Gamelismic clan: Difference between revisions
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{{Technical data page}} | {{Technical data page}} | ||
The [[2.3.7 subgroup|2.3.7-subgroup]] [[comma]] for the '''gamelismic clan''' is the gamelisma, [[1029/1024]], with [[monzo]] {{monzo| -10 1 0 3 }}. For any member of the clan, for the rank-3 [[gamelismic family #Gamelismic|gamelismic temperament]] itself, and for the rank-2 2.3.7 temperament [[slendric]] (a.k.a. gamelic), this means three [[~]][[8/7]] intervals give a fifth, [[3/2]]. In fact, we find that {{nowrap|3/2 {{=}} (8/7)<sup>3</sup> | The [[2.3.7 subgroup|2.3.7-subgroup]] [[comma]] for the '''gamelismic clan''' is the gamelisma, [[1029/1024]], with [[monzo]] {{monzo| -10 1 0 3 }}. For any member of the clan, for the rank-3 [[gamelismic family #Gamelismic|gamelismic temperament]] itself, and for the rank-2 2.3.7 temperament [[slendric]] (a.k.a. gamelic), this means three [[~]][[8/7]] intervals give a fifth, [[3/2]]. In fact, we find that {{nowrap| 3/2 {{=}} (8/7)<sup>3</sup>⋅(1029/1024) }}. From this it follows that gamelismic temperaments tend to flatten both the fifth and the harmonic seventh, or if they do not, the other of the pair must be flattened even more. [[36edo]] is a good tuning for slendric, though if the full 7-limit is desired, [[72edo]], [[77edo]], or [[118edo]] might be preferred. | ||
To the gamelisma itself we need to add the comma which appears next on the modified [[Normal lists #Normal interval list|normal comma list]] for the full 7-limit. The second comma on the list for mothra is [[81/80]], for rodan [[245/243]], for guiron [[32805/32768]], for gorgo [[36/35]], and for gidorah [[256/245]]. These all use ~8/7 as a generator, though in the case of gidorah that is the same as ~6/5. | == Slendric == | ||
{{Main| Slendric }} | |||
[[Subgroup]]: 2.3.7 | |||
[[Comma list]]: 1029/1024 | |||
{{Mapping|legend=3| 1 1 3 | 0 3 -1 }} | |||
{{Mapping|legend=5| 1 1 0 3 | 0 3 0 -1 }} | |||
: mapping generators: ~2, ~8/7 | |||
[[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 }} | |||
{{Optimal ET sequence|legend=1| 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 lists #Normal interval list|normal comma list]] for the full 7-limit. The second comma on the list for mothra is [[81/80]], for rodan [[245/243]], for guiron [[32805/32768]], for gorgo [[36/35]], and for gidorah [[256/245]]. These all use ~8/7 as a generator, though in the case of gidorah that is the same as ~[[6/5]]. | |||
Miracle adds [[33075/32768]] and uses the [[secor]], half an ~8/7, as generator. Lemba adds [[525/512]] to the list, and has a half-octave [[period]]. Valentine adds [[6144/6125]] with a generator of ~21/20 and superkleismic adds [[875/864]] with a generator of ~6/5. Unidec adds [[4375/4374]], and has a generator of ~10/9 with a half-octave period. Hemithirds adds [[65625/65536]] with a generator half of a classical major third. Finally, tritikleismic adds [[15625/15552]] and has a generator of 6/5 with a 1/3-octave period. | 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: | Full 7-limit temperaments discussed elsewhere are: | ||
* [[Blackwood]] (+28/27) → [[Blackwood family #Blackwood|Blackwood family]] | |||
* [[Lemba]] (+50/49) → [[Jubilismic clan #Lemba|Jubilismic clan]] | * [[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]] | * [[Hemithirds]] (+3136/3125) → [[Hemimean clan #Hemithirds|Hemimean clan]] | ||
* ''[[Gamity]]'' (+1071875/1062882) → [[Amity family #Gamity|Amity family]] | * ''[[Gamity]]'' (+1071875/1062882) → [[Amity family #Gamity|Amity family]] | ||
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The rest are considered below. | The rest are considered below. | ||
No-five subgroup extensions of slendric include radon, a 2.3.7.11 extension that may be viewed as no-five rodan, | ==== 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. | |||
== | === 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 | |||
{{Mapping|legend=2| 1 1 3 6 | 0 3 -1 -13 }} | |||
{{Mapping|legend= | {{Mapping|legend=4| 1 1 0 3 6 | 0 3 0 -1 -13 }} | ||
: | Optimal tunings: | ||
* WE: ~2 = 1199.9708{{c}}, ~8/7 = 234.3748{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.3813{{c}} | |||
{{Optimal ET sequence|legend=0| 5, …, 36, 41, 87, 128 }} | |||
Badness (Sintel): 0.619 | |||
== | == Mothra == | ||
{{Main| Mothra }} | |||
Mothra tempers out [[81/80]] and finds the prime 5 at a stack of four fifths as does any temperament in the [[meantone family]]. It also tempers out [[1728/1715]], the orwellisma. It can be described as the {{nowrap| 26 & 31 }}. Using [[31edo]] with a generator of 6\31 is an excellent tuning choice. However, a pure mos mothra scale is often described as directionless and has limited chord-building potential<ref>[https://www.youtube.com/watch?v=uH3ahBzDSrs 31-EDO Music Theory: Supermajor Hexatonic Scale] by [[Zhea Erose]]</ref>, so something other than a mos may be used as a scale to get the most out of mothra. There are examples of non-mos mothra scales in 31edo [[Strictly proper 7-tone 31edo scales|in the article on strictly proper 7-tone 31edo scales]]. | |||
[[ | |||
Note that mothra is also called '''cynder''' in the 7-limit, which can be a little confusing sometimes. | |||
[[ | Its [[S-expression]]-based comma list is {[[1728/1715|S6/S7]], [[1029/1024|S7/S8]], ([[81/80|S6/S8 = S9]])}, taking advantage of the fact that [[81/80]] is a [[semiparticular]]. | ||
[[ | [[Subgroup]]: 2.3.5.7 | ||
[[Comma list]]: 81/80, 1029/1024 | |||
{{Mapping|legend=1| 1 1 0 3 | 0 3 12 -1 }} | |||
== | [[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 }} | |||
[[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 | {{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 | |||
Subgroup: 2.3.7. | |||
Comma list: | Comma list: 81/80, 99/98, 385/384 | ||
{{Mapping|legend=0| 1 1 0 3 5 | 0 3 12 -1 -8 }} | |||
Optimal tunings: | 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= | {{Optimal ET sequence|legend=0| 5, 26, 31, 88, 119be, 150be }} | ||
Badness ( | Badness (Sintel): 0.848 | ||
==== | ==== 13-limit ==== | ||
Subgroup: 2.3.7.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: | Comma list: 81/80, 99/98, 105/104, 144/143 | ||
{{Mapping|legend=0| 1 1 0 3 5 1 | 0 3 12 -1 -8 14 }} | |||
Optimal tunings: | 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= | {{Optimal ET sequence|legend=0| 5, 26, 31, 57, 88 }} | ||
Badness ( | Badness (Sintel): 0.990 | ||
==== | ==== 17-limit ==== | ||
Subgroup: 2.3.7.13.17 | Subgroup: 2.3.5.7.11.13.17 | ||
Comma list: | Comma list: 81/80, 99/98, 105/104, 120/119, 144/143 | ||
{{Mapping|legend=0| 1 1 0 3 5 1 | 0 3 12 -1 -8 14 16 }} | |||
Optimal tunings: | 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= | {{Optimal ET sequence|legend=0| 5g, 26, 31, 57, 88 }} | ||
Badness ( | Badness (Sintel): 1.00 | ||
=== | ==== 19-limit ==== | ||
Subgroup: 2.3.5.7.11.13.17.19 | |||
Comma list: 81/80, 99/98, 105/104, 120/119, 144/143, 153/152 | |||
{{Mapping|legend=0| 1 1 0 3 5 1 | 0 3 12 -1 -8 14 16 22 }} | |||
{{ | Optimal tunings: | ||
* WE: ~2 = 1200.9663{{c}}, ~8/7 = 231.8393{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 231.6842{{c}} | |||
{{ | {{Optimal ET sequence|legend=0| 26, 31, 57 }} | ||
: | 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|legend=0| 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|legend=0| 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|legend=0| 1 1 0 3 -1 7 | 0 3 12 -1 23 -17 -15 }} | |||
{{ | Optimal tunings: | ||
* WE: ~2 = 1199.7124{{c}}, ~8/7 = 232.6376{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 232.6917{{c}} | |||
{{Optimal ET sequence|legend=0| 31, 67, 98 }} | |||
{{ | |||
Badness (Sintel): 1.53 | |||
==== 19-limit ==== | |||
Subgroup: 2.3.5.7.11.13.17.19 | |||
Comma list: 81/80, 96/95, 144/143, 153/152, 176/175, 196/195 | |||
{{Mapping|legend=0| 1 1 0 3 -1 7 | 0 3 12 -1 23 -17 -15 -9 }} | |||
Optimal tunings: | |||
* WE: ~2 = 1199.4885{{c}}, ~8/7 = 232.6310{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 232.7287{{c}} | |||
{{ | {{Optimal ET sequence|legend=0| 31, 67, 98h }} | ||
: | Badness (Sintel): 1.50 | ||
=== Cyndra === | |||
Subgroup: 2.3.5.7.11 | |||
Comma list: 45/44, 81/80, 1029/1024 | |||
{{Mapping|legend=0| 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 ( | Badness (Sintel): 1.84 | ||
=== | ==== 13-limit ==== | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: 81/80, | Comma list: 45/44, 78/77, 81/80, 640/637 | ||
{{Mapping|legend=0| 1 1 0 3 0 1 | 0 3 12 -1 18 14 }} | |||
Optimal tunings: | 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| | {{Optimal ET sequence|legend=0| 5e, 21cef, 26 }} | ||
Badness ( | 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 | ||
Subgroup: 2.3.5.7.11 | |||
=== 11-limit === | |||
Subgroup: 2.3.5.7.11 | |||
Comma list: | Comma list: 245/243, 385/384, 441/440 | ||
{{Mapping|legend=0| 1 1 -1 3 6 | 0 3 17 -1 -13 }} | |||
Optimal tunings: | 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| | {{Optimal ET sequence|legend=0| 41, 87 }} | ||
Badness ( | Badness (Sintel): 0.763 | ||
==== | ==== 13-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: | Comma list: 196/195, 245/243, 352/351, 364/363 | ||
{{Mapping|legend=0| 1 1 -1 3 6 8 | 0 3 17 -1 -13 -22 }} | |||
Optimal tunings: | 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 | |||
Subgroup: 2.3.5.7.11 | ===== 17-limit ===== | ||
Subgroup: 2.3.5.7.11.13.17 | |||
Comma list: | Comma list: 154/153, 196/195, 245/243, 256/255, 273/272 | ||
{{Mapping|legend=0| 1 1 -1 3 6 8 8 | 0 3 17 -1 -13 -22 -20 }} | |||
Optimal tunings: | 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 ( | Badness (Sintel): 0.853 | ||
==== | ==== Aerodactyl ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: | Comma list: 91/90, 245/243, 385/384, 441/440 | ||
{{Mapping|legend=0| 1 1 -1 3 6 -1 | 0 3 17 -1 -13 24 }} | |||
Optimal tunings: | 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| | {{Optimal ET sequence|legend=0| 5, 41f, 46 }} | ||
Badness ( | Badness (Sintel): 1.40 | ||
=== Aerodino === | |||
Subgroup: 2.3.5.7.11 | |||
Comma list: 176/175, 245/243, 1029/1024 | |||
{{Mapping|legend=0| 1 1 -1 3 -3 | 0 3 17 -1 33 }} | |||
Optimal tunings: | 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| | {{Optimal ET sequence|legend=0| 5e, 41e, 46 }} | ||
Badness ( | Badness (Sintel): 1.79 | ||
==== | ==== 13-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: | Comma list: 91/90, 176/175, 245/243, 847/845 | ||
{{Mapping|legend=0| 1 1 -1 3 -3 -1 | 0 3 17 -1 33 24 }} | |||
Optimal tunings: | 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| | {{Optimal ET sequence|legend=0| 5e, 41ef, 46 }} | ||
Badness ( | Badness (Sintel): 1.48 | ||
=== | === Varan === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: | Comma list: 100/99, 245/243, 1029/1024 | ||
{{Mapping|legend=0| 1 1 -1 3 -2 | 0 3 17 -1 28 }} | |||
Optimal tunings: | 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, | {{Optimal ET sequence|legend=0| 5e, 36ce, 41 }} | ||
Badness ( | Badness (Sintel): 1.49 | ||
==== 13-limit ==== | ==== 13-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: | Comma list: 100/99, 105/104, 245/243, 352/351 | ||
{{Mapping|legend=0| 1 1 -1 3 -2 0 | 0 3 17 -1 28 19 }} | |||
Optimal tunings: | 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, | {{Optimal ET sequence|legend=0| 5e, 36ce, 41 }} | ||
Badness ( | 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 | [[Subgroup]]: 2.3.5.7 | ||
[[Comma list]]: | [[Comma list]]: 1029/1024, 10976/10935 | ||
{{Mapping|legend=1| 1 1 | {{Mapping|legend=1| 1 1 7 3 | 0 3 -24 -1 }} | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[ | * [[WE]]: ~2 = 1200.3395{{c}}, ~8/7 = 233.9963{{c}} | ||
: [[error map]]: {{val| 0. | : [[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 | : error map: {{val| 0.000 -0.183 -0.487 -2.750 }} | ||
[[Minimax tuning]]: | [[Minimax tuning]]: | ||
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~8/7 = {{monzo| | * [[7-odd-limit|7-]] and [[9-odd-limit]]: ~8/7 = {{monzo| 7/24 0 -1/24 }} | ||
: {{monzo list| 1 0 0 0 | | : {{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. | : [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5 | ||
{{Optimal ET sequence|legend=1| 36, 41, 77, 118, 277d }} | |||
{{Optimal ET sequence|legend=1| 41, | |||
[[Badness]] ( | [[Badness]] (Sintel): 1.20 | ||
=== 11-limit === | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: | Comma list: 385/384, 441/440, 10976/10935 | ||
{{Mapping|legend=0| 1 1 7 3 -2 | 0 3 -24 -1 28 }} | |||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1200.3453{{c}}, ~8/7 = 233.9988{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.9312{{c}} | ||
Minimax tuning: | Minimax tuning: | ||
* 11-odd-limit: ~8/7 = {{monzo| | * 11-odd-limit: ~8/7 = {{monzo| 7/24 0 -1/24 }} | ||
: [{{monzo| 1 0 0 0 0 }}, {{monzo| | : [{{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. | : 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 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: 196/195, | Comma list: 196/195, 352/351, 385/384, 729/728 | ||
{{Mapping|legend=0| 1 1 7 3 -2 0 | 0 3 -24 -1 28 19 }} | |||
Optimal tunings: | 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=0| | {{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|legend=0| 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 ( | Badness (Sintel): 1.64 | ||
==== | ==== 13-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: | Comma list: 27/26, 36/35, 45/44, 507/500 | ||
{{Mapping|legend=0| 1 1 1 3 1 2 | 0 3 7 -1 13 9 }} | |||
Optimal tunings: | 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| | {{Optimal ET sequence|legend=0| 5e, 16, 21, 37b }} | ||
Badness ( | Badness (Sintel): 1.35 | ||
=== | === Spartan === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: | Comma list: 36/35, 56/55, 1029/1024 | ||
{{Mapping|legend=0| 1 1 1 3 5 | 0 3 7 -1 -8 }} | |||
Optimal tunings: | 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| | {{Optimal ET sequence|legend=0| 5, 16e, 21 }} | ||
Badness ( | Badness (Sintel): 2.07 | ||
==== 13-limit ==== | ==== 13-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: | Comma list: 27/26, 36/35, 56/55, 507/500 | ||
{{Mapping|legend=0| 1 1 1 3 5 2 | 0 3 7 -1 -8 9 }} | |||
Optimal tunings: | 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| | {{Optimal ET sequence|legend=0| 5, 16e, 21 }} | ||
Badness ( | 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|legend=0| 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 | [[Subgroup]]: 2.3.5.7 | ||
[[Comma list]]: | [[Comma list]]: 225/224, 1029/1024 | ||
{{Mapping|legend=1| | {{Mapping|legend=1| 1 1 3 3 | 0 6 -7 -2 }} | ||
: mapping generator: ~2, ~15/14 | |||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[ | * [[WE]]: ~2 = 1200.8209{{c}}, ~15/14 = 116.7550{{c}} | ||
: [[error map]]: {{val| 0. | : [[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 - | : 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 | |||
{{Optimal ET sequence|legend=1| | [[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]] ( | [[Badness]] (Sintel): 0.424 | ||
=== 11-limit === | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: | Comma list: 225/224, 243/242, 385/384 | ||
{{Mapping|legend=0| 1 1 3 3 2 | 0 6 -7 -2 15 }} | |||
Optimal tunings: | 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 | ||
{{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 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: | Comma list: 105/104, 144/143, 196/195, 243/242 | ||
{{Mapping|legend=0| 1 1 3 3 2 4 | 0 6 -7 -2 15 -3 }} | |||
Optimal tunings: | 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| | {{Optimal ET sequence|legend=0| 10, 21e, 31, 41, 72f }} | ||
Badness ( | Badness (Sintel): 0.771 | ||
=== | ===== 17-limit ===== | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11.13.17 | ||
Comma list: | Comma list: 105/104, 120/119, 144/143, 154/153, 170/169 | ||
{{Mapping|legend=0| 1 1 3 3 2 4 4 | 0 6 -7 -2 15 -3 1 }} | |||
Optimal tunings: | 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| | {{Optimal ET sequence|legend=0| 10, 21e, 31, 41, 72fg }} | ||
Badness ( | Badness (Sintel): 0.870 | ||
==== | ===== 19-limit ===== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13.17.19 | ||
Comma list: | 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|legend=0| 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|legend=0| 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|legend=0| 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|legend=0| 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 | Subgroup: 2.3.5.7.11.13.17.19.23 | ||
Comma list: | 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|legend=0| 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|legend=0| 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 | Subgroup: 2.3.5.7.11.13.17.19.23 | ||
Comma list: 225/224, 243/242, | 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|legend=0| 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| | {{Optimal ET sequence|legend=0| 41, 62, 103, 247c, 350bcde }} | ||
Badness ( | Badness (Sintel): 1.06 | ||
==== | ===== 17-limit ===== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13.17 | ||
Comma list: | Comma list: 225/224, 243/242, 273/272, 385/384, 847/845 | ||
{{Mapping|legend=0| 1 1 3 3 2 2 2 | 0 12 -14 -4 30 35 43 }} | |||
Optimal tunings: | 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| | {{Optimal ET sequence|legend=0| 41, 62, 103 }} | ||
Badness ( | Badness (Sintel): 1.15 | ||
===== | ===== 19-limit ===== | ||
Subgroup: 2.3.5.7.11.13.17 | Subgroup: 2.3.5.7.11.13.17.19 | ||
Comma list: | Comma list: | ||
{{Todo|complete temperament data|inline=1}} | |||
===== 23-limit ===== | |||
Subgroup: 2.3.5.7.11.13.17.19.23 | |||
Comma list: | |||
{{Todo|complete temperament data|inline=1}} | |||
==== | ===== Semihemisecordite ===== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13.17 | ||
Comma list: 225/224, 243/242, | Comma list: 225/224, 243/242, 289/288, 385/384, 847/845 | ||
{{Mapping|legend=0| 2 2 6 6 4 4 7 | 0 12 -14 -4 30 35 12 }} | |||
: mapping generators: ~17/12, ~27/26 | |||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~17/12 = 600.3951{{c}}, ~27/26 = 58.3260{{c}} | ||
* | * CWE: ~17/12 = 600.0000{{c}}, ~27/26 = 58.2974{{c}} | ||
{{Optimal ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 62, 144g, 206begg }} | ||
Badness ( | Badness (Sintel): 2.39 | ||
===== | ====== 19-limit ====== | ||
Subgroup: 2.3.5.7.11.13.17 | Subgroup: 2.3.5.7.11.13.17.19 | ||
Comma list: 225/224, 243/242, | Comma list: 209/208, 225/224, 243/242, 289/288, 361/360, 385/384 | ||
{{Mapping|legend=0| 2 2 6 6 4 4 7 8 | 0 12 -14 -4 30 35 12 5 }} | |||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~17/12 = 600.4418{{c}}, ~27/26 = 58.3255{{c}} | ||
* | * CWE: ~17/12 = 600.0000{{c}}, ~27/26 = 58.2928{{c}} | ||
{{Optimal ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 62, 144gh, 206begghh }} | ||
Badness ( | Badness (Sintel): 2.13 | ||
==== | ====== 23-limit ====== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13.17.19.23 | ||
Comma list: 225/224, 243/242, | Comma list: 209/208, 225/224, 243/242, 289/288, 323/322, 361/360, 385/384 | ||
{{Mapping|legend=0| 2 2 6 6 4 4 7 8 7 | 0 12 -14 -4 30 35 12 5 21 }} | |||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~17/12 = 600.4451{{c}}, ~27/26 = 58.3264{{c}} | ||
* | * CWE: ~17/12 = 600.0000{{c}}, ~27/26 = 58.2942{{c}} | ||
{{Optimal ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 62, 144gh, 206begghhi }} | ||
Badness ( | Badness (Sintel): 1.89 | ||
==== | ==== Phicordial ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: 225/224, 243/242, | Comma list: 225/224, 243/242, 385/384, 2200/2197 | ||
{{Mapping|legend=0| 1 -11 17 7 -28 3 | 0 18 -21 -6 45 1 }} | |||
: mapping generators: ~2, ~13/8 | |||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1200.7056{{c}}, ~13/8 = 839.3726{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~13/8 = 838.8831{{c}} | ||
{{Optimal ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 103, 216c, 319bcde, 535bccdef }} | ||
Badness ( | Badness (Sintel): 1.37 | ||
==== | ===== 17-limit ===== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13.17 | ||
Comma list: 225/224, 243/242, 273/272, 385/384, 2200/2197 | |||
{{Mapping|legend=0| 1 -11 17 7 -28 3 -5 | 0 18 -21 -6 45 1 13 }} | |||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1200.5918{{c}}, ~13/8 = 839.2912{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~13/8 = 838.8809{{c}} | ||
{{Optimal ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 103, 216c, 319bcde }} | ||
Badness ( | Badness (Sintel): 1.26 | ||
===== | ===== 19-limit ===== | ||
Subgroup: 2.3.5.7.11.13.17 | Subgroup: 2.3.5.7.11.13.17.19 | ||
Comma list: | Comma list: 210/209, 225/224, 243/242, 273/272, 385/384, 2200/2197 | ||
{{Todo|complete temperament data|inline=1}} | |||
===== 23-limit ===== | |||
Subgroup: 2.3.5.7.11.13.17.19.23 | |||
Comma list: 210/209, 225/224, 243/242, 273/272, 300/299, 385/384, 1105/1104 | |||
{{Todo|complete temperament data|inline=1}} | |||
=== | === Revelation === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: | Comma list: 99/98, 176/175, 1029/1024 | ||
{{Mapping|legend=0| 1 1 3 3 5 | 0 6 -7 -2 -16 }} | |||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1201.3320{{c}}, ~15/14 = 116.4057{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.2524{{c}} | ||
{{Optimal ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 10e, 21, 31 }} | ||
Badness ( | Badness (Sintel): 1.09 | ||
==== | ==== 13-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: | Comma list: 66/65, 99/98, 105/104, 512/507 | ||
{{Mapping|legend=0| 1 1 3 3 5 4 | 0 6 -7 -2 -16 -3 }} | |||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1200.6059{{c}}, ~15/14 = 116.3263{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.2564{{c}} | ||
{{Optimal ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 10e, 21, 31 }} | ||
Badness ( | Badness (Sintel): 1.22 | ||
=== | === Hemimiracle === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: 225/224, | Comma list: 225/224, 245/242, 1029/1024 | ||
{{Mapping|legend=0| 1 1 3 3 4 | 0 12 -14 -4 -11 }} | |||
: mapping generators: ~2, ~33/32 | |||
: mapping generators: ~ | |||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1200.2902{{c}}, ~33/32 = 58.4217{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~33/32 = 58.4062{{c}} | ||
{{Optimal ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 20, 21, 41 }} | ||
Badness ( | Badness (Sintel): 1.96 | ||
==== | ==== 13-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: | Comma list: 105/104, 196/195, 245/242, 512/507 | ||
{{Mapping|legend=0| 1 1 3 3 4 4 | 0 12 -14 -4 -11 -6 }} | |||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1199.8454{{c}}, ~33/32 = 58.4220{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~33/32 = 58.4305{{c}} | ||
{{Optimal ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 20, 21, 41 }} | ||
Badness (Sintel): 1.78 | |||
=== Oracle === | |||
The name is a portmanteau of [[orwell]] and [[miracle]]: Oracle is a weak extension of 7-limit miracle, splitting its ~[[16/15]] generator and an octave into two ~[[16/11]] generators. Additionally, when [[restriction|restricted]] to the 2.15.7.11 subgroup, oracle's generator corresponds to 2 stacked orwell generators. | |||
Subgroup: 2.3.5.7.11 | |||
Subgroup: 2.3.5.7.11 | |||
Comma list: | Comma list: 121/120, 225/224, 1029/1024 | ||
{{Mapping|legend=0| 1 -5 10 5 4 | 0 12 -14 -4 -1 }} | |||
: mapping generators: ~2, ~16/11 | |||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1201.2122{{c}}, ~16/11 = 658.9974{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~16/11 = 658.3320{{c}} | ||
{{Optimal ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 11, 20, 31, 82e, 113e, 144ee }} | ||
Badness ( | Badness (Sintel): 1.41 | ||
== | == 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. | |||
[[Subgroup]]: 2.3.5.7 | |||
[[Comma list]]: 1029/1024, 19683/19600 | |||
: mapping generators: ~2, ~ | {{Mapping|legend=1| 1 -2 -15 4 | 0 6 29 -2 }} | ||
: mapping generators: ~2, ~243/160 | |||
Optimal | [[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 ET sequence|legend= | {{Optimal ET sequence|legend=1| 72, 149, 221, 514bd, 735bcdd }} | ||
Badness ( | [[Badness]] (Sintel): 1.43 | ||
=== | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: | Comma list: 385/384, 441/440, 19683/19600 | ||
{{Mapping|legend=0| 1 -2 -15 4 16 | 0 6 29 -2 -21 }} | |||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1200.6243{{c}}, ~243/160 = 717.0969{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~243/160 = 716.7292{{c}} | ||
{{Optimal ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 72, 149, 221e, 293de }} | ||
Badness ( | Badness (Sintel): 0.941 | ||
=== | === 13-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: | Comma list: 351/350, 385/384, 441/440, 676/675 | ||
{{Mapping|legend=0| 1 -2 -15 4 16 -19 | 0 6 29 -2 -21 38 }} | |||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1200.6781{{c}}, ~91/60 = 717.1496{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~91/60 = 716.7520{{c}} | ||
{{Optimal ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 72, 149, 221ef }} | ||
Badness ( | Badness (Sintel): 0.905 | ||
=== | === 17-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13.17 | ||
Comma list: | Comma list: 273/272, 351/350, 385/384, 441/440, 676/675 | ||
{{Mapping|legend=0| 1 -2 -15 4 16 -19 -21 | 0 6 29 -2 -21 38 42 }} | |||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1200.6635{{c}}, ~68/45 = 717.1354{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~68/45 = 716.7472{{c}} | ||
{{Optimal ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 72, 149, 221ef }} | ||
Badness ( | Badness (Sintel): 0.800 | ||
== | == Valentine == | ||
{{Main| 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 [[7-limit symmetrical lattices|lattice of 7-limit tetrads]]. Valentine can be described as the {{nowrap| 31 & 46 }} temperament; its ploidacot is enneacot. [[77edo]], [[108edo]], or [[185edo]] make for excellent tunings, which also happen to be excellent tunings for [[starling]], the rank-3 temperament tempering out 126/125. Hence 7-limit valentine can be used whenever starling is wanted, with the extra tempering out of 1029/1024 having no discernible effect on tuning accuracy. Another tuning for valentine uses (3/2)<sup>1/9</sup> as a generator, giving pure 3/2 fifths. Valentine extends naturally to the 11-limit, tempering out 121/120 and 441/440; 46edo has a valentine generator 3\46 which is only 0.0117 cents sharp of the minimax generator, ([[11/7]])<sup>1/10</sup>. | |||
Valentine has a very straighforward [[S-expression]]-based comma list in the [[11-limit]] add-23 (i.e. the 2.3.5.7.11.23 subgroup) of {([[176/175|S8/S10 = S22 × S23 × S24]], [[121/120|S11]]), [[441/440|S21]], [[484/483|S22]], [[529/528|S23]], [[576/575|S24]]}, so it is the temperament that equalizes the 20::25 segment of the harmonic series. | |||
: | [[Subgroup]]: 2.3.5.7 | ||
[[Comma list]]: 126/125, 1029/1024 | |||
{{ | {{Mapping|legend=1| 1 1 2 3 | 0 9 5 -3 }} | ||
: mapping generators: ~2, ~21/20 | |||
[[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 }} | |||
[[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 | |||
[[Algebraic generator]]: smaller root of ''x''<sup>2</sup> - 89''x'' + 92, or (89 - sqrt (7553))/2, at 77.8616 cents. | |||
{{Optimal ET sequence|legend=1| 15, 31, 46, 77, 185 }} | |||
[[Badness]] (Sintel): 0.786 | |||
=== 11-limit === | |||
Subgroup: 2.3.5.7.11 | |||
Comma list: 121/120, 126/125, 176/175 | |||
= | {{Mapping|legend=0| 1 1 2 3 3 | 0 9 5 -3 7 }} | ||
Optimal tunings: | |||
* WE: ~2 = 1200.3890{{c}}, ~22/21 = 77.9065{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~22/21 = 77.9007{{c}} | |||
Minimax tuning: | |||
* 11-odd-limit: ~21/20 = {{monzo| 0 0 0 -1/10 1/10 }} | |||
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 1 0 0 -9/10 9/10 }}, {{monzo| 2 0 0 -1/2 1/2 }}, {{monzo| 3 0 0 3/10 -3/10 }}, {{monzo| 3 0 0 -7/10 7/10 }}] | |||
: unchanged-interval (eigenmonzo) basis: 2.11/7 | |||
: | Algebraic generator: positive root of 4''x''<sup>3</sup> + 15''x''<sup>2</sup> - 21, or else Gontrand2, the smallest positive root of 4''x''<sup>7</sup> - 8''x''<sup>6</sup> + 5. | ||
Optimal | {{Optimal ET sequence|legend=0| 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|legend=0| 1 1 2 3 3 5 | 0 9 5 -3 7 -20 }} | |||
Optimal tunings: | |||
* WE: ~2 = 1200.1967{{c}}, ~22/21 = 77.9708{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~22/21 = 77.9594{{c}} | |||
{{ | {{Optimal ET sequence|legend=0| 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|legend=0| 1 1 2 3 3 5 5 | 0 9 5 -3 7 -20 -14 }} | |||
=== | Optimal tunings: | ||
Subgroup: 2.3.5.7.11 | * WE: ~2 = 1200.0404{{c}}, ~22/21 = 78.0055{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~22/21 = 78.0029{{c}} | |||
{{Optimal ET sequence|legend=0| 15f, 31, 46, 77, 123e }} | |||
Badness (Sintel): 0.854 | |||
==== Lupercalia ==== | |||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: | Comma list: 66/65, 105/104, 121/120, 126/125 | ||
{{Mapping|legend=0| 1 1 2 3 3 3 | 0 9 5 -3 7 11 }} | |||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1199.9143{{c}}, ~22/21 = 77.7039{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~22/21 = 77.7049{{c}} | ||
{{Optimal ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 15, 31 }} | ||
Badness ( | Badness (Sintel): 0.881 | ||
=== | ==== Dwynwen ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: | Comma list: 91/90, 121/120, 126/125, 176/175 | ||
{{Mapping|legend=0| 1 1 2 3 3 2 | 0 9 5 -3 7 26 }} | |||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1200.1306{{c}}, ~22/21 = 78.2273{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~22/21 = 78.2241{{c}} | ||
{{Optimal ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 15, 31f, 46 }} | ||
Badness ( | Badness (Sintel): 0.969 | ||
=== | ==== Semivalentine ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: | Comma list: 121/120, 126/125, 169/168, 176/175 | ||
{{Mapping|legend=0| 2 2 4 6 6 7 | 0 9 5 -3 7 3 }} | |||
: mapping generators: ~55/39, ~22/21 | |||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~55/39 = 600.3497{{c}}, ~22/21 = 77.8845{{c}} | ||
* | * CWE: ~55/39 = 600.0000{{c}}, ~22/21 = 77.8715{{c}} | ||
{{Optimal ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 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|legend=0| 1 1 2 3 3 4 | 0 18 10 -6 14 -9 }} | |||
{{ | : mapping generators: ~2, ~40/39 | ||
: | |||
Optimal tunings: | |||
* WE: ~2 = 1199.6529{{c}}, ~40/39 = 39.0323{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~40/39 = 39.0383{{c}} | |||
{{Optimal ET sequence|legend=0| 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 generators: ~2, ~ | {{Mapping|legend=0| 1 -8 -3 6 -4 -16 | 0 18 10 -6 14 37 }} | ||
: mapping generators: ~2, ~13/9 | |||
Optimal tunings: | |||
* | * WE: ~2 = 1200.3929{{c}}, ~13/9 = 639.1320{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~13/9 = 638.9325{{c}} | |||
* | |||
{{Optimal ET sequence|legend=0| 15, 47ef, 62, 77 }} | |||
Badness (Sintel): 1.44 | |||
=== Hemivalentino === | |||
== | |||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: | Comma list: 126/125, 243/242, 1029/1024 | ||
{{Mapping|legend=0| 1 1 2 3 2 | 0 18 10 -6 45 }} | |||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1200.0816{{c}}, ~45/44 = 38.9236{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~45/44 = 38.9228{{c}} | ||
{{Optimal ET sequence|legend=0| 31, 92e, 123, 154, 185 }} | |||
Badness (Sintel): 2.03 | |||
==== 13-limit ==== | |||
=== | |||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: | Comma list: 126/125, 196/195, 243/242, 1029/1024 | ||
{{Mapping|legend=0| 1 1 2 3 2 5 | 0 18 10 -6 45 -40 }} | |||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1199.8782{{c}}, ~45/44 = 38.9440{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~45/44 = 38.9472{{c}} | ||
{{Optimal ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 31, 123, 154 }} | ||
Badness ( | Badness (Sintel): 2.39 | ||
==== | ==== Hemivalentoid ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: | Comma list: 126/125, 144/143, 243/242, 343/338 | ||
{{Mapping|legend=0| 1 1 2 3 2 4 | 0 18 10 -6 45 -9 }} | |||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1199.3614{{c}}, ~45/44 = 38.9721{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~45/44 = 38.9839{{c}} | ||
{{Optimal ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 31, 92ef }} | ||
Badness ( | Badness (Sintel): 2.39 | ||
==== | == Superkleismic == | ||
{{Main| 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 {{nowrap| 15 & 26 }} temperament. It splits the ~[[7/4]] into three ~6/5 generators of around 322 cents. This is noticeably sharper than the [[kleismic]] generator, hence the name. | |||
In the 11-limit, two generator steps can be identified with ~16/11, and in the 13-limit, the same step can be treated as ~13/9. The [[S-expression]]-based comma list of 13-limit superkleismic is {[[875/864|S5/S6]], [[1029/1024|S7/S8]], [[100/99|S10]], [[144/143|S12]], ([[441/440|S21]])}. Through careful observation of the equivalences therein one can derive the mapping of the full 13-limit. | |||
Note that the generator is given as 6/5's octave complement, [[5/3]], in the data that follow, since a stack of 9 such generators octave-reduced is the perfect fifth; the [[ploidacot]] of superkleismic is wau-enneacot. | |||
{{ | Superkleismic also sets two intervals of [[21/20]] equal to [[10/9]]; as {{nowrap| 10/9 {{=}} ([[20/19]])⋅([[19/18]]) }}, we can identify 21/20, 20/19, and 19/18 together to add prime 19, tempering out [[361/360]] ({{S|19}}) and [[400/399]] ({{S|20}}). This structure is preserved within the entire superkleismic tuning range between 15edo and 26edo, while extensions for primes 13 and 17 bifurcate and are of higher complexity and lower accuracy. | ||
41edo gives an obvious tuning in all the subgroups. | |||
[[Subgroup]]: 2.3.5.7 | |||
Subgroup: 2.3.5.7 | |||
Comma list: | [[Comma list]]: 875/864, 1029/1024 | ||
{{Mapping|legend=1| 1 -5 -5 5 | 0 9 10 -3 }} | |||
: mapping generators: ~2, ~5/3 | |||
Optimal | [[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 }} | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=1| 11c, 15, 26, 41 }} | ||
Badness ( | [[Badness]] (Sintel): 1.21 | ||
=== | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: | Comma list: 100/99, 245/242, 385/384 | ||
{{Mapping|legend=0| 1 -5 -5 5 2 | 0 9 10 -3 2 }} | |||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1200.1691{{c}}, ~5/3 = 878.2772{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~5/3 = 878.1606{{c}} | ||
{{Optimal ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 11c, 15, 26, 41, 179cde, 220cde, 261ccdee }} | ||
Badness ( | Badness (Sintel): 0.848 | ||
==== | ==== 2.3.5.7.11.19 subgroup ==== | ||
Subgroup: 2.3.5.7.11. | Subgroup: 2.3.5.7.11.19 | ||
Comma list: | Comma list: 100/99, 133/132, 190/189, 385/384 | ||
{{Mapping|legend=0| 1 -5 -5 5 2 -6 | 0 9 10 -3 2 14 }} | |||
: | Optimal tunings: | ||
* WE: ~2 = 1200.2289{{c}}, ~5/3 = 878.3409{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 878.1840{{c}} | |||
Optimal | {{Optimal ET sequence|legend=0| 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 val]]s 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 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: | Comma list: 100/99, 105/104, 144/143, 245/242 | ||
{{Mapping|legend=0| 1 -5 -5 5 2 -8 | 0 9 10 -3 2 16 }} | |||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1200.0261{{c}}, ~5/3 = 878.0252{{c}} | ||
* CWE: ~2 = 1200. | * CWE: ~2 = 1200.0000{{c}}, ~5/3 = 878.0073{{c}} | ||
{{Optimal ET sequence|legend=0| 15, | {{Optimal ET sequence|legend=0| 11cf, 15, 26, 41 }} | ||
Badness ( | Badness (Sintel): 0.887 | ||
=== | ==== 17-limit ==== | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11.13.17 | ||
Comma list: | Comma list: 100/99, 105/104, 120/119, 144/143, 245/242 | ||
{{Mapping|legend=0| 1 -5 -5 5 2 -8 -12 | 0 9 10 -3 2 16 22 }} | |||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1200.0488{{c}}, ~5/3 = 877.8872{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~5/3 = 877.8537{{c}} | ||
{{Optimal ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 11cfg, 15g, 26, 41 }} | ||
Badness ( | Badness (Sintel): 1.01 | ||
==== | ==== 19-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13.17.19 | ||
Comma list: | Comma list: 100/99, 105/104, 120/119, 144/143, 133/132, 190/189 | ||
{{Mapping|legend=0| 1 -5 -5 5 2 -8 -12 -6 | 0 9 10 -3 2 16 22 14 }} | |||
Optimal tunings: | 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| | {{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 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: | Comma list: 100/99, 196/195, 245/242, 385/384 | ||
{{Mapping|legend=0| 1 -5 -5 5 2 22 | 0 9 10 -3 2 -25 }} | |||
Optimal tunings: | 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| | {{Optimal ET sequence|legend=0| 15f, 41, 97, 138e }} | ||
Badness ( | 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|legend=0| 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|legend=0| 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 }} | ||
Subgroup: 2.3.5.7.11 | [[Badness]] (Sintel): 2.12 | ||
=== 11-limit === | |||
Subgroup: 2.3.5.7.11 | |||
Comma list: | Comma list: 385/384, 441/440, 2460375/2458624 | ||
{{Mapping|legend=0| 1 -5 25 5 -28 | 0 9 -31 -3 43 }} | |||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1200.4874{{c}}, ~224/135 = 878.2543{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~224/135 = 877.8987{{c}} | ||
{{Optimal ET sequence|legend=0| 41, 108e, 149, 190 }} | |||
Badness (Sintel): | Badness (Sintel): 1.35 | ||
== | == Unidec == | ||
{{Main| Unidec }} | |||
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. | |||
[[Subgroup]]: 2.3.5.7 | |||
[[Comma list]]: 1029/1024, 4375/4374 | |||
{{Mapping|legend=1| 2 -1 -3 7 | 0 6 11 -2 }} | |||
{{ | [[Optimal tuning]]s: | ||
* [[WE]]: ~1225/864 = 600.2429{{c}}, ~80/63 = 417.0073{{c}} | |||
: [[error map]]: {{val| +0.486 -0.154 +0.038 -1.140 }} | |||
* [[CWE]]: ~1225/864 = 600.0000{{c}}, ~80/63 = 416.8688{{c}} | |||
: error map: {{val| 0.000 -0.924 -1.090 -2.503 }} | |||
[[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 }} | ||
[[Badness]] (Sintel): 0.972 | |||
=== 11-limit === | |||
Subgroup: 2.3.5.7.11 | |||
Comma list: 385/384, 441/440, 4375/4374 | |||
Comma list: | |||
{{Mapping|legend=0| 2 -1 -3 7 9 | 0 6 11 -2 -3 }} | |||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~99/70 = 600.2497{{c}}, ~14/11 = 417.0085{{c}} | ||
* CWE: ~ | * CWE: ~99/70 = 600.0000{{c}}, ~14/11 = 416.8543{{c}} | ||
{{ | Minimax tuning: | ||
* [[11-odd-limit]]: ~10/9 = {{monzo| 5/28 -1/7 0 1/14 }} | |||
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 10/7 6/7 0 -3/7 0 }}, {{monzo| 57/28 11/7 0 -11/14 0 }}, {{monzo| 20/7 -2/7 0 1/7 0 }}, {{monzo| 99/28 -3/7 0 3/14 0 }}] | |||
: unchanged-interval (eigenmonzo) basis: 2.9/7 | |||
{{Optimal ET sequence|legend=0| 26, 46, 72, 118, 190 }} | |||
Badness (Sintel): 0.512 | |||
==== Ekadash ==== | |||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: | Comma list: 385/384, 441/440, 625/624, 729/728 | ||
{{Mapping|legend=0| 2 -1 -3 7 9 -19 | 0 6 11 -2 -3 38 }} | |||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~99/70 = 600.2497{{c}}, ~14/11 = 417.0085{{c}} | ||
* | * CWE: ~99/70 = 600.0000{{c}}, ~14/11 = 416.8543{{c}} | ||
{{Optimal ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 46f, 72, 118, 190, 262df, 452cdef }} | ||
Badness (Sintel): | Badness (Sintel): 0.842 | ||
==== | ==== Hendec ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: | Comma list: 169/168, 325/324, 364/363, 385/384 | ||
{{Mapping|legend=0| 2 -1 -3 7 9 6 | 0 6 11 -2 -3 2 }} | |||
Optimal tunings: | 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 ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 26, 46, 72, 190ff }} | ||
Badness (Sintel): | Badness (Sintel): 0.732 | ||
==== | ===== 17-limit ===== | ||
Subgroup: 2.3.5.7.11.13.17 | Subgroup: 2.3.5.7.11.13.17 | ||
Comma list: | Comma list: 169/168, 221/220, 273/272, 325/324, 364/363 | ||
{{Mapping|legend=0| 2 -1 -3 7 9 6 4 | 0 6 11 -2 -3 2 6 }} | |||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~17/12 = 600.3991{{c}}, ~14/11 = 417.0809{{c}} | ||
* | * CWE: ~17/12 = 600.0000{{c}}, ~14/11 = 416.8330{{c}} | ||
{{Optimal ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 26, 46, 72, 190ffg }} | ||
Badness (Sintel): | Badness (Sintel): 0.595 | ||
== | == Restles == | ||
{{ | {{See also| Lesser tendoneutralic }} | ||
Restles may be described as the {{nowrap| 77 & 87 }} temperament, and has a [[ploidacot]] signature of gamma-dodecacot. It was named by [[Petr Pařízek]] in 2011 for it is some sort of opposite to [[beatles]]<ref name="petr's long post">[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101780.html Yahoo! Tuning Group | ''Suggested names for the unclasified temperaments'']</ref>. | |||
[[ | |||
[[ | [[Subgroup]]: 2.3.5.7 | ||
[[Comma list]]: 1029/1024, 153664/151875 | |||
: mapping generators: ~ | {{Mapping|legend=1| 1 -2 8 4 | 0 12 -19 -4 }} | ||
: mapping generators: ~2. ~315/256 | |||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[ | * [[WE]]: ~2 = 1200.0322{{c}}, ~315/256 = 358.5581{{c}} | ||
: [[error map]]: {{val| 0. | : [[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 | : error map: {{val| 0.000 +0.626 +1.267 -3.019 }} | ||
{{Optimal ET sequence|legend=1| | {{Optimal ET sequence|legend=1| 77, 87, 164 }} | ||
[[Badness]] ( | [[Badness]] (Sintel): 2.73 | ||
=== | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | |||
Comma list: 385/384, 441/440, 153664/151875 | |||
{{Mapping|legend=1 | {{Mapping|legend=0| 1 -2 8 4 -7 | 0 12 -19 -4 35 }} | ||
Optimal tunings: | |||
* | * WE: ~2 = 1200.1110{{c}}, ~27/22 = 358.6045{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~27/22 = 358.5720{{c}} | |||
* | |||
{{Optimal ET sequence|legend=0| 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|legend=0| 1 -2 8 4 -7 4 | 0 12 -19 -4 35 -1 }} | |||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1200.0482{{c}}, ~~16/13 = 358.5883{{c}} | ||
* CWE: ~ | * CWE: ~2 = 1200.0000{{c}}, ~16/13 = 358.5741{{c}} | ||
{{Optimal ET sequence|legend=0| 77, 87, 164, 251d }} | |||
Badness (Sintel): 1.16 | |||
== Necromanteion == | |||
Necromanteion, named by [[Johannes Werpup]] in 2014<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_106371.html Yahoo! Tuning Group | ''Temperament ideas: A cuckoo, and two oracles'']</ref> may be described as the {{nowrap| 31 & 51c }} temperament. The generator is a subfifth representing ~[[35/24]], four of which minus two octaves make slendric's generator. Therefore, its [[ploidacot]] is wau-dodecacot. | |||
[[Subgroup]]: 2.3.5.7 | |||
Subgroup: 2.3.5.7 | |||
Comma list: | [[Comma list]]: 1029/1024, 5103/5000 | ||
{{Mapping|legend=1| 1 -5 -7 5 | 0 12 17 -4 }} | |||
: mapping generators: ~2, ~35/24 | |||
Optimal | [[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 }} | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=1| 11c, 20c, 31, 144c, 175c }} | ||
Badness ( | [[Badness]] (Sintel): 2.98 | ||
=== | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: | Comma list: 176/175, 243/242, 1029/1024 | ||
{{Mapping|legend=0| 1 -5 -7 5 -13 | 0 12 17 -4 30 }} | |||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1200.2862{{c}}, ~22/15 = 658.4276{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~22/15 = 658.2805{{c}} | ||
{{Optimal ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 20ce, 31, 113c, 144c }} | ||
Badness ( | Badness (Sintel): 1.77 | ||
=== | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: | Comma list: 144/143, 176/175, 243/242, 343/338 | ||
{{Mapping|legend=0| 1 -5 -7 5 -13 7 | 0 12 17 -4 30 -6 }} | |||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1199.3663{{c}}, ~22/15 = 658.0465{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~22/15 = 658.3800{{c}} | ||
{{Optimal ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 20ce, 31, 82cf, 113cf }} | ||
Badness ( | Badness (Sintel): 1.94 | ||
== Lagaca == | == Lagaca == | ||
Cryptically named by [[Petr Pařízek]] in 2011<ref name="petr's long post"/>, lagaca may be described as the {{nowrap| 10 & 118 }} temperament with a [[ploidacot]] signature of diploid wau-enneacot. The name actually refers to the fact that 12 generator steps in this temperament make ~[[7/3]], where "l", "g", "c" are integers alphabetically converted to letters. | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
[[Comma list]]: 1029/1024, 11529602/11390625 | [[Comma list]]: 1029/1024, 11529602/11390625 | ||
{{Mapping|legend=1| 2 | {{Mapping|legend=1| 2 -4 15 8 | 0 9 -13 -3 }} | ||
: mapping generators: ~3375/2401, ~450/343 | |||
: mapping generators: ~3375/2401, ~ | |||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[ | * [[WE]]: ~3375/2401 = 600.1355{{c}}, ~450/343 = 478.0813{{c}} | ||
: [[error map]]: {{val| 0. | : [[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. | : error map: {{val| 0.000 -0.202 +0.043 -2.743 }} | ||
{{Optimal ET sequence|legend=1| 10, 98, 108, 118 }} | {{Optimal ET sequence|legend=1| 10, 98, 108, 118 }} | ||
[[Badness]] ( | [[Badness]] (Sintel): 3.65 | ||
== | == Quartemka == | ||
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Quartemka]].'' | |||
Quartemka may be described as the {{nowrap| 26 & 61 }} temperament. Its [[ploidacot]] is 18-sheared 21-cot. It was named by [[Petr Pařízek]] in 2011 for its generator is close to 1/4 of the generator for [[emka]]<ref name="petr's long post"/>. | |||
[[Subgroup]]: 2.3.5.7 | |||
[[Comma list]]: 1029/1024, 1250000/1240029 | |||
{{Mapping|legend=1| 1 -17 -26 9 | 0 21 32 -7 }} | |||
: mapping generators: ~2, ~50/27 | |||
== | [[Optimal tuning]]s: | ||
* [[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 }} | |||
{{Optimal ET sequence|legend=1| 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|legend=0| 1 -17 -26 9 7 | 0 21 32 -7 -4 }} | ||
Optimal tunings: | |||
* WE: ~2 = 1200.3051{{c}}, ~50/27 = 1062.2805{{c}} | |||
* CWE: ~21 = 1200.0000{{c}}, ~50/27 = 1062.0147{{c}} | |||
{{ | {{Optimal ET sequence|legend=0| 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|legend=0| 1 -17 -26 9 7 -14 | 0 21 32 -7 -4 20 }} | |||
Optimal tunings: | |||
* WE: ~2 = 1200.2708{{c}}, ~24/13 = 1062.2496{{c}} | |||
* CWE: ~21 = 1200.0000{{c}}, ~24/13 = 1062.0139{{c}} | |||
{{Optimal ET sequence|legend=0| 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 {{nowrap| 103 & 118 }} temperament. Its [[ploidacot]] is iota-beta-27-cot. It was named by [[Petr Pařízek]] in 2011 for its generator is 1/9 of the generator for [[slendric]], so that 3×3 generators [[octave reduction|octave reduced]] give slendric's generator, and another ×3 give the [[3/2|perfect fifth]]<ref name="petr's long post"/>. | ||
[[Subgroup]]: 2.3.5.7 | |||
[[Comma list]]: 1029/1024, 1959552/1953125 | |||
{{Mapping|legend=1| 1 -11 -7 7 | 0 27 20 -9 }} | |||
: mapping generators: ~2, ~864/625 | |||
[[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 }} | |||
Optimal | {{Optimal ET sequence|legend=1| 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|legend=0| 1 -11 -7 7 -4 | 0 27 20 -9 16 }} | |||
[[ | Optimal tunings: | ||
* WE: ~2 = 1200.4953{{c}}, ~242/175 = 559.5243{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~242/175 = 559.3016{{c}} | |||
{{Optimal ET sequence|legend=0| 15, …, 88, 103, 118, 221e, 339de }} | |||
Badness (Sintel): 1.17 | |||
== Widefourth == | |||
[[Subgroup]]: 2.3.5.7 | |||
[[Comma list]]: 1029/1024, 48828125/48771072 | |||
{{Mapping|legend=1| 1 -17 -5 9 | 0 33 13 -11 }} | |||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[ | * [[WE]]: ~2 = 1200.4770{{c}}, ~4608/3125 = 676.0584{{c}} | ||
: [[error map]]: {{val| 0. | : [[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 | : error map: {{val| 0.000 -0.705 -0.973 -2.576 }} | ||
{{Optimal ET sequence|legend=1| | {{Optimal ET sequence|legend=1| 16, 71, 87, 103, 190 }} | ||
[[Badness]] ( | [[Badness]] (Sintel): 3.90 | ||
=== 11-limit === | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: 385/384, 441/440, | Comma list: 385/384, 441/440, 234375/234256 | ||
{{Mapping|legend=0| 1 16 8 -2 17 | 0 -33 -13 11 -31 }} | |||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1200.4852{{c}}, ~1250/847 = 676.0634{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~1250/847 = 675.7966{{c}} | ||
{{Optimal ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 16, 71, 87, 103, 190 }} | ||
Badness ( | Badness (Sintel): 1.35 | ||
=== 13-limit === | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: | Comma list: 385/384, 441/440, 625/624, 847/845 | ||
{{Mapping|legend=0| 1 16 8 -2 17 12 | 0 -33 -13 11 -31 -19 }} | |||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1200.4217{{c}}, ~77/52 = 676.0286{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~77/52 = 675.7967{{c}} | ||
{{Optimal ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 16, 71, 87, 103, 190 }} | ||
Badness ( | Badness (Sintel): 0.894 | ||
== | == Other subgroup extensions == | ||
=== Euslendric (2.3.7.13) === | |||
Forms of slendric in the most optimal range for the 2.3.7 temperament ({{nowrap| 36 & 77 }}) lack an obvious strong mapping of prime 5 or prime 11. However, slendric can extend well to the no-fives no-elevens [[29-limit]] by tempering out [[273/272]], [[343/342]], [[378/377]], [[392/391]], [[513/512]], and [[729/728]], or a comma basis defined in terms of [[S-expression]]s as {S7/S8, S14/S16, S15/S20, S24/S26, S27, S28}. [[113edo]] is an obvious tuning. | |||
Subgroup: 2.3.7.13 | |||
Comma list: 729/728, 1029/1024 | |||
{{Mapping|legend=1 | {{Mapping|legend=2| 1 1 3 0 | 0 3 -1 19 }} | ||
{{Mapping|legend=4| 1 1 0 3 0 0 | 0 3 0 -1 0 19 }} | |||
Optimal tunings: | |||
* | * WE: ~2 = 1200.5057{{c}}, ~8/7 = 233.7200{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.6534{{c}} | |||
* | |||
{{Optimal ET sequence|legend=0| 5, 31f, 36, 77, 113, 827bdddff }} | |||
Badness (Sintel): 0.339 | |||
==== 2.3.7.13.17 subgroup ==== | |||
Subgroup: 2.3.7.13.17 | |||
Comma list: 273/272, 729/728, 833/832 | |||
{{Mapping|legend=2| 1 1 3 0 0 | 0 3 -1 19 21 }} | |||
{{Mapping|legend=4| 1 1 0 3 0 0 0 | 0 3 0 -1 0 19 21 }} | |||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1200.5282{{c}}, ~8/7 = 233.6492{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.5776{{c}} | ||
{{Optimal ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 5g, 31fg, 36, 113, 149 }} | ||
Badness (Sintel): 0.332 | |||
==== 2.3.7.13.17.19 subgroup ==== | |||
Subgroup: 2.3.7.13.17.19 | |||
Comma list: 273/272, 343/342, 513/512, 729/728 | |||
{{Mapping|legend=2| 1 1 3 0 0 6 | 0 3 -1 19 21 -9 }} | |||
{{Mapping|legend=4| 1 1 0 3 0 0 0 6 | 0 3 0 -1 0 19 21 -9 }} | |||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1200.3292{{c}}, ~8/7 = 233.6651{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.6106{{c}} | ||
{{Optimal ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 5g, 36, 77, 113, 262df }} | ||
Badness ( | 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 | |||
{{Mapping|legend=2| 1 1 3 0 0 6 9 | 0 3 -1 19 21 -9 -23 }} | |||
{{Mapping|legend=1| 1 | {{Mapping|legend=4| 1 1 0 3 0 0 0 6 9 | 0 3 0 -1 0 19 21 -9 -23 }} | ||
: | Optimal tunings: | ||
* WE: ~2 = 1200.3127{{c}}, ~8/7 = 233.6679{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.6091{{c}} | |||
{{Optimal ET sequence|legend=0| 36, 77, 113, 262df }} | |||
Badness (Sintel): 0.474 | |||
==== 2.3.7.13.17.19.23.29 subgroup ==== | |||
Subgroup: 2.3.7.13.17.19.23.29 | |||
Comma list: 273/272, 343/342, 378/377, 392/391, 513/512, 609/608 | |||
{{Mapping|legend=2| 1 1 3 0 0 6 9 7 | 0 3 -1 19 21 -9 -23 -11 }} | |||
{{Mapping|legend=4| 1 1 0 3 0 0 0 6 9 7 | 0 3 0 -1 0 19 21 -9 -23 -11 }} | |||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1200.2503{{c}}, ~8/7 = 233.6688{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.6208{{c}} | ||
{{Optimal ET sequence|legend=0| 36, 77, 113 }} | |||
Badness (Sintel): 0.473 | |||
=== Baladic (2.3.7.13) === | |||
Baladic is a 2.3.7.13.17-subgroup temperament that attempts to approximate the Maqam Sikah Baladi scale. It tempers out [[169/168]] ({{S|13}}), which splits [[7/6]] in half ([[13/12]]~[[14/13]]) and one finds that the octave is therefore split in half via the interval [[91/64]], which is then equated to [[17/12]]. 36edo is an excellent baladic tuning. | |||
Subgroup: 2.3.7.13 | |||
Comma list: 169/168, 1029/1024 | |||
= | {{Mapping|legend=2| 2 2 6 7 | 0 3 -1 1 }} | ||
{{Mapping|legend=4| 2 2 0 6 0 7 | 0 3 0 -1 0 1 }} | |||
: mapping generators: ~91/64, ~8/7 | |||
{{ | Optimal tunings: | ||
* WE: ~91/64 = 600.4315{{c}}, ~8/7 = 233.7724{{c}} | |||
* CWE: ~91/64 = 600.0000{{c}}, ~8/7 = 233.7039{{c}} | |||
{{Optimal ET sequence|legend=0| 10, 26, 36, 154f, 190ff, 226ff, 262dfff }} | |||
Badness (Sintel): 0.434 | |||
==== 2.3.7.13.17 subgroup ==== | |||
Subgroup: 2.3.7.13.17 | |||
Comma list: 169/168, 273/272, 289/288 | |||
{{Mapping|legend=2| 2 2 6 7 7 | 0 3 -1 1 3 }} | |||
{{Mapping|legend=4| 2 2 0 6 0 7 7 | 0 3 0 -1 0 1 3 }} | |||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~17/12 = 600.4436{{c}}, ~8/7 = 233.7883{{c}} | ||
* | * CWE: ~17/12 = 600.0000{{c}}, ~8/7 = 233.7312{{c}} | ||
{{Optimal ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 10, 26, 36, 154f, 190ffg, 226ffg }} | ||
Badness ( | Badness (Sintel): 0.253 | ||
=== | === Gigapyth (2.3.7.85) === | ||
Subgroup: 2.3 | Subgroup: 2.3.7.85 | ||
Comma list: | Comma list: 1029/1024, 7225/7203 | ||
{{Mapping|legend=2| 1 -2 4 7 | 0 6 -2 -1 }} | |||
Optimal tunings: | Optimal tunings: | ||
* | * WE: ~2 = 1200.8295{{c}}, ~128/85 = 717.2597{{c}} | ||
* | * CWE: ~2 = 1200.0000{{c}}, ~128/85 = 716.7933{{c}} | ||
{{Optimal ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 5, 42*, 47, 52, 57, 62, 67, 72, 149*, 370d***, 519bdd***** }} | ||
<nowiki/>* Wart for 85 | |||
== | == References == | ||
[[Category:Gamelismic clan| ]] <!-- main article --> | |||
[[Category:Temperament clans]] | [[Category:Temperament clans]] | ||
[[Category: | [[Category:Catalogs of rank-2 temperaments]] | ||
[[Category:Listen]] | [[Category:Listen]] | ||