period: 1\1
Mapping: {{mapping| 1 1 3 3 2 2 | 0 12 -14 -4 30 35 }}
: mapping generators: ~2, ~27/26


Optimal ([[POTE]]) generator: ~25/24 = 78.039
Optimal tunings:  
* WE: ~2 = 1200.6969{{c}}, ~27/26 = 58.3217{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~27/26 = 58.2964{{c}}


EDO generators: [[15edo|1\15]], [[16edo|1\16]], [[31edo|2\31]], [[46edo|3\46]], [[77edo|5\77]]
{{Optimal ET sequence|legend=0| 41, 62, 103, 247c, 350bcde }}


Scales (Scala files):  
Badness (Sintel): 1.06


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
===== 17-limit =====
<div style="line-height:1.6;">Technical data</div>
Subgroup: 2.3.5.7.11.13.17
<div class="mw-collapsible-content">


Subgroup: 2.3.5
Comma list: 225/224, 243/242, 273/272, 385/384, 847/845


Comma list: 1990656/1953125
Mapping: {{mapping| 1 1 3 3 2 2 2 | 0 12 -14 -4 30 35 43 }}


Mapping: [{{val| 1 1 2 }}, {{val| 0 9 5 }}]
Optimal tunings:  
* WE: ~2 = 1200.6557{{c}}, ~27/26 = 58.2932{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~27/26 = 58.2702{{c}}


{{Val list|legend=1| 15, 31, 46, 77, 123 }}
{{Optimal ET sequence|legend=0| 41, 62, 103 }}


Badness: 0.1228
Badness (Sintel): 1.15


</div></div>
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


== 7-limit ==
Comma list:


Period: 1\1
{{Todo|complete temperament data|inline=1}}


Optimal ([[POTE]]) generator: ~21/20 = 77.864
===== 23-limit =====
Subgroup: 2.3.5.7.11.13.17.19.23


EDO generators: [[15edo|1\15]], [[16edo|1\16]], [[31edo|2\31]], [[46edo|3\46]], [[77edo|5\77]]
Comma list:  


Scales (Scala files):
{{Todo|complete temperament data|inline=1}}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
===== Semihemisecordite =====
<div style="line-height:1.6;">Technical data</div>
Subgroup: 2.3.5.7.11.13.17
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7
Comma list: 225/224, 243/242, 289/288, 385/384, 847/845


[[Comma list]]: 126/125, 1029/1024
Mapping: {{mapping| 2 2 6 6 4 4 7 | 0 12 -14 -4 30 35 12 }}
: mapping generators: ~17/12, ~27/26


[[Mapping]]: [{{val| 1 1 2 3 }}, {{val| 0 9 5 -3 }}]
Optimal tunings:  
* WE: ~17/12 = 600.3951{{c}}, ~27/26 = 58.3260{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~27/26 = 58.2974{{c}}


Mapping generators: ~2, ~21/20
{{Optimal ET sequence|legend=0| 62, 144g, 206begg }}


[[Minimax tuning]]:
Badness (Sintel): 2.39
* 7-odd-limit:
: [{{monzo| 1 0 0 0 }}, {{monzo| 5/2 3/4 0 -3/4 }}, {{monzo| 17/6 5/12 0 -5/12 }}, {{monzo| 5/2 -1/4 0 1/4 }}]
: [[Eigenmonzo]]s: 2, 7/6
* 9-odd-limit:
: [{{monzo| 1 0 0 0 }}, {{monzo| 10/7 6/7 0 -3/7 }}, {{monzo| 47/21 10/21 0 -5/21 }}, {{monzo| 20/7 -2/7 0 1/7 }}]
: [[Eigenmonzo]]s: 2, 9/7


Algebraic generator: [[Algebraic number|smaller root]] of ''x''<sup>2</sup> - 89''x'' + 92, or (89 - sqrt (7553))/2, at 77.8616 cents.  
====== 19-limit ======
Subgroup: 2.3.5.7.11.13.17.19


{{Val list|legend=1| 15, 31, 46, 77, 185, 262cd }}
Comma list: 209/208, 225/224, 243/242, 289/288, 361/360, 385/384


Badness: 0.0311
Mapping: {{mapping| 2 2 6 6 4 4 7 8 | 0 12 -14 -4 30 35 12 5 }}


</div></div>
Optimal tunings:
* WE: ~17/12 = 600.4418{{c}}, ~27/26 = 58.3255{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~27/26 = 58.2928{{c}}


== 11-limit ==
{{Optimal ET sequence|legend=0| 62, 144gh, 206begghh }}


Period: 1\1
Badness (Sintel): 2.13


Optimal ([[POTE]]) generator: ~21/20 = 77.881
====== 23-limit ======
Subgroup: 2.3.5.7.11.13.17.19.23


EDO generators: [[15edo|1\15]], [[16edo|1\16]], [[31edo|2\31]], [[46edo|3\46]], [[77edo|5\77]]
Comma list: 209/208, 225/224, 243/242, 289/288, 323/322, 361/360, 385/384


Scales (Scala files):  
Mapping: {{mapping| 2 2 6 6 4 4 7 8 7 | 0 12 -14 -4 30 35 12 5 21 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Optimal tunings:
<div style="line-height:1.6;">Technical data</div>
* WE: ~17/12 = 600.4451{{c}}, ~27/26 = 58.3264{{c}}
<div class="mw-collapsible-content">
* CWE: ~17/12 = 600.0000{{c}}, ~27/26 = 58.2942{{c}}


Subgroup: 2.3.5.7.11
{{Optimal ET sequence|legend=0| 62, 144gh, 206begghhi }}


Comma list: 121/120, 126/125, 176/175
Badness (Sintel): 1.89


Mapping: [{{val| 1 1 2 3 3 }}, {{val| 0 9 5 -3 7 }}]
==== Phicordial ====
Subgroup: 2.3.5.7.11.13


Mapping generators: ~2, ~21/20
Comma list: 225/224, 243/242, 385/384, 2200/2197


Minimax tuning:
Mapping: {{mapping| 1 -11 17 7 -28 3 | 0 18 -21 -6 45 1 }}
* 11-odd-limit:
: mapping generators: ~2, ~13/8
: [{{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 }}]
: Eigenmonzos: 2, 11/7


Algebraic generator: [[Algebraic number|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 tunings:  
* WE: ~2 = 1200.7056{{c}}, ~13/8 = 839.3726{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/8 = 838.8831{{c}}


{{Val list|legend=1| 15, 31, 46, 77, 262cdee, 339cdeee }}
{{Optimal ET sequence|legend=0| 103, 216c, 319bcde, 535bccdef }}


Badness: 0.0167
Badness (Sintel): 1.37


</div></div>
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


= Unidec =
Comma list: 225/224, 243/242, 273/272, 385/384, 2200/2197
{{main| Unidec }}


== 5-limit ==
Mapping: {{mapping| 1 -11 17 7 -28 3 -5 | 0 18 -21 -6 45 1 13 }}


Period: 1\2
Optimal tunings:  
* WE: ~2 = 1200.5918{{c}}, ~13/8 = 839.2912{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/8 = 838.8809{{c}}


Optimal ([[POTE]]) generator: ~10/9 = 183.047
{{Optimal ET sequence|legend=0| 103, 216c, 319bcde }}


EDO generator: [[26edo|4\26]], [[46edo|7\46]], [[72edo|11\72]], [[118edo|18\118]]
Badness (Sintel): 1.26


Scales (Scala files): [[unidec26]]
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Comma list: 210/209, 225/224, 243/242, 273/272, 385/384, 2200/2197
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5
{{Todo|complete temperament data|inline=1}}


Comma list: 31381059609/31250000000
===== 23-limit =====
Subgroup: 2.3.5.7.11.13.17.19.23


Mapping: [{{val| 2 5 8 }}, {{val| 0 -6 -11 }}]
Comma list: 210/209, 225/224, 243/242, 273/272, 300/299, 385/384, 1105/1104


Mapping generators: ~177147/125000, ~10/9
{{Todo|complete temperament data|inline=1}}


{{Val list|legend=1| 26, 46, 72, 118, 2524, 2642, 2760, 2878b, …, 5002bc }}
=== Revelation ===
Subgroup: 2.3.5.7.11


Badness: 0.0824
Comma list: 99/98, 176/175, 1029/1024


</div></div>
Mapping: {{mapping| 1 1 3 3 5 | 0 6 -7 -2 -16 }}


== 7-limit ==
Optimal tunings:
* WE: ~2 = 1201.3320{{c}}, ~15/14 = 116.4057{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.2524{{c}}


Period: 1\2
{{Optimal ET sequence|legend=0| 10e, 21, 31 }}


Optimal ([[POTE]]) generator: ~10/9 = 183.161
Badness (Sintel): 1.09


EDO generator: [[26edo|4\26]], [[46edo|7\46]], [[72edo|11\72]], [[118edo|18\118]]
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Scales (Scala files): [[unidec26]]
Comma list: 66/65, 99/98, 105/104, 512/507


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Mapping: {{mapping| 1 1 3 3 5 4 | 0 6 -7 -2 -16 -3 }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7
Optimal tunings:  
* WE: ~2 = 1200.6059{{c}}, ~15/14 = 116.3263{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.2564{{c}}


[[Comma list]]: 1029/1024, 4375/4374
{{Optimal ET sequence|legend=0| 10e, 21, 31 }}


[[Mapping]]: [{{val| 2 5 8 5 }}, {{val| 0 -6 -11 2 }}]
Badness (Sintel): 1.22


Mapping generators: ~1225/864, ~10/9
=== Hemimiracle ===
Subgroup: 2.3.5.7.11


[[Wedgie]]: {{wedgie| 12 22 -4 7 -40 -71 }}
Comma list: 225/224, 245/242, 1029/1024


[[Minimax tuning]]:
Mapping: {{mapping| 1 1 3 3 4 | 0 12 -14 -4 -11 }}
* 7-odd-limit:
: mapping generators: ~2, ~33/32
: [{{monzo| 1 0 0 0 }}, {{monzo| 47/26 0 6/13 -6/13 }}, {{monzo| 71/26 0 11/13 -11/13 }}, {{monzo| 71/26 0 -2/13 2/13 }}]
: [[Eigenmonzo]]s: 2, 7/5
* 9-odd-limit:
: [{{monzo| 1 0 0 0 }}, {{monzo| 10/7 6/7 0 -3/7 }}, {{monzo| 57/28 11/7 0 -11/14 }}, {{monzo| 20/7 -2/7 0 1/7 }}]
: [[Eigenmonzo]]s: 2, 9/7


{{Val list|legend=1| 26, 46, 72, 118, 190 }}
Optimal tunings:
* WE: ~2 = 1200.2902{{c}}, ~33/32 = 58.4217{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~33/32 = 58.4062{{c}}


[[Badness]]: 0.0384
{{Optimal ET sequence|legend=0| 20, 21, 41 }}


</div></div>
Badness (Sintel): 1.96


== 11-limit ==
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Period: 1\2
Comma list: 105/104, 196/195, 245/242, 512/507


Optimal ([[POTE]]) generator: ~10/9 = 183.165
Mapping: {{mapping| 1 1 3 3 4 4 | 0 12 -14 -4 -11 -6 }}


EDO generator: [[26edo|4\26]], [[46edo|7\46]], [[72edo|11\72]], [[118edo|18\118]]
Optimal tunings:  
* WE: ~2 = 1199.8454{{c}}, ~33/32 = 58.4220{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~33/32 = 58.4305{{c}}


Scales (Scala files): [[unidec26]]
{{Optimal ET sequence|legend=0| 20, 21, 41 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Badness (Sintel): 1.78
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


=== Oracle ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 385/384, 441/440, 4375/4374
Comma list: 121/120, 225/224, 1029/1024
 
Mapping: {{mapping| 1 -5 10 5 4 | 0 12 -14 -4 -1 }}
: mapping generators: ~2, ~16/11


Mapping: [{{val| 2 5 8 5 6 }}, {{val| 0 -6 -11 2 3 }}]
Optimal tunings:  
* WE: ~2 = 1201.2122{{c}}, ~16/11 = 658.9974{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/11 = 658.3320{{c}}


Mapping generators: ~99/70, ~10/9
{{Optimal ET sequence|legend=0| 11, 20, 31, 82e, 113e, 144ee }}


Minimax tuning:
Badness (Sintel): 1.41
* 11-odd-limit:
: [{{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]]s: 2, 9/7


{{Val list|legend=1| 26, 46, 72, 118, 190 }}
== Hemiseven ==
Unlike miracle which splits 8/7, hemiseven splits ~16/7, an octave above. It can be described as the {{nowrap| 72 & 77 }} temperament; its ploidacot is gamma-hexacot. [[149edo]] is an obvious tuning.


Badness: 0.0155
[[Subgroup]]: 2.3.5.7


</div></div>
[[Comma list]]: 1029/1024, 19683/19600


=== Ekadash ===
{{Mapping|legend=1| 1 -2 -15 4 | 0 6 29 -2 }}
: mapping generators: ~2, ~243/160


Period: 1\2
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.5612{{c}}, ~243/160 = 717.0687{{c}}
: [[error map]]: {{val| +0.561 -0.665 +0.260 -0.718 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~243/160 = 716.7478{{c}}
: error map: {{val| 0.000 -1.468 -0.629 -2.321 }}


Optimal ([[POTE]]) generator: ~10/9 = 183.187
{{Optimal ET sequence|legend=1| 72, 149, 221, 514bd, 735bcdd }}


EDO generator: [[72edo|11\72]], [[118edo|18\118]], [[190edo|29\190]]
[[Badness]] (Sintel): 1.43


Scales (Scala files): [[unidec26]]
=== 11-limit ===
Subgroup: 2.3.5.7.11


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Comma list: 385/384, 441/440, 19683/19600
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11.13
Mapping: {{mapping| 1 -2 -15 4 16 | 0 6 29 -2 -21 }}


Comma list: 385/384, 441/440, 625/624, 729/728
Optimal tunings:  
* WE: ~2 = 1200.6243{{c}}, ~243/160 = 717.0969{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~243/160 = 716.7292{{c}}


Mapping: [{{val| 2 5 8 5 6 19 }}, {{val| 0 -6 -11 2 3 -38 }}]
{{Optimal ET sequence|legend=0| 72, 149, 221e, 293de }}


Mapping generators: ~99/70, ~10/9
Badness (Sintel): 0.941


{{Val list|legend=1| 26f, 46f, 72, 118, 190, 262df, 452cdef }}
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Badness: 0.0204
Comma list: 351/350, 385/384, 441/440, 676/675


</div></div>
Mapping: {{mapping| 1 -2 -15 4 16 -19 | 0 6 29 -2 -21 38 }}


=== Hendec ===
Optimal tunings:
* WE: ~2 = 1200.6781{{c}}, ~91/60 = 717.1496{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~91/60 = 716.7520{{c}}


Period: 1\2
{{Optimal ET sequence|legend=0| 72, 149, 221ef }}


Optimal ([[POTE]]) generator: ~10/9 = 183.198
Badness (Sintel): 0.905


EDO generator: [[26edo|4\26]], [[46edo|7\46]], [[72edo|11\72]], [[118edo|18\118]]
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Scales (Scala files): [[unidec26]]
Comma list: 273/272, 351/350, 385/384, 441/440, 676/675


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Mapping: {{mapping| 1 -2 -15 4 16 -19 -21 | 0 6 29 -2 -21 38 42 }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11.13
Optimal tunings:  
* WE: ~2 = 1200.6635{{c}}, ~68/45 = 717.1354{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~68/45 = 716.7472{{c}}


Comma list: 169/168, 325/324, 364/363, 1716/1715
{{Optimal ET sequence|legend=0| 72, 149, 221ef }}


Mapping: [{{val| 2 5 8 5 6 8 }}, {{val| 0 -6 -11 2 3 -2 }}]
Badness (Sintel): 0.800


Mapping generators: ~91/64, ~10/9
== Valentine ==
{{Main| Valentine }}
: ''For the 5-limit version, see [[Syntonic–31 equivalence continuum #Valentine (5-limit)]].''


{{Val list|legend=1| 26, 46, 72, 190ff }}
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>.


Badness: 0.0177
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.


</div></div>
[[Subgroup]]: 2.3.5.7


==== 17-limit ====
[[Comma list]]: 126/125, 1029/1024


Period: 1\2
{{Mapping|legend=1| 1 1 2 3 | 0 9 5 -3 }}
: mapping generators: ~2, ~21/20


Optimal ([[POTE]]) generator: ~10/9 = 183.196
[[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 }}


EDO generator: [[26edo|4\26]], [[46edo|7\46]], [[72edo|11\72]], [[118edo|18\118]]
[[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


Scales (Scala files): [[unidec26]]
[[Algebraic generator]]: smaller root of ''x''<sup>2</sup> - 89''x'' + 92, or (89 - sqrt (7553))/2, at 77.8616 cents.


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
{{Optimal ET sequence|legend=1| 15, 31, 46, 77, 185 }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11.13.17
[[Badness]] (Sintel): 0.786


Comma list: 169/168, 221/220, 273/272, 325/324, 364/363
=== 11-limit ===
Subgroup: 2.3.5.7.11


Mapping: [{{val| 2 5 8 5 6 8 10 }}, {{val| 0 -6 -11 2 3 -2 -6 }}]
Comma list: 121/120, 126/125, 176/175


Mapping generators: ~17/12, ~10/9
Mapping: {{mapping| 1 1 2 3 3 | 0 9 5 -3 7 }}


{{Val list|legend=1| 26, 46, 72, 190ffg }}
Optimal tunings:
* WE: ~2 = 1200.3890{{c}}, ~22/21 = 77.9065{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/21 = 77.9007{{c}}


</div></div>
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


= Hemithirds =
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.
{{main| Hemithirds }}
{{see also| Luna family #Hemithirds }}


Period: 1\1
{{Optimal ET sequence|legend=0| 15, 31, 46, 77 }}


Optimal ([[POTE]]) generator: ~28/25 = 193.244
Badness (Sintel): 0.552


EDO generator: [[31edo|5\31]], [[56edo|9\56]], [[87edo|14\87]], [[118edo|23\118]]
==== Valentino ====
Subgroup: 2.3.5.7.11.13


Scales (Scala files):  
Comma list: 121/120, 126/125, 176/175, 196/195


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Mapping: {{mapping| 1 1 2 3 3 5 | 0 9 5 -3 7 -20 }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7
Optimal tunings:  
* WE: ~2 = 1200.1967{{c}}, ~22/21 = 77.9708{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/21 = 77.9594{{c}}


[[Comma list]]: 1029/1024, 3136/3125
{{Optimal ET sequence|legend=0| 15f, 31, 46, 77 }}


[[Mapping]]: [{{val| 1 4 2 2 }}, {{val| 0 -15 2 5 }}]
Badness (Sintel): 0.854


{{Multival|legend=1| 15 -2 -5 -38 -50 -6 }}
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


[[Minimax tuning]]:
Comma list: 121/120, 126/125, 154/153, 176/175, 196/195
* 7-odd-limit:
: [{{monzo| 1 0 0 0 }}, {{monzo| 5/2 3/4 0 -3/4 }}, {{monzo| 11/5 -1/10 0 1/10 }}, {{monzo| 5/2 -1/4 0 1/4 }}]
: [[Eigenmonzo]]s: 2, 7/6
* 9-odd-limit:
: [{{monzo| 1 0 0 0 }}, {{monzo| 10/7 6/7 0 -3/7 }}, {{monzo| 82/35 -4/35 0 2/35 }}, {{monzo| 20/7 -2/7 0 1/7 }}]
: [[Eigenmonzo]]s: 2, 7/6


{{Val list|legend=1| 31, 87, 118 }}
Mapping: {{mapping| 1 1 2 3 3 5 5 | 0 9 5 -3 7 -20 -14 }}


Badness: 0.0443
Optimal tunings:  
* WE: ~2 = 1200.0404{{c}}, ~22/21 = 78.0055{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/21 = 78.0029{{c}}


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


== 11-limit ==
Badness (Sintel): 0.854


Period: 1\1
==== Lupercalia ====
Subgroup: 2.3.5.7.11.13


Optimal ([[POTE]]) generator: ~28/25 = 193.227
Comma list: 66/65, 105/104, 121/120, 126/125


EDO generator: [[31edo|5\31]], [[56edo|9\56]], [[87edo|14\87]], [[118edo|23\118]]
Mapping: {{mapping| 1 1 2 3 3 3 | 0 9 5 -3 7 11 }}


Scales (Scala files):  
Optimal tunings:  
* WE: ~2 = 1199.9143{{c}}, ~22/21 = 77.7039{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/21 = 77.7049{{c}}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
{{Optimal ET sequence|legend=0| 15, 31 }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11
Badness (Sintel): 0.881


Comma list: 385/384, 441/440, 3136/3125
==== Dwynwen ====
Subgroup: 2.3.5.7.11.13


Mapping: [{{val| 1 4 2 2 7 }}, {{val| 0 -15 2 5 -22 }}]
Comma list: 91/90, 121/120, 126/125, 176/175


[[Minimax tuning]]:
Mapping: {{mapping| 1 1 2 3 3 2 | 0 9 5 -3 7 26 }}
* 11-odd-limit:
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 11/9 0 0 -5/9 5/9 }}, {{monzo| 64/27 0 0 2/27 -2/27 }}, {{monzo| 79/27 0 0 5/27 -5/27 }}, {{monzo| 79/27 0 0 -22/27 22/27 }}]
: [[Eigenmonzo]]s: 2, 11/7


{{Val list|legend=1| 31, 87, 118 }}
Optimal tunings:
* WE: ~2 = 1200.1306{{c}}, ~22/21 = 78.2273{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/21 = 78.2241{{c}}


Badness: 0.0190
{{Optimal ET sequence|legend=0| 15, 31f, 46 }}


</div></div>
Badness (Sintel): 0.969


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


Period: 1\1
Comma list: 121/120, 126/125, 169/168, 176/175


Optimal ([[POTE]]) generator: ~28/25 = 193.166
Mapping: {{mapping| 2 2 4 6 6 7 | 0 9 5 -3 7 3 }}
: mapping generators: ~55/39, ~22/21


EDO generator: [[31edo|5\31]], [[56edo|9\56]], [[87edo|14\87]], [[118edo|23\118]]
Optimal tunings:  
* WE: ~55/39 = 600.3497{{c}}, ~22/21 = 77.8845{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~22/21 = 77.8715{{c}}


Scales (Scala files):
{{Optimal ET sequence|legend=0| 16, 30, 46, 62, 108ef }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Badness (Sintel): 1.35
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


==== Hemivalentine ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 196/195, 352/351, 385/384, 625/624
Comma list: 121/120, 126/125, 176/175, 343/338


Mapping: [{{val| 1 4 2 2 7 0 }}, {{val| 0 -15 2 5 -22 23 }}]
Mapping: {{mapping| 1 1 2 3 3 4 | 0 18 10 -6 14 -9 }}
: mapping generators: ~2, ~40/39


{{Val list|legend=1| 31, 56, 87, 118, 205d }}
Optimal tunings:
* WE: ~2 = 1199.6529{{c}}, ~40/39 = 39.0323{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~40/39 = 39.0383{{c}}


Badness: 0.0217
{{Optimal ET sequence|legend=0| 30, 31, 61, 92f }}


</div></div>
Badness (Sintel): 1.94


= Hemiseven =
==== Demivalentine ====
Subgroup: 2.3.5.7.11.13


Period: 1\1
Comma list: 121/120, 126/125, 176/175, 676/675


Optimal ([[POTE]]) generator: ~320/243 = 483.267
Mapping: {{mapping| 1 -8 -3 6 -4 -16 | 0 18 10 -6 14 37 }}
: mapping generators: ~2, ~13/9


EDO generator: [[72edo|29\72]], [[77edo|31\77]], [[149edo|60\149]]
Optimal tunings:  
* WE: ~2 = 1200.3929{{c}}, ~13/9 = 639.1320{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/9 = 638.9325{{c}}


Scales (Scala files):
{{Optimal ET sequence|legend=0| 15, 47ef, 62, 77 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Badness (Sintel): 1.44
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7
=== Hemivalentino ===
Subgroup: 2.3.5.7.11


Comma list: 1029/1024, 19683/19600
Comma list: 126/125, 243/242, 1029/1024


Mapping: [{{val| 1 4 14 2 }}, {{val| 0 -6 -29 2 }}]
Mapping: {{mapping| 1 1 2 3 2 | 0 18 10 -6 45 }}


Wedgie: {{wedgie| 6 29 -2 32 -20 -86 }}
Optimal tunings:  
* WE: ~2 = 1200.0816{{c}}, ~45/44 = 38.9236{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~45/44 = 38.9228{{c}}


{{Val list|legend=1| 72, 77, 149, 221, 514bd, 735bcdd }}
{{Optimal ET sequence|legend=0| 31, 92e, 123, 154, 185 }}


Badness: 0.0566
Badness (Sintel): 2.03


</div></div>
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


== 11-limit ==
Comma list: 126/125, 196/195, 243/242, 1029/1024


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


Optimal ([[POTE]]) generator: ~320/243 = 483.276
Optimal tunings:  
* WE: ~2 = 1199.8782{{c}}, ~45/44 = 38.9440{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~45/44 = 38.9472{{c}}


EDO generator: [[72edo|29\72]], [[77edo|31\77]], [[149edo|60\149]]
{{Optimal ET sequence|legend=0| 31, 123, 154 }}


Scales (Scala files):  
Badness (Sintel): 2.39


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
==== Hemivalentoid ====
<div style="line-height:1.6;">Technical data</div>
Subgroup: 2.3.5.7.11.13
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11
Comma list: 126/125, 144/143, 243/242, 343/338


Comma list: 385/384, 441/440, 19683/19600
Mapping: {{mapping| 1 1 2 3 2 4 | 0 18 10 -6 45 -9 }}


Mapping: [{{val| 1 4 14 2 -5 }}, {{val| 0 -6 -29 2 21 }}]
Optimal tunings:  
* WE: ~2 = 1199.3614{{c}}, ~45/44 = 38.9721{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~45/44 = 38.9839{{c}}


{{Val list|legend=1| 72, 77, 149, 221e, 293de }}
{{Optimal ET sequence|legend=0| 31, 92ef }}


Badness: 0.0285
Badness (Sintel): 2.39


</div></div>
== Superkleismic ==
{{Main| Superkleismic }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Shibboleth]].''


== 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.


Period: 1\1
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.


Optimal ([[POTE]]) generator: ~120/91 = 483.256
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.


EDO generator: [[72edo|29\72]], [[77edo|31\77]], [[149edo|60\149]]
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.


Scales (Scala files):
41edo gives an obvious tuning in all the subgroups.


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
[[Subgroup]]: 2.3.5.7
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11.13
[[Comma list]]: 875/864, 1029/1024


Comma list: 351/350, 385/384, 441/440, 676/675
{{Mapping|legend=1| 1 -5 -5 5 | 0 9 10 -3 }}
: mapping generators: ~2, ~5/3


Mapping: [{{val| 1 4 14 2 -5 19 }}, {{val| 0 -6 -29 2 21 -38 }}]
[[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 }}


{{Val list|legend=1| 72, 77, 149, 221ef }}
{{Optimal ET sequence|legend=1| 11c, 15, 26, 41 }}


</div></div>
[[Badness]] (Sintel): 1.21


== 17-limit ==
=== 11-limit ===
Subgroup: 2.3.5.7.11


Period: 1\1
Comma list: 100/99, 245/242, 385/384


Optimal ([[POTE]]) generator: ~45/34 = 483.261
Mapping: {{mapping| 1 -5 -5 5 2 | 0 9 10 -3 2 }}


EDO generator: [[72edo|29\72]], [[77edo|31\77]], [[149edo|60\149]]
Optimal tunings:  
* WE: ~2 = 1200.1691{{c}}, ~5/3 = 878.2772{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 878.1606{{c}}


Scales (Scala files):
{{Optimal ET sequence|legend=0| 11c, 15, 26, 41, 179cde, 220cde, 261ccdee }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Badness (Sintel): 0.848
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11.13.17
==== 2.3.5.7.11.19 subgroup ====
Subgroup: 2.3.5.7.11.19


Comma list: 273/272, 351/350, 385/384, 441/440, 676/675
Comma list: 100/99, 133/132, 190/189, 385/384


Mapping: [{{val| 1 4 14 2 -5 19 21 }}, {{val| 0 -6 -29 2 21 -38 -42 }}]
Mapping: {{mapping| 1 -5 -5 5 2 -6 | 0 9 10 -3 2 14 }}


{{Val list|legend=1| 72, 77, 149, 221ef }}
Optimal tunings:
* WE: ~2 = 1200.2289{{c}}, ~5/3 = 878.3409{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 878.1840{{c}}


</div></div>
{{Optimal ET sequence|legend=0| 11c, 15, 26, 41, 138e }}


= Tritikleismic =
Badness (Sintel): 0.692
{{see also| Kleismic family #Tritikleismic }}


Period: 1\3
=== 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.


Optimal ([[POTE]]) generator: ~6/5 = 316.872
Subgroup: 2.3.5.7.11.13


EDO generator: [[57edo|15\57]], [[72edo|19\72]], [[87edo|23\87]]
Comma list: 100/99, 105/104, 144/143, 245/242


Scales (Scala files):  
Mapping: {{mapping| 1 -5 -5 5 2 -8 | 0 9 10 -3 2 16 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Optimal tunings:
<div style="line-height:1.6;">Technical data</div>
* WE: ~2 = 1200.0261{{c}}, ~5/3 = 878.0252{{c}}
<div class="mw-collapsible-content">
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 878.0073{{c}}


Subgroup: 2.3.5.7
{{Optimal ET sequence|legend=0| 11cf, 15, 26, 41 }}


[[Comma list]]: 1029/1024, 15625/15552
Badness (Sintel): 0.887


[[Mapping]]: [{{val| 3 0 3 10 }}, {{val| 0 6 5 -2 }}]
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


Mapping generators: ~63/50, ~6/5
Comma list: 100/99, 105/104, 120/119, 144/143, 245/242


[[Minimax tuning]]:
Mapping: {{mapping| 1 -5 -5 5 2 -8 -12 | 0 9 10 -3 2 16 22 }}
* 7-odd-limit:
: [{{monzo| 1 0 0 0 }}, {{monzo| 2 0 6/7 -6/7 }}, {{monzo| 8/3 0 5/7 -5/7 }}, {{monzo| 8/3 0 -2/7 2/7 }}]
: [[Eigenmonzo]]s: 2, 7/5
* 9-odd-limit:
: [{{monzo| 1 0 0 0 }}, {{monzo| 10/7 6/7 0 -3/7 }}, {{monzo| 46/21 5/7 0 -5/14 }}, {{monzo| 20/7 -2/7 0 1/7 }}]
: [[Eigenmonzo]]s: 2, 9/7


{{Val list|legend=1| 15, 42bc, 57, 72, 87, 159, 231 }}
Optimal tunings:
* WE: ~2 = 1200.0488{{c}}, ~5/3 = 877.8872{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 877.8537{{c}}


[[Badness]]: 0.0563
{{Optimal ET sequence|legend=0| 11cfg, 15g, 26, 41 }}


</div></div>
Badness (Sintel): 1.01


== 11-limit ==
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


Period: 1\3
Comma list: 100/99, 105/104, 120/119, 144/143, 133/132, 190/189


Optimal ([[POTE]]) generator: ~6/5 = 316.881
Mapping: {{mapping| 1 -5 -5 5 2 -8 -12 -6 | 0 9 10 -3 2 16 22 14 }}


EDO generator: [[57edo|15\57]], [[72edo|19\72]], [[87edo|23\87]]
Optimal tunings:  
* WE: ~2 = 1200.2120{{c}}, ~5/3 = 878.0243{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 877.8789{{c}}


Scales (Scala files):
{{Optimal ET sequence|legend=0| 11cfgh, 15g, 26, 41 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Badness (Sintel): 0.964
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11
=== Superana ===
This extension ({{nowrap| 41 & 56 }}) is the counterpart of canonical superkleismic on the other side of 41edo.


Comma list: 385/384, 441/440, 4000/3993
Subgroup: 2.3.5.7.11.13


Mapping: [{{val| 3 0 3 10 8 }}, {{val| 0 6 5 -2 3 }}]
Comma list: 100/99, 196/195, 245/242, 385/384


Mapping generators: ~44/35, ~6/5
Mapping: {{mapping| 1 -5 -5 5 2 22 | 0 9 10 -3 2 -25 }}


Minimax tuning:
Optimal tunings:  
* 11-odd-limit:  
* WE: ~2 = 1199.8272{{c}}, ~5/3 = 878.1538{{c}}
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 10/7 6/7 0 -3/7 0 }}, {{monzo| 46/21 5/7 0 -5/14 0 }}, {{monzo| 20/7 -2/7 0 1/7 0 }}, {{monzo| 71/21 3/7 0 -3/14 0 }}]
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 878.2795{{c}}
: [[Eigenmonzo]]s: 2, 9/7


{{Val list|legend=1| 15, 42bc, 57, 72, 159, 231 }}
{{Optimal ET sequence|legend=0| 15f, 41, 97, 138e }}


Badness: 0.0193
Badness (Sintel): 1.40


</div></div>
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


== 13-limit ==
Comma list: 100/99, 154/153, 196/195, 245/242, 256/255


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


Optimal ([[POTE]]) generator: ~6/5 = 316.9585
Optimal tunings:  
* WE: ~2 = 1199.5964{{c}}, ~5/3 = 878.0482{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 878.3444{{c}}


EDO generator: [[72edo|19\72]], [[87edo|23\87]], [[159edo|42\159]]
{{Optimal ET sequence|legend=0| 15f, 41, 56, 97g }}


Scales (Scala files):  
Badness (Sintel): 1.45


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
==== 19-limit ====
<div style="line-height:1.6;">Technical data</div>
Subgroup: 2.3.5.7.11.13.17.19
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11.13
Comma list: 100/99, 133/132, 154/153, 190/189, 196/195, 256/255


Comma list: 325/324, 364/363, 385/384, 625/624
Mapping: {{mapping| 1 -5 -5 5 2 22 18 -6 | 0 9 10 -3 2 -25 -19 14 }}


Mapping: [{{val| 3 0 3 10 8 0 }}, {{val| 0 6 5 -2 3 14 }}]
Optimal tunings:  
* WE: ~2 = 1199.6638{{c}}, ~5/3 = 878.1109{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 878.3566{{c}}


Mapping generators: ~44/35, ~6/5
{{Optimal ET sequence|legend=0| 15f, 41, 56, 97g }}


{{Val list|legend=1| 72, 87, 159 }}
Badness (Sintel): 1.36


</div></div>
== Dee leap week ==
{{Main| Dee leap week }}


== 17-limit ==
[[Subgroup]]: 2.3.5.7


Period: 1\3
[[Comma list]]: 1029/1024, 2460375/2458624


Optimal ([[POTE]]) generator: ~6/5 = 316.9082
{{Mapping|legend=1| 1 -5 25 5 | 0 9 -31 -3 }}


EDO generator: [[72edo|19\72]], [[87edo|23\87]], [[159edo|42\159]]
[[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 }}


Scales (Scala files):
{{Optimal ET sequence|legend=1| 41, 108, 149, 190 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
[[Badness]] (Sintel): 2.12
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11.13.17
=== 11-limit ===
Subgroup: 2.3.5.7.11


Comma list: 273/272, 325/324, 364/363, 375/374, 385/384
Comma list: 385/384, 441/440, 2460375/2458624


Mapping: [{{val| 3 0 3 10 8 0 -2 }}, {{val| 0 6 5 -2 3 14 18 }}]
Mapping: {{mapping| 1 -5 25 5 -28 | 0 9 -31 -3 43 }}


Mapping generators: ~34/27, ~6/5
Optimal tunings:
* WE: ~2 = 1200.4874{{c}}, ~224/135 = 878.2543{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~224/135 = 877.8987{{c}}


{{Val list|legend=1| 72, 159, 231f }}
{{Optimal ET sequence|legend=0| 41, 108e, 149, 190 }}


</div></div>
Badness (Sintel): 1.35


= Superkleismic =
== Unidec ==
{{see also| Shibboleth family #Superkleismic }}
{{Main| Unidec }}


Period: 1\1
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.


Optimal ([[POTE]]) generator: ~6/5 = 321.930
[[Subgroup]]: 2.3.5.7


EDO generator: [[15edo|4\15]], [[26edo|7\26]], [[41edo|11\41]]
[[Comma list]]: 1029/1024, 4375/4374


Scales (Scala files):
{{Mapping|legend=1| 2 -1 -3 7 | 0 6 11 -2 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
[[Optimal tuning]]s:
<div style="line-height:1.6;">Technical data</div>
* [[WE]]: ~1225/864 = 600.2429{{c}}, ~80/63 = 417.0073{{c}}
<div class="mw-collapsible-content">
: [[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 }}


Subgroup: 2.3.5.7
[[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


[[Comma list]]: 875/864, 1029/1024
{{Optimal ET sequence|legend=1| 26, 46, 72, 118, 190 }}


[[Mapping]]: [{{val| 1 4 5 2 }}, {{val| 0 -9 -10 3 }}]
[[Badness]] (Sintel): 0.972


{{Val list|legend=1| 11c, 15, 26, 41 }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


[[Badness]]: 0.0479
Comma list: 385/384, 441/440, 4375/4374


</div></div>
Mapping: {{mapping| 2 -1 -3 7 9 | 0 6 11 -2 -3 }}


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


Period: 1\1
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


Optimal ([[POTE]]) generator: ~6/5 = 321.847
{{Optimal ET sequence|legend=0| 26, 46, 72, 118, 190 }}


EDO generator: [[15edo|4\15]], [[26edo|7\26]], [[41edo|11\41]]
Badness (Sintel): 0.512


Scales (Scala files):  
==== Ekadash ====
Subgroup: 2.3.5.7.11.13


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Comma list: 385/384, 441/440, 625/624, 729/728
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11
Mapping: {{mapping| 2 -1 -3 7 9 -19 | 0 6 11 -2 -3 38 }}


Comma list: 100/99, 245/242, 385/384
Optimal tunings:  
* WE: ~99/70 = 600.2497{{c}}, ~14/11 = 417.0085{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~14/11 = 416.8543{{c}}


Mapping: [{{val| 1 4 5 2 4 }}, {{val| 0 -9 -10 3 -2 }}]
{{Optimal ET sequence|legend=0| 46f, 72, 118, 190, 262df, 452cdef }}


{{Val list|legend=1| 11c, 15, 26, 41, 261ccdee }}
Badness (Sintel): 0.842


Badness: 0.0257
==== Hendec ====
Subgroup: 2.3.5.7.11.13


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


== 13-limit ==
Mapping: {{mapping| 2 -1 -3 7 9 6 | 0 6 11 -2 -3 2 }}


Period: 1\1
Optimal tunings:  
* WE: ~91/64 = 600.3825{{c}}, ~14/11 = 417.0678{{c}}
* CWE: ~91/64 = 600.0000{{c}}, ~14/11 = 416.8290{{c}}


Optimal ([[POTE]]) generator: ~6/5 = 321.994
{{Optimal ET sequence|legend=0| 26, 46, 72, 190ff }}


EDO generator: [[15edo|4\15]], [[26edo|7\26]], [[41edo|11\41]]
Badness (Sintel): 0.732


Scales (Scala files):  
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Comma list: 169/168, 221/220, 273/272, 325/324, 364/363
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11.13
Mapping: {{mapping| 2 -1 -3 7 9 6 4 | 0 6 11 -2 -3 2 6 }}


Comma list: 100/99, 105/104, 245/243, 1188/1183
Optimal tunings:  
* WE: ~17/12 = 600.3991{{c}}, ~14/11 = 417.0809{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~14/11 = 416.8330{{c}}


Mapping: [{{val| 1 4 5 2 4 8 }}, {{val| 0 -9 -10 3 -2 -16 }}]
{{Optimal ET sequence|legend=0| 26, 46, 72, 190ffg }}


{{Val list|legend=1| 11cf, 15, 26, 41 }}
Badness (Sintel): 0.595


Badness: 0.0215
== 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.  


</div></div>
[[Subgroup]]: 2.3.5.7


= Gorgo =
[[Comma list]]: 1029/1024, 5103/5000
{{see also| Laconic family #Gorgo }}


Period: 1\1
{{Mapping|legend=1| 1 -5 -7 5 | 0 12 17 -4 }}
: mapping generators: ~2, ~35/24


Optimal ([[POTE]]) generator: ~8/7 = 228.334
[[Optimal tuning]]s:
* [[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 }}


EDO generator: [[16edo|3\16]], [[21edo|4\21]]
{{Optimal ET sequence|legend=1| 11c, 20c, 31, 144c, 175c }}


Scales (Scala files):  
[[Badness]] (Sintel): 2.98


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
=== 11-limit ===
<div style="line-height:1.6;">Technical data</div>
Subgroup: 2.3.5.7.11
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7
Comma list: 176/175, 243/242, 1029/1024


[[Comma list]]: 36/35, 1029/1024
Mapping: {{mapping| 1 -5 -7 5 -13 | 0 12 17 -4 30 }}


[[Mapping]]: [{{val| 1 1 1 3 }}, {{val| 0 3 7 -1 }}]
Optimal tunings:  
* WE: ~2 = 1200.2862{{c}}, ~22/15 = 658.4276{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/15 = 658.2805{{c}}


[[Wedgie]]: {{wedgie| 3 7 -1 4 -10 -22 }}
{{Optimal ET sequence|legend=0| 20ce, 31, 113c, 144c }}


{{Val list|legend=1| 5, 11c, 16, 21 }}
Badness (Sintel): 1.77


[[Badness]]: 0.0607
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


</div></div>
Comma list: 144/143, 176/175, 243/242, 343/338


== 11-limit ==
Mapping: {{mapping| 1 -5 -7 5 -13 7 | 0 12 17 -4 30 -6 }}


Period: 1\1
Optimal tunings:  
* WE: ~2 = 1199.3663{{c}}, ~22/15 = 658.0465{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/15 = 658.3800{{c}}


Optimal ([[POTE]]) generator: ~8/7 = 229.535
{{Optimal ET sequence|legend=0| 20ce, 31, 82cf, 113cf }}


EDO generator: [[16edo|3\16]], [[21edo|4\21]]
Badness (Sintel): 1.94


Scales (Scala files):
== Restles ==
{{See also| Lesser tendoneutralic }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
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>.
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7


Comma list: 36/35, 56/55, 1029/1024
[[Comma list]]: 1029/1024, 153664/151875


Mapping: [{{val| 1 1 1 3 5 }}, {{val| 0 3 7 -1 -8 }}]
{{Mapping|legend=1| 1 -2 8 4 | 0 12 -19 -4 }}
: mapping generators: ~2. ~315/256


{{Val list|legend=1| 5, 16e, 21, 47c, 68bce }}
[[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 }}


Badness: 0.0627
{{Optimal ET sequence|legend=1| 77, 87, 164 }}


</div></div>
[[Badness]] (Sintel): 2.73


== 13-limit ==
=== 11-limit ===
Subgroup: 2.3.5.7.11


Period: 1\1
Comma list: 385/384, 441/440, 153664/151875


Optimal ([[POTE]]) generator: ~8/7 = 229.059
Mapping: {{mapping| 1 -2 8 4 -7 | 0 12 -19 -4 35 }}


EDO generator: [[16edo|3\16]], [[21edo|4\21]]
Optimal tunings:  
* WE: ~2 = 1200.1110{{c}}, ~27/22 = 358.6045{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~27/22 = 358.5720{{c}}


Scales (Scala files):
{{Optimal ET sequence|legend=0| 77, 87, 164, 251d }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Badness (Sintel): 1.81
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


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


Comma list: 27/26, 36/35, 56/55, 507/500
Comma list: 196/195, 352/351, 385/384, 676/675


Mapping: [{{val| 1 1 1 3 5 2 }}, {{val| 0 3 7 -1 -8 9 }}]
Mapping: {{mapping| 1 -2 8 4 -7 4 | 0 12 -19 -4 35 -1 }}


{{Val list|legend=1| 5, 16e, 21, 68bcef }}
Optimal tunings:
* WE: ~2 = 1200.0482{{c}}, ~~16/13 = 358.5883{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/13 = 358.5741{{c}}


Badness: 0.0471
{{Optimal ET sequence|legend=0| 77, 87, 164, 251d }}


</div></div>
Badness (Sintel): 1.16


== Music ==
== Lagaca ==
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Herman/gorgo-example.mp3 Gorgo Example] by [[Herman Miller]]
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.


= Lemba =
[[Subgroup]]: 2.3.5.7
{{main| Lemba }}
{{see also| Jubilismic clan #Lemba }}


Period: 1\2
[[Comma list]]: 1029/1024, 11529602/11390625


Optimal ([[POTE]]) generator: ~8/7 = 232.089
{{Mapping|legend=1| 2 -4 15 8 | 0 9 -13 -3 }}
: mapping generators: ~3375/2401, ~450/343


EDO generator: [[10edo|2\10]], [[16edo|3\16]], [[26edo|5\26]]
[[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 }}


Scales (Scala files):
{{Optimal ET sequence|legend=1| 10, 98, 108, 118 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
[[Badness]] (Sintel): 3.65
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


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


[[Comma list]]: 50/49, 525/512
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"/>.


[[Mapping]]: [{{val| 2 2 5 6 }}, {{val| 0 3 -1 -1 }}]
[[Subgroup]]: 2.3.5.7


Mapping generators: ~7/5, ~8/7
[[Comma list]]: 1029/1024, 1250000/1240029


{{Multival|legend=1| 6 -2 -2 -17 -20 1 }}
{{Mapping|legend=1| 1 -17 -26 9 | 0 21 32 -7 }}
: mapping generators: ~2, ~50/27


{{Val list|legend=1| 10, 16, 26 }}
[[Optimal tuning]]s:
* [[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 }}


[[Badness]]: 0.0622
{{Optimal ET sequence|legend=1| 26, 61, 87, 113, 200 }}


</div></div>
[[Badness]] (Sintel): 3.85


= Gidorah =
=== 11-limit ===
{{see also|University temperament}}
Subgroup: 2.3.5.7.11


Period: 1\1
Comma list: 385/384, 441/440, 800000/793881


Optimal ([[POTE]]) generator: ~8/7 = 230.762
Mapping: {{mapping| 1 -17 -26 9 7 | 0 21 32 -7 -4 }}


EDO generator: [[5edo|1\5]], [[11edo|2\11]], [[16edo|3\16]]
Optimal tunings:  
* WE: ~2 = 1200.3051{{c}}, ~50/27 = 1062.2805{{c}}
* CWE: ~21 = 1200.0000{{c}}, ~50/27 = 1062.0147{{c}}


Scales (Scala files):
{{Optimal ET sequence|legend=0| 26, 61, 87, 200, 287d }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Badness (Sintel): 1.89
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


[[Comma list]]: 21/20, 144/125
Comma list: 325/324, 364/363, 385/384, 2200/2197


[[Mapping]]: [{{val| 1 1 2 3 }}, {{val| 0 3 2 -1 }}]
Mapping: {{mapping| 1 -17 -26 9 7 -14 | 0 21 32 -7 -4 20 }}


{{Val list|legend=1| 5, 11, 16c, 21cc, 26ccc }}
Optimal tunings:
* WE: ~2 = 1200.2708{{c}}, ~24/13 = 1062.2496{{c}}
* CWE: ~21 = 1200.0000{{c}}, ~24/13 = 1062.0139{{c}}


[[Badness]]: 0.0623
{{Optimal ET sequence|legend=0| 26, 61, 87, 200 }}


</div></div>
Badness (Sintel): 1.17


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


Period: 1\1
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"/>.


Optimal ([[POTE]]) generator: ~8/7 = 226.469
[[Subgroup]]: 2.3.5.7


EDO generator: [[5edo|1\5]], [[11edo|2\11]], [[16edo|3\16]]
[[Comma list]]: 1029/1024, 1959552/1953125


Scales (Scala files):  
{{Mapping|legend=1| 1 -11 -7 7 | 0 27 20 -9 }}
: mapping generators: ~2, ~864/625


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
[[Optimal tuning]]s:
<div style="line-height:1.6;">Technical data</div>
* [[WE]]: ~2 = 1200.4239{{c}}, ~864/625 = 559.4921{{c}}
<div class="mw-collapsible-content">
: [[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 }}


Subgroup: 2.3.5.7
{{Optimal ET sequence|legend=1| 15, …, 88, 103, 118, 221, 339d }}


[[Comma list]]: 135/128, 360/343
[[Badness]] (Sintel): 3.00


[[Mapping]]: [{{val| 1 1 4 3 }}, {{val| 0 3 -9 -1 }}]
=== 11-limit ===
Subgroup: 2.3.5.7.11


{{Val list|legend=1| 5c, 11, 16 }}
Comma list: 385/384, 441/440, 43923/43750


[[Badness]]: 0.1592
Mapping: {{mapping| 1 -11 -7 7 -4 | 0 27 20 -9 16 }}


</div></div>
Optimal tunings:
* WE: ~2 = 1200.4953{{c}}, ~242/175 = 559.5243{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~242/175 = 559.3016{{c}}


== 11-limit ==
{{Optimal ET sequence|legend=0| 15, …, 88, 103, 118, 221e, 339de }}


Period: 1\1
Badness (Sintel): 1.17


Optimal ([[POTE]]) generator: ~8/7 = 226.428
== Widefourth ==
[[Subgroup]]: 2.3.5.7


EDO generator: [[5edo|1\5]], [[11edo|2\11]], [[16edo|3\16]]
[[Comma list]]: 1029/1024, 48828125/48771072
 
Scales (Scala files):


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
{{Mapping|legend=1| 1 -17 -5 9 | 0 33 13 -11 }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11
[[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 }}


Comma list: 33/32, 45/44, 352/343
{{Optimal ET sequence|legend=1| 16, 71, 87, 103, 190 }}


Mapping: [{{val| 1 1 4 3 4 }}, {{val| 0 3 -9 -1 -3 }}]
[[Badness]] (Sintel): 3.90


{{Val list|legend=1| 5c, 11, 16 }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.0697
Comma list: 385/384, 441/440, 234375/234256


</div></div>
Mapping: {{mapping| 1 16 8 -2 17 | 0 -33 -13 11 -31 }}


= Necromanteion =
Optimal tunings:
[[Comma list]]: 1029/1024, 5103/5000
* WE: ~2 = 1200.4852{{c}}, ~1250/847 = 676.0634{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~1250/847 = 675.7966{{c}}


POTE generator: ~48/35 = 541.779
{{Optimal ET sequence|legend=0| 16, 71, 87, 103, 190 }}


[[Mapping]]: [{{val| 1 7 10 1 }}, {{val| 0 -12 -17 4 }}]
Badness (Sintel): 1.35


{{Val list|legend=1| 11c, 20c, 31, 51c, 82c, 113c, 144c, 175c, 206bc, 237bc, 505bcd }}
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


[[Badness]]: 0.1177
Comma list: 385/384, 441/440, 625/624, 847/845


== 11-limit ==
Mapping: {{mapping| 1 16 8 -2 17 12 | 0 -33 -13 11 -31 -19 }}
Comma list: 176/175, 243/242, 1029/1024


POTE generator: ~15/11 = 541.729
Optimal tunings:  
* WE: ~2 = 1200.4217{{c}}, ~77/52 = 676.0286{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~77/52 = 675.7967{{c}}


Mapping: [{{val| 1 7 10 1 17 }}, {{val| 0 -12 -17 4 -30 }}]
{{Optimal ET sequence|legend=0| 16, 71, 87, 103, 190 }}


{{Val list|legend=1| 31, 82c, 113c, 144c, 175c, 350bcde, 381bcde }}
Badness (Sintel): 0.894


Badness: 0.0535
== 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.


== 13-limit ==
Subgroup: 2.3.7.13
Comma list: 144/143, 176/175, 243/242, 343/338


POTE generator: ~15/11 = 541.606
Comma list: 729/728, 1029/1024


Mapping: [{{val| 1 7 10 1 17 1 }}, {{val| 0 -12 -17 4 -30 6 }}]
Subgroup-val mapping: {{mapping| 1 1 3 0 | 0 3 -1 19 }}


{{Val list|legend=1| 31, 51ce, 82cf, 113cf, 144cf }}
Gencom mapping: {{mapping| 1 1 0 3 0 0 | 0 3 0 -1 0 19 }}


Badness: 0.0470
Optimal tunings:  
* WE: ~2 = 1200.5057{{c}}, ~8/7 = 233.7200{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.6534{{c}}


= Widefourth =
{{Optimal ET sequence|legend=0| 5, 31f, 36, 77, 113, 827bdddff }}
[[Comma list]]: 1029/1024, 48828125/48771072


POTE generator: ~3125/2304 = 524.210
Badness (Sintel): 0.339


[[Mapping]]: [{{val| 1 16 8 -2 }}, {{val| 0 -33 -13 11 }}]
==== 2.3.7.13.17 subgroup ====
Subgroup: 2.3.7.13.17


[[Wedgie]]: {{wedgie| 33 13 -11 -56 -110 -62 }}
Comma list: 273/272, 729/728, 833/832


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


[[Badness]]: 0.1541
Gencom mapping: {{mapping| 1 1 0 3 0 0 0 | 0 3 0 -1 0 19 21 }}


== 11-limit ==
Optimal tunings:
Comma list: 385/384, 441/440, 234375/234256
* WE: ~2 = 1200.5282{{c}}, ~8/7 = 233.6492{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.5776{{c}}


POTE generator: ~3125/2304 = 524.210
{{Optimal ET sequence|legend=0| 5g, 31fg, 36, 113, 149 }}


Mapping: [{{val| 1 16 8 -2 17 }}, {{val| 0 -33 -13 11 -31 }}]
Badness (Sintel): 0.332


{{Val list|legend=1| 16, 71, 87, 103, 190 }}
==== 2.3.7.13.17.19 subgroup ====
Subgroup: 2.3.7.13.17.19


Badness: 0.0408
Comma list: 273/272, 343/342, 513/512, 729/728


== 13-limit ==
Subgroup-val mapping: {{mapping| 1 1 3 0 0 6 | 0 3 -1 19 21 -9 }}
Comma list: 385/384, 441/440, 625/624, 847/845


POTE generator: ~65/48 = 524.209
Gencom mapping: {{mapping| 1 1 0 3 0 0 0 6 | 0 3 0 -1 0 19 21 -9 }}


Mapping: [{{val| 1 16 8 -2 17 12 }}, {{val| 0 -33 -13 11 -31 -19 }}]
Optimal tunings:  
* WE: ~2 = 1200.3292{{c}}, ~8/7 = 233.6651{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.6106{{c}}


{{Val list|legend=1| 16, 71, 87, 103, 190 }}
{{Optimal ET sequence|legend=0| 5g, 36, 77, 113, 262df }}


Badness: 0.0216
Badness (Sintel): 0.380


= Tritriple =
==== 2.3.7.13.17.19.23 subgroup ====
== 5-limit ==
Subgroup: 2.3.7.13.17.19.23
Comma list: {{monzo| 31 20 -27 }}


POTE generator: ~864/625 = 559.332
Comma list: 273/272, 343/342, 392/391, 513/512, 729/728


Mapping: [{{val| 1 -11 -7 }}, {{val| 0 27 20 }}]
Subgroup-val mapping: {{mapping| 1 1 3 0 0 6 9 | 0 3 -1 19 21 -9 -23 }}


{{Val list|legend=1| 15, 103, 118, 133, 959, 1077 }}
Gencom mapping: {{mapping| 1 1 0 3 0 0 0 6 9 | 0 3 0 -1 0 19 21 -9 -23 }}


Badness: 0.2836
Optimal tunings:  
* WE: ~2 = 1200.3127{{c}}, ~8/7 = 233.6679{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.6091{{c}}


== 7-limit ==
{{Optimal ET sequence|legend=0| 36, 77, 113, 262df }}
[[Comma list]]: 1029/1024, 1959552/1953125


POTE generator: ~864/625 = 559.295
Badness (Sintel): 0.474


[[Mapping]]: [{{val| 1 -11 -7 7 }}, {{val| 0 27 20 -9 }}]
==== 2.3.7.13.17.19.23.29 subgroup ====
Subgroup: 2.3.7.13.17.19.23.29


{{Val list|legend=1| 15, 103, 118, 133, 339d }}
Comma list: 273/272, 343/342, 378/377, 392/391, 513/512, 609/608


[[Badness]]: 0.1186
Subgroup-val mapping: {{mapping| 1 1 3 0 0 6 9 7 | 0 3 -1 19 21 -9 -23 -11 }}


== 11-limit ==
Gencom mapping: {{mapping| 1 1 0 3 0 0 0 6 9 7 | 0 3 0 -1 0 19 21 -9 -23 -11 }}
Comma list: 385/384, 441/440, 43923/43750


POTE generator: ~242/175 = 559.293
Optimal tunings:  
* WE: ~2 = 1200.2503{{c}}, ~8/7 = 233.6688{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.6208{{c}}


Mapping: [{{val| 1 -11 -7 7 -4 }}, {{val| 0 27 20 -9 16 }}]
{{Optimal ET sequence|legend=0| 36, 77, 113 }}


{{Val list|legend=1| 15, 103, 118, 133, 339de }}
Badness (Sintel): 0.473


Badness: 0.0353
=== 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.


= Restles =
Subgroup: 2.3.7.13
[[Comma list]]: 1029/1024, 153664/151875


POTE generator: ~315/256 = 358.5485
Comma list: 169/168, 1029/1024


[[Mapping]]: [{{val| 1 -2 8 4 }}, {{val| 0 12 -19 -4 }}]
Subgroup-val mapping: {{mapping| 2 2 6 7 | 0 3 -1 1 }}


{{Val list|legend=1| 10, 67, 77, 87, 164 }}
Gencom mapping: {{mapping| 2 2 0 6 0 7 | 0 3 0 -1 0 1 }}
: mapping generators: ~91/64, ~8/7


[[Badness]]: 0.1080
Optimal tunings:  
* WE: ~91/64 = 600.4315{{c}}, ~8/7 = 233.7724{{c}}
* CWE: ~91/64 = 600.0000{{c}}, ~8/7 = 233.7039{{c}}


== 11-limit ==
{{Optimal ET sequence|legend=0| 10, 26, 36, 154f, 190ff, 226ff, 262dfff }}
Comma list: 385/384, 441/440, 153664/151875


POTE generator: ~27/22 = 358.5713
Badness (Sintel): 0.434


Mapping: [{{val| 1 -2 8 4 -7 }}, {{val| 0 12 -19 -4 35 }}]
==== 2.3.7.13.17 subgroup ====
Subgroup: 2.3.7.13.17


{{Val list|legend=1| 10, 77, 87, 164 }}
Comma list: 169/168, 273/272, 289/288


Badness: 0.0547
Subgroup-val mapping: {{mapping| 2 2 6 7 7 | 0 3 -1 1 3 }}


== 13-limit ==
Gencom mapping: {{mapping| 2 2 0 6 0 7 7 | 0 3 0 -1 0 1 3 }}
Comma list: 196/195, 352/351, 385/384, 676/675


POTE generator: ~16/13 = 358.5739
Optimal tunings:  
* WE: ~17/12 = 600.4436{{c}}, ~8/7 = 233.7883{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~8/7 = 233.7312{{c}}


Mapping: [{{val| 1 -2 8 4 -7 4 }}, {{val| 0 12 -19 -4 35 -1 }}]
{{Optimal ET sequence|legend=0| 10, 26, 36, 154f, 190ffg, 226ffg }}


{{Val list|legend=1| 10, 77, 87, 164 }}
Badness (Sintel): 0.253


Badness: 0.0282
=== Gigapyth (2.3.7.85) ===
Subgroup: 2.3.7.85


= Baladic =
Comma list: 1029/1024, 7225/7203
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.


Comma list: 169/168, 273/272, 289/288
Subgroup-val mapping: {{mapping| 1 -2 4 7 | 0 6 -2 -1 }}


Period: 1\2
Optimal tunings:  
* WE: ~2 = 1200.8295{{c}}, ~128/85 = 717.2597{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~128/85 = 716.7933{{c}}


POTE generator: ~8/7 = 233.6155
{{Optimal ET sequence|legend=0| 5, 42*, 47, 52, 57, 62, 67, 72, 149*, 370d***, 519bdd***** }}


Sval mapping: [{{val| 2 2 6 7 7 }}, {{val| 0 3 -1 1 3 }}]
<nowiki/>* Wart for 85


{{Val list|legend=1| 26, 36, 46, 82, 118f }}
== References ==


[[Category:Theory]]
[[Category:Temperament clans]]
[[Category:Temperament clan]]
[[Category:Gamelismic clan| ]] <!-- main article -->
[[Category:Gamelismic]]
[[Category:Miracle]]
[[Category:Rodan]]
[[Category:Rank 2]]
[[Category:Rank 2]]
[[Category:Listen]]
[[Category:Listen]]
{{todo| review | improve readability }}

Latest revision as of 19:51, 4 May 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The 2.3.7-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-3 gamelismic temperament itself, and for the rank-2 2.3.7 temperament slendric (a.k.a. gamelic), this means three ~8/7 intervals give a fifth, 3/2. In fact, we find that 3/2 = (8/7)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.

Slendric

Main article: Slendric

Subgroup: 2.3.7

Comma list: 1029/1024

Subgroup-val mapping: [⟨1 1 3], ⟨0 3 -1]]

Gencom mapping: [⟨1 1 0 3], ⟨0 3 0 -1]]

mapping generators: ~2, ~8/7

Optimal tunings:

  • WE: ~2 = 1200.4859 ¢, ~8/7 = 233.7822 ¢
error map: ⟨+0.486 -0.123 -1.151]
  • CWE: ~2 = 1200.000 ¢, ~8/7 = 233.7474 ¢
error map: ⟨0.000 -0.713 -2.573]

Optimal ET sequence: 5, 21, 26, 31, 36, 77, 113, 190

Badness (Sintel): 0.158

Overview to extensions

Full 7-limit extensions

To the gamelisma itself we need to add the comma which appears next on the modified normal 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:

The rest are considered below.

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 trisect in the 2.3.7.11/5 subgroup.

Radon

See also: Chromatic pairs § Radon

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

Subgroup: 2.3.7.11

Comma list: 896/891, 1029/1024

Subgroup-val mapping: [⟨1 1 3 6], ⟨0 3 -1 -13]]

Gencom mapping: [⟨1 1 0 3 6], ⟨0 3 0 -1 -13]]

Optimal tunings:

  • WE: ~2 = 1199.9708 ¢, ~8/7 = 234.3748 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 234.3813 ¢

Optimal ET sequence: 5, …, 36, 41, 87, 128

Badness (Sintel): 0.619

Mothra

Main article: 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 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[1], 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 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 {S6/S7, S7/S8, (S6/S8 = S9)}, taking advantage of the fact that 81/80 is a semiparticular.

Subgroup: 2.3.5.7

Comma list: 81/80, 1029/1024

Mapping: [⟨1 1 0 3], ⟨0 3 12 -1]]

Optimal tunings:

  • WE: ~2 = 1200.9303 ¢, ~8/7 = 232.3733 ¢
error map: ⟨+0.930 -3.905 +2.165 +1.592]
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 232.2514 ¢
error map: ⟨0.000 -5.520 +0.703 -1.077]

Algebraic generator: Rabrindanath, largest real root of x8 - 3x2 + 1, or 232.0774 cents.

Minimax tuning:

[[1 0 0 0⟩, [1 0 1/4 0⟩, [0 0 1 0⟩, [3 0 -1/12 0⟩]
unchanged-interval (eigenmonzo) basis: 2.5

Optimal ET sequence: 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: [⟨1 1 0 3 5], ⟨0 3 12 -1 -8]]

Optimal tunings:

  • WE: ~2 = 1201.3979 ¢, ~8/7 = 232.3010 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 232.0621 ¢

Optimal ET sequence: 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: [⟨1 1 0 3 5 1], ⟨0 3 12 -1 -8 14]]

Optimal tunings:

  • WE: ~2 = 1201.0985 ¢, ~8/7 = 232.0231 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 231.8425 ¢

Optimal ET sequence: 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: [⟨1 1 0 3 5 1], ⟨0 3 12 -1 -8 14 16]]

Optimal tunings:

  • WE: ~2 = 1200.9734 ¢, ~8/7 = 231.8960 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 231.7392 ¢

Optimal ET sequence: 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: [⟨1 1 0 3 5 1], ⟨0 3 12 -1 -8 14 16 22]]

Optimal tunings:

  • WE: ~2 = 1200.9663 ¢, ~8/7 = 231.8393 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 231.6842 ¢

Optimal ET sequence: 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: {S6/S7, S7/S8, (S6/S8 = S9), S8/S10}.

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 1 0 3 -1], ⟨0 3 12 -1 23]]

Optimal tunings:

  • WE: ~2 = 1200.7675 ¢, ~8/7 = 232.5673 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 232.4567 ¢

Optimal ET sequence: 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: [⟨1 1 0 3 -1 7], ⟨0 3 12 -1 23 -17]]

Optimal tunings:

  • WE: ~2 = 1199.9347 ¢, ~8/7 = 232.6275 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 232.6392 ¢

Optimal ET sequence: 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: [⟨1 1 0 3 -1 7], ⟨0 3 12 -1 23 -17 -15]]

Optimal tunings:

  • WE: ~2 = 1199.7124 ¢, ~8/7 = 232.6376 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 232.6917 ¢

Optimal ET sequence: 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: [⟨1 1 0 3 -1 7], ⟨0 3 12 -1 23 -17 -15 -9]]

Optimal tunings:

  • WE: ~2 = 1199.4885 ¢, ~8/7 = 232.6310 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 232.7287 ¢

Optimal ET sequence: 31, 67, 98h

Badness (Sintel): 1.50

Cyndra

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 1 0 3 0], ⟨0 3 12 -1 18]]

Optimal tunings:

  • WE: ~2 = 1201.1585 ¢, ~8/7 = 231.5404 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 231.3850 ¢

Optimal ET sequence: 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: [⟨1 1 0 3 0 1], ⟨0 3 12 -1 18 14]]

Optimal tunings:

  • WE: ~2 = 1201.1152 ¢, ~8/7 = 231.5079 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 231.3612 ¢

Optimal ET sequence: 5e, 21cef, 26

Badness (Sintel): 1.41

Rodan

Main article: 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 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: [⟨1 1 -1 3], ⟨0 3 17 -1]]

Optimal tunings:

  • WE: ~2 = 1200.2146 ¢, ~8/7 = 234.4587 ¢
error map: ⟨+0.215 +1.636 -0.731 -2.641]
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 234.4259 ¢
error map: ⟨0.000 +1.323 -1.073 -3.252]

Minimax tuning:

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

Algebraic generator: larger root of 20x2 - 36x + 15, or (9 + √6)/10.

Optimal ET sequence: 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: [⟨1 1 -1 3 6], ⟨0 3 17 -1 -13]]

Optimal tunings:

  • WE: ~2 = 1200.0553 ¢, ~8/7 = 234.4695 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 234.4594 ¢

Minimax tuning:

  • 11-odd-limit: ~8/7 = [4/19 2/19 0 0 -1/19⟩
[[1 0 0 0 0⟩, [31/19 6/19 0 0 -3/19⟩, [49/19 34/19 0 0 -17/19⟩, [53/19 -2/19 0 0 1/19⟩, [62/19 -26/19 0 0 13/19⟩]
unchanged-interval (eigenmonzo) basis: 2.11/9

Algebraic generator: positive root of x2 + 16x - 31, or √95 - 8.

Optimal ET sequence: 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: [⟨1 1 -1 3 6 8], ⟨0 3 17 -1 -13 -22]]

Optimal tunings:

  • WE: ~2 = 1199.9868 ¢, ~8/7 = 234.4796 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 234.4822 ¢

Minimax tuning:

  • 13- and 15-odd-limit: ~8/7 = [3/14 1/14 0 0 0 -1/28⟩
unchanged-interval (eigenmonzo) basis: 2.13/9

Algebraic generator: Gatetone, positive root of 4x6 - 7x - 1. Recurrence converges slowly.

Optimal ET sequence: 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: [⟨1 1 -1 3 6 8 8], ⟨0 3 17 -1 -13 -22 -20]]

Optimal tunings:

  • WE: ~2 = 1199.8331 ¢, ~8/7 = 234.4919 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 234.5254 ¢

Minimax tuning:

  • 17-odd-limit: ~8/7 = [3/13 1/13 0 0 0 0 -1/26⟩
unchanged-interval (eigenmonzo) basis: 2.17/9

Optimal ET sequence: 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: [⟨1 1 -1 3 6 -1], ⟨0 3 17 -1 -13 24]]

Optimal tunings:

  • WE: ~2 = 1200.2997 ¢, ~8/7 = 234.6972 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 234.6439 ¢

Optimal ET sequence: 5, 41f, 46

Badness (Sintel): 1.40

Aerodino

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 1 -1 3 -3], ⟨0 3 17 -1 33]]

Optimal tunings:

  • WE: ~2 = 1199.9179 ¢, ~8/7 = 234.7123 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 234.7256 ¢

Optimal ET sequence: 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: [⟨1 1 -1 3 -3 -1], ⟨0 3 17 -1 33 24]]

Optimal tunings:

  • WE: ~2 = 1200.0242 ¢, ~8/7 = 234.7863 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 234.7824 ¢

Optimal ET sequence: 5e, 41ef, 46

Badness (Sintel): 1.48

Varan

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 1 -1 3 -2], ⟨0 3 17 -1 28]]

Optimal tunings:

  • WE: ~2 = 1200.3738 ¢, ~8/7 = 234.2174 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 234.1586 ¢

Optimal ET sequence: 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: [⟨1 1 -1 3 -2 0], ⟨0 3 17 -1 28 19]]

Optimal tunings:

  • WE: ~2 = 1200.1389 ¢, ~8/7 = 234.1162 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 234.0946 ¢

Optimal ET sequence: 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 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: [⟨1 1 7 3], ⟨0 3 -24 -1]]

Optimal tunings:

  • WE: ~2 = 1200.3395 ¢, ~8/7 = 233.9963 ¢
error map: ⟨+0.340 +0.374 +0.151 -1.804]
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 233.9239 ¢
error map: ⟨0.000 -0.183 -0.487 -2.750]

Minimax tuning:

[[1 0 0 0⟩, [15/8 0 -1/8 0⟩, [0 0 1 0⟩, [65/24 0 1/24 0⟩]
unchanged-interval (eigenmonzo) basis: 2.5

Optimal ET sequence: 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: [⟨1 1 7 3 -2], ⟨0 3 -24 -1 28]]

Optimal tunings:

  • WE: ~2 = 1200.3453 ¢, ~8/7 = 233.9988 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 233.9312 ¢

Minimax tuning:

  • 11-odd-limit: ~8/7 = [7/24 0 -1/24⟩
[[1 0 0 0 0⟩, [15/8 0 -1/8 0 0⟩, [0 0 1 0 0⟩, [65/24 0 1/24 0 0⟩, [37/6 0 -7/6 0 0⟩]
unchanged-interval (eigenmonzo) basis: 2.5

Optimal ET sequence: 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: [⟨1 1 7 3 -2 0], ⟨0 3 -24 -1 28 19]]

Optimal tunings:

  • WE: ~2 = 1200.1222 ¢, ~8/7 = 233.9228 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 233.8994 ¢

Optimal ET sequence: 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 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: [⟨1 1 1 3], ⟨0 3 7 -1]]

Optimal tunings:

  • WE: ~2 = 1200.9847 ¢, ~8/7 = 228.5210 ¢
error map: ⟨+0.985 -15.407 +14.318 +5.607]
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 228.4371 ¢
error map: ⟨0.000 -16.644 +12.746 +2.737]

Optimal ET sequence: 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: [⟨1 1 1 3 1], ⟨0 3 7 -1 13]]

Optimal tunings:

  • WE: ~2 = 1201.3609 ¢, ~8/7 = 227.6312 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 227.4955 ¢

Optimal ET sequence: 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: [⟨1 1 1 3 1 2], ⟨0 3 7 -1 13 9]]

Optimal tunings:

  • WE: ~2 = 1201.0996 ¢, ~8/7 = 227.4378 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 227.3327 ¢

Optimal ET sequence: 5e, 16, 21, 37b

Badness (Sintel): 1.35

Spartan

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 1 1 3 5], ⟨0 3 7 -1 -8]]

Optimal tunings:

  • WE: ~2 = 1198.9344 ¢, ~8/7 = 229.3316 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 229.5124 ¢

Optimal ET sequence: 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: [⟨1 1 1 3 5 2], ⟨0 3 7 -1 -8 9]]

Optimal tunings:

  • WE: ~2 = 1198.3002 ¢, ~8/7 = 228.7341 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 229.0044 ¢

Optimal ET sequence: 5, 16e, 21

Badness (Sintel): 1.95

Music

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: [⟨1 1 2 3], ⟨0 3 2 -1]]

Optimal tunings:

  • WE: ~2 = 1192.4932 ¢, ~8/7 = 229.3187 ¢
error map: ⟨-7.507 -21.506 +57.310 -20.665]
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 229.6649 ¢
error map: ⟨0.000 -12.960 +73.016 +1.509]

Optimal ET sequence: 1b, 5

Badness (Sintel): 1.58

Oncle

For the 5-limit version, see Miscellaneous 5-limit temperaments #Oncle.

Oncle can be described as the 31 & 36c temperament.

Subgroup: 2.3.5.7

Comma list: 1029/1024, 2430/2401

Mapping: [⟨1 1 6 3], ⟨0 3 -19 -1]]

Optimal tunings:

  • WE: ~2 = 1201.2246 ¢, ~8/7 = 232.7354 ¢
error map: ⟨+1.225 -2.524 -0.939 +2.112]
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 232.4718 ¢
error map: ⟨0.000 -4.539 -3.279 -1.298]

Optimal ET sequence: 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 21 & 26 temperament.

Subgroup: 2.3.5.7

Comma list: 405/392, 1029/1024

Mapping: [⟨1 1 5 3], ⟨0 3 -14 -1]]

Optimal tunings:

  • WE: ~2 = 1202.7179 ¢, ~8/7 = 230.7800 ¢
error map: ⟨+2.718 -6.897 -3.644 +8.548]
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 230.1909 ¢
error map: ⟨0.000 -11.382 -8.986 +0.983]

Optimal ET sequence: 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 11 & 16 temperament.

Subgroup: 2.3.5.7

Comma list: 135/128, 360/343

Mapping: [⟨1 1 4 3], ⟨0 3 -9 -1]]

Optimal tunings:

  • WE: ~2 = 1205.6135 ¢, ~8/7 = 227.5283 ¢
error map: ⟨+5.613 -13.757 -11.614 +20.486]
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 226.3207 ¢
error map: ⟨0.000 -22.993 -23.200 +4.853]

Optimal ET sequence: 5c, 11, 16

Badness (Sintel): 4.03

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 1 4 3 4], ⟨0 3 -9 -1 -3]]

Optimal tunings:

  • WE: ~2 = 1206.2134 ¢, ~8/7 = 227.6004 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 226.2421 ¢

Optimal ET sequence: 5c, 11, 16

Badness (Sintel): 2.30

Miracle

Main article: 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 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: [⟨1 1 3 3], ⟨0 6 -7 -2]]

mapping generator: ~2, ~15/14

Optimal tunings:

  • WE: ~2 = 1200.8209 ¢, ~15/14 = 116.7550 ¢
error map: ⟨+0.821 -0.604 -1.136 +0.127]
  • CWE: ~2 = 1200.0000 ¢, ~15/14 = 116.6756 ¢
error map: ⟨0.000 -1.901 -3.043 -2.177]

Minimax tuning:

[[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⟩]
unchanged-interval (eigenmonzo) basis: 2.5/3
[[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⟩]
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 15x6 - 8x4 - 12

Optimal ET sequence: 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: [⟨1 1 3 3 2], ⟨0 6 -7 -2 15]]

Optimal tunings:

  • WE: ~2 = 1200.7626 ¢, ~15/14 = 116.7069 ¢
  • CWE: ~2 = 1200.0000 ¢, ~15/14 = 116.6469 ¢

Minimax tuning:

  • 11-odd-limit: ~15/14 = [1/19 2/19 -1/19⟩
[[1 0 0 0 0⟩, [25/19 12/19 -6/19 0 0⟩, [50/19 -14/19 7/19 0 0⟩, [55/19 -4/19 2/19 0 0⟩, [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

Optimal ET sequence: 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: [⟨1 1 3 3 2 4], ⟨0 6 -7 -2 15 -3]]

Optimal tunings:

  • WE: ~2 = 1200.1267 ¢, ~15/14 = 116.7596 ¢
  • CWE: ~2 = 1200.0000 ¢, ~15/14 = 116.7488 ¢

Optimal ET sequence: 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: [⟨1 1 3 3 2 4 4], ⟨0 6 -7 -2 15 -3 1]]

Optimal tunings:

  • WE: ~2 = 1199.6759 ¢, ~15/14 = 116.7378 ¢
  • CWE: ~2 = 1200.0000 ¢, ~15/14 = 116.7657 ¢

Optimal ET sequence: 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
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

Benediction

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 1 3 3 2 7], ⟨0 6 -7 -2 15 -34]]

Optimal tunings:

  • WE: ~2 = 1199.8601 ¢, ~15/14 = 116.6572 ¢
  • CWE: ~2 = 1200.0000 ¢, ~15/14 = 116.5688 ¢

Optimal ET sequence: 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: [⟨1 1 3 3 2 7 7], ⟨0 6 -7 -2 15 -34 -30]]

Optimal tunings:

  • WE: ~2 = 1200.8328 ¢, ~15/14 = 116.6661 ¢
  • CWE: ~2 = 1200.0000 ¢, ~15/14 = 116.5774 ¢

Optimal ET sequence: 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
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

Manna

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 1 3 3 2 0], ⟨0 6 -7 -2 15 38]]

Optimal tunings:

  • WE: ~2 = 1200.7564 ¢, ~15/14 = 116.8129 ¢
  • CWE: ~2 = 1200.0000 ¢, ~15/14 = 116.7528 ¢

Optimal ET sequence: 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: [⟨1 1 3 3 2 0 0], ⟨0 6 -7 -2 15 38 42]]

Optimal tunings:

  • WE: ~2 = 1200.7570 ¢, ~15/14 = 116.8011 ¢
  • CWE: ~2 = 1200.0000 ¢, ~15/14 = 116.7408 ¢

Optimal ET sequence: 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
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

Semimiracle

Subgroup: 2.3.5.7.11.13

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

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 ¢, ~15/14 = 116.7182 ¢
  • CWE: ~55/39 = 600.0000 ¢, ~15/14 = 116.6413 ¢

Optimal ET sequence: 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: [⟨2 2 6 6 4 7 7], ⟨0 6 -7 -2 15 2 6]]

Optimal tunings:

  • WE: ~17/12 = 600.5042 ¢, ~15/14 = 116.7264 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~15/14 = 116.6485 ¢

Optimal ET sequence: 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
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

Hemisecordite

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 243/242, 385/384, 847/845

Mapping: [⟨1 1 3 3 2 2], ⟨0 12 -14 -4 30 35]]

mapping generators: ~2, ~27/26

Optimal tunings:

  • WE: ~2 = 1200.6969 ¢, ~27/26 = 58.3217 ¢
  • CWE: ~2 = 1200.0000 ¢, ~27/26 = 58.2964 ¢

Optimal ET sequence: 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: [⟨1 1 3 3 2 2 2], ⟨0 12 -14 -4 30 35 43]]

Optimal tunings:

  • WE: ~2 = 1200.6557 ¢, ~27/26 = 58.2932 ¢
  • CWE: ~2 = 1200.0000 ¢, ~27/26 = 58.2702 ¢

Optimal ET sequence: 41, 62, 103

Badness (Sintel): 1.15

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list:

Todo: complete temperament data
23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list:

Todo: complete temperament data
Semihemisecordite

Subgroup: 2.3.5.7.11.13.17

Comma list: 225/224, 243/242, 289/288, 385/384, 847/845

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 ¢, ~27/26 = 58.3260 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~27/26 = 58.2974 ¢

Optimal ET sequence: 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: [⟨2 2 6 6 4 4 7 8], ⟨0 12 -14 -4 30 35 12 5]]

Optimal tunings:

  • WE: ~17/12 = 600.4418 ¢, ~27/26 = 58.3255 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~27/26 = 58.2928 ¢

Optimal ET sequence: 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: [⟨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 ¢, ~27/26 = 58.3264 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~27/26 = 58.2942 ¢

Optimal ET sequence: 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: [⟨1 -11 17 7 -28 3], ⟨0 18 -21 -6 45 1]]

mapping generators: ~2, ~13/8

Optimal tunings:

  • WE: ~2 = 1200.7056 ¢, ~13/8 = 839.3726 ¢
  • CWE: ~2 = 1200.0000 ¢, ~13/8 = 838.8831 ¢

Optimal ET sequence: 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: [⟨1 -11 17 7 -28 3 -5], ⟨0 18 -21 -6 45 1 13]]

Optimal tunings:

  • WE: ~2 = 1200.5918 ¢, ~13/8 = 839.2912 ¢
  • CWE: ~2 = 1200.0000 ¢, ~13/8 = 838.8809 ¢

Optimal ET sequence: 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
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

Revelation

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 1 3 3 5], ⟨0 6 -7 -2 -16]]

Optimal tunings:

  • WE: ~2 = 1201.3320 ¢, ~15/14 = 116.4057 ¢
  • CWE: ~2 = 1200.0000 ¢, ~15/14 = 116.2524 ¢

Optimal ET sequence: 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: [⟨1 1 3 3 5 4], ⟨0 6 -7 -2 -16 -3]]

Optimal tunings:

  • WE: ~2 = 1200.6059 ¢, ~15/14 = 116.3263 ¢
  • CWE: ~2 = 1200.0000 ¢, ~15/14 = 116.2564 ¢

Optimal ET sequence: 10e, 21, 31

Badness (Sintel): 1.22

Hemimiracle

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 1 3 3 4], ⟨0 12 -14 -4 -11]]

mapping generators: ~2, ~33/32

Optimal tunings:

  • WE: ~2 = 1200.2902 ¢, ~33/32 = 58.4217 ¢
  • CWE: ~2 = 1200.0000 ¢, ~33/32 = 58.4062 ¢

Optimal ET sequence: 20, 21, 41

Badness (Sintel): 1.96

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 1 3 3 4 4], ⟨0 12 -14 -4 -11 -6]]

Optimal tunings:

  • WE: ~2 = 1199.8454 ¢, ~33/32 = 58.4220 ¢
  • CWE: ~2 = 1200.0000 ¢, ~33/32 = 58.4305 ¢

Optimal ET sequence: 20, 21, 41

Badness (Sintel): 1.78

Oracle

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -5 10 5 4], ⟨0 12 -14 -4 -1]]

mapping generators: ~2, ~16/11

Optimal tunings:

  • WE: ~2 = 1201.2122 ¢, ~16/11 = 658.9974 ¢
  • CWE: ~2 = 1200.0000 ¢, ~16/11 = 658.3320 ¢

Optimal ET sequence: 11, 20, 31, 82e, 113e, 144ee

Badness (Sintel): 1.41

Hemiseven

Unlike miracle which splits 8/7, hemiseven splits ~16/7, an octave above. It can be described as the 72 & 77 temperament; its ploidacot is gamma-hexacot. 149edo is an obvious tuning.

Subgroup: 2.3.5.7

Comma list: 1029/1024, 19683/19600

Mapping: [⟨1 -2 -15 4], ⟨0 6 29 -2]]

mapping generators: ~2, ~243/160

Optimal tunings:

  • WE: ~2 = 1200.5612 ¢, ~243/160 = 717.0687 ¢
error map: ⟨+0.561 -0.665 +0.260 -0.718]
  • CWE: ~2 = 1200.0000 ¢, ~243/160 = 716.7478 ¢
error map: ⟨0.000 -1.468 -0.629 -2.321]

Optimal ET sequence: 72, 149, 221, 514bd, 735bcdd

Badness (Sintel): 1.43

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -2 -15 4 16], ⟨0 6 29 -2 -21]]

Optimal tunings:

  • WE: ~2 = 1200.6243 ¢, ~243/160 = 717.0969 ¢
  • CWE: ~2 = 1200.0000 ¢, ~243/160 = 716.7292 ¢

Optimal ET sequence: 72, 149, 221e, 293de

Badness (Sintel): 0.941

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -2 -15 4 16 -19], ⟨0 6 29 -2 -21 38]]

Optimal tunings:

  • WE: ~2 = 1200.6781 ¢, ~91/60 = 717.1496 ¢
  • CWE: ~2 = 1200.0000 ¢, ~91/60 = 716.7520 ¢

Optimal ET sequence: 72, 149, 221ef

Badness (Sintel): 0.905

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 273/272, 351/350, 385/384, 441/440, 676/675

Mapping: [⟨1 -2 -15 4 16 -19 -21], ⟨0 6 29 -2 -21 38 42]]

Optimal tunings:

  • WE: ~2 = 1200.6635 ¢, ~68/45 = 717.1354 ¢
  • CWE: ~2 = 1200.0000 ¢, ~68/45 = 716.7472 ¢

Optimal ET sequence: 72, 149, 221ef

Badness (Sintel): 0.800

Valentine

Main article: Valentine
For the 5-limit version, see Syntonic–31 equivalence continuum #Valentine (5-limit).

Valentine tempers out 126/125 and 6144/6125 as well as 1029/1024. It has a generator of ~21/20, three of which make the slendric generator ~8/7. 21/20 can be stripped of its 2 and taken as 3 × 7/5. In this respect it resembles miracle, with a generator of 3 × 5/7, and casablanca, with a generator of 5 × 7/3. These three generators are the simplest in terms of the relationship of tetrads in the lattice of 7-limit tetrads. Valentine can be described as the 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 (i.e. the 2.3.5.7.11.23 subgroup) of {(S8/S10 = S22 × S23 × S24, S11), S21, S22, S23, S24}, so it is the temperament that equalizes the 20::25 segment of the harmonic series.

Subgroup: 2.3.5.7

Comma list: 126/125, 1029/1024

Mapping: [⟨1 1 2 3], ⟨0 9 5 -3]]

mapping generators: ~2, ~21/20

Optimal tunings:

  • WE: ~2 = 1200.0749 ¢, ~21/20 = 77.8687 ¢
error map: ⟨+0.075 -1.062 +3.179 -2.207]
  • CWE: ~2 = 1200.0000 ¢, ~21/20 = 77.8673 ¢
error map: ⟨0.000 -1.149 +3.023 -2.428]

Minimax tuning:

[[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⟩]
unchanged-interval (eigenmonzo) basis: 2.7/3
[[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⟩]
unchanged-interval (eigenmonzo) basis: 2.9/7

Algebraic generator: smaller root of x2 - 89x + 92, or (89 - sqrt (7553))/2, at 77.8616 cents.

Optimal ET sequence: 15, 31, 46, 77, 185

Badness (Sintel): 0.786

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 1 2 3 3], ⟨0 9 5 -3 7]]

Optimal tunings:

  • WE: ~2 = 1200.3890 ¢, ~22/21 = 77.9065 ¢
  • CWE: ~2 = 1200.0000 ¢, ~22/21 = 77.9007 ¢

Minimax tuning:

  • 11-odd-limit: ~21/20 = [0 0 0 -1/10 1/10⟩
[[1 0 0 0 0⟩, [1 0 0 -9/10 9/10⟩, [2 0 0 -1/2 1/2⟩, [3 0 0 3/10 -3/10⟩, [3 0 0 -7/10 7/10⟩]
unchanged-interval (eigenmonzo) basis: 2.11/7

Algebraic generator: positive root of 4x3 + 15x2 - 21, or else Gontrand2, the smallest positive root of 4x7 - 8x6 + 5.

Optimal ET sequence: 15, 31, 46, 77

Badness (Sintel): 0.552

Valentino

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 1 2 3 3 5], ⟨0 9 5 -3 7 -20]]

Optimal tunings:

  • WE: ~2 = 1200.1967 ¢, ~22/21 = 77.9708 ¢
  • CWE: ~2 = 1200.0000 ¢, ~22/21 = 77.9594 ¢

Optimal ET sequence: 15f, 31, 46, 77

Badness (Sintel): 0.854

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨1 1 2 3 3 5 5], ⟨0 9 5 -3 7 -20 -14]]

Optimal tunings:

  • WE: ~2 = 1200.0404 ¢, ~22/21 = 78.0055 ¢
  • CWE: ~2 = 1200.0000 ¢, ~22/21 = 78.0029 ¢

Optimal ET sequence: 15f, 31, 46, 77, 123e

Badness (Sintel): 0.854

Lupercalia

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 1 2 3 3 3], ⟨0 9 5 -3 7 11]]

Optimal tunings:

  • WE: ~2 = 1199.9143 ¢, ~22/21 = 77.7039 ¢
  • CWE: ~2 = 1200.0000 ¢, ~22/21 = 77.7049 ¢

Optimal ET sequence: 15, 31

Badness (Sintel): 0.881

Dwynwen

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 1 2 3 3 2], ⟨0 9 5 -3 7 26]]

Optimal tunings:

  • WE: ~2 = 1200.1306 ¢, ~22/21 = 78.2273 ¢
  • CWE: ~2 = 1200.0000 ¢, ~22/21 = 78.2241 ¢

Optimal ET sequence: 15, 31f, 46

Badness (Sintel): 0.969

Semivalentine

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨2 2 4 6 6 7], ⟨0 9 5 -3 7 3]]

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

Optimal tunings:

  • WE: ~55/39 = 600.3497 ¢, ~22/21 = 77.8845 ¢
  • CWE: ~55/39 = 600.0000 ¢, ~22/21 = 77.8715 ¢

Optimal ET sequence: 16, 30, 46, 62, 108ef

Badness (Sintel): 1.35

Hemivalentine

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 1 2 3 3 4], ⟨0 18 10 -6 14 -9]]

mapping generators: ~2, ~40/39

Optimal tunings:

  • WE: ~2 = 1199.6529 ¢, ~40/39 = 39.0323 ¢
  • CWE: ~2 = 1200.0000 ¢, ~40/39 = 39.0383 ¢

Optimal ET sequence: 30, 31, 61, 92f

Badness (Sintel): 1.94

Demivalentine

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -8 -3 6 -4 -16], ⟨0 18 10 -6 14 37]]

mapping generators: ~2, ~13/9

Optimal tunings:

  • WE: ~2 = 1200.3929 ¢, ~13/9 = 639.1320 ¢
  • CWE: ~2 = 1200.0000 ¢, ~13/9 = 638.9325 ¢

Optimal ET sequence: 15, 47ef, 62, 77

Badness (Sintel): 1.44

Hemivalentino

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 1 2 3 2], ⟨0 18 10 -6 45]]

Optimal tunings:

  • WE: ~2 = 1200.0816 ¢, ~45/44 = 38.9236 ¢
  • CWE: ~2 = 1200.0000 ¢, ~45/44 = 38.9228 ¢

Optimal ET sequence: 31, 92e, 123, 154, 185

Badness (Sintel): 2.03

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 1 2 3 2 5], ⟨0 18 10 -6 45 -40]]

Optimal tunings:

  • WE: ~2 = 1199.8782 ¢, ~45/44 = 38.9440 ¢
  • CWE: ~2 = 1200.0000 ¢, ~45/44 = 38.9472 ¢

Optimal ET sequence: 31, 123, 154

Badness (Sintel): 2.39

Hemivalentoid

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 1 2 3 2 4], ⟨0 18 10 -6 45 -9]]

Optimal tunings:

  • WE: ~2 = 1199.3614 ¢, ~45/44 = 38.9721 ¢
  • CWE: ~2 = 1200.0000 ¢, ~45/44 = 38.9839 ¢

Optimal ET sequence: 31, 92ef

Badness (Sintel): 2.39

Superkleismic

Main article: Superkleismic
For the 5-limit version, see Miscellaneous 5-limit temperaments #Shibboleth.

Superkleismic tempers out the keema, 875/864, and can be described as the 15 & 26 temperament. It splits the ~7/4 into three ~6/5 generators of around 322 cents. This is noticeably sharper than the kleismic generator, hence the name.

In the 11-limit, two generator steps can be identified with ~16/11, and in the 13-limit, the same step can be treated as ~13/9. The S-expression-based comma list of 13-limit superkleismic is {S5/S6, S7/S8, S10, S12, (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 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 (S19) and 400/399 (S20). 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.

Subgroup: 2.3.5.7

Comma list: 875/864, 1029/1024

Mapping: [⟨1 -5 -5 5], ⟨0 9 10 -3]]

mapping generators: ~2, ~5/3

Optimal tunings:

  • WE: ~2 = 1200.7640 ¢, ~5/3 = 878.6289 ¢
error map: ⟨+0.764 +1.885 +3.844 -0.893]
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 878.1077 ¢
error map: ⟨0.000 +1.014 -5.237 -3.149]

Optimal ET sequence: 11c, 15, 26, 41

Badness (Sintel): 1.21

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -5 -5 5 2], ⟨0 9 10 -3 2]]

Optimal tunings:

  • WE: ~2 = 1200.1691 ¢, ~5/3 = 878.2772 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 878.1606 ¢

Optimal ET sequence: 11c, 15, 26, 41, 179cde, 220cde, 261ccdee

Badness (Sintel): 0.848

2.3.5.7.11.19 subgroup

Subgroup: 2.3.5.7.11.19

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

Mapping: [⟨1 -5 -5 5 2 -6], ⟨0 9 10 -3 2 14]]

Optimal tunings:

  • WE: ~2 = 1200.2289 ¢, ~5/3 = 878.3409 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 878.1840 ¢

Optimal ET sequence: 11c, 15, 26, 41, 138e

Badness (Sintel): 0.692

13-limit

Superkleismic in the 13-limit does considerably more damage than in the 11-limit, as indicated by being supported by much fewer patent vals and having higher Dirichlet badness than its 11-limit counterpart. However, this remains an obvious canonical mapping for prime 13.

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -5 -5 5 2 -8], ⟨0 9 10 -3 2 16]]

Optimal tunings:

  • WE: ~2 = 1200.0261 ¢, ~5/3 = 878.0252 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 878.0073 ¢

Optimal ET sequence: 11cf, 15, 26, 41

Badness (Sintel): 0.887

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 100/99, 105/104, 120/119, 144/143, 245/242

Mapping: [⟨1 -5 -5 5 2 -8 -12], ⟨0 9 10 -3 2 16 22]]

Optimal tunings:

  • WE: ~2 = 1200.0488 ¢, ~5/3 = 877.8872 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 877.8537 ¢

Optimal ET sequence: 11cfg, 15g, 26, 41

Badness (Sintel): 1.01

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 100/99, 105/104, 120/119, 144/143, 133/132, 190/189

Mapping: [⟨1 -5 -5 5 2 -8 -12 -6], ⟨0 9 10 -3 2 16 22 14]]

Optimal tunings:

  • WE: ~2 = 1200.2120 ¢, ~5/3 = 878.0243 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 877.8789 ¢

Optimal ET sequence: 11cfgh, 15g, 26, 41

Badness (Sintel): 0.964

Superana

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

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -5 -5 5 2 22], ⟨0 9 10 -3 2 -25]]

Optimal tunings:

  • WE: ~2 = 1199.8272 ¢, ~5/3 = 878.1538 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 878.2795 ¢

Optimal ET sequence: 15f, 41, 97, 138e

Badness (Sintel): 1.40

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨1 -5 -5 5 2 22 18], ⟨0 9 10 -3 2 -25 -19]]

Optimal tunings:

  • WE: ~2 = 1199.5964 ¢, ~5/3 = 878.0482 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 878.3444 ¢

Optimal ET sequence: 15f, 41, 56, 97g

Badness (Sintel): 1.45

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 100/99, 133/132, 154/153, 190/189, 196/195, 256/255

Mapping: [⟨1 -5 -5 5 2 22 18 -6], ⟨0 9 10 -3 2 -25 -19 14]]

Optimal tunings:

  • WE: ~2 = 1199.6638 ¢, ~5/3 = 878.1109 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 878.3566 ¢

Optimal ET sequence: 15f, 41, 56, 97g

Badness (Sintel): 1.36

Dee leap week

Main article: Dee leap week

Subgroup: 2.3.5.7

Comma list: 1029/1024, 2460375/2458624

Mapping: [⟨1 -5 25 5], ⟨0 9 -31 -3]]

Optimal tunings:

  • WE: ~2 = 1200.4835 ¢, ~224/135 = 878.2507 ¢
error map: ⟨+0.484 -0.117 +0.004 -1.160]
  • CWE: ~2 = 1200.0000 ¢, ~224/135 = 877.8926 ¢
error map: ⟨0.000 -0.921 -0.985 -2.504]

Optimal ET sequence: 41, 108, 149, 190

Badness (Sintel): 2.12

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -5 25 5 -28], ⟨0 9 -31 -3 43]]

Optimal tunings:

  • WE: ~2 = 1200.4874 ¢, ~224/135 = 878.2543 ¢
  • CWE: ~2 = 1200.0000 ¢, ~224/135 = 877.8987 ¢

Optimal ET sequence: 41, 108e, 149, 190

Badness (Sintel): 1.35

Unidec

Main article: Unidec

Unidec tempers out the ragisma, 4375/4374, and may be described as the 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.

Subgroup: 2.3.5.7

Comma list: 1029/1024, 4375/4374

Mapping: [⟨2 -1 -3 7], ⟨0 6 11 -2]]

Optimal tunings:

  • WE: ~1225/864 = 600.2429 ¢, ~80/63 = 417.0073 ¢
error map: ⟨+0.486 -0.154 +0.038 -1.140]
  • CWE: ~1225/864 = 600.0000 ¢, ~80/63 = 416.8688 ¢
error map: ⟨0.000 -0.924 -1.090 -2.503]

Minimax tuning:

[[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⟩]
unchanged-interval (eigenmonzo) basis: 2.7/5
[[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⟩]
unchanged-interval (eigenmonzo) basis: 2.9/7

Optimal ET sequence: 26, 46, 72, 118, 190

Badness (Sintel): 0.972

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨2 -1 -3 7 9], ⟨0 6 11 -2 -3]]

Optimal tunings:

  • WE: ~99/70 = 600.2497 ¢, ~14/11 = 417.0085 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~14/11 = 416.8543 ¢

Minimax tuning:

[[1 0 0 0 0⟩, [10/7 6/7 0 -3/7 0⟩, [57/28 11/7 0 -11/14 0⟩, [20/7 -2/7 0 1/7 0⟩, [99/28 -3/7 0 3/14 0⟩]
unchanged-interval (eigenmonzo) basis: 2.9/7

Optimal ET sequence: 26, 46, 72, 118, 190

Badness (Sintel): 0.512

Ekadash

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨2 -1 -3 7 9 -19], ⟨0 6 11 -2 -3 38]]

Optimal tunings:

  • WE: ~99/70 = 600.2497 ¢, ~14/11 = 417.0085 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~14/11 = 416.8543 ¢

Optimal ET sequence: 46f, 72, 118, 190, 262df, 452cdef

Badness (Sintel): 0.842

Hendec

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨2 -1 -3 7 9 6], ⟨0 6 11 -2 -3 2]]

Optimal tunings:

  • WE: ~91/64 = 600.3825 ¢, ~14/11 = 417.0678 ¢
  • CWE: ~91/64 = 600.0000 ¢, ~14/11 = 416.8290 ¢

Optimal ET sequence: 26, 46, 72, 190ff

Badness (Sintel): 0.732

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 169/168, 221/220, 273/272, 325/324, 364/363

Mapping: [⟨2 -1 -3 7 9 6 4], ⟨0 6 11 -2 -3 2 6]]

Optimal tunings:

  • WE: ~17/12 = 600.3991 ¢, ~14/11 = 417.0809 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~14/11 = 416.8330 ¢

Optimal ET sequence: 26, 46, 72, 190ffg

Badness (Sintel): 0.595

Necromanteion

Necromanteion, named by Johannes Werpup in 2014[2] may be described as the 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.

Subgroup: 2.3.5.7

Comma list: 1029/1024, 5103/5000

Mapping: [⟨1 -5 -7 5], ⟨0 12 17 -4]]

mapping generators: ~2, ~35/24

Optimal tunings:

  • WE: ~2 = 1200.2959 ¢, ~35/24 = 658.3833 ¢
error map: ⟨+0.296 -2.835 +4.130 -0.879]
  • CWE: ~2 = 1200.0000 ¢, ~35/24 = 658.2313 ¢
error map: ⟨0.000 -3.179 +3.619 -1.751]

Optimal ET sequence: 11c, 20c, 31, 144c, 175c

Badness (Sintel): 2.98

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -5 -7 5 -13], ⟨0 12 17 -4 30]]

Optimal tunings:

  • WE: ~2 = 1200.2862 ¢, ~22/15 = 658.4276 ¢
  • CWE: ~2 = 1200.0000 ¢, ~22/15 = 658.2805 ¢

Optimal ET sequence: 20ce, 31, 113c, 144c

Badness (Sintel): 1.77

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 144/143, 176/175, 243/242, 343/338

Mapping: [⟨1 -5 -7 5 -13 7], ⟨0 12 17 -4 30 -6]]

Optimal tunings:

  • WE: ~2 = 1199.3663 ¢, ~22/15 = 658.0465 ¢
  • CWE: ~2 = 1200.0000 ¢, ~22/15 = 658.3800 ¢

Optimal ET sequence: 20ce, 31, 82cf, 113cf

Badness (Sintel): 1.94

Restles

See also: Lesser tendoneutralic

Restles may be described as the 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[3].

Subgroup: 2.3.5.7

Comma list: 1029/1024, 153664/151875

Mapping: [⟨1 -2 8 4], ⟨0 12 -19 -4]]

mapping generators: ~2. ~315/256

Optimal tunings:

  • WE: ~2 = 1200.0322 ¢, ~315/256 = 358.5581 ¢
error map: ⟨+0.032 +0.678 +1.340 -2.930]
  • CWE: ~2 = 1200.0000 ¢, ~315/256 = 358.5484 ¢
error map: ⟨0.000 +0.626 +1.267 -3.019]

Optimal ET sequence: 77, 87, 164

Badness (Sintel): 2.73

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -2 8 4 -7], ⟨0 12 -19 -4 35]]

Optimal tunings:

  • WE: ~2 = 1200.1110 ¢, ~27/22 = 358.6045 ¢
  • CWE: ~2 = 1200.0000 ¢, ~27/22 = 358.5720 ¢

Optimal ET sequence: 77, 87, 164, 251d

Badness (Sintel): 1.81

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -2 8 4 -7 4], ⟨0 12 -19 -4 35 -1]]

Optimal tunings:

  • WE: ~2 = 1200.0482 ¢, ~~16/13 = 358.5883 ¢
  • CWE: ~2 = 1200.0000 ¢, ~16/13 = 358.5741 ¢

Optimal ET sequence: 77, 87, 164, 251d

Badness (Sintel): 1.16

Lagaca

Cryptically named by Petr Pařízek in 2011[3], lagaca may be described as the 10 & 118 temperament with a ploidacot signature of diploid wau-enneacot. The name actually refers to the fact that 12 generator steps in this temperament make ~7/3, where "l", "g", "c" are integers alphabetically converted to letters.

Subgroup: 2.3.5.7

Comma list: 1029/1024, 11529602/11390625

Mapping: [⟨2 -4 15 8], ⟨0 9 -13 -3]]

mapping generators: ~3375/2401, ~450/343

Optimal tunings:

  • WE: ~3375/2401 = 600.1355 ¢, ~450/343 = 478.0813 ¢
error map: ⟨+0.271 +0.235 +0.662 -1.986]
  • CWE: ~3375/2401 = 600.000 ¢, ~450/343 = 477.9725 ¢
error map: ⟨0.000 -0.202 +0.043 -2.743]

Optimal ET sequence: 10, 98, 108, 118

Badness (Sintel): 3.65

Quartemka

For the 5-limit version, see Miscellaneous 5-limit temperaments #Quartemka.

Quartemka may be described as the 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[3].

Subgroup: 2.3.5.7

Comma list: 1029/1024, 1250000/1240029

Mapping: [⟨1 -17 -26 9], ⟨0 21 32 -7]]

mapping generators: ~2, ~50/27

Optimal tunings:

  • WE: ~2 = 1200.5278 ¢, ~50/27 = 1062.4614 ¢
error map: ⟨+0.528 +0.762 -1.272 -1.305]
  • CWE: ~21 = 1200.0000 ¢, ~50/27 = 1062.0046 ¢
error map: ⟨0.000 +0.142 -2.167 -2.858]

Optimal ET sequence: 26, 61, 87, 113, 200

Badness (Sintel): 3.85

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -17 -26 9 7], ⟨0 21 32 -7 -4]]

Optimal tunings:

  • WE: ~2 = 1200.3051 ¢, ~50/27 = 1062.2805 ¢
  • CWE: ~21 = 1200.0000 ¢, ~50/27 = 1062.0147 ¢

Optimal ET sequence: 26, 61, 87, 200, 287d

Badness (Sintel): 1.89

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 364/363, 385/384, 2200/2197

Mapping: [⟨1 -17 -26 9 7 -14], ⟨0 21 32 -7 -4 20]]

Optimal tunings:

  • WE: ~2 = 1200.2708 ¢, ~24/13 = 1062.2496 ¢
  • CWE: ~21 = 1200.0000 ¢, ~24/13 = 1062.0139 ¢

Optimal ET sequence: 26, 61, 87, 200

Badness (Sintel): 1.17

Tritriple

For the 5-limit version, see Miscellaneous 5-limit temperaments #Tritriple.

Tritriple may be described as the 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 reduced give slendric's generator, and another ×3 give the perfect fifth[3].

Subgroup: 2.3.5.7

Comma list: 1029/1024, 1959552/1953125

Mapping: [⟨1 -11 -7 7], ⟨0 27 20 -9]]

mapping generators: ~2, ~864/625

Optimal tunings:

  • WE: ~2 = 1200.4239 ¢, ~864/625 = 559.4921 ¢
error map: ⟨+0.424 -0.331 +0.561 -1.287]
  • CWE: ~2 = 1200.0000 ¢, ~864/625 = 559.3015 ¢
error map: ⟨0.000 -0.815 -0.284 -2.539]

Optimal ET sequence: 15, …, 88, 103, 118, 221, 339d

Badness (Sintel): 3.00

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -11 -7 7 -4], ⟨0 27 20 -9 16]]

Optimal tunings:

  • WE: ~2 = 1200.4953 ¢, ~242/175 = 559.5243 ¢
  • CWE: ~2 = 1200.0000 ¢, ~242/175 = 559.3016 ¢

Optimal ET sequence: 15, …, 88, 103, 118, 221e, 339de

Badness (Sintel): 1.17

Widefourth

Subgroup: 2.3.5.7

Comma list: 1029/1024, 48828125/48771072

Mapping: [⟨1 -17 -5 9], ⟨0 33 13 -11]]

Optimal tunings:

  • WE: ~2 = 1200.4770 ¢, ~4608/3125 = 676.0584 ¢
error map: ⟨+0.477 -0.137 +0.061 -1.175]
  • CWE: ~2 = 1200.0000 ¢, ~4608/3125 = 675.7954 ¢
error map: ⟨0.000 -0.705 -0.973 -2.576]

Optimal ET sequence: 16, 71, 87, 103, 190

Badness (Sintel): 3.90

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 441/440, 234375/234256

Mapping: [⟨1 16 8 -2 17], ⟨0 -33 -13 11 -31]]

Optimal tunings:

  • WE: ~2 = 1200.4852 ¢, ~1250/847 = 676.0634 ¢
  • CWE: ~2 = 1200.0000 ¢, ~1250/847 = 675.7966 ¢

Optimal ET sequence: 16, 71, 87, 103, 190

Badness (Sintel): 1.35

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 385/384, 441/440, 625/624, 847/845

Mapping: [⟨1 16 8 -2 17 12], ⟨0 -33 -13 11 -31 -19]]

Optimal tunings:

  • WE: ~2 = 1200.4217 ¢, ~77/52 = 676.0286 ¢
  • CWE: ~2 = 1200.0000 ¢, ~77/52 = 675.7967 ¢

Optimal ET sequence: 16, 71, 87, 103, 190

Badness (Sintel): 0.894

Other subgroup extensions

Euslendric (2.3.7.13)

Forms of slendric in the most optimal range for the 2.3.7 temperament (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-expressions 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: [⟨1 1 3 0], ⟨0 3 -1 19]]

Gencom mapping: [⟨1 1 0 3 0 0], ⟨0 3 0 -1 0 19]]

Optimal tunings:

  • WE: ~2 = 1200.5057 ¢, ~8/7 = 233.7200 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 233.6534 ¢

Optimal ET sequence: 5, 31f, 36, 77, 113, 827bdddff

Badness (Sintel): 0.339

2.3.7.13.17 subgroup

Subgroup: 2.3.7.13.17

Comma list: 273/272, 729/728, 833/832

Subgroup-val mapping: [⟨1 1 3 0 0], ⟨0 3 -1 19 21]]

Gencom mapping: [⟨1 1 0 3 0 0 0], ⟨0 3 0 -1 0 19 21]]

Optimal tunings:

  • WE: ~2 = 1200.5282 ¢, ~8/7 = 233.6492 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 233.5776 ¢

Optimal ET sequence: 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: [⟨1 1 3 0 0 6], ⟨0 3 -1 19 21 -9]]

Gencom mapping: [⟨1 1 0 3 0 0 0 6], ⟨0 3 0 -1 0 19 21 -9]]

Optimal tunings:

  • WE: ~2 = 1200.3292 ¢, ~8/7 = 233.6651 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 233.6106 ¢

Optimal ET sequence: 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: [⟨1 1 3 0 0 6 9], ⟨0 3 -1 19 21 -9 -23]]

Gencom 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 ¢, ~8/7 = 233.6679 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 233.6091 ¢

Optimal ET sequence: 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: [⟨1 1 3 0 0 6 9 7], ⟨0 3 -1 19 21 -9 -23 -11]]

Gencom 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 ¢, ~8/7 = 233.6688 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 233.6208 ¢

Optimal ET sequence: 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 (S13), 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: [⟨2 2 6 7], ⟨0 3 -1 1]]

Gencom 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 ¢, ~8/7 = 233.7724 ¢
  • CWE: ~91/64 = 600.0000 ¢, ~8/7 = 233.7039 ¢

Optimal ET sequence: 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: [⟨2 2 6 7 7], ⟨0 3 -1 1 3]]

Gencom mapping: [⟨2 2 0 6 0 7 7], ⟨0 3 0 -1 0 1 3]]

Optimal tunings:

  • WE: ~17/12 = 600.4436 ¢, ~8/7 = 233.7883 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~8/7 = 233.7312 ¢

Optimal ET sequence: 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: [⟨1 -2 4 7], ⟨0 6 -2 -1]]

Optimal tunings:

  • WE: ~2 = 1200.8295 ¢, ~128/85 = 717.2597 ¢
  • CWE: ~2 = 1200.0000 ¢, ~128/85 = 716.7933 ¢

Optimal ET sequence: 5, 42*, 47, 52, 57, 62, 67, 72, 149*, 370d***, 519bdd*****

* Wart for 85

References

  1. ↑ 31-EDO Music Theory: Supermajor Hexatonic Scale by Zhea Erose
  2. ↑ Yahoo! Tuning Group | Temperament ideas: A cuckoo, and two oracles
  3. ↑ 3.0 3.1 3.2 3.3 Yahoo! Tuning Group | Suggested names for the unclasified temperaments
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