Xenharmonic Wiki

Gamelismic clan: Difference between revisions

← Older edit
VisualWikitext
Inthar (talk | contribs)
autopatrolled, emailconfirmed
15,664 edits
Move gigapyth here
 
(244 intermediate revisions by 20 users not shown)
Line 1: Line 1:
The 2.3.7 [[Just_intonation_subgroups|subgroup]] comma for the gamelismic clan is the gamelisma, 1029/1024, with monzo |-10 1 0 3>. For any member of the clan, for the rank three [[Gamelismic_family#Gamelan|gamelan temperament]] itself, and for the rank two 2.3.7 temperament [[Slendric|slendric]], this means three 8/7 intervals give a fifth, 3/2. In fact, we find that 3/2 = (8/7)^3 * 1029/1024. From this it follows that gamelismic temperaments tend to flatten the both fifth and the 7/4, or if they do not, the other of the pair must be flattened even more. [[36edo]] is a good tuning for gamelismic itself, though if the full 7-limit is desired, [[72edo]], [[77edo]] or [[118edo]] might be preferred.
{{Technical data page}}
The [[2.3.7 subgroup|2.3.7-subgroup]] [[comma]] for the '''gamelismic clan''' is the gamelisma, [[1029/1024]], with [[monzo]] {{monzo| -10 1 0 3 }}. For any member of the clan, for the rank-3 [[gamelismic family #Gamelismic|gamelismic temperament]] itself, and for the rank-2 2.3.7 temperament [[slendric]] (a.k.a. gamelic), this means three [[~]][[8/7]] intervals give a fifth, [[3/2]]. In fact, we find that {{nowrap| 3/2 {{=}} (8/7)<sup>3</sup>⋅(1029/1024) }}. From this it follows that gamelismic temperaments tend to flatten both the fifth and the harmonic seventh, or if they do not, the other of the pair must be flattened even more. [[36edo]] is a good tuning for slendric, though if the full 7-limit is desired, [[72edo]], [[77edo]], or [[118edo]] might be preferred.


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


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


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


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


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


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


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


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


Map to lattice: [&lt;0 3 0 -1|, &lt;0 0 1 0|]
=== Overview to extensions ===
==== Full 7-limit extensions ====
To the gamelisma itself we need to add the comma which appears next on the modified [[Normal lists #Normal interval list|normal comma list]] for the full 7-limit. The second comma on the list for mothra is [[81/80]], for rodan [[245/243]], for guiron [[32805/32768]], for gorgo [[36/35]], and for gidorah [[256/245]]. These all use ~8/7 as a generator, though in the case of gidorah that is the same as ~6/5.


Map: [&lt;1 1 0 3|, &lt;0 3 0 -1|, &lt; 0 0 1 0|]
Miracle adds [[33075/32768]] and uses the [[secor]], half an ~8/7, as generator. Lemba adds [[525/512]] to the list, and has a half-octave [[period]]. Valentine adds [[6144/6125]] with a generator of ~21/20 and superkleismic adds [[875/864]] with a generator of ~6/5. Unidec adds [[4375/4374]], and has a generator of ~10/9 with a half-octave period. Hemithirds adds [[65625/65536]] with a generator half of a classical major third. Finally, tritikleismic adds [[15625/15552]] and has a generator of 6/5 with a 1/3-octave period.


Generators: 2, 5, 7
Full 7-limit temperaments discussed elsewhere are:
* [[Blackwood]] (+28/27) → [[Limmic temperaments #Blackwood|Limmic temperaments]]
* [[Lemba]] (+50/49) → [[Jubilismic clan #Lemba|Jubilismic clan]]
* [[Trisected]] (+128/125) → [[Augmented family #Trisected|Augmented family]]
* ''[[Echidnic]]'' (+686/675) → [[Diaschismic family #Echidnic|Diaschismic family]]
* [[Trismegistus]] (+3125/3072) → [[Magic family #Trismegistus|Magic family]]
* [[Hemithirds]] (+3136/3125) → [[Hemimean clan #Hemithirds|Hemimean clan]]
* ''[[Gamity]]'' (+1071875/1062882) → [[Amity family #Gamity|Amity family]]
* ''[[Tritikleismic]]'' (+15625/15552) → [[Kleismic family #Tritikleismic|Kleismic family]]
* ''[[Heinz]]'' (+78732/78125) → [[Sensipent family #Heinz|Sensipent family]]
* ''[[Triwell]]'' (+235298/234375) → [[Semicomma family #Triwell|Semicomma family]]
* ''[[Gamelstearn]]'' (+118098/117649) → [[Compton family #Gamelstearn|Compton family]]


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


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


==Full seven limit children==
=== Radon ===
To the gamelisma itself we need to add the comma which appears next on the modified [[Normal_lists|normal comma list]], which is often a 5-limit comma. The second comma on the list for mothra is 81/80, for rodan 245/243, for guiron 32805/32768, for gorgo 36/35, and for gidorah 256/245. These all use 8/7 as a generator, though in the case of gidorah that's the same as 6/5. Miracle adds 33075/32768 and uses the secor, half an 8/7, as generator. Lemba adds 525/512 to the list, and has a half-octave period. Valentine adds 6144/6125 with a generator of 21/20 and superkleismic adds 875/864 with a generator of 6/5. Unidec adds 4375/4374, and has a generator of 10/9 with a half-octave period. Hemithirds adds 65625/65536 with a generator half of a major third. Finally, tritikleismic adds 15625/15536 and has a generator of 6/5 with a 1/3 octave period.
{{See also|Chromatic pairs #Radon}}


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


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


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


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


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


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


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


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


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


Wedgie: &lt;&lt;6 -7 -2 -25 -20 15||
Mothra tempers out [[81/80]] and finds the prime 5 at a stack of four fifths as does any temperament in the [[meantone family]]. It also tempers out [[1728/1715]], the orwellisma. It can be described as the {{nowrap| 26 & 31 }}. Using [[31edo]] with a generator of 6/31 is an excellent tuning choice. However, a pure mos mothra scale is often described as directionless and has limited chord-building potential<ref>[https://www.youtube.com/watch?v=uH3ahBzDSrs 31-EDO Music Theory: Supermajor Hexatonic Scale] by [[Zhea Erose]]</ref>, so something other than a mos may be used as a scale to get the most out of mothra. There are examples of non-mos mothra scales in 31edo [[Strictly proper 7-tone 31edo scales|in the article on strictly proper 7-tone 31edo scales]].


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


Badness: 0.0167
Its [[S-expression]]-based comma list is {[[1728/1715|S6/S7]], [[1029/1024|S7/S8]], ([[81/80|S6/S8 = S9]])}, taking advantage of the fact that [[81/80]] is a [[semiparticular]].


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


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


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


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


[[POTE_tuning|POTE generator]]: ~15/14 = 116.633
[[Algebraic generator]]: Rabrindanath, largest real root of ''x''<sup>8</sup> - 3''x''<sup>2</sup> + 1, or 232.0774 cents.
 
[[Minimax tuning]]:
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~8/7 = {{monzo| 0 0 1/12 }}
: {{monzo list| 1 0 0 0 | 1 0 1/4 0 | 0 0 1 0 | 3 0 -1/12 0 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5
 
{{Optimal ET sequence|legend=1| 5, 21c, 26, 31 }}
 
[[Badness]] (Sintel): 0.940
 
=== Undecimal mothra ===
Undecimal mothra is the extension of 7-limit cynder which tempers out 385/384 as is natural in slendric temperaments. It is the simplest extension, supported within a reasonable tuning range (between [[26edo]] and 31edo), and is supported by the patent val of [[5edo]], which implies that it is better behaved as a cluster temperament. It is also notable for being supported by the just tuning of 8/7, and has a restriction to the 2.7.11 subgroup, namely [[amaranthine]], that is a microtemperament.
 
Subgroup: 2.3.5.7.11
 
Comma list: 81/80, 99/98, 385/384
 
Mapping: {{mapping| 1 1 0 3 5 | 0 3 12 -1 -8 }}
 
Optimal tunings:
* WE: ~2 = 1201.3979{{c}}, ~8/7 = 232.3010{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 232.0621{{c}}
 
{{Optimal ET sequence|legend=0| 5, 26, 31, 88, 119be, 150be }}
 
Badness (Sintel): 0.848
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 81/80, 99/98, 105/104, 144/143
 
Mapping: {{mapping| 1 1 0 3 5 1 | 0 3 12 -1 -8 14 }}
 
Optimal tunings:
* WE: ~2 = 1201.0985{{c}}, ~8/7 = 232.0231{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 231.8425{{c}}
 
{{Optimal ET sequence|legend=0| 5, 26, 31, 57, 88 }}
 
Badness (Sintel): 0.990
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 81/80, 99/98, 105/104, 120/119, 144/143
 
Mapping: {{mapping| 1 1 0 3 5 1 | 0 3 12 -1 -8 14 16 }}
 
Optimal tunings:
* WE: ~2 = 1200.9734{{c}}, ~8/7 = 231.8960{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 231.7392{{c}}
 
{{Optimal ET sequence|legend=0| 5g, 26, 31, 57, 88 }}
 
Badness (Sintel): 1.00
 
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 81/80, 99/98, 105/104, 120/119, 144/143, 153/152
 
Mapping: {{mapping| 1 1 0 3 5 1 | 0 3 12 -1 -8 14 16 22 }}
 
Optimal tunings:
* WE: ~2 = 1200.9663{{c}}, ~8/7 = 231.8393{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 231.6842{{c}}
 
{{Optimal ET sequence|legend=0| 26, 31, 57 }}
 
Badness (Sintel): 1.05
 
=== Mosura ===
The [[S-expression]]-based comma list of mosura suggests it might be the most natural extension of 7-limit cynder to the 11-limit: {[[1728/1715|S6/S7]], [[1029/1024|S7/S8]], ([[81/80|S6/S8 = S9]]), [[176/175|S8/S10]]}.
 
Subgroup: 2.3.5.7.11
 
Comma list: 81/80, 176/175, 540/539
 
Mapping: {{mapping| 1 1 0 3 -1 | 0 3 12 -1 23 }}
 
Optimal tunings:
* WE: ~2 = 1200.7675{{c}}, ~8/7 = 232.5673{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 232.4567{{c}}
 
{{Optimal ET sequence|legend=0| 5e, 26e, 31, 129 }}
 
Badness (Sintel): 1.04
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 81/80, 144/143, 176/175, 196/195
 
Mapping: {{mapping| 1 1 0 3 -1 7 | 0 3 12 -1 23 -17 }}
 
Optimal tunings:
* WE: ~2 = 1199.9347{{c}}, ~8/7 = 232.6275{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 232.6392{{c}}
 
{{Optimal ET sequence|legend=0| 31, 67, 98 }}
 
Badness (Sintel): 1.52
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 81/80, 144/143, 176/175, 189/187, 196/195
 
Mapping: {{mapping| 1 1 0 3 -1 7 | 0 3 12 -1 23 -17 -15 }}
 
Optimal tunings:
* WE: ~2 = 1199.7124{{c}}, ~8/7 = 232.6376{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 232.6917{{c}}
 
{{Optimal ET sequence|legend=0| 31, 67, 98 }}
 
Badness (Sintel): 1.53
 
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 81/80, 96/95, 144/143, 153/152, 176/175, 196/195
 
Mapping: {{mapping| 1 1 0 3 -1 7 | 0 3 12 -1 23 -17 -15 -9 }}
 
Optimal tunings:
* WE: ~2 = 1199.4885{{c}}, ~8/7 = 232.6310{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 232.7287{{c}}
 
{{Optimal ET sequence|legend=0| 31, 67, 98h }}
 
Badness (Sintel): 1.50
 
=== Cyndra ===
Subgroup: 2.3.5.7.11
 
Comma list: 45/44, 81/80, 1029/1024
 
Mapping: {{mapping| 1 1 0 3 0 | 0 3 12 -1 18 }}
 
Optimal tunings:
* WE: ~2 = 1201.1585{{c}}, ~8/7 = 231.5404{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 231.3850{{c}}
 
{{Optimal ET sequence|legend=0| 5e, 21ce, 26 }}
 
Badness (Sintel): 1.84
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 45/44, 78/77, 81/80, 640/637
 
Mapping: {{mapping| 1 1 0 3 0 1 | 0 3 12 -1 18 14 }}
 
Optimal tunings:
* WE: ~2 = 1201.1152{{c}}, ~8/7 = 231.5079{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 231.3612{{c}}
 
{{Optimal ET sequence|legend=0| 5e, 21cef, 26 }}
 
Badness (Sintel): 1.41
 
== Rodan ==
{{Main| Rodan }}
: ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum #Rodan (5-limit)]].''
 
Rodan tempers out 245/243 and can be described as the {{nowrap| 41 & 46 }} temperament. This temperament is more accurate than mothra and extends neatly to the 13-limit, though the perfect fifth is sharper than ideal for slendric. [[87edo]] is excellent for this, with the 17\87 generator missing the 13-limit CWE tuning by less than a millicent.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 245/243, 1029/1024
 
{{Mapping|legend=1| 1 1 -1 3 | 0 3 17 -1 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.2146{{c}}, ~8/7 = 234.4587{{c}}
: [[error map]]: {{val| +0.215 +1.636 -0.731 -2.641 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 234.4259{{c}}
: error map: {{val| 0.000 +1.323 -1.073 -3.252 }}
 
[[Minimax tuning]]:
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~8/7 = {{monzo| 2/9 0 1/18 -1/18 }}
: {{monzo list| 1 0 0 0 | 5/3 0 1/6 -1/6 | 25/9 0 17/18 -17/18 | 25/9 0 -1/18 1/18 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5
 
[[Algebraic generator]]: larger root of 20''x''<sup>2</sup> - 36''x'' + 15, or (9 + √6)/10.
 
{{Optimal ET sequence|legend=1| 41, 87, 128, 215d }}
 
[[Badness]] (Sintel): 0.939
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 245/243, 385/384, 441/440
 
Mapping: {{mapping| 1 1 -1 3 6 | 0 3 17 -1 -13 }}
 
Optimal tunings:
* WE: ~2 = 1200.0553{{c}}, ~8/7 = 234.4695{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.4594{{c}}
 
Minimax tuning:
* 11-odd-limit: ~8/7 = {{monzo| 4/19 2/19 0 0 -1/19 }}
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 31/19 6/19 0 0 -3/19 }}, {{monzo| 49/19 34/19 0 0 -17/19 }}, {{monzo| 53/19 -2/19 0 0 1/19 }}, {{monzo| 62/19 -26/19 0 0 13/19 }}]
: unchanged-interval (eigenmonzo) basis: 2.11/9
 
Algebraic generator: positive root of ''x''<sup>2</sup> + 16''x'' - 31, or √95 - 8.
 
{{Optimal ET sequence|legend=0| 41, 87 }}
 
Badness (Sintel): 0.763
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 196/195, 245/243, 352/351, 364/363
 
Mapping: {{mapping| 1 1 -1 3 6 8 | 0 3 17 -1 -13 -22 }}
 
Optimal tunings:
* WE: ~2 = 1199.9868{{c}}, ~8/7 = 234.4796{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.4822{{c}}
 
Minimax tuning:
* 13- and 15-odd-limit: ~8/7 = {{monzo| 3/14 1/14 0 0 0 -1/28 }}
: unchanged-interval (eigenmonzo) basis: 2.13/9
 
Algebraic generator: Gatetone, positive root of 4''x''<sup>6</sup> - 7''x'' - 1. Recurrence converges slowly.
 
{{Optimal ET sequence|legend=0| 41, 46, 87 }}
 
Badness (Sintel): 0.762
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 154/153, 196/195, 245/243, 256/255, 273/272
 
Mapping: {{mapping| 1 1 -1 3 6 8 8 | 0 3 17 -1 -13 -22 -20 }}
 
Optimal tunings:
* WE: ~2 = 1199.8331{{c}}, ~8/7 = 234.4919{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.5254{{c}}
 
Minimax tuning:
* 17-odd-limit: ~8/7 = {{monzo| 3/13 1/13 0 0 0 0 -1/26 }}
: unchanged-interval (eigenmonzo) basis: 2.17/9
 
{{Optimal ET sequence|legend=0| 41, 46, 87 }}
 
Badness (Sintel): 0.853
 
==== Aerodactyl ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 91/90, 245/243, 385/384, 441/440
 
Mapping: {{mapping| 1 1 -1 3 6 -1 | 0 3 17 -1 -13 24 }}
 
Optimal tunings:
* WE: ~2 = 1200.2997{{c}}, ~8/7 = 234.6972{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.6439{{c}}
 
{{Optimal ET sequence|legend=0| 5, 41f, 46 }}
 
Badness (Sintel): 1.40
 
=== Aerodino ===
Subgroup: 2.3.5.7.11
 
Comma list: 176/175, 245/243, 1029/1024
 
Mapping: {{mapping| 1 1 -1 3 -3 | 0 3 17 -1 33 }}
 
Optimal tunings:
* WE: ~2 = 1199.9179{{c}}, ~8/7 = 234.7123{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.7256{{c}}
 
{{Optimal ET sequence|legend=0| 5e, 41e, 46 }}
 
Badness (Sintel): 1.79
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 91/90, 176/175, 245/243, 847/845
 
Mapping: {{mapping| 1 1 -1 3 -3 -1 | 0 3 17 -1 33 24 }}
 
Optimal tunings:
* WE: ~2 = 1200.0242{{c}}, ~8/7 = 234.7863{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.7824{{c}}
 
{{Optimal ET sequence|legend=0| 5e, 41ef, 46 }}
 
Badness (Sintel): 1.48
 
=== Varan ===
Subgroup: 2.3.5.7.11
 
Comma list: 100/99, 245/243, 1029/1024
 
Mapping: {{mapping| 1 1 -1 3 -2 | 0 3 17 -1 28 }}
 
Optimal tunings:
* WE: ~2 = 1200.3738{{c}}, ~8/7 = 234.2174{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.1586{{c}}
 
{{Optimal ET sequence|legend=0| 5e, 36ce, 41 }}
 
Badness (Sintel): 1.49
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 100/99, 105/104, 245/243, 352/351
 
Mapping: {{mapping| 1 1 -1 3 -2 0 | 0 3 17 -1 28 19 }}
 
Optimal tunings:
* WE: ~2 = 1200.1389{{c}}, ~8/7 = 234.1162{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.0946{{c}}
 
{{Optimal ET sequence|legend=0| 5e, 36ce, 41 }}
 
Badness (Sintel): 1.33
 
== Guiron ==
Guiron tempers out the [[schisma]], and finds the prime 5 at the diminished fourth as does any temperament in the [[schismatic family]]. It can be described as the {{nowrap| 36 & 41 }} temperament. It is more complex than rodan, but the optimal tuning is closer to optimal slendric.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 1029/1024, 10976/10935
 
{{Mapping|legend=1| 1 1 7 3 | 0 3 -24 -1 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.3395{{c}}, ~8/7 = 233.9963{{c}}
: [[error map]]: {{val| +0.340 +0.374 +0.151 -1.804 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 233.9239{{c}}
: error map: {{val| 0.000 -0.183 -0.487 -2.750 }}
 
[[Minimax tuning]]:
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~8/7 = {{monzo| 7/24 0 -1/24 }}
: {{monzo list| 1 0 0 0 | 15/8 0 -1/8 0 | 0 0 1 0 | 65/24 0 1/24 0 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5
 
{{Optimal ET sequence|legend=1| 36, 41, 77, 118, 277d }}
 
[[Badness]] (Sintel): 1.20
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 385/384, 441/440, 10976/10935
 
Mapping: {{mapping| 1 1 7 3 -2 | 0 3 -24 -1 28 }}
 
Optimal tunings:
* WE: ~2 = 1200.3453{{c}}, ~8/7 = 233.9988{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.9312{{c}}
 
Minimax tuning:
* 11-odd-limit: ~8/7 = {{monzo| 7/24 0 -1/24 }}
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 15/8 0 -1/8 0 0 }}, {{monzo| 0 0 1 0 0 }}, {{monzo| 65/24 0 1/24 0 0 }}, {{monzo| 37/6 0 -7/6 0 0 }}]
: unchanged-interval (eigenmonzo) basis: 2.5
 
{{Optimal ET sequence|legend=0| 36e, 41, 77, 118, 159, 277d }}
 
Badness (Sintel): 0.881
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 196/195, 352/351, 385/384, 729/728
 
Mapping: {{mapping| 1 1 7 3 -2 0 | 0 3 -24 -1 28 19 }}
 
Optimal tunings:
* WE: ~2 = 1200.1222{{c}}, ~8/7 = 233.9228{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.8994{{c}}
 
{{Optimal ET sequence|legend=0| 36e, 41, 77, 118 }}
 
Badness (Sintel): 1.18
 
== Gorgo ==
: ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum #Laconic]].''
{{See also| Llywelynsmic clan }}
 
Gorgo tempers the generator of ~8/7 together with ~10/9. It can be described as the {{nowrap| 16 & 21 }} temperament.
 
If we discard the inaccurate mapping of prime 3, we get [[shoe]], so that the large commas of gorgo are explained practically entirely by the inaccurate 3.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 36/35, 1029/1024
 
{{Mapping|legend=1| 1 1 1 3 | 0 3 7 -1 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.9847{{c}}, ~8/7 = 228.5210{{c}}
: [[error map]]: {{val| +0.985 -15.407 +14.318 +5.607 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 228.4371{{c}}
: error map: {{val| 0.000 -16.644 +12.746 +2.737 }}
 
{{Optimal ET sequence|legend=1| 5, 11c, 16, 21 }}
 
[[Badness]] (Sintel): 1.54
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 36/35, 45/44, 1029/1024
 
Mapping: {{mapping| 1 1 1 3 1 | 0 3 7 -1 13 }}
 
Optimal tunings:
* WE: ~2 = 1201.3609{{c}}, ~8/7 = 227.6312{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 227.4955{{c}}
 
{{Optimal ET sequence|legend=0| 5e, 16, 21, 37b }}
 
Badness (Sintel): 1.64
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 27/26, 36/35, 45/44, 507/500
 
Mapping: {{mapping| 1 1 1 3 1 2 | 0 3 7 -1 13 9 }}
 
Optimal tunings:
* WE: ~2 = 1201.0996{{c}}, ~8/7 = 227.4378{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 227.3327{{c}}
 
{{Optimal ET sequence|legend=0| 5e, 16, 21, 37b }}
 
Badness (Sintel): 1.35
 
=== Spartan ===
Subgroup: 2.3.5.7.11
 
Comma list: 36/35, 56/55, 1029/1024
 
Mapping: {{mapping| 1 1 1 3 5 | 0 3 7 -1 -8 }}
 
Optimal tunings:
* WE: ~2 = 1198.9344{{c}}, ~8/7 = 229.3316{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 229.5124{{c}}
 
{{Optimal ET sequence|legend=0| 5, 16e, 21 }}
 
Badness (Sintel): 2.07
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 27/26, 36/35, 56/55, 507/500
 
Mapping: {{mapping| 1 1 1 3 5 2 | 0 3 7 -1 -8 9 }}
 
Optimal tunings:
* WE: ~2 = 1198.3002{{c}}, ~8/7 = 228.7341{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 229.0044{{c}}
 
{{Optimal ET sequence|legend=0| 5, 16e, 21 }}
 
Badness (Sintel): 1.95
 
; Music
* [https://web.archive.org/web/20201127012514/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Herman/gorgo-example.mp3 ''Gorgo Example''] by [[Herman Miller]]
 
== Gidorah ==
: ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum #University]].''
 
Gidorah is a very low-accuracy temperament where the generator of ~8/7 is lumped together with ~6/5. 16c-, 21cc-, and 26ccc-edo are among the possible tunings.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 21/20, 144/125
 
{{Mapping|legend=1| 1 1 2 3 | 0 3 2 -1 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1192.4932{{c}}, ~8/7 = 229.3187{{c}}
: [[error map]]: {{val| -7.507 -21.506 +57.310 -20.665 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 229.6649{{c}}
: error map: {{val| 0.000 -12.960 +73.016 +1.509 }}
 
{{Optimal ET sequence|legend=1| 1b, 5 }}
 
[[Badness]] (Sintel): 1.58
 
== Oncle ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Oncle]].''
 
Oncle can be described as the {{nowrap| 31 & 36c }} temperament.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 1029/1024, 2430/2401
 
{{Mapping|legend=1| 1 1 6 3 | 0 3 -19 -1 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.2246{{c}}, ~8/7 = 232.7354{{c}}
: [[error map]]: {{val| +1.225 -2.524 -0.939 +2.112 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 232.4718{{c}}
: error map: {{val| 0.000 -4.539 -3.279 -1.298 }}
 
{{Optimal ET sequence|legend=1| 31, 98c, 129c, 160bc }}
 
[[Badness]] (Sintel): 2.24
 
== Archaeotherium ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Archaeotherium]].''
 
Archaeotherium can be described as the {{nowrap| 21 & 26 }} temperament.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 405/392, 1029/1024
 
{{Mapping|legend=1| 1 1 5 3 | 0 3 -14 -1 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1202.7179{{c}}, ~8/7 = 230.7800{{c}}
: [[error map]]: {{val| +2.718 -6.897 -3.644 +8.548 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 230.1909{{c}}
: error map: {{val| 0.000 -11.382 -8.986 +0.983 }}
 
{{Optimal ET sequence|legend=1| 21, 26, 47, 73bc }}
 
[[Badness]] (Sintel): 3.70
 
== Clyndro ==
Clyndro tempers out [[135/128]] and finds the interval class of 5 at a stack of -3 fifths as does any temperament in the [[mavila family]]. It can be described as the {{nowrap| 11 & 16 }} temperament.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 135/128, 360/343
 
{{Mapping|legend=1| 1 1 4 3 | 0 3 -9 -1 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1205.6135{{c}}, ~8/7 = 227.5283{{c}}
: [[error map]]: {{val| +5.613 -13.757 -11.614 +20.486 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 226.3207{{c}}
: error map: {{val| 0.000 -22.993 -23.200 +4.853 }}
 
{{Optimal ET sequence|legend=1| 5c, 11, 16 }}
 
[[Badness]] (Sintel): 4.03
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 33/32, 45/44, 352/343
 
Mapping: {{mapping| 1 1 4 3 4 | 0 3 -9 -1 -3 }}
 
Optimal tunings:
* WE: ~2 = 1206.2134{{c}}, ~8/7 = 227.6004{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 226.2421{{c}}
 
{{Optimal ET sequence|legend=0| 5c, 11, 16 }}
 
Badness (Sintel): 2.30
 
== Miracle ==
{{Main| Miracle }}
: ''For the 5-limit version, see [[Syntonic–31 equivalence continuum #Ampersand]].''
 
Miracle is one of the most important entries of this temperament clan. It tempers out [[225/224]], splitting the ~8/7 generator of slendric into 15/14~16/15, and can be described as the {{nowrap| 31 & 41 }} temperament. Its ploidacot is hexacot. It is then extremely natural to equate the neutral third, three generators up, to [[11/9]] and thereby extend miracle to the full [[11-limit]] with essentially no further damage. [[72edo]] makes for an excellent tuning.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 225/224, 1029/1024
 
{{Mapping|legend=1| 1 1 3 3 | 0 6 -7 -2 }}
: mapping generator: ~2, ~15/14
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.8209{{c}}, ~15/14 = 116.7550{{c}}
: [[error map]]: {{val| +0.821 -0.604 -1.136 +0.127 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~15/14 = 116.6756{{c}}
: error map: {{val| 0.000 -1.901 -3.043 -2.177 }}
 
[[Minimax tuning]]:
* [[7-odd-limit]]: ~15/14 = {{monzo| 2/13 1/13 -1/13 }}
: {{monzo list| 1 0 0 0 | 25/13 6/13 -6/13 0 | 25/13 -7/13 7/13 0 | 35/13 -2/13 2/13 0 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5/3
* [[9-odd-limit]]: ~15/14 = {{monzo| 1/19 2/19 -1/19 }}
: {{monzo list| 1 0 0 0 | 25/19 12/19 -6/19 0 | 50/19 -14/19 7/19 0 | 55/19 -4/19 2/19 0 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5
 
[[Tuning ranges]]:
* 7-odd-limit [[diamond monotone]]: ~15/14 = [114.286, 120.000] (2\21 to 1\10)
* 9-odd-limit diamond monotone: ~15/14 = [116.129, 120.000] (3\31 to 1\10)
* 7- and 9-odd-limit [[diamond tradeoff]]: ~15/14 = [115.587, 116.993]
 
[[Algebraic generator]]: Secor59, positive root of 15''x''<sup>6</sup> - 8''x''<sup>4</sup> - 12
 
{{Optimal ET sequence|legend=1| 10, 21, 31, 41, 72 }}
 
[[Badness]] (Sintel): 0.424
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 243/242, 385/384
 
Mapping: {{mapping| 1 1 3 3 2 | 0 6 -7 -2 15 }}
 
Optimal tunings:
* WE: ~2 = 1200.7626{{c}}, ~15/14 = 116.7069{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.6469{{c}}
 
Minimax tuning:
* 11-odd-limit: ~15/14 = {{monzo| 1/19 2/19 -1/19 }}
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 25/19 12/19 -6/19 0 0 }}, {{monzo| 50/19 -14/19 7/19 0 0 }}, {{monzo| 55/19 -4/19 2/19 0 0 }}, {{monzo| 53/19 30/19 -15/19 0 0 }}]
: unchanged-interval (eigenmonzo) basis: 2.9/5
 
Tuning ranges:
* 11-odd-limit diamond monotone: ~15/14 = [116.129, 117.073] (3\31 to 4\41)
* 11-odd-limit diamond tradeoff: ~15/14 = [115.587, 116.993]


Algebraic generator: Secor59
Algebraic generator: Secor59


Map: [&lt;1 1 3 3 2|, &lt;0 6 -7 -2 15|]
{{Optimal ET sequence|legend=0| 10, 21e, 31, 41, 72, 247c, 319bcde, 391bcde, 463bccde }}
 
Badness (Sintel): 0.353
 
==== Miraculous ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 105/104, 144/143, 196/195, 243/242
 
Mapping: {{mapping| 1 1 3 3 2 4 | 0 6 -7 -2 15 -3 }}
 
Optimal tunings:
* WE: ~2 = 1200.1267{{c}}, ~15/14 = 116.7596{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.7488{{c}}
 
{{Optimal ET sequence|legend=0| 10, 21e, 31, 41, 72f }}
 
Badness (Sintel): 0.771
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 105/104, 120/119, 144/143, 154/153, 170/169
 
Mapping: {{mapping| 1 1 3 3 2 4 4 | 0 6 -7 -2 15 -3 1 }}
 
Optimal tunings:
* WE: ~2 = 1199.6759{{c}}, ~15/14 = 116.7378{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.7657{{c}}
 
{{Optimal ET sequence|legend=0| 10, 21e, 31, 41, 72fg }}
 
Badness (Sintel): 0.870
 
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 105/104, 120/119, 144/143, 154/153, 170/169, 210/209
 
{{Todo|complete temperament data|inline=1}}
 
===== 23-limit =====
Subgroup: 2.3.5.7.11.13.17.19.23
 
Comma list: 105/104, 120/119, 144/143, 154/153, 161/160, 170/169, 210/209
 
{{Todo|complete temperament data|inline=1}}
 
==== Benediction ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 225/224, 243/242, 351/350, 385/384
 
Mapping: {{mapping| 1 1 3 3 2 7 | 0 6 -7 -2 15 -34 }}
 
Optimal tunings:
* WE: ~2 = 1199.8601{{c}}, ~15/14 = 116.6572{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.5688{{c}}
 
{{Optimal ET sequence|legend=0| 31, 72, 103, 175f }}
 
Badness (Sintel): 0.649
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 225/224, 243/242, 273/272, 351/350, 375/374
 
Mapping: {{mapping| 1 1 3 3 2 7 7 | 0 6 -7 -2 15 -34 -30 }}
 
Optimal tunings:
* WE: ~2 = 1200.8328{{c}}, ~15/14 = 116.6661{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.5774{{c}}
 
{{Optimal ET sequence|legend=0| 31, 72, 103, 175f, 422bcdefffg }}
 
Badness (Sintel): 0.639
 
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 210/209, 225/224, 243/242, 273/272, 286/285, 375/374
 
{{Todo|complete temperament data|inline=1}}
 
===== 23-limit =====
Subgroup: 2.3.5.7.11.13.17.19.23
 
Comma list: 162/161, 210/209, 225/224, 231/230, 243/242, 273/272, 286/285
 
{{Todo|complete temperament data|inline=1}}
 
==== Manna ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 225/224, 243/242, 325/324, 385/384
 
Mapping: {{mapping| 1 1 3 3 2 0 | 0 6 -7 -2 15 38 }}
 
Optimal tunings:
* WE: ~2 = 1200.7564{{c}}, ~15/14 = 116.8129{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.7528{{c}}
 
{{Optimal ET sequence|legend=0| 31f, 41, 72, 185cf, 257cff }}
 
Badness (Sintel): 0.703
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 225/224, 243/242, 273/272, 325/324, 385/384
 
Mapping: {{mapping| 1 1 3 3 2 0 0 | 0 6 -7 -2 15 38 42 }}
 
Optimal tunings:
* WE: ~2 = 1200.7570{{c}}, ~15/14 = 116.8011{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.7408{{c}}
 
{{Optimal ET sequence|legend=0| 31fg, 41, 72, 185cf, 257cff }}
 
Badness (Sintel): 0.748
 
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 210/209, 225/224, 243/242, 273/272, 325/324, 343/342
 
{{Todo|complete temperament data|inline=1}}
 
===== 23-limit =====
Subgroup: 2.3.5.7.11.13.17.19.23
 
Comma list: 210/209, 225/224, 243/242, 273/272, 300/299, 325/324, 343/342
 
{{Todo|complete temperament data|inline=1}}
 
==== Semimiracle ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 169/168, 225/224, 243/242, 385/384
 
Mapping: {{mapping| 2 2 6 6 4 7 | 0 6 -7 -2 15 2 }}
: mapping generators: ~55/39, ~15/14
 
Optimal tunings:
* WE: ~55/39 = 600.4844{{c}}, ~15/14 = 116.7182{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~15/14 = 116.6413{{c}}
 
{{Optimal ET sequence|legend=0| 10, 62, 72 }}
 
Badness (Sintel): 1.02
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 169/168, 221/220, 225/224, 243/242, 273/272
 
Mapping: {{mapping| 2 2 6 6 4 7 7 | 0 6 -7 -2 15 2 6 }}
 
Optimal tunings:
* WE: ~17/12 = 600.5042{{c}}, ~15/14 = 116.7264{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~15/14 = 116.6485{{c}}
 
{{Optimal ET sequence|legend=0| 10, 62, 72 }}
 
Badness (Sintel): 0.822
 
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 169/168, 210/209, 221/220, 225/224, 243/242, 273/272
 
{{Todo|complete temperament data|inline=1}}
 
===== 23-limit =====
Subgroup: 2.3.5.7.11.13.17.19.23
 
Comma list: 169/168, 208/207, 210/209, 221/220, 225/224, 243/242, 273/272
 
{{Todo|complete temperament data|inline=1}}
 
==== Hemisecordite ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 225/224, 243/242, 385/384, 847/845
 
Mapping: {{mapping| 1 1 3 3 2 2 | 0 12 -14 -4 30 35 }}
: mapping generators: ~2, ~27/26
 
Optimal tunings:
* WE: ~2 = 1200.6969{{c}}, ~27/26 = 58.3217{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~27/26 = 58.2964{{c}}
 
{{Optimal ET sequence|legend=0| 41, 62, 103, 247c, 350bcde }}
 
Badness (Sintel): 1.06
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 225/224, 243/242, 273/272, 385/384, 847/845
 
Mapping: {{mapping| 1 1 3 3 2 2 2 | 0 12 -14 -4 30 35 43 }}
 
Optimal tunings:
* WE: ~2 = 1200.6557{{c}}, ~27/26 = 58.2932{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~27/26 = 58.2702{{c}}
 
{{Optimal ET sequence|legend=0| 41, 62, 103 }}
 
Badness (Sintel): 1.15
 
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


[[EDO|EDOs]]: 10, 31, 41, [[72edo|72]], 247c, 319bcde, [[391edo|391bcde]], 463bccde
Comma list:  


Badness: 0.0107
{{Todo|complete temperament data|inline=1}}


[[Chords_of_miracle|Chords of miracle]]
===== 23-limit =====
Subgroup: 2.3.5.7.11.13.17.19.23


==Revelation==
Comma list:  
Commas: 99/98, 176/175, 1029/1024


POTE generator: ~15/14 = 116.277
{{Todo|complete temperament data|inline=1}}


Map: [&lt;1 1 3 3 5|, &lt;0 6 -7 -2 -16|]
===== Semihemisecordite =====
Subgroup: 2.3.5.7.11.13.17


EDOs: 10e, 21, 31
Comma list: 225/224, 243/242, 289/288, 385/384, 847/845


Badness: 0.0329
Mapping: {{mapping| 2 2 6 6 4 4 7 | 0 12 -14 -4 30 35 12 }}
: mapping generators: ~17/12, ~27/26


===13-limit===
Optimal tunings:
Commas: 66/65, 99/98, 105/104, 512/507
* WE: ~17/12 = 600.3951{{c}}, ~27/26 = 58.3260{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~27/26 = 58.2974{{c}}


POTE generator: ~15/14 = 116.268
{{Optimal ET sequence|legend=0| 62, 144g, 206begg }}


Map: [&lt;1 1 3 3 5 4|, &lt;0 6 -7 -2 -16 -3|]
Badness (Sintel): 2.39


EDOs: 10e, 21, 31
====== 19-limit ======
Subgroup: 2.3.5.7.11.13.17.19


Badness: 0.0295
Comma list: 209/208, 225/224, 243/242, 289/288, 361/360, 385/384


==Miraculous==
Mapping: {{mapping| 2 2 6 6 4 4 7 8 | 0 12 -14 -4 30 35 12 5 }}
Commas: 105/104, 144/143, 196/195, 243/242


POTE generator: ~15/14 = 116.747
Optimal tunings:  
* WE: ~17/12 = 600.4418{{c}}, ~27/26 = 58.3255{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~27/26 = 58.2928{{c}}


Map: [&lt;1 1 3 3 2 4|, &lt;0 6 -7 -2 15 -3|]
{{Optimal ET sequence|legend=0| 62, 144gh, 206begghh }}


EDOs: 10, 31, 41, 72f, 113f, 185cff
Badness (Sintel): 2.13


Badness: 0.0187
====== 23-limit ======
Subgroup: 2.3.5.7.11.13.17.19.23


==Benediction==
Comma list: 209/208, 225/224, 243/242, 289/288, 323/322, 361/360, 385/384
Commas: 225/224, 243/242, 351/350, 385 /384


POTE generator: ~15/14 = 116.574
Mapping: {{mapping| 2 2 6 6 4 4 7 8 7 | 0 12 -14 -4 30 35 12 5 21 }}


Map: [&lt;1 1 3 3 2 7|, &lt;0 6 -7 -2 15 -34|]
Optimal tunings:  
* WE: ~17/12 = 600.4451{{c}}, ~27/26 = 58.3264{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~27/26 = 58.2942{{c}}


EDOs: 31, 72, 103, [[175edo|175f]]
{{Optimal ET sequence|legend=0| 62, 144gh, 206begghhi }}


Badness: 0.0157
Badness (Sintel): 1.89


===17-limit===
==== Phicordial ====
Commas: 225/224, 243/242, 273/272, 351/350, 375/374
Subgroup: 2.3.5.7.11.13


POTE generator: ~15/14 = 116.585
Comma list: 225/224, 243/242, 385/384, 2200/2197


Map: [&lt;1 1 3 3 2 7 7|, &lt;0 6 -7 -2 15 -34 -30|]
Mapping: {{mapping| 1 -11 17 7 -28 3 | 0 18 -21 -6 45 1 }}
: mapping generators: ~2, ~13/8


EDOs: 31, 72, 103, 175f
Optimal tunings:  
* WE: ~2 = 1200.7056{{c}}, ~13/8 = 839.3726{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/8 = 838.8831{{c}}


==Manna==
{{Optimal ET sequence|legend=0| 103, 216c, 319bcde, 535bccdef }}
Commas: 225/224, 243/242, 325/324, 385/384


POTE generator: ~15/14 = 116.739
Badness (Sintel): 1.37


Map: [&lt;1 1 3 3 2 0|, &lt;0 6 -7 -2 15 38|]
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


EDOs: 10f, 31f, 41, 72, 113, 185cf, 257cff
Comma list: 225/224, 243/242, 273/272, 385/384, 2200/2197


Badness: 0.0170
Mapping: {{mapping| 1 -11 17 7 -28 3 -5 | 0 18 -21 -6 45 1 13 }}


==Semimiracle==
Optimal tunings:
Commas: 169/168, 225/224, 243/242, 385/384
* WE: ~2 = 1200.5918{{c}}, ~13/8 = 839.2912{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/8 = 838.8809{{c}}


POTE generator: ~15/14 = 116.624
{{Optimal ET sequence|legend=0| 103, 216c, 319bcde }}


Map: [&lt;2 2 6 6 4 7|, &lt;0 6 -7 -2 15 2|]
Badness (Sintel): 1.26


EDOs: 10, 62, 72
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


Badness: 0.0246
Comma list: 210/209, 225/224, 243/242, 273/272, 385/384, 2200/2197


===17-limit===
{{Todo|complete temperament data|inline=1}}
Commas: 169/168, 221/220, 225/224, 243/242, 273/272


POTE generator: ~15/14 = 116.628
===== 23-limit =====
Subgroup: 2.3.5.7.11.13.17.19.23


Map: [&lt;2 2 6 6 4 7 7|, &lt;0 6 -7 -2 15 2 6|]
Comma list: 210/209, 225/224, 243/242, 273/272, 300/299, 385/384, 1105/1104


EDOs: 10, 62, 72
{{Todo|complete temperament data|inline=1}}


Badness: 0.0161
=== Revelation ===
Subgroup: 2.3.5.7.11


==Hemimiracle==
Comma list: 99/98, 176/175, 1029/1024
Commas: 225/224, 245/242, 1029/1024


POTE generator: ~33/32 = 58.408
Mapping: {{mapping| 1 1 3 3 5 | 0 6 -7 -2 -16 }}


Map: [&lt;1 1 3 3 4|, &lt;0 12 -14 -4 -11|]
Optimal tunings:  
* WE: ~2 = 1201.3320{{c}}, ~15/14 = 116.4057{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.2524{{c}}


EDOs: 20, 21, 41, 144e, 185cee, 226cee
{{Optimal ET sequence|legend=0| 10e, 21, 31 }}


Badness: 0.0592
Badness (Sintel): 1.09


===13-limit===
==== 13-limit ====
Commas: 105/104, 196/195, 245/242, 512/507
Subgroup: 2.3.5.7.11.13


POTE generator: ~33/32 = 58.430
Comma list: 66/65, 99/98, 105/104, 512/507


Map: [&lt;1 1 3 3 4 4|, &lt;0 12 -14 -4 -11 -6|]
Mapping: {{mapping| 1 1 3 3 5 4 | 0 6 -7 -2 -16 -3 }}


EDOs: 20, 21, 41, 144eff, 185ceeff
Optimal tunings:  
* WE: ~2 = 1200.6059{{c}}, ~15/14 = 116.3263{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.2564{{c}}


Badness: 0.0432
{{Optimal ET sequence|legend=0| 10e, 21, 31 }}


==Oracle==
Badness (Sintel): 1.22
Commas: 121/120, 225/224, 1029/1024


POTE generator: ~11/8 = 541.668
=== Hemimiracle ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 7 -4 1 3|, &lt;0 -12 14 4 1|]
Comma list: 225/224, 245/242, 1029/1024


EDOs: 11, 20, 31, 51, 82e, 113e, 144ee
Mapping: {{mapping| 1 1 3 3 4 | 0 12 -14 -4 -11 }}
: mapping generators: ~2, ~33/32


Badness: 0.0427
Optimal tunings:  
* WE: ~2 = 1200.2902{{c}}, ~33/32 = 58.4217{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~33/32 = 58.4062{{c}}


==Hemisecordite==
{{Optimal ET sequence|legend=0| 20, 21, 41 }}
Commas: 225/224, 243/242, 385/384, 847/845


POTE generator: ~27/26 = 58.288
Badness (Sintel): 1.96


Map: [&lt;1 1 3 3 2 2|, &lt;0 12 -14 -4 30 35|]
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


EDOs: 41, 62, 103, 247c, 350bcde
Comma list: 105/104, 196/195, 245/242, 512/507


Badness: 0.0256
Mapping: {{mapping| 1 1 3 3 4 4 | 0 12 -14 -4 -11 -6 }}


===17-limit===
Optimal tunings:
Commas: 225/224, 243/242, 273/272, 385/384, 847/845
* WE: ~2 = 1199.8454{{c}}, ~33/32 = 58.4220{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~33/32 = 58.4305{{c}}


POTE generator: ~27/26 = 58.261
{{Optimal ET sequence|legend=0| 20, 21, 41 }}


Map: [&lt;1 1 3 3 2 2 2|, &lt;0 12 -14 -4 30 35 43|]
Badness (Sintel): 1.78


EDOs: 41, 62, 103
=== Oracle ===
Subgroup: 2.3.5.7.11


Badness: 0.0225
Comma list: 121/120, 225/224, 1029/1024


==Phicordial==
Mapping: {{mapping| 1 -5 10 5 4 | 0 12 -14 -4 -1 }}
Commas: 225/224, 243/242, 385/384, 2200/2197
: mapping generators: ~2, ~16/11


POTE generator: ~16/13 = 361.121
Optimal tunings:  
* WE: ~2 = 1201.2122{{c}}, ~16/11 = 658.9974{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/11 = 658.3320{{c}}


Map: [&lt;1 7 -4 1 17 4|, &lt;0 -18 21 6 -45 -1|]
{{Optimal ET sequence|legend=0| 11, 20, 31, 82e, 113e, 144ee }}


EDOs: 10, 103, 113, 216c
Badness (Sintel): 1.41


Badness: 0.0332
== 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.  


===17-limit===
[[Subgroup]]: 2.3.5.7
Commas: 225/224, 243/242, 273/272, 441/440, 2200/2197


POTE generator: ~16/13 = 361.123
[[Comma list]]: 1029/1024, 19683/19600


Map: [&lt;1 7 -4 1 17 4 8|, &lt;0 -18 21 6 -45 -1 -13|]
{{Mapping|legend=1| 1 -2 -15 4 | 0 6 29 -2 }}
: mapping generators: ~2, ~243/160


EDOs: 10, 103, 113, 216c
[[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 }}


Badness: 0.0247
{{Optimal ET sequence|legend=1| 72, 149, 221, 514bd, 735bcdd }}


==Music==
[[Badness]] (Sintel): 1.43
By [[Gene_Ward_Smith|Gene Ward Smith]]


[http://www.archive.org/details/RachmaninoffPlaysBlackjack Rachmaninoff Plays Blackjack] [http://www.archive.org/download/RachmaninoffPlaysBlackjack/rachman.mp3 play]
=== 11-limit ===
Subgroup: 2.3.5.7.11


By [[Joseph_Pehrson|Joseph Pehrson]]
Comma list: 385/384, 441/440, 19683/19600


[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Pehrson/blackandjill.mp3 Black and Jill]
Mapping: {{mapping| 1 -2 -15 4 16 | 0 6 29 -2 -21 }}


[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Pehrson/josephpehrson+blacklight.mp3 Blacklight]
Optimal tunings:  
* WE: ~2 = 1200.6243{{c}}, ~243/160 = 717.0969{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~243/160 = 716.7292{{c}}


[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Pehrson/josephpehrson+blackjack.mp3 Blackjack]
{{Optimal ET sequence|legend=0| 72, 149, 221e, 293de }}


=Rodan=
Badness (Sintel): 0.941
{{main|Rodan}}
[[Comma|Commas]]: 245/243, 1029/1024


7+9 limit minimax tuning: [|1 0 0 0&gt;, |5/3 0 1/6 -1/6&gt;,
=== 13-limit ===
|25/9 0 17/18 -17/18&gt;, |25/9 0 -1/18 1/18&gt;<nowiki>]</nowiki>
Subgroup: 2.3.5.7.11.13


[[Eigenmonzo|Eigenmonzos]]: 2, 7/5
Comma list: 351/350, 385/384, 441/440, 676/675


[[POTE_tuning|POTE generator]]: ~8/7 = 234.417
Mapping: {{mapping| 1 -2 -15 4 16 -19 | 0 6 29 -2 -21 38 }}


Algebraic generator: [[Algebraic_number|larger root]] of 20x^2-36x+15, or (9+√6)/10.
Optimal tunings:  
* WE: ~2 = 1200.6781{{c}}, ~91/60 = 717.1496{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~91/60 = 716.7520{{c}}


Map: [&lt;1 1 -1 3|, &lt;0 3 17 -1|]
{{Optimal ET sequence|legend=0| 72, 149, 221ef }}


EDOs: 5, 41, 87, 128, 215d
Badness (Sintel): 0.905


Badness: 0.0371
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


==11-limit==
Comma list: 273/272, 351/350, 385/384, 441/440, 676/675
[[Comma|Commas]]: 245/243, 385/384, 441/440


[[Minimax_tuning|Minimax tuning]]: [|1 0 0 0 0&gt;, |31/19 6/19 0 0 -3/19&gt;,
Mapping: {{mapping| 1 -2 -15 4 16 -19 -21 | 0 6 29 -2 -21 38 42 }}
|49/19 34/19 0 0 -17/19&gt;, |53/19 -2/19 0 0 1/19&gt;,
|62/19 -26/19 0 0 13/19&gt;<nowiki>]</nowiki>


[[Eigenmonzo|Eigenmonzos]]: 2, 11/9
Optimal tunings:  
* WE: ~2 = 1200.6635{{c}}, ~68/45 = 717.1354{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~68/45 = 716.7472{{c}}


[[POTE_tuning|POTE generator]]: ~8/7 = 234.459
{{Optimal ET sequence|legend=0| 72, 149, 221ef }}


Algebraic generator: [[Algebraic_number|positive root]] of x^2+16x-31, or √95-8.
Badness (Sintel): 0.800


Map: [&lt;1 1 -1 3 6|, &lt;0 3 17 -1 -13|]
== Valentine ==
{{Main| Valentine }}
: ''For the 5-limit version, see [[Syntonic–31 equivalence continuum #Valentine (5-limit)]].''


EDOs: 5, 41, 46, 87
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: 0.0231
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.


[[Chords_of_rodan|Chords of rodan]]
[[Subgroup]]: 2.3.5.7


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


13 and 15 limit minimax
{{Mapping|legend=1| 1 1 2 3 | 0 9 5 -3 }}
: mapping generators: ~2, ~21/20


[|1 0 0 0 0 0&gt;, |23/14 3/14 0 0 0 -3/28&gt;,
[[Optimal tuning]]s:
|37/14 17/14 0 0 0 -17/28&gt;, |39/14 -1/14 0 0 0 1/28&gt;,
* [[WE]]: ~2 = 1200.0749{{c}}, ~21/20 = 77.8687{{c}}
|45/14 -13/14 0 0 0 13/28&gt;, |23/7 -11/7 0 0 0 11/14&gt;<nowiki>]</nowiki>
: [[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 }}


Eigenmonzos: 2, 13/9
[[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


[[POTE_tuning|POTE generator]]: ~8/7 = 234.482
[[Algebraic generator]]: smaller root of ''x''<sup>2</sup> - 89''x'' + 92, or (89 - sqrt (7553))/2, at 77.8616 cents.  


Algebraic generator: Gatetone, positive root of 4x^6-7x-1. Recurrence converges slowly.
{{Optimal ET sequence|legend=1| 15, 31, 46, 77, 185 }}


Map: [&lt;1 1 -1 3 6 8|, &lt;0 3 17 -1 -13 -22|]
[[Badness]] (Sintel): 0.786


Generators: 2, 8/7
=== 11-limit ===
Subgroup: 2.3.5.7.11


EDOs: 41, 46, [[87edo|87]]
Comma list: 121/120, 126/125, 176/175


Badness: 0.0184
Mapping: {{mapping| 1 1 2 3 3 | 0 9 5 -3 7 }}


==17-limit==
Optimal tunings:
Commas: 154/153, 196/195, 245/243, 256/255, 273/272
* WE: ~2 = 1200.3890{{c}}, ~22/21 = 77.9065{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/21 = 77.9007{{c}}


Eigenmonzos: 2, 18/17
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


[[POTE_tuning|POTE generator]]: ~8/7 = 234.524
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.


Map: [&lt;1 1 -1 3 6 8 8|, &lt;0 3 17 -1 -13 -22 -20|]
{{Optimal ET sequence|legend=0| 15, 31, 46, 77 }}


Edos: 41, 46, 87, 220dg, 307dgg
Badness (Sintel): 0.552


Badness: 0.0167
==== Valentino ====
Subgroup: 2.3.5.7.11.13


===Music===
Comma list: 121/120, 126/125, 176/175, 196/195
By [[Gene_Ward_Smith|Gene Ward Smith]]


[http://www.archive.org/details/Pianodactyl Pianodactyl] [http://www.archive.org/download/Pianodactyl/pianodactyl.mp3 play]
Mapping: {{mapping| 1 1 2 3 3 5 | 0 9 5 -3 7 -20 }}


==Aerodactyl==
Optimal tunings:
Commas: 91/90, 245/243, 385/384, 441/440
* WE: ~2 = 1200.1967{{c}}, ~22/21 = 77.9708{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/21 = 77.9594{{c}}


[[POTE_tuning|POTE generator]]: ~8/7 = 234.639
{{Optimal ET sequence|legend=0| 15f, 31, 46, 77 }}


Map: [&lt;1 1 -1 3 6 -1|, &lt;0 3 17 -1 -13 24|
Badness (Sintel): 0.854


EDOs: 5, 41f, 46, 51c
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


Badness: 0.0340
Comma list: 121/120, 126/125, 154/153, 176/175, 196/195


==Aerodino==
Mapping: {{mapping| 1 1 2 3 3 5 5 | 0 9 5 -3 7 -20 -14 }}
Commas: 176/175, 245/243, 1029/1024


POTE generator: ~8/7 = 234.728
Optimal tunings:  
* WE: ~2 = 1200.0404{{c}}, ~22/21 = 78.0055{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/21 = 78.0029{{c}}


Map: [&lt;1 1 -1 3 -3|, &lt;0 3 17 -1 33|
{{Optimal ET sequence|legend=0| 15f, 31, 46, 77, 123e }}


EDOS: 5e, 41e, 46
Badness (Sintel): 0.854


Badness: 0.0543
==== Lupercalia ====
Subgroup: 2.3.5.7.11.13


===13-limit===
Comma list: 66/65, 105/104, 121/120, 126/125
Commas: 91/90, 176/175, 245/243, 847/845


POTE generator: ~8/7 = 234.782
Mapping: {{mapping| 1 1 2 3 3 3 | 0 9 5 -3 7 11 }}


Map: [&lt;1 1 -1 3 -3 -1|, &lt;0 3 17 -1 33 24|
Optimal tunings:  
* WE: ~2 = 1199.9143{{c}}, ~22/21 = 77.7039{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/21 = 77.7049{{c}}


EDOs: 5e, 46
{{Optimal ET sequence|legend=0| 15, 31 }}


Badness: 0.0358
Badness (Sintel): 0.881


==Varan==
==== Dwynwen ====
Commas: 100/99, 245/243, 1029/1024
Subgroup: 2.3.5.7.11.13


POTE generator: ~8/7 = 234.145
Comma list: 91/90, 121/120, 126/125, 176/175


Map: [&lt;1 1 -1 3 -2|, &lt;0 3 17 -1 28|
Mapping: {{mapping| 1 1 2 3 3 2 | 0 9 5 -3 7 26 }}


EDOs: 5e, 41, 46e
Optimal tunings:  
* WE: ~2 = 1200.1306{{c}}, ~22/21 = 78.2273{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/21 = 78.2241{{c}}


Badness: 0.0449
{{Optimal ET sequence|legend=0| 15, 31f, 46 }}


===13-limit===
Badness (Sintel): 0.969
Commas: 100/99, 105/104, 245/243, 352/351


POTE generator: ~8/7 = 234.089
==== Semivalentine ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 1 -1 3 -2 0|, &lt;0 3 17 -1 28 19|
Comma list: 121/120, 126/125, 169/168, 176/175


EDOs: 5e, 41
Mapping: {{mapping| 2 2 4 6 6 7 | 0 9 5 -3 7 3 }}
: mapping generators: ~55/39, ~22/21


Badness: 0.0323
Optimal tunings:  
* WE: ~55/39 = 600.3497{{c}}, ~22/21 = 77.8845{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~22/21 = 77.8715{{c}}


= Valentine =
{{Optimal ET sequence|legend=0| 16, 30, 46, 62, 108ef }}
{{main|Valentine}}
{{see also|Starling temperaments #Valentine}}


Comma: 1990656/1953125
Badness (Sintel): 1.35


POTE generator: ~25/24 = 78.039
==== Hemivalentine ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 1 2|, &lt;0 9 5|]
Comma list: 121/120, 126/125, 176/175, 343/338


EDOs: 15, 31, 46, 77, 123
Mapping: {{mapping| 1 1 2 3 3 4 | 0 18 10 -6 14 -9 }}
: mapping generators: ~2, ~40/39


Badness: 0.1228
Optimal tunings:  
* WE: ~2 = 1199.6529{{c}}, ~40/39 = 39.0323{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~40/39 = 39.0383{{c}}


==7-limit==
{{Optimal ET sequence|legend=0| 30, 31, 61, 92f }}
[[Comma|Commas]]: 126/125, 1029/1024


[[Minimax_tuning|Minimax tuning]]:
Badness (Sintel): 1.94


7-limit: [|1 0 0 0&gt;, |5/2 3/4 0 -3/4&gt;,
==== Demivalentine ====
|17/6 5/12 0 -5/12&gt;, [5/2 -1/4 0 1/4&gt;<nowiki>]</nowiki>
Subgroup: 2.3.5.7.11.13


Eigenmonzos: 2, 7/6
Comma list: 121/120, 126/125, 176/175, 676/675


9-limit: [|1 0 0 0&gt;, |10/7 6/7 0 -3/7&gt;,
Mapping: {{mapping| 1 -8 -3 6 -4 -16 | 0 18 10 -6 14 37 }}
|47/21 10/21 0 -5/21&gt;, |20/7 -2/7 0 1/7&gt;<nowiki>]</nowiki>
: mapping generators: ~2, ~13/9


Eigenmonzos: 2, 9/7
Optimal tunings:  
* WE: ~2 = 1200.3929{{c}}, ~13/9 = 639.1320{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/9 = 638.9325{{c}}


[[POTE_tuning|POTE generator]]: ~21/20 = 77.864
{{Optimal ET sequence|legend=0| 15, 47ef, 62, 77 }}


Algebraic generator: [[Algebraic_number|smaller root]] of x^2-89x+92, or (89-√7553)/2, at 77.8616 cents.
Badness (Sintel): 1.44


Map: [&lt;1 1 2 3|, &lt;0 9 5 -3|]
=== Hemivalentino ===
Subgroup: 2.3.5.7.11


Wedgie: &lt;&lt;9 5 -3 -13 -30 -21||
Comma list: 126/125, 243/242, 1029/1024


EDOs: 15, 31, 46, 77, 185, 262cd
Mapping: {{mapping| 1 1 2 3 2 | 0 18 10 -6 45 }}


Badness: 0.0311
Optimal tunings:  
* WE: ~2 = 1200.0816{{c}}, ~45/44 = 38.9236{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~45/44 = 38.9228{{c}}


==11-limit==
{{Optimal ET sequence|legend=0| 31, 92e, 123, 154, 185 }}
[[Comma|Commas]]: 121/120, 126/125, 176/175


[[Minimax_tuning|Minimax tuning]]:
Badness (Sintel): 2.03


[|1 0 0 0 0&gt;, |1 0 0 -9/10 9/10&gt;,
==== 13-limit ====
|2 0 0 -1/2 1/2&gt;, |3 0 0 3/10 -3/10&gt;, |3 0 0 -7/10 7/10&gt;<nowiki>]</nowiki>
Subgroup: 2.3.5.7.11.13


Eigenmonzos: 2, 11/7
Comma list: 126/125, 196/195, 243/242, 1029/1024


[[POTE_tuning|POTE generator]]: ~21/20 = 77.881
Mapping: {{mapping| 1 1 2 3 2 5 | 0 18 10 -6 45 -40 }}


Algebraic generator: [[Algebraic_number|positive root]] of 4x^3+15x^2-21, or else Gontrand2, the smallest positive root of 4x^7-8x^6+5.
Optimal tunings:  
* WE: ~2 = 1199.8782{{c}}, ~45/44 = 38.9440{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~45/44 = 38.9472{{c}}


Map: [&lt;1 1 2 3 3|, &lt;0 9 5 -3 7|]
{{Optimal ET sequence|legend=0| 31, 123, 154 }}


Generators: 2, 21/20
Badness (Sintel): 2.39


[[EDO|EDOs]]: 15, 31, 46, 77, 262cdee, 339cdeee
==== Hemivalentoid ====
Subgroup: 2.3.5.7.11.13


Badness: 0.0167
Comma list: 126/125, 144/143, 243/242, 343/338


[[Chords_of_valentine|Chords of valentine]]
Mapping: {{mapping| 1 1 2 3 2 4 | 0 18 10 -6 45 -9 }}


=Unidec=
Optimal tunings:
Comma: 31381059609/31250000000
* WE: ~2 = 1199.3614{{c}}, ~45/44 = 38.9721{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~45/44 = 38.9839{{c}}


POTE generator: ~10/9 = 183.047
{{Optimal ET sequence|legend=0| 31, 92ef }}


Map: [&lt;2 5 8|, &lt;0 -6 -11|]
Badness (Sintel): 2.39


EDOs: 26, 46, 72, 118, 2524, 2642, 2760, 5002bc
== Superkleismic ==
{{Main| Superkleismic }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Shibboleth]].''


Badness: 0.0824
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.  


==7-limit==
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.
[[Comma|Commas]]: 1029/1024, 4375/4374


[[Minimax_tuning|Minimax tuning]]:
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.


7-limit: [|1 0 0 0&gt;, |47/26 0 6/13 -6/13&gt;,
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.
|71/26 0 11/13 -11/13&gt;, |71/26 0 -2/13 2/13&gt;<nowiki>]</nowiki>


[[Eigenmonzo|Eigenmonzos]]: 2, 7/5
41edo gives an obvious tuning in all the subgroups.


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


[[Eigenmonzo|Eigenmonzos]]: 2, 9/7
[[Comma list]]: 875/864, 1029/1024


[[POTE_tuning|POTE generator]]: ~10/9 = 183.161
{{Mapping|legend=1| 1 -5 -5 5 | 0 9 10 -3 }}
: mapping generators: ~2, ~5/3


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


Wedgie: &lt;&lt;12 22 -4 7 -40 -71||
{{Optimal ET sequence|legend=1| 11c, 15, 26, 41 }}


EDOs: 26, 46, 72, 118, 190
[[Badness]] (Sintel): 1.21


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


==11-limit==
Comma list: 100/99, 245/242, 385/384
[[Comma|Commas]]: 385/384, 441/440, 4375/4374


[[Minimax_tuning|Minimax tuning]]:
Mapping: {{mapping| 1 -5 -5 5 2 | 0 9 10 -3 2 }}


[|1 0 0 0 0&gt;, |10/7 6/7 0 -3/7 0&gt;, |57/28 11/7 0 -11/14 0&gt;,
Optimal tunings:
|20/7 -2/7 0 1/7 0&gt;, |99/28 -3/7 0 3/14 0&gt;<nowiki>]</nowiki>
* WE: ~2 = 1200.1691{{c}}, ~5/3 = 878.2772{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 878.1606{{c}}


[[Eigenmonzo|Eigenmonzos]]: 2, 9/7
{{Optimal ET sequence|legend=0| 11c, 15, 26, 41, 179cde, 220cde, 261ccdee }}


[[POTE_tuning|POTE generator]]: 183.165
Badness (Sintel): 0.848


Map: [&lt;2 5 8 5 6|, &lt;0 -6 -11 2 3|]
==== 2.3.5.7.11.19 subgroup ====
Subgroup: 2.3.5.7.11.19


EDOs: 26, 46, 72, 118, 190
Comma list: 100/99, 133/132, 190/189, 385/384


Badness: 0.0155
Mapping: {{mapping| 1 -5 -5 5 2 -6 | 0 9 10 -3 2 14 }}


[[Chords_of_unidec|Chords of unidec]]
Optimal tunings:
* WE: ~2 = 1200.2289{{c}}, ~5/3 = 878.3409{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 878.1840{{c}}


==Ekadash==
{{Optimal ET sequence|legend=0| 11c, 15, 26, 41, 138e }}
Commas: 385/384, 441/440, 625/624, 729/728


[[POTE_tuning|POTE generator]]: ~10/9 = 183.187
Badness (Sintel): 0.692


Map: [&lt;2 5 8 5 6 19|, &lt;0 -6 -11 2 3 -38|]
=== 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.


EDOs: 20cf, 26f, 46f, 72, 118, 190, 262df, 452cdef
Subgroup: 2.3.5.7.11.13


Badness: 0.0204
Comma list: 100/99, 105/104, 144/143, 245/242


==Hendec==
Mapping: {{mapping| 1 -5 -5 5 2 -8 | 0 9 10 -3 2 16 }}
Commas: 169/168, 325/324, 364/363, 1716/1715


[[POTE_tuning|POTE generator]]: ~10/9 = 183.187
Optimal tunings:  
* WE: ~2 = 1200.0261{{c}}, ~5/3 = 878.0252{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 878.0073{{c}}


Map: [&lt;2 5 8 5 6 8|, &lt;0 -6 -11 2 3 -2|]
{{Optimal ET sequence|legend=0| 11cf, 15, 26, 41 }}


EDOs: 26, 46, 72
Badness (Sintel): 0.887


Badness: 0.0177
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


===17-limit===
Comma list: 100/99, 105/104, 120/119, 144/143, 245/242
Commas: 169/168, 221/220, 273/272, 325/324, 364/363


[[POTE_tuning|POTE generator]]: ~10/9 = 183.196
Mapping: {{mapping| 1 -5 -5 5 2 -8 -12 | 0 9 10 -3 2 16 22 }}


Map: [&lt;2 5 8 5 6 8 10|, &lt;0 -6 -11 2 3 -2 -6|]
Optimal tunings:  
* WE: ~2 = 1200.0488{{c}}, ~5/3 = 877.8872{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 877.8537{{c}}


EDOs: 26, 46, 72
{{Optimal ET sequence|legend=0| 11cfg, 15g, 26, 41 }}


===Music===
Badness (Sintel): 1.01
[[Technical_Notes_for_Newbeams#Track notes:-Hypnocloudsmack 2|Hypnocloudsmack 2]] by [[Andrew_Heathwaite|Andrew Heathwaite]]


= Hemithirds =
==== 19-limit ====
{{main|Hemithirds}}
Subgroup: 2.3.5.7.11.13.17.19
{{see also|Luna family #Hemithirds}}


== 7-limit ==
Comma list: 100/99, 105/104, 120/119, 144/143, 133/132, 190/189
[[Comma|Commas]]: 1029/1024, 3136/3125


[[Minimax tuning]]:
Mapping: {{mapping| 1 -5 -5 5 2 -8 -12 -6 | 0 9 10 -3 2 16 22 14 }}
 
Optimal tunings:
* WE: ~2 = 1200.2120{{c}}, ~5/3 = 878.0243{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 877.8789{{c}}
 
{{Optimal ET sequence|legend=0| 11cfgh, 15g, 26, 41 }}
 
Badness (Sintel): 0.964
 
=== Superana ===
This extension ({{nowrap| 41 & 56 }}) is the counterpart of canonical superkleismic on the other side of 41edo.
 
Subgroup: 2.3.5.7.11.13
 
Comma list: 100/99, 196/195, 245/242, 385/384
 
Mapping: {{mapping| 1 -5 -5 5 2 22 | 0 9 10 -3 2 -25 }}
 
Optimal tunings:
* WE: ~2 = 1199.8272{{c}}, ~5/3 = 878.1538{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 878.2795{{c}}
 
{{Optimal ET sequence|legend=0| 15f, 41, 97, 138e }}
 
Badness (Sintel): 1.40
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 100/99, 154/153, 196/195, 245/242, 256/255
 
Mapping: {{mapping| 1 -5 -5 5 2 22 18 | 0 9 10 -3 2 -25 -19 }}
 
Optimal tunings:
* WE: ~2 = 1199.5964{{c}}, ~5/3 = 878.0482{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 878.3444{{c}}
 
{{Optimal ET sequence|legend=0| 15f, 41, 56, 97g }}
 
Badness (Sintel): 1.45
 
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 100/99, 133/132, 154/153, 190/189, 196/195, 256/255
 
Mapping: {{mapping| 1 -5 -5 5 2 22 18 -6 | 0 9 10 -3 2 -25 -19 14 }}
 
Optimal tunings:
* WE: ~2 = 1199.6638{{c}}, ~5/3 = 878.1109{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 878.3566{{c}}
 
{{Optimal ET sequence|legend=0| 15f, 41, 56, 97g }}
 
Badness (Sintel): 1.36
 
== Dee leap week ==
{{Main| Dee leap week }}
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 1029/1024, 2460375/2458624
 
{{Mapping|legend=1| 1 -5 25 5 | 0 9 -31 -3 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.4835{{c}}, ~224/135 = 878.2507{{c}}
: [[error map]]: {{val| +0.484 -0.117 +0.004 -1.160 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~224/135 = 877.8926{{c}}
: error map: {{val| 0.000 -0.921 -0.985 -2.504 }}
 
{{Optimal ET sequence|legend=1| 41, 108, 149, 190 }}
 
[[Badness]] (Sintel): 2.12
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 385/384, 441/440, 2460375/2458624
 
Mapping: {{mapping| 1 -5 25 5 -28 | 0 9 -31 -3 43 }}


7-limit: [|1 0 0 0&gt;, |5/2 3/4 0 -3/4&gt;,
Optimal tunings:  
|11/5 -1/10 0 1/10&gt;, |5/2 -1/4 0 1/4&gt;<nowiki>]</nowiki>
* WE: ~2 = 1200.4874{{c}}, ~224/135 = 878.2543{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~224/135 = 877.8987{{c}}


[[Eigenmonzo|Eigenmonzos]]: 2, 7/6
{{Optimal ET sequence|legend=0| 41, 108e, 149, 190 }}


9-limit: [|1 0 0 0&gt;, |10/7 6/7 0 -3/7&gt;,
Badness (Sintel): 1.35
|82/35 -4/35 0 2/35&gt;, |20/7 -2/7 0 1/7&gt;<nowiki>]</nowiki>


[[Eigenmonzo|Eigenmonzos]]: 2, 7/6
== Unidec ==
{{Main| Unidec }}


POTE generator: ~28/25 = 193.244
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.  


Map: [&lt;1 4 2 2|, &lt;0 -15 2 5|]
[[Subgroup]]: 2.3.5.7


[[EDO|EDOs]]: {{EDOs|31, 87, 118}}
[[Comma list]]: 1029/1024, 4375/4374


Badness: 0.0443
{{Mapping|legend=1| 2 -1 -3 7 | 0 6 11 -2 }}


== 11-limit ==
[[Optimal tuning]]s:
[[Comma|Commas]]: 385/384, 441/440, 3136/3125
* [[WE]]: ~1225/864 = 600.2429{{c}}, ~80/63 = 417.0073{{c}}
: [[error map]]: {{val| +0.486 -0.154 +0.038 -1.140 }}
* [[CWE]]: ~1225/864 = 600.0000{{c}}, ~80/63 = 416.8688{{c}}
: error map: {{val| 0.000 -0.924 -1.090 -2.503 }}


[[Minimax tuning]]:
[[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
{{Optimal ET sequence|legend=1| 26, 46, 72, 118, 190 }}


[|1 0 0 0 0&gt;, |11/9 0 0 -5/9 5/9&gt;, |64/27 0 0 2/27 -2/27&gt;,
[[Badness]] (Sintel): 0.972
|79/27 0 0 5/27 -5/27&gt;, |79/27 0 0 -22/27 22/27&gt;<nowiki>]</nowiki>


[[Eigenmonzo|Eigenmonzos]]: 2, 11/7
=== 11-limit ===
Subgroup: 2.3.5.7.11


POTE generator: ~28/25 = 193.227
Comma list: 385/384, 441/440, 4375/4374


Map: [&lt;1 4 2 2 7|, &lt;0 -15 2 5 -22|]
Mapping: {{mapping| 2 -1 -3 7 9 | 0 6 11 -2 -3 }}


EDOs: {{EDOs|31, 87, 118}}
Optimal tunings:  
* WE: ~99/70 = 600.2497{{c}}, ~14/11 = 417.0085{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~14/11 = 416.8543{{c}}


Badness: 0.0190
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


{{see also|Chords of hemithirds}}
{{Optimal ET sequence|legend=0| 26, 46, 72, 118, 190 }}


== 13-limit ==
Badness (Sintel): 0.512
Commas: 196/195, 352/351, 1001/1000, 1029/1024


POTE generator: ~28/25 = 193.166
==== Ekadash ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 4 2 2 7 0|, &lt;0 -15 2 5 -22 23|]
Comma list: 385/384, 441/440, 625/624, 729/728


EDOs: {{EDOs|31, 56, 87, 118, 205d}}
Mapping: {{mapping| 2 -1 -3 7 9 -19 | 0 6 11 -2 -3 38 }}


Badness: 0.0217
Optimal tunings:  
* WE: ~99/70 = 600.2497{{c}}, ~14/11 = 417.0085{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~14/11 = 416.8543{{c}}


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


==7-limit==
Badness (Sintel): 0.842
Commas: 1029/1024, 19683/19600


POTE generator: ~320/243 = 483.267
==== Hendec ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 4 14 2|, &lt;0 -6 -29 2|]
Comma list: 169/168, 325/324, 364/363, 385/384


Wedgie: &lt;&lt;6 29 -2 32 -20 -86||
Mapping: {{mapping| 2 -1 -3 7 9 6 | 0 6 11 -2 -3 2 }}


EDOs: 5, 72, 77, 149, 221, 514bd, 735bcd
Optimal tunings:  
* WE: ~91/64 = 600.3825{{c}}, ~14/11 = 417.0678{{c}}
* CWE: ~91/64 = 600.0000{{c}}, ~14/11 = 416.8290{{c}}


Badness: 0.0566
{{Optimal ET sequence|legend=0| 26, 46, 72, 190ff }}


==11-limit==
Badness (Sintel): 0.732
Commas: 385/384, 441/440, 19683/19600


POTE generator: ~320/243 = 483.276
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


Map: [&lt;1 4 14 2 -5|, &lt;0 -6 -29 2 21|]
Comma list: 169/168, 221/220, 273/272, 325/324, 364/363


EDOs: 72, 77, 149, 221e, 293de
Mapping: {{mapping| 2 -1 -3 7 9 6 4 | 0 6 11 -2 -3 2 6 }}


Badness: 0.0285
Optimal tunings:  
* WE: ~17/12 = 600.3991{{c}}, ~14/11 = 417.0809{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~14/11 = 416.8330{{c}}


==13-limit==
{{Optimal ET sequence|legend=0| 26, 46, 72, 190ffg }}
Commas: 351/350, 385/384, 441/440, 676/675


POTE generator: ~320/243 = 483.256
Badness (Sintel): 0.595


Map: [&lt;1 4 14 2 -5 19|, &lt;0 -6 -29 2 21 -38|]
== Necromanteion ==
Necromanteion, named by [[Johannes Werpup]] in 2014<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_106371.html Yahoo! Tuning Group | ''Temperament ideas: A cuckoo, and two oracles'']</ref> may be described as the {{nowrap| 31 & 51c }} temperament. The generator is a subfifth representing 35/24, four of which minus two octaves make slendric's generator, so its [[ploidacot]] is beta-dodecacot.


EDOs: 72, 77, 149, 221ef
[[Subgroup]]: 2.3.5.7


==17-limit==
[[Comma list]]: 1029/1024, 5103/5000
Commas: 273/272, 351/350, 385/384, 441/440, 676/675


POTE generator: ~320/243 = 483.261
{{Mapping|legend=1| 1 -5 -7 5 | 0 12 17 -4 }}
: mapping generators: ~2, ~35/24


Map: [&lt;1 4 14 2 -5 19 21|, &lt;0 -6 -29 2 21 -38 -42|]
[[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 }}


EDOs: 72, 77, 149, 221ef
{{Optimal ET sequence|legend=1| 11c, 20c, 31, 144c, 175c }}


=Tritikleismic=
[[Badness]] (Sintel): 2.98


==7-limit==
=== 11-limit ===
[[Comma|Commas]]: 1029/1024, 15625/15552
Subgroup: 2.3.5.7.11


[[Minimax_tunings|Minimax tunings]]:
Comma list: 176/175, 243/242, 1029/1024


7-limit: [|1 0 0 0&gt;, |2 0 6/7 -6/7&gt;,
Mapping: {{mapping| 1 -5 -7 5 -13 | 0 12 17 -4 30 }}
|8/3 0 5/7 -5/7&gt;, |8/3 0 -2/7 2/7&gt;<nowiki>]</nowiki>


[[Eigenmonzo|Eigenmonzos]]: 2, 7/5
Optimal tunings:  
* WE: ~2 = 1200.2862{{c}}, ~22/15 = 658.4276{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/15 = 658.2805{{c}}


9-limit: [|1 0 0 0&gt;, |10/7 6/7 0 -3/7&gt;,
{{Optimal ET sequence|legend=0| 20ce, 31, 113c, 144c }}
|46/21 5/7 0 -5/14&gt;, |20/7 -2/7 0 1/7&gt;<nowiki>]</nowiki>


[[Eigenmonzo|Eigenmonzos]]: 2, 9/7
Badness (Sintel): 1.77


POTE generator: ~6/5 = 316.872
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Map: [&lt;3 0 3 10|, &lt;0 6 5 -2|]
Comma list: 144/143, 176/175, 243/242, 343/338


[[EDO|EDOs]]: [[15edo|15]], [[72edo|72]], [[87edo|87]], [[159edo|159]], [[231edo|231]]
Mapping: {{mapping| 1 -5 -7 5 -13 7 | 0 12 17 -4 30 -6 }}


Badness: 0.0563
Optimal tunings:  
* WE: ~2 = 1199.3663{{c}}, ~22/15 = 658.0465{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/15 = 658.3800{{c}}


==11-limit==
{{Optimal ET sequence|legend=0| 20ce, 31, 82cf, 113cf }}
[[Comma|Commas]]: 385/384, 441/440, 4000/3993


[[Minimax_tuning|Minimax tuning]]:
Badness (Sintel): 1.94


[|1 0 0 0 0&gt;, |10/7 6/7 0 -3/7 0&gt;, |46/21 5/7 0 -5/14 0&gt;,
== Restles ==
|20/7 -2/7 0 1/7 0&gt;, |71/21 3/7 0 -3/14 0&gt;<nowiki>]</nowiki>
{{See also| Lesser tendoneutralic }}


[[Eigenmonzo|Eigenmonzos]]: 2, 9/7
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>.


POTE generator: ~6/5 = 316.881
[[Subgroup]]: 2.3.5.7


Map: [&lt;3 0 3 10 8|, &lt;0 6 5 -2 3|]
[[Comma list]]: 1029/1024, 153664/151875


EDOs: 72, 159, 231
{{Mapping|legend=1| 1 -2 8 4 | 0 12 -19 -4 }}
: mapping generators: ~2. ~315/256


Badness: 0.0193
[[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 }}


==13-limit==
{{Optimal ET sequence|legend=1| 77, 87, 164 }}
Commas: 325/324, 364/363, 441/440, 625/624


Map: [&lt;3 0 3 10 8 0|, &lt;0 6 5 -2 3 14|]
[[Badness]] (Sintel): 2.73


EDOs: 15, 72, 87, 159
=== 11-limit ===
Subgroup: 2.3.5.7.11


==17-limit==
Comma list: 385/384, 441/440, 153664/151875
Commas: 273/272, 325/324, 364/363, 375/374, 385/384


Map: [&lt;3 0 3 10 8 0 -2|, &lt;0 6 5 -2 3 14 18|]
Mapping: {{mapping| 1 -2 8 4 -7 | 0 12 -19 -4 35 }}


EDOs: 15g, 72, 87, 159
Optimal tunings:  
* WE: ~2 = 1200.1110{{c}}, ~27/22 = 358.6045{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~27/22 = 358.5720{{c}}


=Superkleismic=
{{Optimal ET sequence|legend=0| 77, 87, 164, 251d }}
Commas: 1029/1024, 875/864


POTE generator: ~6/5 = 321.930
Badness (Sintel): 1.81


Map: [&lt;1 4 5 2|, &lt;0 -9 -10 3|]
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


EDOs: 11, 15, 26, 41
Comma list: 196/195, 352/351, 385/384, 676/675


Badness: 0.0479
Mapping: {{mapping| 1 -2 8 4 -7 4 | 0 12 -19 -4 35 -1 }}


==11-limit==
Optimal tunings:
Commas: 100/99, 245/242, 385/384
* WE: ~2 = 1200.0482{{c}}, ~~16/13 = 358.5883{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/13 = 358.5741{{c}}


POTE generator: ~6/5 = 321.847
{{Optimal ET sequence|legend=0| 77, 87, 164, 251d }}


Map: [&lt;1 4 5 2 4|, &lt;0 -9 -10 3 -2|]
Badness (Sintel): 1.16


EDOs: 15, 26, 41, 261
== Lagaca ==
Cryptically named by [[Petr Pařízek]] in 2011<ref name="petr's long post"/>, lagaca may be described as the {{nowrap| 10 & 118 }} temperament with a [[ploidacot]] signature of diploid wau-enneacot. The name actually refers to the fact that 12 generator steps in this temperament make ~7/3, where "l", "g", "c" are integers alphabetically converted to letters.


Badness: 0.0257
[[Subgroup]]: 2.3.5.7


==13-limit==
[[Comma list]]: 1029/1024, 11529602/11390625
Commas: 100/99, 105/104, 245/243, 1188/1183


POTE generator: ~6/5 = 321.994
{{Mapping|legend=1| 2 -4 15 8 | 0 9 -13 -3 }}
: mapping generators: ~3375/2401, ~450/343


Map: [&lt;1 4 5 2 4 8|, &lt;0 -9 -10 3 -2 -16|]
[[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 }}


EDOs: 11, 15, 26, 41
{{Optimal ET sequence|legend=1| 10, 98, 108, 118 }}


Badness: 0.0215
[[Badness]] (Sintel): 3.65


=Gorgo=
== Quartemka ==
Commas: 36/35, 1029/1024
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Quartemka]].''


[[POTE_tuning|POTE generator]]: ~8/7 = 228.334
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"/>.  


Map: [&lt;1 1 1 3|, &lt;0 3 7 -1|]
[[Subgroup]]: 2.3.5.7


Wedgie: &lt;&lt;3 7 -1 4 -10 -22||
[[Comma list]]: 1029/1024, 1250000/1240029


EDOs: 5, 16, 21
{{Mapping|legend=1| 1 -17 -26 9 | 0 21 32 -7 }}
: mapping generators: ~2, ~50/27


Badness: 0.0607
[[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 }}


==11-limit==
{{Optimal ET sequence|legend=1| 26, 61, 87, 113, 200 }}
Commas: 36/35, 56/55, 1029/1024


POTE generator: ~8/7 = 229.535
[[Badness]] (Sintel): 3.85


Map: [&lt;1 1 1 3 5|, &lt;0 3 7 -1 -8|]
=== 11-limit ===
Subgroup: 2.3.5.7.11


EDOs: 5, 16e, 21, 47c, 68bce
Comma list: 385/384, 441/440, 800000/793881


Badness: 0.0627
Mapping: {{mapping| 1 -17 -26 9 7 | 0 21 32 -7 -4 }}


==13-limit==
Optimal tunings:
Commas: 27/26, 36/35, 56/55, 507/500
* WE: ~2 = 1200.3051{{c}}, ~50/27 = 1062.2805{{c}}
* CWE: ~21 = 1200.0000{{c}}, ~50/27 = 1062.0147{{c}}


POTE generator: ~8/7 = 229.059
{{Optimal ET sequence|legend=0| 26, 61, 87, 200, 287d }}


Map: [&lt;1 1 1 3 5 2|, &lt;0 3 7 -1 -8 9|]
Badness (Sintel): 1.89


EDOs 5, 21, 68bcef
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Badness: 0.0471
Comma list: 325/324, 364/363, 385/384, 2200/2197


==Music==
Mapping: {{mapping| 1 -17 -26 9 7 -14 | 0 21 32 -7 -4 20 }}
[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Herman/gorgo-example.mp3 Gorgo Example] by [[Herman_Miller|Herman Miller]]


=Lemba=
Optimal tunings:
Commas: 50/49, 525/512
* WE: ~2 = 1200.2708{{c}}, ~24/13 = 1062.2496{{c}}
* CWE: ~21 = 1200.0000{{c}}, ~24/13 = 1062.0139{{c}}


[[POTE_tuning|POTE generator]]: ~8/7 = 232.089
{{Optimal ET sequence|legend=0| 26, 61, 87, 200 }}


Map: [&lt;2 2 5 6|, &lt;0 3 -1 -1|]
Badness (Sintel): 1.17


Wedgie: &lt;&lt;6 -2 -2 -17 -20 1||
== Tritriple ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tritriple]].''


EDOs: 10, 16, 26
Tritriple may be described as the {{nowrap| 103 & 118 }} temperament. Its [[ploidacot]] is iota-beta-27-cot. It was named by [[Petr Pařízek]] in 2011 for its generator is 1/9 of the generator for [[slendric]], so that 3×3 generators [[octave reduction|octave reduced]] give slendric's generator, and another ×3 give the [[3/2|perfect fifth]]<ref name="petr's long post"/>.


Badness: 0.0622
[[Subgroup]]: 2.3.5.7


==Music==
[[Comma list]]: 1029/1024, 1959552/1953125
By [[Herman_Miller|Herman Miller]]


[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Herman/LembaGalatsia.mp3 Lemba Galatsia]
{{Mapping|legend=1| 1 -11 -7 7 | 0 27 20 -9 }}
: mapping generators: ~2, ~864/625


[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Herman/lemba-gpo-test.mp3 GPO Lemba]
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.4239{{c}}, ~864/625 = 559.4921{{c}}
: [[error map]]: {{val| +0.424 -0.331 +0.561 -1.287 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~864/625 = 559.3015{{c}}
: error map: {{val| 0.000 -0.815 -0.284 -2.539 }}


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


Commas: 21/20, 144/125
[[Badness]] (Sintel): 3.00


POTE generator: ~8/7 = 230.762
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 1 2 3|, &lt;0 3 2 -1|]
Comma list: 385/384, 441/440, 43923/43750


EDOs: 5, 11, 16c, 21cc, 26ccc
Mapping: {{mapping| 1 -11 -7 7 -4 | 0 27 20 -9 16 }}


Badness: 0.0623
Optimal tunings:  
* WE: ~2 = 1200.4953{{c}}, ~242/175 = 559.5243{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~242/175 = 559.3016{{c}}


: ''see also [[University temperament]]''
{{Optimal ET sequence|legend=0| 15, …, 88, 103, 118, 221e, 339de }}


=Clyndro=
Badness (Sintel): 1.17
Commas: 135/128, 360/343


POTE generator: ~8/7 = 226.469
== Widefourth ==
[[Subgroup]]: 2.3.5.7


Map: [&lt;1 1 4 3|, &lt;0 3 -9 -1|]
[[Comma list]]: 1029/1024, 48828125/48771072


EDOs: 5c, 11, 16
{{Mapping|legend=1| 1 -17 -5 9 | 0 33 13 -11 }}


Badness: 0.1592
[[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 }}


==11-limit==
{{Optimal ET sequence|legend=1| 16, 71, 87, 103, 190 }}
Commas: 33/32, 45/44, 352/343


POTE generator: ~8/7 = 226.428
[[Badness]] (Sintel): 3.90


Map: [&lt;1 1 4 3 4|, &lt;0 3 -9 -1 -3|]
=== 11-limit ===
Subgroup: 2.3.5.7.11


EDOs: 5c, 11, 16
Comma list: 385/384, 441/440, 234375/234256


Badness: 0.0697
Mapping: {{mapping| 1 16 8 -2 17 | 0 -33 -13 11 -31 }}


=Necromanteion=
Optimal tunings:
Commas: 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 }}


Map: [&lt;1 7 10 1|, &lt;0 -12 -17 4|]
Badness (Sintel): 1.35


EDOs: 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 }}
Commas: 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}}


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


EDOs: 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
Commas: 144/143, 176/175, 243/242, 343/338


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


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


EDOs: 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 }}
Commas: 1029/1024, 48828125/48771072


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


Map: [&lt;1 16 8 -2|, &lt;0 -33 -13 11|]
==== 2.3.7.13.17 subgroup ====
Subgroup: 2.3.7.13.17


Wedgie: &lt;&lt;33 13 -11 -56 -110 -62||
Comma list: 273/272, 729/728, 833/832


EDOs: 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:
Commas: 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 }}


Map: [&lt;1 16 8 -2 17|, &lt;0 -33 -13 11 -31|]
Badness (Sintel): 0.332


EDOs: 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 }}
Commas: 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 }}


Map: [&lt;1 16 8 -2 17 12|, &lt;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}}


EDOs: 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 ====
Comma: |31 20 -27&gt;
Subgroup: 2.3.7.13.17.19.23


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


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


EDOs: 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 }}
Commas: 1029/1024, 1959552/1953125


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


Map: [&lt;1 -11 -7 7|, &lt;0 27 20 -9|]
==== 2.3.7.13.17.19.23.29 subgroup ====
Subgroup: 2.3.7.13.17.19.23.29


EDOs: 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 }}
Commas: 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}}


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


EDOs: 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
Commas: 1029/1024, 153664/151875


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


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


EDOs: 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 }}
Commas: 385/384, 441/440, 132055/131072


POTE generator: ~315/256 = 358.5713
Badness (Sintel): 0.434


Map: [<1 -2 8 4 -7|, <0 12 -19 -4 35|]
==== 2.3.7.13.17 subgroup ====
Subgroup: 2.3.7.13.17


EDOs: 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 }}
Commas: 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}}


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


EDOs: 10, 77, 87, 164
Badness (Sintel): 0.253


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


Commas: 169/168, 273/272, 289/288
Comma list: 1029/1024, 7225/7203


Period: 1\2
Subgroup-val mapping: {{mapping| 1 -2 4 7 | 0 6 -2 -1 }}


POTE generator: ~8/7 = 233.6155
Optimal tunings:  
* WE: ~2 = 1200.8295{{c}}, ~128/85 = 717.2597{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~128/85 = 716.7933{{c}}


Map: [<2 2 6 7 7|, <0 3 -1 1 3|]
{{Optimal ET sequence|legend=0| 5, 42*, 47, 52, 57, 62, 67, 72, 149*, 370d***, 519bdd***** }}


EDOs: 36, 46, 82, 118
<nowiki/>* Wart for 85


== References ==


[[Category:Theory]]
[[Category:Temperament clans]]
[[Category:Gamelismic]]
[[Category:Gamelismic clan| ]] <!-- main article -->
[[Category:Clan]]
[[Category:Rank 2]]
[[Category:Miracle]]
[[Category:Listen]]
[[Category:Listen]]
Retrieved from "https://en.xen.wiki/w/Gamelismic_clan"