Gamelismic clan: Difference between revisions
mNo edit summary |
m Units; -gencom (trivial); misc. cleanup |
||
| Line 1: | Line 1: | ||
{{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. | 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. | ||
| Line 30: | Line 30: | ||
{{Mapping|legend=2| 1 1 3 | 0 3 -1 }} | {{Mapping|legend=2| 1 1 3 | 0 3 -1 }} | ||
{{Mapping|legend=3| 1 1 0 3 | 0 3 0 -1 }} | {{Mapping|legend=3| 1 1 0 3 | 0 3 0 -1 }} | ||
: mapping generators: ~2, ~8/7 | |||
: | |||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[CTE]]: ~2 = 1200.000, ~8/7 = 233.889 | * [[CTE]]: ~2 = 1200.000{{c}}, ~8/7 = 233.889{{c}} | ||
: [[error map]]: {{val| 0.000 -0.288 -2.715 }} | : [[error map]]: {{val| 0.000 -0.288 -2.715 }} | ||
* [[POTE]]: ~2 = 1200.000, ~8/7 = 233.688 | * [[POTE]]: ~2 = 1200.000{{c}}, ~8/7 = 233.688{{c}} | ||
: error map: {{val| 0.000 -0.892 -2.513 }} | : error map: {{val| 0.000 -0.892 -2.513 }} | ||
| Line 46: | Line 43: | ||
=== Euslendric === | === Euslendric === | ||
Forms of slendric in the most optimal range for the 2.3.7 temperament (36 & 77) lack an obvious strong mapping of prime 5 or prime 11. However, slendric can extend well to the no-fives no-elevens [[29-limit]] by tempering out [[273/272]], [[343/342]], [[378/377]], [[392/391]], [[513/512]], and [[729/728]], or a comma basis defined in terms of [[S-expression]]s as {S7/S8, S14/S16, S15/S20, S24/S26, S27, S28}. [[113edo]] is an obvious tuning. | 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. | ||
==== 2.3.7.13 ==== | ==== 2.3.7.13 subgroup ==== | ||
Subgroup: 2.3.7.13 | |||
Comma list: 729/728, 1029/1024 | |||
Sval mapping: {{mapping| 1 1 3 0 | 0 3 -1 19 }} | |||
Optimal tunings: | |||
* | * CTE: ~2 = 1200.000{{c}}, ~8/7 = 233.734{{c}} | ||
* | * POTE: ~2 = 1200.000{{c}}, ~8/7 = 233.622{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 5, …, 31f, 36, 77, 113 }} | ||
Badness ( | Badness (Sintel): 0.339 | ||
==== 2.3.7.13.17 ==== | ==== 2.3.7.13.17 subgroup ==== | ||
Subgroup: 2.3.7.13.17 | Subgroup: 2.3.7.13.17 | ||
Comma list: 273/272, 729/728, 833/832 | Comma list: 273/272, 729/728, 833/832 | ||
Sval mapping: | Sval mapping: {{mapping| 1 1 3 0 0 | 0 3 -1 19 21 }} | ||
Optimal tunings: | Optimal tunings: | ||
* | * CTE: ~2 = 1200.000{{c}}, ~8/7 = 233.657{{c}} | ||
* | * POTE: ~2 = 1200.000{{c}}, ~8/7 = 233.546{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 5g, …, 31fg, 36, 113, 149 }} | ||
Badness ( | Badness (Sintel): 0.332 | ||
==== 2.3.7.13.17.19 ==== | ==== 2.3.7.13.17.19 subgroup ==== | ||
Subgroup: 2.3.7.13.17.19 | Subgroup: 2.3.7.13.17.19 | ||
Comma list: 273/272, 343/342, 513/512, 729/728 | Comma list: 273/272, 343/342, 513/512, 729/728 | ||
Sval mapping: | Sval mapping: {{mapping| 1 1 3 0 0 6 | 0 3 -1 19 21 -9 }} | ||
Optimal tunings: | Optimal tunings: | ||
* | * CTE: ~2 = 1200.000{{c}}, ~8/7 = 233.657{{c}} | ||
* | * POTE: ~2 = 1200.000{{c}}, ~8/7 = 233.601{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 5g, …, 36, 77, 113 }} | ||
Badness ( | Badness (Sintel): 0.380 | ||
==== 2.3.7.13.17.19.23 ==== | ==== 2.3.7.13.17.19.23 subgroup ==== | ||
Subgroup: 2.3.7.13.17.19.23 | Subgroup: 2.3.7.13.17.19.23 | ||
Comma list: 273/272, 343/342, 392/391, 513/512, 729/728 | Comma list: 273/272, 343/342, 392/391, 513/512, 729/728 | ||
Sval mapping: | Sval mapping: {{mapping| 1 1 3 0 0 6 9 | 0 3 -1 19 21 -9 -23 }} | ||
Optimal tunings: | Optimal tunings: | ||
* | * CTE: ~2 = 1200.000{{c}}, ~8/7 = 233.624{{c}} | ||
* | * POTE: ~2 = 1200.000{{c}}, ~8/7 = 233.607{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 5gi, …, 36, 77, 113 }} | ||
Badness ( | Badness (Sintel): 0.474 | ||
==== 2.3.7.13.17.19.23.29 ==== | ==== 2.3.7.13.17.19.23.29 subgroup ==== | ||
Subgroup: 2.3.7.13.17.19.23.29 | Subgroup: 2.3.7.13.17.19.23.29 | ||
Comma list: 273/272, 343/342, 378/377, 392/391, 513/512, 609/608 | Comma list: 273/272, 343/342, 378/377, 392/391, 513/512, 609/608 | ||
Sval mapping: | Sval mapping: {{mapping| 1 1 3 0 0 6 9 7 | 0 3 -1 19 21 -9 -23 -11 }} | ||
Optimal tunings: | Optimal tunings: | ||
* | * CTE: ~2 = 1200.000{{c}}, ~8/7 = 233.626{{c}} | ||
* | * POTE: ~2 = 1200.000{{c}}, ~8/7 = 233.620{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 5gi, …, 36, 77, 113 }} | ||
Badness ( | Badness (Sintel): 0.473 | ||
=== Radon === | === Radon === | ||
Radon is the no-fives version of [[rodan]], equating the diatonic major third to [[14/11]]. | 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 | |||
{{ | Sval mapping: {{mapping| 1 1 3 6 | 0 3 -1 -13 }} | ||
{{ | Gencom mapping: {{mapping| 1 1 0 3 6 | 0 3 0 -1 -13 }} | ||
Optimal tunings: | |||
* CTE: ~2 = 1200.000{{c}}, ~8/7 = 234.384{{c}} | |||
: error map: {{val| 0.000 +1.197 -3.210 +1.691 }} | |||
* | * POTE: ~2 = 1200.000{{c}}, ~8/7 = 234.381{{c}} | ||
: | |||
* | |||
: error map: {{val| 0.000 +1.187 -3.206 +1.735 }} | : error map: {{val| 0.000 +1.187 -3.206 +1.735 }} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 36, 41, 87, 128 }} | ||
=== Baladic === | === Baladic === | ||
Baladic is a 2.3.7.13.17 subgroup temperament that attempts to approximate the Maqam Sikah Baladi scale. It tempers out {{S|13}} = [[169/168]], 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. | Baladic is a 2.3.7.13.17 subgroup temperament that attempts to approximate the Maqam Sikah Baladi scale. It tempers out {{S|13}} = [[169/168]], 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. | ||
==== 2.3.7.13 ==== | ==== 2.3.7.13 subgroup ==== | ||
Subgroup: 2.3.7.13 | |||
Comma list: 169/168, 1029/1024 | |||
Sval mapping: {{mapping| 2 2 6 7 | 0 3 -1 1 }} | |||
: mapping generators: ~91/64, ~8/7 | |||
Optimal tunings: | |||
* POTE: ~91/64 = 600.0000{{c}}, ~8/7 = 233.6044{{c}} | |||
{{Optimal ET sequence|legend=0| 10, 26, 36, 154f, 190ff, 226ff, 262dfff }} | |||
RMS error: 0.5452 cents | |||
==== 2.3.7.13.17 subgroup ==== | |||
==== 2.3.7.13.17 ==== | |||
Subgroup: 2.3.7.13.17 | Subgroup: 2.3.7.13.17 | ||
Comma list: 169/168, 273/272, 289/288 | |||
{{ | Sval mapping: {{mapping| 2 2 6 7 7 | 0 3 -1 1 3 }} | ||
: | Optimal tunings: | ||
* CTE: ~17/12 = 600.000{{c}}, ~8/7 = 234.138{{c}} | |||
* POTE: ~17/12 = 600.000{{c}}, ~8/7 = 233.616{{c}} | |||
{{Optimal ET sequence|legend=0| 10, 26, 36, 154f, 190ffg, 226ffg }} | |||
{{Optimal ET sequence|legend= | |||
== Mothra == | == 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 & | 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. | 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]]( | 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 | [[Subgroup]]: 2.3.5.7 | ||
| Line 194: | Line 184: | ||
{{Mapping|legend=1| 1 1 0 3 | 0 3 12 -1 }} | {{Mapping|legend=1| 1 1 0 3 | 0 3 12 -1 }} | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[CTE]]: ~2 = 1200.000, ~8/7 = 232.400 | * [[CTE]]: ~2 = 1200.000{{c}}, ~8/7 = 232.400{{c}} | ||
: [[error map]]: {{val| 0.000 -4.756 +2.482 -1.226 }} | : [[error map]]: {{val| 0.000 -4.756 +2.482 -1.226 }} | ||
* [[POTE]]: ~2 = 1200.000, ~8/7 = 232.193 | * [[POTE]]: ~2 = 1200.000{{c}}, ~8/7 = 232.193{{c}} | ||
: error map: {{val| 0.000 -5.375 +0.005 -1.019 }} | : error map: {{val| 0.000 -5.375 +0.005 -1.019 }} | ||
| Line 207: | Line 195: | ||
[[Minimax tuning]]: | [[Minimax tuning]]: | ||
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~8/7 = {{monzo| 0 0 1/12 }} | * [[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 | : [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5 | ||
{{Optimal ET sequence|legend=1| 5, 21c, 26, 31 }} | {{Optimal ET sequence|legend=1| 5, 21c, 26, 31 }} | ||
[[Badness]] | [[Badness]]: | ||
* Smith: 0.037146 | |||
* Sintel: 0.940 | |||
=== Undecimal mothra === | === Undecimal mothra === | ||
| Line 226: | Line 214: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200.000, ~8/7 = 232.203 | * CTE: ~2 = 1200.000{{c}}, ~8/7 = 232.203{{c}} | ||
* POTE: ~2 = 1200.000, ~8/7 = 232.031 | * POTE: ~2 = 1200.000{{c}}, ~8/7 = 232.031{{c}} | ||
{{Optimal ET sequence|legend=0| 5, 26, 31, 88, 119be, 150be }} | {{Optimal ET sequence|legend=0| 5, 26, 31, 88, 119be, 150be }} | ||
Badness | Badness: | ||
* Smith: 0.025642 | |||
* Sintel: 0.848 | |||
==== 13-limit ==== | ==== 13-limit ==== | ||
| Line 243: | Line 231: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200.000, ~8/7 = 231.993 | * CTE: ~2 = 1200.000{{c}}, ~8/7 = 231.993{{c}} | ||
* POTE: ~2 = 1200.000, ~8/7 = 231.811 | * POTE: ~2 = 1200.000{{c}}, ~8/7 = 231.811{{c}} | ||
{{Optimal ET sequence|legend=0| 5, 26, 31, 57, 88 }} | {{Optimal ET sequence|legend=0| 5, 26, 31, 57, 88 }} | ||
Badness | Badness: | ||
* Smith: 0.023954 | |||
* Sintel: 0.990 | |||
; Music | ; Music | ||
| Line 263: | Line 251: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200.000, ~8/7 = 231.891 | * CTE: ~2 = 1200.000{{c}}, ~8/7 = 231.891{{c}} | ||
* POTE: ~2 = 1200.000, ~8/7 = 231.708 | * POTE: ~2 = 1200.000{{c}}, ~8/7 = 231.708{{c}} | ||
{{Optimal ET sequence|legend=0| 5g, 26, 31, 57, 88 }} | {{Optimal ET sequence|legend=0| 5g, 26, 31, 57, 88 }} | ||
Badness ( | Badness (Sintel): 1.001 | ||
==== 19-limit ==== | ==== 19-limit ==== | ||
| Line 278: | Line 266: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200.000, ~8/7 = 231.837 | * CTE: ~2 = 1200.000{{c}}, ~8/7 = 231.837{{c}} | ||
* POTE: ~2 = 1200.000, ~8/7 = 231.653 | * POTE: ~2 = 1200.000{{c}}, ~8/7 = 231.653{{c}} | ||
{{Optimal ET sequence|legend=0| 5gh, 26, 31, 57 }} | {{Optimal ET sequence|legend=0| 5gh, 26, 31, 57 }} | ||
Badness ( | Badness (Sintel): 1.053 | ||
=== Mosura === | === 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]], | 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 | Subgroup: 2.3.5.7.11 | ||
| Line 295: | Line 283: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200.000, ~8/7 = 232.557 | * CTE: ~2 = 1200.000{{c}}, ~8/7 = 232.557{{c}} | ||
* POTE: ~2 = 1200.000, ~8/7 = 232.419 | * POTE: ~2 = 1200.000{{c}}, ~8/7 = 232.419{{c}} | ||
{{Optimal ET sequence|legend=0| 5e, 26e, 31, 129 }} | {{Optimal ET sequence|legend=0| 5e, 26e, 31, 129 }} | ||
Badness | Badness: | ||
* Smith: 0.031334 | |||
* Sintel: 1.036 | |||
==== 13-limit ==== | ==== 13-limit ==== | ||
| Line 312: | Line 300: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200.000, ~8/7 = 232.635 | * CTE: ~2 = 1200.000{{c}}, ~8/7 = 232.635{{c}} | ||
* POTE: ~2 = 1200.000, ~8/7 = 232.640 | * POTE: ~2 = 1200.000{{c}}, ~8/7 = 232.640{{c}} | ||
{{Optimal ET sequence|legend=0| 31, 67, 98 }} | {{Optimal ET sequence|legend=0| 31, 67, 98 }} | ||
Badness | Badness: | ||
* Smith: 0.036857 | |||
* Sintel: 1.523 | |||
==== 17-limit ==== | ==== 17-limit ==== | ||
| Line 329: | Line 317: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200.000, ~8/7 = 232.681 | * CTE: ~2 = 1200.000{{c}}, ~8/7 = 232.681{{c}} | ||
* POTE: ~2 = 1200.000, ~8/7 = 232.693 | * POTE: ~2 = 1200.000{{c}}, ~8/7 = 232.693{{c}} | ||
{{Optimal ET sequence|legend=0| 31, 67, 98 }} | {{Optimal ET sequence|legend=0| 31, 67, 98 }} | ||
Badness ( | Badness (Sintel): 1.527 | ||
==== 19-limit ==== | ==== 19-limit ==== | ||
| Line 344: | Line 332: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200.000, ~8/7 = 232.717 | * CTE: ~2 = 1200.000{{c}}, ~8/7 = 232.717{{c}} | ||
* POTE: ~2 = 1200.000, ~8/7 = 232.730 | * POTE: ~2 = 1200.000{{c}}, ~8/7 = 232.730{{c}} | ||
{{Optimal ET sequence|legend=0| 31, 67, 98h }} | {{Optimal ET sequence|legend=0| 31, 67, 98h }} | ||
Badness ( | Badness (Sintel): 1.496 | ||
=== Cyndra === | === Cyndra === | ||
| Line 359: | Line 347: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200.000, ~8/7 = 231.566 | * CTE: ~2 = 1200.000{{c}}, ~8/7 = 231.566{{c}} | ||
* POTE: ~2 = 1200.000, ~8/7 = 231.317 | * POTE: ~2 = 1200.000{{c}}, ~8/7 = 231.317{{c}} | ||
{{Optimal ET sequence|legend=0| 5e, 21ce, 26 }} | {{Optimal ET sequence|legend=0| 5e, 21ce, 26 }} | ||
| Line 374: | Line 362: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200.000, ~8/7 = 231.546 | * CTE: ~2 = 1200.000{{c}}, ~8/7 = 231.546{{c}} | ||
* POTE: ~2 = 1200.000, ~8/7 = 231.293 | * POTE: ~2 = 1200.000{{c}}, ~8/7 = 231.293{{c}} | ||
{{Optimal ET sequence|legend=0| 5e, 21cef, 26 }} | {{Optimal ET sequence|legend=0| 5e, 21cef, 26 }} | ||
| Line 385: | Line 373: | ||
: ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum #Rodan (5-limit)]].'' | : ''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 & | Rodan tempers out 245/243 and can be described as the {{nowrap| 41 & 46 }} temperament. This temperament extends neatly to the 13-limit, though the perfect fifth is sharper than ideal for slendric. | ||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
| Line 394: | Line 382: | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[CTE]]: ~2 = 1200.000, ~8/7 = 234.450 | * [[CTE]]: ~2 = 1200.000{{c}}, ~8/7 = 234.450{{c}} | ||
: [[error map]]: {{val| 0.000 +1.396 -0.660 -3.276 }} | : [[error map]]: {{val| 0.000 +1.396 -0.660 -3.276 }} | ||
* [[POTE]]: ~2 = 1200.000, ~8/7 = 234.417 | * [[POTE]]: ~2 = 1200.000{{c}}, ~8/7 = 234.417{{c}} | ||
: error map: {{val| 0.000 +1.295 -1.229 -3.243 }} | : error map: {{val| 0.000 +1.295 -1.229 -3.243 }} | ||
| Line 418: | Line 406: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200.000, ~8/7 = 234.463 | * CTE: ~2 = 1200.000{{c}}, ~8/7 = 234.463{{c}} | ||
* POTE: ~2 = 1200.000, ~8/7 = 234.459 | * POTE: ~2 = 1200.000{{c}}, ~8/7 = 234.459{{c}} | ||
Minimax tuning: | Minimax tuning: | ||
| Line 440: | Line 428: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200.000, ~8/7 = 234.482 | * CTE: ~2 = 1200.000{{c}}, ~8/7 = 234.482{{c}} | ||
* POTE: ~2 = 1200.000, ~8/7 = 234.482 | * POTE: ~2 = 1200.000{{c}}, ~8/7 = 234.482{{c}} | ||
Minimax tuning: | Minimax tuning: | ||
| Line 461: | Line 449: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200.000, ~8/7 = 234.532 | * CTE: ~2 = 1200.000{{c}}, ~8/7 = 234.532{{c}} | ||
* POTE: ~2 = 1200.000, ~8/7 = 234.524 | * POTE: ~2 = 1200.000{{c}}, ~8/7 = 234.524{{c}} | ||
Minimax tuning: | Minimax tuning: | ||
| Line 480: | Line 468: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200.000, ~8/7 = 234.670 | * CTE: ~2 = 1200.000{{c}}, ~8/7 = 234.670{{c}} | ||
* POTE: ~2 = 1200.000, ~8/7 = 234.639 | * POTE: ~2 = 1200.000{{c}}, ~8/7 = 234.639{{c}} | ||
{{Optimal ET sequence|legend=0| 5, 41f, 46 }} | {{Optimal ET sequence|legend=0| 5, 41f, 46 }} | ||
| Line 495: | Line 483: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200.000, ~8/7 = 234.719 | * CTE: ~2 = 1200.000{{c}}, ~8/7 = 234.719{{c}} | ||
* POTE: ~2 = 1200.000, ~8/7 = 234.728 | * POTE: ~2 = 1200.000{{c}}, ~8/7 = 234.728{{c}} | ||
{{Optimal ET sequence|legend=0| 5e, 41e, 46 }} | {{Optimal ET sequence|legend=0| 5e, 41e, 46 }} | ||
| Line 510: | Line 498: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200.000, ~8/7 = 234.786 | * CTE: ~2 = 1200.000{{c}}, ~8/7 = 234.786{{c}} | ||
* POTE: ~2 = 1200.000, ~8/7 = 234.782 | * POTE: ~2 = 1200.000{{c}}, ~8/7 = 234.782{{c}} | ||
{{Optimal ET sequence|legend=0| 5e, 41ef, 46 }} | {{Optimal ET sequence|legend=0| 5e, 41ef, 46 }} | ||
| Line 525: | Line 513: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200.000, ~8/7 = 234.197 | * CTE: ~2 = 1200.000{{c}}, ~8/7 = 234.197{{c}} | ||
* POTE: ~2 = 1200.000, ~8/7 = 234.145 | * POTE: ~2 = 1200.000{{c}}, ~8/7 = 234.145{{c}} | ||
{{Optimal ET sequence|legend=0| 5e, 36ce, 41 }} | {{Optimal ET sequence|legend=0| 5e, 36ce, 41 }} | ||
| Line 540: | Line 528: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200.000, ~8/7 = 234.111 | * CTE: ~2 = 1200.000{{c}}, ~8/7 = 234.111{{c}} | ||
* POTE: ~2 = 1200.000, ~8/7 = 234.089 | * POTE: ~2 = 1200.000{{c}}, ~8/7 = 234.089{{c}} | ||
{{Optimal ET sequence|legend=0| 5e, 36ce, 41 }} | {{Optimal ET sequence|legend=0| 5e, 36ce, 41 }} | ||
| Line 548: | Line 536: | ||
== Guiron == | == 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 & | 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 | ||
| Line 555: | Line 543: | ||
{{Mapping|legend=1| 1 1 7 3 | 0 3 -24 -1 }} | {{Mapping|legend=1| 1 1 7 3 | 0 3 -24 -1 }} | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[CTE]]: ~2 = 1200.000, ~8/7 = 233.903 | * [[CTE]]: ~2 = 1200.000{{c}}, ~8/7 = 233.903{{c}} | ||
: [[error map]]: {{val| 0.000 -0.246 +0.012 -2.729 }} | : [[error map]]: {{val| 0.000 -0.246 +0.012 -2.729 }} | ||
* [[POTE]]: ~2 = 1200.000, ~8/7 = 233.930 | * [[POTE]]: ~2 = 1200.000{{c}}, ~8/7 = 233.930{{c}} | ||
: error map: {{val| 0.000 -0.165 -0.637 -2.756 }} | : error map: {{val| 0.000 -0.165 -0.637 -2.756 }} | ||
| Line 579: | Line 565: | ||
Mapping: {{mapping| 1 1 7 3 -2 | 0 3 -24 -1 28 }} | Mapping: {{mapping| 1 1 7 3 -2 | 0 3 -24 -1 28 }} | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200.000, ~8/7 = 233.930 | * CTE: ~2 = 1200.000{{c}}, ~8/7 = 233.930{{c}} | ||
* POTE: ~2 = 1200.000, ~8/7 = 233.931 | * POTE: ~2 = 1200.000{{c}}, ~8/7 = 233.931{{c}} | ||
Minimax tuning: | Minimax tuning: | ||
| Line 601: | Line 585: | ||
Mapping: {{mapping| 1 1 7 3 -2 0 | 0 3 -24 -1 28 19 }} | Mapping: {{mapping| 1 1 7 3 -2 0 | 0 3 -24 -1 28 19 }} | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200.000, ~8/7 = 233.902 | * CTE: ~2 = 1200.000{{c}}, ~8/7 = 233.902{{c}} | ||
* POTE: ~2 = 1200.000, ~8/7 = 233.899 | * POTE: ~2 = 1200.000{{c}}, ~8/7 = 233.899{{c}} | ||
{{Optimal ET sequence|legend=0| 36e, 41, 77, 118 }} | {{Optimal ET sequence|legend=0| 36e, 41, 77, 118 }} | ||
| Line 615: | Line 597: | ||
: ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum #Laconic]].'' | : ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum #Laconic]].'' | ||
{{ See also | Shoe }} | {{See also| Shoe }} | ||
Gorgo tempers the generator of ~8/7 together with ~10/9. It can be described as the {{nowrap|16 & | 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, meaning that this temperament is much more accurate than its comma list suggests. | 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, meaning that this temperament is much more accurate than its comma list suggests. | ||
| Line 628: | Line 610: | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[CTE]]: ~2 = 1200.000, ~8/7 = 228.724 | * [[CTE]]: ~2 = 1200.000{{c}}, ~8/7 = 228.724{{c}} | ||
: [[error map]]: {{val| 0.000 -15.782 +14.756 +2.450 }} | : [[error map]]: {{val| 0.000 -15.782 +14.756 +2.450 }} | ||
* [[POTE]]: ~2 = 1200.000, ~8/7 = 228.334 | * [[POTE]]: ~2 = 1200.000{{c}}, ~8/7 = 228.334{{c}} | ||
: error map: {{val| 0.000 -16.954 +12.022 +2.840 }} | : error map: {{val| 0.000 -16.954 +12.022 +2.840 }} | ||
| Line 645: | Line 627: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200.000, ~8/7 = 227.833 | * CTE: ~2 = 1200.000{{c}}, ~8/7 = 227.833{{c}} | ||
* POTE: ~2 = 1200.000, ~8/7 = 227.373 | * POTE: ~2 = 1200.000{{c}}, ~8/7 = 227.373{{c}} | ||
{{Optimal ET sequence|legend=0| 5e, 16, 21, 37b }} | {{Optimal ET sequence|legend=0| 5e, 16, 21, 37b }} | ||
| Line 660: | Line 642: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200.000, ~8/7 = 227.633 | * CTE: ~2 = 1200.000{{c}}, ~8/7 = 227.633{{c}} | ||
* POTE: ~2 = 1200.000, ~8/7 = 227.230 | * POTE: ~2 = 1200.000{{c}}, ~8/7 = 227.230{{c}} | ||
{{Optimal ET sequence|legend=0| 5e, 16, 21, 37b }} | {{Optimal ET sequence|legend=0| 5e, 16, 21, 37b }} | ||
| Line 675: | Line 657: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200.000, ~8/7 = 229.420 | * CTE: ~2 = 1200.000{{c}}, ~8/7 = 229.420{{c}} | ||
* POTE: ~2 = 1200.000, ~8/7 = 229.535 | * POTE: ~2 = 1200.000{{c}}, ~8/7 = 229.535{{c}} | ||
{{Optimal ET sequence|legend=0| 5, 16e, 21 }} | {{Optimal ET sequence|legend=0| 5, 16e, 21 }} | ||
| Line 690: | Line 672: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200.000, ~8/7 = 228.758 | * CTE: ~2 = 1200.000{{c}}, ~8/7 = 228.758{{c}} | ||
* POTE: ~2 = 1200.000, ~8/7 = 229.059 | * POTE: ~2 = 1200.000{{c}}, ~8/7 = 229.059{{c}} | ||
{{Optimal ET sequence|legend=0| 5, 16e, 21 }} | {{Optimal ET sequence|legend=0| 5, 16e, 21 }} | ||
| Line 712: | Line 694: | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[CTE]]: ~2 = 1200.000, ~8/7 = 227.100 | * [[CTE]]: ~2 = 1200.000{{c}}, ~8/7 = 227.100{{c}} | ||
: [[error map]]: {{val| 0.000 -20.655 +67.886 +4.074 }} | : [[error map]]: {{val| 0.000 -20.655 +67.886 +4.074 }} | ||
* [[POTE]]: ~2 = 1200.000, ~8/7 = 230.762 | * [[POTE]]: ~2 = 1200.000{{c}}, ~8/7 = 230.762{{c}} | ||
: error map: {{val| 0.000 -9.668 +75.211 +0.412 }} | : error map: {{val| 0.000 -9.668 +75.211 +0.412 }} | ||
| Line 724: | Line 706: | ||
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Oncle]].'' | : ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Oncle]].'' | ||
Oncle can be described as the {{nowrap|31 & | Oncle can be described as the {{nowrap|31 & 36c}} temperament. | ||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
| Line 733: | Line 715: | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[CTE]]: ~2 = 1200.000, ~8/7 = 232.383 | * [[CTE]]: ~2 = 1200.000{{c}}, ~8/7 = 232.383{{c}} | ||
: [[error map]]: {{val| 0.000 -4.807 -1.585 -1.209 }} | : [[error map]]: {{val| 0.000 -4.807 -1.585 -1.209 }} | ||
* [[POTE]]: ~2 = 1200.000, ~8/7 = 232.498 | * [[POTE]]: ~2 = 1200.000{{c}}, ~8/7 = 232.498{{c}} | ||
: error map: {{val| 0.000 -4.461 -3.778 -1.324 }} | : error map: {{val| 0.000 -4.461 -3.778 -1.324 }} | ||
| Line 745: | Line 727: | ||
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Archaeotherium]].'' | : ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Archaeotherium]].'' | ||
Archaeotherium can be described as the {{nowrap|21 & | Archaeotherium can be described as the {{nowrap|21 & 26}} temperament. | ||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
| Line 754: | Line 736: | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[CTE]]: ~2 = 1200.000, ~8/7 = 229.951 | * [[CTE]]: ~2 = 1200.000{{c}}, ~8/7 = 229.951{{c}} | ||
: [[error map]]: {{val| 0.000 -12.102 -5.626 +1.223 }} | : [[error map]]: {{val| 0.000 -12.102 -5.626 +1.223 }} | ||
* [[POTE]]: ~2 = 1200.000, ~8/7 = 230.258 | * [[POTE]]: ~2 = 1200.000{{c}}, ~8/7 = 230.258{{c}} | ||
: error map: {{val| 0.000 -11.180 -9.933 +0.916 }} | : error map: {{val| 0.000 -11.180 -9.933 +0.916 }} | ||
| Line 775: | Line 757: | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[CTE]]: ~2 = 1200.000, ~8/7 = 225.752 | * [[CTE]]: ~2 = 1200.000{{c}}, ~8/7 = 225.752{{c}} | ||
: [[error map]]: {{val| 0.000 -24.699 -18.081 +5.422 }} | : [[error map]]: {{val| 0.000 -24.699 -18.081 +5.422 }} | ||
* [[POTE]]: ~2 = 1200.000, ~8/7 = 226.469 | * [[POTE]]: ~2 = 1200.000{{c}}, ~8/7 = 226.469{{c}} | ||
: error map: {{val| 0.000 -22.548 -24.534 +4.705 }} | : error map: {{val| 0.000 -22.548 -24.534 +4.705 }} | ||
| Line 792: | Line 774: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200.000, ~8/7 = 225.384 | * CTE: ~2 = 1200.000{{c}}, ~8/7 = 225.384{{c}} | ||
* POTE: ~2 = 1200.000, ~8/7 = 226.428 | * POTE: ~2 = 1200.000{{c}}, ~8/7 = 226.428{{c}} | ||
{{Optimal ET sequence|legend=0| 5c, 11, 16 }} | {{Optimal ET sequence|legend=0| 5c, 11, 16 }} | ||
| Line 803: | Line 785: | ||
: ''For the 5-limit version, see [[Syntonic–31 equivalence continuum #Ampersand]].'' | : ''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 & | 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. 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 | ||
| Line 814: | Line 796: | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[CTE]]: ~2 = 1200.000, ~15/14 = 116.677 | * [[CTE]]: ~2 = 1200.000{{c}}, ~15/14 = 116.677{{c}} | ||
: [[error map]]: {{val| 0.000 -1.892 -3.054 -2.180 }} | : [[error map]]: {{val| 0.000 -1.892 -3.054 -2.180 }} | ||
* [[POTE]]: ~2 = 1200.000, ~15/14 = 116.675 | * [[POTE]]: ~2 = 1200.000{{c}}, ~15/14 = 116.675{{c}} | ||
: error map: {{val| 0.000 -1.904 -3.040 -2.176 }} | : error map: {{val| 0.000 -1.904 -3.040 -2.176 }} | ||
| Line 846: | Line 828: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200.000, ~15/14 = 116.711 | * CTE: ~2 = 1200.000{{c}}, ~15/14 = 116.711{{c}} | ||
* POTE: ~2 = 1200.000, ~15/14 = 116.633 | * POTE: ~2 = 1200.000{{c}}, ~15/14 = 116.633{{c}} | ||
Minimax tuning: | Minimax tuning: | ||
| Line 872: | Line 854: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200.000, ~15/14 = 116.758 | * CTE: ~2 = 1200.000{{c}}, ~15/14 = 116.758{{c}} | ||
* POTE: ~2 = 1200.000, ~15/14 = 116.747 | * POTE: ~2 = 1200.000{{c}}, ~15/14 = 116.747{{c}} | ||
{{Optimal ET sequence|legend=0| 10, 21e, 31, 41, 72f, 113f, 185cff }} | {{Optimal ET sequence|legend=0| 10, 21e, 31, 41, 72f, 113f, 185cff }} | ||
| Line 887: | Line 869: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200.000, ~15/14 = 116.742 | * CTE: ~2 = 1200.000{{c}}, ~15/14 = 116.742{{c}} | ||
* POTE: ~2 = 1200.000, ~15/14 = 116.769 | * POTE: ~2 = 1200.000{{c}}, ~15/14 = 116.769{{c}} | ||
{{Optimal ET sequence|legend=0| 10, 21e, 31, 41, 72fg }} | {{Optimal ET sequence|legend=0| 10, 21e, 31, 41, 72fg }} | ||
| Line 902: | Line 884: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200.000, ~15/14 = 116.541 | * CTE: ~2 = 1200.000{{c}}, ~15/14 = 116.541{{c}} | ||
* POTE: ~2 = 1200.000, ~15/14 = 116.574 | * POTE: ~2 = 1200.000{{c}}, ~15/14 = 116.574{{c}} | ||
{{Optimal ET sequence|legend=0| 31, 72, 103, 175f }} | {{Optimal ET sequence|legend=0| 31, 72, 103, 175f }} | ||
| Line 917: | Line 899: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200.000, ~15/14 = 116.529 | * CTE: ~2 = 1200.000{{c}}, ~15/14 = 116.529{{c}} | ||
* POTE: ~2 = 1200.000, ~15/14 = 116.585 | * POTE: ~2 = 1200.000{{c}}, ~15/14 = 116.585{{c}} | ||
{{Optimal ET sequence|legend=0| 31, 72, 103, 175f, 422bcdefffg }} | {{Optimal ET sequence|legend=0| 31, 72, 103, 175f, 422bcdefffg }} | ||
| Line 932: | Line 914: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200.000, ~15/14 = 116.814 | * CTE: ~2 = 1200.000{{c}}, ~15/14 = 116.814{{c}} | ||
* POTE: ~2 = 1200.000, ~15/14 = 116.739 | * POTE: ~2 = 1200.000{{c}}, ~15/14 = 116.739{{c}} | ||
{{Optimal ET sequence|legend=0| 31f, 41, 72, 185cf, 257cff }} | {{Optimal ET sequence|legend=0| 31f, 41, 72, 185cf, 257cff }} | ||
| Line 947: | Line 929: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200.000, ~15/14 = 116.802 | * CTE: ~2 = 1200.000{{c}}, ~15/14 = 116.802{{c}} | ||
* POTE: ~2 = 1200.000, ~15/14 = 116.727 | * POTE: ~2 = 1200.000{{c}}, ~15/14 = 116.727{{c}} | ||
{{Optimal ET sequence|legend=0| 31fg, 41, 72, 185cf, 257cff }} | {{Optimal ET sequence|legend=0| 31fg, 41, 72, 185cf, 257cff }} | ||
| Line 964: | Line 946: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~55/39 = 600.000, ~15/14 = 116.735 | * CTE: ~55/39 = 600.000{{c}}, ~15/14 = 116.735{{c}} | ||
* POTE: ~55/39 = 600.000, ~15/14 = 116.624 | * POTE: ~55/39 = 600.000{{c}}, ~15/14 = 116.624{{c}} | ||
{{Optimal ET sequence|legend=0| 10, 62, 72 }} | {{Optimal ET sequence|legend=0| 10, 62, 72 }} | ||
| Line 979: | Line 961: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~17/12 = 600.000, ~15/14 = 116.771 | * CTE: ~17/12 = 600.000{{c}}, ~15/14 = 116.771{{c}} | ||
* POTE: ~17/12 = 600.000, ~15/14 = 116.628 | * POTE: ~17/12 = 600.000{{c}}, ~15/14 = 116.628{{c}} | ||
{{Optimal ET sequence|legend=0| 10, 62, 72 }} | {{Optimal ET sequence|legend=0| 10, 62, 72 }} | ||
| Line 996: | Line 978: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200.000, ~27/26 = 58.337 | * CTE: ~2 = 1200.000{{c}}, ~27/26 = 58.337{{c}} | ||
* POTE: ~2 = 1200.000, ~27/26 = 58.288 | * POTE: ~2 = 1200.000{{c}}, ~27/26 = 58.288{{c}} | ||
{{Optimal ET sequence|legend=0| 41, 62, 103, 247c, 350bcde }} | {{Optimal ET sequence|legend=0| 41, 62, 103, 247c, 350bcde }} | ||
| Line 1,011: | Line 993: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200.000, ~27/26 = 58.312 | * CTE: ~2 = 1200.000{{c}}, ~27/26 = 58.312{{c}} | ||
* POTE: ~2 = 1200.000, ~27/26 = 58.261 | * POTE: ~2 = 1200.000{{c}}, ~27/26 = 58.261{{c}} | ||
{{Optimal ET sequence|legend=0| 41, 62, 103 }} | {{Optimal ET sequence|legend=0| 41, 62, 103 }} | ||
| Line 1,028: | Line 1,010: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~17/12 = 600.000, ~27/26 = 58.350 | * CTE: ~17/12 = 600.000{{c}}, ~27/26 = 58.350{{c}} | ||
* POTE: ~17/12 = 600.000, ~27/26 = 58.288 | * POTE: ~17/12 = 600.000{{c}}, ~27/26 = 58.288{{c}} | ||
{{Optimal ET sequence|legend=0| 62, 144g, 206begg }} | {{Optimal ET sequence|legend=0| 62, 144g, 206begg }} | ||
| Line 1,043: | Line 1,025: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~17/12 = 600.000, ~27/26 = 58.356 | * CTE: ~17/12 = 600.000{{c}}, ~27/26 = 58.356{{c}} | ||
* POTE: ~17/12 = 600.000, ~27/26 = 58.283 | * POTE: ~17/12 = 600.000{{c}}, ~27/26 = 58.283{{c}} | ||
{{Optimal ET sequence|legend=0| 62, 144gh, 206begghh }} | {{Optimal ET sequence|legend=0| 62, 144gh, 206begghh }} | ||
| Line 1,058: | Line 1,040: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~17/12 = 600.000, ~27/26 = 58.366 | * CTE: ~17/12 = 600.000{{c}}, ~27/26 = 58.366{{c}} | ||
* POTE: ~17/12 = 600.000, ~27/26 = 58.283 | * POTE: ~17/12 = 600.000{{c}}, ~27/26 = 58.283{{c}} | ||
{{Optimal ET sequence|legend=0| 62, 144gh, 206begghhi }} | {{Optimal ET sequence|legend=0| 62, 144gh, 206begghhi }} | ||
| Line 1,075: | Line 1,057: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200.000, ~16/13 = 361.096 | * CTE: ~2 = 1200.000{{c}}, ~16/13 = 361.096{{c}} | ||
* POTE: ~2 = 1200.000, ~16/13 = 361.121 | * POTE: ~2 = 1200.000{{c}}, ~16/13 = 361.121{{c}} | ||
{{Optimal ET sequence|legend=0| 103, 216c, 319bcde, 535bccdef }} | {{Optimal ET sequence|legend=0| 103, 216c, 319bcde, 535bccdef }} | ||
| Line 1,090: | Line 1,072: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200.000, ~16/13 = 361.098 | * CTE: ~2 = 1200.000{{c}}, ~16/13 = 361.098{{c}} | ||
* POTE: ~2 = 1200.000, ~16/13 = 361.123 | * POTE: ~2 = 1200.000{{c}}, ~16/13 = 361.123{{c}} | ||
{{Optimal ET sequence|legend=0| 103, 216c, 319bcde }} | {{Optimal ET sequence|legend=0| 103, 216c, 319bcde }} | ||
| Line 1,105: | Line 1,087: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200.000, ~15/14 = 116.142 | * CTE: ~2 = 1200.000{{c}}, ~15/14 = 116.142{{c}} | ||
* POTE: ~2 = 1200.000, ~15/14 = 116.277 | * POTE: ~2 = 1200.000{{c}}, ~15/14 = 116.277{{c}} | ||
{{Optimal ET sequence|legend=0| 10e, 21, 31 }} | {{Optimal ET sequence|legend=0| 10e, 21, 31 }} | ||
| Line 1,120: | Line 1,102: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200.000, ~15/14 = 116.194 | * CTE: ~2 = 1200.000{{c}}, ~15/14 = 116.194{{c}} | ||
* POTE: ~2 = 1200.000, ~15/14 = 116.268 | * POTE: ~2 = 1200.000{{c}}, ~15/14 = 116.268{{c}} | ||
{{Optimal ET sequence|legend=0| 10e, 21, 31 }} | {{Optimal ET sequence|legend=0| 10e, 21, 31 }} | ||
| Line 1,137: | Line 1,119: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200.000, ~33/32 = 58.399 | * CTE: ~2 = 1200.000{{c}}, ~33/32 = 58.399{{c}} | ||
* POTE: ~2 = 1200.000, ~33/32 = 58.408 | * POTE: ~2 = 1200.000{{c}}, ~33/32 = 58.408{{c}} | ||
{{Optimal ET sequence|legend=0| 20, 21, 41 }} | {{Optimal ET sequence|legend=0| 20, 21, 41 }} | ||
| Line 1,152: | Line 1,134: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200.000, ~33/32 = 58.436 | * CTE: ~2 = 1200.000{{c}}, ~33/32 = 58.436{{c}} | ||
* POTE: ~2 = 1200.000, ~33/32 = 58.430 | * POTE: ~2 = 1200.000{{c}}, ~33/32 = 58.430{{c}} | ||
{{Optimal ET sequence|legend=0| 20, 21, 41 }} | {{Optimal ET sequence|legend=0| 20, 21, 41 }} | ||
| Line 1,169: | Line 1,151: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200.000, ~11/8 = 541.670 | * CTE: ~2 = 1200.000{{c}}, ~11/8 = 541.670{{c}} | ||
* POTE: ~2 = 1200.000, ~11/8 = 541.668 | * POTE: ~2 = 1200.000{{c}}, ~11/8 = 541.668{{c}} | ||
{{Optimal ET sequence|legend=0| 11, 20, 31, 82e, 113e, 144ee }} | {{Optimal ET sequence|legend=0| 11, 20, 31, 82e, 113e, 144ee }} | ||
| Line 1,177: | Line 1,159: | ||
== Hemiseven == | == Hemiseven == | ||
Unlike miracle which splits ~8/7, hemiseven splits ~7/4. It can be described as the {{nowrap|72 & | Unlike miracle which splits ~8/7, hemiseven splits ~7/4. It can be described as the {{nowrap| 72 & 77 }} temperament. [[149edo]] is an obvious tuning. | ||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
| Line 1,184: | Line 1,166: | ||
{{Mapping|legend=1| 1 4 14 2 | 0 -6 -29 2 }} | {{Mapping|legend=1| 1 4 14 2 | 0 -6 -29 2 }} | ||
: mapping generators: ~2, ~320/243 | : mapping generators: ~2, ~320/243 | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[CTE]]: ~2 = 1200.000, ~320/243 = 483.215 | * [[CTE]]: ~2 = 1200.000{{c}}, ~320/243 = 483.215{{c}} | ||
: [[error map]]: {{val| 0.000 -1.247 +0.441 -2.395 }} | : [[error map]]: {{val| 0.000 -1.247 +0.441 -2.395 }} | ||
* [[POTE]]: ~2 = 1200.000, ~320/243 = 483.267 | * [[POTE]]: ~2 = 1200.000{{c}}, ~320/243 = 483.267{{c}} | ||
: error map: {{val| 0.000 -1.554 -1.043 -2.293 }} | : error map: {{val| 0.000 -1.554 -1.043 -2.293 }} | ||
| Line 1,205: | Line 1,186: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200.000, ~320/243 = 483.247 | * CTE: ~2 = 1200.000{{c}}, ~320/243 = 483.247{{c}} | ||
* POTE: ~2 = 1200.000, ~320/243 = 483.276 | * POTE: ~2 = 1200.000{{c}}, ~320/243 = 483.276{{c}} | ||
{{Optimal ET sequence|legend=0| 72, 149, 221e, 293de }} | {{Optimal ET sequence|legend=0| 72, 149, 221e, 293de }} | ||
| Line 1,220: | Line 1,201: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200.000, ~120/91 = 483.213 | * CTE: ~2 = 1200.000{{c}}, ~120/91 = 483.213{{c}} | ||
* POTE: ~2 = 1200.000, ~120/91 = 483.255 | * POTE: ~2 = 1200.000{{c}}, ~120/91 = 483.255{{c}} | ||
{{Optimal ET sequence|legend=0| 72, 149, 221ef }} | {{Optimal ET sequence|legend=0| 72, 149, 221ef }} | ||
| Line 1,235: | Line 1,216: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200.000, ~45/34 = 483.213 | * CTE: ~2 = 1200.000{{c}}, ~45/34 = 483.213{{c}} | ||
* POTE: ~2 = 1200.000, ~45/34 = 483.261 | * POTE: ~2 = 1200.000{{c}}, ~45/34 = 483.261{{c}} | ||
{{Optimal ET sequence|legend=0| 72, 149, 221ef }} | {{Optimal ET sequence|legend=0| 72, 149, 221ef }} | ||
| Line 1,246: | Line 1,227: | ||
: ''For the 5-limit version, see [[Syntonic–31 equivalence continuum #Valentine (5-limit)]].'' | : ''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 [[ | 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 also be described as the {{nowrap| 31 & 46 }} temperament, and [[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 ( | 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 | [[Subgroup]]: 2.3.5.7 | ||
| Line 1,255: | Line 1,236: | ||
{{Mapping|legend=1| 1 1 2 3 | 0 9 5 -3 }} | {{Mapping|legend=1| 1 1 2 3 | 0 9 5 -3 }} | ||
: mapping generators: ~2, ~21/20 | : mapping generators: ~2, ~21/20 | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[CTE]]: ~2 = 1200.000, ~21/20 = 77.878 | * [[CTE]]: ~2 = 1200.000{{c}}, ~21/20 = 77.878{{c}} | ||
: [[error map]]: {{val| 0.000 -1.057 +3.074 -2.459 }} | : [[error map]]: {{val| 0.000 -1.057 +3.074 -2.459 }} | ||
* [[POTE]]: ~2 = 1200.000, ~21/20 = 77.864 | * [[POTE]]: ~2 = 1200.000{{c}}, ~21/20 = 77.864{{c}} | ||
: error map: {{val| 0.000 -1.181 +3.005 -2.417 }} | : error map: {{val| 0.000 -1.181 +3.005 -2.417 }} | ||
| Line 1,288: | Line 1,268: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200.000, ~22/21 = 77.963 | * CTE: ~2 = 1200.000{{c}}, ~22/21 = 77.963{{c}} | ||
* POTE: ~2 = 1200.000, ~22/21 = 77.881 | * POTE: ~2 = 1200.000{{c}}, ~22/21 = 77.881{{c}} | ||
Minimax tuning: | Minimax tuning: | ||
| Line 1,310: | Line 1,290: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200.000, ~22/21 = 77.968 | * CTE: ~2 = 1200.000{{c}}, ~22/21 = 77.968{{c}} | ||
* POTE: ~2 = 1200.000, ~22/21 = 77.958 | * POTE: ~2 = 1200.000{{c}}, ~22/21 = 77.958{{c}} | ||
{{Optimal ET sequence|legend=0| 15f, 31, 46, 77 }} | {{Optimal ET sequence|legend=0| 15f, 31, 46, 77 }} | ||
| Line 1,325: | Line 1,305: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200.000, ~22/21 = 78.003 | * CTE: ~2 = 1200.000{{c}}, ~22/21 = 78.003{{c}} | ||
* POTE: ~2 = 1200.000, ~22/21 = 78.003 | * POTE: ~2 = 1200.000{{c}}, ~22/21 = 78.003{{c}} | ||
{{Optimal ET sequence|legend=0| 15f, 31, 46, 77, 123e }} | {{Optimal ET sequence|legend=0| 15f, 31, 46, 77, 123e }} | ||
| Line 1,340: | Line 1,320: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200.000, ~22/21 = 77.694 | * CTE: ~2 = 1200.000{{c}}, ~22/21 = 77.694{{c}} | ||
* POTE: ~2 = 1200.000, ~22/21 = 77.709 | * POTE: ~2 = 1200.000{{c}}, ~22/21 = 77.709{{c}} | ||
{{Optimal ET sequence|legend=0| 15, 31 }} | {{Optimal ET sequence|legend=0| 15, 31 }} | ||
| Line 1,355: | Line 1,335: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200.000, ~22/21 = 78.243 | * CTE: ~2 = 1200.000{{c}}, ~22/21 = 78.243{{c}} | ||
* POTE: ~2 = 1200.000, ~22/21 = 78.219 | * POTE: ~2 = 1200.000{{c}}, ~22/21 = 78.219{{c}} | ||
{{Optimal ET sequence|legend=0| 15, 31f, 46 }} | {{Optimal ET sequence|legend=0| 15, 31f, 46 }} | ||
| Line 1,372: | Line 1,352: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~55/39 = 600.000, ~22/21 = 77.997 | * CTE: ~55/39 = 600.000{{c}}, ~22/21 = 77.997{{c}} | ||
* POTE: ~55/39 = 600.000, ~22/21 = 77.839 | * POTE: ~55/39 = 600.000{{c}}, ~22/21 = 77.839{{c}} | ||
{{Optimal ET sequence|legend=0| 16, 30, 46, 62, 108ef }} | {{Optimal ET sequence|legend=0| 16, 30, 46, 62, 108ef }} | ||
| Line 1,389: | Line 1,369: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200.000, ~40/39 = 39.014 | * CTE: ~2 = 1200.000{{c}}, ~40/39 = 39.014{{c}} | ||
* POTE: ~2 = 1200.000, ~40/39 = 39.044 | * POTE: ~2 = 1200.000{{c}}, ~40/39 = 39.044{{c}} | ||
{{Optimal ET sequence|legend=0| 30, 31, 61, 92f }} | {{Optimal ET sequence|legend=0| 30, 31, 61, 92f }} | ||
| Line 1,406: | Line 1,386: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200.000, ~13/9 = 638.964 | * CTE: ~2 = 1200.000{{c}}, ~13/9 = 638.964{{c}} | ||
* CWE: ~2 = 1200.000, ~13/9 = 638.932 | * CWE: ~2 = 1200.000{{c}}, ~13/9 = 638.932{{c}} | ||
{{Optimal ET sequence|legend=0| 15, 47ef, 62, 77 }} | {{Optimal ET sequence|legend=0| 15, 47ef, 62, 77 }} | ||
| Line 1,421: | Line 1,401: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200.000, ~45/44 = 38.928 | * CTE: ~2 = 1200.000{{c}}, ~45/44 = 38.928{{c}} | ||
* POTE: ~2 = 1200.000, ~45/44 = 38.921 | * POTE: ~2 = 1200.000{{c}}, ~45/44 = 38.921{{c}} | ||
{{Optimal ET sequence|legend=0| 31, 92e, 123, 154, 185 }} | {{Optimal ET sequence|legend=0| 31, 92e, 123, 154, 185 }} | ||
| Line 1,436: | Line 1,416: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200.000, ~45/44 = 38.944 | * CTE: ~2 = 1200.000{{c}}, ~45/44 = 38.944{{c}} | ||
* POTE: ~2 = 1200.000, ~45/44 = 38.948 | * POTE: ~2 = 1200.000{{c}}, ~45/44 = 38.948{{c}} | ||
{{Optimal ET sequence|legend=0| 31, 123, 154 }} | {{Optimal ET sequence|legend=0| 31, 123, 154 }} | ||
| Line 1,451: | Line 1,431: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200.000, ~40/39 = 38.946 | * CTE: ~2 = 1200.000{{c}}, ~40/39 = 38.946{{c}} | ||
* POTE: ~2 = 1200.000, ~40/39 = 38.993 | * POTE: ~2 = 1200.000{{c}}, ~40/39 = 38.993{{c}} | ||
{{Optimal ET sequence|legend=0| 31, 92ef }} | {{Optimal ET sequence|legend=0| 31, 92ef }} | ||
| Line 1,462: | Line 1,442: | ||
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Shibboleth]].'' | : ''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. Its [[ploidacot]] is wau-enneacot. 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]]( | 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. Its [[ploidacot]] is wau-enneacot. 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. | ||
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 {{nowrap| S19 {{=}} [[361/360]] }} and {{nowrap| S20 {{=}} [[400/399]] }}. | 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 {{nowrap| S19 {{=}} [[361/360]] }} and {{nowrap| S20 {{=}} [[400/399]] }}. | ||
| Line 1,477: | Line 1,457: | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[CTE]]: ~2 = 1200.000, ~6/5 = 321.798 | * [[CTE]]: ~2 = 1200.000{{c}}, ~6/5 = 321.798{{c}} | ||
* [[POTE]]: ~2 = 1200.000, ~6/5 = 321.930 | * [[POTE]]: ~2 = 1200.000{{c}}, ~6/5 = 321.930{{c}} | ||
{{Optimal ET sequence|legend=1| 11c, 15, 26, 41 }} | {{Optimal ET sequence|legend=1| 11c, 15, 26, 41 }} | ||
| Line 1,484: | Line 1,464: | ||
[[Badness]]: | [[Badness]]: | ||
* Smith: 0.0479 | * Smith: 0.0479 | ||
* | * Sintel: 1.21 | ||
=== 11-limit === | === 11-limit === | ||
| Line 1,494: | Line 1,474: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200.000, ~6/5 = 321.815 | * CTE: ~2 = 1200.000{{c}}, ~6/5 = 321.815{{c}} | ||
* POTE: ~2 = 1200.000, ~6/5 = 321.847 | * POTE: ~2 = 1200.000{{c}}, ~6/5 = 321.847{{c}} | ||
{{Optimal ET sequence|legend=0| 11c, 15, 26, 41, 179cde, 220cde, 261ccdee }} | {{Optimal ET sequence|legend=0| 11c, 15, 26, 41, 179cde, 220cde, 261ccdee }} | ||
| Line 1,501: | Line 1,481: | ||
Badness: | Badness: | ||
* Smith: 0.0257 | * Smith: 0.0257 | ||
* | * Sintel: 0.848 | ||
==== 2.3.5.7.11.19 subgroup ==== | ==== 2.3.5.7.11.19 subgroup ==== | ||
| Line 1,513: | Line 1,493: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = | * CTE: ~2 = 1200.000{{c}}, ~6/5 = 321.779{{c}} | ||
* POTE: ~2 = | * POTE: ~2 = 1200.000{{c}}, ~6/5 = 321.827{{c}} | ||
Optimal ET sequence: {{Optimal ET sequence| 11c, 15, 26, 41, 138e, 179cde, 220cdeh }} | Optimal ET sequence: {{Optimal ET sequence| 11c, 15, 26, 41, 138e, 179cde, 220cdeh }} | ||
| Line 1,530: | Line 1,510: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200.000, ~6/5 = 321.986 | * CTE: ~2 = 1200.000{{c}}, ~6/5 = 321.986{{c}} | ||
* POTE: ~2 = 1200.000, ~6/5 = 321.994 | * POTE: ~2 = 1200.000{{c}}, ~6/5 = 321.994{{c}} | ||
{{Optimal ET sequence|legend=0| 11cf, 15, 26, 41 }} | {{Optimal ET sequence|legend=0| 11cf, 15, 26, 41 }} | ||
| Line 1,537: | Line 1,517: | ||
Badness: | Badness: | ||
* Smith: 0.0215 | * Smith: 0.0215 | ||
* | * Sintel: 0.887 | ||
==== 17-limit ==== | ==== 17-limit ==== | ||
| Line 1,547: | Line 1,527: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200.000, ~6/5 = 322.136 | * CTE: ~2 = 1200.000{{c}}, ~6/5 = 322.136{{c}} | ||
* POTE: ~2 = 1200.000, ~6/5 = 322.149 | * POTE: ~2 = 1200.000{{c}}, ~6/5 = 322.149{{c}} | ||
{{Optimal ET sequence|legend=0| 11cfg, 15g, 26, 41 }} | {{Optimal ET sequence|legend=0| 11cfg, 15g, 26, 41 }} | ||
| Line 1,562: | Line 1,542: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200.000, ~6/5 = 322.084 | * CTE: ~2 = 1200.000{{c}}, ~6/5 = 322.084{{c}} | ||
* CWE: ~2 = 1200.000, ~6/5 = 322.121 | * CWE: ~2 = 1200.000{{c}}, ~6/5 = 322.121{{c}} | ||
{{Optimal ET sequence|legend=0| 11cfgh, 15g, 26, 41 }} | {{Optimal ET sequence|legend=0| 11cfgh, 15g, 26, 41 }} | ||
| Line 1,570: | Line 1,550: | ||
=== Superana === | === Superana === | ||
This extension (41 & 56) is the counterpart of canonical superkleismic on the other side of 41edo. | 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 | ||
| Line 1,579: | Line 1,559: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200.000, ~6/5 = 321.724 | * CTE: ~2 = 1200.000{{c}}, ~6/5 = 321.724{{c}} | ||
* POTE: ~2 = 1200.000, ~6/5 = 321.719 | * POTE: ~2 = 1200.000{{c}}, ~6/5 = 321.719{{c}} | ||
{{Optimal ET sequence|legend=0| 15f, 26f, 41, 97, 138e }} | {{Optimal ET sequence|legend=0| 15f, 26f, 41, 97, 138e }} | ||
| Line 1,594: | Line 1,574: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200.000, ~6/5 = 321.650 | * CTE: ~2 = 1200.000{{c}}, ~6/5 = 321.650{{c}} | ||
* POTE: ~2 = 1200.000, ~6/5 = 321.657 | * POTE: ~2 = 1200.000{{c}}, ~6/5 = 321.657{{c}} | ||
{{Optimal ET sequence|legend=0| 15f, 26fg, 41, 56, 97g }} | {{Optimal ET sequence|legend=0| 15f, 26fg, 41, 56, 97g }} | ||
| Line 1,609: | Line 1,589: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200.000, ~6/5 = 321.646 | * CTE: ~2 = 1200.000{{c}}, ~6/5 = 321.646{{c}} | ||
* POTE: ~2 = 1200.000, ~6/5 = 321.643 | * POTE: ~2 = 1200.000{{c}}, ~6/5 = 321.643{{c}} | ||
{{Optimal ET sequence|legend=0| 15f, 26fg, 41, 56, 97g }} | {{Optimal ET sequence|legend=0| 15f, 26fg, 41, 56, 97g }} | ||
| Line 1,625: | Line 1,605: | ||
{{Mapping|legend=1| 2 5 8 | 0 -6 -11 }} | {{Mapping|legend=1| 2 5 8 | 0 -6 -11 }} | ||
: mapping generators: ~177147/125000, ~10/9 | : mapping generators: ~177147/125000, ~10/9 | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[CTE]]: ~177147/125000 = 600.000, ~10/9 = 183.041 | * [[CTE]]: ~177147/125000 = 600.000{{c}}, ~10/9 = 183.041{{c}} | ||
: [[error map]]: {{val| 0.000 -0.201 +0.235 }} | : [[error map]]: {{val| 0.000 -0.201 +0.235 }} | ||
* [[POTE]]: ~177147/125000 = 600.000, ~10/9 = 183.047 | * [[POTE]]: ~177147/125000 = 600.000{{c}}, ~10/9 = 183.047{{c}} | ||
: error map: {{val| 0.000 -0.236 +0.172 }} | : error map: {{val| 0.000 -0.236 +0.172 }} | ||
| Line 1,646: | Line 1,625: | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[CTE]]: ~1225/864 = 600.000, ~10/9 = 183.060 | * [[CTE]]: ~1225/864 = 600.000{{c}}, ~10/9 = 183.060{{c}} | ||
: [[error map]]: {{val| 0.000 -0.313 +0.030 -2.707 }} | : [[error map]]: {{val| 0.000 -0.313 +0.030 -2.707 }} | ||
* [[POTE]]: ~1225/864 = 600.000, ~10/9 = 183.161 | * [[POTE]]: ~1225/864 = 600.000{{c}}, ~10/9 = 183.161{{c}} | ||
: error map: {{val| 0.000 -0.924 -1.090 -2.503 }} | : error map: {{val| 0.000 -0.924 -1.090 -2.503 }} | ||
| Line 1,671: | Line 1,650: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~99/70 = 600.000, ~10/9 = 183.074 | * CTE: ~99/70 = 600.000{{c}}, ~10/9 = 183.074{{c}} | ||
* CWE: ~99/70 = 600.000, ~10/9 = 183.146 | * CWE: ~99/70 = 600.000{{c}}, ~10/9 = 183.146{{c}} | ||
Minimax tuning: | Minimax tuning: | ||
| Line 1,691: | Line 1,670: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~99/70 = 600.000, ~10/9 = 183.125 | * CTE: ~99/70 = 600.000{{c}}, ~10/9 = 183.125{{c}} | ||
* POTE: ~99/70 = 600.000, ~10/9 = 183.187 | * POTE: ~99/70 = 600.000{{c}}, ~10/9 = 183.187{{c}} | ||
{{Optimal ET sequence|legend=0| 46f, 72, 118, 190, 262df, 452cdef }} | {{Optimal ET sequence|legend=0| 46f, 72, 118, 190, 262df, 452cdef }} | ||
| Line 1,706: | Line 1,685: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~91/64 = 600.000, ~10/9 = 183.048 | * CTE: ~91/64 = 600.000{{c}}, ~10/9 = 183.048{{c}} | ||
* POTE: ~91/64 = 600.000, ~10/9 = 183.198 | * POTE: ~91/64 = 600.000{{c}}, ~10/9 = 183.198{{c}} | ||
{{Optimal ET sequence|legend=0| 26, 46, 72, 190ff }} | {{Optimal ET sequence|legend=0| 26, 46, 72, 190ff }} | ||
| Line 1,721: | Line 1,700: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~17/12 = 600.000, ~10/9 = 183.020 | * CTE: ~17/12 = 600.000{{c}}, ~10/9 = 183.020{{c}} | ||
* POTE: ~17/12 = 600.000, ~10/9 = 183.196 | * POTE: ~17/12 = 600.000{{c}}, ~10/9 = 183.196{{c}} | ||
{{Optimal ET sequence|legend=0| 26, 46, 72, 190ffg }} | {{Optimal ET sequence|legend=0| 26, 46, 72, 190ffg }} | ||
| Line 1,738: | Line 1,717: | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[CTE]]: ~3375/2401 = 600.000, ~15/14 = 122.031 | * [[CTE]]: ~3375/2401 = 600.000{{c}}, ~15/14 = 122.031{{c}} | ||
: [[error map]]: {{val| 0.000 -0.232 +0.087 -2.734 }} | : [[error map]]: {{val| 0.000 -0.232 +0.087 -2.734 }} | ||
* [[POTE]]: ~3375/2401 = 600.000, ~15/14 = 122.027 | * [[POTE]]: ~3375/2401 = 600.000{{c}}, ~15/14 = 122.027{{c}} | ||
: error map: {{val| 0.000 -0.195 +0.033 -2.746 }} | : error map: {{val| 0.000 -0.195 +0.033 -2.746 }} | ||
| Line 1,750: | Line 1,729: | ||
{{Main|Dee leap week}} | {{Main|Dee leap week}} | ||
Subgroup: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
Comma list: 1029/1024, 2460375/2458624 | [[Comma list]]: 1029/1024, 2460375/2458624 | ||
{{Mapping|legend=1| 1 4 -6 2 | 0 -9 31 3 }} | |||
Optimal tuning (CTE): ~135/112 = 322.123 | [[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~135/112 = 322.123{{c}} | ||
=== 11-limit === | === 11-limit === | ||
| Line 1,763: | Line 1,742: | ||
Comma list: 385/384, 441/440, 2460375/2458624 | Comma list: 385/384, 441/440, 2460375/2458624 | ||
Mapping: | Mapping: {{mapping| 1 4 -6 2 15 | 0 -9 31 3 43 }} | ||
Optimal tuning (CTE): ~135/112 = 322.097 | Optimal tuning (CTE): ~2 = 1200.000{{c}}, ~135/112 = 322.097 | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 41, 149, 190, 231 }} | ||
== Necromanteion == | == Necromanteion == | ||
| Line 1,775: | Line 1,754: | ||
{{Mapping|legend=1| 1 7 10 1 | 0 -12 -17 4 }} | {{Mapping|legend=1| 1 7 10 1 | 0 -12 -17 4 }} | ||
: mapping generators: ~2, ~48/35 | : mapping generators: ~2, ~48/35 | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[CTE]]: ~2 = 1200.000, ~48/35 = 541.743 | * [[CTE]]: ~2 = 1200.000{{c}}, ~48/35 = 541.743{{c}} | ||
: [[error map]]: {{val| 0.000 -2.872 +4.053 -1.853 }} | : [[error map]]: {{val| 0.000 -2.872 +4.053 -1.853 }} | ||
* [[POTE]]: ~2 = 1200.000, ~48/35 = 541.779 | * [[POTE]]: ~2 = 1200.000{{c}}, ~48/35 = 541.779{{c}} | ||
: error map: {{val| 0.000 -3.304 +3.442 -1.710 }} | : error map: {{val| 0.000 -3.304 +3.442 -1.710 }} | ||
| Line 1,796: | Line 1,774: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200.000, ~15/11 = 541.695 | * CTE: ~2 = 1200.000{{c}}, ~15/11 = 541.695{{c}} | ||
* POTE: ~2 = 1200.000, ~15/11 = 541.729 | * POTE: ~2 = 1200.000{{c}}, ~15/11 = 541.729{{c}} | ||
{{Optimal ET sequence|legend=0| 20ce, 31, 113c, 144c }} | {{Optimal ET sequence|legend=0| 20ce, 31, 113c, 144c }} | ||
| Line 1,811: | Line 1,789: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200.000, ~15/11 = 541.673 | * CTE: ~2 = 1200.000{{c}}, ~15/11 = 541.673{{c}} | ||
* POTE: ~2 = 1200.000, ~15/11 = 541.606 | * POTE: ~2 = 1200.000{{c}}, ~15/11 = 541.606{{c}} | ||
{{Optimal ET sequence|legend=0| 20ce, 31, 82cf, 113cf }} | {{Optimal ET sequence|legend=0| 20ce, 31, 82cf, 113cf }} | ||
| Line 1,826: | Line 1,804: | ||
{{Mapping|legend=1| 1 -2 8 4 | 0 12 -19 -4 }} | {{Mapping|legend=1| 1 -2 8 4 | 0 12 -19 -4 }} | ||
: mapping generators: ~2. ~315/256 | : mapping generators: ~2. ~315/256 | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[CTE]]: ~2 = 1200.000, ~315/256 = 358.548 | * [[CTE]]: ~2 = 1200.000{{c}}, ~315/256 = 358.548{{c}} | ||
: [[error map]]: {{val| 0.000 +0.620 +1.275 -3.018 }} | : [[error map]]: {{val| 0.000 +0.620 +1.275 -3.018 }} | ||
* [[POTE]]: ~2 = 1200.000, ~315/256 = 358.548 | * [[POTE]]: ~2 = 1200.000{{c}}, ~315/256 = 358.548{{c}} | ||
: error map: {{val| 0.000 +0.627 +1.265 -3.020 }} | : error map: {{val| 0.000 +0.627 +1.265 -3.020 }} | ||
| Line 1,847: | Line 1,824: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200.000, ~27/22 = 358.575 | * CTE: ~2 = 1200.000{{c}}, ~27/22 = 358.575{{c}} | ||
* POTE: ~2 = 1200.000, ~27/22 = 358.571 | * POTE: ~2 = 1200.000{{c}}, ~27/22 = 358.571{{c}} | ||
{{Optimal ET sequence|legend=0| 77, 87, 164, 251d }} | {{Optimal ET sequence|legend=0| 77, 87, 164, 251d }} | ||
| Line 1,862: | Line 1,839: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200.000, ~16/13 = 358.576 | * CTE: ~2 = 1200.000{{c}}, ~16/13 = 358.576{{c}} | ||
* POTE: ~2 = 1200.000, ~16/13 = 358.574 | * POTE: ~2 = 1200.000{{c}}, ~16/13 = 358.574{{c}} | ||
{{Optimal ET sequence|legend=0| 77, 87, 164, 251d }} | {{Optimal ET sequence|legend=0| 77, 87, 164, 251d }} | ||
| Line 1,877: | Line 1,854: | ||
{{Mapping|legend=1| 1 4 6 2 | 0 -21 -32 7 }} | {{Mapping|legend=1| 1 4 6 2 | 0 -21 -32 7 }} | ||
: mapping generators: ~2, ~27/25 | : mapping generators: ~2, ~27/25 | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[CTE]]: ~2 = 1200.000, ~27/25 = 137.971 | * [[CTE]]: ~2 = 1200.000{{c}}, ~27/25 = 137.971{{c}} | ||
: [[error map]]: {{val| 0.000 +0.658 -1.380 -3.030 }} | : [[error map]]: {{val| 0.000 +0.658 -1.380 -3.030 }} | ||
* [[POTE]]: ~2 = 1200.000, ~27/25 = 138.006 | * [[POTE]]: ~2 = 1200.000{{c}}, ~27/25 = 138.006{{c}} | ||
: error map: {{val| 0.000 -0.075 -2.496 -2.786 }} | : error map: {{val| 0.000 -0.075 -2.496 -2.786 }} | ||
| Line 1,898: | Line 1,874: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200.000, ~27/25 = 137.970 | * CTE: ~2 = 1200.000{{c}}, ~27/25 = 137.970{{c}} | ||
* POTE: ~2 = 1200.000, ~27/25 = 137.990 | * POTE: ~2 = 1200.000{{c}}, ~27/25 = 137.990{{c}} | ||
{{Optimal ET sequence|legend=0| 26, 61, 87, 200, 287d }} | {{Optimal ET sequence|legend=0| 26, 61, 87, 200, 287d }} | ||
| Line 1,913: | Line 1,889: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200.000, ~13/12 = 137.971 | * CTE: ~2 = 1200.000{{c}}, ~13/12 = 137.971{{c}} | ||
* POTE: ~2 = 1200.000, ~13/12 = 137.990 | * POTE: ~2 = 1200.000{{c}}, ~13/12 = 137.990{{c}} | ||
{{Optimal ET sequence|legend=0| 26, 61, 87, 200 }} | {{Optimal ET sequence|legend=0| 26, 61, 87, 200 }} | ||
| Line 1,928: | Line 1,904: | ||
{{Mapping|legend=1| 1 -11 -7 7 | 0 27 20 -9 }} | {{Mapping|legend=1| 1 -11 -7 7 | 0 27 20 -9 }} | ||
: mapping generators: ~2, ~864/625 | : mapping generators: ~2, ~864/625 | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[CTE]]: ~2 = 1200.000, ~864/625 = 559.320 | * [[CTE]]: ~2 = 1200.000{{c}}, ~864/625 = 559.320{{c}} | ||
: [[error map]]: {{val| 0.000 -0.317 +0.085 -2.705 }} | : [[error map]]: {{val| 0.000 -0.317 +0.085 -2.705 }} | ||
* [[POTE]]: ~2 = 1200.000, ~864/625 = 559.295 | * [[POTE]]: ~2 = 1200.000{{c}}, ~864/625 = 559.295{{c}} | ||
: error map: {{val| 0.000 -1.003 -0.423 -2.477 }} | : error map: {{val| 0.000 -1.003 -0.423 -2.477 }} | ||
| Line 1,949: | Line 1,924: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200.000, ~242/175 = 559.327 | * CTE: ~2 = 1200.000{{c}}, ~242/175 = 559.327{{c}} | ||
* POTE: ~2 = 1200.000, ~242/175 = 559.293 | * POTE: ~2 = 1200.000{{c}}, ~242/175 = 559.293{{c}} | ||
{{Optimal ET sequence|legend=0| 15, …, 88, 103, 118, 221e, 339de }} | {{Optimal ET sequence|legend=0| 15, …, 88, 103, 118, 221e, 339de }} | ||
| Line 1,964: | Line 1,939: | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[CTE]]: ~2 = 1200.000, ~3125/2304 = 524.188 | * [[CTE]]: ~2 = 1200.000{{c}}, ~3125/2304 = 524.188{{c}} | ||
: [[error map]]: {{val| 0.000 -0.154 -0.756 -2.759 }} | : [[error map]]: {{val| 0.000 -0.154 -0.756 -2.759 }} | ||
* [[POTE]]: ~2 = 1200.000, ~3125/2304 = 524.210 | * [[POTE]]: ~2 = 1200.000{{c}}, ~3125/2304 = 524.210{{c}} | ||
: error map: {{val| 0.000 -0.892 -1.047 -2.513 }} | : error map: {{val| 0.000 -0.892 -1.047 -2.513 }} | ||
| Line 1,981: | Line 1,956: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200.000, ~847/625 = 524.183 | * CTE: ~2 = 1200.000{{c}}, ~847/625 = 524.183{{c}} | ||
* POTE: ~2 = 1200.000, ~847/625 = 524.210 | * POTE: ~2 = 1200.000{{c}}, ~847/625 = 524.210{{c}} | ||
{{Optimal ET sequence|legend=0| 16, 71, 87, 103, 190 }} | {{Optimal ET sequence|legend=0| 16, 71, 87, 103, 190 }} | ||
| Line 1,996: | Line 1,971: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = 1200.000, ~65/48 = 524.183 | * CTE: ~2 = 1200.000{{c}}, ~65/48 = 524.183{{c}} | ||
* POTE: ~2 = 1200.000, ~65/48 = 524.209 | * POTE: ~2 = 1200.000{{c}}, ~65/48 = 524.209{{c}} | ||
{{Optimal ET sequence|legend=0| 16, 71, 87, 103, 190 }} | {{Optimal ET sequence|legend=0| 16, 71, 87, 103, 190 }} | ||
| Line 2,003: | Line 1,978: | ||
Badness (Smith): 0.021636 | Badness (Smith): 0.021636 | ||
== | == References == | ||
[[Category:Temperament clans]] | [[Category:Temperament clans]] | ||