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A '''subgroup temperament''' is a regular temperament defined on a [[just intonation subgroup]] that is not a full ''p-limit'' group.  
{{Technical data page}}
A '''subgroup temperament''' is a regular temperament defined on a [[just intonation subgroup]] that is not a full ''p''-limit group.  


Below are some 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]].
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]]).


== No-elevens subgroup ==
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]].


=== [[Ultrapyth]] ===
= Composite subgroup temperaments =
: <small>''For full 13-limit extensions, see [[Archytas clan #Ultrapyth]].''</small>
== 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]].


Subgroup: 2.3.5.7.13
==== 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.


[[Comma list]]: 64/63, 91/90, 4394/4375
[[Subgroup]]: 2.3.11.19


[[Gencom]]: [2 4/3; 64/63 91/90 4394/4375]
{{Mapping|legend=2| 4 5 13 18 | 0 8 5 -6 }}


[[Gencom|Gencom mapping]]: [{{val|1 2 8 2 0 11}}, {{val|0 -1 -14 2 0 -18}}]
: sval mapping generators: ~6291456/5285401, ~25289/24576


[[Mapping|Sval mapping]]: [{{val|1 2 8 2 11}}, {{val|0 -1 -14 2 -18}}]
[[Optimal tuning]] ([[CTE]]): ~6291456/5285401 = 1\4, ~25289/24576 = 50.257


[[Tp tuning|POL2 generator]]: ~4/3 = 486.255
[[Support]]ing [[ET]]s: {{EDOs|24, 596, 620, 644, 668, 692, 716}}, ...


{{Val list|legend=1| 5, 32, 37 }}
==== 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.


[[Tp tuning #T2 tuning|RMS error]]: 2.318 cents
Subgroup: 2.3.35.11.19


=== Sensi (aka Sensation) ===
Sval mapping: {{mapping| 4 0 5 13 18 | 0 1 8 5 -6 }}
: <small>''For full 13-limit extensions, see [[Sensipent family]] or [[Sensi extensions]].''</small>


Subgroup: 2.3.5.7.13
: sval mapping generators: ~2240/1881, ~36/35


[[Comma list]]: 91/90, 126/125, 169/168
Optimal tuning (CTE): ~2240/1881 = 1\4, ~36/35 = 50.288


[[Gencom]]: [2 9/7; 91/90 126/125 169/168]
[[Support]]ing [[ET]]s: {{EDOs|24, 668}}, ...


[[Gencom|Gencom mapping]]: [{{val|1 6 8 11 0 10}}, {{val|0 -7 -9 -13 0 -10}}]
== 2.9.5.7 subgroup ==
See also [[Jubilismic clan #Antikythera|antikythera]] and [[Hemimean clan #Isra|isra]].


[[Mapping|Sval mapping]]: [{{val|1 6 8 11 10}}, {{val|0 -7 -9 -13 -10}}]
=== 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|POL2 generator]]: ~9/7 = 443.322
[[Subgroup]]: 2.9.5.7


{{Val list|legend=1| 19, 27, 46, 111de, 157de }}
[[Comma list]]: {{monzo| 28 -2 -19 8 }}, {{monzo| 9 -25 23 6 }}


[[Tp tuning #T2 tuning|RMS error]]: 1.321 cents
{{Mapping|legend=2| 1 9 6 13 | 0 -298 -188 -521 }}


=== Quartonic ===
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~531441/524288 = 23.4765
: <small>''For full 13-limit extensions, see [[Orwellismic temperaments #Quartonic]].''</small>


Subgroup: 2.3.5.7.13
{{Optimal ET sequence|legend=1| 460, 869, 1329 }}


[[Comma list]]: 169/168, 325/324, 640/637
[[Badness]]: 0.611


[[Gencom]]: [2 36/35; 169/168 325/324 640/637]
==== 2.9.5.7.11 ====
Subgroup: 2.9.5.7.11


[[Gencom|Gencom mapping]]: [{{val|1 2 3 3 0 4}}, {{val|0 -11 -18 -5 0 -8}}]
Comma list: {{monzo| -7 7 -3 2 -4 }}, {{monzo| 17 0 -13 1 3 }}, {{monzo| 11 -2 -6 7 -3 }}


[[Mapping|Sval mapping]]: [{{val|1 2 3 3 4}}, {{val|0 -11 -18 -5 -8}}]
Sval mapping: {{mapping| 1 9 6 13 16 | 0 -298 -188 -521 -641 }}


[[Tp tuning|POL2 generator]]: ~36/35 = 45.1821
Optimal tuning (CTE): ~2 = 1\1, ~531441/524288 = 23.4767


{{Val list|legend=1| 26, 27, 53 }}
{{Optimal ET sequence|legend=1| 460, 869e, 1329, 1789, 3118 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.7439 cents
Badness: 0.165


=== [[Septidiasemi]] ===
==== 2.9.5.7.11.13 ====
: <small>''For full 13- and 17-limit extensions, see [[Breedsmic temperaments #Septidiasemi]].''</small>
Subgroup: 2.9.5.7.11.13


Subgroup: 2.3.5.7.13
Comma list: 123201/123200, 1016064/1015625, 2250423/2249390, 2599051/2598156


[[Comma list]]: 2205/2197, 2401/2400, 4096/4095
Sval mapping: {{mapping| 0 9 6 13 16 10 | -298 -188 -521 -641 -322 }}


[[Gencom]]: [2 15/14; 2205/2197 2401/2400 4096/4095]
Optimal tuning (CTE): ~2 = 1\1, ~3575/3528 = 23.4767


[[Gencom|Gencom mapping]]: [{{val|1 -1 6 4 0 4}}, {{val|0 26 -37 -12 0 -3}}]
{{Optimal ET sequence|legend=1| 460, 869e, 1329, 1789, 3118 }}


[[Mapping|Sval mapping]]: [{{val|1 -1 6 4 4}}, {{val|0 26 -37 -12 -3}}]
Badness: 0.0564


[[Tp tuning|POL2 generator]]: ~15/14 = 119.297
=== 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.


{{Val list|legend=1| 10, 151, 161, 171 }}
Subgroup: 2.9.5.7.33.13.17


[[Tp tuning #T2 tuning|RMS error]]: 0.2002 cents
Comma list: 325/324, 375/374, 385/384, 595/594, 10985/10976


==== 2.3.5.7.13.17 ====
{{Mapping|legend=2|1 1 11 -16 13 -18 20|0 3 -12 26 -11 30 -22}}
Subgroup: 2.3.5.7.13.17


[[Comma list]]: 833/832, 1275/1274, 2025/2023, 2205/2197
Optimal tuning (CTE): ~2 = 1\1, 33/20 = 867.982


[[Gencom]]: [2 15/14; 833/832 1275/1274 2025/2023 2205/2197]
[[Support]]ing [[ET]]s: {{Optimal ET sequence|47, 65f, 112, 159, 206, 253}}


[[Gencom|Gencom mapping]]: [{{val|1 -1 6 4 0 4 2}}, {{val|0 26 -37 -12 0 -3 21}}]
=== Baldy ===
{{See also|Schismatic family #Garibaldi}}
{{See also|No-threes subgroup temperaments #Frostburn}}


[[Mapping|Sval mapping]]: [{{val|1 -1 6 4 4 2}}, {{val|0 26 -37 -12 -3 21}}]
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|POL2 generator]]: ~15/14 = 119.297
[[Subgroup]]: 2.9.5.7


{{Val list|legend=1| 10, 151, 161, 171 }}
[[Comma list]]: 225/224, 3125/3087


[[Tp tuning #T2 tuning|RMS error]]: 0.1867 cents
{{Mapping|legend=2| 1 3 3 4 | 0 1 -4 -7 }}


=== [[Pontiac]] ===
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 204.170
: <small>''For full 13- and 17-limit extensions, see [[Schismatic family #Pontiac]].''</small>


Subgroup: 2.3.5.7.13
{{Optimal ET sequence|legend=1| 6, 29, 35, 41, 47 }}


[[Comma list]]: 625/624 729/728 4096/4095
Related temperament: [[Schismatic family #Garibaldi|Garibaldi]]


[[Gencom]]: [2 4/3; 625/624 729/728 4096/4095]
==== 2.9.5.7.13 ====
{{See also|Chromatic pairs #Baldy}}


[[Gencom|Gencom mapping]]: [{{val|1 2 -1 19 0 -10}}, {{val|0 -1 8 -39 0 33}}]
Baldy is every other step of [[garibaldi]], without the mapping of prime 11. It can be described as the 6 &amp; 35 temperament.


[[Mapping|Sval mapping]]: [{{val|1 2 -1 19 -10}}, {{val|0 -1 8 -39 33}}]
[[Subgroup]]: 2.9.5.7.13


[[Tp tuning|POL2 generator]]: ~3/2 = 701.773
[[Comma list]]: [[225/224]], [[325/324]], [[640/637]]


{{Val list|legend=1| 53, 118, 171, 224, 395 }}
{{Mapping|legend=2| 1 0 15 25 -28 | 0 1 -4 -7 10 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.1525 cents
{{Mapping|legend=3| 1 3/2 3 4 0 2 | 0 1/2 -4 -7 0 10 }}


==== 2.3.5.7.13.17 ====
: [[gencom]]: [2 9/8; 225/224 325/324 640/637]
Subgroup: 2.3.5.7.13.17


[[Comma list]]: 625/624, 729/728, 1225/1224, 2880/2873
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 204.090


[[Gencom]]: [2 4/3; 625/624 729/728 1225/1224 2880/2873]
{{Optimal ET sequence|legend=1| 6, 11, 17, 23, 29, 35, 41, 47, 100, 147, 488cd, 635cd }}


[[Gencom|Gencom mapping]]: [{{val|1 2 -1 19 0 -10 29}}, {{val|0 -1 8 -39 0 33 -60}}]
[[Tp tuning #T2 tuning|RMS error]]: 0.5999 cents


[[Mapping|Sval mapping]]: [{{val|1 2 -1 19 -10 29}}, {{val|0 -1 8 -39 33 -60}}]
Related temperament: [[Schismatic family #Garibaldi|Cassandra]]


[[Tp tuning|POL2 generator]]: ~3/2 = 701.764
==== Baldanders ====
Baldanders results from taking every other generator of the andromeda, with mapping 11/8 to -9 whole tones.


{{Val list|legend=1| 53, 118, 171, 395, 566… }}
Subgroup: 2.9.5.7.11


[[Tp tuning #T2 tuning|RMS error]]: 0.1696 cents
Comma list: 100/99, 225/224, 245/242


=== [[Mitonic]] ===
{{Mapping|legend=2| 1 3 3 4 5 | 0 1 -4 -7 -9 }}
: <small>''For full 13- and 17-limit extensions, see [[Minortonic family #Mitonic]].''</small>


Subgroup: 2.3.5.7.13
Optimal tuning (POTE): ~2 = 1\1, ~9/8 = 204.743


[[Comma list]]: 4096/4095, 4375/4374, 13720/13689
{{Optimal ET sequence|legend=1| 6, 23de, 29, 35, 41 }}


[[Gencom]]: [2 10/9; 4096/4095 4375/4374 13720/13689]
Related temperament: [[Schismatic family #Garibaldi|Andromeda]]


[[Gencom|Gencom mapping]]: [{{val|1 -1 -3 6 0 11}}, {{val|0 17 35 -21 0 -48}}]
===== 2.9.5.7.11.13 =====
Subgroup: 2.9.5.7.11.13


[[Mapping|Sval mapping]]: [{{val|1 -1 -3 6 11}}, {{val|0 17 35 -21 -48}}]
Comma list: 100/99, 144/143, 225/224, 245/242


[[Tp tuning|POL2 generator]]: ~10/9 = 182.471
{{Mapping|legend=2| 1 3 3 4 5 2 | 0 1 -4 -7 -9 10 }}


{{Val list|legend=1| 46, 125, 171, 388 }}
Optimal tuning (POTE): ~2 = 1\1, ~9/8 = 204.414


[[Tp tuning #T2 tuning|RMS error]]: 0.1442 cents
{{Optimal ET sequence|legend=1| 6, 23def, 29f, 35, 41, 47 }}


==== 2.3.5.7.13.17 ====
== 2.3.25 subgroup ==
Subgroup: 2.3.5.7.13.17


[[Comma list]]: 833/832, 1225/1224, 1701/1700, 4096/4095
=== 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]]: [2 10/9; 833/832 1225/1224 1701/1700 4096/4095]
Subgroup: 2.3.25


[[Gencom|Gencom mapping]]: [{{val|1 -1 -3 6 0 11 5}}, {{val|0 17 35 -21 0 -48 -6}}]
Edo join: 17 & 12


[[Mapping|Sval mapping]]: [{{val|1 -1 -3 6 11 5}}, {{val|0 17 35 -21 -48 -6}}]
Comma list: [[2048/2025]]


[[Tp tuning|POL2 generator]]: ~10/9 = 182.471
{{Mapping|legend=2| 1 1 7| 0 1 -4}}


{{Val list|legend=1| 46, 125, 171, 388 }}
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 705.136


[[Tp tuning #T2 tuning|RMS error]]: 0.1341 cents
==== 2.3.23.25.41 subgroup ====
''See also: [[Reversed meantone]]''


=== [[Tertiaseptal]] ===
Edo join: 17 & 12
: <small>''For full 13- and 17-limit extensions, see [[Breedsmic temperaments #Tertiaseptal]].''</small>


Subgroup: 2.3.5.7.13
Comma list: 2048/2025, 576/575, 82/81


[[Comma list]]: 625/624, 2401/2400, 4096/4095
{{Mapping|legend=2| 1 1 1 7 3| 0 1 6 -4 4}}


[[Gencom]]: [2 117/112; 625/624 2401/2400 4096/4095]
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 705.264


[[Gencom|Gencom mapping]]: [{{val|1 3 2 3 0 1}}, {{val|0 -22 5 -3 0 42}}]
===== Sburb =====
This temperament sets the [[octave reduction|octave-reduced]] 413th harmonic (413/256, 827.998{{c}}) to the diminished seventh.


[[Mapping|Sval mapping]]: [{{val|1 3 2 3 1}}, {{val|0 -22 5 -3 42}}]
Subgroup: 2.3.7.23.25.41.59


[[Tp tuning|POL2 generator]]: ~117/112 = 77.173
Edo join: 17 & 12


{{Val list|legend=1| 31, 109, 140, 171, 311 }}
Comma list: 64/63, 225/224, 162/161, 82/81, 177/175


[[Tp tuning #T2 tuning|RMS error]]: 0.1383 cents
{{Mapping|legend=2| 1 1 4 1 7 3 10| 0 1 -2 6 -4 4 -7}}


==== 2.3.5.7.13.17 ====
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 706.387
Subgroup: 2.3.5.7.13.17


[[Comma list]]: 625/624, 833/832, 1225/1224, 4096/4095
== 2.9.5.11 subgroup ==
=== Glacial ===
{{See also| Chromatic pairs #Glacial }}


[[Gencom]]: [2 68/65; 625/624 833/832 1225/1224 4096/4095]
[[Subgroup]]: 2.9.5.11.13


[[Gencom|Gencom mapping]]: [{{val|1 3 2 3 0 1 1}}, {{val|0 -22 5 -3 0 42 48}}]
[[Comma list]]: 45/44, 65/64, 81/80


[[Mapping|Sval mapping]]: [{{val|1 3 2 3 1 1}}, {{val|0 -22 5 -3 42 48}}]
{{Mapping|legend=2| 1 0 -4 -6 10 | 0 1 2 3 -2 }}


[[Tp tuning|POL2 generator]]: ~68/65 = 77.177
{{Mapping|legend=3| 1 3/2 2 0 3 4 | 0 1/2 2 0 3 -2 }}


{{Val list|legend=1| 31, 109f, 140, 171, 311 }}
: [[gencom]]: [2 9/8; 45/44 65/64 81/80]


[[Tp tuning #T2 tuning|RMS error]]: 0.1367 cents
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 186.151


=== Oquatonic ===
{{Optimal ET sequence|legend=1| 6, 13, 45be, 58bce, 71bce, 84bce }}
: <small>''For full 13-limit extensions, see [[Horwell temperaments #Oquatonic]].''</small>


Subgroup: 2.3.5.7.13
[[Tp tuning #T2 tuning|RMS error]]: 2.887 cents


[[Comma list]]: 625/624, 4096/4095, 10985/10976
Music:
* ''[[Thundersnow]]'' - [[Sevish]] (2021)


[[Gencom]]: [40/39 105/104; 625/624 4096/4095 10985/10976]
== 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.


[[Gencom|Gencom mapping]]: [{{val|28 44 65 79 0 104}}, {{val|0 1 0 -1 0 -1}}]
Subgroup: 2.9.7


[[Mapping|Sval mapping]]: [{{val|28 44 65 79 104}}, {{val|0 1 0 -1 -1}}]
Comma basis: 44957696/43046721


[[Tp tuning|POL2 generator]]: ~105/104 = 16.4240
Sval mapping: [{{val|1 1 -3}}, {{val|0 3 8}}]


{{Val list|legend=1| 28, 56, 84, 140, 224, 364 }}
Optimal tuning (CTE): ~729/448 = 870.792


[[Tp tuning #T2 tuning|RMS error]]: 0.1347 cents
{{Optimal ET sequence|legend=1|7d, 11, 18d, 29, 40, 62}}, ...


== No-sevens subgroup ==
==== 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| 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=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
 
[[Comma list]]: 99/98, 9317/9216
 
{{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 }}


=== Porkypine ===
: [[gencom]]: [2 8/7; 64/63 99/98]
Related temperament: [[Porcupine]]


Subgroup: 2.3.5.11
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~9/8 = 216.9128
* [[POTE]]: ~2 = 1\1, ~9/8 = 214.3843


[[Comma list]]: 55/54, 100/99
{{Optimal ET sequence|legend=1| 5, 6, 11, 17, 28 }}


[[Gencom]]: [2 10/9; 55/54, 100/99]
[[Badness]]: 0.00233


[[Gencom|Gencom mapping]]: [{{val|1 2 3 0 4}}, {{val|0 -3 -5 0 -4}}]
=== Penta a.k.a. mechanism ===
Penta or mechanism is the 8 &amp; 11 temperament in the 2.9.7.11 subgroup.


[[Mapping|Sval mapping]]: [{{val|1 2 3 4}}, {{val|0 -3 -5 -4}}]
[[Subgroup]]: 2.9.7.11


[[Tp tuning|POL2 generator]]: ~11/10 = 164.078
[[Comma list]]: 896/891, 26411/26244


{{Val list|legend=1| 7, 15, 22, 37, 73cd, 95cd, 117bcd }}
{{Mapping|legend=2| 1 0 -1 6 | 0 5 6 -4 }}


[[Tp tuning #T2 tuning|RMS error]]: 2.287 cents
: sval mapping generators: ~2, ~14/9


=== Mohaha ===
{{Mapping|legend=3| 1 5/2 0 5 2 | 0 -5/2 0 -6 4 }}
Related temperament: [[mohajira]], [[Meantone family #Migration|migration]]


Subgroup: 2.3.5.11
: [[gencom]]: [2 9/7; 896/891 26411/26244]


[[Comma list]]: 81/80, 121/120
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~14/9 = 761.3782


[[Gencom]]: [2 11/9; 81/80 121/120]
{{Optimal ET sequence|legend=1| 8, 11, 30, 41, 52 }}


[[Gencom|Gencom mapping]]: [{{val|1 1 0 0 2}}, {{val|0 2 8 0 5}}]
[[Tp tuning #T2 tuning|RMS error]]: 0.4262 cents


[[Mapping|Sval mapping]]: [{{val|1 1 0 2}}, {{val|0 2 8 5}}]
[[Badness]]: 0.00439


[[Tp tuning|POL2 generator]]: ~11/9 = 348.094
Scales: [[penta5]], [[penta8]], [[penta11]], [[penta19]]


{{Val list|legend=1| 7, 10, 17, 24, 31, 55, 69d, 100d, 131bd }}
== 2.9.7.13.17 subgroup ==


[[Tp tuning #T2 tuning|RMS error]]: 1.392 cents
=== 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]].


; Music
[[Subgroup]]: 2.9.7.13.17
[http://micro.soonlabel.com/gene_ward_smith/Others/Stiltner/mohaha10ping2.mp3 Mohaha10ping2] by [[Billy Stiltner]]


=== Tetracot ===
[[Comma list]]: 729/728, 442/441, 833/832
: <small>''For full 11- and 13-limit extensions, see [[Tetracot family]].''</small>


Subgroup: 2.3.5.11
{{Mapping|legend=2| 1 1 1 -1 3| 0 6 5 13 3 }}


[[Comma list]]: 100/99, 243/242
[[Optimal tuning]] ([[CWE]]): ~2 = 1\1, ~9/7 = 433.836


[[Gencom]]: [2 10/9; 100/99 243/242]
Badness (Dirichlet): 0.142


[[Gencom|Gencom mapping]]: [{{val|1 1 1 0 2}}, {{val|0 4 9 0 10}}]
== 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.


[[Mapping|Sval mapping]]: [{{val|1 1 1 2}}, {{val|0 4 9 10}}]
[[Subgroup]]: 2.9.11


[[Tp tuning|POL2 generator]]: ~10/9 = 175.985
[[Comma list]]: [[1331/1296]]


{{Val list|legend=1| 7, 27d, 34, 41, 75d }}
{{Mapping|legend=2|1 1 2|0 3 2}}


[[Tp tuning #T2 tuning|RMS error]]: 1.182 cents
[[Optimal tuning]] ([[CTE]]): ~[[18/11]] = 870.060


==== 2.3.5.11.13 ====
{{Optimal ET sequence|legend=1|4, 7, 11, 18, 29, 76e}}
Subgroup: 2.3.5.11.13


[[Comma list]]: 100/99, 144/143, 243/242
=== Genius ===


[[Gencom]]: [2 10/9; 100/99 243/242]
Named after the genius in Roman religion, following the demon (daimon) in Greek mythology.


[[Gencom|Gencom mapping]]: [{{val|1 1 1 0 2 4}}, {{val|0 4 9 0 10 -2}}]
[[Subgroup]]: 2.9.11


[[Mapping|Sval mapping]]: [{{val|1 1 1 2 4}}, {{val|0 4 9 10 -2}}]
[[Comma list]]: [[131769/131072]]


[[Tp tuning|POL2 generator]]: ~10/9 = 176.196
{{Mapping|legend=2|1 1 4|0 4 -1}}


{{Val list|legend=1| 7, 27d, 34, 41, 75d }}
[[Optimal tuning]] ([[CTE]]): ~[[16/11]] = 650.863


[[Tp tuning #T2 tuning|RMS error]]: 1.140 cents
{{Optimal ET sequence|legend=1|9, 11, 24, 59, 83, 142, 225, 367}}[-11], 592[-11], 959[-9, --11], 1326[-9, --11]


=== Larry ===
== 2.9.15.7 subgroup ==
Subgroup: 2.3.5.11
=== Stacks (a.k.a. 2magic) ===
Stacks, the 11 &amp; 30 temperament in the 2.9.15.7.11.13 subgroup, is every other step of [[magic]].


[[Comma list]]: 243/242, 4000/3993
[[Subgroup]]: 2.9.15.7


Related temperaments: [[Gravity family|gravity]], [[Gravity family #Harry|harry]]
[[Comma list]]: 225/224, 245/243


[[Gencom]]: [2 40/27; 243/242 4000/3993]
{{Mapping|legend=2| 1 0 2 -1 | 0 5 3 6 }}


[[Gencom|Gencom mapping]]: [{{val|1 5 12 0 12}}, {{val|0 -6 -17 0 -15}}]
: sval mapping generators: ~2, ~14/9


[[Mapping|Sval mapping]]: [{{val|1 5 12 12}}, {{val|0 -6 -17 -15}}]
{{Mapping|legend=3| 1 5/2 5/2 5 | 0 -5/2 -1/2 -6 }}


[[Tp tuning|POL2 generator]]: ~40/27 = 683.166
: [[gencom]]: [2 9/7; 225/224 245/243]


{{Val list|legend=1| 7, 58, 65, 137, 202 }}
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~2 = 1\1, ~14/9 = 760.704


[[Tp tuning #T2 tuning|RMS error]]: 0.3025 cents
{{Optimal ET sequence|legend=1| 8, 11, 30, 41, 71, 93, 112c, 134c, 175c }}
 
[[Tp tuning #T2 tuning|RMS error]]: 1.074 cents


=== Twentcufo ===
==== 2.9.15.7.11 ====
: <small>''For full 11- and 13-limit extensions, see [[Hemimean clan #Undetrita]].''</small>
Subgroup: 2.9.15.7.11


Subgroup: 2.3.5.11
Comma list: 100/99, 225/224, 245/243


[[Comma list]]: 8019/8000, 14641/14580
Sval mapping: {{mapping| 1 0 2 -1 6 | 0 5 3 6 -4 }}


[[Gencom]]: [2 400/363; 8019/8000 14641/14580]
Gencom mapping: {{mapping| 1 5/2 5/2 5 2 | 0 -5/2 -1/2 -6 4 }}


[[Gencom|Gencom mapping]]: [{{val|1 0 -2 0 0}}, {{val|0 11 30 0 24}}]
: gencom: [2 9/7; 100/99 225/224 245/243]


[[Mapping|Sval mapping]]: [{{val|1 0 -2 0}}, {{val|0 11 30 24}}]
Optimal tuning (subgroup POTE): ~2 = 1\1, ~14/9 = 761.393


[[Tp tuning|POL2 generator]]: ~400/363 = 172.8796
Optimal ET sequence: {{Optimal ET sequence| 8, 11, 30, 41, 52, 93, 145, 342bce }}


{{Val list|legend=1| 7, 111, 118 }}
RMS error: 1.226 cents


[[Tp tuning #T2 tuning|RMS error]]: 0.2393 cents
==== 2.9.15.7.11.13 ====
Subgroup: 2.9.15.7.11.13


=== Emka ===
Comma list: 100/99, 105/104, 144/143, 196/195
: <small>''For full 11- and 13-limit extensions, see [[Hemimean clan #Emka]] or [[Horwell temperaments #Emkay]].''</small>


Subgroup: 2.3.5.11
Sval mapping: {{mapping| 1 0 2 -1 6 -2 | 0 5 3 6 -4 9 }}


[[Comma list]]: 4000/3993, 9453125/9437184
Gencom mapping: {{mapping| 1 5/2 5/2 5 2 7 | 0 -5/2 -1/2 -6 4 -9 }}


[[Gencom]]: [2 11/8; 4000/3993 9453125/9437184]
: gencom: [2 9/7; 100/99 105/104 144/143 196/195]


[[Gencom|Gencom mapping]]: [{{val|1 14 6 0 3}}, {{val|0 -27 -8 0 1}}]
Optimal tuning (subgroup POTE): ~2 = 1\1, ~14/9 = 761.023


[[Mapping|Sval mapping]]: [{{val|1 14 6 3}}, {{val|0 -27 -8 1}}]
Optimal ET sequence: {{Optimal ET sequence| 11, 30, 41, 153cdef, 194cdef, 235cdef }}


[[Tp tuning|POL2 generator]]: ~11/8 = 551.7778
RMS error: 1.540 cents


{{Val list|legend=1| 37, 50, 87, 137, 224 }}
== 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]].


[[Tp tuning #T2 tuning|RMS error]]: 0.1188 cents
[[Subgroup]]: 2.9.21


==== 2.3.5.11.13 ====
[[Comma list]]: 1029/1024
Subgroup: 2.3.5.11.13


[[Comma list]]: 625/624, 2200/2197, 4000/3993
{{Mapping|legend=2| 1 2 4 | 0 3 1 }}


[[Gencom]]: [2 11/8; 625/624 2200/2197 4000/3993]
: sval mapping generators: ~2, ~21/16


[[Gencom|Gencom mapping]]: [{{val|1 14 6 0 3 6}}, {{val|0 -27 -8 0 1 -5}}]
{{Mapping|legend=3| 1 1 0 3 | 0 3/2 0 -1/2 }}


[[Mapping|Sval mapping]]: [{{val|1 14 6 3 6}}, {{val|0 -27 -8 1 -5}}]
: [[gencom]]: [2 21/16; 1029/1024]


[[Tp tuning|POL2 generator]]: ~11/8 = 551.7753
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~2 = 1\1, ~21/16 = 467.375


{{Val list|legend=1| 37, 50, 87, 137, 224 }}
{{Optimal ET sequence|legend=1| 5, 13, 18, 41, 59, 77, 95 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.1250 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.3202 cents


=== Majvam ===
==== 2.9.5.21 ====
: <small>''For full 13- and 17-limit extensions, see [[Parkleiness temperaments #Majvamic]] or [[Cataharry temperaments #Majvamoid]].''</small>
''Lookalike temperament: [[Dual-fifth_temperaments#Dual-3_A-Team|Dual-3 A-Team]]''


Subgroup: 2.3.5.13
Subgroup: 2.9.5.21


[[Comma list]]: 676/675, 127401984/126953125
[[Comma]] list: 81/80, 1029/1024


[[Gencom]]: [2 125/96; 676/675 127401984/126953125]
Sval mapping: {{mapping| 1 2 0 4 | 0 3 6 1 }}


[[Gencom|Gencom mapping]]: [{{val|1 10 5 0 0 19}}, {{val|0 -22 -7 0 0 -40}}]
Mapping generators: ~2, ~21/16


[[Mapping|Sval mapping]]: [{{val|1 10 5 19}}, {{val|0 -22 -7 -40}}]
Optimal ([[Lp tuning|POL2]]) generator: 464.3865


[[Tp tuning|POL2 generator]]: ~125/96 = 458.988
{{Optimal ET sequence|legend=1| 13, 18, 31, 44 }}


{{Val list|legend=1| 34, 149, 183, 217, 400 }}
===== 2.9.5.21.11 =====
Subgroup: 2.9.5.21.11


[[Tp tuning #T2 tuning|RMS error]]: 0.1171 cents
Comma list: 81/80, 99/98, 385/384


==== 2.3.5.13.17 ====
Sval mapping: {{mapping| 1 2 0 4 5 | 0 3 6 1 -4 }}
Subgroup: 2.3.5.13.17


[[Comma list]]: 676/675, 2601/2600, 24576/24565
Gencom mapping: {{mapping| 1 1 0 3 5 | 0 3/2 6 -1/2 -4 }}


[[Gencom]]: [2 125/96; 676/675 2601/2600 24576/24565]
: gencom: [2 21/16; 81/80 99/98 385/384]


[[Gencom|Gencom mapping]]: [{{val|1 10 5 0 0 19 6}}, {{val|0 -22 -7 0 0 -40 -5}}]
Optimal tuning (subgroup POTE): ~2 = 1\1, ~21/16 = 463.956


[[Mapping|Sval mapping]]: [{{val|1 10 5 19 6}}, {{val|0 -22 -7 -40 -5}}]
{{Optimal ET sequence|legend=1| 5, 13, 31 }}


[[Tp tuning|POL2 generator]]: ~125/96 = 458.991
==== B-team ====
B-team (23 & 41) is every other step of [[rodan]].


{{Val list|legend=1| 34, 149, 183, 217, 400 }}
Subgroup: 2.9.15.21.33


[[Tp tuning #T2 tuning|RMS error]]: 0.1129 cents
Comma list: 245/243, 385/384, 441/440


=== Photia ===
Sval mapping: {{mapping| 1 2 0 4 7 | 0 3 10 1 -5 }}
Related temperament: [[Schismatic family|Schismic]]


Subgroup: 2.3.5.17
Optimal tuning (subgroup POTE): ~2 = 1\1, ~21/16 = 468.918


[[Comma list]]: 256/255, 1458/1445
{{Optimal ET sequence|legend=1| 5, 13c, 18, 23, 41, 64, 87, 151 }}


[[Gencom]]: [2 4/3; 256/255 1458/1445]
== 4.3.5 subgroup ==
=== Tetrahanson ===
{{Main| Tetrahanson }}


[[Gencom|Gencom mapping]]: [{{val|1 2 -1 0 0 0 7}}, {{val|0 -1 8 0 0 0 -7}}]
[[Subgroup]]: 4.3.5


[[Mapping|Sval mapping]]: [{{val|1 2 -1 7}}, {{val|0 -1 8 -7}}]
[[Comma list]]: 15625/15552


[[Tp tuning|POL2 generator]]: ~3/2 = 701.491
{{Mapping|legend=2| 1 3 3 | 0 -6 -5 }}


{{Val list|legend=1| 12, 41, 53, 65 }}
: Mapping generators: ~4, ~5/3


[[Tp tuning #T2 tuning|RMS error]]: 0.4842 cents
[[Optimal tuning]] ([[CTE]]): ~4 = 2\1, ~5/3 = 882.941


==== 2.3.5.17.19 ====
[[Support]]ing [[ET]]s: {{EDs|19, 106, 87, 68, 11, 8, 125, 49, 30, 27, 117, 46, 41b, 79|equave=4}}
Subgroup: 2.3.5.17.19


[[Comma list]]: 171/170, 256/255, 324/323
=== Tetrameantone ===
{{Main| Tetrameantone }}


[[Gencom]]: [2 4/3; 171/170 256/255 324/323]
[[Subgroup]]: 4.3.5


[[Gencom|Gencom mapping]]: [{{val|1 2 -1 0 0 0 7 3}}, {{val|0 -1 8 0 0 0 -7 3}}]
[[Comma list]]: 81/80


[[Mapping|Sval mapping]]: [{{val|1 2 -1 7 3}}, {{val|0 -1 8 -7 3}}]
{{Mapping|legend=2| 1 1 2 | 0 -1 -4 }}


[[Tp tuning|POL2 generator]]: ~3/2 = 701.470
: Mapping generators: ~4, ~4/3


{{Val list|legend=1| 12, 41, 53, 65 }}
[[Optimal tuning]] ([[POTE]]): 4 = 2400.0, ~4/3 = 503.761


[[Tp tuning #T2 tuning|RMS error]]: 0.5374 cents
[[Support]]ing [[ET]]s: {{EDs|5, 9, 14, 19, 24, 43, 62, 81, 100|equave=4}}


=== Nestoria ===
=== Tetramagic ===
Related temperament: [[Schismatic family|Schismic]]


Subgroup: 2.3.5.19
[[Subgroup]]: 4.3.5


[[Comma list]]: 361/360, 513/512
[[Comma list]]: 3125/3072


[[Gencom]]: [2 4/3; 361/360 513/512]
{{Mapping|legend=2| 1 0 1 | 0 5 1 }}


[[Gencom|Gencom mapping]]: [{{val|1 2 -1 0 0 0 0 3}}, {{val|0 -1 8 0 0 0 0 3}}]
: Mapping generators: ~4, ~5/4


[[Mapping|Sval mapping]]: [{{val|1 2 -1 3}}, {{val|0 -1 8 3}}]
[[Optimal tuning]] ([[POTE]]): 4 = 2400.0, ~5/4 = 380.059


[[Tp tuning|POL2 generator]]: ~3/2 = 701.746
[[Support]]ing [[ET]]s: {{EDs|6, 13, 19, 25, 38, 44, 63, 82|equave=4}}


{{Val list|legend=1| 12, 29, 41, 53, 118, 171 }}
=== Blacktetra ===


[[Tp tuning #T2 tuning|RMS error]]: 0.1763 cents
[[Subgroup]]: 4.3.5


== No-fives subgroup ==
[[Comma list]]: 256/243


=== [[Semaphore and Godzilla|Semaphore]] ===
{{Mapping|legend=2| 5 4 6 | 0 0 -1 }}
Subgroup: 2.3.7


[[Comma]]: 49/48
: Mapping generators: ~4, ~16/15


[[Gencom]]: [2 8/7; 49/48]
[[Optimal tuning]] ([[POTE]]): 1\5ed4 = 480.0, ~16/15 = 80.4062


[[Gencom|Gencom mapping]]: [{{val|1 2 0 3}}, {{val|0 -2 0 -1}}]
[[Support]]ing [[ET]]s: {{EDs|5, 10, 15, 20, 25, 30, 55, 85, 115|equave=4}}


[[Mapping|Sval mapping]]: [{{val|1 2 3}}, {{val|0 -2 -1}}]
== 4.6.5 subgroup ==
=== Meanquad ===
{{Main| Meanquad }}


[[Tp tuning|POL2 generator]]: ~7/6 = 250.385
[[Subgroup]]: 4.6.5


{{Val list|legend=1| 5, 14, 19, 24, 67c, 91c }}
[[Comma list]]: [[81/80]] = {{monzo| -4 4 -1 }}


[[Tp tuning# T2 tuning|RMS error]]: 2.523 cents
{{Mapping|legend=2| 1 0 -4| 0 1 4 }}


=== Bleu ===
: mapping generators: ~4, ~6
Subgroup: 2.3.7


[[Comma]]: 17496/16807
[[Optimal tuning]] (subgroup [[CTE]]): ~4 = 2\1, ~3/2 = 697.214


[[Gencom]]: [2 54/49; 17496/16807]
[[Support]]ing [[ET]]s: *7, *10, *11[-5], *13[+5], *17, *24, *27[+5], *31, *38, *41, *45, *52, *55, *69


[[Gencom|Gencom mapping]]: [{{val|1 1 0 2}}, {{val|0 5 0 7}}]
<nowiki />* Wart for 4


[[Mapping|Sval mapping]]: [{{val|1 1 2}}, {{val|0 5 7}}]
==== 4.6.5.7 subgroup (tetrominant) ====
[[Subgroup]]: 4.6.5.7


[[Tp tuning|POL2 generator]]: ~54/49 = 139.848
[[Comma list]]: [[36/35]] = {{monzo| 0 2 -1 -1 }}, [[64/63]] = {{monzo| 4 -2 0 -1 }}


{{Val list| 9, 17, 43, 60c }}
{{Mapping|legend=2| 1 0 -4 4 | 0 1 4 -2 }}


[[Tp tuning #T2 tuning|RMS error]]: 1.917 cents
[[Optimal tuning]] (subgroup [[CTE]]): ~4 = 2\1, ~3/2 = 699.622


==== 2.3.7.11 ====
[[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]
Subgroup: 2.3.7.11


[[Comma list]]: 99/98, 864/847
<nowiki />* Wart for 4


[[Gencom]]: [2 12/11; 99/98 864/847]
=== 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.


[[Gencom|Gencom mapping]]: [{{val|1 1 0 2 3}}, {{val|0 5 0 7 4}}]
Fourwar is named after the closely related [[hemiwar]] temperament.


[[Mapping|Sval mapping]]: [{{val|1 1 2 3}}, {{val|0 5 7 4}}]
{{Todo|inline=1|cleanup}}


[[Tp tuning|POL2 generator]]: ~12/11 = 140.005
<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)


{{Val list| 9, 17, 43, 60c }}
Subsets
q99, q62, q37, q161, q136, q198, q25, q124, q74, q235
</pre>


[[Tp tuning #T2 tuning|RMS error]]: 1.829 cents
==== 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)


==== 2.3.7.11.13 ====
Subsets
Subgroup: 2.3.7.11.13
q99, q62, q37, q161, q136, q198, q25, q124, q74, q235
</pre>


[[Comma list]]: 78/77, 99/98, 144/143
==== 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)


[[Gencom]]: [2 12/11; 78/77 99/98 144/143]
Subsets
q37, q25, q62, q12, q74, q99, q87, q49r, q50r, q124
</pre>


[[Gencom|Gencom mapping]]: [{{val|1 1 0 2 3 3}}, {{val|0 5 0 7 4 6}}]
==== 4.6.5.7.11.13 ====


[[Mapping|Sval mapping]]: [{{val|1 1 2 3 3}}, {{val|0 5 7 4 6}}]
<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)


[[Tp tuning|POL2 generator]]: ~12/11 = 139.990
Subsets
q25, q37f, q12f, q62, q50rf, q13rff, q49rff, q87, q74ff, q24rfff
</pre>


{{Val list|legend=1| 17, 43, 60c }}
==== 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)


[[Tp tuning #T2 tuning|RMS error]]: 1.752 cents
Subsets
q25, q12f, q37f, q13rffg, q50rf, q62, q49rffg, q24rfffg, q38rreffg, q74ffg
</pre>


=== Archy ===
==== 4.6.5.7.11.13.17.19 ====
Archy (properly pronounced "arky", after the Greek theorist Archytas) can be thought of as "no-fives [[dominant]]" or "no-fives [[superpyth]]". The name comes from the fact that it tempers out 64/63, the Archytas comma.
<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)


Subgroup: 2.3.7
Subsets
q25, q12fh, q37f, q13rffgh, q50rf, q62, q49rffgh, q24rfffghh, q38rreffgh, q74ffgh
</pre>


[[Comma]]: 64/63
==== 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)


[[Gencom]]: [2 3/2; 64/63]
Subsets
q25i, q12fhi, q37f, q13rffghii, q62, q50rfii, q49rffghii, q24rfffghhiii, q74ffghi, q38rreffghiii
</pre>


[[Gencom|Gencom mapping]]: [{{val|1 1 0 4}}, {{val|0 1 0 -2}}]
== 4.9.25 subgroup ==
=== Meansquared ===
[[Subgroup]]: 4.9.25


[[Mapping|Sval mapping]]: [{{val|1 2 2}}, {{val|0 -1 2}}]
[[Comma list]]: [[6561/6400]]


[[Tp tuning|POL2 generator]]: ~3/2 = 709.321
{{Mapping|legend=2| 1 3 4 | 0 1 4 }}


{{Val list|legend=1| 5, 12, 17, 22, 27, 137bc }}
Mapping generators: ~4, ~9/64


[[Tp tuning #T2 tuning|RMS error]]: 1.856 cents
[[Optimal tuning]] ([[CTE]]): ~4 = 2\1, ~9/4 = 1394.429


==== Supra ====
[[Support]]ing [[ET]]s: 12, 7, 19, 5, 31, 26, 17[+25], 43, 9[-25], 33[-25], 45, 29[+25], 8[+25], 22[+25]
Subgroup: 2.3.7.11


[[Comma list]]: 64/63, 99/98
== 4.9.49 subgroup ==
=== Archsquared ===
[[Subgroup]]: 4.9.49


[[Gencom]]: [2 3/2; 64/63 99/98]
[[Comma list]]: 4096/3969


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


[[Mapping|Sval mapping]]: [{{val|1 0 6 13}}, {{val|0 1 -2 -6}}]
Mapping generators: ~4, ~9/64


[[Tp tuning|POL2 generator]]: ~3/2 = 707.192
[[Optimal tuning]] ([[CTE]]): ~9/4 = 1419.190


{{Val list|legend=1| 5, 12, 17, 39c, 56c }}
[[Support]]ing [[ET]]s: 5, 17, 22, 12, 7, 27, 32, 8, 39[+49], 29[+49], 9[+49], 19[+49], 37, 49


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


===== Supraphon =====
[[Subgroup]]: 8.9.7
Subgroup: 2.3.7.11.13


[[Comma list]]: 64/63, 78/77, 99/98
[[Comma list]]: 64/63


[[Gencom]]: [2 3/2; 64/63 78/77 99/98]
{{Mapping|legend=2| 1 0 2 | 0 1 -1 }}


[[Gencom|Gencom mapping]]: [{{val|1 1 0 4 7 9}}, {{val|0 1 0 -2 -6 -9}}]
: sval mapping generators: ~8, ~9


[[Mapping|Sval mapping]]: [{{val|1 0 6 13 18}}, {{val|0 1 -2 -6 -9}}]
: [[gencom]]: [8 9/8; 64/63]


[[Tp tuning|POL2 generator]]: ~3/2 = 706.137
[[Optimal tuning]] ([[CTE]]): ~9/8 = 219.1898


{{Val list|legend=1| 5e, 12e, 17, 22, 39c, 56c }}
[[Optimal ET sequence]]: {{val| 16 17 15 }}, {{val| 33 35 31 }}, {{val| 148 … }}, {{val| 181 … }}, {{val| 214 … }}, {{val| 247 … }}


[[Tp tuning #T2 tuning|RMS error]]: 2.095 cents
[[Badness]]: 0.0215 × 10<sup>-3</sup>


==== Suhajira ====
= Fractional subgroup temperaments =
Subgroup: 2.3.7.11
== 2.5/3.… subgroups ==
=== Magicaltet ===
{{See also| Chromatic pairs #Magicaltet }}


[[Comma list]]: 64/63, 243/242
Magicaltet is related to [[keemic]], [[superkleismic]], and [[magic]]. The tonic and the first three generator steps make a [[magical seventh chord]], hence the name.


[[Gencom]]: [2 11/9; 64/63 243/242]
[[Subgroup]]: 2.5/3.7.11


[[Gencom|Gencom mapping]]: [{{val|1 1 0 4 2}}, {{val|0 2 0 -4 5}}]
[[Comma list]]: 100/99 ({{monzo| 2 2 0 -1 }}), 385/384 ({{monzo| -7 1 1 1 }})


[[Mapping|Sval mapping]]: [{{val|1 1 4 2}}, {{val|0 2 -4 5}}]
{{Mapping|legend=2| 1 0 5 2 | 0 1 -3 2 }}
: mapping generators: ~2, ~5/3


[[Tp tuning|POL2 generator]]: ~11/9 = 353.958
{{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]


{{Val list|legend=1| 7, 10, 17, 44d, 61cd, 78cd }}
[[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


[[Tp tuning #T2 tuning|RMS error]]: 1.968 cents
{{Optimal ET sequence|legend=1| 4, 7, 11, 15, 26, 67, 93* }}
: <nowiki/>* wart for 5/3


===== 2.3.7.11.13 =====
[[Tp tuning #T2 tuning|RMS error]]: 1.206 cents
Subgroup: 2.3.7.11.13


[[Comma list]]: 64/63, 78/77, 144/143
=== Starlingtet ===
{{See also | Chromatic pairs #Starlingtet }}


[[Gencom]]: [2 11/9; 64/63 78/77 144/143]
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.


[[Gencom|Gencom mapping]]: [{{val|1 1 0 4 2 4}}, {{val|0 2 0 -4 5 -1}}]
[[Subgroup]]: 2.5/3.7/3


[[Mapping|Sval mapping]]: [{{val|1 1 4 2 4}}, {{val|0 2 -4 5 -1}}]
[[Comma list]]: [[126/125]] ({{monzo| 1 -3 1 }})


[[Tp tuning|POL2 generator]]: ~11/9 = 353.775
{{Mapping|legend=2| 1 0 -1 | 0 1 3 }}


{{Val list|legend=1| 7, 10, 17, 44d, 61cd, 78cd }}
: mapping generators: ~2, ~5/3


[[Tp tuning #T2 tuning|RMS error]]: 1.953 cents
{{Mapping|legend=3| 1 -1 0 1 | 0 4/3 1/3 -5/3 }}
: [[gencom]]: [2 6/5; 126/125]


=== Skwares ===
[[Optimal tuning]]s:
Related temperament: [[Meantone family #Squares|squares]]
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 888.759
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 888.846


Subgroup: 2.3.7
{{Optimal ET sequence|legend=1| 4, 15, 19, 23, 27 }}


[[Comma]]: 19683/19208
[[Tp tuning #T2 tuning|RMS error]]: 0.8398 cents


[[Gencom]]: [2 9/7; 19683/19208]
==== Greeley ====
{{See also| Chromatic pairs #Greeley }}


[[Gencom|Gencom mapping]]: [{{val|1 3 6}}, {{val|0 -4 -9}}]
Greeley is related to [[opossum]] as well as to [[nusecond]].


[[Mapping|Sval mapping]]: [{{val|1 3 6}}, {{val|0 -4 -9}}]
[[Subgroup]]: 2.5/3.7/3.11/3


[[Tp tuning|POL2 generator]]: ~9/7 = 425.365
[[Comma list]]: 121/120 ({{monzo| -3 -1 0 2 }}), 126/125 ({{monzo| 1 -3 1 }})


{{Val list|legend=1| 14, 17, 31, 48, 79, 189b, 268bc, 347bc }}
{{Mapping|legend=2| 1 1 2 2 | 0 -2 -6 -1 }}


[[Tp tuning #T2 tuning|RMS error]]: 1.149 cents
{{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]


==== 2.3.7.11 ====
[[Optimal tuning]]s:
Subgroup: 2.3.7.11
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~11/10 = 155.696
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~11/10 = 155.776


[[Comma list]]: 99/98, 243/242
{{Optimal ET sequence|legend=1| 8, 15, 23, 54, 77, 100, 131* }}
: <nowiki/>* wart for 11/3


[[Gencom]]: [2 9/7; 99/98 243/242]
[[Tp tuning #T2 tuning|RMS error]]: 1.034 cents


[[Gencom|Gencom mapping]]: [{{val|1 3 0 6 7}}, {{val|0 -4 0 -9 -10}}]
==== Skateboard ====
{{See also| Chromatic pairs #Skateboard }}


[[Mapping|Sval mapping]]: [{{val|1 3 6 7}}, {{val|0 -4 -9 -10}}]
Skateboard is related to [[thrasher]].


[[Tp tuning|POL2 generator]]: ~9/7 = 425.244
[[Subgroup]]: 2.5/3.7/3.11.13/9


{{Val list|legend=1| 14, 17, 31, 48, 79, 127, 206bcd }}
[[Comma list]]: 56/55 ({{monzo| 3 -1 1 -1 }}), 91/90 ({{monzo| -1 -1 1 0 1 }}), 100/99 ({{monzo| 2 2 0 -1 }})


[[Tp tuning #T2 tuning|RMS error]]: 1.099 cents
{{Mapping|legend=2| 1 0 -1 2 2 | 0 1 3 2 -2 }}


==== 2.3.7.11.13 ====
{{Mapping|legend=3| 1 -3/7 4/7 11/7 4 -6/7 | 0 0 -1 -3 -2 2 }}
Subgroup: 2.3.7.11.13
: [[gencom]]: [2 6/5; 56/55 91/90 100/99]


[[Comma list]]: 78/77, 99/98, 243/242
[[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


[[Gencom]]: [2 9/7; 78/77, 99/98, 243/242]
{{Optimal ET sequence|legend=1| 11, 15, 19, 23, 42d, 65d }}


[[Gencom|Gencom mapping]]: [{{val|1 3 0 6 7 9}}, {{val|0 -4 0 -9 -10 -15}}]
[[Tp tuning #T2 tuning|RMS error]]: 2.396 cents


[[Mapping|Sval mapping]]: [{{val|1 3 6 7 9}}, {{val|0 -4 -9 -10 -15}}]
=== Gariberttet ===
Gariberttet is the 2.5/3.7/3 [[Subgroup temperament families, relationships, and genes|altergene]] of [[sirius]].


[[Tp tuning|POL2 generator]]: ~9/7 = 424.457
==== Gariberttet (2.5/3.7/3.13/11 subgroup) ====
{{See also | Chromatic pairs #Gariberttet }}


{{Val list|legend=1| 17, 48e, 65de, 82c, 147ce }}
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]].


[[Tp tuning #T2 tuning|RMS error]]: 1.769 cents
[[Subgroup]]: 2.5/3.7/3.13/11


==== Skwairs ====
[[Comma list]]: [[275/273]] ({{monzo| 0 2 -1 -1 }}), [[847/845]] ({{monzo| 0 -1 1 -2 }})
Subgroup: 2.3.7.11.13


[[Comma list]]: 99/98, 144/143, 243/242
{{Mapping|legend=2| 1 0 0 0 | 0 3 5 1 }}


[[Gencom]]: [2 9/7; 99/98, 144/143, 243/242]
{{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]


[[Gencom|Gencom mapping]]: [{{val|1 3 0 6 7 3}}, {{val|0 -4 0 -9 -10 2}}]
[[Optimal tuning]]s:  
* [[Tp tuning|subgroup]] [[CTE]] and [[POTE]]: ~2 = 1200.000, ~13/11 = 293.679


[[Mapping|Sval mapping]]: [{{val|1 3 6 7 3}}, {{val|0 -4 -9 -10 2}}]
{{Optimal ET sequence|legend=1| 29, 33, 37, 41, 45, 49, 78, 94, 143* }}
: <nowiki/>* wart for 13/11


[[Tp tuning|POL2 generator]]: ~9/7 = 424.702
[[Tp tuning #T2 tuning|RMS error]]: 0.6914 cents


{{Val list|legend=1| 14, 17, 31 }}
==== Indium ====
{{See also | Chromatic pairs #Indium }}


[[Tp tuning #T2 tuning|RMS error]]: 1.290 cents
Indium can be described as the {{nowrap| 8 & 33 }} temperament in the 2.5/3.7/3.11/3 subgroup.  


=== Leapfrog ===
[[Subgroup]]: 2.5/3.7/3.11/3
{{see also|Gentle region}}


Subgroup: 2.3.7
[[Comma list]]: [[3025/3024]] ({{monzo| -4 2 -1 2 }}), [[3125/3087]] ({{monzo| 0 5 -3 }})


[[Comma list]]: 14680064/14348907
{{Mapping|legend=2| 1 0 0 2 | 0 6 10 -1 }}


[[Gencom]]: [2 3/2; 14680064/14348907]
{{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]


[[Gencom|Gencom mapping]]: [{{val|1 1 0 -6}}, {{val|0 1 0 15}}]
[[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


[[Mapping|Sval mapping]]: [{{val|1 0 -21}}, {{val|0 1 15}}]
{{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|POL2 generator]]: ~3/2 = 704.721 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.7788 cents


{{Val list|legend=1| 17, 46, 63 }}
==== Ammon ====
{{See also| Chromatic pairs #Ammon }}


[[Tp tuning #T2 tuning|RMS error]]: 0.6202 cents
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.


==== 2.3.7.11 ====
[[Subgroup]]: 2.5/3.7/3.11/3.13/3
Subgroup: 2.3.7.11


[[Comma list]]: 896/891, 1331/1323
[[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 }})


[[Gencom]]: [2 3/2; 896/891 1331/1323]
{{Mapping|legend=2| 1 3 5 3 4 | 0 -6 -10 -3 -5 }}


[[Gencom|Gencom mapping]]: [{{val|1 1 0 -6 -3}}, {{val|0 1 0 15 11}}]
{{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]


[[Mapping|Sval mapping]]: [{{val|1 0 -21 -14}}, {{val|0 1 15 11}}]
[[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


[[Tp tuning|POL2 generator]]: ~3/2 = 704.753 cents
{{Optimal ET sequence|legend=1| 8, 29, 37, 45 }}


{{Val list|legend=1| 17, 46, 63 }}
[[Tp tuning #T2 tuning|RMS error]]: 1.052 cents


[[Tp tuning #T2 tuning|RMS error]]: 0.6047 cents
=== Sentry ===
{{See also | Chromatic pairs #Sentry }}


==== 2.3.7.11.13 ====
Sentry, the {{nowrap| 3 & 5 }} temperament in the 2.5/3.9/7 subgroup, is related to [[sensi]].  
Subgroup: 2.3.7.11.13


[[Comma list]]: 169/168, 352/351, 364/363
[[Subgroup]]: 2.5/3.9/7


[[Gencom]]: [2 3/2; 169/169 352/351 364/363]
[[Comma list]]: [[245/243]] ({{monzo| 0 1 -2 }})


[[Gencom|Gencom mapping]]: [{{val|1 1 0 -6 -3 -1}}, {{val|0 1 0 15 11 8}}]
{{Mapping|legend=2| 1 0 0 | 0 2 1 }}


[[Mapping|Sval mapping]]: [{{val|1 0 -21 -14 -9}}, {{val|0 1 15 11 8}}]
{{Mapping|legend=3| 1 0 0 0 | 0 0 2 -1 }}
: [[gencom]]: [2 9/7; 245/243]


[[Tp tuning|POL2 generator]]: ~3/2 = 704.745 cents
[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]] and [[POTE]]: ~2 = 1200.000, ~9/7 = 440.902


{{Val list|legend=1| 17, 46, 63 }}
{{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]]: 0.7541 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.7105 cents


; Music
=== Marveltwintri ===
* Suite for Harpsichord in A Locrian, tuning: Eb-G# in [[46edo|46EDO]] by [[User:IlL|IlL]] (in progress):
{{See also| Chromatic pairs #Marveltwintri }}
** I. Prelude
** II. Allemande
** III. Courante
** [[:File:Locrian Suite Sarabande.mp3|IV. Sarabande]] ([[:File:Locrian Suite Sarabande Score.pdf|score]], [[:File:Locrian Suite Sarabande 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]]


=== Lee ===
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: 2.3.7


[[Comma]]: 177147/175616
[[Subgroup]]: 2.5/3.13/9


[[Gencom]]: [2 81/56; 177147/175616]
[[Comma list]]: [[325/324]] ({{monzo| -2 2 1 }})


[[Gencom|Gencom mapping]]: [{{val|1 0 0 -3}}, {{val|0 3 0 11}}]
{{Mapping|legend=2| 1 0 2 | 0 1 -2 }}


[[Mapping|Sval mapping]]: [{{val|1 0 -3}}, {{val|0 3 11}}]
{{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]


[[Tp tuning|POL2 generator]]: ~81/56 = 633.525
[[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


{{Val list|legend=1| 17, 36, 89, 125, 161, 358, 519b }}
{{Optimal ET sequence|legend=1| 3, 4, 11, 15, 19, 34, 53, 87, 140 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.3519 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.2444 cents


=== [[Slendric]] ===
== 2.….7/3.… subgroups ==
Subgroup: 2.3.7
=== Guanyintet ===
{{See also | Chromatic pairs #Guanyintet }}


[[Comma]]: 1029/1024
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.


[[Gencom]]: [2 8/7; 1029/1024]
[[Subgroup]]: 2.5.7/3.11/3


[[Gencom|Gencom mapping]]: [{{val|1 1 0 3}}, {{val|0 3 0 -1}}]
[[Comma list]]: [[176/175]] ({{monzo| 4 -2 -1 1 }}), [[540/539]] ({{monzo| 2 1 -2 -1 }})


[[Mapping|Sval mapping]]: [{{val|1 1 3}}, {{val|0 3 -1}}]
{{Mapping|legend=2| 1 0 1 3 | 0 -3 1 -5 }}
: mapping generators: ~2, ~7/6


[[Tp tuning|POL2 generator]]: ~8/7 = 233.688
{{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]


{{Val list|legend=1| 5, 21, 26, 31, 36, 77, 113, 190 }}
[[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


[[Tp tuning #T2 tuning|RMS error]]: 0.3202 cents
{{Optimal ET sequence|legend=1| 9, 22, 31, 40, 191c*, 231c*, 271c*, 311c* }}
: <nowiki/>* wart for 7/3


==== [[Gamelismic clan #Baladic|Baladic]] ====
[[Tp tuning #T2 tuning|RMS error]]: 0.6028 cents
Subgroup: 2.3.7.13


[[Comma list]]: 169/168, 1029/1024
==== 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.


[[Gencom]]: [91/64 8/7; 169/168 1029/1024]
[[Subgroup]]: 2.5.7/3.11/3.13


[[Mapping|Sval mapping]]: [{{val|2 2 6 7}}, {{val|0 3 -1 1}}]
[[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 }})


[[Tp tuning|POL2 generator]]: ~8/7 = 233.6044
{{Mapping|legend=2| 1 0 1 3 1 | 0 -3 1 -5 12 }}
: mapping generators: ~2, ~12/7


{{Val list|legend=1| 10, 26, 36, 154…, 190…, 226…, 262… }}
[[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


[[Tp tuning #T2 tuning|RMS error]]: 0.5452 cents
{{Optimal ET sequence|legend=1| 9, 22, 31, 40, 71, 111, 151, 262c*}} <small> using subgroup TE </small>
: <nowiki/>* wart for 7/3


==== 2.3.7.13.17 ====
Badness (Sintel): 0.329
Subgroup: 2.3.7.13.17


[[Comma list]]: 169/168, 273/272, 289/288
==== Laz ====
{{See also | Chromatic pairs #Laz }}


[[Gencom]]: [17/12 8/7; 169/168 273/272 289/288]
Laz is related to [[avalokita]] as well as to [[winston]].


[[Mapping|Sval mapping]]: [{{val|2 2 6 7 7}}, {{val|0 3 -1 1 3}}]
[[Subgroup]]: 2.5.7/3.11/3.13/3


[[Tp tuning|POL2 generator]]: ~8/7 = 233.6155
[[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 }}


{{Val list|legend=1| 10, 26, 36, 154…, 190…, 226… }}
{{Mapping|legend=2| 1 0 2 -2 6 | 0 3 -1 5 -5 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.5073 cents
{{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]


=== Hemif ===
[[Optimal tuning]]s:
Related temperament: [[Breedsmic temperaments #Hemififths|hemififths]], namo
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~12/7 = 930.598
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~12/7 = 930.700


Subgroup: 2.3.7
{{Optimal ET sequence|legend=1| 9, 31, 40, 49, 156c*†, 205c*† }}
: <nowiki/>* wart for 7/3
: † wart for 11/3


[[Comma]]: 1605632/1594323
[[Tp tuning #T2 tuning|RMS error]]: 0.8790 cents


[[Gencom]]: [2 2187/1792; 1605632/1594323]
=== Kryptonite ===
{{See also| Chromatic pairs #Kryptonite }}


[[Gencom|Gencom mapping]]: [{{val|1 1 0 -1}}, {{val|0 2 0 13}}]
Kryptonite is related to [[krypton]].


[[Mapping|Sval mapping]]: [{{val|1 1 -1}}, {{val|0 2 13}}]
[[Subgroup]]: 2.5.7/3.11/3.13/3


[[Tp tuning|POL2 generator]]: ~2187/1792 = 351.485
[[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 }})


{{Val list|legend=1| 7, 17, 41, 58, 99 }}
{{Mapping|legend=2| 1 2 1 2 2 | 0 3 2 -1 1 }}
: mapping generators: ~2, ~13/12


[[Tp tuning #T2 tuning|RMS error]]: 0.2344 cents
{{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]


==== 2.3.7.11 ====
[[Optimal tuning]]s:
Subgroup: 2.3.7.11
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~13/12 = 130.945
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~13/12 = 132.428


[[Comma list]]: 243/242, 896/891
{{Optimal ET sequence|legend=1| 1, …, 8, 9 }}


[[Gencom]]: [2 11/9; 243/242 896/891]
[[Tp tuning #T2 tuning|RMS error]]: 2.545 cents


[[Gencom|Gencom mapping]]: [{{val|1 1 0 -1 2}}, {{val|0 2 0 13 5}}]
=== Kiribati ===
{{See also| Chromatic pairs #Kiribati }}


[[Mapping|Sval mapping]]: [{{val|1 1 -1 2}}, {{val|0 2 13 5}}]
Kiribati is related to [[nakika]] as well as to [[octacot]].


[[Tp tuning|POL2 generator]]: ~11/9 = 351.535
[[Subgroup]]: 2.9/5.7/3.11/9


{{Val list|legend=1| 7, 17, 41, 58, 99d }}
[[Comma list]]: 100/99 ({{monzo| 2 -2 0 -1 }}), 245/242 ({{monzo| -1 -1 2 -2 }})


[[Tp tuning #T2 tuning|RMS error]]: 0.6108 cents
{{Mapping|legend=2| 1 1 1 0 | 0 -2 3 4 }}
: mapping generators: ~2, ~21/20


==== 2.3.7.11.13 ====
{{Mapping|legend=3| 1 1/10 -4/5 11/10 1/5 | 0 -3/2 -1 3/2 1 }}
Subgroup: 2.3.7.11.13
: [[gencom]]: [2 21/20; 100/99 245/242]


[[Comma list]]: 144/143, 243/242, 364/363
[[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


[[Gencom]]: [2 11/9; 144/143 243/242 364/363]
{{Optimal ET sequence|legend=1| 13, 14, 27, 41 }}


[[Gencom|Gencom mapping]]: [{{val|1 1 0 -1 2 4}}, {{val|0 2 0 13 5 -1}}]
[[Tp tuning #T2 tuning|RMS error]]: 1.245 cents


[[Mapping|Sval mapping]]: [{{val|1 1 -1 2 4}}, {{val|0 2 13 5 -1}}]
=== Mothwelltri ===
{{See also| Chromatic pairs #Mothwelltri }}


[[Tp tuning|POL2 generator]]: ~11/9 = 351.691
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.  


{{Val list|legend=1| 7, 10, 17, 24, 41, 58 }}
[[Subgroup]]: 2.7/3.11


[[Tp tuning #T2 tuning|RMS error]]: 0.7167 cents
[[Comma list]]: [[99/98]] ({{monzo| -1 -2 1 }})


=== Ennea ===
{{Mapping|legend=2| 1 0 1 | 0 1 2 }}
Subgroup: 2.3.7.11
: mapping generators: ~2, ~7/3


[[Comma list]]: 41503/41472, 43923/43904
{{Mapping|legend=3| 1 -1/2 0 1/2 3 | 0 -1/2 0 1/2 2 }}
: [[gencom]]: [2 7/6; 99/98]


[[Gencom]]: [2 99/98; 41503/41472, 43923/43904]
[[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


[[Gencom|Gencom mapping]]: [{{val|1 14/9 0 25/9 31/9}}, {{val|0 2 0 2 1}}]
{{Optimal ET sequence|legend=1| 4, 9, 13, 22, 79 }}


[[Mapping|Sval mapping]]: [{{val|9 0 11 24}}, {{val|0 2 2 1}}]
[[Tp tuning #T2 tuning|RMS error]]: 1.064 cents


[[Tp tuning|POL2 generator]]: ~99/98 = 17.6258
== 2.….9/7.… subgroups ==
=== Marveltri ===
{{See also| Chromatic pairs #Marveltri }}


{{Val list|legend=1| 54, 63, 72, 135, 342, 477, 1089, 1566 }}
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.


[[Tp tuning #T2 tuning|RMS error]]: 0.0383 cents
[[Subgroup]]: 2.5.9/7


=== Parapyth (Rank 3) ===
[[Comma list]]: 225/224 ({{monzo| -5 2 1 }})
{{see also| Pentacircle temperaments #Parapyth }}


Subgroup: 2.3.7.11
{{Mapping|legend=2| 1 0 5 | 0 1 -2 }}
: mapping generators: ~2, ~5


[[Comma list]]: 896/891
{{Mapping|legend=3| 1 2 0 -1 | 0 -4/5 1 2/5 }}
: [[gencom]]: [2 5; 225/224]


[[Gencom]]: [2 3/2 28/27; 896/891]
[[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


[[Gencom]] [[mapping]]: [{{val| 1 1 0 1 4 }}, {{val| 0 1 0 3 -1 }}, {{val| 0 0 0 1 1 }}]
{{Optimal ET sequence|legend=1| 3, 13, 16, 19, 22, 25, 72, 97, 122, 269c* }}
: <nowiki/>* wart for 9/7


[[Sval]] [[mapping]]: [{{val| 1 0 0 7 }}, {{val| 0 1 0 -4 }}, {{val| 0 0 1 1 }}]
[[Tp tuning #T2 tuning|RMS error]]: 0.4801 cents


[[Tp tuning|POL2 tuning]]: ~3 = 1903.834, ~7 = 3369.872
==== Sulis ====
Sulis is related to [[minerva]] and [[würschmidt]].  


{{Val list|legend=1| 17, 36, 41, 58, 63, 104 }}
[[Subgroup]]: 2.5.9/7.11/9


[[Tp tuning #T2 tuning|RMS error]]: 0.4149 cents
[[Comma list]]: 99/98 ({{monzo| -1 0 2 1 }}), 176/175 ({{monzo| 4 -2 1 1 }})


==== 2.3.7.11.13 ====
{{Mapping|legend=2| 1 0 5 -9 | 0 1 -2 4 }}]
Subgroup: 2.3.7.11.13


[[Comma list]]: 352/351, 364/363
[[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


The gencom below gives [[Margo Schulter]]'s favored basis
{{Optimal ET sequence|legend=1| 3, …, 22, 25, 28, 31, 59 }}


[[Gencom]]: [2 3/2 28/27; 352/351 364/363]
[[Tp tuning #T2 tuning|RMS error]]: 1.074 cents


[[Gencom]] [[mapping]]: [{{val| 1 1 0 1 4 6 }}, {{val| 0 1 0 3 -1 -4 }}, {{val| 0 0 0 1 1 1 }}]
== 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.


[[Sval]] [[mapping]]: [{{val| 1 0 0 7 12 }}, {{val| 0 1 0 -4 -7 }}, {{val| 0 0 1 1 1 }}]
[[Subgroup]]: 2.3.7/5


[[Tp tuning|POL2 tuning]]: ~3 = 1903.856, ~7 = 3369.907
[[Comma list]]: [[50/49]]


{{Val list|legend=1| 17, 41, 46, 58, 87, 104 }}
{{Mapping|legend=2| 2 3 1 | 0 1 0 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.3789 cents
[[Optimal tuning]] (inharmonic [[TE]]): ~1\2 = 590.998, ~[[10/7]]-1\2 = 128.962


=== Neutral ===
[[Support]]ing [[ET]]s: {{EDOs|4, 6, 8, 10, 18, 28, 46, 64, 110}}
Neutral can be thought of as the 2.3.11 version of either [[mohajira]] or [[maqamic]], as well as suhajira and ringo. Among other things, it is the temperament optimizing the [[neutral tetrad]].


Subgroup: 2.3.11
=== Argentic ===
Argentic is the 2.3.7/5 subgroup temperament tempering out [[5120/5103]].


[[Comma]]: 243/242
[[Subgroup]]: 2.3.7/5


[[Gencom]]: [2 11/9; 243/242]
[[Comma list]]: [[5120/5103]] = {{monzo| 10 -6 -1 }}


[[Gencom|Gencom mapping]]: [{{val|1 1 0 0 2}}, {{val|0 2 0 0 5}}]
{{Mapping|legend=2| 1 0 10 | 0 1 -6 }}
: mapping generators: ~2, ~3


[[Mapping|Sval mapping]]: [{{val|1 1 2}}, {{val|0 2 5}}]
[[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


[[Tp tuning|POL2 generator]]: ~11/9 = 350.525
{{Optimal ET sequence|legend=1| 12, 29, 41, 70, 321, 391, 461, 531, 601 }}
<small> based on subgroup TE </small>


{{Val list|legend=1| 7, 10, 17, 24, 41, 65, 89, 202, 291, 380 }}
Badness (Sintel): 0.119


[[Tp tuning #T2 tuning|RMS error]]: 0.3021 cents
==== Edson (2.3.7/5.11/5.13/5 subgroup) ====
{{See also| Chromatic pairs #Edson }}


[[neutraltet7|Seven note albitonic scale]]
Edson is related to [[pele]] and [[andromeda]].


[[neutraltet10|Ten note chromatic scale]]
[[Subgroup]]: 2.3.7/5.11/5.13/5


[[neutraltet17|Seventeen note mega chromatic scale]]
[[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 }}


==== Namo ====
{{Mapping|legend=2| 1 0 10 17 22 | 0 1 -6 -10 -13 }}
Subgroup: 2.3.11.13
: mapping generators: ~2, ~3


[[Comma list]]: 144/143, 243/242
{{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]


[[Gencom]]: [2 11/9; 144/143 243/242]
[[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


[[Gencom|Gencom mapping]]: [{{val|1 1 0 0 2 4}}, {{val|0 2 0 0 5 -1}}]
{{Optimal ET sequence|legend=1| 12, 17, 29 }}


[[Mapping|Sval mapping]]: [{{val|1 1 2 4}}, {{val|0 2 5 -1}}]
[[Tp tuning #T2 tuning|RMS error]]: 0.5102 cents


[[Tp tuning|POL2 generator]]: ~11/9 = 351.488
==== Haumea ====
{{See also| Chromatic pairs #Haumea }}


{{Val list|legend=1| 7, 10, 17, 24, 41 }}
Related temperaments include [[#Bridgetown|bridgetown]], [[namaka]], [[hemigari]], [[#Barbados|barbados]], and [[parizekmic]].


[[Tp tuning #T2 tuning|RMS error]]: 0.7038 cents
[[Subgroup]]: 2.3.7/5.11/5.13/5


== No-threes subgroup ==
[[Comma list]]: [[352/351]], [[676/675]], [[847/845]]
=== Llywelyn ===
Subgroup: 2.5.7


[[Comma]]: 4194304/4117715
{{Mapping|legend=2| 1 0 10 -6 -1 | 0 2 -12 9 3 }}


[[Gencom]]: [2 8/7; 4194304/4117715]
{{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]


[[Gencom|Gencom mapping]]: [{{val|1 0 1 3}}, {{val|0 0 7 -1}}]
[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~15/13 = 248.491


[[Mapping|Sval mapping]]: [{{val|1 1 3}}, {{val|0 7 -1}}]
{{Optimal ET sequence|legend=1| 24, 29, 111, 140, 169, 198, 565d, 763bd, 961bd }}


[[Tp tuning|POL2 generator]]: ~8/7 = 226.910
[[Tp tuning #T2 tuning|RMS error]]: 0.2668 cents


{{Val list|legend=1| 16, 37 }}
=== Historical ===
{{distinguish|Historical temperaments}}
{{distinguish|History (temperament)}}, which is the rank-3 version of this temperament in the full 13-limit.


[[Tp tuning #T2 tuning|RMS error]]: 0.5391 cents
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.


=== Didacus ===
[[Subgroup]]: 2.3.7/5.11/5.13/5
{{See also| Hemimean clan #Didacus }}


Related temperaments: [[Chromatic pairs #Roulette|roulette]], [[Gamelismic clan #Hemithirds|hemithirds]]
[[Comma list]]: 364/363, 441/440, 1001/1000


Subgroup: 2.5.7
{{Mapping|legend=2| 1 2 0 1 2 | 0 -6 7 2 -9 }}


[[Comma]]: 3136/3125
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~21/20 = 83.016


[[Gencom]]: [2 28/25; 3136/3125]
{{Optimal ET sequence|legend=1| 14, 29, 72, 101, 130, 159 }}


[[Gencom|Gencom map]]: [{{val|1 0 2 2}}, {{val|0 0 2 5}}]
[[Tp tuning #T2 tuning|RMS error]]: 0.2562 cents


[[Mapping|Sval mapping]]: [{{val|1 2 2}}, {{val|0 2 5}}]
=== Terrain ===
{{Redirect|Terrain|the scale|Terrain (scale)}}
{{See also| Chromatic pairs #Terrain }}


[[Tp tuning|POL2 generator]]: ~28/25 = 93.772
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.


{{Val list|legend=1| 6, 19, 25, 31, 37, 99, 130, 161, 353 }}
[[Subgroup]]: 2.7/5.9/5


[[Tp tuning #T2 tuning|RMS error]]: 0.2138 cents
[[Comma list]]: [[250047/250000]]


=== Rainy ===
{{Mapping|legend=2| 3 1 3 | 0 1 -1 }}


Three generators make an [[8/7]]; five generators make a [[5/4]]. This is the no-threes version of [[tertiaseptal]].
{{Mapping|legend=3| 3 10/9 -7/9 2/9 | 0 -2/3 -1/3 2/3 }}
: [[gencom]]: [63/50 10/9; 250047/250000]


Subgroup: 2.5.7
[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~63/50 = 1\3, ~10/9 = 182.461


[[Comma]]: [[2100875/2097152]]
{{Optimal ET sequence|legend=1| 6, 21, 27, 33, 105, 138, 171, 1848, 2019, 2190, 2361, 2532, 2703, 2874, 3045, 3216, 3387, 3558 }}


[[Gencom]]: [2 256/245; 2100875/2097152]
[[Tp tuning #T2 tuning|RMS error]]: 0.00844 cents


[[Gencom|Gencom mapping]]: [{{val|1 0 2 3 }}, {{val|0 0 5 -3}}]
=== Tridec ===
{{See also| Chromatic pairs #Tridec }}
{{See also| Non-over-1 temperament #Tridec }}


[[Mapping|Sval mapping]]: [{{val|1 2 3 }}, {{val|0 5 -3}}]
Tridec, the 5 &amp; 8 temperament in the 2.7/5.11/5.13/5 subgroup, extends [[#Petrtri]].


[[Tp tuning|POL2 generator]]: ~256/245 = 77.205
[[Subgroup]]: 2.7/5.11/5.13/5


{{Val list|legend=1| 31, 47, 78, 109, 140, 171, 202, 233 }}
[[Comma list]]: [[847/845]], [[1001/1000]]


[[Tp tuning #T2 tuning|RMS error]]: 0.0586 cents
{{Mapping|legend=2| 1 2 0 1 | 0 -4 3 1 }}


=== Mercy ===
{{Mapping|legend=3| 1 0 -3/4 5/4 -3/4 1/4 | 0 0 0 -4 3 1 }}
{{See also| Quince clan #Mercy }}
: [[gencom]]: [2 13/10; 847/845 1001/1000]


Two generators make an [[8/7]]; seven generators make an [[8/5]]. Mercy can be thought of as a way to conceptualize the 2.5.7.13.17.19 subgroup of [[31edo]], and is the no-threes or elevens version of [[miracle]].
[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~13/10 = 454.556


Subgroup: 2.5.7
{{Optimal ET sequence|legend=1| 5, 8, 21, 29, 37, 66, 169, 235, 404c, 639c, 953bc }}


[[Comma list]]: 823543/819200
[[Tp tuning #T2 tuning|RMS error]]: 0.1613 cents


[[Gencom]]: [2 2744/2560; 823543/819200]
==== Naiadec ====
[[Subgroup]]: 2.7/5.11/5.13/5.17/5


[[Gencom|Gencom mapping]]: [{{val| 1 0 3 3 }}, {{val| 0 0 -7 -2 }}]
[[Comma list]]: [[170/169]], [[221/220]], [[847/845]]


[[Sval]] [[mapping]]: [{{val| 1 3 3 }}, {{val| 0 -7 -2 }}]
{{Mapping|legend=2| 1 2 0 1 1 | 0 -4 3 1 2 }}


[[Tp tuning|POL2 generator]]: ~343/320 = 116.291
{{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]


{{Val list|legend=1| 10, 21, 31, 134, 165, 196, 227, 485d, 712d, 1197dd }}
[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~13/10 = 454.882


==== 2.5.7.13 ====
{{Optimal ET sequence|legend=1| 5, 8, 21, 29, 95<sup>t</sup>, 124<sup>t</sup> }}
Subgroup: 2.5.7.13
: <sup>t</sup> wart for 17/5


[[Comma list]]: 343/338, 640/637
[[Tp tuning #T2 tuning|RMS error]]: 0.7521 cents


[[Gencom]]: [2 14/13; 343/338 640/637]
== 2.….11/5.… subgroups ==
=== Petrtri ===
{{See also| Chromatic pairs #Petrtri }}
{{See also| 5L 3s/Temperaments #Petrtri }}


[[Gencom|Gencom mapping]]: [{{val|1 0 3 3 4}}, {{val|0 0 -7 -2 -3}}]
Petrtri can be described as 3 &amp; 5 temperament in the 2.11/5.13/5 subgroup.


[[Mapping|Sval mapping]]: [{{val|1 3 3 4}}, {{val|0 -7 -2 -3}}]
[[Subgroup]]: 2.11/5.13/5


[[Tp tuning|POL2 generator]]: ~14/13 = 116.094
[[Comma list]]: [[2200/2197]]


{{Val list|legend=1| 10, 21, 31}}
{{Mapping|legend=2| 1 0 1| 0 3 1 }}


==== 2.5.7.13.17 ====
{{Mapping|legend=3| 1 0 -1/3 0 -1/3 2/3 | 0 0 -4/3 0 5/3 -1/3 }}
Subgroup: 2.5.7.13.17
: [[gencom]]: [2 13/10; 2200/2197]


[[Comma list]]: 170/169, 224/221, 640/637
[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~13/10 = 455.012


[[Gencom]]: [2 14/13; 170/169 224/221 640/637]
{{Optimal ET sequence|legend=1| 21, 29, 153, 182, 211, 240, 269, 298, 327, 356, 385, 509, 741c, 1126c }}


[[Gencom|Gencom mapping]]: [{{val|1 0 3 3 4 4}}, {{val|0 0 -7 -2 -3 1}}]
[[Tp tuning #T2 tuning|RMS error]]: 0.0749 cents


[[Mapping|Sval mapping]]: [{{val|1 3 3 4 4}}, {{val|0 -7 -2 -3 1}}]
==== Bridgetown ====
{{See also| Chromatic pairs #Bridgetown }}


[[Tp tuning|POL2 generator]]: ~14/13 = 115.769
Bridgetown, the 5 &amp; 24 temperament in the 2.3.11/5.13/5 subgroup, is related to [[#Haumea|haumea]] and [[#Barbados|barbados]].  


{{Val list|legend=1| 10, 21, 31}}
[[Subgroup]]: 2.3.11/5.13/5


==== 2.5.7.13.17.19 ====
[[Comma list]]: [[352/351]], [[676/675]]
Subgroup: 2.5.7.13.17.19


[[Comma list]]: 170/169, 343/338, 640/637, 16384/16055
{{Mapping|legend=2| 1 0 -6 -1 | 0 2 9 3 }}


[[Gencom]]: [2 14/13; 170/169 343/338 640/637 16384/16055]
{{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]


[[Gencom|Gencom mapping]]: [{{val|1 0 3 3 4 4 3}}, {{val|0 0 -7 -2 -3 1 13}}]
[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~15/13 = 248.399


[[Mapping|Sval mapping]]: [{{val|1 3 3 4 4 3}}, {{val|0 -7 -2 -3 1 13}}]
{{Optimal ET sequence|legend=1| 5, 9, 14, 19, 24, 29, 169, 198, 227, 256, 285, 314 }}


[[Tp tuning|POL2 generator]]: ~14/13 = 115.716
[[Tp tuning #T2 tuning|RMS error]]: 0.2513 cents


{{Val list|legend=1| 10, 21, 31, 52f}}
=== Hypnosis ===
Related temperaments: [[Swetismic temperaments #Hypnos|hypnos]], [[Alphatricot family #Alphatricot|alphatricot]]


== 2.9.7.11 subgroup ==
[[Subgroup]]: 2.3.7.11/5.13


=== Machine ===
[[Comma list]]: 169/168, 540/539, 729/728
Subgroup: 2.9.7.11


Commas: 64/63, 99/98
{{Mapping|legend=2| 1 0 -3 8 0 | 0 3 11 -13 7 }}


[[Gencom]]: [2 8/7; 64/63 99/98]
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~13/9 = 633.518


[[Gencom|Gencom mapping]]: [{{val|1 3/2 0 3 4}}, {{val|0 1/2 0 -1 -3}}]
{{Optimal ET sequence|legend=1| 17, 36, 118f, 125f, 161f, 197f }}


[[Mapping|Sval mapping]]: [{{val|1 0 6 13}}, {{val|0 1 -1 -3}}]
[[Tp tuning #T2 tuning|RMS error]]: 0.5379 cents


[[Tp tuning|POL2 generator]]: ~8/7 = 214.384
=== Trisect ===
Trisect divides every Pythagorean interval into three, and is the much more accurate subgroup restriction of [[Augmented family #Trisected|trisected]].


{{Val list|legend=1| 5, 6, 11, 17, 28 }}
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]].


[[Tp tuning #T2 tuning|RMS error]]: 1.977 cents
[[Subgroup]]: 2.3.7.11/5


=== Mechanism ===
[[Comma list]]: 1029/1024, 4000/3993
Subgroup: 2.9.7.11


[[Comma list]]: 896/891, 26411/26244
{{Mapping|legend=2| 3 0 10 5 | 0 3 -1 -1 }}


[[Gencom]]: [2 9/7; 896/891 26411/26244]
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~44/35 = 1\3, ~13/9 = 633.742


[[Gencom|Gencom mapping]]: [{{val|1 5/2 0 5 2}}, {{val|0 -5/2 0 -6 4}}]
{{Optimal ET sequence|legend=1| 15, 21, 36, 123, 159, 195, 231 }}


[[Mapping|Sval mapping]]: [{{val|1 5 5 2}}, {{val|0 -5 -6 4}}]
[[Tp tuning #T2 tuning|RMS error]]: ???


[[Tp tuning|POL2 generator]]: ~9/7 = 438.465
==== 2.3.7.11/5.13 subgroup ====
[[Subgroup]]: 2.3.7.11/5.13


{{Val list|legend=1| 8, 11, 30, 41, 52 }}
[[Comma list]]: 1029/1024, 1575/1573, 2080/2079


[[Tp tuning #T2 tuning|RMS error]]: 0.4262 cents
{{Mapping|legend=2| 3 0 10 5 0 | 0 3 -1 -1 7 }}


=== Apparatus ===
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~44/35 = 1\3, ~13/9 = 633.918
Subgroup: 2.9.7.11


[[Comma list]]: 41503/41472, 322102/321489
{{Optimal ET sequence|legend=1| 15, 21f, 36, 87, 123, 159 }}


[[Gencom]]: [2 77/72; 41503/41472 322102/321489]
[[Tp tuning #T2 tuning|RMS error]]: ???


[[Gencom|Gencom mapping]]: [{{val|1 5/2 0 3 5}}, {{val|0 -19/2 0 -2 -16}}]
==== 2.3.7.11/5.13.17 subgroup ====
[[Subgroup]]: 2.3.7.11/5.13.17


[[Mapping|Sval mapping]]: [{{val|1 5 3 5}}, {{val|0 -19 -2 -16}}]
[[Comma list]]: 273/272, 833/832, 1575/1573, 2080/2079


[[Tp tuning|POL2 generator]]: ~77/72 = 115.570
{{Mapping|legend=2| 3 0 10 5 0 -2 | 0 3 -1 -1 7 9 }}


{{Val list|legend=1| 10, 21, 31, 52, 83, 135, 353, 488, 623 }}
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~34/27 = 1\3, ~13/9 = 633.820


[[Tp tuning #T2 tuning|RMS error]]: 0.0673 cents
{{Optimal ET sequence|legend=1| 15, 21fg, 36, 123, 159 }}


== 2.9.15.7 subgroup ==
[[Tp tuning #T2 tuning|RMS error]]: ???


=== Stacks (aka 2magic) ===
===== Trisector =====
Subgroup: 2.9.15.7
[[Subgroup]]: 2.3.7.11/5.13.17.19


[[Comma list]]: 225/224, 245/243
[[Comma list]]: 210/209, 273/272, 286/285, 595/594, 2080/2079


[[Gencom]]: [2 9/7; 225/224 245/243]
{{Mapping|legend=2| 3 0 10 5 0 -2 8 | 0 3 -1 -1 7 9 3 }}


[[Gencom|Gencom mapping]]: [{{val|1 5/2 5/2 5}}, {{val|0 -5/2 -1/2 -6}}]
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~34/27 = 1\3, ~13/9 = 633.894


[[Mapping|Sval mapping]]: [{{val|1 0 2 -1}}, {{val|0 5 3 6}}]
{{Optimal ET sequence|legend=1| 15, 21fg, 36, 123h, 159h }}


[[Tp tuning|POL2 generator]]: ~9/7 = 439.296
[[Tp tuning #T2 tuning|RMS error]]: ???


{{Val list|legend=1| 8, 11, 30, 41, 71, 93, 112c, 134c, 175c }}
===== 2.3.7.11/5.13.17.19.23 subgroup =====
[[Subgroup]]: 2.3.7.11/5.13.17.19.23


[[Tp tuning #T2 tuning|RMS error]]: 1.074 cents
[[Comma list]]: 210/209, 231/230, 273/272, 286/285, 595/594, 2080/2079


==== 2.9.15.7.11 ====
{{Mapping|legend=2| 3 0 10 5 0 -2 8 12 | 0 3 -1 -1 7 9 3 1 }}
Subgroup: 2.9.15.7.11


Comma list: 100/99, 225/224, 245/243
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~34/27 = 1\3, ~13/9 = 634.038


Gencom: [2 9/7; 100/99 225/224 245/243]
{{Optimal ET sequence|legend=1| 15g, 21fg, 36, 87, 123hi }}


Gencom mapping: [{{val|1 5/2 5/2 5 2}}, {{val|0 -5/2 -1/2 -6 4}}]
[[Tp tuning #T2 tuning|RMS error]]: ???


Sval mapping: [{{val|1 0 2 -1 6}}, {{val|0 5 3 6 -4}}]
===== 2.3.7.11/5.13.17.19.23.29 subgroup =====
[[Subgroup]]: 2.3.7.11/5.13.17.19.23.29


POL2 generator: ~9/7 = 438.607
[[Comma list]]: 210/209, 231/230, 273/272, 286/285, 320/319, 595/594, 2080/2079


Vals: {{Val list| 8, 11, 30, 41, 52, 93, 145, 342bce }}
{{Mapping|legend=2| 3 0 10 5 0 -2 8 12 13 | 0 3 -1 -1 7 9 3 1 1 }}


RMS error: 1.226 cents
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~29/23 = 1\3, ~13/9 = 634.102


==== 2.9.15.7.11.13 ====
{{Optimal ET sequence|legend=1| 15g, 21fg, 36, 87, 123hi }}
Subgroup: 2.9.15.7.11.13


Comma list: 100/99, 105/104, 144/143, 196/195
[[Tp tuning #T2 tuning|RMS error]]: ???


Gencom: [2 9/7; 100/99 105/104 144/143 196/195]
== 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.


Gencom mapping: [{{val|1 5/2 5/2 5 2 7}}, {{val|0 -5/2 -1/2 -6 4 -9}}]
[[Subgroup]]: 2.3.11/7


Sval map: [{{val|1 0 2 -1 6 -2}}, {{val|0 5 3 6 -4 9}}]
[[Comma list]]: {{monzo| 8 -5 }} (256/243)


POL2 generator: ~9/7 = 438.977
{{Mapping|legend=2| 5 8 0 | 0 0 1 }}
: mapping generators: ~9/8, ~11/7


Vals: {{Val list| 11, 30, 41, 153cdef, 194cdef, 235cdef }}
[[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 }}


RMS error: 1.540 cents
{{Optimal ET sequence|legend=1| 15, 20, 35b, 55b }}


== 2.9.21 subgroup ==
=== Pepperoni ===
{{Main| Parapyth }}
{{See also| Chromatic pairs #Pepperoni }}


=== A-team ===
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: 2.9.21


[[Comma]]: 1029/1024
[[Subgroup]]: 2.3.11/7.13/7


[[Gencom]]: [2 21/16; 1029/1024]
[[Comma list]]: 352/351, 364/363


[[Gencom|Gencom mapping]]: [{{val|1 1 0 3}}, {{val|0 3/2 0 -1/2}}]
{{Mapping|legend=2| 1 0 7 12 | 0 1 -4 -7 }}


[[Mapping|Sval mapping]]: [{{val|1 2 4}}, {{val|0 3 1}}]
{{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]


[[Tp tuning|POL2 generator]]: ~21/16 = 467.375
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~3/2 = 703.856


{{Val list|legend=1| 5, 13, 18, 41, 59, 77, 95 }}
{{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]]: 0.3202 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.3789 cents


== 2.11.13.17.19 subgroup ==
== 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.


=== Yamablu ===
[[Subgroup]]: 2.3.13/5
Yamablu, with a generator of ~17/13, is named for it's tempering of the yama comma (209/208) and the blume comma (2057/2048), which also implies the blumeyer comma (2432/2431). The [[Kite's Method of Naming Rank-2 Scales using Mode Numbers|13th Yamablu[13]]] scale is a linear-temperament version of [[Gjaeck]].


Subgroup: 2.11.13.17.19
[[Comma list]]: 676/675 = {{monzo| 2 -3 2 }}


[[Comma list]]: 209/208, 2057/2048, 83521/83486
[[Sval]] [[mapping]]: [{{val| 1 0 -1 }}, {{val| 0 2 3 }}]


[[Mapping|Sval mapping]]: [{{val|1 5 1 1 0}}, {{val|0 -4 7 8 11}}]
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~2 = 1\1, ~15/13 = 248.621


[[Tp tuning|POL2 generator]]: ~17/13 = 462.9606
{{Optimal ET sequence|legend=1| 5, 9, 14, 19, 24, 29, 53, 82, 111, 140, 251, 362 }}


{{Val list|legend=1| 13, 44, 57, 70}}
[[Badness]]: 0.002335


[[Tp tuning #T2 tuning|RMS error]]: 0.4898 cents
; 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]


== Miscellaneous subgroup temperaments ==
==== Tobago ====
{{See also| Chromatic pairs #Tobago }}


=== Historical ===
Tobago, the 10 &amp; 14 temperament in the 2.3.11.13/5 subgroup, extends [[neutral]] and [[barbados]].
Subgroup: 2.3.7/5.11/5.13/5


[[Comma list]]: 364/363, 441/440, 1001/1000
[[Subgroup]]: 2.3.11.13/5


[[Mapping|Sval mapping]]: [{{val|1 2 0 1 2}}, {{val|0 -6 7 2 -9}}]
[[Comma list]]: [[243/242]], [[676/675]]


[[Tp tuning|POL2 generator]]: ~21/20 = 83.016
{{Mapping|legend=2| 2 0 -1 -2 | 0 2 5 3 }}


{{Val list|legend=1| 14, 29, 72, 101, 130, 159 }}
{{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]


[[Tp tuning #T2 tuning|RMS error]]: 0.2562 cents
[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~55/39 = 1\2, ~15/13 = 249.312


=== Hypnosis ===
{{Optimal ET sequence|legend=1| 10, 14, 24, 58, 82, 130 }}
Related temperament: [[Swetismic temperaments #Hypnos|hypnos]], [[Hemifamity temperaments #Tricot|tricot]]


Subgroup: 2.3.7.11/5.13
[[Tp tuning #T2 tuning|RMS error]]: 0.3533 cents


[[Comma list]]: 169/168, 540/539, 729/728
==== Pakkanian hemipyth ====
[[Subgroup]]: 2.3.11.13/5.17


[[Mapping|Sval mapping]]: [{{val|1 0 -3 8 0}}, {{val|0 3 11 -13 7}}]
[[Comma list]]: 221/220, 243/242, 289/288


[[Tp tuning|POL2 generator]]: ~13/9 = 633.518
{{Mapping|legend=2| 2 0 -1 -2 5 | 0 2 5 3 2 }}


{{Val list|legend=1| 17, 36, 118e, 125e, 161e, 197e }}
[[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)


[[Tp tuning #T2 tuning|RMS error]]: 0.5379 cents
{{Optimal ET sequence|legend=1| 10, 14, 24, 106, 130, 154, 178*, 202* }}
: <nowiki />* wart for 13/5


=== Oceanfront ===
=== Oceanfront ===
Subgroup: 2.3.7.13/5
Related temperaments: [[Archytas clan #Superpyth|superpyth]], [[Archytas clan #Ultrapyth|ultrapyth]]


Related temperament: [[Archytas clan #Superpyth|superpyth]], [[Archytas clan #Ultrapyth|ultrapyth]]
[[Subgroup]]: 2.3.7.13/5


[[Comma list]]: 64/63, 91/90
[[Comma list]]: 64/63, 91/90


[[Mapping|Sval mapping]]: [{{val|1 0 6 -5}}, {{val|0 1 -2 4}}]
{{Mapping|legend=2| 1 0 6 -5 | 0 1 -2 4 }}


[[Tp tuning|POL2 generator]]: ~3/2 = 713.910
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~3/2 = 713.910


{{Val list|legend=1| 5, 22, 27, 32, 37 }}
{{Optimal ET sequence|legend=1| 5, 22, 27, 32, 37 }}


[[Tp tuning #T2 tuning|RMS error]]: 2.063 cents
[[Tp tuning #T2 tuning|RMS error]]: 2.063 cents


[[Category:Regular temperament theory]]
Scales: [[Oceanfront scales]]
[[Category:Temperament clan]]
 
[[Category:Subgroup]]
=== 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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