Gamelismic clan: Difference between revisions
m Terminology |
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==== Subgroup extensions ==== | ==== Subgroup extensions ==== | ||
No-five subgroup extensions of slendric include | 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, and baladic, a weak 2.3.7.13.17-subgroup extension, considered in [[#Other subgroup extensions]]. Dicussed elsewhere is [[No-fives subgroup temperaments #Gigapyth|gigapyth]] in the 2.3.7.85 subgroup. | ||
=== | === Radon === | ||
Radon is the no-fives version of [[rodan]], equating the diatonic major third to [[14/11]]. | |||
Subgroup: 2.3.7. | Subgroup: 2.3.7.11 | ||
Comma list: | Comma list: 896/891, 1029/1024 | ||
Subgroup-val mapping: {{mapping| 1 1 3 | Subgroup-val mapping: {{mapping| 1 1 3 6 | 0 3 -1 -13 }} | ||
Gencom mapping: {{mapping| 1 1 0 3 | Gencom mapping: {{mapping| 1 1 0 3 6 | 0 3 0 -1 -13 }} | ||
Optimal tunings: | Optimal tunings: | ||
* WE: ~2 = | * WE: ~2 = 1199.9708{{c}}, ~8/7 = 234.3748{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = | * CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.3813{{c}} | ||
{{Optimal ET sequence|legend=0| 5, | {{Optimal ET sequence|legend=0| 5, …, 36, 41, 87, 128 }} | ||
Badness (Sintel): 0. | 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 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: | Optimal tunings: | ||
* WE: ~2 = | * WE: ~2 = 1201.3979{{c}}, ~8/7 = 232.3010{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = | * CWE: ~2 = 1200.0000{{c}}, ~8/7 = 232.0621{{c}} | ||
{{Optimal ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 5, 26, 31, 88, 119be, 150be }} | ||
Badness (Sintel): 0. | 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: {{mapping| 1 1 0 3 5 1 | 0 3 12 -1 -8 14 }} | |||
Optimal tunings: | Optimal tunings: | ||
* WE: ~2 = | * WE: ~2 = 1201.0985{{c}}, ~8/7 = 232.0231{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = | * CWE: ~2 = 1200.0000{{c}}, ~8/7 = 231.8425{{c}} | ||
{{Optimal ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 5, 26, 31, 57, 88 }} | ||
Badness (Sintel): 0. | Badness (Sintel): 0.990 | ||
; Music | |||
* ''Prelude for solo piano'' (2014) by [[Chris Vaisvil]] – [https://web.archive.org/web/20201127013310/http://micro.soonlabel.com/16-ET/mothra/20141028_mothra16br4.mp3 play] | [https://www.chrisvaisvil.com/prelude-for-solo-piano-in-mothra16-brat-4-tuning/ blog] – in Mothra[16], brat 4 tuning | |||
Subgroup: 2.3.7.11 | ==== 17-limit ==== | ||
Subgroup: 2.3.5.7.11.13.17 | |||
Comma list: | 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: | Optimal tunings: | ||
* WE: ~2 = | * WE: ~2 = 1200.9734{{c}}, ~8/7 = 231.8960{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = | * CWE: ~2 = 1200.0000{{c}}, ~8/7 = 231.7392{{c}} | ||
{{Optimal ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 5g, 26, 31, 57, 88 }} | ||
Badness (Sintel): | 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: | Optimal tunings: | ||
* WE: ~ | * WE: ~2 = 1200.9663{{c}}, ~8/7 = 231.8393{{c}} | ||
* CWE: ~ | * CWE: ~2 = 1200.0000{{c}}, ~8/7 = 231.6842{{c}} | ||
{{Optimal ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 26, 31, 57 }} | ||
Badness (Sintel): | 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: | Optimal tunings: | ||
* WE: ~ | * WE: ~2 = 1200.7675{{c}}, ~8/7 = 232.5673{{c}} | ||
* CWE: ~ | * 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}} | |||
Mapping: {{mapping| 1 1 0 3 | {{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: | Optimal tunings: | ||
* WE: ~2 = 1201. | * WE: ~2 = 1201.1585{{c}}, ~8/7 = 231.5404{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = | * CWE: ~2 = 1200.0000{{c}}, ~8/7 = 231.3850{{c}} | ||
{{Optimal ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 5e, 21ce, 26 }} | ||
Badness (Sintel): | Badness (Sintel): 1.84 | ||
==== 13-limit ==== | ==== 13-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: | Comma list: 45/44, 78/77, 81/80, 640/637 | ||
Mapping: {{mapping| 1 1 0 3 | Mapping: {{mapping| 1 1 0 3 0 1 | 0 3 12 -1 18 14 }} | ||
Optimal tunings: | Optimal tunings: | ||
* WE: ~2 = 1201. | * WE: ~2 = 1201.1152{{c}}, ~8/7 = 231.5079{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 231. | * CWE: ~2 = 1200.0000{{c}}, ~8/7 = 231.3612{{c}} | ||
{{Optimal ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 5e, 21cef, 26 }} | ||
Badness (Sintel): | 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 microcent. | |||
[[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 | |||
Badness (Sintel): | |||
==== 13-limit ==== | ==== 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: {{mapping| 1 1 | Mapping: {{mapping| 1 1 -1 3 6 8 | 0 3 17 -1 -13 -22 }} | ||
Optimal tunings: | Optimal tunings: | ||
* WE: ~2 = 1199. | * WE: ~2 = 1199.9868{{c}}, ~8/7 = 234.4796{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = | * 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| | {{Optimal ET sequence|legend=0| 41, 46, 87 }} | ||
Badness (Sintel): | Badness (Sintel): 0.762 | ||
==== 17-limit ==== | ===== 17-limit ===== | ||
Subgroup: 2.3.5.7.11.13.17 | Subgroup: 2.3.5.7.11.13.17 | ||
Comma list: | Comma list: 154/153, 196/195, 245/243, 256/255, 273/272 | ||
Mapping: {{mapping| 1 1 | Mapping: {{mapping| 1 1 -1 3 6 8 8 | 0 3 17 -1 -13 -22 -20 }} | ||
Optimal tunings: | Optimal tunings: | ||
* WE: ~2 = 1199. | * WE: ~2 = 1199.8331{{c}}, ~8/7 = 234.4919{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = | * 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| | {{Optimal ET sequence|legend=0| 41, 46, 87 }} | ||
Badness (Sintel): | 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: {{mapping| 1 1 | Mapping: {{mapping| 1 1 -1 3 6 -1 | 0 3 17 -1 -13 24 }} | ||
Optimal tunings: | Optimal tunings: | ||
* WE: ~2 = | * WE: ~2 = 1200.2997{{c}}, ~8/7 = 234.6972{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = | * CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.6439{{c}} | ||
{{Optimal ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 5, 41f, 46 }} | ||
Badness (Sintel): 1. | Badness (Sintel): 1.40 | ||
=== | === Aerodino === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: | Comma list: 176/175, 245/243, 1029/1024 | ||
Mapping: {{mapping| 1 1 | Mapping: {{mapping| 1 1 -1 3 -3 | 0 3 17 -1 33 }} | ||
Optimal tunings: | Optimal tunings: | ||
* WE: ~2 = | * WE: ~2 = 1199.9179{{c}}, ~8/7 = 234.7123{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = | * CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.7256{{c}} | ||
{{Optimal ET sequence|legend=0| 5e, | {{Optimal ET sequence|legend=0| 5e, 41e, 46 }} | ||
Badness (Sintel): 1. | Badness (Sintel): 1.79 | ||
==== 13-limit ==== | ==== 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: {{mapping| 1 1 | Mapping: {{mapping| 1 1 -1 3 -3 -1 | 0 3 17 -1 33 24 }} | ||
Optimal tunings: | Optimal tunings: | ||
* WE: ~2 = | * WE: ~2 = 1200.0242{{c}}, ~8/7 = 234.7863{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = | * CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.7824{{c}} | ||
{{Optimal ET sequence|legend=0| 5e, | {{Optimal ET sequence|legend=0| 5e, 41ef, 46 }} | ||
Badness (Sintel): 1. | 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 }} | ||
Comma list: | [[Badness]] (Sintel): 1.20 | ||
=== 11-limit === | |||
Subgroup: 2.3.5.7.11 | |||
Comma list: 385/384, 441/440, 10976/10935 | |||
Mapping: {{mapping| 1 1 - | Mapping: {{mapping| 1 1 7 3 -2 | 0 3 -24 -1 28 }} | ||
Optimal tunings: | Optimal tunings: | ||
* WE: ~2 = | * WE: ~2 = 1200.3453{{c}}, ~8/7 = 233.9988{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = | * CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.9312{{c}} | ||
Minimax tuning: | Minimax tuning: | ||
* | * 11-odd-limit: ~8/7 = {{monzo| 7/24 0 -1/24 }} | ||
: unchanged-interval (eigenmonzo) basis: 2. | : [{{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 }} | |||
Mapping: {{mapping| 1 1 - | |||
Optimal tunings: | Optimal tunings: | ||
* WE: ~2 = | * WE: ~2 = 1200.1222{{c}}, ~8/7 = 233.9228{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = | * 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 | |||
{{Optimal ET sequence|legend= | {{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. | [[Badness]] (Sintel): 1.54 | ||
=== | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: | Comma list: 36/35, 45/44, 1029/1024 | ||
Mapping: {{mapping| 1 1 | Mapping: {{mapping| 1 1 1 3 1 | 0 3 7 -1 13 }} | ||
Optimal tunings: | Optimal tunings: | ||
* WE: ~2 = | * WE: ~2 = 1201.3609{{c}}, ~8/7 = 227.6312{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = | * CWE: ~2 = 1200.0000{{c}}, ~8/7 = 227.4955{{c}} | ||
{{Optimal ET sequence|legend=0| 5e, | {{Optimal ET sequence|legend=0| 5e, 16, 21, 37b }} | ||
Badness (Sintel): 1. | Badness (Sintel): 1.64 | ||
==== 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, 45/44, 507/500 | ||
Mapping: {{mapping| 1 1 | Mapping: {{mapping| 1 1 1 3 1 2 | 0 3 7 -1 13 9 }} | ||
Optimal tunings: | Optimal tunings: | ||
* WE: ~2 = | * WE: ~2 = 1201.0996{{c}}, ~8/7 = 227.4378{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = | * CWE: ~2 = 1200.0000{{c}}, ~8/7 = 227.3327{{c}} | ||
{{Optimal ET sequence|legend=0| 5e, | {{Optimal ET sequence|legend=0| 5e, 16, 21, 37b }} | ||
Badness (Sintel): 1. | 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: {{mapping| 1 1 | Mapping: {{mapping| 1 1 1 3 5 | 0 3 7 -1 -8 }} | ||
Optimal tunings: | Optimal tunings: | ||
* WE: ~2 = | * WE: ~2 = 1198.9344{{c}}, ~8/7 = 229.3316{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = | * CWE: ~2 = 1200.0000{{c}}, ~8/7 = 229.5124{{c}} | ||
{{Optimal ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 5, 16e, 21 }} | ||
Badness (Sintel): | 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: {{mapping| 1 1 | Mapping: {{mapping| 1 1 1 3 5 2 | 0 3 7 -1 -8 9 }} | ||
Optimal tunings: | Optimal tunings: | ||
* WE: ~2 = | * WE: ~2 = 1198.3002{{c}}, ~8/7 = 228.7341{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = | * CWE: ~2 = 1200.0000{{c}}, ~8/7 = 229.0044{{c}} | ||
{{Optimal ET sequence|legend=0| | {{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 | [[Subgroup]]: 2.3.5.7 | ||
[[Comma list]]: | [[Comma list]]: 21/20, 144/125 | ||
{{Mapping|legend=1| 1 1 | {{Mapping|legend=1| 1 1 2 3 | 0 3 2 -1 }} | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[WE]]: ~2 = | * [[WE]]: ~2 = 1192.4932{{c}}, ~8/7 = 229.3187{{c}} | ||
: [[error map]]: {{val| | : [[error map]]: {{val| -7.507 -21.506 +57.310 -20.665 }} | ||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = | * [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 229.6649{{c}} | ||
: error map: {{val| 0.000 - | : 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= | {{Optimal ET sequence|legend=1| 31, 98c, 129c, 160bc }} | ||
Badness (Sintel): | [[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 | [[Subgroup]]: 2.3.5.7 | ||
[[Comma list]]: | [[Comma list]]: 135/128, 360/343 | ||
{{Mapping|legend=1| 1 1 | {{Mapping|legend=1| 1 1 4 3 | 0 3 -9 -1 }} | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[WE]]: ~2 = | * [[WE]]: ~2 = 1205.6135{{c}}, ~8/7 = 227.5283{{c}} | ||
: [[error map]]: {{val| + | : [[error map]]: {{val| +5.613 -13.757 -11.614 +20.486 }} | ||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = | * [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 226.3207{{c}} | ||
: error map: {{val| 0.000 - | : error map: {{val| 0.000 -22.993 -23.200 +4.853 }} | ||
{{Optimal ET sequence|legend=1| | {{Optimal ET sequence|legend=1| 5c, 11, 16 }} | ||
[[Badness]] (Sintel): | [[Badness]] (Sintel): 4.03 | ||
=== 11-limit === | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: | Comma list: 33/32, 45/44, 352/343 | ||
Mapping: {{mapping| 1 1 | Mapping: {{mapping| 1 1 4 3 4 | 0 3 -9 -1 -3 }} | ||
Optimal tunings: | Optimal tunings: | ||
* WE: ~2 = | * WE: ~2 = 1206.2134{{c}}, ~8/7 = 227.6004{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = | * CWE: ~2 = 1200.0000{{c}}, ~8/7 = 226.2421{{c}} | ||
{{Optimal ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 5c, 11, 16 }} | ||
Badness (Sintel): | 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= | {{Optimal ET sequence|legend=1| 10, 21, 31, 41, 72 }} | ||
Badness (Sintel): | [[Badness]] (Sintel): 0.424 | ||
=== | === 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: {{mapping| | Mapping: {{mapping| 1 1 3 3 2 | 0 6 -7 -2 15 }} | ||
Optimal tunings: | Optimal tunings: | ||
* WE: ~2 = | * WE: ~2 = 1200.7626{{c}}, ~15/14 = 116.7069{{c}} | ||
* CWE: ~2 = 1200.0000{{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 | |||
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= | {{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 | |||
==== 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 | |||
==== 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= | {{Optimal ET sequence|legend=0| 31f, 41, 72, 185cf, 257cff }} | ||
Badness (Sintel): 0.703 | |||
=== | ===== 17-limit ===== | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11.13.17 | ||
Comma list: | Comma list: 225/224, 243/242, 273/272, 325/324, 385/384 | ||
Mapping: {{mapping| 1 1 | Mapping: {{mapping| 1 1 3 3 2 0 0 | 0 6 -7 -2 15 38 42 }} | ||
Optimal tunings: | Optimal tunings: | ||
* WE: ~2 = | * WE: ~2 = 1200.7570{{c}}, ~15/14 = 116.8011{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~ | * CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.7408{{c}} | ||
{{Optimal ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 31fg, 41, 72, 185cf, 257cff }} | ||
Badness (Sintel): | Badness (Sintel): 0.748 | ||
== | ==== 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= | {{Optimal ET sequence|legend=0| 10, 62, 72 }} | ||
Badness (Sintel): 0.822 | |||
=== | ==== Hemisecordite ==== | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: 225/224, 243/242, 385/384 | Comma list: 225/224, 243/242, 385/384, 847/845 | ||
Mapping: {{mapping| 1 1 3 3 2 | 0 | Mapping: {{mapping| 1 1 3 3 2 2 | 0 12 -14 -4 30 35 }} | ||
: mapping generators: ~2, ~27/26 | |||
Optimal tunings: | Optimal tunings: | ||
* WE: ~2 = 1200. | * WE: ~2 = 1200.6969{{c}}, ~27/26 = 58.3217{{c}} | ||
* CWE: ~2 = 1200.0000{{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 | |||
===== Semihemisecordite ===== | |||
Subgroup: 2.3.5.7.11.13.17 | |||
Comma list: 225/224, 243/242, 289/288, 385/384, 847/845 | |||
Mapping: {{mapping| 2 2 6 6 4 4 7 | 0 12 -14 -4 30 35 12 }} | |||
: mapping generators: ~17/12, ~27/26 | |||
Mapping: {{mapping| | |||
Optimal tunings: | Optimal tunings: | ||
* WE: ~ | * WE: ~17/12 = 600.3951{{c}}, ~27/26 = 58.3260{{c}} | ||
* CWE: ~ | * 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 (Sintel): | Badness (Sintel): 2.39 | ||
==== | ====== 19-limit ====== | ||
Subgroup: 2.3.5.7.11.13 | 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: {{mapping| | Mapping: {{mapping| 2 2 6 6 4 4 7 8 | 0 12 -14 -4 30 35 12 5 }} | ||
Optimal tunings: | Optimal tunings: | ||
* WE: ~ | * WE: ~17/12 = 600.4418{{c}}, ~27/26 = 58.3255{{c}} | ||
* CWE: ~ | * 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 (Sintel): | Badness (Sintel): 2.13 | ||
===== | ====== 23-limit ====== | ||
Subgroup: 2.3.5.7.11.13.17 | 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: {{mapping| | Mapping: {{mapping| 2 2 6 6 4 4 7 8 7 | 0 12 -14 -4 30 35 12 5 21 }} | ||
Optimal tunings: | Optimal tunings: | ||
* WE: ~ | * WE: ~17/12 = 600.4451{{c}}, ~27/26 = 58.3264{{c}} | ||
* CWE: ~ | * 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 (Sintel): | 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: {{mapping| 1 | Mapping: {{mapping| 1 -11 17 7 -28 3 | 0 18 -21 -6 45 1 }} | ||
: mapping generators: ~2, ~13/8 | |||
Optimal tunings: | Optimal tunings: | ||
* WE: ~2 = 1200. | * WE: ~2 = 1200.7056{{c}}, ~13/8 = 839.3726{{c}} | ||
* CWE: ~2 = 1200.0000{{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 (Sintel): | Badness (Sintel): 1.37 | ||
===== 17-limit ===== | ===== 17-limit ===== | ||
Subgroup: 2.3.5.7.11.13.17 | Subgroup: 2.3.5.7.11.13.17 | ||
Comma list: 225/224, 243/242, 273/272, | Comma list: 225/224, 243/242, 273/272, 441/440, 2200/2197 | ||
Mapping: {{mapping| 1 | Mapping: {{mapping| 1 -11 17 7 -28 3 -5 | 0 18 -21 -6 45 1 13 }} | ||
Optimal tunings: | Optimal tunings: | ||
* WE: ~2 = 1200. | * WE: ~2 = 1200.5918{{c}}, ~13/8 = 839.2912{{c}} | ||
* CWE: ~2 = 1200.0000{{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 (Sintel): | Badness (Sintel): 1.26 | ||
=== | === Revelation === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: | Comma list: 99/98, 176/175, 1029/1024 | ||
Mapping: {{mapping| | Mapping: {{mapping| 1 1 3 3 5 | 0 6 -7 -2 -16 }} | ||
Optimal tunings: | Optimal tunings: | ||
* WE: ~ | * WE: ~2 = 1201.3320{{c}}, ~15/14 = 116.4057{{c}} | ||
* CWE: ~ | * CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.2524{{c}} | ||
{{Optimal ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 10e, 21, 31 }} | ||
Badness (Sintel): 1. | 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: {{mapping| | Mapping: {{mapping| 1 1 3 3 5 4 | 0 6 -7 -2 -16 -3 }} | ||
Optimal tunings: | Optimal tunings: | ||
* WE: ~ | * WE: ~2 = 1200.6059{{c}}, ~15/14 = 116.3263{{c}} | ||
* CWE: ~ | * CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.2564{{c}} | ||
{{Optimal ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 10e, 21, 31 }} | ||
Badness (Sintel): | 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: {{mapping| 1 1 3 3 | Mapping: {{mapping| 1 1 3 3 4 | 0 12 -14 -4 -11 }} | ||
: mapping generators: ~2, ~ | : mapping generators: ~2, ~33/32 | ||
Optimal tunings: | Optimal tunings: | ||
* WE: ~2 = 1200. | * WE: ~2 = 1200.2902{{c}}, ~33/32 = 58.4217{{c}} | ||
* CWE: ~2 = 1200.0000{{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 (Sintel): 1. | 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: {{mapping| 1 1 3 3 | Mapping: {{mapping| 1 1 3 3 4 4 | 0 12 -14 -4 -11 -6 }} | ||
Optimal tunings: | Optimal tunings: | ||
* WE: ~2 = | * WE: ~2 = 1199.8454{{c}}, ~33/32 = 58.4220{{c}} | ||
* CWE: ~2 = 1200.0000{{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. | Badness (Sintel): 1.78 | ||
=== | === Oracle === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: 225/224, | Comma list: 121/120, 225/224, 1029/1024 | ||
Mapping: {{mapping| | Mapping: {{mapping| 1 -5 10 5 4 | 0 12 -14 -4 -1 }} | ||
: mapping generators: ~ | : mapping generators: ~2, ~16/11 | ||
Optimal tunings: | Optimal tunings: | ||
* WE: ~ | * WE: ~2 = 1201.2122{{c}}, ~16/11 = 658.9974{{c}} | ||
* CWE: ~ | * 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 (Sintel): | 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|legend=1| 1 -2 -15 4 | 0 6 29 -2 }} | |||
: mapping generators: ~2, ~243/160 | |||
{{Optimal ET sequence|legend= | [[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=1| 72, 149, 221, 514bd, 735bcdd }} | |||
Badness (Sintel): | [[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: {{mapping| 2 | Mapping: {{mapping| 1 -2 -15 4 16 | 0 6 29 -2 -21 }} | ||
Optimal tunings: | Optimal tunings: | ||
* WE: ~ | * WE: ~2 = 1200.6243{{c}}, ~243/160 = 717.0969{{c}} | ||
* CWE: ~ | * 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 (Sintel): | Badness (Sintel): 0.941 | ||
=== | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: | Comma list: 351/350, 385/384, 441/440, 676/675 | ||
Mapping: {{mapping| 1 - | Mapping: {{mapping| 1 -2 -15 4 16 -19 | 0 6 29 -2 -21 38 }} | ||
Optimal tunings: | Optimal tunings: | ||
* WE: ~2 = 1200. | * WE: ~2 = 1200.6781{{c}}, ~91/60 = 717.1496{{c}} | ||
* CWE: ~2 = 1200.0000{{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 (Sintel): | Badness (Sintel): 0.905 | ||
=== 17-limit === | |||
Subgroup: 2.3.5.7.11.13.17 | Subgroup: 2.3.5.7.11.13.17 | ||
Comma list: | Comma list: 273/272, 351/350, 385/384, 441/440, 676/675 | ||
Mapping: {{mapping| 1 - | Mapping: {{mapping| 1 -2 -15 4 16 -19 -21 | 0 6 29 -2 -21 38 42 }} | ||
Optimal tunings: | Optimal tunings: | ||
* WE: ~2 = 1200. | * WE: ~2 = 1200.6635{{c}}, ~68/45 = 717.1354{{c}} | ||
* CWE: ~2 = 1200.0000{{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 (Sintel): | 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 | Subgroup: 2.3.5.7.11 | ||
Comma list: | Comma list: 121/120, 126/125, 176/175 | ||
Mapping: {{mapping| 1 1 3 3 | Mapping: {{mapping| 1 1 2 3 3 | 0 9 5 -3 7 }} | ||
Optimal tunings: | Optimal tunings: | ||
* WE: ~2 = 1200. | * WE: ~2 = 1200.3890{{c}}, ~22/21 = 77.9065{{c}} | ||
* CWE: ~2 = 1200.0000{{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 ET sequence|legend=0| 15, 31, 46, 77 }} | ||
Badness (Sintel): 0.552 | |||
Mapping: {{mapping| 1 1 3 3 | ==== Valentino ==== | ||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 121/120, 126/125, 176/175, 196/195 | |||
Mapping: {{mapping| 1 1 2 3 3 5 | 0 9 5 -3 7 -20 }} | |||
Optimal tunings: | Optimal tunings: | ||
* WE: ~2 = | * WE: ~2 = 1200.1967{{c}}, ~22/21 = 77.9708{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~ | * CWE: ~2 = 1200.0000{{c}}, ~22/21 = 77.9594{{c}} | ||
{{Optimal ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 15f, 31, 46, 77 }} | ||
Badness (Sintel): | Badness (Sintel): 0.854 | ||
=== | ===== 17-limit ===== | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11.13.17 | ||
Comma list: 121/120, | Comma list: 121/120, 126/125, 154/153, 176/175, 196/195 | ||
Mapping: {{mapping| 1 | Mapping: {{mapping| 1 1 2 3 3 5 5 | 0 9 5 -3 7 -20 -14 }} | ||
Optimal tunings: | Optimal tunings: | ||
* WE: ~2 = | * WE: ~2 = 1200.0404{{c}}, ~22/21 = 78.0055{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~ | * CWE: ~2 = 1200.0000{{c}}, ~22/21 = 78.0029{{c}} | ||
{{Optimal ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 15f, 31, 46, 77, 123e }} | ||
Badness (Sintel): | Badness (Sintel): 0.854 | ||
== | ==== Lupercalia ==== | ||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 66/65, 105/104, 121/120, 126/125 | |||
Mapping: {{mapping| 1 1 2 3 3 3 | 0 9 5 -3 7 11 }} | |||
{{ | Optimal tunings: | ||
* 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| 15, 31 }} | |||
Badness (Sintel): 0.881 | |||
==== Dwynwen ==== | |||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 91/90, 121/120, 126/125, 176/175 | |||
Mapping: {{mapping| 1 1 2 3 3 2 | 0 9 5 -3 7 26 }} | |||
Mapping: {{mapping| 1 | |||
Optimal tunings: | Optimal tunings: | ||
* WE: ~2 = 1200. | * WE: ~2 = 1200.1306{{c}}, ~22/21 = 78.2273{{c}} | ||
* CWE: ~2 = 1200.0000{{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 (Sintel): 0. | 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: {{mapping| | Mapping: {{mapping| 2 2 4 6 6 7 | 0 9 5 -3 7 3 }} | ||
: mapping generators: ~55/39, ~22/21 | |||
Optimal tunings: | Optimal tunings: | ||
* WE: ~ | * WE: ~55/39 = 600.3497{{c}}, ~22/21 = 77.8845{{c}} | ||
* CWE: ~ | * 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): | Badness (Sintel): 1.35 | ||
=== | ==== Hemivalentine ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: | Comma list: 121/120, 126/125, 176/175, 343/338 | ||
Mapping: {{mapping| 1 | Mapping: {{mapping| 1 1 2 3 3 4 | 0 18 10 -6 14 -9 }} | ||
: mapping generators: ~2, ~40/39 | |||
Optimal tunings: | Optimal tunings: | ||
* WE: ~2 = | * WE: ~2 = 1199.6529{{c}}, ~40/39 = 39.0323{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~ | * CWE: ~2 = 1200.0000{{c}}, ~40/39 = 39.0383{{c}} | ||
{{Optimal ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 30, 31, 61, 92f }} | ||
Badness (Sintel): | Badness (Sintel): 1.94 | ||
== | ==== Demivalentine ==== | ||
Subgroup: 2.3.5.7.11.13 | |||
: | |||
Comma list: 121/120, 126/125, 176/175, 676/675 | |||
Mapping: {{mapping| 1 -8 -3 6 -4 -16 | 0 18 10 -6 14 37 }} | |||
: mapping generators: ~2, ~13/9 | |||
Optimal tunings: | |||
* 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 | |||
: | |||
Comma list: 126/125, 243/242, 1029/1024 | |||
Mapping: {{mapping| 1 1 2 3 2 | 0 18 10 -6 45 }} | |||
{{ | 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 | |||
Comma list: | ==== 13-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 126/125, 196/195, 243/242, 1029/1024 | |||
Mapping: {{mapping| 1 1 2 3 | Mapping: {{mapping| 1 1 2 3 2 5 | 0 18 10 -6 45 -40 }} | ||
Optimal tunings: | Optimal tunings: | ||
* WE: ~2 = | * WE: ~2 = 1199.8782{{c}}, ~45/44 = 38.9440{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~ | * CWE: ~2 = 1200.0000{{c}}, ~45/44 = 38.9472{{c}} | ||
{{Optimal ET sequence|legend=0| 31, 123, 154 }} | |||
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: {{mapping| 1 1 2 3 | Mapping: {{mapping| 1 1 2 3 2 4 | 0 18 10 -6 45 -9 }} | ||
Optimal tunings: | Optimal tunings: | ||
* WE: ~2 = | * WE: ~2 = 1199.3614{{c}}, ~45/44 = 38.9721{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~ | * CWE: ~2 = 1200.0000{{c}}, ~45/44 = 38.9839{{c}} | ||
{{Optimal ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 31, 92ef }} | ||
Badness (Sintel): | 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 = | * [[WE]]: ~2 = 1200.7640{{c}}, ~5/3 = 878.6289{{c}} | ||
* CWE: ~2 = 1200.0000{{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 (Sintel): | [[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: {{mapping| 1 | Mapping: {{mapping| 1 -5 -5 5 2 | 0 9 10 -3 2 }} | ||
Optimal tunings: | Optimal tunings: | ||
* WE: ~2 = 1200. | * WE: ~2 = 1200.1691{{c}}, ~5/3 = 878.2772{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~ | * CWE: ~2 = 1200.0000{{c}}, ~5/3 = 878.1606{{c}} | ||
{{Optimal ET sequence|legend=0| 15, | {{Optimal ET sequence|legend=0| 11c, 15, 26, 41, 179cde, 220cde, 261ccdee }} | ||
Badness (Sintel): 0. | 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: {{mapping| 2 | Mapping: {{mapping| 1 -5 -5 5 2 -6 | 0 9 10 -3 2 14 }} | ||
Optimal tunings: | Optimal tunings: | ||
* WE: ~ | * WE: ~2 = 1200.2289{{c}}, ~5/3 = 878.3409{{c}} | ||
* CWE: ~ | * CWE: ~2 = 1200.0000{{c}}, ~5/3 = 878.1840{{c}} | ||
{{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: {{mapping| 1 | Mapping: {{mapping| 1 -5 -5 5 2 -8 | 0 9 10 -3 2 16 }} | ||
Optimal tunings: | Optimal tunings: | ||
* WE: ~2 = | * WE: ~2 = 1200.0261{{c}}, ~5/3 = 878.0252{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~ | * CWE: ~2 = 1200.0000{{c}}, ~5/3 = 878.0073{{c}} | ||
{{Optimal ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 11cf, 15, 26, 41 }} | ||
Badness (Sintel): | Badness (Sintel): 0.887 | ||
==== | ==== 17-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13.17 | ||
Comma list: | Comma list: 100/99, 105/104, 120/119, 144/143, 245/242 | ||
Mapping: {{mapping| 1 - | Mapping: {{mapping| 1 -5 -5 5 2 -8 -12 | 0 9 10 -3 2 16 22 }} | ||
Optimal tunings: | Optimal tunings: | ||
* WE: ~2 = 1200. | * WE: ~2 = 1200.0488{{c}}, ~5/3 = 877.8872{{c}} | ||
* CWE: ~2 = 1200.0000{{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 (Sintel): 1. | Badness (Sintel): 1.01 | ||
=== | ==== 19-limit ==== | ||
Subgroup: 2.3.5.7.11 | 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: {{mapping| 1 | Mapping: {{mapping| 1 -5 -5 5 2 -8 -12 -6 | 0 9 10 -3 2 16 22 14 }} | ||
Optimal tunings: | Optimal tunings: | ||
* WE: ~2 = 1200. | * WE: ~2 = 1200.2120{{c}}, ~5/3 = 878.0243{{c}} | ||
* CWE: ~2 = 1200.0000{{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: {{mapping| 1 | Mapping: {{mapping| 1 -5 -5 5 2 22 | 0 9 10 -3 2 -25 }} | ||
Optimal tunings: | Optimal tunings: | ||
* WE: ~2 = 1199. | * WE: ~2 = 1199.8272{{c}}, ~5/3 = 878.1538{{c}} | ||
* CWE: ~2 = 1200.0000{{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 (Sintel): | Badness (Sintel): 1.40 | ||
==== | ==== 17-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13.17 | ||
Comma list: | Comma list: 100/99, 154/153, 196/195, 245/242, 256/255 | ||
Mapping: {{mapping| 1 | Mapping: {{mapping| 1 -5 -5 5 2 22 18 | 0 9 10 -3 2 -25 -19 }} | ||
Optimal tunings: | Optimal tunings: | ||
* WE: ~2 = 1199. | * WE: ~2 = 1199.5964{{c}}, ~5/3 = 878.0482{{c}} | ||
* CWE: ~2 = 1200.0000{{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 | [[Subgroup]]: 2.3.5.7 | ||
[[Comma list]]: | [[Comma list]]: 1029/1024, 2460375/2458624 | ||
{{Mapping|legend=1| 1 -5 | {{Mapping|legend=1| 1 -5 25 5 | 0 9 -31 -3 }} | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[WE]]: ~2 = 1200. | * [[WE]]: ~2 = 1200.4835{{c}}, ~224/135 = 878.2507{{c}} | ||
: [[error map]]: {{val| +0. | : [[error map]]: {{val| +0.484 -0.117 +0.004 -1.160 }} | ||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~ | * [[CWE]]: ~2 = 1200.0000{{c}}, ~224/135 = 877.8926{{c}} | ||
: error map: {{val| 0.000 | : error map: {{val| 0.000 -0.921 -0.985 -2.504 }} | ||
{{Optimal ET sequence|legend=1| | {{Optimal ET sequence|legend=1| 41, 108, 149, 190 }} | ||
[[Badness]] (Sintel): | [[Badness]] (Sintel): 2.12 | ||
=== 11-limit === | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: | Comma list: 385/384, 441/440, 2460375/2458624 | ||
Mapping: {{mapping| 1 -5 - | Mapping: {{mapping| 1 -5 25 5 -28 | 0 9 -31 -3 43 }} | ||
Optimal tunings: | Optimal tunings: | ||
* WE: ~2 = 1200. | * WE: ~2 = 1200.4874{{c}}, ~224/135 = 878.2543{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~ | * CWE: ~2 = 1200.0000{{c}}, ~224/135 = 877.8987{{c}} | ||
{{Optimal ET sequence|legend=0| | {{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 | |||
Mapping: {{mapping| 1 - | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | |||
Comma list: 385/384, 441/440, 4375/4374 | |||
Mapping: {{mapping| 2 -1 -3 7 9 | 0 6 11 -2 -3 }} | |||
Optimal tunings: | Optimal tunings: | ||
* WE: ~ | * 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| | {{Optimal ET sequence|legend=0| 26, 46, 72, 118, 190 }} | ||
Badness (Sintel): 0. | 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: {{mapping| 1 - | Mapping: {{mapping| 2 -1 -3 7 9 -19 | 0 6 11 -2 -3 38 }} | ||
Optimal tunings: | Optimal tunings: | ||
* WE: ~ | * WE: ~99/70 = 600.2497{{c}}, ~14/11 = 417.0085{{c}} | ||
* CWE: ~ | * 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: {{mapping| | Mapping: {{mapping| 2 -1 -3 7 9 6 | 0 6 11 -2 -3 2 }} | ||
Optimal tunings: | Optimal tunings: | ||
* WE: ~ | * WE: ~91/64 = 600.3825{{c}}, ~14/11 = 417.0678{{c}} | ||
* CWE: ~ | * 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): 0. | Badness (Sintel): 0.732 | ||
=== | ===== 17-limit ===== | ||
Subgroup: 2.3.5.7.11.13.17 | |||
Comma list: 169/168, 221/220, 273/272, 325/324, 364/363 | |||
Mapping: {{mapping| 2 -1 -3 7 9 6 4 | 0 6 11 -2 -3 2 6 }} | |||
Mapping: {{mapping| 1 - | |||
Optimal tunings: | Optimal tunings: | ||
* WE: ~ | * WE: ~17/12 = 600.3991{{c}}, ~14/11 = 417.0809{{c}} | ||
* CWE: ~ | * 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 | ||
== | == 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. | |||
[[Subgroup]]: 2.3.5.7 | |||
[[Comma list]]: 1029/1024, 5103/5000 | |||
{{Mapping|legend=1| 1 -5 -7 5 | 0 12 17 -4 }} | |||
: mapping generators: ~2, ~35/24 | |||
{{ | [[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=1| 11c, 20c, 31, 144c, 175c }} | |||
=== | [[Badness]] (Sintel): 2.98 | ||
Subgroup: 2.3.5.7.11 | |||
=== 11-limit === | |||
Subgroup: 2.3.5.7.11 | |||
Comma list: | Comma list: 176/175, 243/242, 1029/1024 | ||
Mapping: {{mapping| 1 -5 -5 | Mapping: {{mapping| 1 -5 -7 5 -13 | 0 12 17 -4 30 }} | ||
Optimal tunings: | Optimal tunings: | ||
* WE: ~2 = | * WE: ~2 = 1200.2862{{c}}, ~22/15 = 658.4276{{c}} | ||
* CWE: ~2 = 1200.0000{{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 (Sintel): 1. | Badness (Sintel): 1.77 | ||
== | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 144/143, 176/175, 243/242, 343/338 | |||
Mapping: {{mapping| 1 -5 -7 5 -13 7 | 0 12 17 -4 30 -6 }} | |||
{{Mapping|legend=1| 1 - | Optimal tunings: | ||
* WE: ~2 = 1199.3663{{c}}, ~22/15 = 658.0465{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~22/15 = 658.3800{{c}} | |||
{{Optimal ET sequence|legend=0| 20ce, 31, 82cf, 113cf }} | |||
Badness (Sintel): 1.94 | |||
== Restles == | |||
{{See also| Lesser tendoneutralic }} | |||
[[Subgroup]]: 2.3.5.7 | |||
[[Comma list]]: 1029/1024, 153664/151875 | |||
{{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. | * [[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}}, ~ | * [[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]] (Sintel): 2. | [[Badness]] (Sintel): 2.73 | ||
=== 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, 153664/151875 | ||
Mapping: {{mapping| 1 - | Mapping: {{mapping| 1 -2 8 4 -7 | 0 12 -19 -4 35 }} | ||
Optimal tunings: | Optimal tunings: | ||
* WE: ~2 = 1200. | * WE: ~2 = 1200.1110{{c}}, ~27/22 = 358.6045{{c}} | ||
* CWE: ~2 = 1200.0000{{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: {{mapping| 1 -2 8 4 -7 4 | 0 12 -19 -4 35 -1 }} | ||
Optimal tunings: | |||
* WE: ~2 = 1200.0482{{c}}, ~~16/13 = 358.5883{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~16/13 = 358.5741{{c}} | |||
{{Optimal ET sequence|legend=0| 77, 87, 164, 251d }} | |||
{{ | |||
Badness (Sintel): 1.16 | |||
== Lagaca == | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
[[Comma list]]: 1029/1024, | [[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 | |||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[WE]]: ~ | * [[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]]: ~ | * [[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 }} | |||
[[Badness]] (Sintel): 3.65 | |||
[[ | == Quartemka == | ||
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Quartemka]].'' | |||
[[Subgroup]]: 2.3.5.7 | |||
Subgroup: 2.3.5.7 | |||
Comma list: | [[Comma list]]: 1029/1024, 1250000/1240029 | ||
{{Mapping|legend=1| 1 -17 -26 9 | 0 21 32 -7 }} | |||
: mapping generators: ~2, ~50/27 | |||
Optimal | [[Optimal tuning]]s: | ||
* WE: ~ | * [[WE]]: ~2 = 1200.5278{{c}}, ~50/27 = 1062.4614{{c}} | ||
* CWE: ~ | : [[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: {{mapping| 1 -17 -26 9 7 | 0 21 32 -7 -4 }} | |||
Mapping: {{mapping| | |||
Optimal tunings: | Optimal tunings: | ||
* WE: ~ | * WE: ~2 = 1200.3051{{c}}, ~50/27 = 1062.2805{{c}} | ||
* CWE: ~ | * CWE: ~21 = 1200.0000{{c}}, ~50/27 = 1062.0147{{c}} | ||
{{Optimal ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 26, 61, 87, 200, 287d }} | ||
Badness (Sintel): | Badness (Sintel): 1.89 | ||
=== | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: | Comma list: 325/324, 364/363, 385/384, 2200/2197 | ||
Mapping: {{mapping| | Mapping: {{mapping| 1 -17 -26 9 7 -14 | 0 21 32 -7 -4 20 }} | ||
Optimal tunings: | Optimal tunings: | ||
* WE: ~ | * WE: ~2 = 1200.2708{{c}}, ~24/13 = 1062.2496{{c}} | ||
* CWE: ~ | * CWE: ~21 = 1200.0000{{c}}, ~24/13 = 1062.0139{{c}} | ||
{{Optimal ET sequence|legend=0| 26, | {{Optimal ET sequence|legend=0| 26, 61, 87, 200 }} | ||
Badness (Sintel): | Badness (Sintel): 1.17 | ||
==== | == Tritriple == | ||
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tritriple]].'' | |||
[[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 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: {{mapping| 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= | {{Optimal ET sequence|legend=0| 15, …, 88, 103, 118, 221e, 339de }} | ||
Badness (Sintel): 1.17 | |||
=== 11-limit === | == 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: | |||
* [[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 }} | |||
{{Optimal ET sequence|legend=1| 16, 71, 87, 103, 190 }} | |||
[[Badness]] (Sintel): 3.90 | |||
=== 11-limit === | |||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: | Comma list: 385/384, 441/440, 234375/234256 | ||
Mapping: {{mapping| 1 - | Mapping: {{mapping| 1 16 8 -2 17 | 0 -33 -13 11 -31 }} | ||
Optimal tunings: | Optimal tunings: | ||
* WE: ~2 = 1200. | * WE: ~2 = 1200.4852{{c}}, ~1250/847 = 676.0634{{c}} | ||
* CWE: ~2 = 1200.0000{{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 (Sintel): 1. | 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: {{mapping| 1 - | Mapping: {{mapping| 1 16 8 -2 17 12 | 0 -33 -13 11 -31 -19 }} | ||
Optimal tunings: | Optimal tunings: | ||
* WE: ~2 = | * WE: ~2 = 1200.4217{{c}}, ~77/52 = 676.0286{{c}} | ||
* CWE: ~2 = 1200.0000{{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 (Sintel): | Badness (Sintel): 0.894 | ||
== | == Other subgroup extensions == | ||
{{ | === Euslendric === | ||
Forms of slendric in the most optimal range for the 2.3.7 temperament ({{nowrap| 36 & 77 }}) lack an obvious strong mapping of prime 5 or prime 11. However, slendric can extend well to the no-fives no-elevens [[29-limit]] by tempering out [[273/272]], [[343/342]], [[378/377]], [[392/391]], [[513/512]], and [[729/728]], or a comma basis defined in terms of [[S-expression]]s as {S7/S8, S14/S16, S15/S20, S24/S26, S27, S28}. [[113edo]] is an obvious tuning. | |||
Subgroup: 2.3.7.13 | |||
Comma list: 729/728, 1029/1024 | |||
{{ | Subgroup-val mapping: {{mapping| 1 1 3 0 | 0 3 -1 19 }} | ||
Gencom mapping: {{mapping| 1 1 0 3 0 0 | 0 3 0 -1 0 19 }} | |||
{{ | Optimal tunings: | ||
* WE: ~2 = 1200.5057{{c}}, ~8/7 = 233.7200{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.6534{{c}} | |||
{{Optimal ET sequence|legend=0| 5, 31f, 36, 77, 113, 827bdddff }} | |||
Badness (Sintel): 0.339 | |||
==== 2.3.7.13.17 subgroup ==== | |||
Subgroup: 2.3.7.13.17 | |||
Comma list: 273/272, 729/728, 833/832 | |||
Subgroup-val mapping: {{mapping| 1 1 3 0 0 | 0 3 -1 19 21 }} | |||
{{ | Gencom mapping: {{mapping| 1 1 0 3 0 0 0 | 0 3 0 -1 0 19 21 }} | ||
Optimal tunings: | |||
* WE: ~2 = 1200.5282{{c}}, ~8/7 = 233.6492{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.5776{{c}} | |||
= | {{Optimal ET sequence|legend=0| 5g, 31fg, 36, 113, 149 }} | ||
Badness (Sintel): 0.332 | |||
==== 2.3.7.13.17.19 subgroup ==== | |||
Subgroup: 2.3.7.13.17.19 | |||
Comma list: 273/272, 343/342, 513/512, 729/728 | |||
{{ | Subgroup-val mapping: {{mapping| 1 1 3 0 0 6 | 0 3 -1 19 21 -9 }} | ||
Gencom mapping: {{mapping| 1 1 0 3 0 0 0 6 | 0 3 0 -1 0 19 21 -9 }} | |||
== | Optimal tunings: | ||
* WE: ~2 = 1200.3292{{c}}, ~8/7 = 233.6651{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.6106{{c}} | |||
{{Optimal ET sequence|legend=0| 5g, 36, 77, 113, 262df }} | |||
Badness (Sintel): 0.380 | |||
==== 2.3.7.13.17.19.23 subgroup ==== | |||
Subgroup: 2.3.7.13.17.19.23 | |||
Comma list: 273/272, 343/342, 392/391, 513/512, 729/728 | |||
Subgroup-val mapping: {{mapping| 1 1 3 0 0 6 9 | 0 3 -1 19 21 -9 -23 }} | |||
Gencom mapping: {{mapping| 1 1 0 3 0 0 0 6 9 | 0 3 0 -1 0 19 21 -9 -23 }} | |||
: | |||
Optimal tunings: | |||
* WE: ~2 = 1200.3127{{c}}, ~8/7 = 233.6679{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.6091{{c}} | |||
{{Optimal ET sequence|legend=0| 36, 77, 113, 262df }} | |||
Badness (Sintel): 0.474 | |||
==== 2.3.7.13.17.19.23.29 subgroup ==== | |||
Subgroup: 2.3.7.13.17.19.23.29 | |||
Comma list: 273/272, 343/342, 378/377, 392/391, 513/512, 609/608 | |||
Subgroup-val mapping: {{mapping| 1 1 3 0 0 6 9 7 | 0 3 -1 19 21 -9 -23 -11 }} | |||
Gencom mapping: {{mapping| 1 1 0 3 0 0 0 6 9 7 | 0 3 0 -1 0 19 21 -9 -23 -11 }} | |||
Optimal tunings: | |||
* WE: ~2 = 1200.2503{{c}}, ~8/7 = 233.6688{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.6208{{c}} | |||
{{Optimal ET sequence|legend=0| 36, 77, 113 }} | |||
Badness (Sintel): 0.473 | |||
{{ | === Baladic === | ||
Baladic is a 2.3.7.13.17-subgroup temperament that attempts to approximate the Maqam Sikah Baladi scale. It tempers out [[169/168]] ({{S|13}}), which splits [[7/6]] in half ([[13/12]]~[[14/13]]) and one finds that the octave is therefore split in half via the interval [[91/64]], which is then equated to [[17/12]]. 36edo is an excellent baladic tuning. | |||
Subgroup: 2.3.7.13 | |||
Comma list: 169/168, 1029/1024 | |||
Subgroup-val mapping: {{mapping| 2 2 6 7 | 0 3 -1 1 }} | |||
Gencom mapping: {{mapping| 2 2 0 6 0 7 | 0 3 0 -1 0 1 }} | |||
: mapping generators: ~91/64, ~8/7 | |||
Optimal tunings: | Optimal tunings: | ||
* WE: ~ | * WE: ~91/64 = 600.4315{{c}}, ~8/7 = 233.7724{{c}} | ||
* CWE: ~ | * CWE: ~91/64 = 600.0000{{c}}, ~8/7 = 233.7039{{c}} | ||
{{Optimal ET sequence|legend=0| 26, | {{Optimal ET sequence|legend=0| 10, 26, 36, 154f, 190ff, 226ff, 262dfff }} | ||
Badness (Sintel): | Badness (Sintel): 0.434 | ||
== | ==== 2.3.7.13.17 subgroup ==== | ||
Subgroup: 2.3.7.13.17 | |||
= | |||
Subgroup: 2.3 | |||
Comma list: 169/168, 273/272, 289/288 | |||
Subgroup-val mapping: {{mapping| 2 2 6 7 7 | 0 3 -1 1 3 }} | |||
Gencom mapping: {{mapping| 2 2 0 6 0 7 7 | 0 3 0 -1 0 1 3 }} | |||
Optimal tunings: | Optimal tunings: | ||
* WE: ~ | * WE: ~17/12 = 600.4436{{c}}, ~8/7 = 233.7883{{c}} | ||
* CWE: ~ | * 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 (Sintel): 0.253 | |||
Badness (Sintel): 0. | |||
== References == | == References == | ||