Subgroup temperaments: Difference between revisions
m →2.3.13/5 subgroup: More specific todo category (not a perfect match, but a good spot to be found by someone who can help) |
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For temperaments that omit various prime harmonics, see: | For temperaments that omit various prime harmonics, see: | ||
* [[No-elevens subgroup temperaments]] | * [[No-elevens subgroup temperaments]] | ||
* [[No-sevens subgroup temperaments]] | * [[No-sevens subgroup temperaments]] | ||
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= Composite subgroup temperaments = | = Composite subgroup temperaments = | ||
== 2.3.35 subgroup == | |||
=== Darian calendar === | |||
Darian calendar is described as 24 & 668 temperament in the 2.3.11.19 [[subgroup]] and is named after a certain calendar layout by the same name. The generator is close to the [[36/35]] quartertone, and this allows an extension to the 2.3.35.11.19 subgroup. 5 of them make [[11/8]], 8 of them make [[3/2]], and 6 of them make [[32/19]]. | |||
==== 2.3.11.19 subgroup ==== | |||
The temperament is simplest in this subgroup, although there is a tradeoff of breaking up the simplicity of the 36/35 quartertone. | |||
[[Subgroup]]: 2.3.11.19 | |||
{{Mapping|legend=2| 4 5 13 18 | 0 8 5 -6 }} | |||
: sval mapping generators: ~6291456/5285401, ~25289/24576 | |||
[[Optimal tuning]] ([[CTE]]): ~6291456/5285401 = 1\4, ~25289/24576 = 50.257 | |||
[[Support]]ing [[ET]]s: {{EDOs|24, 596, 620, 644, 668, 692, 716}}, ... | |||
==== 2.3.35.11.19 subgroup ==== | |||
668edo does not map 36/35 consistently, with its own [[direct approximation]] being 27 steps while the direct approximations of its constituent odd harmonics do not sum to that same amount: 3/2, 8/5, and 8/7 are 391, 453, and 129 steps, respectively, and 391 + 391 + 453 + 129 - 668 - 668 = 28, ≠ 27. | |||
Subgroup: 2.3.35.11.19 | |||
Sval mapping: {{mapping| 4 0 5 13 18 | 0 1 8 5 -6 }} | |||
: sval mapping generators: ~2240/1881, ~36/35 | |||
Optimal tuning (CTE): ~2240/1881 = 1\4, ~36/35 = 50.288 | |||
[[Support]]ing [[ET]]s: {{EDOs|24, 668}}, ... | |||
== 2.9.5.7 subgroup == | == 2.9.5.7 subgroup == | ||
See also [[Jubilismic clan #Antikythera|antikythera]] and [[Hemimean clan #Isra|isra]]. | See also [[Jubilismic clan #Antikythera|antikythera]] and [[Hemimean clan #Isra|isra]]. | ||
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{{Optimal ET sequence|legend=1| 6, 23def, 29f, 35, 41, 47 }} | {{Optimal ET sequence|legend=1| 6, 23def, 29f, 35, 41, 47 }} | ||
== 2. | == 2.3.25 subgroup == | ||
[[ | === Shrub === | ||
This is a restriction of diaschismic which omits the tritone to produce a diatonic scale. True to its name, it generates a [[shrubmajor]] third (~425c) in quarter-comma tuning. | |||
Subgroup: 2.3.25 | |||
Edo join: 17 & 12 | |||
Comma list: [[2048/2025]] | |||
{{Mapping|legend=2| 1 1 7| 0 1 -4}} | |||
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 705.136 | |||
==== 2.3.23.25.41 subgroup ==== | |||
''See also: [[Reversed meantone]]'' | |||
Edo join: 17 & 12 | |||
Comma list: 2048/2025, 576/575, 82/81 | |||
{{Mapping|legend=2| 1 1 1 7 3| 0 1 6 -4 4}} | |||
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 705.264 | |||
===== Sburb ===== | |||
This temperament sets the [[octave reduction|octave-reduced]] 413th harmonic (413/256, 827.998{{c}}) to the diminished seventh. | |||
Subgroup: 2.3.7.23.25.41.59 | |||
Edo join: 17 & 12 | |||
Comma list: 64/63, 225/224, 162/161, 82/81, 177/175 | |||
{{Mapping|legend=2| 1 1 4 1 7 3 10| 0 1 -2 6 -4 4 -7}} | |||
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 706.387 | |||
== 2.9.5.11 subgroup == | |||
=== Glacial === | |||
{{See also| Chromatic pairs #Glacial }} | |||
[[Subgroup]]: 2.9.5.11.13 | |||
[[Comma list]]: 45/44, 65/64, 81/80 | |||
= | {{Mapping|legend=2| 1 0 -4 -6 10 | 0 1 2 3 -2 }} | ||
{{Mapping|legend=3| 1 3/2 2 0 3 4 | 0 1/2 2 0 3 -2 }} | |||
: [[gencom]]: [2 9/8; 45/44 65/64 81/80] | |||
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 186.151 | |||
{{ | {{Optimal ET sequence|legend=1| 6, 13, 45be, 58bce, 71bce, 84bce }} | ||
[[Tp tuning #T2 tuning|RMS error]]: 2.887 cents | |||
[[ | Music: | ||
* ''[[Thundersnow]]'' - [[Sevish]] (2021) | |||
== 2.9.7 subgroup == | |||
=== Mabon === | |||
Derived from a [http://individual.utoronto.ca/kalendis/leap/index.htm#se calendar leap cycle built for the autumn equinox], hence the name. Defined as the 11 & 62 temperament. | |||
Subgroup: 2.9.7 | |||
Comma basis: 44957696/43046721 | |||
Sval mapping: [{{val|1 1 -3}}, {{val|0 3 8}}] | |||
Optimal tuning (CTE): ~729/448 = 870.792 | |||
{{Optimal ET sequence|legend=1|7d, 11, 18d, 29, 40, 62}}, ... | |||
==== 2.9.7.11 subgroup ==== | |||
Subgroup: 2.9.7.11 | |||
Comma basis: 896/891, 1331/1296 | |||
: [ | Sval mapping: [{{val|1 1 -3 2}}, {{val|0 3 8 2}}] | ||
Optimal tuning (CTE): ~16/11 = 870.966 | |||
{{Optimal ET sequence|legend=1| 11, | {{Optimal ET sequence|legend=1| 7d, 11, 40, 51, 62 }} | ||
[[ | == 2.9.7.11 subgroup == | ||
=== Apparatus === | |||
[[Subgroup]]: 2.9.7.11 | |||
[[Comma list]]: 41503/41472, 322102/321489 | |||
{{Mapping|legend=2| 1 5 3 5 | 0 -19 -2 -16 }} | |||
: mapping generators: ~2, ~77/72 | |||
{{Mapping|legend= | {{Mapping|legend=3| 1 5/2 0 3 5 | 0 -19/2 0 -2 -16 }} | ||
: | : [[gencom]]: [2 77/72; 41503/41472 322102/321489] | ||
[[Optimal tuning]] ([[CTE]]): ~77/72 = 115.5685 | |||
{{Optimal ET sequence|legend=1| 10e, 21, 31, 52, 83, 135, 353, 488, 623 }} | |||
[[ | [[Badness]]: 0.00263 | ||
{{ | === Joan === | ||
{{See also| Chromatic pairs #Joan }} | |||
[[ | Joan is related to [[casablanca]] as well as to [[orwell]]. | ||
[[Subgroup]]: 2.9.7.11 | [[Subgroup]]: 2.9.7.11 | ||
[[Comma list]]: | [[Comma list]]: 99/98, 9317/9216 | ||
{{Mapping|legend=2| 1 0 | {{Mapping|legend=2| 1 0 1 3 | 0 7 4 1 }} | ||
{{Mapping|legend=3| 1 0 0 1 3 | 0 7/2 0 4 1 }} | |||
: [[gencom]]: [2 11/8; 99/98 9317/9216] | |||
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~11/8 = 542.672 cents | |||
{{Optimal ET sequence|legend=1| 11, 20, 31, 42, 115bd, 157bd }} | |||
[[Tp tuning #T2 tuning|RMS error]]: 1.424 cents | |||
[[ | === Machine === | ||
Machine is every other step of [[supra]], most interesting for its scale patterns. | |||
[[ | [[Subgroup]]: 2.9.7.11 | ||
[[Comma list]]: 64/63, 99/98 | |||
{{Mapping|legend=2| 1 0 6 13 | 0 1 -1 -3 }} | |||
: sval mapping generators: ~2, ~9 | |||
{{Mapping|legend=3| 1 3/2 0 3 4 | 0 1/2 0 -1 -3 }} | |||
[[ | : [[gencom]]: [2 8/7; 64/63 99/98] | ||
[[Optimal tuning]]s: | |||
* [[CTE]]: ~2 = 1\1, ~9/8 = 216.9128 | |||
* [[POTE]]: ~2 = 1\1, ~9/8 = 214.3843 | |||
{{Optimal ET sequence|legend=1| 5, 6, 11, 17, 28 }} | |||
[[Badness]]: 0.00233 | |||
=== | === Penta a.k.a. mechanism === | ||
Penta or mechanism is the 8 & 11 temperament in the 2.9.7.11 subgroup. | |||
[[Subgroup]]: 2.9.7.11 | |||
[[ | [[Comma list]]: 896/891, 26411/26244 | ||
{{Mapping|legend=2| 1 0 -1 6 | 0 5 6 -4 }} | |||
: sval mapping generators: ~2, ~14/9 | |||
{{Mapping|legend=3| 1 5/2 0 5 2 | 0 -5/2 0 -6 4 }} | |||
: [[gencom]]: [2 9/7; 896/891 26411/26244] | |||
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~14/9 = 761.3782 | |||
{{Optimal ET sequence|legend=1| 8, 11, 30, 41, 52 }} | |||
[[ | [[Tp tuning #T2 tuning|RMS error]]: 0.4262 cents | ||
[[Badness]]: 0.00439 | |||
: | Scales: [[penta5]], [[penta8]], [[penta11]], [[penta19]] | ||
== 2.9.7.13.17 subgroup == | |||
=== Novisept === | |||
Novisept is generated by a one-cent-flat 9/7, such that stacking 5 of them gives you 7/4. It can be formed by doubling both generator and period of [[gizzard]]. | |||
[[ | [[Subgroup]]: 2.9.7.13.17 | ||
[[Comma list]]: 729/728, 442/441, 833/832 | |||
{{Mapping|legend=2| 1 1 1 -1 3| 0 6 5 13 3 }} | |||
= | [[Optimal tuning]] ([[CWE]]): ~2 = 1\1, ~9/7 = 433.836 | ||
Badness (Dirichlet): 0.142 | |||
== 2.9.11 subgroup == | |||
=== Demon === | |||
Demon is a temperament which equates 3 [[11/9]] with [[16/9]], or equivalently 3 [[18/11]] with [[9/8]], tempering out [[1331/1296]]. This results in [[11/9]] being tuned flat to a supraminor third, and [[27/22]] being tuned sharp to a submajor third. It was discovered by [[User:CompactStar|CompactStar]] while searching for temperaments assosciated with the [[7L 4s]] ("daemotonic") MOS, known for its lack of representation of simple temperaments. The optimal tuning for demon temperament is near the basic tuning of 7L 4s (13\18), and indeed [[18edo]] supports demon temperament. | |||
[[Subgroup]]: 2.9.11 | |||
: | [[Comma list]]: [[1331/1296]] | ||
{{Mapping|legend=2|1 1 2|0 3 2}} | |||
Optimal | [[Optimal tuning]] ([[CTE]]): ~[[18/11]] = 870.060 | ||
{{Optimal ET sequence|legend=1|4, 7, 11, 18, 29, 76e}} | |||
==== | === Genius === | ||
Named after the genius in Roman religion, following the demon (daimon) in Greek mythology. | |||
[[Subgroup]]: 2.9.11 | |||
[[Comma list]]: [[131769/131072]] | |||
{{Mapping|legend=2|1 1 4|0 4 -1}} | |||
Optimal tuning ( | [[Optimal tuning]] ([[CTE]]): ~[[16/11]] = 650.863 | ||
{{Optimal ET sequence|legend=1|9, 11, 24, 59, 83, 142, 225, 367}}[-11], 592[-11], 959[-9, --11], 1326[-9, --11] | |||
== 2.9.15.7 subgroup == | |||
=== Stacks (a.k.a. 2magic) === | |||
Stacks, the 11 & 30 temperament in the 2.9.15.7.11.13 subgroup, is every other step of [[magic]]. | |||
[[Subgroup]]: 2.9.15.7 | |||
[[ | [[Comma list]]: 225/224, 245/243 | ||
{{Mapping|legend=2| 1 0 2 -1 | 0 5 3 6 }} | |||
: sval mapping generators: ~2, ~14/9 | |||
{{Mapping|legend=3| 1 5/2 5/2 5 | 0 -5/2 -1/2 -6 }} | |||
: [[gencom]]: [2 9/7; 225/224 245/243] | |||
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~2 = 1\1, ~14/9 = 760.704 | |||
{{Optimal ET sequence|legend=1| 8, 11, 30, 41, 71, 93, 112c, 134c, 175c }} | |||
[[Tp tuning #T2 tuning|RMS error]]: 1.074 cents | |||
==== 2.9.15.7.11 ==== | |||
Subgroup: 2.9.15.7.11 | |||
Comma list: 100/99, 225/224, 245/243 | |||
Sval mapping: {{mapping| 1 0 2 -1 6 | 0 5 3 6 -4 }} | |||
Gencom mapping: {{mapping| 1 5/2 5/2 5 2 | 0 -5/2 -1/2 -6 4 }} | |||
: gencom: [2 9/7; 100/99 225/224 245/243] | |||
Optimal tuning (subgroup POTE): ~2 = 1\1, ~14/9 = 761.393 | |||
Optimal | Optimal ET sequence: {{Optimal ET sequence| 8, 11, 30, 41, 52, 93, 145, 342bce }} | ||
RMS error: 1.226 cents | |||
==== 2.9.15.7.11.13 ==== | |||
Subgroup: 2.9. | Subgroup: 2.9.15.7.11.13 | ||
Comma list: | Comma list: 100/99, 105/104, 144/143, 196/195 | ||
Sval mapping: {{mapping| 1 2 0 | Sval mapping: {{mapping| 1 0 2 -1 6 -2 | 0 5 3 6 -4 9 }} | ||
Gencom mapping: {{mapping| 1 | Gencom mapping: {{mapping| 1 5/2 5/2 5 2 7 | 0 -5/2 -1/2 -6 4 -9 }} | ||
: gencom: [2 | : gencom: [2 9/7; 100/99 105/104 144/143 196/195] | ||
Optimal tuning (subgroup POTE): ~2 = 1\1, ~ | Optimal tuning (subgroup POTE): ~2 = 1\1, ~14/9 = 761.023 | ||
{{Optimal ET sequence| | Optimal ET sequence: {{Optimal ET sequence| 11, 30, 41, 153cdef, 194cdef, 235cdef }} | ||
RMS error: 1.540 cents | |||
== 2.9.21 subgroup == | |||
=== A-team === | |||
A-team is every other step of [[slendric]]; the 2.9.5.21.11 extension below specifically restricts [[mothra]]. | |||
[[Subgroup]]: 2.9.21 | |||
[[Comma list]]: 1029/1024 | |||
{{Mapping|legend=2| 1 2 4 | 0 3 1 }} | |||
: sval mapping generators: ~2, ~21/16 | |||
= | {{Mapping|legend=3| 1 1 0 3 | 0 3/2 0 -1/2 }} | ||
[[ | : [[gencom]]: [2 21/16; 1029/1024] | ||
[[ | [[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~2 = 1\1, ~21/16 = 467.375 | ||
{{ | {{Optimal ET sequence|legend=1| 5, 13, 18, 41, 59, 77, 95 }} | ||
: | [[Tp tuning #T2 tuning|RMS error]]: 0.3202 cents | ||
[[ | ==== 2.9.5.21 ==== | ||
''Lookalike temperament: [[Dual-fifth_temperaments#Dual-3_A-Team|Dual-3 A-Team]]'' | |||
Subgroup: 2.9.5.21 | |||
[[Comma]] list: 81/80, 1029/1024 | |||
Sval mapping: {{mapping| 1 2 0 4 | 0 3 6 1 }} | |||
Mapping generators: ~2, ~21/16 | |||
Optimal ([[Lp tuning|POL2]]) generator: 464.3865 | |||
{{Optimal ET sequence|legend=1| 13, 18, 31, 44 }} | |||
===== 2.9.5.21.11 ===== | |||
Subgroup: 2.9.5.21.11 | |||
Comma list: 81/80, 99/98, 385/384 | |||
Sval mapping: {{mapping| 1 2 0 4 5 | 0 3 6 1 -4 }} | |||
Gencom mapping: {{mapping| 1 1 0 3 5 | 0 3/2 6 -1/2 -4 }} | |||
[ | : gencom: [2 21/16; 81/80 99/98 385/384] | ||
Optimal tuning (subgroup POTE): ~2 = 1\1, ~21/16 = 463.956 | |||
{{Optimal ET sequence|legend=1| 5, 13, 31 }} | |||
==== B-team ==== | |||
B-team (23 & 41) is every other step of [[rodan]]. | |||
Subgroup: 2.9.15.21.33 | |||
Comma list: 245/243, 385/384, 441/440 | |||
Sval mapping: {{mapping| 1 2 0 4 7 | 0 3 10 1 -5 }} | |||
Optimal tuning (subgroup POTE): ~2 = 1\1, ~21/16 = 468.918 | |||
{{ | {{Optimal ET sequence|legend=1| 5, 13c, 18, 23, 41, 64, 87, 151 }} | ||
== 4.3.5 subgroup == | |||
=== Tetrahanson === | |||
{{Main| Tetrahanson }} | |||
[[ | [[Subgroup]]: 4.3.5 | ||
[[ | [[Comma list]]: 15625/15552 | ||
= | {{Mapping|legend=2| 1 3 3 | 0 -6 -5 }} | ||
: Mapping generators: ~4, ~5/3 | |||
[[ | [[Optimal tuning]] ([[CTE]]): ~4 = 2\1, ~5/3 = 882.941 | ||
{{ | [[Support]]ing [[ET]]s: {{EDs|19, 106, 87, 68, 11, 8, 125, 49, 30, 27, 117, 46, 41b, 79|equave=4}} | ||
=== Tetrameantone === | |||
{{Main| Tetrameantone }} | |||
[[ | [[Subgroup]]: 4.3.5 | ||
[[ | [[Comma list]]: 81/80 | ||
{{Mapping|legend=2| 1 1 2 | 0 -1 -4 }} | |||
: Mapping generators: ~4, ~4/3 | |||
[[Optimal tuning]] ([[POTE]]): 4 = 2400.0, ~4/3 = 503.761 | |||
[[Support]]ing [[ET]]s: {{EDs|5, 9, 14, 19, 24, 43, 62, 81, 100|equave=4}} | |||
[[ | |||
=== Tetramagic === | |||
[[Subgroup]]: 4.3.5 | |||
[[ | [[Comma list]]: 3125/3072 | ||
{{Mapping|legend=2| 1 0 1 | 0 5 1 }} | |||
: Mapping generators: ~4, ~5/4 | |||
= | [[Optimal tuning]] ([[POTE]]): 4 = 2400.0, ~5/4 = 380.059 | ||
[[Support]]ing [[ET]]s: {{EDs|6, 13, 19, 25, 38, 44, 63, 82|equave=4}} | |||
=== Blacktetra === | |||
[[Subgroup]]: 4.3.5 | |||
[[Comma list]]: 256/243 | |||
[ | |||
{{Mapping|legend=2| 5 4 6 | 0 0 -1 }} | |||
: Mapping generators: ~4, ~16/15 | |||
4 | |||
[[Optimal tuning]] ([[POTE]]): 1\5ed4 = 480.0, ~16/15 = 80.4062 | |||
[[Support]]ing [[ET]]s: {{EDs|5, 10, 15, 20, 25, 30, 55, 85, 115|equave=4}} | |||
== 4.6.5 subgroup == | |||
=== Meanquad === | |||
{{Main| Meanquad }} | |||
[[Subgroup]]: 4.6.5 | |||
[[Comma list]]: [[81/80]] = {{monzo| -4 4 -1 }} | |||
{{Mapping|legend=2| 1 0 -4| 0 1 4 }} | |||
: mapping generators: ~4, ~6 | |||
[[Optimal tuning]] (subgroup [[CTE]]): ~4 = 2\1, ~3/2 = 697.214 | |||
[[Support]]ing [[ET]]s: *7, *10, *11[-5], *13[+5], *17, *24, *27[+5], *31, *38, *41, *45, *52, *55, *69 | |||
[ | |||
[ | |||
[- | |||
[ | |||
[ | |||
<nowiki />* Wart for 4 | |||
</ | |||
==== 4.6.5.7 | ==== 4.6.5.7 subgroup (tetrominant) ==== | ||
[[Subgroup]]: 4.6.5.7 | |||
<pre> | [[Comma list]]: [[36/35]] = {{monzo| 0 2 -1 -1 }}, [[64/63]] = {{monzo| 4 -2 0 -1 }} | ||
Reduced Mapping | |||
4 6 5 | {{Mapping|legend=2| 1 0 -4 4 | 0 1 4 -2 }} | ||
[ ⟨ 1 0 1 | |||
⟨ 0 16 2 | [[Optimal tuning]] (subgroup [[CTE]]): ~4 = 2\1, ~3/2 = 699.622 | ||
[[Support]]ing [[ET]]s: *7, *10, *17, *24, *27[+5], *31, *38[+7], *41, *44[+5], *55[+7], *58[+5, +7], *65[+5, +7], *75[+5, +7] | |||
<nowiki />* Wart for 4 | |||
=== Fourwar === | |||
The 23-limit version of Fourwar was created first, as an attempt to approximate subgroup 4.6.5.7.11.13.17.19.23 as accurately as possible using 25 to 35 notes per equave. Then the lower limit versions were created by simply extrapolating the temperament downwards. | |||
Fourwar is named after the closely related [[hemiwar]] temperament. | |||
{{Todo|inline=1|cleanup}} | |||
<pre> | |||
Reduced Mapping | |||
4 6 5 | |||
[ ⟨ 1 0 1 ] | |||
⟨ 0 16 2 ] ⟩ | |||
TE Generator Tunings (cents) | TE Generator Tunings (cents) | ||
⟨2399.3973, 193.8643] | |||
TE Step Tunings (cents) | TE Step Tunings (cents) | ||
⟨25.21211, 47.81337] | |||
TE Tuning Map (cents) | TE Tuning Map (cents) | ||
⟨2399.397, 3101.829, 2787.126] | |||
TE Mistunings (cents) | TE Mistunings (cents) | ||
⟨-0.603, -0.126, 0.812] | |||
Complexity 1. | Complexity 1.369085 | ||
Adjusted Error | Adjusted Error 0.692892 cents | ||
TE Error | TE Error 0.268047 cents/octave | ||
Unison | Unison Vector | ||
[ | [8, 1, -8⟩ (393216:390625) | ||
Subsets | Subsets | ||
q99, q62, q37, q161, q136, q198, q25, q124, q74, q235 | |||
</pre> | </pre> | ||
==== 4.6.5.7 | ==== 4.6.5.7 ==== | ||
<pre> | <pre> | ||
Reduced Mapping | Reduced Mapping | ||
4 6 5 7 | 4 6 5 7 | ||
[ ⟨ 1 0 1 | [ ⟨ 1 0 1 1 ] | ||
⟨ 0 16 2 5 | ⟨ 0 16 2 5 ] ⟩ | ||
TE Generator Tunings (cents) | TE Generator Tunings (cents) | ||
⟨2399.4195, 193.8654] | |||
TE Step Tunings (cents) | TE Step Tunings (cents) | ||
⟨25.23883, 47.79592] | |||
TE Tuning Map (cents) | TE Tuning Map (cents) | ||
⟨2399.420, 3101.846, 2787.150, 3368.747] | |||
TE Mistunings (cents) | TE Mistunings (cents) | ||
⟨-0.580, -0.109, 0.837, -0.079] | |||
Complexity 1. | Complexity 1.192044 | ||
Adjusted Error | Adjusted Error 0.653313 cents | ||
TE Error | TE Error 0.232715 cents/octave | ||
Unison Vectors | Unison Vectors | ||
[ | [-2, -1, -2, 4⟩ (2401:2400) | ||
[3, 0, -5, 2⟩ (3136:3125) | |||
[ | [5, 1, -3, -2⟩ (6144:6125) | ||
[ | [8, 1, -8, 0⟩ (393216:390625) | ||
[ | |||
Subsets | Subsets | ||
q99, q62, q37, q161, q136, q198, q25, q124, q74, q235 | |||
</pre> | </pre> | ||
==== 4.6.5.7.11 | ==== 4.6.5.7.11 ==== | ||
<pre> | <pre> | ||
Reduced Mapping | Reduced Mapping | ||
4 6 5 7 11 | 4 6 5 7 11 | ||
[ ⟨ 1 0 1 | [ ⟨ 1 0 1 1 1 ] | ||
⟨ 0 16 2 5 9 | ⟨ 0 16 2 5 9 ] ⟩ | ||
TE Generator Tunings (cents) | TE Generator Tunings (cents) | ||
⟨2400.1097, 193.9498] | |||
TE Step Tunings (cents) | TE Step Tunings (cents) | ||
⟨24.18752, 48.52491] | |||
TE Tuning Map (cents) | TE Tuning Map (cents) | ||
⟨2400.110, 3103.196, 2788.009, 3369.859, 4145.658] | |||
TE Mistunings (cents) | TE Mistunings (cents) | ||
⟨0.110, 1.241, 1.696, 1.033, -5.660] | |||
Complexity 1. | Complexity 1.068792 | ||
Adjusted Error | Adjusted Error 2.926965 cents | ||
TE Error | TE Error 0.846083 cents/octave | ||
Unison Vectors | Unison Vectors | ||
[ | [-1, -1, -1, 0, 2⟩ (121:120) | ||
[ | [2, 0, -2, -1, 1⟩ (176:175) | ||
[-3, -1, 1, 1, 1⟩ (385:384) | |||
[ | [-1, 0, 3, -3, 1⟩ (1375:1372) | ||
[ | [-2, -1, -2, 4, 0⟩ (2401:2400) | ||
[ | [1, 0, 1, -4, 2⟩ (2420:2401) | ||
[ | |||
Subsets | Subsets | ||
q25, | q37, q25, q62, q12, q74, q99, q87, q49r, q50r, q124 | ||
</pre> | </pre> | ||
==== 4.6.5.7.11.13 | ==== 4.6.5.7.11.13 ==== | ||
<pre> | <pre> | ||
Reduced Mapping | Reduced Mapping | ||
4 6 5 7 11 13 | 4 6 5 7 11 13 | ||
[ ⟨ 1 0 1 | [ ⟨ 1 0 1 1 1 0 ] | ||
⟨ 0 16 2 5 9 23 | ⟨ 0 16 2 5 9 23 ] ⟩ | ||
TE Generator Tunings (cents) | TE Generator Tunings (cents) | ||
⟨2401.2305, 193.5378] | |||
TE Step Tunings (cents) | TE Step Tunings (cents) | ||
⟨42.79107, 35.98524] | |||
TE Tuning Map (cents) | TE Tuning Map (cents) | ||
⟨2401.230, 3096.606, 2788.306, 3368.920, 4143.071, 4451.371] | |||
TE Mistunings (cents) | TE Mistunings (cents) | ||
⟨1.230, -5.349, 1.992, 0.094, -8.247, 10.843] | |||
Complexity 1. | Complexity 1.219191 | ||
Adjusted Error | Adjusted Error 6.699599 cents | ||
TE Error | TE Error 1.810487 cents/octave | ||
Unison Vectors | Unison Vectors | ||
[0, 1, -1, 0, 1, - | [0, 1, -1, 0, 1, -1⟩ (66:65) | ||
[1 | [-1, -1, -1, 0, 2, 0⟩ (121:120) | ||
[ | [1, 2, 0, 0, -1, -1⟩ (144:143) | ||
[ | [2, 0, -2, -1, 1, 0⟩ (176:175) | ||
[ | [-2, 1, 1, 1, 0, -1⟩ (105:104) | ||
[-3, -1, 1, 1, 1 | [-3, -1, 1, 1, 1, 0⟩ (385:384) | ||
[ | [-3, 0, 0, 1, 2, -1⟩ (847:832) | ||
[1, | [1, 3, -1, 0, 0, -2⟩ (864:845) | ||
[-1, 0, 3, -3, 1, 0⟩ (1375:1372) | |||
[ | |||
Subsets | Subsets | ||
q25, q37f, q12f, q62, q50rf, q13rff, q49rff, q87, q74ff, q24rfff | |||
</pre> | </pre> | ||
== 4. | ==== 4.6.5.7.11.13.17 ==== | ||
<pre> | |||
[ | Reduced Mapping | ||
4 6 5 7 11 13 17 | |||
[ ⟨ 1 0 1 1 1 0 1 ] | |||
⟨ 0 16 2 5 9 23 13 ] ⟩ | |||
TE Generator Tunings (cents) | |||
⟨2400.4701, 193.4599] | |||
TE Step Tunings (cents) | |||
⟨43.39350, 35.55764] | |||
TE Tuning Map (cents) | |||
⟨2400.470, 3095.359, 2787.390, 3367.770, 4141.609, 4449.578, 4915.449] | |||
TE Mistunings (cents) | |||
⟨0.470, -6.596, 1.076, -1.056, -9.709, 9.050, 10.494] | |||
Complexity 1.129881 | |||
Adjusted Error 8.082725 cents | |||
TE Error 1.977443 cents/octave | |||
Unison Vectors | |||
[0, 1, -1, 0, 1, -1, 0⟩ (66:65) | |||
[1, 1, 1, -1, 0, 0, -1⟩ (120:119) | |||
[1, 2, 0, 0, -1, -1, 0⟩ (144:143) | |||
[-2, 1, 1, 1, 0, -1, 0⟩ (105:104) | |||
[-1, 2, 2, 0, 0, -1, -1⟩ (225:221) | |||
[-1, 1, 2, -2, 0, -1, 1⟩ (1275:1274) | |||
Subsets | |||
q25, q12f, q37f, q13rffg, q50rf, q62, q49rffg, q24rfffg, q38rreffg, q74ffg | |||
</pre> | |||
==== 4.6.5.7.11.13.17.19 ==== | |||
<pre> | |||
Mapping | Reduced Mapping | ||
4 6 5 7 11 13 17 19 | |||
[ | [ ⟨ 1 0 1 1 1 0 1 1 ] | ||
⟨ 0 16 2 5 9 23 13 14 ] ⟩ | |||
TE Generator Tunings (cents) | |||
⟨2399.9219, 193.3952] | |||
[[ | TE Step Tunings (cents) | ||
⟨44.14256, 35.03670] | |||
[[ | |||
TE Tuning Map (cents) | |||
⟨2399.922, 3094.324, 2786.712, 3366.898, 4140.479, 4448.090, 4914.060, 5107.455] | |||
TE Mistunings (cents) | |||
⟨-0.078, -7.631, 0.399, -1.928, -10.839, 7.562, 9.104, 9.942] | |||
Complexity 1.058472 | |||
Adjusted Error 8.712222 cents | |||
TE Error 2.050935 cents/octave | |||
Unison Vectors | |||
[0, 1, -1, 0, 1, -1, 0, 0⟩ (66:65) | |||
[-1, 0, 0, 1, 1, 0, 0, -1⟩ (77:76) | |||
[2, 1, -1, 0, 0, 0, 0, -1⟩ (96:95) | |||
[1, 1, 1, -1, 0, 0, -1, 0⟩ (120:119) | |||
[0, 1, 1, 1, -1, 0, 0, -1⟩ (210:209) | |||
[0, 0, 1, -2, 1, 0, 1, -1⟩ (935:931) | |||
[2, 0, -3, 1, 0, 0, -1, 1⟩ (2128:2125) | |||
Subsets | |||
q25, q12fh, q37f, q13rffgh, q50rf, q62, q49rffgh, q24rfffghh, q38rreffgh, q74ffgh | |||
</pre> | |||
Mapping | ==== 4.6.5.7.11.13.17.19.23 ==== | ||
<pre> | |||
[ | Reduced Mapping | ||
4 6 5 7 11 13 17 19 23 | |||
[ ⟨ 1 0 1 1 1 0 1 1 0 ] | |||
⟨ 0 16 2 5 9 23 13 14 28 ] ⟩ | |||
TE Generator Tunings (cents) | |||
⟨2399.3286, 193.5316] | |||
[[ | TE Step Tunings (cents) | ||
⟨37.31613, 39.63311] | |||
[[ | |||
TE Tuning Map (cents) | |||
⟨2399.329, 3096.506, 2786.392, 3366.987, 4141.113, 4451.227, 4915.240, 5108.771, 5418.885] | |||
TE Mistunings (cents) | |||
⟨-0.671, -5.449, 0.078, -1.839, -10.205, 10.699, 10.284, 11.258, -9.389] | |||
Complexity 1.115920 | |||
Adjusted Error 9.502017 cents | |||
TE Error 2.100561 cents/octave | |||
Unison Vectors | |||
[0, 1, -1, 0, 1, -1, 0, 0, 0⟩ (66:65) | |||
[1, 0, 0, -1, 0, -1, 0, 0, 1⟩ (92:91) | |||
[0, -1, 1, 0, 0, 0, 0, -1, 1⟩ (115:114) | |||
[1, 1, 1, -1, 0, 0, -1, 0, 0⟩ (120:119) | |||
[2, 0, -2, -1, 1, 0, 0, 0, 0⟩ (176:175) | |||
[-3, -1, 1, 1, 1, 0, 0, 0, 0⟩ (385:384) | |||
[1, 0, -2, 1, 0, 0, 1, -1, 0⟩ (476:475) | |||
[1, 0, 0, -2, 1, 0, -1, 1, 0⟩ (836:833) | |||
[0, 0, 1, -2, 1, 0, 1, -1, 0⟩ (935:931) | |||
[1, -1, 0, 0, 0, 0, -2, 1, 1⟩ (874:867) | |||
Subsets | |||
q25i, q12fhi, q37f, q13rffghii, q62, q50rfii, q49rffghii, q24rfffghhiii, q74ffghi, q38rreffghiii | |||
</pre> | |||
[[ | == 4.9.25 subgroup == | ||
=== Meansquared === | |||
[[Subgroup]]: 4.9.25 | |||
[[ | [[Comma list]]: [[6561/6400]] | ||
{{Mapping|legend=2| 1 3 4 | 0 1 4 }} | |||
Mapping generators: ~4, ~9/64 | |||
[[Optimal tuning]] ([[CTE]]): ~4 = 2\1, ~9/4 = 1394.429 | |||
[[ | [[Support]]ing [[ET]]s: 12, 7, 19, 5, 31, 26, 17[+25], 43, 9[-25], 33[-25], 45, 29[+25], 8[+25], 22[+25] | ||
[[ | == 4.9.49 subgroup == | ||
=== Archsquared === | |||
[[Subgroup]]: 4.9.49 | |||
[[Comma list]]: 4096/3969 | |||
: | |||
{{Mapping|legend= | {{Mapping|legend=2| 1 3 0 | 0 1 -2 }} | ||
Mapping generators: ~4, ~9/64 | |||
[[Optimal tuning]] ([[CTE]]): ~9/4 = 1419.190 | |||
: | |||
[[ | [[Support]]ing [[ET]]s: 5, 17, 22, 12, 7, 27, 32, 8, 39[+49], 29[+49], 9[+49], 19[+49], 37, 49 | ||
=== | == 8.9.7 subgroup == | ||
=== Sixscared === | |||
Sixscared is a tuning which still maintains some consonance, while eviscerating the rules of conventional 12-tone harmony. The familiar major, minor and perfect intervals are nowhere to be found, and octaves are far and few between, so the seventh harmonic becomes the backbone of harmony. Approximating the harmonics 7, 8, 9, Sixscared is named for the classic dad joke: "Why was six scared? Because seven ate nine." | |||
[[Subgroup]]: 8.9.7 | |||
[[ | [[Comma list]]: 64/63 | ||
{{Mapping|legend=2| 1 0 2 | 0 1 -1 }} | |||
: sval mapping generators: ~8, ~9 | |||
: | : [[gencom]]: [8 9/8; 64/63] | ||
[[Optimal tuning]] ([[CTE]]): ~9/8 = 219.1898 | |||
[[Optimal | [[Optimal ET sequence]]: {{val| 16 17 15 }}, {{val| 33 35 31 }}, {{val| 148 … }}, {{val| 181 … }}, {{val| 214 … }}, {{val| 247 … }} | ||
[[Badness]]: 0.0215 × 10<sup>-3</sup> | |||
= Fractional subgroup temperaments = | |||
== 2.5/3.… subgroups == | |||
=== Magicaltet === | |||
{{See also| Chromatic pairs #Magicaltet }} | |||
Magicaltet is related to [[keemic]], [[superkleismic]], and [[magic]]. The tonic and the first three generator steps make a [[magical seventh chord]], hence the name. | |||
[[Subgroup]]: 2.5/3.7.11 | |||
[[ | [[Comma list]]: 100/99 ({{monzo| 2 2 0 -1 }}), 385/384 ({{monzo| -7 1 1 1 }}) | ||
{{Mapping|legend=2| 1 0 5 2 | 0 1 -3 2 }} | |||
: mapping generators: ~2, ~5/3 | |||
{{Mapping|legend= | {{Mapping|legend=3| 1 -1/2 1/2 2 4 | 0 1/2 -1/2 3 -2 }} | ||
: [[gencom]]: [2 6/5; 100/99 385/384] | |||
[[Optimal tuning]]s: | |||
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 877.343 | |||
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 877.351 | |||
{{Optimal ET sequence|legend=1| 4, 7, 11, 15, 26, 67, 93* }} | |||
: <nowiki/>* wart for 5/3 | |||
* | |||
[[Tp tuning #T2 tuning|RMS error]]: 1.206 cents | |||
=== Starlingtet === | |||
{{See also | Chromatic pairs #Starlingtet }} | |||
Starlingtet, the {{nowrap| 4 & 15 }} temperament in the 2.5/3.7/3 subgroup, is related to [[starling]] as well as to [[myna]]. The tonic and the first three generator steps make a [[starling tetrad]], hence the name. | |||
{{ | |||
[[Subgroup]]: 2.5/3.7/3 | |||
[[ | [[Comma list]]: [[126/125]] ({{monzo| 1 -3 1 }}) | ||
{{Mapping|legend=2| 1 0 -1 | 0 1 3 }} | |||
: mapping generators: ~2, ~5/3 | |||
{{Mapping|legend=3| 1 - | {{Mapping|legend=3| 1 -1 0 1 | 0 4/3 1/3 -5/3 }} | ||
: [[gencom]]: [2 6/5; | : [[gencom]]: [2 6/5; 126/125] | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = | * [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 888.759 | ||
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = | * [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 888.846 | ||
{{Optimal ET sequence|legend=1| | {{Optimal ET sequence|legend=1| 4, 15, 19, 23, 27 }} | ||
[[Tp tuning #T2 tuning|RMS error]]: | [[Tp tuning #T2 tuning|RMS error]]: 0.8398 cents | ||
=== | ==== Greeley ==== | ||
{{See also| Chromatic pairs #Greeley }} | |||
Greeley is related to [[opossum]] as well as to [[nusecond]]. | |||
[[Subgroup]]: 2.5/3.7/3.11/3 | |||
[[ | [[Comma list]]: 121/120 ({{monzo| -3 -1 0 2 }}), 126/125 ({{monzo| 1 -3 1 }}) | ||
{{Mapping|legend=2| 1 1 2 2 | 0 -2 -6 -1 }} | |||
{{Mapping|legend= | {{Mapping|legend=3| 1 -5/4 -1/4 3/4 3/4 | 0 9/4 1/4 -15/4 5/4 }} | ||
: [[gencom]]: [2 11/10; 121/120 126/125] | |||
: [[gencom]]: [2 | |||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[Tp tuning|subgroup]] [[CTE]] | * [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~11/10 = 155.696 | ||
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~11/10 = 155.776 | |||
{{Optimal ET sequence|legend=1| | {{Optimal ET sequence|legend=1| 8, 15, 23, 54, 77, 100, 131* }} | ||
: <nowiki/>* wart for | : <nowiki/>* wart for 11/3 | ||
[[Tp tuning #T2 tuning|RMS error]]: | [[Tp tuning #T2 tuning|RMS error]]: 1.034 cents | ||
==== | ==== Skateboard ==== | ||
{{See also | Chromatic pairs # | {{See also| Chromatic pairs #Skateboard }} | ||
Skateboard is related to [[thrasher]]. | |||
[[Subgroup]]: 2.5/3.7/3.11/ | [[Subgroup]]: 2.5/3.7/3.11.13/9 | ||
[[Comma list]]: | [[Comma list]]: 56/55 ({{monzo| 3 -1 1 -1 }}), 91/90 ({{monzo| -1 -1 1 0 1 }}), 100/99 ({{monzo| 2 2 0 -1 }}) | ||
{{Mapping|legend=2| 1 0 | {{Mapping|legend=2| 1 0 -1 2 2 | 0 1 3 2 -2 }} | ||
{{Mapping|legend=3| 1 - | {{Mapping|legend=3| 1 -3/7 4/7 11/7 4 -6/7 | 0 0 -1 -3 -2 2 }} | ||
: [[gencom]]: [2 | : [[gencom]]: [2 6/5; 56/55 91/90 100/99] | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~ | * [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 886.158 | ||
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~ | * [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 886.158 | ||
{{Optimal ET sequence|legend=1| | {{Optimal ET sequence|legend=1| 11, 15, 19, 23, 42d, 65d }} | ||
[[Tp tuning #T2 tuning|RMS error]]: | [[Tp tuning #T2 tuning|RMS error]]: 2.396 cents | ||
=== | === Gariberttet === | ||
Gariberttet is the 2.5/3.7/3 [[Subgroup temperament families, relationships, and genes|altergene]] of [[sirius]]. | |||
==== Gariberttet (2.5/3.7/3.13/11 subgroup) ==== | |||
{{See also | Chromatic pairs #Gariberttet }} | |||
Gariberttet can be described as the {{nowrap| 4 & 29 }} temperament in the 2.5/3.7/3.13/11 subgroup. Extensions to the full 7-, 11-, and 13-limits include [[quasitemp]]. | |||
[[ | [[Subgroup]]: 2.5/3.7/3.13/11 | ||
{{Mapping|legend=2| 1 | [[Comma list]]: [[275/273]] ({{monzo| 0 2 -1 -1 }}), [[847/845]] ({{monzo| 0 -1 1 -2 }}) | ||
{{Mapping|legend=2| 1 0 0 0 | 0 3 5 1 }} | |||
{{Mapping|legend=3| 1 | {{Mapping|legend=3| 1 0 0 0 0 0 | 0 -8/3 1/3 7/3 -1/2 1/2 }} | ||
: [[gencom]]: [2 13/ | : [[gencom]]: [2 13/11; 275/273 847/845] | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[Tp tuning|subgroup]] [[CTE]] | * [[Tp tuning|subgroup]] [[CTE]] and [[POTE]]: ~2 = 1200.000, ~13/11 = 293.679 | ||
{{Optimal ET sequence|legend=1| | {{Optimal ET sequence|legend=1| 29, 33, 37, 41, 45, 49, 78, 94, 143* }} | ||
: <nowiki/>* wart for 13/11 | |||
[[Tp tuning #T2 tuning|RMS error]]: | [[Tp tuning #T2 tuning|RMS error]]: 0.6914 cents | ||
=== | ==== Indium ==== | ||
{{See also | Chromatic pairs # | {{See also | Chromatic pairs #Indium }} | ||
Indium can be described as the {{nowrap| 8 & 33 }} temperament in the 2.5/3.7/3.11/3 subgroup. | |||
[[Subgroup]]: 2.5/3. | [[Subgroup]]: 2.5/3.7/3.11/3 | ||
[[Comma list]]: [[ | [[Comma list]]: [[3025/3024]] ({{monzo| -4 2 -1 2 }}), [[3125/3087]] ({{monzo| 0 5 -3 }}) | ||
{{Mapping|legend=2| 1 0 0 | 0 | {{Mapping|legend=2| 1 0 0 2 | 0 6 10 -1 }} | ||
{{Mapping|legend=3| 1 | {{Mapping|legend=3| 1 -1/2 -1/2 -1/2 3/2 | 0 -15/4 9/4 25/4 -19/4 }} | ||
: [[gencom]]: [2 | : [[gencom]]: [2 12/11; 3025/3024 3125/3087] | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[Tp tuning|subgroup]] [[CTE]] | * [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~12/11 = 146.978 | ||
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~12/11 = 147.010 | |||
{{Optimal ET sequence|legend=1| 8, | {{Optimal ET sequence|legend=1| 8, 33, 41, 49, 204*<sup>†</sup> }} | ||
: <nowiki/>* wart for | : <nowiki/>* wart for 7/3 | ||
: <sup>†</sup> wart for | : <sup>†</sup> wart for 11/3 | ||
[[Tp tuning #T2 tuning|RMS error]]: 0. | [[Tp tuning #T2 tuning|RMS error]]: 0.7788 cents | ||
=== | ==== Ammon ==== | ||
{{See also| Chromatic pairs # | {{See also| Chromatic pairs #Ammon }} | ||
Ammon can be described as the {{nowrap| 8 & 29 }} temperament in the 2.5/3.7/3.11/3.13/3 subgroup. It extends [[tridec]], and is related to [[ammonite]]. It is generated by a semidiminished fourth, hence the old name ''semidim'', which has been rejected since 2025 to avoid confusion with another temperament of the same name. | |||
[[Subgroup]]: 2.5/3.13/ | [[Subgroup]]: 2.5/3.7/3.11/3.13/3 | ||
[[Comma list]]: [[ | [[Comma list]]: [[121/120]] ({{monzo| -3 -1 0 2 }}), [[169/168]] ({{monzo| -3 0 -1 0 2 }}), [[275/273]] ({{monzo| 0 2 -1 1 -1 }}) | ||
{{Mapping|legend=2| 1 | {{Mapping|legend=2| 1 3 5 3 4 | 0 -6 -10 -3 -5 }} | ||
{{Mapping|legend=3| 1 - | {{Mapping|legend=3| 1 -3 0 2 0 1 | 0 24/5 -6/5 -26/5 9/5 -1/5 }} | ||
: [[gencom]]: [2 | : [[gencom]]: [2 13/10; 121/120 169/168 275/273] | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~ | * [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~13/10 = 453.121 | ||
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~ | * [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~13/10 = 453.242 | ||
{{Optimal ET sequence|legend=1| | {{Optimal ET sequence|legend=1| 8, 29, 37, 45 }} | ||
[[Tp tuning #T2 tuning|RMS error]]: | [[Tp tuning #T2 tuning|RMS error]]: 1.052 cents | ||
== | === Sentry === | ||
{{See also | Chromatic pairs #Sentry }} | |||
{{See also | Chromatic pairs # | |||
Sentry, the {{nowrap| 3 & 5 }} temperament in the 2.5/3.9/7 subgroup, is related to [[sensi]]. | |||
[[Subgroup]]: 2.5 | [[Subgroup]]: 2.5/3.9/7 | ||
[[Comma list]]: [[ | [[Comma list]]: [[245/243]] ({{monzo| 0 1 -2 }}) | ||
{{Mapping|legend=2| 1 0 | {{Mapping|legend=2| 1 0 0 | 0 2 1 }} | ||
{{Mapping|legend=3| 1 | {{Mapping|legend=3| 1 0 0 0 | 0 0 2 -1 }} | ||
: [[gencom]]: [2 7 | : [[gencom]]: [2 9/7; 245/243] | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* | * [[Tp tuning|subgroup]] [[CTE]] and [[POTE]]: ~2 = 1200.000, ~9/7 = 440.902 | ||
{{Optimal ET sequence|legend=1| | {{Optimal ET sequence|legend=1| 8, 11, 19, 30, 41, 49, 52, 145*, 166<sup>†</sup>, 197*<sup>†</sup>, 215<sup>†</sup>, 264*<sup>†</sup> }} | ||
: <nowiki/>* wart for | : <nowiki/>* wart for 5/3 | ||
: <sup>†</sup> wart for 9/7 | |||
[[Tp tuning #T2 tuning|RMS error]]: 0.7105 cents | |||
=== Marveltwintri === | |||
{{See also| Chromatic pairs #Marveltwintri }} | |||
[[ | Marveltwintri can be described as the {{nowrap| 3 & 4 }} temperament in the 2.5/3.13/9 subgroup. The tonic and the first two generator steps make a [[marveltwin triad]], hence the name. [[Cata]] is a very natural extension of this temperament to the [[2.3.5.13 subgroup|2.3.5.13-subgroup]]. | ||
[[Subgroup]]: 2.5/3.13/9 | |||
[[ | [[Comma list]]: [[325/324]] ({{monzo| -2 2 1 }}) | ||
{{Mapping|legend=2| 1 0 2 | 0 1 -2 }} | |||
{{Mapping|legend= | {{Mapping|legend=3| 1 -1/6 5/6 0 0 -1/3 | 0 -1/2 -3/2 0 0 1 }} | ||
: | : [[gencom]]: [2 6/5; 325/324] | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* | * [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 882.886 | ||
* | * [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 882.861 | ||
{{Optimal ET sequence|legend=1| | {{Optimal ET sequence|legend=1| 3, 4, 11, 15, 19, 34, 53, 87, 140 }} | ||
[[Tp tuning #T2 tuning|RMS error]]: 0.2444 cents | |||
==== | == 2.….7/3.… subgroups == | ||
{{See also | Chromatic pairs # | === Guanyintet === | ||
{{See also | Chromatic pairs #Guanyintet }} | |||
Guanyintet, the {{nowrap| 4 & 9 }} temperament in the 2.5.7/3.11/3 subgroup, is the main rank-2 chain of [[guanyin]] and a restriction of [[orwell]]. It is defined by tempering out [[1728/1715]] ({{S|6/S7}}) and [[540/539]] (S12/S14), which imply [[176/175]] (S8/S10) as well as S11/S15 being tempered out. The tonic and the first three generator steps make a [[guanyin tetrad]], hence the name. | |||
[[Subgroup]]: 2.5.7/3.11 | [[Subgroup]]: 2.5.7/3.11/3 | ||
[[Comma list]]: | [[Comma list]]: [[176/175]] ({{monzo| 4 -2 -1 1 }}), [[540/539]] ({{monzo| 2 1 -2 -1 }}) | ||
{{Mapping|legend=2| 1 0 | {{Mapping|legend=2| 1 0 1 3 | 0 -3 1 -5 }} | ||
: mapping generators: ~2, ~7/6 | |||
{{Mapping|legend=3| 1 - | {{Mapping|legend=3| 1 -4/3 3 -1/3 5/3 | 0 4/3 -3 7/3 -11/3 }} | ||
: [[gencom]]: [2 7/6; | : [[gencom]]: [2 7/6; 176/175 540/539] | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~ | * ([[Tp tuning|subgroup]] [[CTE]]): ~2 = 1200.000, ~7/6 = 270.455 | ||
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~ | * ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1200.000, ~7/6 = 270.093 | ||
{{Optimal ET sequence|legend=1| 9, 31, 40, | {{Optimal ET sequence|legend=1| 9, 22, 31, 40, 191c*, 231c*, 271c*, 311c* }} | ||
: <nowiki/>* wart for 7/3 | : <nowiki/>* wart for 7/3 | ||
[[Tp tuning #T2 tuning|RMS error]]: 0. | [[Tp tuning #T2 tuning|RMS error]]: 0.6028 cents | ||
=== | ==== Tridecimal guanyintet ==== | ||
{ | Guanyintet can extend to the 13th harmonic by the equivalences ([[12/11]])<sup>3</sup> = [[13/10]] and ([[15/14]])<sup>3</sup> = [[16/13]], therefore tempering out {S11/S12/S14/S15}. However, note that it is not supported by the 31 & 53 orwell extension dubbed "tridecimal orwell", but instead the less accurate [[winston]] (22f & 31), as orwell prefers slightly sharper tunings than guanyintet. [[40edo]] remains an excellent tuning. | ||
[[Subgroup]]: 2.5.7/3.11/3.13 | |||
[[ | [[Comma list]]: [[176/175]] ({{monzo| 4 -2 -1 1 0 }}), [[540/539]] ({{monzo| 2 1 -2 -1 0 }}), [[1573/1568]] ({{monzo| -5 0 -2 2 1 }}) | ||
{{Mapping|legend=2| 1 0 1 3 1 | 0 -3 1 -5 12 }} | |||
: mapping generators: ~2, ~12/7 | |||
[[Optimal tuning]]s: | |||
* ([[Tp tuning|subgroup]] [[CTE]]): ~2 = 1200.000, ~7/6 = 270.152 | |||
* ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1200.000, ~7/6 = 270.218 | |||
{{ | {{Optimal ET sequence|legend=1| 9, 22, 31, 40, 71, 111, 151, 262c*}} <small> using subgroup TE </small> | ||
: | : <nowiki/>* wart for 7/3 | ||
Badness (Sintel): 0.329 | |||
{{ | ==== Laz ==== | ||
{{See also | Chromatic pairs #Laz }} | |||
[[ | Laz is related to [[avalokita]] as well as to [[winston]]. | ||
[[Subgroup]]: 2.5.7/3.11/3.13/3 | |||
[[Comma list]]: [[144/143]] ({{monzo| 4 0 0 -1 -1 }}), [[176/175]] ({{monzo| 4 -2 -1 1 }}), [[196/195]] ({{monzo| 2 -1 2 0 -1 }} | |||
{{Mapping|legend=2| 1 0 2 -2 6 | 0 3 -1 5 -5 }} | |||
{{Mapping|legend=3| 1 -5/4 3 -1/4 7/4 -1/4 | 0 -1/4 -3 3/4 -21/4 19/4 }} | |||
: [[gencom]]: [2 7/6; 144/143 176/175 196/195] | |||
[[Optimal tuning]]s: | |||
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~12/7 = 930.598 | |||
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~12/7 = 930.700 | |||
{{ | {{Optimal ET sequence|legend=1| 9, 31, 40, 49, 156c*†, 205c*† }} | ||
: | : <nowiki/>* wart for 7/3 | ||
: † wart for 11/3 | |||
[[Tp tuning #T2 tuning|RMS error]]: 0.8790 cents | |||
{{ | === Kryptonite === | ||
{{See also| Chromatic pairs #Kryptonite }} | |||
[[ | Kryptonite is related to [[krypton]]. | ||
[[Subgroup]]: 2.5.7/3.11/3.13/3 | |||
[[Comma list]]: 56/55 ({{monzo| 3 -1 1 -1 }}), 78/77 ({{monzo| 1 0 -1 -1 1 }}), 91/90 ({{monzo| -1 -2 1 0 1 }}) | |||
{{Mapping|legend=2| 1 2 1 2 2 | 0 3 2 -1 1 }} | |||
: mapping generators: ~2, ~13/12 | |||
{{Mapping|legend=3| 1 -5/4 2 -1/4 3/4 3/4 | 0 -1/2 3 3/2 -3/2 1/2 }} | |||
: [[gencom]]: [2 13/12; 56/55 78/77 91/90] | |||
[[Optimal tuning]]s: | |||
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~13/12 = 130.945 | |||
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~13/12 = 132.428 | |||
{{ | {{Optimal ET sequence|legend=1| 1, …, 8, 9 }} | ||
[[Tp tuning #T2 tuning|RMS error]]: 2.545 cents | |||
{{ | === Kiribati === | ||
{{See also| Chromatic pairs #Kiribati }} | |||
[[ | Kiribati is related to [[nakika]] as well as to [[octacot]]. | ||
[[Subgroup]]: 2.9/5.7/3.11/9 | |||
[[Comma list]]: 100/99 ({{monzo| 2 -2 0 -1 }}), 245/242 ({{monzo| -1 -1 2 -2 }}) | |||
{{Mapping|legend=2| 1 1 1 0 | 0 -2 3 4 }} | |||
: mapping generators: ~2, ~21/20 | |||
{{Mapping|legend=3| 1 1/10 -4/5 11/10 1/5 | 0 -3/2 -1 3/2 1 }} | |||
: [[gencom]]: [2 21/20; 100/99 245/242] | |||
[[Optimal tuning]]s: | |||
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~21/20 = 87.776 | |||
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~21/20 = 87.892 | |||
{{ | {{Optimal ET sequence|legend=1| 13, 14, 27, 41 }} | ||
[[Tp tuning #T2 tuning|RMS error]]: 1.245 cents | |||
{{ | === Mothwelltri === | ||
{{See also| Chromatic pairs #Mothwelltri }} | |||
[[ | Mothwelltri, the {{nowrap| 1 & 4 }} temperament in the 2.7/3.11 subgroup, is related to [[orwell]]. The tonic and the first two generator steps make a [[mothwellsmic triad]], hence the name. | ||
[[Subgroup]]: 2.7/3.11 | |||
[[ | [[Comma list]]: [[99/98]] ({{monzo| -1 -2 1 }}) | ||
{{Mapping|legend=2| 1 0 1 | 0 1 2 }} | |||
: mapping generators: ~2, ~7/3 | |||
{{Mapping|legend=2| | {{Mapping|legend=3| 1 -1/2 0 1/2 3 | 0 -1/2 0 1/2 2 }} | ||
: [[gencom]]: [2 7/6; 99/98] | |||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~ | * [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~7/6 = 273.695 | ||
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~ | * [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~7/6 = 273.174 | ||
{{Optimal ET sequence|legend=1| | {{Optimal ET sequence|legend=1| 4, 9, 13, 22, 79 }} | ||
[[Tp tuning #T2 tuning|RMS error]]: 1. | [[Tp tuning #T2 tuning|RMS error]]: 1.064 cents | ||
== 2.….7 | == 2.….9/7.… subgroups == | ||
=== | === Marveltri === | ||
{{See also| Chromatic pairs #Marveltri }} | |||
[[ | Marveltri, the {{nowrap| 3 & 13 }} temperament in the 2.5.9/7 subgroup, is related to [[marvel]], [[magic]], and the unnamed {{nowrap| 22 & 47 }} temperament. The tonic and the first two generator steps make a [[marvel triad]], hence the name. | ||
[[ | [[Subgroup]]: 2.5.9/7 | ||
{{ | [[Comma list]]: 225/224 ({{monzo| -5 2 1 }}) | ||
{{Mapping|legend=2| 1 0 5 | 0 1 -2 }} | |||
: mapping generators: ~2, ~5 | |||
[[ | {{Mapping|legend=3| 1 2 0 -1 | 0 -4/5 1 2/5 }} | ||
: [[gencom]]: [2 5; 225/224] | |||
[[Optimal tuning]]s: | |||
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/4 = 384.208 | |||
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/4 = 383.638 | |||
{{Optimal ET sequence|legend=1| 3, 13, 16, 19, 22, 25, 72, 97, 122, 269c* }} | |||
: <nowiki/>* wart for 9/7 | |||
[[ | [[Tp tuning #T2 tuning|RMS error]]: 0.4801 cents | ||
==== Sulis ==== | |||
Sulis is related to [[minerva]] and [[würschmidt]]. | |||
[[ | [[Subgroup]]: 2.5.9/7.11/9 | ||
{{ | [[Comma list]]: 99/98 ({{monzo| -1 0 2 1 }}), 176/175 ({{monzo| 4 -2 1 1 }}) | ||
{{Mapping|legend=2| 1 0 5 -9 | 0 1 -2 4 }}] | |||
= | [[Optimal tuning]]s: | ||
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/4 = 386.617 | |||
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/4 = 386.558 | |||
{{Optimal ET sequence|legend=1| 3, …, 22, 25, 28, 31, 59 }} | |||
[[ | [[Tp tuning #T2 tuning|RMS error]]: 1.074 cents | ||
== 2.….7/5.… subgroups == | |||
=== Hydrothermal === | |||
A tuning whose distinctively sharp (but still consonant) fifth, and flat (but still consonant) octave, lend it a mysterious, heavy atmosphere. The 6-tone (hexatonic) MOS is melodically interesting and flavorful. The 18-tone MOS is a useful 'chromatic' scale for taking subsets of. | |||
[[Subgroup]]: 2.3.7/5 | |||
[[Comma list]]: [[50/49]] | |||
{{Mapping|legend=2| 2 3 1 | 0 1 0 }} | |||
[[Optimal tuning]] (inharmonic [[TE]]): ~1\2 = 590.998, ~[[10/7]]-1\2 = 128.962 | |||
[[ | [[Support]]ing [[ET]]s: {{EDOs|4, 6, 8, 10, 18, 28, 46, 64, 110}} | ||
=== | === Argentic === | ||
Argentic is the 2.3.7/5 subgroup temperament tempering out [[5120/5103]]. | |||
[[Subgroup]]: 2.3.7/5 | |||
[[ | [[Comma list]]: [[5120/5103]] = {{monzo| 10 -6 -1 }} | ||
{{Mapping|legend=2| 1 0 10 | 0 1 -6 }} | |||
: mapping generators: ~2, ~3 | |||
[[Optimal tuning]]s: | |||
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1\1, ~3/2 = 702.792 | |||
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1\1, ~3/2 = 702.830 | |||
{{ | {{Optimal ET sequence|legend=1| 12, 29, 41, 70, 321, 391, 461, 531, 601 }} | ||
<small> based on subgroup TE </small> | |||
Badness (Sintel): 0.119 | |||
{{ | ==== Edson (2.3.7/5.11/5.13/5 subgroup) ==== | ||
{{See also| Chromatic pairs #Edson }} | |||
[[ | Edson is related to [[pele]] and [[andromeda]]. | ||
[[Subgroup]]: 2.3.7/5.11/5.13/5 | |||
[[Comma list]]: [[196/195]] = {{monzo| 2 -1 2 0 -1 }}, [[352/351]] = {{monzo| 5 -3 0 1 -1 }}, [[364/363]] = {{monzo| 2 -1 1 -2 1 }} | |||
{{Mapping|legend=2| 1 0 10 17 22 | 0 1 -6 -10 -13 }} | |||
: mapping generators: ~2, ~3 | |||
[[ | {{Mapping|legend=3| 1 1 -5 -1 2 4 | 0 1 29/4 5/4 -11/4 -23/4 }} | ||
: [[gencom]]: [2 3/2; 196/195, 352/351, 364/363] | |||
[[Optimal tuning]]s: | |||
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1\1, ~3/2 = 703.4398 | |||
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1\1, ~3/2 = 703.414 | |||
{{Optimal ET sequence|legend=1| 12, 17, 29 }} | |||
[[Tp tuning #T2 tuning|RMS error]]: 0.5102 cents | |||
==== Haumea ==== | |||
{{See also| Chromatic pairs #Haumea }} | |||
Related temperaments include [[#Bridgetown|bridgetown]], [[namaka]], [[hemigari]], [[#Barbados|barbados]], and [[parizekmic]]. | |||
[[Subgroup]]: 2.3.7/5.11/5.13/5 | |||
[[ | [[Comma list]]: [[352/351]], [[676/675]], [[847/845]] | ||
{{Mapping|legend=2| 1 0 10 -6 -1 | 0 2 -12 9 3 }} | |||
{{Mapping|legend= | {{Mapping|legend=3| 1 2 -3/4 -11/4 9/4 5/4 | 0 -2 0 12 -9 -3 }} | ||
: [[gencom]]: [2 15/13; 352/351 676/675 847/845] | |||
[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~15/13 = 248.491 | |||
{{Optimal ET sequence|legend=1| 24, 29, 111, 140, 169, 198, 565d, 763bd, 961bd }} | |||
[[Tp tuning #T2 tuning|RMS error]]: 0.2668 cents | |||
=== Historical === | |||
{{distinguish|Historical temperaments}} | |||
{{distinguish|History (temperament)}}, which is the rank-3 version of this temperament in the full 13-limit. | |||
=== | Historical is essentially an analogue of [[miracle]] that splits [[4/3]] in six rather than [[3/2]]. It tempers out the comma S10/S11 = [[4000/3993]] to set [[11/10]] equal to one-third of 4/3, and S13/S15 = [[676/675]] to equate [[15/13]] to one-half of 4/3, and tempers out S21 = [[441/440]] to split 11/10 into two instances of [[22/21]]~[[21/20]]. [[Sextilifourths]] adds the [[schismic]] mapping of prime 5 (reached by eight fourths) to complete the 13-limit. | ||
[[Subgroup]]: 2.3.7/5.11/5.13/5 | |||
[[ | [[Comma list]]: 364/363, 441/440, 1001/1000 | ||
{{Mapping|legend=2| 1 2 0 1 2 | 0 -6 7 2 -9 }} | |||
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~21/20 = 83.016 | |||
{{ | {{Optimal ET sequence|legend=1| 14, 29, 72, 101, 130, 159 }} | ||
[[ | [[Tp tuning #T2 tuning|RMS error]]: 0.2562 cents | ||
{{ | === Terrain === | ||
{{Redirect|Terrain|the scale|Terrain (scale)}} | |||
{{See also| Chromatic pairs #Terrain }} | |||
[[ | Terrain, the 6 & 21 temperament in the 2.7/5.9/5 subgroup, is related to [[domain (temperament)|domain]]. It is a remarkable temperament, in that while its complexity is low, it has no discernible error. The 1–7/5–9/5 and 1–9/7–9/5 chords are characteristic. | ||
[[Subgroup]]: 2.7/5.9/5 | |||
[[Subgroup]]: 2.7/5. | |||
[[Comma list]]: [[ | [[Comma list]]: [[250047/250000]] | ||
{{Mapping|legend=2| 1 | {{Mapping|legend=2| 3 1 3 | 0 1 -1 }} | ||
{{Mapping|legend=3| | {{Mapping|legend=3| 3 10/9 -7/9 2/9 | 0 -2/3 -1/3 2/3 }} | ||
: [[gencom]]: [ | : [[gencom]]: [63/50 10/9; 250047/250000] | ||
[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~ | [[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~63/50 = 1\3, ~10/9 = 182.461 | ||
{{Optimal ET sequence|legend=1| | {{Optimal ET sequence|legend=1| 6, 21, 27, 33, 105, 138, 171, 1848, 2019, 2190, 2361, 2532, 2703, 2874, 3045, 3216, 3387, 3558 }} | ||
[[Tp tuning #T2 tuning|RMS error]]: 0. | [[Tp tuning #T2 tuning|RMS error]]: 0.00844 cents | ||
== | === Tridec === | ||
{{See also| Chromatic pairs #Tridec }} | |||
{{See also| Chromatic pairs # | {{See also| Non-over-1 temperament #Tridec }} | ||
{{See also| | |||
Tridec, the 5 & 8 temperament in the 2.7/5.11/5.13/5 subgroup, extends [[#Petrtri]]. | |||
[[Subgroup]]: 2.11/5.13/5 | [[Subgroup]]: 2.7/5.11/5.13/5 | ||
[[Comma list]]: [[ | [[Comma list]]: [[847/845]], [[1001/1000]] | ||
{{Mapping|legend=2| 1 0 1| 0 3 1 }} | {{Mapping|legend=2| 1 2 0 1 | 0 -4 3 1 }} | ||
{{Mapping|legend=3| 1 0 - | {{Mapping|legend=3| 1 0 -3/4 5/4 -3/4 1/4 | 0 0 0 -4 3 1 }} | ||
: [[gencom]]: [2 13/10; | : [[gencom]]: [2 13/10; 847/845 1001/1000] | ||
[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~13/10 = | [[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~13/10 = 454.556 | ||
{{Optimal ET sequence|legend=1| 21, 29, | {{Optimal ET sequence|legend=1| 5, 8, 21, 29, 37, 66, 169, 235, 404c, 639c, 953bc }} | ||
[[Tp tuning #T2 tuning|RMS error]]: 0. | [[Tp tuning #T2 tuning|RMS error]]: 0.1613 cents | ||
==== | ==== Naiadec ==== | ||
[[Subgroup]]: 2.7/5.11/5.13/5.17/5 | |||
[[Comma list]]: [[170/169]], [[221/220]], [[847/845]] | |||
{{Mapping|legend=2| 1 2 0 1 1 | 0 -4 3 1 2 }} | |||
[[ | {{Mapping|legend=3| 1 0 -3/4 5/4 -3/4 1/4 1/4 | 0 0 0 -4 3 1 2 }} | ||
: [[gencom]]: [2 13/10; 170/169 221/220 847/845] | |||
[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~13/10 = 454.882 | |||
{{ | {{Optimal ET sequence|legend=1| 5, 8, 21, 29, 95<sup>t</sup>, 124<sup>t</sup> }} | ||
: | : <sup>t</sup> wart for 17/5 | ||
[[ | [[Tp tuning #T2 tuning|RMS error]]: 0.7521 cents | ||
{{ | == 2.….11/5.… subgroups == | ||
=== Petrtri === | |||
{{See also| Chromatic pairs #Petrtri }} | |||
{{See also| 5L 3s/Temperaments #Petrtri }} | |||
Petrtri can be described as 3 & 5 temperament in the 2.11/5.13/5 subgroup. | |||
[[Subgroup]]: 2.11/5.13/5 | |||
[[ | [[Comma list]]: [[2200/2197]] | ||
{{Mapping|legend=2| 1 0 1| 0 3 1 }} | |||
{{Mapping|legend= | {{Mapping|legend=3| 1 0 -1/3 0 -1/3 2/3 | 0 0 -4/3 0 5/3 -1/3 }} | ||
: [[gencom]]: [2 13/10; 2200/2197] | |||
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~13/ | [[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~13/10 = 455.012 | ||
{{Optimal ET sequence|legend=1| | {{Optimal ET sequence|legend=1| 21, 29, 153, 182, 211, 240, 269, 298, 327, 356, 385, 509, 741c, 1126c }} | ||
[[Tp tuning #T2 tuning|RMS error]]: 0. | [[Tp tuning #T2 tuning|RMS error]]: 0.0749 cents | ||
=== | ==== Bridgetown ==== | ||
{{See also| Chromatic pairs #Bridgetown }} | |||
Bridgetown, the 5 & 24 temperament in the 2.3.11/5.13/5 subgroup, is related to [[#Haumea|haumea]] and [[#Barbados|barbados]]. | |||
[[Subgroup]]: 2.3. | [[Subgroup]]: 2.3.11/5.13/5 | ||
[[Comma list]]: | [[Comma list]]: [[352/351]], [[676/675]] | ||
{{Mapping|legend=2| | {{Mapping|legend=2| 1 0 -6 -1 | 0 2 9 3 }} | ||
[[ | {{Mapping|legend=3| 1 2 -5/3 0 4/3 1/3 | 0 -2 4 0 -5 1 }} | ||
: [[gencom]]: [2 15/13; 352/351 676/675] | |||
[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~15/13 = 248.399 | |||
{{Optimal ET sequence|legend=1| 5, 9, 14, 19, 24, 29, 169, 198, 227, 256, 285, 314 }} | |||
[[Tp tuning #T2 tuning|RMS error]]: 0.2513 cents | |||
[[ | |||
[[ | === Hypnosis === | ||
Related temperaments: [[Swetismic temperaments #Hypnos|hypnos]], [[Alphatricot family #Alphatricot|alphatricot]] | |||
[[Subgroup]]: 2.3.7.11/5.13 | |||
[[ | [[Comma list]]: 169/168, 540/539, 729/728 | ||
{{ | {{Mapping|legend=2| 1 0 -3 8 0 | 0 3 11 -13 7 }} | ||
[[Tp | [[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~13/9 = 633.518 | ||
= | {{Optimal ET sequence|legend=1| 17, 36, 118f, 125f, 161f, 197f }} | ||
[[ | [[Tp tuning #T2 tuning|RMS error]]: 0.5379 cents | ||
=== Trisect === | |||
Trisect divides every Pythagorean interval into three, and is the much more accurate subgroup restriction of [[Augmented family #Trisected|trisected]]. | |||
[[ | Extending this temperament to the full [[11-limit|11-]], [[13-limit|13-]], or [[17-limit]] through [[portent]] or [[landscape]] results in the [[weak extension]] known as [[tritikleismic]]. | ||
[[Subgroup]]: 2.3.7.11/5 | |||
[[ | [[Comma list]]: 1029/1024, 4000/3993 | ||
= | {{Mapping|legend=2| 3 0 10 5 | 0 3 -1 -1 }} | ||
[[ | [[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~44/35 = 1\3, ~13/9 = 633.742 | ||
{{Optimal ET sequence|legend=1| 15, 21, 36, 123, 159, 195, 231 }} | |||
{{Optimal ET sequence|legend=1| 15, | |||
[[Tp tuning #T2 tuning|RMS error]]: ??? | [[Tp tuning #T2 tuning|RMS error]]: ??? | ||
==== 2.3.7.11/5.13 subgroup ==== | |||
[[Subgroup]]: 2.3.7.11/5.13 | [[Subgroup]]: 2.3.7.11/5.13 | ||
[[Comma list]]: | [[Comma list]]: 1029/1024, 1575/1573, 2080/2079 | ||
{{Mapping|legend=2| 3 0 10 5 0 | {{Mapping|legend=2| 3 0 10 5 0 | 0 3 -1 -1 7 }} | ||
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~ | [[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~44/35 = 1\3, ~13/9 = 633.918 | ||
{{Optimal ET sequence|legend=1| | {{Optimal ET sequence|legend=1| 15, 21f, 36, 87, 123, 159 }} | ||
[[Tp tuning #T2 tuning|RMS error]]: ??? | [[Tp tuning #T2 tuning|RMS error]]: ??? | ||
==== 2.3.7.11/5.13.17 subgroup ==== | |||
[[Subgroup]]: 2.3.7.11/5.13.17 | [[Subgroup]]: 2.3.7.11/5.13.17 | ||
[[Comma list]]: | [[Comma list]]: 273/272, 833/832, 1575/1573, 2080/2079 | ||
{{Mapping|legend=2| 3 0 10 5 0 -2 | {{Mapping|legend=2| 3 0 10 5 0 -2 | 0 3 -1 -1 7 9 }} | ||
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~ | [[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~34/27 = 1\3, ~13/9 = 633.820 | ||
{{Optimal ET sequence|legend=1| | {{Optimal ET sequence|legend=1| 15, 21fg, 36, 123, 159 }} | ||
[[Tp tuning #T2 tuning|RMS error]]: ??? | [[Tp tuning #T2 tuning|RMS error]]: ??? | ||
== | ===== Trisector ===== | ||
=== | [[Subgroup]]: 2.3.7.11/5.13.17.19 | ||
[[Comma list]]: 210/209, 273/272, 286/285, 595/594, 2080/2079 | |||
{{Mapping|legend=2| 3 0 10 5 0 -2 8 | 0 3 -1 -1 7 9 3 }} | |||
[[ | [[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~34/27 = 1\3, ~13/9 = 633.894 | ||
{{ | {{Optimal ET sequence|legend=1| 15, 21fg, 36, 123h, 159h }} | ||
[[Tp tuning #T2 tuning|RMS error]]: ??? | |||
[[ | ===== 2.3.7.11/5.13.17.19.23 subgroup ===== | ||
[[Subgroup]]: 2.3.7.11/5.13.17.19.23 | |||
[[Comma list]]: 210/209, 231/230, 273/272, 286/285, 595/594, 2080/2079 | |||
{{Mapping|legend=2| 3 0 10 5 0 -2 8 12 | 0 3 -1 -1 7 9 3 1 }} | |||
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~34/27 = 1\3, ~13/9 = 634.038 | |||
{{Optimal ET sequence|legend=1| 15g, 21fg, 36, 87, 123hi }} | |||
[[ | [[Tp tuning #T2 tuning|RMS error]]: ??? | ||
[[ | ===== 2.3.7.11/5.13.17.19.23.29 subgroup ===== | ||
[[Subgroup]]: 2.3.7.11/5.13.17.19.23.29 | |||
[[ | [[Comma list]]: 210/209, 231/230, 273/272, 286/285, 320/319, 595/594, 2080/2079 | ||
{{ | {{Mapping|legend=2| 3 0 10 5 0 -2 8 12 13 | 0 3 -1 -1 7 9 3 1 1 }} | ||
[[ | [[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~29/23 = 1\3, ~13/9 = 634.102 | ||
{{Optimal ET sequence|legend=1| 15g, 21fg, 36, 87, 123hi }} | |||
[[Tp tuning #T2 tuning|RMS error]]: ??? | |||
== 2.….11/7.… subgroups == | |||
=== Blackweed === | |||
Blackweed is a [[restriction]] of undecimal [[blackwood]] as it tempers out 256/243 alike but in the 2.3.11/7 subgroup. 20edo is close to the optimum, which has 4\20 as the period and 420{{c}} as the generator. | |||
[[Subgroup]]: 2.3.11 | [[Subgroup]]: 2.3.11/7 | ||
[[Comma list]]: | [[Comma list]]: {{monzo| 8 -5 }} (256/243) | ||
{{Mapping|legend=2| | {{Mapping|legend=2| 5 8 0 | 0 0 1 }} | ||
: mapping generators: ~9/8, ~11/7 | |||
{{ | [[Optimal tuning]]s: | ||
* [[Tp tuning|subgroup]] [[WE]]: ~8/7 = 238.851{{c}}, ~11/7 = 782.457{{c}} | |||
: [[error map]]: {{val| -5.746 +8.852 -0.035 }} | |||
* [[Tp tuning|subgroup]] [[CWE]]: ~8/7 = 240.000{{c}}, ~11/7 = 784.967{{c}} | |||
: error map: {{val| 0.000 +18.045 +2.475 }} | |||
{{Optimal ET sequence|legend=1| 15, 20, 35b, 55b }} | |||
{{ | === Pepperoni === | ||
{{Main| Parapyth }} | |||
{{See also| Chromatic pairs #Pepperoni }} | |||
[[ | Pepperoni is generated by a fifth and can be described as the 5 & 12 temperament in the 2.3.11/7.13/7 subgroup. It is the single-chain retraction of [[parapyth]]. The [[Peppermint-24|Pepper fifth]], which is (40200 + 600 sqrt(5))/59 = 704.096 cents, is a good pepperoni generator, hence the name. | ||
[[Subgroup]]: 2.3.11/7.13/7 | |||
[[Subgroup]]: 2.3.11.13/ | |||
[[Comma list]]: | [[Comma list]]: 352/351, 364/363 | ||
{{Mapping|legend=2| | {{Mapping|legend=2| 1 0 7 12 | 0 1 -4 -7 }} | ||
{{Mapping|legend=3| 1 1 0 -8/3 1/3 7/3 | 0 1 0 11/3 -1/3 -10/3 }} | |||
: [[gencom]]: [2 3/2; 352/351 364/363] | |||
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~3/2 = 703.856 | |||
: | |||
= | {{Optimal ET sequence|legend=1| 5, 7, 12f, 17, 29, 46, 58, 75, 80, 87, 104, 121, 167, 196, 208, 271, 595b*<sup>†</sup> }} | ||
: <nowiki />* wart for 11/7 | |||
: <sup>†</sup> wart for 13/7 | |||
[[ | [[Tp tuning #T2 tuning|RMS error]]: 0.3789 cents | ||
[[Comma list]]: | == 2.….13/5.… subgroups == | ||
=== Barbados === | |||
The [[minimax tuning]] for this makes the generator the cube root of 20/13, or 248.5953 cents. Edos which may be used for it are [[24edo]], [[29edo]], [[53edo]] and [[111edo]], with [[mos scale]]s of size 5, 9, 14, 19, 24 and 29 making for a good variety of scales. | |||
[[Subgroup]]: 2.3.13/5 | |||
[[Comma list]]: 676/675 = {{monzo| 2 -3 2 }} | |||
[[Sval]] [[mapping]]: [{{val| 1 0 -1 }}, {{val| 0 2 3 }}] | |||
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~2 = 1\1, ~15/13 = 248.621 | |||
{{Optimal ET sequence|legend=1| 5, 9, 14, 19, 24, 29, 53, 82, 111, 140, 251, 362 }} | |||
[[Badness]]: 0.002335 | |||
[[ | ; Music | ||
* [http://micro.soonlabel.com/gene_ward_smith/Others/Sevish/Sevish%20-%20Desert%20Island%20Rain.mp3 ''Desert Island Rain''] in 313edo tuned Barbados[9], by [https://soundcloud.com/sevish/desert-island-rain Sevish] | |||
==== Tobago ==== | |||
{{See also| Chromatic pairs #Tobago }} | |||
Tobago, the 10 & 14 temperament in the 2.3.11.13/5 subgroup, extends [[neutral]] and [[barbados]]. | |||
[[Subgroup]]: 2.3.49/5 | [[Subgroup]]: 2.3.11.13/5 | ||
[[Comma list]]: [[243/242]], [[676/675]] | |||
{{Mapping|legend=2| 2 0 -1 -2 | 0 2 5 3 }} | |||
{{Mapping|legend=3| 2 4 -2 0 9 2 | 0 -2 3/2 0 -5 -3/2 }} | |||
: [[gencom]]: [55/39 15/13; 243/242 676/675] | |||
[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~55/39 = 1\2, ~15/13 = 249.312 | |||
{{Optimal ET sequence|legend=1| 10, 14, 24, 58, 82, 130 }} | |||
[[Tp tuning #T2 tuning|RMS error]]: 0.3533 cents | |||
==== Pakkanian hemipyth ==== | |||
[[Subgroup]]: 2.3.11.13/5.17 | |||
[[Comma list]]: 221/220, 243/242, 289/288 | |||
{{Mapping|legend=2| 2 0 -1 -2 5 | 0 2 5 3 2 }} | |||
[[Optimal tuning]]s: | |||
* [[Tp tuning|subgroup CTE]]: ~17/12 = 1\2, ~26/15 = 950.7656 (~15/13 = 249.2344) | |||
* [[Tp tuning|subgroup CWE]]: ~17/12 = 1\2, ~26/15 = 950.6011 (~15/13 = 249.3989) | |||
{{Optimal ET sequence|legend=1| 10, 14, 24, 106, 130, 154, 178*, 202* }} | |||
: <nowiki />* wart for 13/5 | |||
=== Oceanfront === | |||
Related temperaments: [[Archytas clan #Superpyth|superpyth]], [[Archytas clan #Ultrapyth|ultrapyth]] | |||
[[Subgroup]]: 2.3.7.13/5 | |||
[[Comma list]]: 64/63, 91/90 | |||
{{Mapping|legend=2| 1 0 6 -5 | 0 1 -2 4 }} | |||
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~3/2 = 713.910 | |||
{{Optimal ET sequence|legend=1| 5, 22, 27, 32, 37 }} | |||
[[Tp tuning #T2 tuning|RMS error]]: 2.063 cents | |||
Scales: [[Oceanfront scales]] | |||
=== Seventeen-cot === | |||
Seventeen-cot is a rank-2 temperament in the 2.3.13/5 and 2.3.11/5.13/5 subgroups. It tempers out the [[Tendoartisma]] in the 2.3.13/5 subgroup. It can be generated with a ~2/1 octave and a ~2250/2197 or ~169/165 generator which is a 17th of a ~3/2 perfect fifth. It can be described as the 29 & 146 temperament in these subgroups. | |||
====2.3.13/5 subgroup==== | |||
Comma basis: {{monzo| -6 -11 17 }} (2.3.13/5) | |||
edo join: 29 & 146 | |||
{{Mapping|legend=2| 1 1 1 | 0 17 11 }} | |||
: mapping generators: ~2, ~2250/2197 | |||
{{Todo|inline=1|correct maths|comment=Optimal tunings and error maps showed below is not yet precise enough.}} | |||
Optimal tunings: | |||
* WE: ~2 = 1200.000{{c}}, ~2250/2197 = 41.291{{c}} | |||
: error map: {{val| +0.000 +0.001 -0.007}} | |||
* CWE: ~2 = 1200.000{{c}}, ~2250/2197 = 41.292{{c}} | |||
: error map: {{val| +0.000 +0.001 -0.007}} | |||
edos: 29, 465, 494, 436, 523, 407, 378, 349, 30[-3], 28[+3], 320, 291, 59[-3], 262 | |||
Badness (Sintel): 0.064 | |||
====2.3.11/5.13/5 subgroup==== | |||
Comma basis: 225000/224939, 43940/43923 | |||
edo join: 29 & 146 | |||
{{Mapping|legend=2| 1 1 1 1 | 0 17 4 11 }} | |||
: mapping generators: ~2, ~169/165 | |||
Optimal tunings: | |||
* WE: ~2 = 1199.993{{c}}, ~169/165 = 41.292{{c}} | |||
: error map: {{val| -0.007 -0.000 +0.157 -0.010}} | |||
* CWE: ~2 = 1200.000{{c}}, ~169/165 = 41.292{{c}} | |||
: error map: {{val| +0.000 +0.005 +0.163 -0.005}} | |||
edos: 29, 465, 494, 436, 523, 407, 378, 349, 320, 291, 30[-3], 262, 28[+3], 233 | |||
Badness (Sintel): 0.080 | |||
== 2.….49/5.… subgroups == | |||
=== Direct breedsmic === | |||
Related temperament: [[hemithirds]], [[newt]] | |||
[[Subgroup]]: 2.3.49/5 | |||
[[Comma list]]: 2401/2400 | [[Comma list]]: 2401/2400 | ||
{{Mapping|legend=2| 1 1 3 | 0 2 1 }} | {{Mapping|legend=2| 1 1 3 | 0 2 1 }} | ||
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~49/40 = 350.966 | |||
{{Optimal ET sequence|legend=1|7, 10, 17}} | |||
[[Tp tuning #T2 tuning|RMS error]]: ? | |||
== 2.….17/5.… subgroups == | |||
=== Fiventeen === | |||
Fiventeen tempers out [[136/135]] ({{monzo| 3 -3 1 }}) in 2.3.17/5. It equates [[17/15]] with [[9/8]], so it implies a [[supersoft]] [[pentic]] [[pentad]] of [[~]]30:34:40:45:51. [[17edo]] makes a good tuning especially for its size, which gives a [[supersoft]] pentic scale corresponding approximately to a just [[20/17]] tuning, although [[80edo]] might be preferred for an approximately just [[51/40]] to optimize plausibility slightly more, and [[97edo]] (= 80 + 17) and [[114edo]] (= 97 + 17) do even better in striking a balance between 80edo's more stable tuning and that having 20/17 more accurate (as in 17edo) is useful because of the more convincing suggestion of the two 15:17:20 chords present in the fiventeen pentad. The same is true of the related rank-3 temperament diatic, for which the [[optimal ET sequence]] is much more characteristic of optimized tunings, finding [[34edo]], then [[80edo]], then [[114edo]] (= 34 + 80) and even [[194edo|194bc-edo]] (= 80 + 114), though because of its focus on primes 5 and 17 it misses 97edo as a tuning, and slightly less optimized though still interesting [[63edo]] and [[143edo]] (= 63 + 80) tunings are found in the optimal ET sequence for fiventeen. | |||
[[Subgroup]]: 2.3.17/5 | |||
[[Comma list]]: 136/135 ({{monzo| 3 -3 1 }}) | |||
{{Mapping|legend=2| 1 0 -3 | 0 1 3 }} | |||
: mapping generators: ~2, ~3 | |||
[[Optimal tuning]]s: | |||
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.2838{{c}}, ~3/2 = 704.4600{{c}} | |||
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.5286{{c}} | |||
{{Optimal ET sequence|legend=1| 5, 12, 17, 46, 63, 143 }} | |||
== 2.….19/7.… subgroups == | |||
=== Surprise === | |||
This temperament was named by [[User:VectorGraphics|Vector]] in 2025, as he was surprised that the temperament of [[57/56]] did not have a name. This is the [[rank-2 temperament|rank-2]] version of the temperament; Vector surmises that the name ''hendrix'' would be more thoughtfully given to the [[rank-3]] version. | |||
[[Subgroup]]: 2.3.19/7 | |||
[[Comma list]]: [[57/56]] ({{Monzo| -3 1 1 }}) | |||
{{Mapping|legend=2| 1 0 3 | 0 1 -1 }} | |||
: mapping generators: ~2, ~3 | |||
[[Optimal tuning]]s: | |||
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1202.4345{{c}}, ~3/2 = 697.4314{{c}} | |||
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 697.3981{{c}} | |||
{{Optimal ET sequence|legend=1| 5, 7, 12, 19, 31*, 50* }} | |||
<nowiki/>* wart for 19/7 | |||
[[Badness]] (Sintel): 0.082 | |||
=== Supramin === | |||
This is a remarkable low-complexity microtemperament that contains the 14:17:19 triad within just four generator steps. An excellent tuning is [[25edo]], which provides an accurate yet tone-efficient tuning of this temperament. It was named by [[User:Overthink|Overthink]] in 2026 after the fact that the generator is a [[17/14]] supraminor third, two of which reach [[28/19]]. | |||
[[Subgroup]]: 2.17/7.19/7 | |||
[[Comma list]]: [[5491/5488]] ({{Monzo| -4 2 1 }}) | |||
{{Mapping|legend=2| 1 0 4 | 0 1 -2 }} | |||
: mapping generators: ~2, ~17/7 | |||
[[Optimal tuning]]s: | |||
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1200.022{{c}}, ~17/14 = 335.793{{c}} | |||
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.000{{c}}, ~17/14 = 335.785{{c}} | |||
{{Optimal ET sequence|legend=1| 7, 18, 25 }} | |||
[[ | [[Badness]] (Sintel): 0.005 | ||
==== Supramine ==== | |||
This extension approximates the 14:17:19:23:25 pentad in just six generator steps, at the cost of some accuracy. 25edo remains a strong tuning. | |||
Subgroup: 2.17/7.19/7.23/7 | |||
Comma list: [[323/322]], [[392/391]] | |||
Subgroup-val mapping: {{Mapping| 1 0 4 3 | 0 1 -2 -1 }} | |||
Optimal tunings: | |||
* Subgroup WE: ~2 = 1199.871{{c}}, ~17/14 = 336.243{{c}} | |||
* Subgroup CWE: ~2 = 1200.000{{c}}, ~17/14 = 336.296{{c}} | |||
{{ | {{Optimal ET sequence|legend=0| 7, 18, 25 }} | ||
Badness (Sintel): 0.029 | |||
==== 2.25/7.17/7.19/7.23/7 subgroup ==== | |||
Subgroup: 2.25/7.17/7.19/7.23/7 | |||
[[ | Comma list: [[323/322]], [[392/391]], [[476/475]] | ||
[[ | |||
Subgroup-val mapping: {{Mapping| 1 -2 0 4 3 | 0 3 1 -2 -1 }} | |||
{{ | Optimal tunings: | ||
* Subgroup WE: ~2 = 1199.757{{c}}, ~17/14 = 335.428{{c}} | |||
* Subgroup CWE: ~2 = 1200.000{{c}}, ~17/14 = 335.479{{c}} | |||
{{Optimal ET sequence|legend=0| 7, 18, 25 }} | |||
Badness (Sintel): 0.053 | |||
== 3/2.5/2.… subgroups == | == 3/2.5/2.… subgroups == | ||
| Line 1,801: | Line 1,997: | ||
[[Category:Subgroup temperaments| ]] <!-- main article --> | [[Category:Subgroup temperaments| ]] <!-- main article --> | ||
[[Category:Temperament collections]] | [[Category:Temperament collections]] | ||
[[Category:Rank 2]] | |||
{{Todo| review | cleanup }} | {{Todo| review | cleanup }} | ||