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Subgroup temperaments: Difference between revisions

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:''todo: Add definition of the term "subgroup temperament"''
{{Technical data page}}
A '''subgroup temperament''' is a regular temperament defined on a [[just intonation subgroup]] that is not a full ''p''-limit group.


__FORCETOC__
For temperaments that omit various prime harmonics, see:
* [[No-elevens subgroup temperaments]]
* [[No-sevens subgroup temperaments]]
* [[No-fives subgroup temperaments]]
* [[No-threes subgroup temperaments]]
* [[No-twos subgroup temperaments]] (additionally, [[Catalog of 3.5.7 subgroup rank two temperaments]]).


Below are some [[JI subgroup|subgroups]] and temperaments for them. Obviously, no attempt has been made at completeness; attention is focused on subgroups containing interesting chords. The reader may also want to consult the page on [[Chromatic pairs]].
Below are some temperaments for composite subgroups and fractional subgroups. Obviously, no attempt has been made at completeness; attention is focused on subgroups containing interesting chords. The reader may also want to consult the page on [[Chromatic pairs]].


=2.3.7=
= 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]].


==[[Semaphore_and_Godzilla|Semaphore]]==
==== 2.3.11.19 subgroup ====
Comma: 49/48
The temperament is simplest in this subgroup, although there is a tradeoff of breaking up the simplicity of the 36/35 quartertone.


[[Lp_tuning|POL2 generator]]: ~15/13= 250.385
[[Subgroup]]: 2.3.11.19


[[Gencom]]: [2 8/7; 49/48]
{{Mapping|legend=2| 4 5 13 18 | 0 8 5 -6 }}


Gencom mapping: [<1 2 0 3|, <0 -2 0 -1|]
: sval mapping generators: ~6291456/5285401, ~25289/24576


Map: [<1 2 3|, <0 -2 -1|]
[[Optimal tuning]] ([[CTE]]): ~6291456/5285401 = 1\4, ~25289/24576 = 50.257


EDOs: 5, 14, 19, 24, 67c, 91c
[[Support]]ing [[ET]]s: {{EDOs|24, 596, 620, 644, 668, 692, 716}}, ...


[[Tp_tuning#T2 tuning|RMS error]]: 2.523 cents
==== 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.


==Archy==
Subgroup: 2.3.35.11.19
Archy can be thought of as "no-5's [[dominant]]" or "no-5's [[superpyth]]". The name comes from the fact that it tempers out 64/63, the Archytas comma.


Comma: 64/63
Sval mapping: {{mapping| 4 0 5 13 18 | 0 1 8 5 -6 }}


[[Lp_tuning|POL2 generator]]: ~3/2 = 709.321
: sval mapping generators: ~2240/1881, ~36/35


Gencom: [2 3/2; 64/63]
Optimal tuning (CTE): ~2240/1881 = 1\4, ~36/35 = 50.288


Gencom mapping: [<1 1 0 4|, <0 1 0 -2|]
[[Support]]ing [[ET]]s: {{EDOs|24, 668}}, ...


Map: [<1 2 2|, <0 -1 2|]
== 2.9.5.7 subgroup ==
See also [[Jubilismic clan #Antikythera|antikythera]] and [[Hemimean clan #Isra|isra]].


EDOs: 5, 12, 17, 22, 27, 137bc
=== Commatose ===
Commatose is a [[Dual-fifth temperaments|dual-fifth temperament]] which uses the Pythagorean comma as a generator. It was developed by [[Eliora]] to highlight the near-perfect expression of 9/8 by [[1789edo]], while at the same time the fact that it completely misses 3/2. It is described as the 460 & 1329 temperament. In the 13-limit extension 24 generators are equal to [[~]][[13/9]].


[[Tp_tuning#T2 tuning|RMS error]]: 1.856 cents
[[Subgroup]]: 2.9.5.7


==[[Slendric]]==
[[Comma list]]: {{monzo| 28 -2 -19 8 }}, {{monzo| 9 -25 23 6 }}
Comma: 1029/1024


[[Lp_tuning|POL2 generator]]: ~8/7 = 233.688
{{Mapping|legend=2| 1 9 6 13 | 0 -298 -188 -521 }}


Gencom: [2 8/7; 1029/1024]
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~531441/524288 = 23.4765


Gencom mapping: [<1 1 0 3|, <0 3 0 -1|]
{{Optimal ET sequence|legend=1| 460, 869, 1329 }}


Map: [<1 1 3|, <0 3 -1|]
[[Badness]]: 0.611


EDOs: 5, 21, 26, 31, 36, 77, 113, 190
==== 2.9.5.7.11 ====
Subgroup: 2.9.5.7.11


[[Tp_tuning#T2 tuning|RMS error]]: 0.3202 cents
Comma list: {{monzo| -7 7 -3 2 -4 }}, {{monzo| 17 0 -13 1 3 }}, {{monzo| 11 -2 -6 7 -3 }}


==Lee==
Sval mapping: {{mapping| 1 9 6 13 16 | 0 -298 -188 -521 -641 }}
Comma: 177147/175616


[[Lp_tuning|POL2 generator]]: ~13/9 = 633.525
Optimal tuning (CTE): ~2 = 1\1, ~531441/524288 = 23.4767


Map: [<1 0 -3|, <0 3 11|]
{{Optimal ET sequence|legend=1| 460, 869e, 1329, 1789, 3118 }}


EDOs: 17, 36, 89, 125, 161, 358, 519b
Badness: 0.165


[[Tp_tuning#T2 tuning|RMS error]]: 0.3519 cents
==== 2.9.5.7.11.13 ====
Subgroup: 2.9.5.7.11.13


==Skwares==
Comma list: 123201/123200, 1016064/1015625, 2250423/2249390, 2599051/2598156
Comma: 19683/19208


[[Lp_tuning|POL2 generator]]: ~23/18 = 425.365 cents
Sval mapping: {{mapping| 0 9 6 13 16 10 | -298 -188 -521 -641 -322 }}


Map: [<1 3 6|, <0 -4 -9|]
Optimal tuning (CTE): ~2 = 1\1, ~3575/3528 = 23.4767


EDOs: 14, 17, 31, 48, 79, 189b, 268bc, 347bc
{{Optimal ET sequence|legend=1| 460, 869e, 1329, 1789, 3118 }}


[[Tp_tuning#T2 tuning|RMS error]]: 1.149 cents
Badness: 0.0564


=2.3.11=
=== Daemotertiaschis ===
{{See also|Schismatic family#Tertiaschis}}
Daemotertiaschis is produced by taking every other generator of tertiaschis, and the subgroup is chosen so it tempers out exactly the same commas. It is notable due to offering a [[7L 4s|daemotonic 7L 4s]] scale of reasonable hardness, which is notoriously difficult to approximate with simple JI or RTT methods.


==Neutral==
Subgroup: 2.9.5.7.33.13.17
Neutral can be thought of as the 2.3.11 version of either [[mohajira]] or [[maqamic]], as well as suhajira (further down on this page) and ringo. Among other things, it is the temperament optimizing the [[neutral tetrad]].


Comma: 243/242
Comma list: 325/324, 375/374, 385/384, 595/594, 10985/10976


[[Lp_tuning|POL2 generator]]: ~11/9 = 350.525
{{Mapping|legend=2|1 1 11 -16 13 -18 20|0 3 -12 26 -11 30 -22}}


Gencom: [2 11/9; 243/242]
Optimal tuning (CTE): ~2 = 1\1, 33/20 = 867.982


Gencom mapping: [<1 1 0 0 2|, <0 2 0 0 5|]
[[Support]]ing [[ET]]s: {{Optimal ET sequence|47, 65f, 112, 159, 206, 253}}


Map: [<1 1 2|, <0 2 5|]
=== Baldy ===
{{See also|Schismatic family #Garibaldi}}
{{See also|No-threes subgroup temperaments #Frostburn}}


EDOs: 7, 10, 17, 24, 41, 65, 89, 202, 291, 380
Baldy results from taking every other generator of the [[garibaldi]] temperament. One of the best extension is 2.9.5.7.13 subgroup with mapping 13/8 to +10 whole tones, as well as the cassandra temperament.


[[Tp_tuning#T2 tuning|RMS error]]: 0.3021 cents
[[Subgroup]]: 2.9.5.7


[[neutraltet7|Seven note albitonic scale]]
[[Comma list]]: 225/224, 3125/3087


[[neutraltet10|Ten note chromatic scale]]
{{Mapping|legend=2| 1 3 3 4 | 0 1 -4 -7 }}


[[neutraltet17|Seventeen note mega chromatic scale]]
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 204.170


=2.3.5.11=
{{Optimal ET sequence|legend=1| 6, 29, 35, 41, 47 }}


==Porkypine==
Related temperament: [[Schismatic family #Garibaldi|Garibaldi]]
Commas: 55/54, 100/99


[[Lp_tuning|POL2 generator]]: ~11/10 = 164.078
==== 2.9.5.7.13 ====
{{See also|Chromatic pairs #Baldy}}


Gencom: [2 10/9; 55/54, 100/99]
Baldy is every other step of [[garibaldi]], without the mapping of prime 11. It can be described as the 6 & 35 temperament.


Gencom mapping: [<1 2 3 0 4|, <0 -3 -5 0 -4|]
[[Subgroup]]: 2.9.5.7.13


Map: [<1 2 3 4|, <0 -3 -5 -4|]
[[Comma list]]: [[225/224]], [[325/324]], [[640/637]]


EDOs: 7, 15, 22, 37, 73cd, 95cd, 117bcd
{{Mapping|legend=2| 1 0 15 25 -28 | 0 1 -4 -7 10 }}


[[Tp_tuning#T2 tuning|RMS error]]: 2.287 cents
{{Mapping|legend=3| 1 3/2 3 4 0 2 | 0 1/2 -4 -7 0 10 }}


==Mohaha==
: [[gencom]]: [2 9/8; 225/224 325/324 640/637]
Commas: 81/80, 121/120


[[Lp_tuning|POL2 generator]]: ~11/9 = 348.094
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 204.090


Gencom: [2 11/9; 81/80 121/120]
{{Optimal ET sequence|legend=1| 6, 11, 17, 23, 29, 35, 41, 47, 100, 147, 488cd, 635cd }}


Gencom mapping: [<1 1 0 0 2|, <0 2 8 0 5|]
[[Tp tuning #T2 tuning|RMS error]]: 0.5999 cents


Map: [<1 1 0 2|, <0 2 8 5|]
Related temperament: [[Schismatic family #Garibaldi|Cassandra]]


EDOs: 7, 10, 17, 24, 31, 55, 69d, 100d, 131bd
==== Baldanders ====
Baldanders results from taking every other generator of the andromeda, with mapping 11/8 to -9 whole tones.


[[Tp_tuning#T2 tuning|RMS error]]: 1.392 cents
Subgroup: 2.9.5.7.11


===Music===
Comma list: 100/99, 225/224, 245/242
[http://micro.soonlabel.com/gene_ward_smith/Others/Stiltner/mohaha10ping2.mp3 Mohaha10ping2] by [[Billy Stiltner]]


=2.3.7.11=
{{Mapping|legend=2| 1 3 3 4 5 | 0 1 -4 -7 -9 }}


==Ennea==
Optimal tuning (POTE): ~2 = 1\1, ~9/8 = 204.743
Commas: 41503/41472, 43923/43904


[[Lp_tuning|POL2 generator]]: ~99/98 = 17.6258
{{Optimal ET sequence|legend=1| 6, 23de, 29, 35, 41 }}


Gencom: [2 99/98; 41503/41472, 43923/43904]
Related temperament: [[Schismatic family #Garibaldi|Andromeda]]


Gencom mapping: [<1 14/9 0 25/9 31/9|, <0 2 0 2 1|]
===== 2.9.5.7.11.13 =====
Subgroup: 2.9.5.7.11.13


Sval map: [<9 0 11 24|, <0 2 2 1|]
Comma list: 100/99, 144/143, 225/224, 245/242


EDOs: 54, 63, 68, 72, 135, 342, 477, 525, 613, 681, 749, 817, 2159bcd, 3336bcd, 4153bcd
{{Mapping|legend=2| 1 3 3 4 5 2 | 0 1 -4 -7 -9 10 }}


[[Tp_tuning#T2 tuning|RMS error]]: 0.0383 cents
Optimal tuning (POTE): ~2 = 1\1, ~9/8 = 204.414


==Supra==
{{Optimal ET sequence|legend=1| 6, 23def, 29f, 35, 41, 47 }}
Commas: 64/63, 99/98


[[POTE_tuning|POTE generator]]: ~3/2 = 707.192
== 2.3.25 subgroup ==


Gencom: [2 3/2; 64/63 99/98]
=== 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.


Gencom mapping: [<1 1 0 4 7|, <0 1 0 -2 -6|]
Subgroup: 2.3.25


Sval map: [<1 0 6 13|, <0 1 -2 -6|]
Edo join: 17 & 12


EDOs: 5, 12, 17, 39c, 56c
Comma list: [[2048/2025]]


[[Tp_tuning#T2 tuning|RMS error]]: 1.977 cents
{{Mapping|legend=2| 1 1 7| 0 1 -4}}


==Skwares==
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 705.136
Commas: 99/98, 243/242


[[POTE_tuning|POTE generator]]: ~23/18 = 425.244
==== 2.3.23.25.41 subgroup ====
''See also: [[Reversed meantone]]''


Gencom: [2 9/7; 99/98 243/242]
Edo join: 17 & 12


Gencom mapping: [<1 3 0 6 7|, <0 -4 0 -9 -10]]
Comma list: 2048/2025, 576/575, 82/81


Sval map: [<1 3 6 7|, <0 -4 -9 -10|]
{{Mapping|legend=2| 1 1 1 7 3| 0 1 6 -4 4}}


EDOs: 14, 17, 31, 48, 79, 127, 206bcd
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 705.264


[[Tp_tuning#T2 tuning|RMS error]]: 1.099 cents
===== Sburb =====
This temperament sets the [[octave reduction|octave-reduced]] 413th harmonic (413/256, 827.998{{c}}) to the diminished seventh.


==Hemif==
Subgroup: 2.3.7.23.25.41.59
Commas: 243/242, 896/891


POTE generator: ~11/9 = 351.535
Edo join: 17 & 12


Gencom: [2 11/9; 243/242 896/891]
Comma list: 64/63, 225/224, 162/161, 82/81, 177/175


Gencom mapping: [<1 1 0 -1 2|, <0 2 0 13 5|]
{{Mapping|legend=2| 1 1 4 1 7 3 10| 0 1 -2 6 -4 4 -7}}


Sval map: [<1 1 -1 2|, <0 2 13 5|]
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 706.387


EDOs: 7, 17, 41, 58, 99d
== 2.9.5.11 subgroup ==
=== Glacial ===
{{See also| Chromatic pairs #Glacial }}


[[Tp_tuning#T2 tuning|RMS error]]: 0.6108 cents
[[Subgroup]]: 2.9.5.11.13


==Bleu==
[[Comma list]]: 45/44, 65/64, 81/80
Commas: 99/98, 864/847


POTE generator: ~13/12 = 140.005
{{Mapping|legend=2| 1 0 -4 -6 10 | 0 1 2 3 -2 }}


Map: [<1 1 2 3|, <0 5 7 4|]
{{Mapping|legend=3| 1 3/2 2 0 3 4 | 0 1/2 2 0 3 -2 }}


EDOs: 9, 17, 43, 60c
: [[gencom]]: [2 9/8; 45/44 65/64 81/80]


RMS error: 1.829 cents
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 186.151


=2.3.7.11.13=
{{Optimal ET sequence|legend=1| 6, 13, 45be, 58bce, 71bce, 84bce }}


==Suhajira==
[[Tp tuning #T2 tuning|RMS error]]: 2.887 cents
Commas: 64/63, 78/77, 144/143


Gencom: [2 11/9; 64/63 78/77 144/143]
Music:
* ''[[Thundersnow]]'' - [[Sevish]] (2021)


Gencom map: [<1 1 0 4 2 4|, <0 2 0 -4 5 -1|]
== 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.


[[Tp_tuning|POT2 generator]]: ~11/9 = 353.775
Subgroup: 2.9.7


Map: [<1 1 4 2 4|, <0 2 -4 5 -1|]
Comma basis: 44957696/43046721


EDOs: 7, 10, 17, 44d, 61cd, 78cd
Sval mapping: [{{val|1 1 -3}}, {{val|0 3 8}}]


[[Tp_tuning#T2 tuning|RMS error]]: 1.953 cents
Optimal tuning (CTE): ~729/448 = 870.792


==Hemif==
{{Optimal ET sequence|legend=1|7d, 11, 18d, 29, 40, 62}}, ...
Commas: 144/143, 243/242, 364/363


Related temperament: [[Breedsmic_temperaments|Hemififths]], namo
==== 2.9.7.11 subgroup ====
Subgroup: 2.9.7.11


[[Tp_tuning|POT2 generator]]: ~11/9 = 351.691
Comma basis: 896/891, 1331/1296


[[Gencom]]: [2 11/9; 144/143 243/242 364/363]
Sval mapping: [{{val|1 1 -3 2}}, {{val|0 3 8 2}}]


Gencom map: [<1 1 0 -1 2 4|, <0 2 0 13 5 -1|]
Optimal tuning (CTE): ~16/11 = 870.966


Map: [<1 1 -1 2 4|, <0 2 13 5 -1|]
{{Optimal ET sequence|legend=1| 7d, 11, 40, 51, 62 }}


EDOs: 7, 10, 17, 24, 41, 58
== 2.9.7.11 subgroup ==
=== Apparatus ===
[[Subgroup]]: 2.9.7.11


[[Tp_tuning#T2 tuning|RMS error]]: 0.7167 cents
[[Comma list]]: 41503/41472, 322102/321489


==Skwares==
{{Mapping|legend=2| 1 5 3 5 | 0 -19 -2 -16 }}
Commas: 78/77, 99/98, 243/242


[[Tp_tuning|POT2 generator]]: ~23/18 = 424.457
: mapping generators: ~2, ~77/72


Map: [<1 3 6 7 9|, <0 -4 -9 -10 -15|]
{{Mapping|legend=3| 1 5/2 0 3 5 | 0 -19/2 0 -2 -16 }}


EDOs: 17, 48e, 65de, 82c, 147ce
: [[gencom]]: [2 77/72; 41503/41472 322102/321489]


[[Tp_tuning#T2 tuning|RMS error]]: 1.769 cents
[[Optimal tuning]] ([[CTE]]): ~77/72 = 115.5685


==Leapfrog==
{{Optimal ET sequence|legend=1| 10e, 21, 31, 52, 83, 135, 353, 488, 623 }}
Commas: 169/168, 352/351, 364/363


[[Lp_tuning|POL2 generator]]: ~3/2 = 704.745 cents
[[Badness]]: 0.00263


Gencom: [2 3/2; 169/169 352/351 364/363]
=== Joan ===
{{See also| Chromatic pairs #Joan }}


Gencom mapping: [<1 1 0 -6 -3 -1|, <0 1 0 15 11 8|]
Joan is related to [[casablanca]] as well as to [[orwell]].


Sval map: [<1 0 -21 -14 -9|, <0 1 15 11 8|]
[[Subgroup]]: 2.9.7.11


EDOs: 17, 46, 63
[[Comma list]]: 99/98, 9317/9216


[[Tp_tuning#T2 tuning|RMS error]]: 0.7541 cents
{{Mapping|legend=2| 1 0 1 3 | 0 7 4 1 }}
===Music===


*Suite for Harpsichord in A Locrian, tuning: Eb-G# in [[46edo]] by [[User:IlL|IlL]] (in progress):
{{Mapping|legend=3| 1 0 0 1 3 | 0 7/2 0 4 1 }}
**I. Prelude
**II. Allemande
**III. Courante
**[[:File:Locrian Suite Sarabande.mp3|IV. Sarabande]] ([[:File:Locrian Suite 17edo.mp3|17edo version]])
**[[:File:Locrian Suite Menuet.mp3|V. Menuet and Trio]]
**[[:File:Locrian Suite Gavotte.mp3|VI. Gavotte I and II]]
**[[:File:Locrian Suite Gigue.mp3|VII. Gigue]]


==Bleu==
: [[gencom]]: [2 11/8; 99/98 9317/9216]
Group: 2.3.7.11.13


Commas: 78/77, 99/98, 144/143
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~11/8 = 542.672 cents


Sval name: 9&17
{{Optimal ET sequence|legend=1| 11, 20, 31, 42, 115bd, 157bd }}


[[Tp_tuning|POL2 generator]]: ~13/12 = 139.990
[[Tp tuning #T2 tuning|RMS error]]: 1.424 cents


[[Gencom]]: [2 13/12; 78/77 99/98 144/143]
=== Machine ===
Machine is every other step of [[supra]], most interesting for its scale patterns.


Gencom map: [<1 1 0 2 3 3|, <0 5 0 7 4 6|]
[[Subgroup]]: 2.9.7.11


Map: [<1 1 2 3 3|, <0 5 7 4 6|]
[[Comma list]]: 64/63, 99/98


EDOs: [[17edo|17]], [[43edo|43]], [[60edo|60c]]
{{Mapping|legend=2| 1 0 6 13 | 0 1 -1 -3 }}


[[Tp_tuning#T2 tuning|RMS error]]: 1.752 cents
: sval mapping generators: ~2, ~9


==Parapyth (Rank 3)==
{{Mapping|legend=3| 1 3/2 0 3 4 | 0 1/2 0 -1 -3 }}
Commas: 352/351, 364/363


[[Lp_tuning|POL2 tuning]]: <1200.000 1903.856 (2787.774) 3369.907 4154.482 4442.913|
: [[gencom]]: [2 8/7; 64/63 99/98]


The gencom below gives [[Margo Schulter]]'s favored basis
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~9/8 = 216.9128
* [[POTE]]: ~2 = 1\1, ~9/8 = 214.3843


Gencom: [2 3/2 28/27; 352/351 364/363]
{{Optimal ET sequence|legend=1| 5, 6, 11, 17, 28 }}


Gencom mapping: [<1 1 0 1 4 6|, <0 1 0 3 -1 -4|, <0 0 0 1 1 1|]
[[Badness]]: 0.00233


Sval map: [<1 0 0 7 12|, <0 1 0 -4 -7|, <0 0 1 1 1|]
=== Penta a.k.a. mechanism ===
Penta or mechanism is the 8 & 11 temperament in the 2.9.7.11 subgroup.


EDOs: 17, 41, 46, 58, 87, 104
[[Subgroup]]: 2.9.7.11


[[Tp_tuning#T2 tuning|RMS error]]: 0.3789 cents
[[Comma list]]: 896/891, 26411/26244


=2.3.5.11=
{{Mapping|legend=2| 1 0 -1 6 | 0 5 6 -4 }}


==Larry==
: sval mapping generators: ~2, ~14/9
Commas: 243/242, 4000/3993


Related temperaments: [[Gravity_family#Gravity|gravity]], [[Gravity_family#Harry|harry]]
{{Mapping|legend=3| 1 5/2 0 5 2 | 0 -5/2 0 -6 4 }}


POTE generator: ~40/27 = 683.166
: [[gencom]]: [2 9/7; 896/891 26411/26244]


Gencom: [2 40/27; 243/242 4000/3993]
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~14/9 = 761.3782


Gencom mapping: [<1 5 12 0 12|, <0 -6 -17 0 -15|]
{{Optimal ET sequence|legend=1| 8, 11, 30, 41, 52 }}


Map: [<1 5 12 12|, <0 -6 -17 -15|]
[[Tp tuning #T2 tuning|RMS error]]: 0.4262 cents


EDOs: 7, 58, 65, 137, 202
[[Badness]]: 0.00439


[[Tp_tuning#T2 tuning|RMS error]]: 0.3025 cents
Scales: [[penta5]], [[penta8]], [[penta11]], [[penta19]]


=2.9.7.11=
== 2.9.7.13.17 subgroup ==


==Machine==
=== Novisept ===
Commas: 64/63, 99/98
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]].


[[POTE_tuning|POTE generator]]: ~17/15 = 214.384
[[Subgroup]]: 2.9.7.13.17


Gencom: [2 8/7; 64/63 99/98]
[[Comma list]]: 729/728, 442/441, 833/832


Gencom mapping: [<1 3/2 0 3 4|, <0 1/2 0 -1 -3|]
{{Mapping|legend=2| 1 1 1 -1 3| 0 6 5 13 3 }}


[[Smonzos_and_Svals|Sval map]]: [<1 0 6 13|, <0 1 -1 -3|]
[[Optimal tuning]] ([[CWE]]): ~2 = 1\1, ~9/7 = 433.836


EDOs: 5, 6, 11, 17, 28
Badness (Dirichlet): 0.142


[[Tp_tuning#T2 tuning|RMS error]]: 1.977 cents
== 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.


==Apparatus==
[[Subgroup]]: 2.9.11
Commas: 41503/41472, 322102/321489


[[POTE_tuning|POTE generator]]: ~31/29 = 115.570
[[Comma list]]: [[1331/1296]]


Gencom: [2 77/72; 41503/41472 322102/321489]
{{Mapping|legend=2|1 1 2|0 3 2}}


Gencom mapping: [<1 5/2 0 3 5|, <0 -19/2 0 -2 -16|]
[[Optimal tuning]] ([[CTE]]): ~[[18/11]] = 870.060


[[Smonzos_and_Svals|Sval map]]: [<1 5 3 5|, <0 -19 -2 -16|]
{{Optimal ET sequence|legend=1|4, 7, 11, 18, 29, 76e}}


EDOs: 10, 21, 31, 52, 83, 135, 353, 488, 623
=== Genius ===


[[Tp_tuning#T2 tuning|RMS error]]: 0.0673 cents
Named after the genius in Roman religion, following the demon (daimon) in Greek mythology.


==Mechanism==
[[Subgroup]]: 2.9.11
Commas: 896/891, 26411/26244


[[POTE_tuning|POTE generator]]: ~9/7 = 438.465
[[Comma list]]: [[131769/131072]]


Gencom: [2 9/7; 896/891 26411/26244]
{{Mapping|legend=2|1 1 4|0 4 -1}}


Gencom mapping: [<1 5/2 0 5 2|, <0 -5/2 0 -6 4|]
[[Optimal tuning]] ([[CTE]]): ~[[16/11]] = 650.863


[[Smonzos_and_Svals|Sval map]]: [<1 5 5 2|, <0 -5 -6 4|]
{{Optimal ET sequence|legend=1|9, 11, 24, 59, 83, 142, 225, 367}}[-11], 592[-11], 959[-9, --11], 1326[-9, --11]


EDOs: 8, 11, 30, 41, 52
== 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]].


[[Tp_tuning#T2 tuning|RMS error]]: 0.4262 cents
[[Subgroup]]: 2.9.15.7


=2.9.15.7.11.13=
[[Comma list]]: 225/224, 245/243


==Stacks (aka 2magic)==
{{Mapping|legend=2| 1 0 2 -1 | 0 5 3 6 }}
Commas: 100/99, 105/104, 144/143, 196/195


[[Lp_tuning|POL2 generator]]: ~9/7 = 438.977
: sval mapping generators: ~2, ~14/9


Gencom: [2 9/7; 100/99 105/104 144/143 196/195]
{{Mapping|legend=3| 1 5/2 5/2 5 | 0 -5/2 -1/2 -6 }}


Gencom mapping: [<1 5/2 5/2 5 2 7|, <0 -5/2 -1/2 -6 4 -9|]
: [[gencom]]: [2 9/7; 225/224 245/243]


Sval map: [<1 0 2 -1 6 -2|, <0 5 3 6 -4 9|]
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~2 = 1\1, ~14/9 = 760.704


EDOs: 11, 30, 41, 153cdef, 194cdef, 235cdef
{{Optimal ET sequence|legend=1| 8, 11, 30, 41, 71, 93, 112c, 134c, 175c }}


[[Tp_tuning#T2 tuning|RMS error]]: 1.540 cents
[[Tp tuning #T2 tuning|RMS error]]: 1.074 cents


===2.9.15.7.11 stacks===
==== 2.9.15.7.11 ====
Commas: 100/99, 225/224, 245/243
Subgroup: 2.9.15.7.11


[[Lp_tuning|POL2 generator]]: ~9/7 = 438.607
Comma list: 100/99, 225/224, 245/243


Gencom: [2 9/7; 100/99 225/224 245/243]
Sval mapping: {{mapping| 1 0 2 -1 6 | 0 5 3 6 -4 }}


Gencom mapping: [<1 5/2 5/2 5 2|, <0 -5/2 -1/2 -6 4|]
Gencom mapping: {{mapping| 1 5/2 5/2 5 2 | 0 -5/2 -1/2 -6 4 }}


Sval map: [<1 0 2 -1 6|, <0 5 3 6 -4|]
: gencom: [2 9/7; 100/99 225/224 245/243]


EDOs: 8, 11, 30, 41, 52, 93, 145, 342bce,
Optimal tuning (subgroup POTE): ~2 = 1\1, ~14/9 = 761.393


[[Tp_tuning#T2 tuning|RMS error]]: 1.226 cents
Optimal ET sequence: {{Optimal ET sequence| 8, 11, 30, 41, 52, 93, 145, 342bce }}


===2.9.15.7 stacks===
RMS error: 1.226 cents
Commas: 225/224, 245/243


[[Lp_tuning|POL2 generator]]: ~9/7 = 439.296
==== 2.9.15.7.11.13 ====
Subgroup: 2.9.15.7.11.13


Gencom: [2 9/7; 225/224 245/243]
Comma list: 100/99, 105/104, 144/143, 196/195


Gencom mapping: [<1 5/2 5/2 5|, <0 -5/2 -1/2 -6|]
Sval mapping: {{mapping| 1 0 2 -1 6 -2 | 0 5 3 6 -4 9 }}


Sval map: [<1 0 2 -1|, <0 5 3 6|]
Gencom mapping: {{mapping| 1 5/2 5/2 5 2 7 | 0 -5/2 -1/2 -6 4 -9 }}


EDOs: 8, 11, 30, 41, 71, 93, 112c, 134c, 175c
: gencom: [2 9/7; 100/99 105/104 144/143 196/195]


[[Tp_tuning#T2 tuning|RMS error]]: 1.074 cents
Optimal tuning (subgroup POTE): ~2 = 1\1, ~14/9 = 761.023


=2.9.21=
Optimal ET sequence: {{Optimal ET sequence| 11, 30, 41, 153cdef, 194cdef, 235cdef }}


==A-team==
RMS error: 1.540 cents
Commas: 1029/1024


[[Lp_tuning|POL2 generator]]: ~17/13 = 467.375
== 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]].  


Gencom: [2 21/16; 1029/1024]
[[Subgroup]]: 2.9.21


Gencom mapping: [<1 1 0 3|, <0 3/2 0 -1/2|]
[[Comma list]]: 1029/1024


Sval map: [<1 2 4|, <0 3 1|]
{{Mapping|legend=2| 1 2 4 | 0 3 1 }}


EDOs: 5, 13, 18, 41, 59, 77, 95
: sval mapping generators: ~2, ~21/16


[[Tp_tuning#T2 tuning|RMS error]]: 0.3202 cents
{{Mapping|legend=3| 1 1 0 3 | 0 3/2 0 -1/2 }}


=2.3.7/5.11/5.13/5=
: [[gencom]]: [2 21/16; 1029/1024]


==Historical==
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~2 = 1\1, ~21/16 = 467.375
Subgroup: 2.3.7/5.11/5.13/5


Commas: 364/363, 441/440, 1001/1000
{{Optimal ET sequence|legend=1| 5, 13, 18, 41, 59, 77, 95 }}


[[Lp_tuning|POL2 generator]]: ~21/20 = 83.016
[[Tp tuning #T2 tuning|RMS error]]: 0.3202 cents


[[Gencom]]: [2 21/20; 364/363 441/440 1001/1000]
==== 2.9.5.21 ====
''Lookalike temperament: [[Dual-fifth_temperaments#Dual-3_A-Team|Dual-3 A-Team]]''


Gencom map: [<1 2 -3/4 -3/4 1/4 5/4|, <0 -6 0 7 2 -9|]
Subgroup: 2.9.5.21


Sval map: [<1 2 0 1 2|, <0 -6 7 2 -9|]
[[Comma]] list: 81/80, 1029/1024


EDOs: 14, 29, 72, 101, 130, 159
Sval mapping: {{mapping| 1 2 0 4 | 0 3 6 1 }}


[[Tp_tuning#T2 tuning|RMS error]]: 0.2562 cents
Mapping generators: ~2, ~21/16


=2.3.7.11/5.13=
Optimal ([[Lp tuning|POL2]]) generator: 464.3865


==Hypnosis==
{{Optimal ET sequence|legend=1| 13, 18, 31, 44 }}
Subgroup: 2.3.7.11/5.13


Commas: 169/168, 540/539, 729/728
===== 2.9.5.21.11 =====
Subgroup: 2.9.5.21.11


Related temperament: [[Swetismic_temperaments#Hypnos-13-limit|hypnos]], [[Hemifamity_temperaments#Tricot-13-limit|tricot]]
Comma list: 81/80, 99/98, 385/384


[[Gencom]]: [2 13/9; 169/168, 540/539, 729/728]
Sval mapping: {{mapping| 1 2 0 4 5 | 0 3 6 1 -4 }}


Gencom map: [<1 0 -4 -3 4 0|, <0 3 13/2 11 -13/2 7|]
Gencom mapping: {{mapping| 1 1 0 3 5 | 0 3/2 6 -1/2 -4 }}


Sval map: [<1 0 -3 8 0|, <0 3 11 -13 7|]
: gencom: [2 21/16; 81/80 99/98 385/384]


EDOs: 17, 36, 118e, 125e, 161e, 197e
Optimal tuning (subgroup POTE): ~2 = 1\1, ~21/16 = 463.956


[[Tp_tuning#T2 tuning|RMS error]]: 0.5379 cents
{{Optimal ET sequence|legend=1| 5, 13, 31 }}


=2.5.7=
==== B-team ====
B-team (23 & 41) is every other step of [[rodan]].


==Didacus==
Subgroup: 2.9.15.21.33
Subgroup: 2.5.7


Commas: 3136/3125
Comma list: 245/243, 385/384, 441/440


Related temperaments: [[Chromatic_pairs#Roulette|roulette]]
Sval mapping: {{mapping| 1 2 0 4 7 | 0 3 10 1 -5 }}


[[Gencom]]: [2 28/25 3136/3125]
Optimal tuning (subgroup POTE): ~2 = 1\1, ~21/16 = 468.918


Gencom map: [<1 0 2 2|, <0 0 2 5|]
{{Optimal ET sequence|legend=1| 5, 13c, 18, 23, 41, 64, 87, 151 }}


Sval map: [<1 2 2|, <0 2 5|]
== 4.3.5 subgroup ==
=== Tetrahanson ===
{{Main| Tetrahanson }}


EDOs: 6, 19, 25, 31, 37, 99, 130, 161, 353
[[Subgroup]]: 4.3.5


[[Tp_tuning#T2 tuning|RMS error]]: 0.2138 cents
[[Comma list]]: 15625/15552


[[Category:Theory]]
{{Mapping|legend=2| 1 3 3 | 0 -6 -5 }}
[[Category:Temperament clan]]
[[Category:Subgroup]]


[[Category:Todo:add definition]]
: 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
 
[[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 subgroup (tetrominant) ====
[[Subgroup]]: 4.6.5.7
 
[[Comma list]]: [[36/35]] = {{monzo| 0 2 -1 -1 }}, [[64/63]] = {{monzo| 4 -2 0 -1 }}
 
{{Mapping|legend=2| 1 0 -4 4 | 0 1 4 -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)
⟨2399.3973, 193.8643]
TE Step Tunings (cents)
⟨25.21211, 47.81337]
TE Tuning Map (cents)
⟨2399.397, 3101.829, 2787.126]
TE Mistunings (cents)
⟨-0.603, -0.126, 0.812]
Complexity 1.369085
Adjusted Error 0.692892 cents
TE Error 0.268047 cents/octave
Unison Vector
[8, 1, -8⟩ (393216:390625)
 
Subsets
q99, q62, q37, q161, q136, q198, q25, q124, q74, q235
</pre>
 
==== 4.6.5.7 ====
<pre>
Reduced Mapping
4 6 5 7
[ ⟨ 1 0 1 1 ]
⟨ 0 16 2 5 ] ⟩
TE Generator Tunings (cents)
⟨2399.4195, 193.8654]
TE Step Tunings (cents)
⟨25.23883, 47.79592]
TE Tuning Map (cents)
⟨2399.420, 3101.846, 2787.150, 3368.747]
TE Mistunings (cents)
⟨-0.580, -0.109, 0.837, -0.079]
Complexity 1.192044
Adjusted Error 0.653313 cents
TE Error 0.232715 cents/octave
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
q99, q62, q37, q161, q136, q198, q25, q124, q74, q235
</pre>
 
==== 4.6.5.7.11 ====
<pre>
Reduced Mapping
4 6 5 7 11
[ ⟨ 1 0 1 1 1 ]
⟨ 0 16 2 5 9 ] ⟩
TE Generator Tunings (cents)
⟨2400.1097, 193.9498]
TE Step Tunings (cents)
⟨24.18752, 48.52491]
TE Tuning Map (cents)
⟨2400.110, 3103.196, 2788.009, 3369.859, 4145.658]
TE Mistunings (cents)
⟨0.110, 1.241, 1.696, 1.033, -5.660]
Complexity 1.068792
Adjusted Error 2.926965 cents
TE Error 0.846083 cents/octave
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
q37, q25, q62, q12, q74, q99, q87, q49r, q50r, q124
</pre>
 
==== 4.6.5.7.11.13 ====
 
<pre>
Reduced Mapping
4 6 5 7 11 13
[ ⟨ 1 0 1 1 1 0 ]
⟨ 0 16 2 5 9 23 ] ⟩
TE Generator Tunings (cents)
⟨2401.2305, 193.5378]
TE Step Tunings (cents)
⟨42.79107, 35.98524]
TE Tuning Map (cents)
⟨2401.230, 3096.606, 2788.306, 3368.920, 4143.071, 4451.371]
TE Mistunings (cents)
⟨1.230, -5.349, 1.992, 0.094, -8.247, 10.843]
Complexity 1.219191
Adjusted Error 6.699599 cents
TE Error 1.810487 cents/octave
Unison Vectors
[0, 1, -1, 0, 1, -1⟩ (66:65)
[-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, 0⟩ (385:384)
[-3, 0, 0, 1, 2, -1⟩ (847:832)
[1, 3, -1, 0, 0, -2⟩ (864:845)
[-1, 0, 3, -3, 1, 0⟩ (1375:1372)
 
Subsets
q25, q37f, q12f, q62, q50rf, q13rff, q49rff, q87, q74ff, q24rfff
</pre>
 
==== 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>
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>
 
==== 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=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 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=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 -1 0 1 | 0 4/3 1/3 -5/3 }}
: [[gencom]]: [2 6/5; 126/125]
 
[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 888.759
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 888.846
 
{{Optimal ET sequence|legend=1| 4, 15, 19, 23, 27 }}
 
[[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=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]
 
[[Optimal tuning]]s:
* [[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| 8, 15, 23, 54, 77, 100, 131* }}
: <nowiki/>* wart for 11/3
 
[[Tp tuning #T2 tuning|RMS error]]: 1.034 cents
 
==== Skateboard ====
{{See also| Chromatic pairs #Skateboard }}
 
Skateboard is related to [[thrasher]].
 
[[Subgroup]]: 2.5/3.7/3.11.13/9
 
[[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 -1 2 2 | 0 1 3 2 -2 }}
 
{{Mapping|legend=3| 1 -3/7 4/7 11/7 4 -6/7 | 0 0 -1 -3 -2 2 }}
: [[gencom]]: [2 6/5; 56/55 91/90 100/99]
 
[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 886.158
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 886.158
 
{{Optimal ET sequence|legend=1| 11, 15, 19, 23, 42d, 65d }}
 
[[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
 
[[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 0 0 0 0 0 | 0 -8/3 1/3 7/3 -1/2 1/2 }}
: [[gencom]]: [2 13/11; 275/273 847/845]
 
[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]] and [[POTE]]: ~2 = 1200.000, ~13/11 = 293.679
 
{{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]]: 0.6914 cents
 
==== Indium ====
{{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]]: 2.5/3.7/3.11/3
 
[[Comma list]]: [[3025/3024]] ({{monzo| -4 2 -1 2 }}), [[3125/3087]] ({{monzo| 0 5 -3 }})
 
{{Mapping|legend=2| 1 0 0 2 | 0 6 10 -1 }}
 
{{Mapping|legend=3| 1 -1/2 -1/2 -1/2 3/2 | 0 -15/4 9/4 25/4 -19/4 }}
: [[gencom]]: [2 12/11; 3025/3024 3125/3087]
 
[[Optimal tuning]]s:
* [[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, 33, 41, 49, 204*<sup>†</sup> }}
: <nowiki/>* wart for 7/3
: <sup>†</sup> wart for 11/3
 
[[Tp tuning #T2 tuning|RMS error]]: 0.7788 cents
 
==== Ammon ====
{{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]]: 2.5/3.7/3.11/3.13/3
 
[[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 3 5 3 4 | 0 -6 -10 -3 -5 }}
 
{{Mapping|legend=3| 1 -3 0 2 0 1 | 0 24/5 -6/5 -26/5 9/5 -1/5 }}
: [[gencom]]: [2 13/10; 121/120 169/168 275/273]
 
[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~13/10 = 453.121
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~13/10 = 453.242
 
{{Optimal ET sequence|legend=1| 8, 29, 37, 45 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 1.052 cents
 
=== Sentry ===
{{See also | Chromatic pairs #Sentry }}
 
Sentry, the {{nowrap| 3 & 5 }} temperament in the 2.5/3.9/7 subgroup, is related to [[sensi]].
 
[[Subgroup]]: 2.5/3.9/7
 
[[Comma list]]: [[245/243]] ({{monzo| 0 1 -2 }})
 
{{Mapping|legend=2| 1 0 0 | 0 2 1 }}
 
{{Mapping|legend=3| 1 0 0 0 | 0 0 2 -1 }}
: [[gencom]]: [2 9/7; 245/243]
 
[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]] and [[POTE]]: ~2 = 1200.000, ~9/7 = 440.902
 
{{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 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=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:
* [[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| 3, 4, 11, 15, 19, 34, 53, 87, 140 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.2444 cents
 
== 2.….7/3.… subgroups ==
=== 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]]: 2.5.7/3.11/3
 
[[Comma list]]: [[176/175]] ({{monzo| 4 -2 -1 1 }}), [[540/539]] ({{monzo| 2 1 -2 -1 }})
 
{{Mapping|legend=2| 1 0 1 3 | 0 -3 1 -5 }}
: mapping generators: ~2, ~7/6
 
{{Mapping|legend=3| 1 -4/3 3 -1/3 5/3 | 0 4/3 -3 7/3 -11/3 }}
: [[gencom]]: [2 7/6; 176/175 540/539]
 
[[Optimal tuning]]s:
* ([[Tp tuning|subgroup]] [[CTE]]): ~2 = 1200.000, ~7/6 = 270.455
* ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1200.000, ~7/6 = 270.093
 
{{Optimal ET sequence|legend=1| 9, 22, 31, 40, 191c*, 231c*, 271c*, 311c* }}
: <nowiki/>* wart for 7/3
 
[[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=3| 1 -1/2 0 1/2 3 | 0 -1/2 0 1/2 2 }}
: [[gencom]]: [2 7/6; 99/98]
 
[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~7/6 = 273.695
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~7/6 = 273.174
 
{{Optimal ET sequence|legend=1| 4, 9, 13, 22, 79 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 1.064 cents
 
== 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=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 &amp; 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
 
[[Comma list]]: [[250047/250000]]
 
{{Mapping|legend=2| 3 1 3 | 0 1 -1 }}
 
{{Mapping|legend=3| 3 10/9 -7/9 2/9 | 0 -2/3 -1/3 2/3 }}
: [[gencom]]: [63/50 10/9; 250047/250000]
 
[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~63/50 = 1\3, ~10/9 = 182.461
 
{{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]]: 0.00844 cents
 
=== Tridec ===
{{See also| Chromatic pairs #Tridec }}
{{See also| Non-over-1 temperament #Tridec }}
 
Tridec, the 5 &amp; 8 temperament in the 2.7/5.11/5.13/5 subgroup, extends [[#Petrtri]].
 
[[Subgroup]]: 2.7/5.11/5.13/5
 
[[Comma list]]: [[847/845]], [[1001/1000]]
 
{{Mapping|legend=2| 1 2 0 1 | 0 -4 3 1 }}
 
{{Mapping|legend=3| 1 0 -3/4 5/4 -3/4 1/4 | 0 0 0 -4 3 1 }}
: [[gencom]]: [2 13/10; 847/845 1001/1000]
 
[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~13/10 = 454.556
 
{{Optimal ET sequence|legend=1| 5, 8, 21, 29, 37, 66, 169, 235, 404c, 639c, 953bc }}
 
[[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 &amp; 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=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]]): ~2 = 1\1, ~13/10 = 455.012
 
{{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]]: 0.0749 cents
 
==== Bridgetown ====
{{See also| Chromatic pairs #Bridgetown }}
 
Bridgetown, the 5 &amp; 24 temperament in the 2.3.11/5.13/5 subgroup, is related to [[#Haumea|haumea]] and [[#Barbados|barbados]].
 
[[Subgroup]]: 2.3.11/5.13/5
 
[[Comma list]]: [[352/351]], [[676/675]]
 
{{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 }}
 
[[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 }}
 
[[Tp tuning #T2 tuning|RMS error]]: ???
 
==== 2.3.7.11/5.13 subgroup ====
[[Subgroup]]: 2.3.7.11/5.13
 
[[Comma list]]: 1029/1024, 1575/1573, 2080/2079
 
{{Mapping|legend=2| 3 0 10 5 0 | 0 3 -1 -1 7 }}
 
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~44/35 = 1\3, ~13/9 = 633.918
 
{{Optimal ET sequence|legend=1| 15, 21f, 36, 87, 123, 159 }}
 
[[Tp tuning #T2 tuning|RMS error]]: ???
 
==== 2.3.7.11/5.13.17 subgroup ====
[[Subgroup]]: 2.3.7.11/5.13.17
 
[[Comma list]]: 273/272, 833/832, 1575/1573, 2080/2079
 
{{Mapping|legend=2| 3 0 10 5 0 -2 | 0 3 -1 -1 7 9 }}
 
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~34/27 = 1\3, ~13/9 = 633.820
 
{{Optimal ET sequence|legend=1| 15, 21fg, 36, 123, 159 }}
 
[[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]]: 2.3.11/7
 
[[Comma list]]: {{monzo| 8 -5 }} (256/243)
 
{{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 &amp; 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
 
[[Comma list]]: 352/351, 364/363
 
{{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
 
== 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 &amp; 14 temperament in the 2.3.11.13/5 subgroup, extends [[neutral]] and [[barbados]].
 
[[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
 
{{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 ==
{{Main|Half-prime subgroup}}
 
=== Hemihemi ===
[[Subgroup]]: 3/2.5/2.7/2
 
[[Comma list]]: [[10976/10935]]
 
{{Mapping|legend=2| 1 2 3 | 0 3 1  }}
 
[[Optimal tuning]] (subgroup [[CTE]]): ~[[3/2]] = 1\[[1edf]], ~[[28/27]] = 60.909
 
[[Support]]ing [[ET]]s: *23, *12, *11, *35, *34, *10, *13, *47, *9[+5/2], *14[-5/2], *45, *25, *21[+5/2], *8[+5/2]
 
=== Halftone ===
{{Main| Halftone }}
 
[[Subgroup]]: 3/2.5/2.7/2
 
[[Comma list]]: 9604/9375
 
{{Mapping|legend=2| 1 3 4 | 0 -4 -5 }}
: sval mapping generators: ~3/2, ~15/14
 
[[Optimal tuning]] (subgroup [[CTE]]): ~3/2 = 1\1edf, ~15/14 = 128.783
 
Supporting ETs: *5, *6, *7[+5/2, +7/2], *9[-5/2, --7/2], *11, *16, *17[+5/2], *23[+5/2, +7/2], *21[-7/2], *27, *28[+5/2], *38, *43[-7/2], *49
: <nowiki />* wart for 3/2
 
==== 3/2.5/2.7/2.11/2 ====
[[Subgroup]]: 3/2.5/2.7/2.11/2
 
[[Comma list]]: 1232/1215, 27783/27500
 
{{Mapping|legend=2| 1 3 4 4 | 0 -4 -5 1 }}
: sval mapping generators: ~3/2, ~15/14
 
[[Optimal tuning]] (subgroup [[CTE]]): ~3/2 = 1\1edf, ~15/14 = 129.186
 
[[Support]]ing [[ET]]s: *11, *5, *16, *6, *27[-11/2], *21[-7/2], *38[-11/2], *43[-7/2, -11/2], *59[-7/2, -11/2], *70[-7/2, -11/2], *75[--7/2, -11/2]
: <nowiki />* wart for 3/2
 
==== 3/2.5/2.7/2.11/2.13/2 ====
[[Subgroup]]: 3/2.5/2.7/2.11/2.13/2
 
[[Comma list]]: 275/273, 1232/1215, 1323/1300
 
{{Mapping|legend=2| 1 3 4 4 5 | 0 -4 -5 1 -2 }}
 
[[Optimal tuning]] (subgroup [[CTE]]): ~3/2 = 1\1edf, ~15/14 = 129.381
 
[[Support]]ing [[ET]]s: *11, *5, *16, *6, *27[-11/2]
: <nowiki />* wart for 3/2
 
=== Semiwolf ===
[[Subgroup]]: 3/2.5/2.7/4
 
[[Comma list]]: 245/243
 
{{Mapping|legend=2| 1 1 2 | 0 2 -1 }}
 
: sval mapping generators: ~3/2, ~9/7
 
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~7/6 = 262.1728
 
[[Optimal ET sequence]]: [[3edf]], [[5edf]], [[8edf]]
 
==== Semilupine ====
[[Subgroup]]: 3/2.5/2.7/4.11/4
 
[[Comma list]]: 100/99, 245/243
 
{{Mapping|legend=2| 1 1 2 0 | 0 2 -1 4 }}
 
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~7/6 = 264.3771
 
[[Optimal ET sequence]]: [[8edf]], [[13edf]]
 
==== Hemilycan ====
[[Subgroup]]: 3/2.5/2.7/4.11/4
 
[[Comma list]]: 245/243, 441/440
 
{{Mapping|legend=2| 1 1 2 5 | 0 2 -1 -4 }}
 
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~7/6 = 261.5939
 
[[Optimal ET sequence]]: [[8edf]], [[11edf]]
 
== 3/2.5/4.… subgroups ==
=== Poseidon ===
'''This temperament will be subjected to renaming due to a conflict.'''
 
[[Subgroup]]: 3/2.5/4.11/8
 
[[Comma list]]: 121/120
 
{{Mapping|legend=2| 1 1 1 | 0 2 -1 }}]
 
: [[gencom]]: [3/2 12/11; 121/120]
 
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~3/2, ~12/11 = 158.29
 
{{Optimal ET sequence|legend=1|9, 5, 13, 22, 14, 31, 17, 6[+5/4], 23, 40, 35, 21[-5/4], 19[+5/4], 49}}
 
== Other 3/2-equave subgroups ==
=== Auk ===
[[Subgroup]]: 3/2.7.13
 
[[Comma list]]: 87808/85293
 
{{Mapping|legend=2| 1 0 -8 | 0 1 3 }}
 
: sval mapping generators: ~3/2, ~7
 
[[Optimal tuning]] (subgroup [[CTE]]): ~3/2 = 1\1edf, ~28/9 = 1950.859
 
Supporting ETs: *5, *6[+13], *7[-7, -13], *9, *11[+13], *13, *14, *17[-7, -13], *19[+13], *21[-7, -13], *22[-7], *23[+13], *25[-7, -13], *31[-7]
: <nowiki />* wart for 3/2
 
=== Doubleton ===
[[Subgroup]]: 3/2.7.13
 
[[Comma list]]: 1352/1323
 
{{Mapping|legend=2| 2 0 3 | 0 1 1 }}
 
: sval mapping generators: ~26/21, ~7
 
[[Optimal tuning]] (subgroup [[CTE]]): ~26/21 = 1\2edf, ~28/9 = 1971.772
 
Supporting ETs: *6, *10, *16, *14[-13], *8[+7], *22, *18[-13], *26, *24[-13], *28[+7], *20[+7], *36[-13], *12[+7, +13], *34[-13]
: <nowiki />* wart for 3/2
 
== 5/2-equave subgroups ==
=== Hyperion ===
[[Subgroup]]: 5/2.7.11
 
[[Comma list]]: {{monzo| 11 1 -5 }}
 
{{Mapping|legend=2| 1 4 3 | 0 -5 -1 }}
 
: [[gencom]]: [5/2 125/88; 341796875/329832448]
 
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~5/2 = 1586.3137, ~125/88 = 593.6668
 
Supporting ETs: *5[-7], *8, *19[+7], *21[-7], *27[+7], *29[-7], *35[+7], *43[+7], *37[-7], *51[+7, +11], *45[-7], *59[+7, +11]
: <nowiki />* wart for 5/2
 
= Related temperament collections =
* [[Dual-fifth temperaments]]
* [[Equalizer subgroup]] temperaments
* [[Substitute harmonic]] temperaments
 
[[Category:Subgroup temperaments| ]] <!-- main article -->
[[Category:Temperament collections]]
[[Category:Rank 2]]
{{Todo| review | cleanup }}
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