@@ Line 1: / Line 1: @@
-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]].
-=== Sensi (aka Sensation) ===
+= Composite subgroup temperaments =
-: <small>''For full 13-limit extensions, see [[Sensipent family]] or [[Sensi extensions]].''</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]]: 91/90, 126/125, 169/168
+[[Subgroup]]: 2.3.11.19
-[[Gencom]]: [2 9/7; 91/90 126/125 169/168]
+{{Mapping|legend=2| 4 5 13 18 | 0 8 5 -6 }}
-[[Gencom|Gencom mapping]]: [{{val|1 6 8 11 0 10}}, {{val|0 -7 -9 -13 0 -10}}]
+: sval mapping generators: ~6291456/5285401, ~25289/24576
-[[Mapping|Sval mapping]]: [{{val|1 6 8 11 10}}, {{val|0 -7 -9 -13 -10}}]
+[[Optimal tuning]] ([[CTE]]): ~6291456/5285401 = 1\4, ~25289/24576 = 50.257
-[[Tp tuning|POL2 generator]]: ~9/7 = 443.322
+[[Support]]ing [[ET]]s: {{EDOs|24, 596, 620, 644, 668, 692, 716}}, ...
-{{Val list|legend=1| 19, 27, 46, 111de, 157de }}
+==== 2.3.35.11.19 subgroup ====
+edo 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]]: 1.321 cents
+Subgroup: 2.3.35.11.19
-=== [[Tertiaseptal]] ===
+Sval mapping: {{mapping| 4 0 5 13 18 | 0 1 8 5 -6 }}
-: <small>''For full 13- and 17-limit extensions, see [[Breedsmic temperaments #Tertiaseptal]].''</small>
-Subgroup: 2.3.5.7.13
+: sval mapping generators: ~2240/1881, ~36/35
-[[Comma list]]: 625/624, 2401/2400, 4096/4095
+Optimal tuning (CTE): ~2240/1881 = 1\4, ~36/35 = 50.288
-[[Gencom]]: [2 117/112; 625/624 2401/2400 4096/4095]
+[[Support]]ing [[ET]]s: {{EDOs|24, 668}}, ...
-[[Gencom|Gencom mapping]]: [{{val|1 3 2 3 0 1}}, {{val|0 -22 5 -3 0 42}}]
+== 2.9.5.7 subgroup ==
+See also [[Jubilismic clan #Antikythera|antikythera]] and [[Hemimean clan #Isra|isra]].
-[[Mapping|Sval mapping]]: [{{val|1 3 2 3 1}}, {{val|0 -22 5 -3 42}}]
+=== 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]]: ~117/112 = 77.173
+[[Subgroup]]: 2.9.5.7
-{{Val list|legend=1| 31, 109, 140, 171, 311 }}
+[[Comma list]]: {{monzo| 28 -2 -19 8 }}, {{monzo| 9 -25 23 6 }}
-[[Tp tuning #T2 tuning|RMS error]]: 0.1383 cents
+{{Mapping|legend=2| 1 9 6 13 | 0 -298 -188 -521 }}
-==== 2.3.5.7.13.17 ====
+[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~531441/524288 = 23.4765
-Subgroup: 2.3.5.7.13.17
-[[Comma list]]: 625/624, 833/832, 1225/1224, 4096/4095
+{{Optimal ET sequence|legend=1| 460, 869, 1329 }}
-[[Gencom]]: [2 68/65; 625/624 833/832 1225/1224 4096/4095]
+[[Badness]]: 0.611
-[[Gencom|Gencom mapping]]: [{{val|1 3 2 3 0 1 1}}, {{val|0 -22 5 -3 0 42 48}}]
+==== 2.9.5.7.11 ====
+Subgroup: 2.9.5.7.11
-[[Mapping|Sval mapping]]: [{{val|1 3 2 3 1 1}}, {{val|0 -22 5 -3 42 48}}]
+Comma list: {{monzo| -7 7 -3 2 -4 }}, {{monzo| 17 0 -13 1 3 }}, {{monzo| 11 -2 -6 7 -3 }}
-[[Tp tuning|POL2 generator]]: ~68/65 = 77.177
+Sval mapping: {{mapping| 1 9 6 13 16 | 0 -298 -188 -521 -641 }}
-{{Val list|legend=1| 31, 109f, 140, 171, 311 }}
+Optimal tuning (CTE): ~2 = 1\1, ~531441/524288 = 23.4767
-[[Tp tuning #T2 tuning|RMS error]]: 0.1367 cents
+{{Optimal ET sequence|legend=1| 460, 869e, 1329, 1789, 3118 }}
-=== [[Pontiac]] ===
+Badness: 0.165
-: <small>''For full 13- and 17-limit extensions, see [[Schismatic family #Pontiac]].''</small>
-Subgroup: 2.3.5.7.13
+==== 2.9.5.7.11.13 ====
+Subgroup: 2.9.5.7.11.13
-[[Comma list]]: 625/624 729/728 4096/4095
+Comma list: 123201/123200, 1016064/1015625, 2250423/2249390, 2599051/2598156
-[[Gencom]]: [2 4/3; 625/624 729/728 4096/4095]
+Sval mapping: {{mapping| 0 9 6 13 16 10 | -298 -188 -521 -641 -322 }}
-[[Gencom|Gencom mapping]]: [{{val|1 2 -1 19 0 -10}}, {{val|0 -1 8 -39 0 33}}]
+Optimal tuning (CTE): ~2 = 1\1, ~3575/3528 = 23.4767
-[[Mapping|Sval mapping]]: [{{val|1 2 -1 19 -10}}, {{val|0 -1 8 -39 33}}]
+{{Optimal ET sequence|legend=1| 460, 869e, 1329, 1789, 3118 }}
-[[Tp tuning|POL2 generator]]: ~3/2 = 701.773
+Badness: 0.0564
-{{Val list|legend=1| 53, 118, 171, 224, 395 }}
+=== 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.
-[[Tp tuning #T2 tuning|RMS error]]: 0.1525 cents
+Subgroup: 2.9.5.7.33.13.17
-==== 2.3.5.7.13.17 ====
+Comma list: 325/324, 375/374, 385/384, 595/594, 10985/10976
-Subgroup: 2.3.5.7.13.17
-[[Comma list]]: 625/624, 729/728, 1225/1224, 2880/2873
+{{Mapping|legend=2|1 1 11 -16 13 -18 20|0 3 -12 26 -11 30 -22}}
-[[Gencom]]: [2 4/3; 625/624 729/728 1225/1224 2880/2873]
+Optimal tuning (CTE): ~2 = 1\1, 33/20 = 867.982
-[[Gencom|Gencom mapping]]: [{{val|1 2 -1 19 0 -10 29}}, {{val|0 -1 8 -39 0 33 -60}}]
+[[Support]]ing [[ET]]s: {{Optimal ET sequence|47, 65f, 112, 159, 206, 253}}
-[[Mapping|Sval mapping]]: [{{val|1 2 -1 19 -10 29}}, {{val|0 -1 8 -39 33 -60}}]
+=== Baldy ===
+{{See also|Schismatic family #Garibaldi}}
+{{See also|No-threes subgroup temperaments #Frostburn}}
-[[Tp tuning|POL2 generator]]: ~3/2 = 701.764
+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.
-{{Val list|legend=1| 53, 118, 171, 395, 566… }}
+[[Subgroup]]: 2.9.5.7
-[[Tp tuning #T2 tuning|RMS error]]: 0.1696 cents
+[[Comma list]]: 225/224, 3125/3087
-=== [[Septidiasemi]] ===
+{{Mapping|legend=2| 1 3 3 4 | 0 1 -4 -7 }}
-: <small>''For full 13- and 17-limit extensions, see [[Breedsmic temperaments #Septidiasemi]].''</small>
-Subgroup: 2.3.5.7.13
+[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 204.170
-[[Comma list]]: 2205/2197, 2401/2400, 4096/4095
+{{Optimal ET sequence|legend=1| 6, 29, 35, 41, 47 }}
-[[Gencom]]: [2 15/14; 2205/2197 2401/2400 4096/4095]
+Related temperament: [[Schismatic family #Garibaldi|Garibaldi]]
-[[Gencom|Gencom mapping]]: [{{val|1 -1 6 4 0 4}}, {{val|0 26 -37 -12 0 -3}}]
+==== 2.9.5.7.13 ====
+{{See also|Chromatic pairs #Baldy}}
-[[Mapping|Sval mapping]]: [{{val|1 -1 6 4 4}}, {{val|0 26 -37 -12 -3}}]
+Baldy is every other step of [[garibaldi]], without the mapping of prime 11. It can be described as the 6 &amp; 35 temperament.
-[[Tp tuning|POL2 generator]]: ~15/14 = 119.297
+[[Subgroup]]: 2.9.5.7.13
-{{Val list|legend=1| 10, 151, 161, 171 }}
+[[Comma list]]: [[225/224]], [[325/324]], [[640/637]]
-[[Tp tuning #T2 tuning|RMS error]]: 0.2002 cents
+{{Mapping|legend=2| 1 0 15 25 -28 | 0 1 -4 -7 10 }}
-==== 2.3.5.7.13.17 ====
+{{Mapping|legend=3| 1 3/2 3 4 0 2 | 0 1/2 -4 -7 0 10 }}
-Subgroup: 2.3.5.7.13.17
-[[Comma list]]: 833/832, 1275/1274, 2025/2023, 2205/2197
+: [[gencom]]: [2 9/8; 225/224 325/324 640/637]
-[[Gencom]]: [2 15/14; 833/832 1275/1274 2025/2023 2205/2197]
+[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 204.090
-[[Gencom|Gencom mapping]]: [{{val|1 -1 6 4 0 4 2}}, {{val|0 26 -37 -12 0 -3 21}}]
+{{Optimal ET sequence|legend=1| 6, 11, 17, 23, 29, 35, 41, 47, 100, 147, 488cd, 635cd }}
-[[Mapping|Sval mapping]]: [{{val|1 -1 6 4 4 2}}, {{val|0 26 -37 -12 -3 21}}]
+[[Tp tuning #T2 tuning|RMS error]]: 0.5999 cents
-[[Tp tuning|POL2 generator]]: ~15/14 = 119.297
+Related temperament: [[Schismatic family #Garibaldi|Cassandra]]
-{{Val list|legend=1| 10, 151, 161, 171 }}
+==== Baldanders ====
+Baldanders results from taking every other generator of the andromeda, with mapping 11/8 to -9 whole tones.
-[[Tp tuning #T2 tuning|RMS error]]: 0.1867 cents
+Subgroup: 2.9.5.7.11
-=== [[Mitonic]] ===
+Comma list: 100/99, 225/224, 245/242
-: <small>''For full 13- and 17-limit extensions, see [[Minortonic family #Mitonic]].''</small>
-Subgroup: 2.3.5.7.13
+{{Mapping|legend=2| 1 3 3 4 5 | 0 1 -4 -7 -9 }}
-[[Comma list]]: 4096/4095, 4375/4374, 13720/13689
+Optimal tuning (POTE): ~2 = 1\1, ~9/8 = 204.743
-[[Gencom]]: [2 10/9; 4096/4095 4375/4374 13720/13689]
+{{Optimal ET sequence|legend=1| 6, 23de, 29, 35, 41 }}
-[[Gencom|Gencom mapping]]: [{{val|1 -1 -3 6 0 11}}, {{val|0 17 35 -21 0 -48}}]
+Related temperament: [[Schismatic family #Garibaldi|Andromeda]]
-[[Mapping|Sval mapping]]: [{{val|1 -1 -3 6 11}}, {{val|0 17 35 -21 -48}}]
+===== 2.9.5.7.11.13 =====
+Subgroup: 2.9.5.7.11.13
-[[Tp tuning|POL2 generator]]: ~10/9 = 182.471
+Comma list: 100/99, 144/143, 225/224, 245/242
-{{Val list|legend=1| 46, 125, 171, 388 }}
+{{Mapping|legend=2| 1 3 3 4 5 2 | 0 1 -4 -7 -9 10 }}
-[[Tp tuning #T2 tuning|RMS error]]: 0.1442 cents
+Optimal tuning (POTE): ~2 = 1\1, ~9/8 = 204.414
-==== 2.3.5.7.13.17 ====
+{{Optimal ET sequence|legend=1| 6, 23def, 29f, 35, 41, 47 }}
-Subgroup: 2.3.5.7.13.17
-[[Comma list]]: 833/832, 1225/1224, 1701/1700, 4096/4095
+== 2.3.25 subgroup ==
-[[Gencom]]: [2 10/9; 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|Gencom mapping]]: [{{val|1 -1 -3 6 0 11 5}}, {{val|0 17 35 -21 0 -48 -6}}]
+Subgroup: 2.3.25
-[[Mapping|Sval mapping]]: [{{val|1 -1 -3 6 11 5}}, {{val|0 17 35 -21 -48 -6}}]
+Edo join: 17 & 12
-[[Tp tuning|POL2 generator]]: ~10/9 = 182.471
+Comma list: [[2048/2025]]
-{{Val list|legend=1| 46, 125, 171, 388 }}
+{{Mapping|legend=2| 1 1 7| 0 1 -4}}
-[[Tp tuning #T2 tuning|RMS error]]: 0.1341 cents
+Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 705.136
-== No-sevens subgroup ==
+==== 2.3.23.25.41 subgroup ====
+''See also: [[Reversed meantone]]''
-=== Porkypine ===
+Edo join: 17 & 12
-Related temperament: [[Porcupine]]
-Subgroup: 2.3.5.11
+Comma list: 2048/2025, 576/575, 82/81
-[[Comma list]]: 55/54, 100/99
+{{Mapping|legend=2| 1 1 1 7 3| 0 1 6 -4 4}}
-[[Gencom]]: [2 10/9; 55/54, 100/99]
+Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 705.264
-[[Gencom|Gencom mapping]]: [{{val|1 2 3 0 4}}, {{val|0 -3 -5 0 -4}}]
+===== 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 2 3 4}}, {{val|0 -3 -5 -4}}]
+Subgroup: 2.3.7.23.25.41.59
-[[Tp tuning|POL2 generator]]: ~11/10 = 164.078
+Edo join: 17 & 12
-{{Val list|legend=1| 7, 15, 22, 37, 73cd, 95cd, 117bcd }}
+Comma list: 64/63, 225/224, 162/161, 82/81, 177/175
-[[Tp tuning #T2 tuning|RMS error]]: 2.287 cents
+{{Mapping|legend=2| 1 1 4 1 7 3 10| 0 1 -2 6 -4 4 -7}}
-=== Mohaha ===
+Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 706.387
-Related temperament: [[mohajira]], [[Meantone family #Migration|migration]]
-Subgroup: 2.3.5.11
+== 2.9.5.11 subgroup ==
+=== Glacial ===
+{{See also| Chromatic pairs #Glacial }}
-[[Comma list]]: 81/80, 121/120
+[[Subgroup]]: 2.9.5.11.13
-[[Gencom]]: [2 11/9; 81/80 121/120]
+[[Comma list]]: 45/44, 65/64, 81/80
-[[Gencom|Gencom mapping]]: [{{val|1 1 0 0 2}}, {{val|0 2 8 0 5}}]
+{{Mapping|legend=2| 1 0 -4 -6 10 | 0 1 2 3 -2 }}
-[[Mapping|Sval mapping]]: [{{val|1 1 0 2}}, {{val|0 2 8 5}}]
+{{Mapping|legend=3| 1 3/2 2 0 3 4 | 0 1/2 2 0 3 -2 }}
-[[Tp tuning|POL2 generator]]: ~11/9 = 348.094
+: [[gencom]]: [2 9/8; 45/44 65/64 81/80]
-{{Val list|legend=1| 7, 10, 17, 24, 31, 55, 69d, 100d, 131bd }}
+[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 186.151
-[[Tp tuning #T2 tuning|RMS error]]: 1.392 cents
+{{Optimal ET sequence|legend=1| 6, 13, 45be, 58bce, 71bce, 84bce }}
-; Music
+[[Tp tuning #T2 tuning|RMS error]]: 2.887 cents
-[http://micro.soonlabel.com/gene_ward_smith/Others/Stiltner/mohaha10ping2.mp3 Mohaha10ping2] by [[Billy Stiltner]]
-=== Larry ===
+Music:
-Subgroup: 2.3.5.11
+* ''[[Thundersnow]]'' - [[Sevish]] (2021)
-[[Comma list]]: 243/242, 4000/3993
+== 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.
-Related temperaments: [[Gravity family|gravity]], [[Gravity family #Harry|harry]]
+Subgroup: 2.9.7
-[[Gencom]]: [2 40/27; 243/242 4000/3993]
+Comma basis: 44957696/43046721
-[[Gencom|Gencom mapping]]: [{{val|1 5 12 0 12}}, {{val|0 -6 -17 0 -15}}]
+Sval mapping: [{{val|1 1 -3}}, {{val|0 3 8}}]
-[[Mapping|Sval mapping]]: [{{val|1 5 12 12}}, {{val|0 -6 -17 -15}}]
+Optimal tuning (CTE): ~729/448 = 870.792
-[[Tp tuning|POL2 generator]]: ~40/27 = 683.166
+{{Optimal ET sequence|legend=1|7d, 11, 18d, 29, 40, 62}}, ...
-{{Val list|legend=1| 7, 58, 65, 137, 202 }}
+==== 2.9.7.11 subgroup ====
+Subgroup: 2.9.7.11
-[[Tp tuning #T2 tuning|RMS error]]: 0.3025 cents
+Comma basis: 896/891, 1331/1296
-=== Tetracot ===
+Sval mapping: [{{val|1 1 -3 2}}, {{val|0 3 8 2}}]
-: <small>''For full 11- and 13-limit extensions, see [[Tetracot family]].''</small>
-Subgroup: 2.3.5.11
+Optimal tuning (CTE): ~16/11 = 870.966
-[[Comma list]]: 100/99, 243/242
+{{Optimal ET sequence|legend=1| 7d, 11, 40, 51, 62 }}
-[[Gencom]]: [2 10/9; 100/99 243/242]
+== 2.9.7.11 subgroup ==
+=== Apparatus ===
+[[Subgroup]]: 2.9.7.11
-[[Gencom|Gencom mapping]]: [{{val|1 1 1 0 2}}, {{val|0 4 9 0 10}}]
+[[Comma list]]: 41503/41472, 322102/321489
-[[Mapping|Sval mapping]]: [{{val|1 1 1 2}}, {{val|0 4 9 10}}]
+{{Mapping|legend=2| 1 5 3 5 | 0 -19 -2 -16 }}
-[[Tp tuning|POL2 generator]]: ~10/9 = 175.985
+: mapping generators: ~2, ~77/72
-{{Val list|legend=1| 7, 27d, 34, 41, 75d }}
+{{Mapping|legend=3| 1 5/2 0 3 5 | 0 -19/2 0 -2 -16 }}
-[[Tp tuning #T2 tuning|RMS error]]: 1.182 cents
+: [[gencom]]: [2 77/72; 41503/41472 322102/321489]
-==== 2.3.5.11.13 ====
+[[Optimal tuning]] ([[CTE]]): ~77/72 = 115.5685
-Subgroup: 2.3.5.11.13
-[[Comma list]]: 100/99, 144/143, 243/242
+{{Optimal ET sequence|legend=1| 10e, 21, 31, 52, 83, 135, 353, 488, 623 }}
-[[Gencom]]: [2 10/9; 100/99 243/242]
+[[Badness]]: 0.00263
-[[Gencom|Gencom mapping]]: [{{val|1 1 1 0 2 4}}, {{val|0 4 9 0 10 -2}}]
+=== Joan ===
+{{See also| Chromatic pairs #Joan }}
-[[Mapping|Sval mapping]]: [{{val|1 1 1 2 4}}, {{val|0 4 9 10 -2}}]
+Joan is related to [[casablanca]] as well as to [[orwell]].
-[[Tp tuning|POL2 generator]]: ~10/9 = 176.196
+[[Subgroup]]: 2.9.7.11
-{{Val list|legend=1| 7, 27d, 34, 41, 75d }}
+[[Comma list]]: 99/98, 9317/9216
-[[Tp tuning #T2 tuning|RMS error]]: 1.140 cents
+{{Mapping|legend=2| 1 0 1 3 | 0 7 4 1 }}
-=== Emka ===
+{{Mapping|legend=3| 1 0 0 1 3 | 0 7/2 0 4 1 }}
-: <small>''For full 11- and 13-limit extensions, see [[Hemimean clan #Emka]] or [[Horwell temperaments #Emkay]].''</small>
-Subgroup: 2.3.5.11
+: [[gencom]]: [2 11/8; 99/98 9317/9216]
-[[Comma list]]: 4000/3993, 9453125/9437184
+[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~11/8 = 542.672 cents
-[[Gencom]]: [2 11/8; 4000/3993 9453125/9437184]
+{{Optimal ET sequence|legend=1| 11, 20, 31, 42, 115bd, 157bd }}
-[[Gencom|Gencom mapping]]: [{{val|1 14 6 0 3}}, {{val|0 -27 -8 0 1}}]
+[[Tp tuning #T2 tuning|RMS error]]: 1.424 cents
-[[Mapping|Sval mapping]]: [{{val|1 14 6 3}}, {{val|0 -27 -8 1}}]
+=== Machine ===
+Machine is every other step of [[supra]], most interesting for its scale patterns.
-[[Tp tuning|POL2 generator]]: ~11/8 = 551.7778
+[[Subgroup]]: 2.9.7.11
-{{Val list|legend=1| 37, 50, 87, 137, 224 }}
+[[Comma list]]: 64/63, 99/98
-[[Tp tuning #T2 tuning|RMS error]]: 0.1188 cents
+{{Mapping|legend=2| 1 0 6 13 | 0 1 -1 -3 }}
-==== 2.3.5.11.13 ====
+: sval mapping generators: ~2, ~9
-Subgroup: 2.3.5.11.13
-[[Comma list]]: 625/624, 2200/2197, 4000/3993
+{{Mapping|legend=3| 1 3/2 0 3 4 | 0 1/2 0 -1 -3 }}
-[[Gencom]]: [2 11/8; 625/624, 2200/2197, 4000/3993]
+: [[gencom]]: [2 8/7; 64/63 99/98]
-[[Gencom|Gencom mapping]]: [{{val|1 14 6 0 3 6}}, {{val|0 -27 -8 0 1 -5}}]
+[[Optimal tuning]]s:
+* [[CTE]]: ~2 = 1\1, ~9/8 = 216.9128
+* [[POTE]]: ~2 = 1\1, ~9/8 = 214.3843
-[[Mapping|Sval mapping]]: [{{val|1 14 6 3 6}}, {{val|0 -27 -8 1 -5}}]
+{{Optimal ET sequence|legend=1| 5, 6, 11, 17, 28 }}
-[[Tp tuning|POL2 generator]]: ~11/8 = 551.7753
+[[Badness]]: 0.00233
-{{Val list|legend=1| 37, 50, 87, 137, 224 }}
+=== Penta a.k.a. mechanism ===
+Penta or mechanism is the 8 &amp; 11 temperament in the 2.9.7.11 subgroup.
-[[Tp tuning #T2 tuning|RMS error]]: 0.1250 cents
+[[Subgroup]]: 2.9.7.11
-== No-fives subgroup ==
+[[Comma list]]: 896/891, 26411/26244
-=== [[Semaphore and Godzilla|Semaphore]] ===
+{{Mapping|legend=2| 1 0 -1 6 | 0 5 6 -4 }}
-Subgroup: 2.3.7
-[[Comma]]: 49/48
+: sval mapping generators: ~2, ~14/9
-[[Gencom]]: [2 8/7; 49/48]
+{{Mapping|legend=3| 1 5/2 0 5 2 | 0 -5/2 0 -6 4 }}
-[[Gencom|Gencom mapping]]: [{{val|1 2 0 3}}, {{val|0 -2 0 -1}}]
+: [[gencom]]: [2 9/7; 896/891 26411/26244]
-[[Mapping|Sval mapping]]: [{{val|1 2 3}}, {{val|0 -2 -1}}]
+[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~14/9 = 761.3782
-[[Tp tuning|POL2 generator]]: ~7/6 = 250.385
+{{Optimal ET sequence|legend=1| 8, 11, 30, 41, 52 }}
-{{Val list|legend=1| 5, 14, 19, 24, 67c, 91c }}
+[[Tp tuning #T2 tuning|RMS error]]: 0.4262 cents
-[[Tp tuning# T2 tuning|RMS error]]: 2.523 cents
+[[Badness]]: 0.00439
-=== Archy ===
+Scales: [[penta5]], [[penta8]], [[penta11]], [[penta19]]
-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.
-Subgroup: 2.3.7
+== 2.9.7.13.17 subgroup ==
-[[Comma]]: 64/63
+=== 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]].
-[[Gencom]]: [2 3/2; 64/63]
+[[Subgroup]]: 2.9.7.13.17
-[[Gencom|Gencom mapping]]: [{{val|1 1 0 4}}, {{val|0 1 0 -2}}]
+[[Comma list]]: 729/728, 442/441, 833/832
-[[Mapping|Sval mapping]]: [{{val|1 2 2}}, {{val|0 -1 2}}]
+{{Mapping|legend=2| 1 1 1 -1 3| 0 6 5 13 3 }}
-[[Tp tuning|POL2 generator]]: ~3/2 = 709.321
+[[Optimal tuning]] ([[CWE]]): ~2 = 1\1, ~9/7 = 433.836
-{{Val list|legend=1| 5, 12, 17, 22, 27, 137bc }}
+Badness (Dirichlet): 0.142
-[[Tp tuning #T2 tuning|RMS error]]: 1.856 cents
+== 2.9.11 subgroup ==
+=== Demon ===
+Demon is a temperament which equates 3 [[11/9]] with [[16/9]], or equivalently 3 [[18/11]] with [[9/8]], tempering out [[1331/1296]]. This results in [[11/9]] being tuned flat to a supraminor third, and [[27/22]] being tuned sharp to a submajor third. It was discovered by [[User:CompactStar|CompactStar]] while searching for temperaments assosciated with the [[7L 4s]] ("daemotonic") MOS, known for its lack of representation of simple temperaments. The optimal tuning for demon temperament is near the basic tuning of 7L 4s (13\18), and indeed [[18edo]] supports demon temperament.
-==== Supra ====
+[[Subgroup]]: 2.9.11
-Subgroup: 2.3.7.11
-[[Comma list]]: 64/63, 99/98
+[[Comma list]]: [[1331/1296]]
-[[Gencom]]: [2 3/2; 64/63 99/98]
+{{Mapping|legend=2|1 1 2|0 3 2}}
-[[Gencom|Gencom mapping]]: [{{val|1 1 0 4 7}}, {{val|0 1 0 -2 -6}}]
+[[Optimal tuning]] ([[CTE]]): ~[[18/11]] = 870.060
-[[Mapping|Sval mapping]]: [{{val|1 0 6 13}}, {{val|0 1 -2 -6}}]
+{{Optimal ET sequence|legend=1|4, 7, 11, 18, 29, 76e}}
-[[Tp tuning|POL2 generator]]: ~3/2 = 707.192
+=== Genius ===
-{{Val list|legend=1| 5, 12, 17, 39c, 56c }}
+Named after the genius in Roman religion, following the demon (daimon) in Greek mythology.
-[[Tp tuning #T2 tuning|RMS error]]: 1.977 cents
+[[Subgroup]]: 2.9.11
-===== Supraphon =====
+[[Comma list]]: [[131769/131072]]
-Subgroup: 2.3.7.11.13
-[[Comma list]]: 64/63, 78/77, 99/98
+{{Mapping|legend=2|1 1 4|0 4 -1}}
-[[Gencom]]: [2 3/2; 64/63 78/77 99/98]
+[[Optimal tuning]] ([[CTE]]): ~[[16/11]] = 650.863
-[[Gencom|Gencom mapping]]: [{{val|1 1 0 4 7 9}}, {{val|0 1 0 -2 -6 -9}}]
+{{Optimal ET sequence|legend=1|9, 11, 24, 59, 83, 142, 225, 367}}[-11], 592[-11], 959[-9, --11], 1326[-9, --11]
-[[Mapping|Sval mapping]]: [{{val|1 0 6 13 18}}, {{val|0 1 -2 -6 -9}}]
+== 2.9.15.7 subgroup ==
+=== 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]].
-[[Tp tuning|POL2 generator]]: ~3/2 = 706.137
+[[Subgroup]]: 2.9.15.7
-{{Val list|legend=1| 5e, 12e, 17, 22, 39c, 56c }}
+[[Comma list]]: 225/224, 245/243
-[[Tp tuning #T2 tuning|RMS error]]: 2.095 cents
+{{Mapping|legend=2| 1 0 2 -1 | 0 5 3 6 }}
-==== Suhajira ====
+: sval mapping generators: ~2, ~14/9
-Subgroup: 2.3.7.11
-[[Comma list]]: 64/63, 243/242
+{{Mapping|legend=3| 1 5/2 5/2 5 | 0 -5/2 -1/2 -6 }}
-[[Gencom]]: [2 11/9; 64/63 243/242]
+: [[gencom]]: [2 9/7; 225/224 245/243]
-[[Gencom|Gencom mapping]]: [{{val|1 1 0 4 2}}, {{val|0 2 0 -4 5}}]
+[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~2 = 1\1, ~14/9 = 760.704
-[[Mapping|Sval mapping]]: [{{val|1 1 4 2}}, {{val|0 2 -4 5}}]
+{{Optimal ET sequence|legend=1| 8, 11, 30, 41, 71, 93, 112c, 134c, 175c }}
-[[Tp tuning|POL2 generator]]: ~11/9 = 353.958
+[[Tp tuning #T2 tuning|RMS error]]: 1.074 cents
-{{Val list|legend=1| 7, 10, 17, 44d, 61cd, 78cd }}
+==== 2.9.15.7.11 ====
+Subgroup: 2.9.15.7.11
-[[Tp tuning #T2 tuning|RMS error]]: 1.968 cents
+Comma list: 100/99, 225/224, 245/243
-===== 2.3.7.11.13 =====
+Sval mapping: {{mapping| 1 0 2 -1 6 | 0 5 3 6 -4 }}
-Subgroup: 2.3.7.11.13
-[[Comma list]]: 64/63, 78/77, 144/143
+Gencom mapping: {{mapping| 1 5/2 5/2 5 2 | 0 -5/2 -1/2 -6 4 }}
-[[Gencom]]: [2 11/9; 64/63 78/77 144/143]
+: gencom: [2 9/7; 100/99 225/224 245/243]
-[[Gencom|Gencom mapping]]: [{{val|1 1 0 4 2 4}}, {{val|0 2 0 -4 5 -1}}]
+Optimal tuning (subgroup POTE): ~2 = 1\1, ~14/9 = 761.393
-[[Mapping|Sval mapping]]: [{{val|1 1 4 2 4}}, {{val|0 2 -4 5 -1}}]
+Optimal ET sequence: {{Optimal ET sequence| 8, 11, 30, 41, 52, 93, 145, 342bce }}
-[[Tp tuning|POL2 generator]]: ~11/9 = 353.775
+RMS error: 1.226 cents
-{{Val list|legend=1| 7, 10, 17, 44d, 61cd, 78cd }}
+==== 2.9.15.7.11.13 ====
+Subgroup: 2.9.15.7.11.13
-[[Tp tuning #T2 tuning|RMS error]]: 1.953 cents
+Comma list: 100/99, 105/104, 144/143, 196/195
-=== [[Slendric]] ===
+Sval mapping: {{mapping| 1 0 2 -1 6 -2 | 0 5 3 6 -4 9 }}
-Subgroup: 2.3.7
-[[Comma]]: 1029/1024
+Gencom mapping: {{mapping| 1 5/2 5/2 5 2 7 | 0 -5/2 -1/2 -6 4 -9 }}
-[[Gencom]]: [2 8/7; 1029/1024]
+: gencom: [2 9/7; 100/99 105/104 144/143 196/195]
-[[Gencom|Gencom mapping]]: [{{val|1 1 0 3}}, {{val|0 3 0 -1}}]
+Optimal tuning (subgroup POTE): ~2 = 1\1, ~14/9 = 761.023
-[[Mapping|Sval mapping]]: [{{val|1 1 3}}, {{val|0 3 -1}}]
+Optimal ET sequence: {{Optimal ET sequence| 11, 30, 41, 153cdef, 194cdef, 235cdef }}
-[[Tp tuning|POL2 generator]]: ~8/7 = 233.688
+RMS error: 1.540 cents
-{{Val list|legend=1| 5, 21, 26, 31, 36, 77, 113, 190 }}
+== 2.9.21 subgroup ==
+=== A-team ===
+A-team is every other step of [[slendric]]; the 2.9.5.21.11 extension below specifically restricts [[mothra]].
+[[Subgroup]]: 2.9.21
+[[Comma list]]: 1029/1024
+{{Mapping|legend=2| 1 2 4 | 0 3 1 }}
+: sval mapping generators: ~2, ~21/16
+{{Mapping|legend=3| 1 1 0 3 | 0 3/2 0 -1/2 }}
+: [[gencom]]: [2 21/16; 1029/1024]
+[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~2 = 1\1, ~21/16 = 467.375
+{{Optimal ET sequence|legend=1| 5, 13, 18, 41, 59, 77, 95 }}
 [[Tp tuning #T2 tuning|RMS error]]: 0.3202 cents
-=== [[Gamelismic clan #Baladic|Baladic]] ===
+==== 2.9.5.21 ====
-Subgroup: 2.3.7.13
+''Lookalike temperament: [[Dual-fifth_temperaments#Dual-3_A-Team|Dual-3 A-Team]]''
+Subgroup: 2.9.5.21
+[[Comma]] list: 81/80, 1029/1024
+Sval mapping: {{mapping| 1 2 0 4 | 0 3 6 1 }}
+Mapping generators: ~2, ~21/16
+Optimal ([[Lp tuning|POL2]]) generator: 464.3865
+{{Optimal ET sequence|legend=1| 13, 18, 31, 44 }}
+===== 2.9.5.21.11 =====
+Subgroup: 2.9.5.21.11
+Comma list: 81/80, 99/98, 385/384
+Sval mapping: {{mapping| 1 2 0 4 5 | 0 3 6 1 -4 }}
+Gencom mapping: {{mapping| 1 1 0 3 5 | 0 3/2 6 -1/2 -4 }}
+: gencom: [2 21/16; 81/80 99/98 385/384]
+Optimal tuning (subgroup POTE): ~2 = 1\1, ~21/16 = 463.956
+{{Optimal ET sequence|legend=1| 5, 13, 31 }}
+==== B-team ====
+B-team (23 & 41) is every other step of [[rodan]].
-[[Comma list]]: 169/168, 1029/1024
+Subgroup: 2.9.15.21.33
-[[Gencom]]: [91/64 8/7; 169/168 1029/1024]
+Comma list: 245/243, 385/384, 441/440
-[[Mapping|Sval mapping]]: [{{val|2 2 6 7}}, {{val|0 3 -1 1}}]
+Sval mapping: {{mapping| 1 2 0 4 7 | 0 3 10 1 -5 }}
-[[Tp tuning|POL2 generator]]: ~8/7 = 233.6044
+Optimal tuning (subgroup POTE): ~2 = 1\1, ~21/16 = 468.918
-{{Val list|legend=1| 10, 26, 36, 154…, 190…, 226…, 262… }}
+{{Optimal ET sequence|legend=1| 5, 13c, 18, 23, 41, 64, 87, 151 }}
-[[Tp tuning #T2 tuning|RMS error]]: 0.5452 cents
+== 4.3.5 subgroup ==
+=== Tetrahanson ===
+{{Main| Tetrahanson }}
-==== 2.3.7.13.17 ====
+[[Subgroup]]: 4.3.5
-Subgroup: 2.3.7.13.17
-[[Comma list]]: 169/168, 273/272, 289/288
+[[Comma list]]: 15625/15552
-[[Gencom]]: [17/12 8/7; 169/168 273/272 289/288]
+{{Mapping|legend=2| 1 3 3 | 0 -6 -5 }}
-[[Mapping|Sval mapping]]: [{{val|2 2 6 7 7}}, {{val|0 3 -1 1 3}}]
+: Mapping generators: ~4, ~5/3
-[[Tp tuning|POL2 generator]]: ~8/7 = 233.6155
+[[Optimal tuning]] ([[CTE]]): ~4 = 2\1, ~5/3 = 882.941
-{{Val list|legend=1| 10, 26, 36, 154…, 190…, 226… }}
+[[Support]]ing [[ET]]s: {{EDs|19, 106, 87, 68, 11, 8, 125, 49, 30, 27, 117, 46, 41b, 79|equave=4}}
-[[Tp tuning #T2 tuning|RMS error]]: 0.5073 cents
+=== Tetrameantone ===
+{{Main| Tetrameantone }}
-=== Lee ===
+[[Subgroup]]: 4.3.5
-Subgroup: 2.3.7
-[[Comma]]: 177147/175616
+[[Comma list]]: 81/80
-[[Gencom]]: [2 81/56; 177147/175616]
+{{Mapping|legend=2| 1 1 2 | 0 -1 -4 }}
-[[Gencom|Gencom mapping]]: [{{val|1 0 0 -3}}, {{val|0 3 0 11}}]
+: Mapping generators: ~4, ~4/3
-[[Mapping|Sval mapping]]: [{{val|1 0 -3}}, {{val|0 3 11}}]
+[[Optimal tuning]] ([[POTE]]): 4 = 2400.0, ~4/3 = 503.761
-[[Tp tuning|POL2 generator]]: ~81/56 = 633.525
+[[Support]]ing [[ET]]s: {{EDs|5, 9, 14, 19, 24, 43, 62, 81, 100|equave=4}}
-{{Val list|legend=1| 17, 36, 89, 125, 161, 358, 519b }}
+=== Tetramagic ===
-[[Tp tuning #T2 tuning|RMS error]]: 0.3519 cents
+[[Subgroup]]: 4.3.5
-=== Skwares ===
+[[Comma list]]: 3125/3072
-Related temperament: [[Meantone family #Squares|squares]]
-Subgroup: 2.3.7
+{{Mapping|legend=2| 1 0 1 | 0 5 1 }}
-[[Comma]]: 19683/19208
+: Mapping generators: ~4, ~5/4
-[[Gencom]]: [2 9/7; 19683/19208]
+[[Optimal tuning]] ([[POTE]]): 4 = 2400.0, ~5/4 = 380.059
-[[Gencom|Gencom mapping]]: [{{val|1 3 6}}, {{val|0 -4 -9}}]
+[[Support]]ing [[ET]]s: {{EDs|6, 13, 19, 25, 38, 44, 63, 82|equave=4}}
-[[Mapping|Sval mapping]]: [{{val|1 3 6}}, {{val|0 -4 -9}}]
+=== Blacktetra ===
-[[Tp tuning|POL2 generator]]: ~9/7 = 425.365
+[[Subgroup]]: 4.3.5
-{{Val list|legend=1| 14, 17, 31, 48, 79, 189b, 268bc, 347bc }}
+[[Comma list]]: 256/243
-[[Tp tuning #T2 tuning|RMS error]]: 1.149 cents
+{{Mapping|legend=2| 5 4 6 | 0 0 -1 }}
-==== 2.3.7.11 ====
+: Mapping generators: ~4, ~16/15
-Subgroup: 2.3.7.11
-[[Comma list]]: 99/98, 243/242
+[[Optimal tuning]] ([[POTE]]): 1\5ed4 = 480.0, ~16/15 = 80.4062
-[[Gencom]]: [2 9/7; 99/98 243/242]
+[[Support]]ing [[ET]]s: {{EDs|5, 10, 15, 20, 25, 30, 55, 85, 115|equave=4}}
-[[Gencom|Gencom mapping]]: [{{val|1 3 0 6 7}}, {{val|0 -4 0 -9 -10}}]
+== 4.6.5 subgroup ==
+=== Meanquad ===
+{{Main| Meanquad }}
-[[Mapping|Sval mapping]]: [{{val|1 3 6 7}}, {{val|0 -4 -9 -10}}]
+[[Subgroup]]: 4.6.5
-[[Tp tuning|POL2 generator]]: ~9/7 = 425.244
+[[Comma list]]: [[81/80]] = {{monzo| -4 4 -1 }}
-{{Val list|legend=1| 14, 17, 31, 48, 79, 127, 206bcd }}
+{{Mapping|legend=2| 1 0 -4| 0 1 4 }}
-[[Tp tuning #T2 tuning|RMS error]]: 1.099 cents
+: mapping generators: ~4, ~6
-==== 2.3.7.11.13 ====
+[[Optimal tuning]] (subgroup [[CTE]]): ~4 = 2\1, ~3/2 = 697.214
-Subgroup: 2.3.7.11.13
-[[Comma list]]: 78/77, 99/98, 243/242
+[[Support]]ing [[ET]]s: *7, *10, *11[-5], *13[+5], *17, *24, *27[+5], *31, *38, *41, *45, *52, *55, *69
-[[Gencom]]: [2 9/7; 78/77, 99/98, 243/242]
+<nowiki />* Wart for 4
-[[Gencom|Gencom mapping]]: [{{val|1 3 0 6 7 9}}, {{val|0 -4 0 -9 -10 -15}}]
+==== 4.6.5.7 subgroup (tetrominant) ====
+[[Subgroup]]: 4.6.5.7
-[[Mapping|Sval mapping]]: [{{val|1 3 6 7 9}}, {{val|0 -4 -9 -10 -15}}]
+[[Comma list]]: [[36/35]] = {{monzo| 0 2 -1 -1 }}, [[64/63]] = {{monzo| 4 -2 0 -1 }}
-[[Tp tuning|POL2 generator]]: ~9/7 = 424.457
+{{Mapping|legend=2| 1 0 -4 4 | 0 1 4 -2 }}
-{{Val list|legend=1| 17, 48e, 65de, 82c, 147ce }}
+[[Optimal tuning]] (subgroup [[CTE]]): ~4 = 2\1, ~3/2 = 699.622
-[[Tp tuning #T2 tuning|RMS error]]: 1.769 cents
+[[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]
-==== Skwairs ====
+<nowiki />* Wart for 4
-Subgroup: 2.3.7.11.13
-[[Comma list]]: 99/98, 144/143, 243/242
+=== 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]]: [2 9/7; 99/98, 144/143, 243/242]
+Fourwar is named after the closely related [[hemiwar]] temperament.
-[[Gencom|Gencom mapping]]: [{{val|1 3 0 6 7 3}}, {{val|0 -4 0 -9 -10 2}}]
+{{Todo|inline=1|cleanup}}
-[[Mapping|Sval mapping]]: [{{val|1 3 6 7 3}}, {{val|0 -4 -9 -10 2}}]
+<pre>
+Reduced Mapping
+	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)
-[[Tp tuning|POL2 generator]]: ~9/7 = 424.702
+Subsets
+q99, q62, q37, q161, q136, q198, q25, q124, q74, q235
+</pre>
-{{Val list|legend=1| 14, 17, 31 }}
+==== 4.6.5.7 ====
+<pre>
+Reduced Mapping
+	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)
-[[Tp tuning #T2 tuning|RMS error]]: 1.290 cents
+Subsets
+q99, q62, q37, q161, q136, q198, q25, q124, q74, q235
+</pre>
-=== Bleu ===
+==== 4.6.5.7.11 ====
-Subgroup: 2.3.7
+<pre>
+Reduced Mapping
+	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)
-[[Comma]]: 17496/16807
+Subsets
+q37, q25, q62, q12, q74, q99, q87, q49r, q50r, q124
+</pre>
-[[Gencom]]: [2 54/49; 17496/16807]
+==== 4.6.5.7.11.13 ====
-[[Gencom|Gencom mapping]]: [{{val|1 1 0 2}}, {{val|0 5 0 7}}]
+<pre>
+Reduced Mapping
+	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)
-[[Mapping|Sval mapping]]: [{{val|1 1 2}}, {{val|0 5 7}}]
+Subsets
+q25, q37f, q12f, q62, q50rf, q13rff, q49rff, q87, q74ff, q24rfff
+</pre>
-[[Tp tuning|POL2 generator]]: ~54/49 = 139.848
+==== 4.6.5.7.11.13.17 ====
+<pre>
+Reduced Mapping
+	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)
-{{Val list| 9, 17, 43, 60c }}
+Subsets
+q25, q12f, q37f, q13rffg, q50rf, q62, q49rffg, q24rfffg, q38rreffg, q74ffg
+</pre>
-[[Tp tuning #T2 tuning|RMS error]]: 1.917 cents
+==== 4.6.5.7.11.13.17.19 ====
+<pre>
+Reduced Mapping
+	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)
-==== 2.3.7.11 ====
+Subsets
-Subgroup: 2.3.7.11
+q25, q12fh, q37f, q13rffgh, q50rf, q62, q49rffgh, q24rfffghh, q38rreffgh, q74ffgh
+</pre>
-[[Comma list]]: 99/98, 864/847
+==== 4.6.5.7.11.13.17.19.23 ====
+<pre>
+Reduced Mapping
+	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 12/11; 99/98 864/847]
+Subsets
+q25i, q12fhi, q37f, q13rffghii, q62, q50rfii, q49rffghii, q24rfffghhiii, q74ffghi, q38rreffghiii
+</pre>
-[[Gencom|Gencom mapping]]: [{{val|1 1 0 2 3}}, {{val|0 5 0 7 4}}]
+== 4.9.25 subgroup ==
+=== Meansquared ===
+[[Subgroup]]: 4.9.25
-[[Mapping|Sval mapping]]: [{{val|1 1 2 3}}, {{val|0 5 7 4}}]
+[[Comma list]]: [[6561/6400]]
-[[Tp tuning|POL2 generator]]: ~12/11 = 140.005
+{{Mapping|legend=2| 1 3 4 | 0 1 4 }}
-{{Val list| 9, 17, 43, 60c }}
+Mapping generators: ~4, ~9/64
-[[Tp tuning #T2 tuning|RMS error]]: 1.829 cents
+[[Optimal tuning]] ([[CTE]]): ~4 = 2\1, ~9/4 = 1394.429
-==== 2.3.7.11.13 ====
+[[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: 2.3.7.11.13
-[[Comma list]]: 78/77, 99/98, 144/143
+== 4.9.49 subgroup ==
+=== Archsquared ===
+[[Subgroup]]: 4.9.49
-[[Gencom]]: [2 12/11; 78/77 99/98 144/143]
+[[Comma list]]: 4096/3969
-[[Gencom|Gencom mapping]]: [{{val|1 1 0 2 3 3}}, {{val|0 5 0 7 4 6}}]
+{{Mapping|legend=2| 1 3 0 | 0 1 -2 }}
-[[Mapping|Sval mapping]]: [{{val|1 1 2 3 3}}, {{val|0 5 7 4 6}}]
+Mapping generators: ~4, ~9/64
-[[Tp tuning|POL2 generator]]: ~12/11 = 139.990
+[[Optimal tuning]] ([[CTE]]): ~9/4 = 1419.190
-{{Val list|legend=1| 17, 43, 60c }}
+[[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]]: 1.752 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."
-=== Hemif ===
+[[Subgroup]]: 8.9.7
-Related temperament: [[Breedsmic temperaments #Hemififths|hemififths]], namo
-Subgroup: 2.3.7
+[[Comma list]]: 64/63
-[[Comma]]: 1605632/1594323
+{{Mapping|legend=2| 1 0 2 | 0 1 -1 }}
-[[Gencom]]: [2 2187/1792; 1605632/1594323]
+: sval mapping generators: ~8, ~9
-[[Gencom|Gencom mapping]]: [{{val|1 1 0 -1}}, {{val|0 2 0 13}}]
+: [[gencom]]: [8 9/8; 64/63]
-[[Mapping|Sval mapping]]: [{{val|1 1 -1}}, {{val|0 2 13}}]
+[[Optimal tuning]] ([[CTE]]): ~9/8 = 219.1898
-[[Tp tuning|POL2 generator]]: ~2187/1792 = 351.485
+[[Optimal ET sequence]]: {{val| 16 17 15 }}, {{val| 33 35 31 }}, {{val| 148 … }}, {{val| 181 … }}, {{val| 214 … }}, {{val| 247 … }}
-{{Val list|legend=1| 7, 17, 41, 58, 99 }}
+[[Badness]]: 0.0215 × 10<sup>-3</sup>
-[[Tp tuning #T2 tuning|RMS error]]: 0.2344 cents
+= Fractional subgroup temperaments =
+== 2.5/3.… subgroups ==
+=== Magicaltet ===
+{{See also| Chromatic pairs #Magicaltet }}
-==== 2.3.7.11 ====
+Magicaltet is related to [[keemic]], [[superkleismic]], and [[magic]]. The tonic and the first three generator steps make a [[magical seventh chord]], hence the name.
-Subgroup: 2.3.7.11
-[[Comma list]]: 243/242, 896/891
+[[Subgroup]]: 2.5/3.7.11
-[[Gencom]]: [2 11/9; 243/242 896/891]
+[[Comma list]]: 100/99 ({{monzo| 2 2 0 -1 }}), 385/384 ({{monzo| -7 1 1 1 }})
-[[Gencom|Gencom mapping]]: [{{val|1 1 0 -1 2}}, {{val|0 2 0 13 5}}]
+{{Mapping|legend=2| 1 0 5 2 | 0 1 -3 2 }}
+: mapping generators: ~2, ~5/3
-[[Mapping|Sval mapping]]: [{{val|1 1 -1 2}}, {{val|0 2 13 5}}]
+{{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]
-[[Tp tuning|POL2 generator]]: ~11/9 = 351.535
+[[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
-{{Val list|legend=1| 7, 17, 41, 58, 99d }}
+{{Optimal ET sequence|legend=1| 4, 7, 11, 15, 26, 67, 93* }}
+: <nowiki/>* wart for 5/3
-[[Tp tuning #T2 tuning|RMS error]]: 0.6108 cents
+[[Tp tuning #T2 tuning|RMS error]]: 1.206 cents
-==== 2.3.7.11.13 ====
+=== Starlingtet ===
-Subgroup: 2.3.7.11.13
+{{See also | Chromatic pairs #Starlingtet }}
-[[Comma list]]: 144/143, 243/242, 364/363
+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]]: [2 11/9; 144/143 243/242 364/363]
+[[Subgroup]]: 2.5/3.7/3
-[[Gencom|Gencom mapping]]: [{{val|1 1 0 -1 2 4}}, {{val|0 2 0 13 5 -1}}]
+[[Comma list]]: [[126/125]] ({{monzo| 1 -3 1 }})
-[[Mapping|Sval mapping]]: [{{val|1 1 -1 2 4}}, {{val|0 2 13 5 -1}}]
+{{Mapping|legend=2| 1 0 -1 | 0 1 3 }}
-[[Tp tuning|POL2 generator]]: ~11/9 = 351.691
+: mapping generators: ~2, ~5/3
-{{Val list|legend=1| 7, 10, 17, 24, 41, 58 }}
+{{Mapping|legend=3| 1 -1 0 1 | 0 4/3 1/3 -5/3 }}
+: [[gencom]]: [2 6/5; 126/125]
-[[Tp tuning #T2 tuning|RMS error]]: 0.7167 cents
+[[Optimal tuning]]s:
+* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 888.759
+* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 888.846
-=== Ennea ===
+{{Optimal ET sequence|legend=1| 4, 15, 19, 23, 27 }}
-Subgroup: 2.3.7.11
-[[Comma list]]: 41503/41472, 43923/43904
+[[Tp tuning #T2 tuning|RMS error]]: 0.8398 cents
-[[Gencom]]: [2 99/98; 41503/41472, 43923/43904]
+==== Greeley ====
+{{See also| Chromatic pairs #Greeley }}
-[[Gencom|Gencom mapping]]: [{{val|1 14/9 0 25/9 31/9}}, {{val|0 2 0 2 1}}]
+Greeley is related to [[opossum]] as well as to [[nusecond]].
-[[Mapping|Sval mapping]]: [{{val|9 0 11 24}}, {{val|0 2 2 1}}]
+[[Subgroup]]: 2.5/3.7/3.11/3
-[[Tp tuning|POL2 generator]]: ~99/98 = 17.6258
+[[Comma list]]: 121/120 ({{monzo| -3 -1 0 2 }}), 126/125 ({{monzo| 1 -3 1 }})
-{{Val list|legend=1| 54, 63, 72, 135, 342, 477, 1089, 1566 }}
+{{Mapping|legend=2| 1 1 2 2 | 0 -2 -6 -1 }}
-[[Tp tuning #T2 tuning|RMS error]]: 0.0383 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]
-=== Leapfrog ===
+[[Optimal tuning]]s:
-{{see also|Gentle region}}
+* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~11/10 = 155.696
+* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~11/10 = 155.776
-Subgroup: 2.3.7
+{{Optimal ET sequence|legend=1| 8, 15, 23, 54, 77, 100, 131* }}
+: <nowiki/>* wart for 11/3
-[[Comma list]]: 14680064/14348907
+[[Tp tuning #T2 tuning|RMS error]]: 1.034 cents
-[[Gencom]]: [2 3/2; 14680064/14348907]
+==== Skateboard ====
+{{See also| Chromatic pairs #Skateboard }}
-[[Gencom|Gencom mapping]]: [{{val|1 1 0 -6}}, {{val|0 1 0 15}}]
+Skateboard is related to [[thrasher]].
-[[Mapping|Sval mapping]]: [{{val|1 0 -21}}, {{val|0 1 15}}]
+[[Subgroup]]: 2.5/3.7/3.11.13/9
-[[Tp tuning|POL2 generator]]: ~3/2 = 704.721 cents
+[[Comma list]]: 56/55 ({{monzo| 3 -1 1 -1 }}), 91/90 ({{monzo| -1 -1 1 0 1 }}), 100/99 ({{monzo| 2 2 0 -1 }})
-{{Val list|legend=1| 17, 46, 63 }}
+{{Mapping|legend=2| 1 0 -1 2 2 | 0 1 3 2 -2 }}
-[[Tp tuning #T2 tuning|RMS error]]: 0.6202 cents
+{{Mapping|legend=3| 1 -3/7 4/7 11/7 4 -6/7 | 0 0 -1 -3 -2 2 }}
+: [[gencom]]: [2 6/5; 56/55 91/90 100/99]
-==== 2.3.7.11 ====
+[[Optimal tuning]]s:
-Subgroup: 2.3.7.11
+* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 886.158
+* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 886.158
-[[Comma list]]: 896/891, 1331/1323
+{{Optimal ET sequence|legend=1| 11, 15, 19, 23, 42d, 65d }}
-[[Gencom]]: [2 3/2; 896/891 1331/1323]
+[[Tp tuning #T2 tuning|RMS error]]: 2.396 cents
-[[Gencom|Gencom mapping]]: [{{val|1 1 0 -6 -3}}, {{val|0 1 0 15 11}}]
+=== Gariberttet ===
+Gariberttet is the 2.5/3.7/3 [[Subgroup temperament families, relationships, and genes|altergene]] of [[sirius]].
-[[Mapping|Sval mapping]]: [{{val|1 0 -21 -14}}, {{val|0 1 15 11}}]
+==== Gariberttet (2.5/3.7/3.13/11 subgroup) ====
+{{See also | Chromatic pairs #Gariberttet }}
-[[Tp tuning|POL2 generator]]: ~3/2 = 704.753 cents
+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]].
-{{Val list|legend=1| 17, 46, 63 }}
+[[Subgroup]]: 2.5/3.7/3.13/11
-[[Tp tuning #T2 tuning|RMS error]]: 0.6047 cents
+[[Comma list]]: [[275/273]] ({{monzo| 0 2 -1 -1 }}), [[847/845]] ({{monzo| 0 -1 1 -2 }})
-==== 2.3.7.11.13 ====
+{{Mapping|legend=2| 1 0 0 0 | 0 3 5 1 }}
-Subgroup: 2.3.7.11.13
-[[Comma list]]: 169/168, 352/351, 364/363
+{{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]]: [2 3/2; 169/169 352/351 364/363]
+[[Optimal tuning]]s:
+* [[Tp tuning|subgroup]] [[CTE]] and [[POTE]]: ~2 = 1200.000, ~13/11 = 293.679
-[[Gencom|Gencom mapping]]: [{{val|1 1 0 -6 -3 -1}}, {{val|0 1 0 15 11 8}}]
+{{Optimal ET sequence|legend=1| 29, 33, 37, 41, 45, 49, 78, 94, 143* }}
+: <nowiki/>* wart for 13/11
-[[Mapping|Sval mapping]]: [{{val|1 0 -21 -14 -9}}, {{val|0 1 15 11 8}}]
+[[Tp tuning #T2 tuning|RMS error]]: 0.6914 cents
-[[Tp tuning|POL2 generator]]: ~3/2 = 704.745 cents
+==== Indium ====
+{{See also | Chromatic pairs #Indium }}
-{{Val list|legend=1| 17, 46, 63 }}
+Indium can be described as the {{nowrap| 8 & 33 }} temperament in the 2.5/3.7/3.11/3 subgroup.
-[[Tp tuning #T2 tuning|RMS error]]: 0.7541 cents
+[[Subgroup]]: 2.5/3.7/3.11/3
-; Music
+[[Comma list]]: [[3025/3024]] ({{monzo| -4 2 -1 2 }}), [[3125/3087]] ({{monzo| 0 5 -3 }})
-* Suite for Harpsichord in A Locrian, tuning: Eb-G# in [[46edo|46EDO]] by [[User:IlL|IlL]] (in progress):
-** I. Prelude
+{{Mapping|legend=2| 1 0 0 2 | 0 6 10 -1 }}
-** II. Allemande
-** III. Courante
+{{Mapping|legend=3| 1 -1/2 -1/2 -1/2 3/2 | 0 -15/4 9/4 25/4 -19/4 }}
-** [[:File:Locrian Suite Sarabande.mp3|IV. Sarabande]] ([[:File:Locrian Suite Sarabande Score.pdf|score]], [[:File:Locrian Suite Sarabande 17edo.mp3|17EDO version]])
+: [[gencom]]: [2 12/11; 3025/3024 3125/3087]
-** [[: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]]
-=== Parapyth (Rank 3) ===
+[[Optimal tuning]]s:
-Subgroup: 2.3.7.11
+* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~12/11 = 146.978
+* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~12/11 = 147.010
-[[Comma list]]: 896/891
+{{Optimal ET sequence|legend=1| 8, 33, 41, 49, 204*<sup>†</sup> }}
+: <nowiki/>* wart for 7/3
+: <sup>†</sup> wart for 11/3
-[[Gencom]]: [2 3/2 28/27; 896/891]
+[[Tp tuning #T2 tuning|RMS error]]: 0.7788 cents
-[[Gencom|Gencom mapping]]: [{{val|1 1 0 1 4}}, {{val|0 1 0 3 -1}}, {{val|0 0 0 1 1}}]
+==== Ammon ====
+{{See also| Chromatic pairs #Ammon }}
-[[Mapping|Sval mapping]]: [{{val|1 0 0 7}}, {{val|0 1 0 -4}}, {{val|0 0 1 1}}]
+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.
-[[Tp tuning|POL2 tuning]]: ~3 = 1903.834, ~7 = 3369.872
+[[Subgroup]]: 2.5/3.7/3.11/3.13/3
-{{Val list|legend=1| 17, 41, 46, 58, 87, 104 }}
+[[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 }})
-[[Tp tuning #T2 tuning|RMS error]]: 0.4149 cents
+{{Mapping|legend=2| 1 3 5 3 4 | 0 -6 -10 -3 -5 }}
-==== 2.3.7.11.13 ====
+{{Mapping|legend=3| 1 -3 0 2 0 1 | 0 24/5 -6/5 -26/5 9/5 -1/5 }}
-Subgroup: 2.3.7.11.13
+: [[gencom]]: [2 13/10; 121/120 169/168 275/273]
-[[Comma list]]: 352/351, 364/363
+[[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
-The gencom below gives [[Margo Schulter]]'s favored basis
+{{Optimal ET sequence|legend=1| 8, 29, 37, 45 }}
-[[Gencom]]: [2 3/2 28/27; 352/351 364/363]
+[[Tp tuning #T2 tuning|RMS error]]: 1.052 cents
-[[Gencom|Gencom mapping]]: [{{val|1 1 0 1 4 6}}, {{val|0 1 0 3 -1 -4}}, {{val|0 0 0 1 1 1}}]
+=== Sentry ===
+{{See also | Chromatic pairs #Sentry }}
-[[Mapping|Sval mapping]]: [{{val|1 0 0 7 12}}, {{val|0 1 0 -4 -7}}, {{val|0 0 1 1 1}}]
+Sentry, the {{nowrap| 3 & 5 }} temperament in the 2.5/3.9/7 subgroup, is related to [[sensi]].
-[[Tp tuning|POL2 tuning]]: ~3 = 1903.856, ~7 = 3369.907
+[[Subgroup]]: 2.5/3.9/7
-{{Val list|legend=1| 17, 41, 46, 58, 87, 104 }}
+[[Comma list]]: [[245/243]] ({{monzo| 0 1 -2 }})
-[[Tp tuning #T2 tuning|RMS error]]: 0.3789 cents
+{{Mapping|legend=2| 1 0 0 | 0 2 1 }}
-=== Neutral ===
+{{Mapping|legend=3| 1 0 0 0 | 0 0 2 -1 }}
-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]].
+: [[gencom]]: [2 9/7; 245/243]
-Subgroup: 2.3.11
+[[Optimal tuning]]s:
+* [[Tp tuning|subgroup]] [[CTE]] and [[POTE]]: ~2 = 1200.000, ~9/7 = 440.902
-[[Comma]]: 243/242
+{{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
-[[Gencom]]: [2 11/9; 243/242]
+[[Tp tuning #T2 tuning|RMS error]]: 0.7105 cents
-[[Gencom|Gencom mapping]]: [{{val|1 1 0 0 2}}, {{val|0 2 0 0 5}}]
+=== Marveltwintri ===
+{{See also| Chromatic pairs #Marveltwintri }}
-[[Mapping|Sval mapping]]: [{{val|1 1 2}}, {{val|0 2 5}}]
+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]].
-[[Tp tuning|POL2 generator]]: ~11/9 = 350.525
+[[Subgroup]]: 2.5/3.13/9
-{{Val list|legend=1| 7, 10, 17, 24, 41, 65, 89, 202, 291, 380 }}
+[[Comma list]]: [[325/324]] ({{monzo| -2 2 1 }})
-[[Tp tuning #T2 tuning|RMS error]]: 0.3021 cents
+{{Mapping|legend=2| 1 0 2 | 0 1 -2 }}
-[[neutraltet7|Seven note albitonic scale]]
+{{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]
-[[neutraltet10|Ten note chromatic scale]]
+[[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
-[[neutraltet17|Seventeen note mega chromatic scale]]
+{{Optimal ET sequence|legend=1| 3, 4, 11, 15, 19, 34, 53, 87, 140 }}
-==== Namo ====
+[[Tp tuning #T2 tuning|RMS error]]: 0.2444 cents
-Subgroup: 2.3.11.13
-[[Comma list]]: 144/143, 243/242
+== 2.….7/3.… subgroups ==
+=== Guanyintet ===
+{{See also | Chromatic pairs #Guanyintet }}
-[[Gencom]]: [2 11/9; 144/143 243/242]
+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|Gencom mapping]]: [{{val|1 1 0 0 2 4}}, {{val|0 2 0 0 5 -1}}]
+[[Subgroup]]: 2.5.7/3.11/3
-[[Mapping|Sval mapping]]: [{{val|1 1 2 4}}, {{val|0 2 5 -1}}]
+[[Comma list]]: [[176/175]] ({{monzo| 4 -2 -1 1 }}), [[540/539]] ({{monzo| 2 1 -2 -1 }})
-[[Tp tuning|POL2 generator]]: ~11/9 = 351.488
+{{Mapping|legend=2| 1 0 1 3 | 0 -3 1 -5 }}
+: mapping generators: ~2, ~7/6
-{{Val list|legend=1| 7, 10, 17, 24, 41 }}
+{{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]
-[[Tp tuning #T2 tuning|RMS error]]: 0.7038 cents
+[[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
-== No-threes subgroup ==
+{{Optimal ET sequence|legend=1| 9, 22, 31, 40, 191c*, 231c*, 271c*, 311c* }}
+: <nowiki/>* wart for 7/3
-=== Didacus ===
+[[Tp tuning #T2 tuning|RMS error]]: 0.6028 cents
-Related temperaments: [[Chromatic pairs #Roulette|roulette]], [[Gamelismic clan #Hemithirds|hemithirds]]
-Subgroup: 2.5.7
+==== 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.
-[[Comma]]: 3136/3125
+[[Subgroup]]: 2.5.7/3.11/3.13
-[[Gencom]]: [2 28/25; 3136/3125]
+[[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 }})
-[[Gencom|Gencom map]]: [{{val|1 0 2 2}}, {{val|0 0 2 5}}]
+{{Mapping|legend=2| 1 0 1 3 1 | 0 -3 1 -5 12 }}
+: mapping generators: ~2, ~12/7
-[[Mapping|Sval mapping]]: [{{val|1 2 2}}, {{val|0 2 5}}]
+[[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|POL2 generator]]: ~28/25 = 93.772
+{{Optimal ET sequence|legend=1| 9, 22, 31, 40, 71, 111, 151, 262c*}} <small> using subgroup TE </small>
+: <nowiki/>* wart for 7/3
-{{Val list|legend=1| 6, 19, 25, 31, 37, 99, 130, 161, 353 }}
+Badness (Sintel): 0.329
-[[Tp tuning #T2 tuning|RMS error]]: 0.2138 cents
+==== Laz ====
+{{See also | Chromatic pairs #Laz }}
-=== Llywelyn ===
+Laz is related to [[avalokita]] as well as to [[winston]].
-Subgroup: 2.5.7
-[[Comma]]: 4194304/4117715
+[[Subgroup]]: 2.5.7/3.11/3.13/3
-[[Gencom]]: [2 8/7; 4194304/4117715]
+[[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 }}
-[[Gencom|Gencom mapping]]: [{{val|1 0 1 3}}, {{val|0 0 7 -1}}]
+{{Mapping|legend=2| 1 0 2 -2 6 | 0 3 -1 5 -5 }}
-[[Mapping|Sval mapping]]: [{{val|1 1 3}}, {{val|0 7 -1}}]
+{{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]
-[[Tp tuning|POL2 generator]]: ~8/7 = 226.910
+[[Optimal tuning]]s:
+* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~12/7 = 930.598
+* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~12/7 = 930.700
-{{Val list|legend=1| 16, 37 }}
+{{Optimal ET sequence|legend=1| 9, 31, 40, 49, 156c*†, 205c*† }}
+: <nowiki/>* wart for 7/3
+: † wart for 11/3
-[[Tp tuning #T2 tuning|RMS error]]: 0.5391 cents
+[[Tp tuning #T2 tuning|RMS error]]: 0.8790 cents
-=== Rainy ===
+=== Kryptonite ===
-Three generators make an [[8/7]]; five generators make a [[5/4]]. This is the no-threes version of [[tertiaseptal]].
+{{See also| Chromatic pairs #Kryptonite }}
-Subgroup: 2.5.7
+Kryptonite is related to [[krypton]].
-[[Comma]]: [[2100875/2097152]]
+[[Subgroup]]: 2.5.7/3.11/3.13/3
-[[Gencom]]: [2 256/245; 2100875/2097152]
+[[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 }})
-[[Gencom|Gencom mapping]]: [{{val|1 0 2 3 }}, {{val|0 0 5 -3}}]
+{{Mapping|legend=2| 1 2 1 2 2 | 0 3 2 -1 1 }}
+: mapping generators: ~2, ~13/12
-[[Mapping|Sval mapping]]: [{{val|1 2 3 }}, {{val|0 5 -3}}]
+{{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]
-[[Tp tuning|POL2 generator]]: ~256/245 = 77.205
+[[Optimal tuning]]s:
+* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~13/12 = 130.945
+* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~13/12 = 132.428
-{{Val list|legend=1| 31, 47, 78, 109, 140, 171, 202, 233 }}
+{{Optimal ET sequence|legend=1| 1, …, 8, 9 }}
-[[Tp tuning #T2 tuning|RMS error]]: 0.058596 cents
+[[Tp tuning #T2 tuning|RMS error]]: 2.545 cents
-== 2.9.7.11 subgroup ==
+=== Kiribati ===
+{{See also| Chromatic pairs #Kiribati }}
-=== Machine ===
+Kiribati is related to [[nakika]] as well as to [[octacot]].
-Subgroup: 2.9.7.11
-Commas: 64/63, 99/98
+[[Subgroup]]: 2.9/5.7/3.11/9
-[[Gencom]]: [2 8/7; 64/63 99/98]
+[[Comma list]]: 100/99 ({{monzo| 2 -2 0 -1 }}), 245/242 ({{monzo| -1 -1 2 -2 }})
-[[Gencom|Gencom mapping]]: [{{val|1 3/2 0 3 4}}, {{val|0 1/2 0 -1 -3}}]
+{{Mapping|legend=2| 1 1 1 0 | 0 -2 3 4 }}
+: mapping generators: ~2, ~21/20
-[[Mapping|Sval mapping]]: [{{val|1 0 6 13}}, {{val|0 1 -1 -3}}]
+{{Mapping|legend=3| 1 1/10 -4/5 11/10 1/5 | 0 -3/2 -1 3/2 1 }}
+: [[gencom]]: [2 21/20; 100/99 245/242]
-[[Tp tuning|POL2 generator]]: ~8/7 = 214.384
+[[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
-{{Val list|legend=1| 5, 6, 11, 17, 28 }}
+{{Optimal ET sequence|legend=1| 13, 14, 27, 41 }}
-[[Tp tuning #T2 tuning|RMS error]]: 1.977 cents
+[[Tp tuning #T2 tuning|RMS error]]: 1.245 cents
-=== Apparatus ===
+=== Mothwelltri ===
-Subgroup: 2.9.7.11
+{{See also| Chromatic pairs #Mothwelltri }}
-[[Comma list]]: 41503/41472, 322102/321489
+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.
-[[Gencom]]: [2 77/72; 41503/41472 322102/321489]
+[[Subgroup]]: 2.7/3.11
-[[Gencom|Gencom mapping]]: [{{val|1 5/2 0 3 5}}, {{val|0 -19/2 0 -2 -16}}]
+[[Comma list]]: [[99/98]] ({{monzo| -1 -2 1 }})
-[[Mapping|Sval mapping]]: [{{val|1 5 3 5}}, {{val|0 -19 -2 -16}}]
+{{Mapping|legend=2| 1 0 1 | 0 1 2 }}
+: mapping generators: ~2, ~7/3
-[[Tp tuning|POL2 generator]]: ~77/72 = 115.570
+{{Mapping|legend=3| 1 -1/2 0 1/2 3 | 0 -1/2 0 1/2 2 }}
+: [[gencom]]: [2 7/6; 99/98]
-{{Val list|legend=1| 10, 21, 31, 52, 83, 135, 353, 488, 623 }}
+[[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
-[[Tp tuning #T2 tuning|RMS error]]: 0.0673 cents
+{{Optimal ET sequence|legend=1| 4, 9, 13, 22, 79 }}
-=== Mechanism ===
+[[Tp tuning #T2 tuning|RMS error]]: 1.064 cents
-Subgroup: 2.9.7.11
-[[Comma list]]: 896/891, 26411/26244
+== 2.….9/7.… subgroups ==
+=== Marveltri ===
+{{See also| Chromatic pairs #Marveltri }}
-[[Gencom]]: [2 9/7; 896/891 26411/26244]
+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.
-[[Gencom|Gencom mapping]]: [{{val|1 5/2 0 5 2}}, {{val|0 -5/2 0 -6 4}}]
+[[Subgroup]]: 2.5.9/7
-[[Mapping|Sval mapping]]: [{{val|1 5 5 2}}, {{val|0 -5 -6 4}}]
+[[Comma list]]: 225/224 ({{monzo| -5 2 1 }})
-[[Tp tuning|POL2 generator]]: ~9/7 = 438.465
+{{Mapping|legend=2| 1 0 5 | 0 1 -2 }}
+: mapping generators: ~2, ~5
-{{Val list|legend=1| 8, 11, 30, 41, 52 }}
+{{Mapping|legend=3| 1 2 0 -1 | 0 -4/5 1 2/5 }}
+: [[gencom]]: [2 5; 225/224]
-[[Tp tuning #T2 tuning|RMS error]]: 0.4262 cents
+[[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
-== 2.9.15.7 subgroup ==
+{{Optimal ET sequence|legend=1| 3, 13, 16, 19, 22, 25, 72, 97, 122, 269c* }}
+: <nowiki/>* wart for 9/7
-=== Stacks (aka 2magic) ===
+[[Tp tuning #T2 tuning|RMS error]]: 0.4801 cents
-Subgroup: 2.9.15.7
-[[Comma list]]: 225/224, 245/243
+==== Sulis ====
+Sulis is related to [[minerva]] and [[würschmidt]].
-[[Gencom]]: [2 9/7; 225/224 245/243]
+[[Subgroup]]: 2.5.9/7.11/9
-[[Gencom|Gencom mapping]]: [{{val|1 5/2 5/2 5}}, {{val|0 -5/2 -1/2 -6}}]
+[[Comma list]]: 99/98 ({{monzo| -1 0 2 1 }}), 176/175 ({{monzo| 4 -2 1 1 }})
-[[Mapping|Sval mapping]]: [{{val|1 0 2 -1}}, {{val|0 5 3 6}}]
+{{Mapping|legend=2| 1 0 5 -9 | 0 1 -2 4 }}]
-[[Tp tuning|POL2 generator]]: ~9/7 = 439.296
+[[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
-{{Val list|legend=1| 8, 11, 30, 41, 71, 93, 112c, 134c, 175c }}
+{{Optimal ET sequence|legend=1| 3, …, 22, 25, 28, 31, 59 }}
 [[Tp tuning #T2 tuning|RMS error]]: 1.074 cents
-==== 2.9.15.7.11 ====
+== 2.….7/5.… subgroups ==
-Subgroup: 2.9.15.7.11
+=== Hydrothermal ===
+A tuning whose distinctively sharp (but still consonant) fifth, and flat (but still consonant) octave, lend it a mysterious, heavy atmosphere. The 6-tone (hexatonic) MOS is melodically interesting and flavorful. The 18-tone MOS is a useful 'chromatic' scale for taking subsets of.
+[[Subgroup]]: 2.3.7/5
+[[Comma list]]: [[50/49]]
+{{Mapping|legend=2| 2 3 1 | 0 1 0 }}
+[[Optimal tuning]] (inharmonic [[TE]]): ~1\2 = 590.998, ~[[10/7]]-1\2 = 128.962
+[[Support]]ing [[ET]]s: {{EDOs|4, 6, 8, 10, 18, 28, 46, 64, 110}}
-Comma list: 100/99, 225/224, 245/243
+=== Argentic ===
+Argentic is the 2.3.7/5 subgroup temperament tempering out [[5120/5103]].
-Gencom: [2 9/7; 100/99 225/224 245/243]
+[[Subgroup]]: 2.3.7/5
-Gencom mapping: [{{val|1 5/2 5/2 5 2}}, {{val|0 -5/2 -1/2 -6 4}}]
+[[Comma list]]: [[5120/5103]] = {{monzo| 10 -6 -1 }}
-Sval mapping: [{{val|1 0 2 -1 6}}, {{val|0 5 3 6 -4}}]
+{{Mapping|legend=2| 1 0 10 | 0 1 -6 }}
+: mapping generators: ~2, ~3
-POL2 generator: ~9/7 = 438.607
+[[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
-Vals: {{Val list| 8, 11, 30, 41, 52, 93, 145, 342bce }}
+{{Optimal ET sequence|legend=1| 12, 29, 41, 70, 321, 391, 461, 531, 601 }}
+<small> based on subgroup TE </small>
-RMS error: 1.226 cents
+Badness (Sintel): 0.119
-==== 2.9.15.7.11.13 ====
+==== Edson (2.3.7/5.11/5.13/5 subgroup) ====
-Subgroup: 2.9.15.7.11.13
+{{See also| Chromatic pairs #Edson }}
-Comma list: 100/99, 105/104, 144/143, 196/195
+Edson is related to [[pele]] and [[andromeda]].
-Gencom: [2 9/7; 100/99 105/104 144/143 196/195]
+[[Subgroup]]: 2.3.7/5.11/5.13/5
-Gencom mapping: [{{val|1 5/2 5/2 5 2 7}}, {{val|0 -5/2 -1/2 -6 4 -9}}]
+[[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 }}
-Sval map: [{{val|1 0 2 -1 6 -2}}, {{val|0 5 3 6 -4 9}}]
+{{Mapping|legend=2| 1 0 10 17 22 | 0 1 -6 -10 -13 }}
+: mapping generators: ~2, ~3
-POL2 generator: ~9/7 = 438.977
+{{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]
-Vals: {{Val list| 11, 30, 41, 153cdef, 194cdef, 235cdef }}
+[[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
-RMS error: 1.540 cents
+{{Optimal ET sequence|legend=1| 12, 17, 29 }}
-== 2.9.21 subgroup ==
+[[Tp tuning #T2 tuning|RMS error]]: 0.5102 cents
-=== A-team ===
+==== Haumea ====
-Subgroup: 2.9.21
+{{See also| Chromatic pairs #Haumea }}
-[[Comma]]: 1029/1024
+Related temperaments include [[#Bridgetown|bridgetown]], [[namaka]], [[hemigari]], [[#Barbados|barbados]], and [[parizekmic]].
-[[Gencom]]: [2 21/16; 1029/1024]
+[[Subgroup]]: 2.3.7/5.11/5.13/5
-[[Gencom|Gencom mapping]]: [{{val|1 1 0 3}}, {{val|0 3/2 0 -1/2}}]
+[[Comma list]]: [[352/351]], [[676/675]], [[847/845]]
-[[Mapping|Sval mapping]]: [{{val|1 2 4}}, {{val|0 3 1}}]
+{{Mapping|legend=2| 1 0 10 -6 -1 | 0 2 -12 9 3 }}
-[[Tp tuning|POL2 generator]]: ~21/16 = 467.375
+{{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]
-{{Val list|legend=1| 5, 13, 18, 41, 59, 77, 95 }}
+[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~15/13 = 248.491
-[[Tp tuning #T2 tuning|RMS error]]: 0.3202 cents
+{{Optimal ET sequence|legend=1| 24, 29, 111, 140, 169, 198, 565d, 763bd, 961bd }}
-== Miscellaneous subgroup temperaments ==
+[[Tp tuning #T2 tuning|RMS error]]: 0.2668 cents
 === Historical ===
-Subgroup: 2.3.7/5.11/5.13/5
+{{distinguish|Historical temperaments}}
+{{distinguish|History (temperament)}}, which is the rank-3 version of this temperament in the full 13-limit.
+Historical is essentially an analogue of [[miracle]] that splits [[4/3]] in six rather than [[3/2]]. It tempers out the comma S10/S11 = [[4000/3993]] to set [[11/10]] equal to one-third of 4/3, and S13/S15 = [[676/675]] to equate [[15/13]] to one-half of 4/3, and tempers out S21 = [[441/440]] to split 11/10 into two instances of [[22/21]]~[[21/20]]. [[Sextilifourths]] adds the [[schismic]] mapping of prime 5 (reached by eight fourths) to complete the 13-limit.
+[[Subgroup]]: 2.3.7/5.11/5.13/5
 [[Comma list]]: 364/363, 441/440, 1001/1000
-[[Mapping|Sval mapping]]: [{{val|1 2 0 1 2}}, {{val|0 -6 7 2 -9}}]
+{{Mapping|legend=2| 1 2 0 1 2 | 0 -6 7 2 -9 }}
-[[Tp tuning|POL2 generator]]: ~21/20 = 83.016
+[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~21/20 = 83.016
-{{Val list|legend=1| 14, 29, 72, 101, 130, 159 }}
+{{Optimal ET sequence|legend=1| 14, 29, 72, 101, 130, 159 }}
 [[Tp tuning #T2 tuning|RMS error]]: 0.2562 cents
+=== Terrain ===
+{{Redirect|Terrain|the scale|Terrain (scale)}}
+{{See also| Chromatic pairs #Terrain }}
+Terrain, the 6 &amp; 21 temperament in the 2.7/5.9/5 subgroup, is related to [[domain (temperament)|domain]]. It is a remarkable temperament, in that while its complexity is low, it has no discernible error. The 1–7/5–9/5 and 1–9/7–9/5 chords are characteristic.
+[[Subgroup]]: 2.7/5.9/5
+[[Comma list]]: [[250047/250000]]
+{{Mapping|legend=2| 3 1 3 | 0 1 -1 }}
+{{Mapping|legend=3| 3 10/9 -7/9 2/9 | 0 -2/3 -1/3 2/3 }}
+: [[gencom]]: [63/50 10/9; 250047/250000]
+[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~63/50 = 1\3, ~10/9 = 182.461
+{{Optimal ET sequence|legend=1| 6, 21, 27, 33, 105, 138, 171, 1848, 2019, 2190, 2361, 2532, 2703, 2874, 3045, 3216, 3387, 3558 }}
+[[Tp tuning #T2 tuning|RMS error]]: 0.00844 cents
+=== Tridec ===
+{{See also| Chromatic pairs #Tridec }}
+{{See also| Non-over-1 temperament #Tridec }}
+Tridec, the 5 &amp; 8 temperament in the 2.7/5.11/5.13/5 subgroup, extends [[#Petrtri]].
+[[Subgroup]]: 2.7/5.11/5.13/5
+[[Comma list]]: [[847/845]], [[1001/1000]]
+{{Mapping|legend=2| 1 2 0 1 | 0 -4 3 1 }}
+{{Mapping|legend=3| 1 0 -3/4 5/4 -3/4 1/4 | 0 0 0 -4 3 1 }}
+: [[gencom]]: [2 13/10; 847/845 1001/1000]
+[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~13/10 = 454.556
+{{Optimal ET sequence|legend=1| 5, 8, 21, 29, 37, 66, 169, 235, 404c, 639c, 953bc }}
+[[Tp tuning #T2 tuning|RMS error]]: 0.1613 cents
+==== Naiadec ====
+[[Subgroup]]: 2.7/5.11/5.13/5.17/5
+[[Comma list]]: [[170/169]], [[221/220]], [[847/845]]
+{{Mapping|legend=2| 1 2 0 1 1 | 0 -4 3 1 2 }}
+{{Mapping|legend=3| 1 0 -3/4 5/4 -3/4 1/4 1/4 | 0 0 0 -4 3 1 2 }}
+: [[gencom]]: [2 13/10; 170/169 221/220 847/845]
+[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~13/10 = 454.882
+{{Optimal ET sequence|legend=1| 5, 8, 21, 29, 95<sup>t</sup>, 124<sup>t</sup> }}
+: <sup>t</sup> wart for 17/5
+[[Tp tuning #T2 tuning|RMS error]]: 0.7521 cents
+== 2.….11/5.… subgroups ==
+=== Petrtri ===
+{{See also| Chromatic pairs #Petrtri }}
+{{See also| 5L 3s/Temperaments #Petrtri }}
+Petrtri can be described as 3 &amp; 5 temperament in the 2.11/5.13/5 subgroup.
+[[Subgroup]]: 2.11/5.13/5
+[[Comma list]]: [[2200/2197]]
+{{Mapping|legend=2| 1 0 1| 0 3 1 }}
+{{Mapping|legend=3| 1 0 -1/3 0 -1/3 2/3 | 0 0 -4/3 0 5/3 -1/3 }}
+: [[gencom]]: [2 13/10; 2200/2197]
+[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~13/10 = 455.012
+{{Optimal ET sequence|legend=1| 21, 29, 153, 182, 211, 240, 269, 298, 327, 356, 385, 509, 741c, 1126c }}
+[[Tp tuning #T2 tuning|RMS error]]: 0.0749 cents
+==== Bridgetown ====
+{{See also| Chromatic pairs #Bridgetown }}
+Bridgetown, the 5 &amp; 24 temperament in the 2.3.11/5.13/5 subgroup, is related to [[#Haumea|haumea]] and [[#Barbados|barbados]].
+[[Subgroup]]: 2.3.11/5.13/5
+[[Comma list]]: [[352/351]], [[676/675]]
+{{Mapping|legend=2| 1 0 -6 -1 | 0 2 9 3 }}
+{{Mapping|legend=3| 1 2 -5/3 0 4/3 1/3 | 0 -2 4 0 -5 1 }}
+: [[gencom]]: [2 15/13; 352/351 676/675]
+[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~15/13 = 248.399
+{{Optimal ET sequence|legend=1| 5, 9, 14, 19, 24, 29, 169, 198, 227, 256, 285, 314 }}
+[[Tp tuning #T2 tuning|RMS error]]: 0.2513 cents
 === Hypnosis ===
-Related temperament: [[Swetismic temperaments #Hypnos|hypnos]], [[Hemifamity temperaments #Tricot|tricot]]
+Related temperaments: [[Swetismic temperaments #Hypnos|hypnos]], [[Alphatricot family #Alphatricot|alphatricot]]
-Subgroup: 2.3.7.11/5.13
+[[Subgroup]]: 2.3.7.11/5.13
 [[Comma list]]: 169/168, 540/539, 729/728
-[[Mapping|Sval mapping]]: [{{val|1 0 -3 8 0}}, {{val|0 3 11 -13 7}}]
+{{Mapping|legend=2| 1 0 -3 8 0 | 0 3 11 -13 7 }}
-[[Tp tuning|POL2 generator]]: ~13/9 = 633.518
+[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~13/9 = 633.518
-{{Val list|legend=1| 17, 36, 118e, 125e, 161e, 197e }}
+{{Optimal ET sequence|legend=1| 17, 36, 118f, 125f, 161f, 197f }}
 [[Tp tuning #T2 tuning|RMS error]]: 0.5379 cents
-=== Ultrapyth ===
+=== Trisect ===
-Subgroup: 2.3.7.13/10
+Trisect divides every Pythagorean interval into three, and is the much more accurate subgroup restriction of [[Augmented family #Trisected|trisected]].
+Extending this temperament to the full [[11-limit|11-]], [[13-limit|13-]], or [[17-limit]] through [[portent]] or [[landscape]] results in the [[weak extension]] known as [[tritikleismic]].
+[[Subgroup]]: 2.3.7.11/5
+[[Comma list]]: 1029/1024, 4000/3993
+{{Mapping|legend=2| 3 0 10 5 | 0 3 -1 -1 }}
+[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~44/35 = 1\3, ~13/9 = 633.742
+{{Optimal ET sequence|legend=1| 15, 21, 36, 123, 159, 195, 231 }}
+[[Tp tuning #T2 tuning|RMS error]]: ???
+==== 2.3.7.11/5.13 subgroup ====
+[[Subgroup]]: 2.3.7.11/5.13
+[[Comma list]]: 1029/1024, 1575/1573, 2080/2079
+{{Mapping|legend=2| 3 0 10 5 0 | 0 3 -1 -1 7 }}
+[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~44/35 = 1\3, ~13/9 = 633.918
+{{Optimal ET sequence|legend=1| 15, 21f, 36, 87, 123, 159 }}
+[[Tp tuning #T2 tuning|RMS error]]: ???
+==== 2.3.7.11/5.13.17 subgroup ====
+[[Subgroup]]: 2.3.7.11/5.13.17
+[[Comma list]]: 273/272, 833/832, 1575/1573, 2080/2079
+{{Mapping|legend=2| 3 0 10 5 0 -2 | 0 3 -1 -1 7 9 }}
+[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~34/27 = 1\3, ~13/9 = 633.820
+{{Optimal ET sequence|legend=1| 15, 21fg, 36, 123, 159 }}
+[[Tp tuning #T2 tuning|RMS error]]: ???
+===== Trisector =====
+[[Subgroup]]: 2.3.7.11/5.13.17.19
+[[Comma list]]: 210/209, 273/272, 286/285, 595/594, 2080/2079
+{{Mapping|legend=2| 3 0 10 5 0 -2 8 | 0 3 -1 -1 7 9 3 }}
+[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~34/27 = 1\3, ~13/9 = 633.894
+{{Optimal ET sequence|legend=1| 15, 21fg, 36, 123h, 159h }}
+[[Tp tuning #T2 tuning|RMS error]]: ???
+===== 2.3.7.11/5.13.17.19.23 subgroup =====
+[[Subgroup]]: 2.3.7.11/5.13.17.19.23
+[[Comma list]]: 210/209, 231/230, 273/272, 286/285, 595/594, 2080/2079
+{{Mapping|legend=2| 3 0 10 5 0 -2 8 12 | 0 3 -1 -1 7 9 3 1 }}
+[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~34/27 = 1\3, ~13/9 = 634.038
+{{Optimal ET sequence|legend=1| 15g, 21fg, 36, 87, 123hi }}
+[[Tp tuning #T2 tuning|RMS error]]: ???
+===== 2.3.7.11/5.13.17.19.23.29 subgroup =====
+[[Subgroup]]: 2.3.7.11/5.13.17.19.23.29
+[[Comma list]]: 210/209, 231/230, 273/272, 286/285, 320/319, 595/594, 2080/2079
+{{Mapping|legend=2| 3 0 10 5 0 -2 8 12 13 | 0 3 -1 -1 7 9 3 1 1 }}
+[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~29/23 = 1\3, ~13/9 = 634.102
+{{Optimal ET sequence|legend=1| 15g, 21fg, 36, 87, 123hi }}
+[[Tp tuning #T2 tuning|RMS error]]: ???
+== 2.….11/7.… subgroups ==
+=== Blackweed ===
+Blackweed is a [[restriction]] of undecimal [[blackwood]] as it tempers out 256/243 alike but in the 2.3.11/7 subgroup. 20edo is close to the optimum, which has 4\20 as the period and 420{{c}} as the generator.
+[[Subgroup]]: 2.3.11/7
+[[Comma list]]: {{monzo| 8 -5 }} (256/243)
+{{Mapping|legend=2| 5 8 0 | 0 0 1 }}
+: mapping generators: ~9/8, ~11/7
+[[Optimal tuning]]s:
+* [[Tp tuning|subgroup]] [[WE]]: ~8/7 = 238.851{{c}}, ~11/7 = 782.457{{c}}
+: [[error map]]: {{val| -5.746 +8.852 -0.035 }}
+* [[Tp tuning|subgroup]] [[CWE]]: ~8/7 = 240.000{{c}}, ~11/7 = 784.967{{c}}
+: error map: {{val| 0.000 +18.045 +2.475 }}
+{{Optimal ET sequence|legend=1| 15, 20, 35b, 55b }}
+=== Pepperoni ===
+{{Main| Parapyth }}
+{{See also| Chromatic pairs #Pepperoni }}
+Pepperoni is generated by a fifth and can be described as the 5 &amp; 12 temperament in the 2.3.11/7.13/7 subgroup. It is the single-chain retraction of [[parapyth]]. The [[Peppermint-24|Pepper fifth]], which is (40200 + 600 sqrt(5))/59 = 704.096 cents, is a good pepperoni generator, hence the name.
+[[Subgroup]]: 2.3.11/7.13/7
+[[Comma list]]: 352/351, 364/363
+{{Mapping|legend=2| 1 0 7 12 | 0 1 -4 -7 }}
+{{Mapping|legend=3| 1 1 0 -8/3 1/3 7/3 | 0 1 0 11/3 -1/3 -10/3 }}
+: [[gencom]]: [2 3/2; 352/351 364/363]
+[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~3/2 = 703.856
+{{Optimal ET sequence|legend=1| 5, 7, 12f, 17, 29, 46, 58, 75, 80, 87, 104, 121, 167, 196, 208, 271, 595b*<sup>†</sup> }}
+: <nowiki />* wart for 11/7
+: <sup>†</sup> wart for 13/7
+[[Tp tuning #T2 tuning|RMS error]]: 0.3789 cents
+== 2.….13/5.… subgroups ==
+=== Barbados ===
+The [[minimax tuning]] for this makes the generator the cube root of 20/13, or 248.5953 cents. Edos which may be used for it are [[24edo]], [[29edo]], [[53edo]] and [[111edo]], with [[mos scale]]s of size 5, 9, 14, 19, 24 and 29 making for a good variety of scales.
+[[Subgroup]]: 2.3.13/5
+[[Comma list]]: 676/675 = {{monzo| 2 -3 2 }}
+[[Sval]] [[mapping]]: [{{val| 1 0 -1 }}, {{val| 0 2 3 }}]
+[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~2 = 1\1, ~15/13 = 248.621
+{{Optimal ET sequence|legend=1| 5, 9, 14, 19, 24, 29, 53, 82, 111, 140, 251, 362 }}
+[[Badness]]: 0.002335
+; Music
+* [http://micro.soonlabel.com/gene_ward_smith/Others/Sevish/Sevish%20-%20Desert%20Island%20Rain.mp3 ''Desert Island Rain''] in 313edo tuned Barbados[9], by [https://soundcloud.com/sevish/desert-island-rain Sevish]
+==== Tobago ====
+{{See also| Chromatic pairs #Tobago }}
+Tobago, the 10 &amp; 14 temperament in the 2.3.11.13/5 subgroup, extends [[neutral]] and [[barbados]].
+[[Subgroup]]: 2.3.11.13/5
+[[Comma list]]: [[243/242]], [[676/675]]
+{{Mapping|legend=2| 2 0 -1 -2 | 0 2 5 3 }}
+{{Mapping|legend=3| 2 4 -2 0 9 2 | 0 -2 3/2 0 -5 -3/2 }}
+: [[gencom]]: [55/39 15/13; 243/242 676/675]
+[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~55/39 = 1\2, ~15/13 = 249.312
+{{Optimal ET sequence|legend=1| 10, 14, 24, 58, 82, 130 }}
+[[Tp tuning #T2 tuning|RMS error]]: 0.3533 cents
+==== Pakkanian hemipyth ====
+[[Subgroup]]: 2.3.11.13/5.17
+[[Comma list]]: 221/220, 243/242, 289/288
+{{Mapping|legend=2| 2 0 -1 -2 5 | 0 2 5 3 2 }}
+[[Optimal tuning]]s:
+* [[Tp tuning|subgroup CTE]]: ~17/12 = 1\2, ~26/15 = 950.7656 (~15/13 = 249.2344)
+* [[Tp tuning|subgroup CWE]]: ~17/12 = 1\2, ~26/15 = 950.6011 (~15/13 = 249.3989)
+{{Optimal ET sequence|legend=1| 10, 14, 24, 106, 130, 154, 178*, 202* }}
+: <nowiki />* wart for 13/5
+=== Oceanfront ===
+Related temperaments: [[Archytas clan #Superpyth|superpyth]], [[Archytas clan #Ultrapyth|ultrapyth]]
+[[Subgroup]]: 2.3.7.13/5
 [[Comma list]]: 64/63, 91/90
-[[Mapping|Sval mapping]]: [{{val|1 0 6 -6}}, {{val|0 1 -2 4}}]
+{{Mapping|legend=2| 1 0 6 -5 | 0 1 -2 4 }}
-[[Tp tuning|POL2 generator]]: ~3/2 = 713.7823
+[[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
-[[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 }}

Subgroup temperaments: Difference between revisions