Subgroup temperaments: Difference between revisions

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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]].
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]].


= Composite subgroup temperaments =
This page uses an extended [[wart notation]] where squares and cubes of a prime basis element are denoted by ′ and ″, respectively, to reduce the use of temporary warts.
 
== 2.3.… subgroups ==
== 2.3.… subgroups ==
=== Darian calendar ===
=== Shrub ===
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]].
Named by [[User: VectorGraphics|Vector]] in 2026, shrub is a [[restriction]] of [[diaschismic family #Diaschismic|diaschismic]] which omits the tritone to produce a [[5L 2s|diatonic]] scale. True to its name, it generates a [[neogothic major and minor|shrubmajor]]{{idio}} third (~425{{c}}) in quarter-comma tuning. It has an equal-temperament join of 12 & 17.  


==== 2.3.11.19 subgroup ====
It can also admit a [[41/1|41st harmonic]] by tempering out [[82/81]] (same mapping as [[reversed meantone]]). However, its tuning tends to be flat, and with the rest of the harmonics tending sharp, this will increase the overall error.  
The temperament is simplest in this subgroup, although there is a tradeoff of breaking up the simplicity of the 36/35 quartertone.


[[Subgroup]]: 2.3.11.19
==== 2.3.25 subgroup ====
[[Subgroup]]: 2.3.25


{{Mapping|legend=3| 4 5 13 18 | 0 8 5 -6 }}
[[Comma list]]: 2048/2025


: sval mapping generators: ~6291456/5285401, ~25289/24576
{{Mapping|legend=3| 1 0 11 | 0 1 -4 }}
: mapping generators: ~2, ~3


[[Optimal tuning]] ([[CTE]]): ~6291456/5285401 = 1\4, ~25289/24576 = 50.257
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1198.8214{{c}}, ~3/2 = 704.2059{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.9585{{c}}


[[Support]]ing [[ET]]s: {{EDOs|24, 596, 620, 644, 668, 692, 716}}, ...
{{Optimal ET sequence|legend=1| 12, 17, 46, 63, 80, 143c′ }}


==== 2.3.35.11.19 subgroup ====
[[Badness]] (Sintel): 0.234
668edo does not map 36/35 consistently, with its own [[direct approximation]] being 27 steps while the direct approximations of its constituent odd harmonics do not sum to that same amount: 3/2, 8/5, and 8/7 are 391, 453, and 129 steps, respectively, and 391 + 391 + 453 + 129 - 668 - 668 = 28, ≠ 27.


Subgroup: 2.3.35.11.19
===== 2.3.25.23 subgroup =====
Subgroup: 2.3.25.23


Sval mapping: {{mapping| 4 0 5 13 18 | 0 1 8 5 -6 }}
Comma list: 576/575, 736/729


: sval mapping generators: ~2240/1881, ~36/35
{{Mapping|legend=2| 1 0 11 -5 | 0 1 -4 6 }}


Optimal tuning (CTE): ~2240/1881 = 1\4, ~36/35 = 50.288
Optimal tunings:  
* WE: ~2 = 1198.9504{{c}}, ~3/2 = 704.6585{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 705.1888{{c}}


[[Support]]ing [[ET]]s: {{EDOs|24, 668}}, ...
{{Optimal ET sequence|legend=0| 12, 17, 63, 80, 97c′, 114c′ }}


=== Shrub ===
Badness (Sintel): 0.223
This is a restriction of diaschismic which omits the tritone to produce a diatonic scale. True to its name, it generates a [[shrubmajor]] third (~425c) in quarter-comma tuning.


Subgroup: 2.3.25
==== Sburb ====
Sburb is every other step of [[pajara]], extended to the add-23 add-41 subgroup in the same way as shrub. It can also admit a [[59/1|59th harmonic]] by setting the [[octave reduction|octave-reduced]] 413th harmonic (413/256, 827.998{{c}}) to the diminished seventh.


Edo join: 17 & 12
Subgroup: 2.3.25.7


Comma list: [[2048/2025]]
Comma list: 50/49, 64/63


{{Mapping|legend=3| 1 1 7| 0 1 -4}}
{{Mapping|legend=2| 1 0 11 6 | 0 1 -4 -2 }}


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 705.136
Optimal tunings:
* WE: ~2 = 1197.6967{{c}}, ~3/2 = 705.6906{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 707.3438{{c}}


==== 2.3.23.25.41 subgroup ====
{{Optimal ET sequence|legend=0| 5, 12, 17, 39d }}
''See also: [[Reversed meantone]]''


Edo join: 17 & 12
Badness (Sintel): 0.253


Comma list: 2048/2025, 576/575, 82/81
===== 2.3.25.7.23 subgroup =====
Subgroup: 2.3.25.7.23


{{Mapping|legend=3| 1 1 1 7 3| 0 1 6 -4 4}}
Comma list: 50/49, 64/63, 162/161


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 705.264
{{Mapping|legend=2| 1 0 11 6 -5 | 0 1 -4 -2 6 }}


===== Sburb =====
Optimal tunings:
This temperament sets the [[octave reduction|octave-reduced]] 413th harmonic (413/256, 827.998{{c}}) to the diminished seventh.
* WE: ~2 = 1197.5577{{c}}, ~3/2 = 705.3089{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 706.7010{{c}}


Subgroup: 2.3.7.23.25.41.59
{{Optimal ET sequence|legend=0| 5, 12, 17 }}


Edo join: 17 & 12
Badness (Sintel): 0.419


Comma list: 64/63, 225/224, 162/161, 82/81, 177/175
=== Hypnosis ===
Related temperaments: [[swetismic temperaments #Hypnos|hypnos]], [[alphatricot family #Alphatrimot|alphatrimot]].


{{Mapping|legend=3| 1 1 4 1 7 3 10| 0 1 -2 6 -4 4 -7}}
[[Subgroup]]: 2.3.7.11/5.13


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 706.387
[[Comma list]]: 169/168 ({{monzo| -3 -1 -1 0 2 }}), 540/539 ({{monzo| 2 3 -2 -1 0 }}), 729/728 ({{monzo| -3 6 -1 0 -1 }})


== 2.9.… subgroups ==
{{Mapping|legend=3| 1 0 -3 8 0 | 0 3 11 -13 7 }}
See also [[Jubilismic clan #Antikythera|antikythera]] and [[Hemimean clan #Isra|isra]].
: mapping generators: ~2, ~13/9


=== Commatose ===
[[Optimal tuning]]s:
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|Subgroup]] [[WE]]: ~2 = 1200.6306{{c}}, ~13/9 = 633.8505{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~13/9 = 633.5546{{c}}


[[Subgroup]]: 2.9.5.7
{{Optimal ET sequence|legend=1| 17, 36, 125f, 161f, 197f }}


[[Comma list]]: {{monzo| 28 -2 -19 8 }}, {{monzo| 9 -25 23 6 }}
[[Tp tuning #T2 tuning|RMS error]]: 0.5379 cents


{{Mapping|legend=3| 1 9 6 13 | 0 -298 -188 -521 }}
=== Darian calendar ===
Darian calendar is described as the 24 & 668 temperament in the 2.3.35.11.19 subgroup. The generator is close to [[36/35]]. Five of them make [[11/8]], six of them make [[32/19]], and eight of them make [[3/2]].


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~531441/524288 = 23.4765
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.


{{Optimal ET sequence|legend=1| 460, 869, 1329 }}
Darian calendar was named by [[Eliora]] in 2023 after a certain calendar layout by the same name.


[[Badness]]: 0.611
[[Subgroup]]: 2.3.35.11.19


==== 2.9.5.7.11 ====
[[Comma list]]: 29282/29241 ({{monzo| 1 -4 0 4 -2 }}), 42875/42768 ({{monzo| -4 -5 3 -1 0 }}), 885115/884736 ({{monzo| -15 -3 1  3 1 }})
Subgroup: 2.9.5.7.11


Comma list: {{monzo| -7 7 -3 2 -4 }}, {{monzo| 17 0 -13 1 3 }}, {{monzo| 11 -2 -6 7 -3 }}
{{Mapping|legend=3| 4 5 18 13 18 | 0 8 15 5 -6 }}
: mapping generators: ~2240/1881, ~36/35


Sval mapping: {{mapping| 1 9 6 13 16 | 0 -298 -188 -521 -641 }}
[[Optimal tuning]]s:  
* [[Tp tuning|Subgroup]] [[WE]]: ~2240/1881 = 299.9912{{c}}, ~36/35 = 50.2951{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2240/1881 = 300.0000{{c}}, ~36/35 = 50.2947{{c}}


Optimal tuning (CTE): ~2 = 1\1, ~531441/524288 = 23.4767
{{Optimal ET sequence|legend=1| 24, …, 524, 548, 572, 596, 620, 644, 668 }}


{{Optimal ET sequence|legend=1| 460, 869e, 1329, 1789, 3118 }}
=== Hydrothermal ===
Named by [[Budjarn Lambeth]] in 2024, hydrothermal tempers out 50/49, the [[jubilisma]], in the 2.3.7/5 subgroup. A tuning whose distinctively sharp (but still consonant) fifth, and flat (but still consonant) octave, will lend it a mysterious, heavy atmosphere. The 6-tone [[mos]] is melodically interesting and flavorful. The 18-tone mos is a useful "chromatic" scale for taking subsets of.


Badness: 0.165
[[Subgroup]]: 2.3.7/5


==== 2.9.5.7.11.13 ====
[[Comma list]]: 50/49 ({{monzo| 1 0 -2 }})
Subgroup: 2.9.5.7.11.13


Comma list: 123201/123200, 1016064/1015625, 2250423/2249390, 2599051/2598156
{{Mapping|legend=3| 2 0 1 | 0 1 0 }}
: mapping generators: ~7/5, ~3


Sval mapping: {{mapping| 0 9 6 13 16 10 | -298 -188 -521 -641 -322 }}
[[Optimal tuning]]s:  
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 599.6673{{c}}, ~3/2 = 702.5906{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 600.0000{{c}}, ~3/2 = 702.4574{{c}}


Optimal tuning (CTE): ~2 = 1\1, ~3575/3528 = 23.4767
{{Optimal ET sequence|legend=1| 2, 6, 8, 10, 12, 34, 46* }}
: <nowiki/>* Wart for 7/5


{{Optimal ET sequence|legend=1| 460, 869e, 1329, 1789, 3118 }}
=== Argentic ===
Argentic is the 2.3.7/5-subgroup temperament tempering out 5120/5103, the [[aberschisma]].


Badness: 0.0564
[[Subgroup]]: 2.3.7/5


=== Daemotertiaschis ===
[[Comma list]]: 5120/5103 ({{monzo| 10 -6 -1 }})
{{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 (hence the name ― daemo- + tertiaschis), which is notoriously difficult to approximate with simple JI or RTT methods.


Subgroup: 2.9.5.7.33.13.17
{{Mapping|legend=3| 1 0 10 | 0 1 -6 }}
: mapping generators: ~2, ~3


Comma list: 325/324, 375/374, 385/384, 595/594, 10985/10976
[[Optimal tuning]]s:  
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.7172{{c}}, ~3/2 = 702.6636{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.8166{{c}}


{{Mapping|legend=2|1 1 11 -16 13 -18 20|0 3 -12 26 -11 30 -22}}
{{Optimal ET sequence|legend=1| 12, 29, 41, 70, 321, 391, 461, 531, 601 }}


Optimal tuning (CTE): ~2 = 1\1, 33/20 = 867.982
[[Badness]] (Sintel): 0.119


[[Support]]ing [[ET]]s: {{Optimal ET sequence|47, 65f, 112, 159, 206, 253}}
==== Edson ====
{{See also| Chromatic pairs #Edson }}


=== Baldy ===
Edson is related to [[aberschismic family #Pele|pele]] and [[schismatic family #Andromeda|andromeda]].
{{See also|Schismatic family #Garibaldi}}
{{See also|No-threes subgroup temperaments #Frostburn}}


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.
[[Subgroup]]: 2.3.7/5.11/5.13/5


[[Subgroup]]: 2.9.5.7
[[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 }})


[[Comma list]]: 225/224, 3125/3087
{{Mapping|legend=3| 1 0 10 17 22 | 0 1 -6 -10 -13 }}


{{Mapping|legend=3| 1 3 3 4 | 0 1 -4 -7 }}
{{Mapping|legend=5| 1 0 -49/4 -9/4 19/4 39/4 | 0 1 29/4 5/4 -11/4 -23/4 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 204.170
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.4965{{c}}, ~3/2 = 703.1192{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.4225{{c}}


{{Optimal ET sequence|legend=1| 6, 29, 35, 41, 47 }}
{{Optimal ET sequence|legend=1| 12, 17, 29 }}


Related temperament: [[Schismatic family #Garibaldi|Garibaldi]]
[[Tp tuning #T2 tuning|RMS error]]: 0.5102 cents


==== 2.9.5.7.13 ====
==== Haumea ====
{{See also|Chromatic pairs #Baldy}}
{{See also| Chromatic pairs #Haumea }}


Baldy is every other step of [[garibaldi]], without the mapping of prime 11. It can be described as the 6 &amp; 35 temperament.  
Haumea is the add-7/5 extension of [[#Bridgetown|bridgetown]], which is in turn the add-11/5 extension of [[#Barbados|barbados]]. Other related temperamnets include [[schismatic family #Hemigari|hemigari]], [[the Archipelago #Parizekmic|parizekmic]], and [[aberschismic family #Namaka|namaka]].  


[[Subgroup]]: 2.9.5.7.13
[[Subgroup]]: 2.3.7/5.11/5.13/5


[[Comma list]]: [[225/224]], [[325/324]], [[640/637]]
[[Comma list]]: [[352/351]] ({{monzo| 5 -3 0 1 -1 }}), [[676/675]] ({{monzo| 2 -3 0 0 2 }}), [[847/845]] ({{monzo| 0 0 1 2 -2 }})


{{Mapping|legend=3| 1 0 15 25 -28 | 0 1 -4 -7 10 }}
{{Mapping|legend=3| 1 0 10 -6 -1 | 0 2 -12 9 3 }}


{{Mapping|legend=5| 1 3/2 3 4 0 2 | 0 1/2 -4 -7 0 10 }}
{{Mapping|legend=5| 1 0 -3/4 37/4 -27/4 -7/4 | 0 2 0 -12 9 3 }}
: mapping generators: ~2, ~26/15


: [[gencom]]: [2 9/8; 225/224 325/324 640/637]
[[Optimal tuning]]s:  
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.7072{{c}}, ~26/15 = 951.2727{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~26/15 = 951.5016{{c}}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 204.090
{{Optimal ET sequence|legend=1| 24, 29, 111, 140, 169, 198 }}


{{Optimal ET sequence|legend=1| 6, 11, 17, 23, 29, 35, 41, 47, 100, 147, 488cd, 635cd }}
[[Tp tuning #T2 tuning|RMS error]]: 0.2668 cents


[[Tp tuning #T2 tuning|RMS error]]: 0.5999 cents
=== Historical ===
{{Distinguish|Historical temperaments}}
{{Distinguish|History (temperament)}}


Related temperament: [[Schismatic family #Garibaldi|Cassandra]]
Historical is essentially an analogue of [[miracle]] that splits [[4/3]] in six rather than [[3/2]]. It tempers out the comma [[4000/3993]] ([[S-expression|S10/S11]]) to set [[11/10]] equal to one-third of 4/3, and [[676/675]] ([[S-expression|S13/S15]]) to equate [[15/13]] to one-half of 4/3, and tempers out [[441/440]] ({{S|21}}) to split 11/10 into two instances of [[22/21]][[~]][[21/20]]. [[Schismatic family #Sextilifourths|Sextilifourths]] adds the [[schismatic family #Schismic|schismic]] mapping of prime 5 (reached by eight fourths) to complete the 13-limit.


==== Baldanders ====
[[Subgroup]]: 2.3.7/5.11/5.13/5
Baldanders results from taking every other generator of the andromeda, with mapping 11/8 to -9 whole tones.


Subgroup: 2.9.5.7.11
[[Comma list]]: 364/363 ({{monzo| 2 -1 1 -2 1 }}), 441/440 ({{monzo| -3 2 2 -1 0 }}, 1001/1000 ({{monzo| -3 0 1 1 1 }})


Comma list: 100/99, 225/224, 245/242
{{Mapping|legend=3| 1 2 0 1 2 | 0 -6 7 2 -9 }}
: mapping generators: ~2, ~21/20


{{Mapping|legend=3| 1 3 3 4 5 | 0 1 -4 -7 -9 }}
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1200.0242{{c}}, ~21/20 = 83.0177{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 83.0144{{c}}


Optimal tuning (POTE): ~2 = 1\1, ~9/8 = 204.743
{{Optimal ET sequence|legend=1| 14, 29, 101, 130, 159 }}


{{Optimal ET sequence|legend=1| 6, 23de, 29, 35, 41 }}
[[Tp tuning #T2 tuning|RMS error]]: 0.2562 cents


Related temperament: [[Schismatic family #Garibaldi|Andromeda]]
=== Direct breedsmic ===
This temperament was first proposed by {{u|Royalmilktea}}. The name was established by [[Lériendil]] in 2024. Related temperaments: [[breedsmic temperaments #Hemififths|hemififths]] and [[septischismic clan #Newt|newt]].


===== 2.9.5.7.11.13 =====
[[Subgroup]]: 2.3.49/5
Subgroup: 2.9.5.7.11.13


Comma list: 100/99, 144/143, 225/224, 245/242
[[Comma list]]: 2401/2400 ({{monzo| -5 -1 2 }})


{{Mapping|legend=3| 1 3 3 4 5 2 | 0 1 -4 -7 -9 10 }}
{{Mapping|legend=3| 1 1 3 | 0 2 1 }}
: mapping generators: ~2, ~49/40


Optimal tuning (POTE): ~2 = 1\1, ~9/8 = 204.414
[[Optimal tuning]]:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1200.0206{{c}}, ~49/40 = 350.9724{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~49/40 = 350.9743{{c}}


{{Optimal ET sequence|legend=1| 6, 23def, 29f, 35, 41, 47 }}
{{Optimal ET sequence|legend=1| 7, 17, 24, 41, 65, 106, 253, 359, 465, 1036, 1501, 1966*, 5433* }}
: <nowiki/>* Wart for 49/5


=== Glacial ===
==== Tobago ====
{{See also| Chromatic pairs #Glacial }}
{{See also| Chromatic pairs #Tobago }}


[[Subgroup]]: 2.9.5.11.13
Tobago, the 10 & 14 temperament in the 2.3.11.13/5 subgroup, extends [[rastmic clan #Neutral|neutral]] and [[#Barbados|barbados]].


[[Comma list]]: 45/44, 65/64, 81/80
[[Subgroup]]: 2.3.11.13/5


{{Mapping|legend=3| 1 0 -4 -6 10 | 0 1 2 3 -2 }}
[[Comma list]]: [[243/242]] ({{monzo| -1 5 -2 0 }}), [[676/675]] ({{monzo| 2 -3 0 2 }})


{{Mapping|legend=5| 1 3/2 2 0 3 4 | 0 1/2 2 0 3 -2 }}
{{Mapping|legend=3| 2 0 -1 -2 | 0 2 5 3 }}


: [[gencom]]: [2 9/8; 45/44 65/64 81/80]
{{Mapping|legend=5| 2 0 -5 0 -1 -1 | 0 2 -3/2 0 5 3/2 }}
: mapping generators: ~55/39, ~26/15


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 186.151
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~55/39 = 599.9845{{c}}, ~26/15 = 950.6637{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~55/39 = 600.0000{{c}}, ~26/15 = 950.6776{{c}}


{{Optimal ET sequence|legend=1| 6, 13, 45be, 58bce, 71bce, 84bce }}
{{Optimal ET sequence|legend=1| 10, 14, 24, 82, 106, 130, 154 }}


[[Tp tuning #T2 tuning|RMS error]]: 2.887 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.3533 cents


Music:
==== Pakkanian hemipyth ====
* ''[[Thundersnow]]'' - [[Sevish]] (2021)
[[Subgroup]]: 2.3.11.13/5.17


=== Mabon ===
[[Comma list]]: 221/220 ({{monzo| -2 0 -1 1 1 }}), 243/242 ({{monzo| -1 5 -2 0 0 }}), 289/288 ({{monzo| -5 -2 0 0 2 }})
Derived from a [http://individual.utoronto.ca/kalendis/leap/index.htm#se calendar leap cycle built for the autumn equinox], hence the name. Defined as the 11 & 62 temperament.


Subgroup: 2.9.7
{{Mapping|legend=3| 2 0 -1 -2 5 | 0 2 5 3 2 }}


Comma basis: 44957696/43046721
[[Optimal tuning]]s:  
* [[Tp tuning|subgroup WE]]: ~17/12 = 600.1781{{c}}, ~26/15 = 950.7656{{c}} (~15/13 = 249.2344{{c}})
* [[Tp tuning|subgroup CWE]]: ~17/12 = 600.0000{{c}}, ~26/15 = 950.6011{{c}} (~15/13 = 249.3989{{c}})


Sval mapping: [{{val|1 1 -3}}, {{val|0 3 8}}]
{{Optimal ET sequence|legend=1| 10, 14, 24, 106, 130, 154, 178*, 202* }}
: <nowiki/>* Wart for 13/5


Optimal tuning (CTE): ~729/448 = 870.792
=== Bridgetown ===
{{See also| Chromatic pairs #Bridgetown }}


{{Optimal ET sequence|legend=1|7d, 11, 18d, 29, 40, 62}}, ...
Bridgetown, the 5 & 24 temperament in the 2.3.11/5.13/5 subgroup, is the no-7/5 restriction of [[#Haumea|haumea]] and the add-11/5 [[extension]] of [[#Barbados|barbados]].  


==== 2.9.7.11 subgroup ====
[[Subgroup]]: 2.3.11/5.13/5
Subgroup: 2.9.7.11


Comma basis: 896/891, 1331/1296
[[Comma list]]: [[352/351]] ({{monzo| 5 -3 1 -1 }}), [[676/675]] ({{monzo| 2 -3 0 2 }})


Sval mapping: [{{val|1 1 -3 2}}, {{val|0 3 8 2}}]
{{Mapping|legend=3| 1 0 -6 -1 | 0 2 9 3 }}


Optimal tuning (CTE): ~16/11 = 870.966
{{Mapping|legend=5| 1 0 7/3 0 -11/3 4/3 | 0 2 -4 0 5 -1 }}
: mapping generators: ~2, ~26/15


{{Optimal ET sequence|legend=1| 7d, 11, 40, 51, 62 }}
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.6281{{c}}, ~26/15 = 951.3060{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~26/15 = 951.5580{{c}}


=== Apparatus ===
{{Optimal ET sequence|legend=1| 5, 19*, 24, 29, 169, 198, 227, 256, 285, 314, 343 }}
[[Subgroup]]: 2.9.7.11
: <nowiki/>* Wart for 11/5


[[Comma list]]: 41503/41472, 322102/321489
[[Tp tuning #T2 tuning|RMS error]]: 0.2513 cents


{{Mapping|legend=3| 1 5 3 5 | 0 -19 -2 -16 }}
=== Seventeen-cot ===
Seventeen-cot may be described as the 29 & 465 temperament in the 2.3.11/5.13/5 subgroup. It tempers out the [[tendoartisma]] in the 2.3.13/5 subgroup. It can be generated with a ~2/1 octave and a ~169/165 generator which is 1/17 of a ~3/2 perfect fifth. It was named by {{u|FilterNashi}} in 2026.


: mapping generators: ~2, ~77/72
[[Subgroup]]: 2.3.11/5.13/5


{{Mapping|legend=5| 1 5/2 0 3 5 | 0 -19/2 0 -2 -16 }}
[[Comma basis]]: 43940/43923 ({{monzo| 2 -1 -4 3 }}), 225000/224939 ({{monzo| 3 2 -3 -2 }})


: [[gencom]]: [2 77/72; 41503/41472 322102/321489]
{{Mapping|legend=3| 1 1 1 1 | 0 17 4 11 }}
: mapping generators: ~2, ~169/165


[[Optimal tuning]] ([[CTE]]): ~77/72 = 115.5685
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.993492{{c}}, ~169/165 = 41.291827{{c}}
: [[error map]]: {{val| -0.0065 -0.0004 +0.1566 -0.0104 }}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.000000, ~169/165 = 41.291746{{c}}
: error map: {{val| 0.0000 +0.0047 +0.1628 -0.0047 }}


{{Optimal ET sequence|legend=1| 10e, 21, 31, 52, 83, 135, 353, 488, 623 }}
{{Optimal ET sequence|legend=1| 29, 262, 291, 320, 349, 378, 407, 436, 465, 959, 1424, 6161… }}


[[Badness]]: 0.00263
[[Badness]] (Sintel): 0.080


=== Joan ===
=== Blackweed ===
{{See also| Chromatic pairs #Joan }}
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.


Joan is related to [[casablanca]] as well as to [[orwell]].  
[[Subgroup]]: 2.3.11/7


[[Subgroup]]: 2.9.7.11
[[Comma list]]: 256/243 ({{monzo| 8 -5 0 }})


[[Comma list]]: 99/98, 9317/9216
{{Mapping|legend=3| 5 8 0 | 0 0 1 }}
: mapping generators: ~9/8, ~11/7


{{Mapping|legend=3| 1 0 1 3 | 0 7 4 1 }}
[[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 }}


{{Mapping|legend=5| 1 0 0 1 3 | 0 7/2 0 4 1 }}
{{Optimal ET sequence|legend=1| 15, 20, 35b, 55b }}


: [[gencom]]: [2 11/8; 99/98 9317/9216]
=== Pepperoni ===
{{Main| Parapyth }}
{{See also| Chromatic pairs #Pepperoni }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~11/8 = 542.672 cents
Pepperoni is generated by a fifth and can be described as the 5 & 12 temperament in the 2.3.11/7.13/7 subgroup. It is the single-chain [[retraction]] of [[parapyth]]. The [[Peppermint-24|Pepper fifth]], which is (40200 + 600 sqrt(5))/59 = 704.096 cents, is a good pepperoni generator, hence the name.


{{Optimal ET sequence|legend=1| 11, 20, 31, 42, 115bd, 157bd }}
[[Subgroup]]: 2.3.11/7.13/7


[[Tp tuning #T2 tuning|RMS error]]: 1.424 cents
[[Comma list]]: 352/351 ({{monzo| 5 -3 1 -1 }}), 364/363 ({{monzo| 2 -1 -2 1 }})


=== Machine ===
{{Mapping|legend=3| 1 0 7 12 | 0 1 -4 -7 }}
Machine is every other step of [[supra]], most interesting for its scale patterns.


[[Subgroup]]: 2.9.7.11
{{Mapping|legend=5| 1 0 0 -19/3 2/3 17/3 | 0 1 0 11/3 -1/3 -10/3 }}
: mapping generators: ~2, ~3


[[Comma list]]: 64/63, 99/98
[[Optimal tuning]]s:  
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.3706{{c}}, ~11/7 = 703.4872{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~11/7 = 703.8328{{c}}


{{Mapping|legend=3| 1 0 6 13 | 0 1 -1 -3 }}
{{Optimal ET sequence|legend=1| 12, 17, 29, 75, 104 }}


: sval mapping generators: ~2, ~9
[[Tp tuning #T2 tuning|RMS error]]: 0.3789 cents


{{Mapping|legend=5| 1 3/2 0 3 4 | 0 1/2 0 -1 -3 }}
=== Fendo ===
Fendo tempers out [[40/39]] in the 2.3.13/5 subgroup. This equates [[4/3]] with the tendo third [[13/10]]. Fendo can be viewed as an alternative interpretation of scales, chords, etc. commonly associated with [[father family #Father|father]], as it is sometimes perceived that father's characteristics tend to suggest harmonic function more than to actually serve as a temperament. It was named by [[User: VectorGraphics|Vector]] in 2025 as a contraction of ''fourth'' and ''tendo third''.


: [[gencom]]: [2 8/7; 64/63 99/98]
In fendo, the fourth / tendo third serves the role of both, making the main chords of triadic harmony in fendo essentially suspended chords.


[[Optimal tuning]]s:
[[5edo]] and [[13edo]] make good fendo tunings.  
* [[CTE]]: ~2 = 1\1, ~9/8 = 216.9128
* [[POTE]]: ~2 = 1\1, ~9/8 = 214.3843


{{Optimal ET sequence|legend=1| 5, 6, 11, 17, 28 }}
[[Subgroup]]: 2.3.13/5


[[Badness]]: 0.00233
[[Comma list]]: 40/39 ({{monzo| 3 -1 -1 }})


=== Penta a.k.a. mechanism ===
{{Mapping|legend=3| 1 0 3 | 0 1 -1 }}
Penta or mechanism is the 8 &amp; 11 temperament in the 2.9.7.11 subgroup.
: mapping generators: ~2, ~3


[[Subgroup]]: 2.9.7.11
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1195.6502{{c}}, ~3/2 = 709.8204{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 710.1529{{c}}


[[Comma list]]: 896/891, 26411/26244
{{Optimal ET sequence|legend=1| 1, 2, 3, 5, 17*, 22*, 27*, 32* }}
: <nowiki/>* Wart for 13/5


{{Mapping|legend=3| 1 0 -1 6 | 0 5 6 -4 }}
=== Barbados ===
Barbados is the restriction of [[#Bridgetown|bridgetown]] to the 2.3.13/5 subgroup, and tempers out [[676/675]] alone. 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.


: sval mapping generators: ~2, ~14/9
[[Subgroup]]: 2.3.13/5


{{Mapping|legend=5| 1 5/2 0 5 2 | 0 -5/2 0 -6 4 }}
[[Comma list]]: 676/675 ({{monzo| 2 -3 2 }})


: [[gencom]]: [2 9/7; 896/891 26411/26244]
{{Mapping|legend=3| 1 0 -1 | 0 2 3 }}
: mapping generators: ~2, ~26/15


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~14/9 = 761.3782
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.9502{{c}}, ~26/15 = 951.0713{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~26/15 = 951.0933{{c}}


{{Optimal ET sequence|legend=1| 8, 11, 30, 41, 52 }}
{{Optimal ET sequence|legend=1| 5, 14, 19, 24, 29, 53, 82, 135 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.4262 cents
[[Badness]] (Smith): 0.002335


[[Badness]]: 0.00439
; Music
* ''[[Desert Island Rain]]'' by [[Sevish]] – [https://web.archive.org/web/20201127013711/http://micro.soonlabel.com/gene_ward_smith/Others/Sevish/Sevish%20-%20Desert%20Island%20Rain.mp3 play] | [https://sevish.bandcamp.com/track/desert-island-rain Bandcamp] | [https://soundcloud.com/sevish/desert-island-rain SoundCloud] | [https://www.youtube.com/watch?v=Gcgawrr2xao YouTube] – in 313edo tuned Barbados[9]


Scales: [[penta5]], [[penta8]], [[penta11]], [[penta19]]
=== 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.


=== Novisept ===
[[Subgroup]]: 2.3.17/5
Novisept is generated by a one-cent-flat 9/7, such that stacking 5 of them gives you 7/4. It can be formed by doubling both generator and period of [[gizzard]].


[[Subgroup]]: 2.9.7.13.17
[[Comma list]]: 136/135 ({{monzo| 3 -3 1 }})


[[Comma list]]: 729/728, 442/441, 833/832
{{Mapping|legend=3| 1 0 -3 | 0 1 3 }}
: mapping generators: ~2, ~3


{{Mapping|legend=3| 1 1 1 -1 3| 0 6 5 13 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 tuning]] ([[CWE]]): ~2 = 1\1, ~9/7 = 433.836
{{Optimal ET sequence|legend=1| 5, 12, 17, 46, 63, 143 }}


Badness (Dirichlet): 0.142
=== 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.  


=== Demon ===
[[Subgroup]]: 2.3.19/7
Demon is a temperament which equates 3 [[11/9]] with [[16/9]], or equivalently 3 [[18/11]] with [[9/8]], tempering out [[1331/1296]]. This results in [[11/9]] being tuned flat to a supraminor third, and [[27/22]] being tuned sharp to a submajor third. It was discovered by [[User:CompactStar|CompactStar]] while searching for temperaments assosciated with the [[7L 4s]] ("daemotonic") MOS, known for its lack of representation of simple temperaments. The optimal tuning for demon temperament is near the basic tuning of 7L 4s (13\18), and indeed [[18edo]] supports demon temperament.


[[Subgroup]]: 2.9.11
[[Comma list]]: [[57/56]] ({{monzo| -3 1 1 }})


[[Comma list]]: [[1331/1296]]
{{Mapping|legend=3| 1 0 3 | 0 1 -1 }}
: mapping generators: ~2, ~3


{{Mapping|legend=2|1 1 2|0 3 2}}
[[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 tuning]] ([[CTE]]): ~[[18/11]] = 870.060
{{Optimal ET sequence|legend=1| 5, 7, 12, 19, 31*, 50* }}
: <nowiki/>* Wart for 19/7


{{Optimal ET sequence|legend=1|4, 7, 11, 18, 29, 76e}}
[[Badness]] (Sintel): 0.082


=== Genius ===
== 2.9.… subgroups ==
See also [[Jubilismic clan #Antikythera|antikythera]] and [[Hemimean clan #Isra|isra]].


Named after the genius in Roman religion, following the demon (daimon) in Greek mythology.
=== Baldy ===
{{See also| Chromatic pairs #Baldy }}


[[Subgroup]]: 2.9.11
Baldy is every other step of [[schismatic family #Garibaldi|garibaldi]]. It is also the add-9 extension of [[no-threes subgroup temperaments #Frostburn|frostburn]]. One of the best extension is to the 2.9.5.7.13 subgroup, mapping 13/8 to +10 whole tones, the same as the cassandra temperament.


[[Comma list]]: [[131769/131072]]
[[Subgroup]]: 2.9.5.7


{{Mapping|legend=2|1 1 4|0 4 -1}}
[[Comma list]]: 225/224, 3125/3087


[[Optimal tuning]] ([[CTE]]): ~[[16/11]] = 650.863
{{Mapping|legend=3| 1 0 15 25 | 0 1 -4 -7 }}
: mapping generators: ~2, ~9


{{Optimal ET sequence|legend=1|9, 11, 24, 59, 83, 142, 225, 367}}[-11], 592[-11], 959[-9, --11], 1326[-9, --11]
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.1233{{c}}, ~9/8 = 204.1913{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~9/8 = 204.1549{{c}}


=== Stacks (a.k.a. 2magic) ===
{{Optimal ET sequence|legend=1| 6, 29, 35, 41, 47, 194 }}
Stacks, the 11 &amp; 30 temperament in the 2.9.15.7.11.13 subgroup, is every other step of [[magic]].


[[Subgroup]]: 2.9.15.7
[[Badness]] (Sintel): 0.274


[[Comma list]]: 225/224, 245/243
==== 2.9.5.7.13 subgroup ====
Subgroup: 2.9.5.7.13


{{Mapping|legend=3| 1 0 2 -1 | 0 5 3 6 }}
Comma list: 225/224, 325/324, 640/637


: sval mapping generators: ~2, ~14/9
{{Mapping|legend=2| 1 0 15 25 -28 | 0 1 -4 -7 10 }}


{{Mapping|legend=5| 1 5/2 5/2 5 | 0 -5/2 -1/2 -6 }}
{{Mapping|legend=4| 1 0 15 25 0 28 | 0 1/2 -4 -7 0 10 }}


: [[gencom]]: [2 9/7; 225/224 245/243]
Optimal tunings:  
* WE: ~2 = 1200.0155{{c}}, ~9/8 = 204.0931{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/8 = 204.0902{{c}}


[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~2 = 1\1, ~14/9 = 760.704
{{Optimal ET sequence|legend=0| 6, 35, 41, 47, 100, 147 }}


{{Optimal ET sequence|legend=1| 8, 11, 30, 41, 71, 93, 112c, 134c, 175c }}
Badness (Sintel): 0.386


[[Tp tuning #T2 tuning|RMS error]]: 1.074 cents
RMS error: 0.5999 cents


==== 2.9.15.7.11 ====
==== Baldanders ====
Subgroup: 2.9.15.7.11
Baldanders results from taking every other generator of the [[schismatic family #Andromeda|andromeda]], mapping 11/8 to -9 whole tones.


Comma list: 100/99, 225/224, 245/243
Subgroup: 2.9.5.7.11


Sval mapping: {{mapping| 1 0 2 -1 6 | 0 5 3 6 -4 }}
Comma list: 100/99, 225/224, 245/242


{{Mapping|legend=4| 1 5/2 5/2 5 2 | 0 -5/2 -1/2 -6 4 }}
{{Mapping|legend=2| 1 0 15 25 32 | 0 1 -4 -7 -9 }}


: gencom: [2 9/7; 100/99 225/224 245/243]
Optimal tunings:  
* WE: ~2 = 1200.1917{{c}}, ~9/8 = 204.7755{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/8 = 204.7197{{c}}


Optimal tuning (subgroup POTE): ~2 = 1\1, ~14/9 = 761.393
{{Optimal ET sequence|legend=0| 6, 23de, 29, 35, 41 }}


Optimal ET sequence: {{Optimal ET sequence| 8, 11, 30, 41, 52, 93, 145, 342bce }}
Badness (Sintel): 0.389


RMS error: 1.226 cents
===== 2.9.5.7.11.13 =====
Subgroup: 2.9.5.7.11.13


==== 2.9.15.7.11.13 ====
Comma list: 100/99, 144/143, 225/224, 245/242
Subgroup: 2.9.15.7.11.13


Comma list: 100/99, 105/104, 144/143, 196/195
{{Mapping|legend=2| 1 0 15 25 32 -28 | 0 1 -4 -7 -9 10 }}


Sval mapping: {{mapping| 1 0 2 -1 6 -2 | 0 5 3 6 -4 9 }}
Optimal tunings:  
* WE: ~2 = 1199.6385{{c}}, ~9/8 = 204.3526{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/8 = 204.4234{{c}}


{{Mapping|legend=4| 1 5/2 5/2 5 2 7 | 0 -5/2 -1/2 -6 4 -9 }}
{{Optimal ET sequence|legend=0| 6, 35, 41, 47, 88e }}


: gencom: [2 9/7; 100/99 105/104 144/143 196/195]
Badness (Sintel): 0.653


Optimal tuning (subgroup POTE): ~2 = 1\1, ~14/9 = 761.023
=== Daemotertiaschis ===
{{See also| Schismatic family #Tertiaschis }}


Optimal ET sequence: {{Optimal ET sequence| 11, 30, 41, 153cdef, 194cdef, 235cdef }}
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 (hence the name – daemo- + tertiaschis), which is notoriously difficult to approximate with simple JI or RTT methods.


RMS error: 1.540 cents
Subgroup: 2.9.5.7.33.13.17


=== A-team ===
Comma list: 325/324, 375/374, 385/384, 595/594, 10985/10976
A-team is every other step of [[slendric]]; the 2.9.5.21.11 extension below specifically restricts [[mothra]].


[[Subgroup]]: 2.9.21
{{Mapping|legend=2| 1 1 11 -16 13 -18 20 | 0 3 -12 26 -11 30 -22 }}
: mapping generators: ~2, ~33/20


[[Comma list]]: 1029/1024
Optimal tunings:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1200.3019{{c}}, 33/20 = 868.1949{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, 33/20 = 867.9773{{c}}


{{Mapping|legend=3| 1 2 4 | 0 3 1 }}
{{Optimal ET sequence|legend=1| 47, 159, 206, 253, 459… }}


: sval mapping generators: ~2, ~21/16
[[Badness]] (Sintel): 0.439


{{Mapping|legend=5| 1 1 0 3 | 0 3/2 0 -1/2 }}
=== A-team ===
{{See also| Dual-fifth temperaments #Dual-3 A-Team }}


: [[gencom]]: [2 21/16; 1029/1024]
A-team is every other step of [[gamelismic clan #Mothra|mothra]].


[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~2 = 1\1, ~21/16 = 467.375
[[Subgroup]]: 2.9.5.21


{{Optimal ET sequence|legend=1| 5, 13, 18, 41, 59, 77, 95 }}
[[Comma list]]: 81/80 ({{monzo| -4 2 -1 0 }}), 1029/1024 ({{monzo| -10 -1 0 3 }})


[[Tp tuning #T2 tuning|RMS error]]: 0.3202 cents
{{Mapping|legend=3| 1 2 0 4 | 0 3 6 1 }}


==== 2.9.5.21 ====
{{Mapping|legend=5| 1 1 0 3 | 0 3/2 6 -1/2 }}
''Lookalike temperament: [[Dual-fifth_temperaments#Dual-3_A-Team|Dual-3 A-Team]]''
: mapping generators: ~2, ~21/16


Subgroup: 2.9.5.21
[[Optimal tuning]]s:
 
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1200.9303{{c}}, ~21/16 = 464.7465{{c}}
[[Comma]] list: 81/80, 1029/1024
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~21/16 = 464.5029{{c}}
 
Sval mapping: {{mapping| 1 2 0 4 | 0 3 6 1 }}


Mapping generators: ~2, ~21/16
{{Optimal ET sequence|legend=1| 13, 18, 31 }}


Optimal ([[Lp tuning|POL2]]) generator: 464.3865
[[Badness]] (Sintel): 0.177


{{Optimal ET sequence|legend=1| 13, 18, 31, 44 }}
===== 2.9.5.21.11 subgroup =====
 
===== 2.9.5.21.11 =====
Subgroup: 2.9.5.21.11
Subgroup: 2.9.5.21.11


Comma list: 81/80, 99/98, 385/384
Comma list: 81/80 ({{monzo| -4 2 -1 0 }}), 99/98 ({{monzo| -1 2 0 -2 1 }}), 385/384 ({{monzo| -7 -1 1 1 1 }})


Sval mapping: {{mapping| 1 2 0 4 5 | 0 3 6 1 -4 }}
{{Mapping|legend=2| 1 2 0 4 5 | 0 3 6 1 -4 }}


{{Mapping|legend=4| 1 1 0 3 5 | 0 3/2 6 -1/2 -4 }}
{{Mapping|legend=4| 1 1 0 3 5 | 0 3/2 6 -1/2 -4 }}


: gencom: [2 21/16; 81/80 99/98 385/384]
Optimal tunings:  
* Subgroup WE: ~2 = 1201.3979{{c}}, ~21/16 = 464.6020{{c}}
* Subgroup CWE: ~2 = 1200.0000{{c}}, ~21/16 = 464.1242{{c}}


Optimal tuning (subgroup POTE): ~2 = 1\1, ~21/16 = 463.956
{{Optimal ET sequence|legend=0| 13, 31, 44, 75b′ }}


{{Optimal ET sequence|legend=1| 5, 13, 31 }}
Badness (Sintel): 0.213


==== B-team ====
=== Glacial ===
B-team (23 & 41) is every other step of [[rodan]].
{{See also| Chromatic pairs #Glacial }}


Subgroup: 2.9.15.21.33
[[Subgroup]]: 2.9.5.11.13


Comma list: 245/243, 385/384, 441/440
[[Comma list]]: 45/44, 65/64, 81/80


Sval mapping: {{mapping| 1 2 0 4 7 | 0 3 10 1 -5 }}
{{Mapping|legend=3| 1 0 -4 -6 10 | 0 1 2 3 -2 }}
 
Optimal tuning (subgroup POTE): ~2 = 1\1, ~21/16 = 468.918
 
{{Optimal ET sequence|legend=1| 5, 13c, 18, 23, 41, 64, 87, 151 }}
 
== 2.75.… subgroups ==
=== Archagall ===
{{See also| Fifthplus }}
 
By tempering out the comma [[24576/24565]] in the 2.75.85 subgroup, we have three [[85/64]]'s up and one octave down as a [[75/64]] and we have two [[128/85]]'s up and one octave down as a [[17/15]] whole tone. It is because of this combination of accuracy, efficiency and mapping-wise simplicity and its corresponding explanatory power in what this comma does that the comma has been named the ''archagallisma''. The ''MVP'' stands for ''minimum viable product'', as this is the core of what the archagall logic achieves, with further extensions adding to the subgroup while avoiding significantly impacting its accuracy. This is a highly accurate temperament that could be considered to be encoding the "high-accuracy logic" of [[superpyth]] and which is inescapably related to the [[17L 5s]] scale form as it is the 17 & 22 temperament (or less accurately, the 5 & 17 temperament) in the 2.75.85 subgroup.
 
It is perhaps worth noting that [[83edo]] is approximately an optimal tuning for getting all of [[75/64]], [[85/64]], and the interval between them, [[17/15]], close to just, in that larger tunings compromise on tone-efficiency.
 
[[Subgroup]]: 2.75.85


[[Comma list]]: [[24576/24565]] ({{monzo| 13 1 -3 }})
{{Mapping|legend=5| 1 0 -4 0 -6 10 | 0 1/2 2 0 3 -2 }}
 
: mapping generators: ~2, ~9
{{Mapping|legend=3| 1 2 5 | 0 3 1 }}
: mapping generators: ~2, ~85/32


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.9692{{c}}, ~85/64 = 491.5853{{c}}
* [[WE]]: ~2 = 1202.6916{{c}}, ~9/8 = 186.1446{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~85/64 = 491.5794{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~9/8 = 186.1740{{c}}


{{Optimal ET sequence|legend=1| 5, 12, 17, 22, 61, 83, 310, 393, 476, 1345, 1821, 4118*, 5939*, 7760* }}
{{Optimal ET sequence|legend=1| 6, 13, 45b′f }}


==== 2.75.9/7.85 subgroup ====
[[Badness]] (Sintel): 0.281
A fairly natural way to extend archagall is by tempering out [[2025/2023]] ([[S-expression|S15/S17]]), which equates a stack of two [[17/15]]'s with [[9/7]] without much damage. As 9/7 was not previously in the subgroup, this does not decrease the rank of the temperament and qualifies a proper and natural extension. We can equally get the same temperament by tempering out S15/S16 instead (equating a stack of three [[16/15]]'s with [[17/14]]); however, [[16/15]] is not in the subgroup, so it is preferred to think of it as adding 2025/2023.  


Subgroup: 2.75.9/7.85
[[Tp tuning #T2 tuning|RMS error]]: 2.887 cents


Comma list: 2025/2023 ({{monzo| 2 -2 1 0 }}), 24576/24565 ({{monzo| 13 1 0 -3 }})
Music:
* ''[[Thundersnow]]'' by [[Sevish]] (2021)


{{Mapping|legend=2| 1 2 6 5 | 0 3 -4 1 }}
=== Stacks (a.k.a. 2magic) ===
Stacks, the 11 & 30 temperament in the 2.9.15.7.11.13 subgroup, is every other step of [[Magic family #Magic|magic]].


Optimal tunings:
[[Subgroup]]: 2.9.15.7
* Subgroup WE: ~2 = 1200.0241{{c}}, ~85/64 = 491.3358{{c}}
* Subgroup CWE: ~2 = 1200.0000{{c}}, ~85/64 = 491.3290{{c}}


{{Optimal ET sequence|legend=0| 5, 12, 17, 22, 83, 105, 127, 403, 530, 657, 784, 1441* }}
[[Comma list]]: 225/224 ({{monzo| -5 0 2 -1 }}), 245/243 ({{monzo| 0 -3 1 2 }})


== 4.3.… subgroups ==
{{Mapping|legend=3| 1 0 2 -1 | 0 5 3 6 }}
=== Tetrahanson ===
{{Main| Tetrahanson }}


[[Subgroup]]: 4.3.5
{{Mapping|legend=5| 1 0 2 -1 | 0 5/2 1/2 6 }}
: mapping generators: ~2, ~14/9


[[Comma list]]: 15625/15552
[[Optimal tuning]]s:  
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1201.0786{{c}}, ~14/9 = 761.3878{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~14/9 = 760.9151{{c}}


{{Mapping|legend=3| 1 3 3 | 0 -6 -5 }}
{{Optimal ET sequence|legend=1| 8, 11, 19, 30, 41, 71, 112* }}
: <nowiki/>* Wart for 15


: Mapping generators: ~4, ~5/3
[[Badness]] (Sintel): 0.0884


[[Optimal tuning]] ([[CTE]]): ~4 = 2\1, ~5/3 = 882.941
[[Tp tuning #T2 tuning|RMS error]]: 1.074 cents


[[Support]]ing [[ET]]s: {{EDs|19, 106, 87, 68, 11, 8, 125, 49, 30, 27, 117, 46, 41b, 79|equave=4}}
==== 2.9.15.7.11 ====
Subgroup: 2.9.15.7.11


=== Tetrameantone ===
Comma list: 100/99 ({{monzo| 2 -2 2 0 -1 }}), 225/224 ({{monzo| -5 0 2 -1 0 }}), 245/243 ({{monzo| 0 -3 1 2 0 }})
{{Main| Tetrameantone }}


[[Subgroup]]: 4.3.5
{{Mapping|legend=2| 1 0 2 -1 6 | 0 5 3 6 -4 }}


[[Comma list]]: 81/80
{{Mapping|legend=4| 1 0 2 -1 6 | 0 5/2 1/2 6 -4 }}


{{Mapping|legend=3| 1 1 2 | 0 -1 -4 }}
Optimal tunings:
* Subgroup WE: ~2 = 1200.1372{{c}}, ~14/9 = 761.4799{{c}}
* Subgroup CWE: ~2 = 1200.0000{{c}}, ~14/9 = 761.4017{{c}}


: Mapping generators: ~4, ~4/3
{{Optimal ET sequence|legend=0| 8, 11, 30, 41, 52, 93, 145 }}


[[Optimal tuning]] ([[POTE]]): 4 = 2400.0, ~4/3 = 503.761
Badness (Sintel): 0.166


[[Support]]ing [[ET]]s: {{EDs|5, 9, 14, 19, 24, 43, 62, 81, 100|equave=4}}
RMS error: 1.226 cents


=== Tetramagic ===
==== 2.9.15.7.11.13 ====
Subgroup: 2.9.15.7.11.13


[[Subgroup]]: 4.3.5
Comma list: 100/99 ({{monzo| 2 -2 2 0 -1 0 }}), 105/104 ({{monzo| -3 0 1 1 0 -1 }}), 144/143 ({{monzo| 4 1 0 0 -1 -1 }}), 196/195 ({{monzo| 2 0 -1 2 0 -1 }})


[[Comma list]]: 3125/3072
{{Mapping|legend=2| 1 0 2 -1 6 -2 | 0 5 3 6 -4 9 }}


{{Mapping|legend=3| 1 0 1 | 0 5 1 }}
{{Mapping|legend=4| 1 0 2 -1 6 -2 | 0 5/2 1/2 6 -4 9 }}


: Mapping generators: ~4, ~5/4
Optimal tunings:  
* Subgroup WE: ~2 = 1200.0331{{c}}, ~14/9 = 760.8754{{c}}
* Subgroup CWE: ~2 = 1200.0000{{c}}, ~14/9 = 760.8568{{c}}


[[Optimal tuning]] ([[POTE]]): 4 = 2400.0, ~5/4 = 380.059
{{Optimal ET sequence|legend=0| 11, 30, 41 }}


[[Support]]ing [[ET]]s: {{EDs|6, 13, 19, 25, 38, 44, 63, 82|equave=4}}
Badness (Sintel): 0.257


=== Blacktetra ===
RMS error: 1.540 cents


[[Subgroup]]: 4.3.5
=== B-team ===
B-team may be described as 23 & 41 and is every other step of [[gamelismic clan #Rodan|rodan]].


[[Comma list]]: 256/243
[[Subgroup]]: 2.9.15.21


{{Mapping|legend=3| 5 4 6 | 0 0 -1 }}
[[Comma list]]: 245/243 ({{monzo| 0 -4 1 2 }}), 1029/1024 ({{monzo| -10 -1 0 3 }})


: Mapping generators: ~4, ~16/15
{{Mapping|legend=3| 1 2 0 4 | 0 3 10 1 }}


[[Optimal tuning]] ([[POTE]]): 1\5ed4 = 480.0, ~16/15 = 80.4062
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1200.2146{{c}}, ~21/16 = 468.9174{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~21/16 = 468.8518{{c}}


[[Support]]ing [[ET]]s: {{EDs|5, 10, 15, 20, 25, 30, 55, 85, 115|equave=4}}
{{Optimal ET sequence|legend=1| 18, 23, 41, 64, 151 }}


== 4.6.… subgroups ==
[[Badness]] (Sintel): 0.155
=== Meanquad ===
{{Main| Meanquad }}


[[Subgroup]]: 4.6.5
==== 2.9.15.21.33 subgroup ====
Subgroup: 2.9.15.21.33


[[Comma list]]: [[81/80]] = {{monzo| -4 4 -1 }}
Comma list: 245/243 ({{monzo| 0 -4 1 2 0 }}), 385/384 ({{monzo| -7 -2 1 1 1 }}), 441/440 ({{monzo| -3 1 -1 2 -1 }})


{{Mapping|legend=3| 1 0 -4| 0 1 4 }}
{{Mapping|legend=2| 1 2 0 4 7 | 0 3 10 1 -5 }}


: mapping generators: ~4, ~6
Optimal tunings:  
* Subgroup WE: ~2 = 1200.0553{{c}}, ~21/16 = 468.9391{{c}}
* Subgroup CWE: ~2 = 1200.0000{{c}}, ~21/16 = 468.9189{{c}}


[[Optimal tuning]] (subgroup [[CTE]]): ~4 = 2\1, ~3/2 = 697.214
{{Optimal ET sequence|legend=0| 18, 23, 41, 64, 87, 151, 238b′ }}


[[Support]]ing [[ET]]s: *7, *10, *11[-5], *13[+5], *17, *24, *27[+5], *31, *38, *41, *45, *52, *55, *69
Badness (Sintel): 0.160


<nowiki />* Wart for 4
=== Mabon ===
Mabon may be described as the 11 & 62 temperament in the 2.9.7 subgroup. It was derived from a [http://individual.utoronto.ca/kalendis/leap/index.htm#se calendar leap cycle built for the autumn equinox], and named so by {{u|Dummy index}} in 2024.


==== 4.6.5.7 subgroup (tetrominant) ====
[[Subgroup]]: 2.9.7
[[Subgroup]]: 4.6.5.7


[[Comma list]]: [[36/35]] = {{monzo| 0 2 -1 -1 }}, [[64/63]] = {{monzo| 4 -2 0 -1 }}
[[Comma basis]]: 44957696/43046721


{{Mapping|legend=3| 1 0 -4 4 | 0 1 4 -2 }}
{{Mapping|legend=3| 1 1 -3 | 0 3 8 }}
: mapping generators: ~2, ~729/448


[[Optimal tuning]] (subgroup [[CTE]]): ~4 = 2\1, ~3/2 = 699.622
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0000{{c}}, ~729/448 = 870.4003{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~729/448 = 871.1047{{c}}


[[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]
{{Optimal ET sequence|legend=1| 7d, 11, 40, 51, 62, 73b′, 157b′b′ }}


<nowiki />* Wart for 4
[[Badness]] (Sintel): 4.08


=== Fourwar ===
==== 2.9.7.11 subgroup ====
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.
Subgroup: 2.9.7.11


Fourwar is named after the closely related [[hemiwar]] temperament.
Comma basis: 896/891, 1331/1296


{{Todo|inline=1|cleanup}}
{{Mapping|legend=2| 1 1 -3 2 | 0 3 8 2 }}


<pre>
Optimal tunings:
Reduced Mapping
* WE: ~2 = 1200.1182{{c}}, ~18/11 = 870.9904{{c}}
4 6 5
* CWE: ~2 = 1200.0000{{c}}, ~18/11 = 870.9234{{c}}
[ ⟨ 1 0 1 ]
⟨ 0 16 2 ] ⟩
TE Generator Tunings (cents)
⟨2399.3973, 193.8643]
TE Step Tunings (cents)
⟨25.21211, 47.81337]
TE Tuning Map (cents)
⟨2399.397, 3101.829, 2787.126]
TE Mistunings (cents)
⟨-0.603, -0.126, 0.812]
Complexity 1.369085
Adjusted Error 0.692892 cents
TE Error 0.268047 cents/octave
Unison Vector
[8, 1, -8⟩ (393216:390625)


Subsets
{{Optimal ET sequence|legend=0| 7d, 11, 40, 51, 62 }}
q99, q62, q37, q161, q136, q198, q25, q124, q74, q235
</pre>


==== 4.6.5.7 ====
Badness (Sintel): 4.08
<pre>
Reduced Mapping
4 6 5 7
[ ⟨ 1 0 1 1 ]
⟨ 0 16 2 5 ] ⟩
TE Generator Tunings (cents)
⟨2399.4195, 193.8654]
TE Step Tunings (cents)
⟨25.23883, 47.79592]
TE Tuning Map (cents)
⟨2399.420, 3101.846, 2787.150, 3368.747]
TE Mistunings (cents)
⟨-0.580, -0.109, 0.837, -0.079]
Complexity 1.192044
Adjusted Error 0.653313 cents
TE Error 0.232715 cents/octave
Unison Vectors
[-2, -1, -2, 4⟩ (2401:2400)
[3, 0, -5, 2⟩ (3136:3125)
[5, 1, -3, -2⟩ (6144:6125)
[8, 1, -8, 0⟩ (393216:390625)


Subsets
=== Machine ===
q99, q62, q37, q161, q136, q198, q25, q124, q74, q235
Machine is every other step of [[supra]], most interesting for its scale patterns.
</pre>


==== 4.6.5.7.11 ====
It was named by [[Mike Battaglia]] in 2011, who spoke highly of this temperament, describing it as "stimulating, like a well-oiled machine".<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_96074.html Yahoo! Tuning Group | ''Ridiculously awesome 6&11 2.3.7.11.13 subgroup temperament'']</ref>
<pre>
Reduced Mapping
4 6 5 7 11
[ ⟨ 1 0 1 1 1 ]
⟨ 0 16 2 5 9 ] ⟩
TE Generator Tunings (cents)
⟨2400.1097, 193.9498]
TE Step Tunings (cents)
⟨24.18752, 48.52491]
TE Tuning Map (cents)
⟨2400.110, 3103.196, 2788.009, 3369.859, 4145.658]
TE Mistunings (cents)
⟨0.110, 1.241, 1.696, 1.033, -5.660]
Complexity 1.068792
Adjusted Error 2.926965 cents
TE Error 0.846083 cents/octave
Unison Vectors
[-1, -1, -1, 0, 2⟩ (121:120)
[2, 0, -2, -1, 1⟩ (176:175)
[-3, -1, 1, 1, 1⟩ (385:384)
[-1, 0, 3, -3, 1⟩ (1375:1372)
[-2, -1, -2, 4, 0⟩ (2401:2400)
[1, 0, 1, -4, 2⟩ (2420:2401)


Subsets
[[Subgroup]]: 2.9.7.11
q37, q25, q62, q12, q74, q99, q87, q49r, q50r, q124
</pre>


==== 4.6.5.7.11.13 ====
[[Comma list]]: 64/63, 99/98


<pre>
{{Mapping|legend=3| 1 0 6 13 | 0 1 -1 -3 }}
Reduced Mapping
4 6 5 7 11 13
[ ⟨ 1 0 1 1 1 0 ]
⟨ 0 16 2 5 9 23 ] ⟩
TE Generator Tunings (cents)
⟨2401.2305, 193.5378]
TE Step Tunings (cents)
⟨42.79107, 35.98524]
TE Tuning Map (cents)
⟨2401.230, 3096.606, 2788.306, 3368.920, 4143.071, 4451.371]
TE Mistunings (cents)
⟨1.230, -5.349, 1.992, 0.094, -8.247, 10.843]
Complexity 1.219191
Adjusted Error 6.699599 cents
TE Error 1.810487 cents/octave
Unison Vectors
[0, 1, -1, 0, 1, -1⟩ (66:65)
[-1, -1, -1, 0, 2, 0⟩ (121:120)
[1, 2, 0, 0, -1, -1⟩ (144:143)
[2, 0, -2, -1, 1, 0⟩ (176:175)
[-2, 1, 1, 1, 0, -1⟩ (105:104)
[-3, -1, 1, 1, 1, 0⟩ (385:384)
[-3, 0, 0, 1, 2, -1⟩ (847:832)
[1, 3, -1, 0, 0, -2⟩ (864:845)
[-1, 0, 3, -3, 1, 0⟩ (1375:1372)


Subsets
{{Mapping|legend=5| 1 0 0 6 13 | 0 1/2 0 -1 -3 }}
q25, q37f, q12f, q62, q50rf, q13rff, q49rff, q87, q74ff, q24rfff
: mapping generators: ~2, ~9
</pre>


==== 4.6.5.7.11.13.17 ====
[[Optimal tuning]]s:
<pre>
* [[WE]]: ~2 = 1197.2650{{c}}, ~9/8 = 213.8957{{c}}
Reduced Mapping
* [[CWE]]: ~2 = 1200.0000{{c}}, ~9/8 = 214.9961{{c}}
4 6 5 7 11 13 17
[ ⟨ 1 0 1 1 1 0 1 ]
⟨ 0 16 2 5 9 23 13 ] ⟩
TE Generator Tunings (cents)
⟨2400.4701, 193.4599]
TE Step Tunings (cents)
⟨43.39350, 35.55764]
TE Tuning Map (cents)
⟨2400.470, 3095.359, 2787.390, 3367.770, 4141.609, 4449.578, 4915.449]
TE Mistunings (cents)
⟨0.470, -6.596, 1.076, -1.056, -9.709, 9.050, 10.494]
Complexity 1.129881
Adjusted Error 8.082725 cents
TE Error 1.977443 cents/octave
Unison Vectors
[0, 1, -1, 0, 1, -1, 0⟩ (66:65)
[1, 1, 1, -1, 0, 0, -1⟩ (120:119)
[1, 2, 0, 0, -1, -1, 0⟩ (144:143)
[-2, 1, 1, 1, 0, -1, 0⟩ (105:104)
[-1, 2, 2, 0, 0, -1, -1⟩ (225:221)
[-1, 1, 2, -2, 0, -1, 1⟩ (1275:1274)


Subsets
{{Optimal ET sequence|legend=1| 5, 6, 11, 17, 28 }}
q25, q12f, q37f, q13rffg, q50rf, q62, q49rffg, q24rfffg, q38rreffg, q74ffg
</pre>


==== 4.6.5.7.11.13.17.19 ====
[[Badness]] (Sintel): 0.176
<pre>
Reduced Mapping
4 6 5 7 11 13 17 19
[ ⟨ 1 0 1 1 1 0 1 1 ]
⟨ 0 16 2 5 9 23 13 14 ] ⟩
TE Generator Tunings (cents)
⟨2399.9219, 193.3952]
TE Step Tunings (cents)
⟨44.14256, 35.03670]
TE Tuning Map (cents)
⟨2399.922, 3094.324, 2786.712, 3366.898, 4140.479, 4448.090, 4914.060, 5107.455]
TE Mistunings (cents)
⟨-0.078, -7.631, 0.399, -1.928, -10.839, 7.562, 9.104, 9.942]
Complexity 1.058472
Adjusted Error 8.712222 cents
TE Error 2.050935 cents/octave
Unison Vectors
[0, 1, -1, 0, 1, -1, 0, 0⟩ (66:65)
[-1, 0, 0, 1, 1, 0, 0, -1⟩ (77:76)
[2, 1, -1, 0, 0, 0, 0, -1⟩ (96:95)
[1, 1, 1, -1, 0, 0, -1, 0⟩ (120:119)
[0, 1, 1, 1, -1, 0, 0, -1⟩ (210:209)
[0, 0, 1, -2, 1, 0, 1, -1⟩ (935:931)
[2, 0, -3, 1, 0, 0, -1, 1⟩ (2128:2125)


Subsets
=== Apparatus ===
q25, q12fh, q37f, q13rffgh, q50rf, q62, q49rffgh, q24rfffghh, q38rreffgh, q74ffgh
Apparatus is the restriction of [[gamelismic clan #Revelation|revelation]] to the 2.9.7.11 subgroup, and every fourth step of [[breedsmic temperaments #Quasiorwell|quasiorwell]], without prime 5.
</pre>


==== 4.6.5.7.11.13.17.19.23 ====
[[Subgroup]]: 2.9.7.11
<pre>
Reduced Mapping
4 6 5 7 11 13 17 19 23
[ ⟨ 1 0 1 1 1 0 1 1 0 ]
⟨ 0 16 2 5 9 23 13 14 28 ] ⟩
TE Generator Tunings (cents)
⟨2399.3286, 193.5316]
TE Step Tunings (cents)
⟨37.31613, 39.63311]
TE Tuning Map (cents)
⟨2399.329, 3096.506, 2786.392, 3366.987, 4141.113, 4451.227, 4915.240, 5108.771, 5418.885]
TE Mistunings (cents)
⟨-0.671, -5.449, 0.078, -1.839, -10.205, 10.699, 10.284, 11.258, -9.389]
Complexity 1.115920
Adjusted Error 9.502017 cents
TE Error 2.100561 cents/octave
Unison Vectors
[0, 1, -1, 0, 1, -1, 0, 0, 0⟩ (66:65)
[1, 0, 0, -1, 0, -1, 0, 0, 1⟩ (92:91)
[0, -1, 1, 0, 0, 0, 0, -1, 1⟩ (115:114)
[1, 1, 1, -1, 0, 0, -1, 0, 0⟩ (120:119)
[2, 0, -2, -1, 1, 0, 0, 0, 0⟩ (176:175)
[-3, -1, 1, 1, 1, 0, 0, 0, 0⟩ (385:384)
[1, 0, -2, 1, 0, 0, 1, -1, 0⟩ (476:475)
[1, 0, 0, -2, 1, 0, -1, 1, 0⟩ (836:833)
[0, 0, 1, -2, 1, 0, 1, -1, 0⟩ (935:931)
[1, -1, 0, 0, 0, 0, -2, 1, 1⟩ (874:867)


Subsets
[[Comma list]]: 41503/41472, 322102/321489
q25i, q12fhi, q37f, q13rffghii, q62, q50rfii, q49rffghii, q24rfffghhiii, q74ffghi, q38rreffghiii
</pre>


== 4.9.… subgroups ==
{{Mapping|legend=3| 1 -14 1 -11 | 0 19 2 16 }}
=== Meansquared ===
[[Subgroup]]: 4.9.25


[[Comma list]]: [[6561/6400]]
{{Mapping|legend=5| 1 -7 0 1 -11 | 0 19/2 0 2 16 }}
: mapping generators: ~2, ~144/77


{{Mapping|legend=3| 1 3 4 | 0 1 4 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0088{{c}}, ~144/77 = 1084.4377{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~144/77 = 1084.4304{{c}}


Mapping generators: ~4, ~9/64
{{Optimal ET sequence|legend=1| 10e, 21, 31, 52, 83, 135, 353, 488, 623 }}


[[Optimal tuning]] ([[CTE]]): ~4 = 2\1, ~9/4 = 1394.429
[[Badness]] (Sintel): 0.199


[[Support]]ing [[ET]]s: 12, 7, 19, 5, 31, 26, 17[+25], 43, 9[-25], 33[-25], 45, 29[+25], 8[+25], 22[+25]
=== Penta a.k.a. mechanism ===
Penta or mechanism is the 8 & 11 temperament in the 2.9.7.11 subgroup.


=== Archsquared ===
[[Subgroup]]: 2.9.7.11
[[Subgroup]]: 4.9.49


[[Comma list]]: 4096/3969
[[Comma list]]: 896/891, 26411/26244


{{Mapping|legend=3| 1 3 0 | 0 1 -2 }}
{{Mapping|legend=3| 1 0 -1 6 | 0 5 6 -4 }}


Mapping generators: ~4, ~9/64
{{Mapping|legend=5| 1 0 0 -1 6 | 0 5/2 0 6 -4 }}
: mapping generators: ~2, ~14/9


[[Optimal tuning]] ([[CTE]]): ~9/4 = 1419.190
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.4843{{c}}, ~14/9 = 761.2079{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~14/9 = 761.4979{{c}}


[[Support]]ing [[ET]]s: 5, 17, 22, 12, 7, 27, 32, 8, 39[+49], 29[+49], 9[+49], 19[+49], 37, 49
{{Optimal ET sequence|legend=1| 8, 11, 30, 41, 52 }}


== 8.9.… subgroups ==
[[Badness]] (Sintel): 0.331
=== Sixscared ===
Sixscared is a tuning which still maintains some consonance, while eviscerating the rules of conventional 12-tone harmony. The familiar major, minor and perfect intervals are nowhere to be found, and octaves are far and few between, so the seventh harmonic becomes the backbone of harmony. Approximating the harmonics 7, 8, 9, Sixscared is named for the classic dad joke: "Why was six scared? Because seven ate nine."


[[Subgroup]]: 8.9.7
[[Tp tuning #T2 tuning|RMS error]]: 0.4262 cents


[[Comma list]]: 64/63
Scales: [[penta5]], [[penta8]], [[penta11]], [[penta19]]


{{Mapping|legend=3| 1 0 2 | 0 1 -1 }}
=== Joan ===
{{See also| Chromatic pairs #Joan }}


: sval mapping generators: ~8, ~9
Joan is the restriction of [[starling temperaments #Casablanca|casablanca]] to the 2.9.7.11 subgroup, as well as every other step of [[orwell]], without prime 5.


: [[gencom]]: [8 9/8; 64/63]
[[Subgroup]]: 2.9.7.11


[[Optimal tuning]] ([[CTE]]): ~9/8 = 219.1898
[[Comma list]]: 99/98, 9317/9216


[[Optimal ET sequence]]: {{val| 16 17 15 }}, {{val| 33 35 31 }}, {{val| 148 … }}, {{val| 181 … }}, {{val| 214 … }}, {{val| 247 … }}
{{Mapping|legend=3| 1 0 1 3 | 0 7 4 1 }}


[[Badness]]: 0.0215 × 10<sup>-3</sup>
{{Mapping|legend=5| 1 0 0 1 3 | 0 7/2 0 4 1 }}
: mapping generators: ~2, ~11/8


= Fractional subgroup temperaments =
[[Optimal tuning]]s:
== 2.3.… subgroups ==
* [[WE]]: ~2 = 1200.8640{{c}}, ~11/8 = 543.0623{{c}}
=== Hypnosis ===
* [[CWE]]: ~2 = 1200.0000{{c}}, ~11/8 = 542.8446{{c}}
Related temperaments: [[Swetismic temperaments #Hypnos|hypnos]], [[Alphatricot family #Alphatricot|alphatricot]]


[[Subgroup]]: 2.3.7.11/5.13
{{Optimal ET sequence|legend=1| 11, 20, 31, 42, 115b′d }}


[[Comma list]]: 169/168, 540/539, 729/728
[[Badness]] (Sintel): 0.407


{{Mapping|legend=3| 1 0 -3 8 0 | 0 3 11 -13 7 }}
[[Tp tuning #T2 tuning|RMS error]]: 1.424 cents


[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~13/9 = 633.518
=== Novisept ===
Named by {{u|Pailiaq}} in 2026, novisept is generated by a one-cent-flat [[9/7]], such that stacking 5 of them gives [[7/4]]. It can be formed by doubling both generator and period of [[gizzard]].


{{Optimal ET sequence|legend=1| 17, 36, 118f, 125f, 161f, 197f }}
[[Subgroup]]: 2.9.7.13.17


[[Tp tuning #T2 tuning|RMS error]]: 0.5379 cents
[[Comma list]]: 442/441, 729/728, 833/832


=== Hydrothermal ===
{{Mapping|legend=3| 1 1 1 -1 3 | 0 6 5 13 3 }}
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.
: mapping generators: ~2, ~9/7


[[Subgroup]]: 2.3.7/5
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.3044{{c}}, ~9/7 = 433.9023{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~9/7 = 433.8360{{c}}


[[Comma list]]: [[50/49]]
{{Optimal ET sequence|legend=1| 11, 25f, 36, 47, 83, 130, 213g, 296g }}


{{Mapping|legend=3| 2 3 1 | 0 1 0 }}
[[Badness]] (Sintel): 0.142


[[Optimal tuning]] (inharmonic [[TE]]): ~1\2 = 590.998, ~[[10/7]]-1\2 = 128.962
=== Genius ===
Genius tempers out the [[Alpharabian comma]] in the 2.9.11 subgroup; it is therefore every other step of [[no-fives subgroup temperaments #Alphaxenean|alphaxenean]]. It was named by {{u|Akselai}} in 2024 after the genius in Roman religion, following the demon (daimon) in Greek mythology.


[[Support]]ing [[ET]]s: {{EDOs|4, 6, 8, 10, 18, 28, 46, 64, 110}}
[[Subgroup]]: 2.9.11
 
=== Argentic ===
Argentic is the 2.3.7/5 subgroup temperament tempering out [[5120/5103]].
 
[[Subgroup]]: 2.3.7/5


[[Comma list]]: [[5120/5103]] = {{monzo| 10 -6 -1 }}
[[Comma list]]: 131769/131072


{{Mapping|legend=3| 1 0 10 | 0 1 -6 }}
{{Mapping|legend=2| 1 1 4 | 0 4 -1 }}
: mapping generators: ~2, ~3
: mapping generators: ~2, ~16/11


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1\1, ~3/2 = 702.792
* [[WE]]: ~2 = 1200.3181{{c}}, ~16/11 = 650.8508{{c}}
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1\1, ~3/2 = 702.830
* [[CWE]]: ~2 = 1200.0000{{c}}, ~16/11 = 650.7321{{c}}


{{Optimal ET sequence|legend=1| 12, 29, 41, 70, 321, 391, 461, 531, 601 }}
{{Optimal ET sequence|legend=1| 11, 24, 59, 83, 142, 225, 367e, 592b′e, 959b′eee, 1326b′b′eeee }}
<small> based on subgroup TE </small>


Badness (Sintel): 0.119
[[Badness]] (Sintel): 0.198


==== Edson ====
=== Demon ===
{{See also| Chromatic pairs #Edson }}
Demon is a temperament which equates three [[11/9]]'s with [[16/9]], or equivalently three [[18/11]]'s 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 {{u|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 the demon temperament is near the basic tuning of 7L 4s (13\18), and indeed [[18edo]] supports this temperament.


Edson is related to [[pele]] and [[andromeda]].  
[[#Mabon|Mabon]] is a strong extension of this temperament to the 2.9.7.11 subgroup.  


[[Subgroup]]: 2.3.7/5.11/5.13/5
[[Subgroup]]: 2.9.11
 
[[Comma list]]: [[196/195]] = {{monzo| 2 -1 2 0 -1 }}, [[352/351]] = {{monzo| 5 -3 0 1 -1 }}, [[364/363]] = {{monzo| 2 -1 1 -2 1 }}


{{Mapping|legend=3| 1 0 10 17 22 | 0 1 -6 -10 -13 }}
[[Comma list]]: 1331/1296
: mapping generators: ~2, ~3


{{Mapping|legend=5| 1 1 -5 -1 2 4 | 0 1 29/4 5/4 -11/4 -23/4 }}
{{Mapping|legend=3| 1 1 2 | 0 3 2 }}
: [[gencom]]: [2 3/2; 196/195, 352/351, 364/363]
: mapping generators: ~2, ~18/11


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1\1, ~3/2 = 703.4398
* [[WE]]: ~2 = 1201.1151{{c}}, ~18/11 = 869.4724{{c}}
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1\1, ~3/2 = 703.414
* [[CWE]]: ~2 = 1200.0000{{c}}, ~18/11 = 869.3196{{c}}


{{Optimal ET sequence|legend=1| 12, 17, 29 }}
{{Optimal ET sequence|legend=1| 4, 7, 11, 18, 29, 76e }}


[[Tp tuning #T2 tuning|RMS error]]: 0.5102 cents
[[Badness]] (Sintel): 0.332


==== Haumea ====
== 4.3.… subgroups ==
{{See also| Chromatic pairs #Haumea }}
=== Tetrameantone ===
{{Main| Tetrameantone }}


Related temperaments include [[#Bridgetown|bridgetown]], [[namaka]], [[hemigari]], [[#Barbados|barbados]], and [[parizekmic]].  
Named by [[Budjarn Lambeth]] in 2024, tetrameantone is the double-octave restriction of [[meantone family #Meantone|meantone]].  


[[Subgroup]]: 2.3.7/5.11/5.13/5
[[Subgroup]]: 4.3.5


[[Comma list]]: [[352/351]], [[676/675]], [[847/845]]
[[Comma list]]: 81/80


{{Mapping|legend=3| 1 0 10 -6 -1 | 0 2 -12 9 3 }}
{{Mapping|legend=3| 1 0 -2 | 0 1 4 }}
: mapping generators: ~4, ~3


{{Mapping|legend=5| 1 2 -3/4 -11/4 9/4 5/4 | 0 -2 0 12 -9 -3 }}
[[Optimal tuning]]s:
: [[gencom]]: [2 15/13; 352/351 676/675 847/845]
* [[WE]]: ~4 = 2402.7812{{c}}, ~3 = 1898.4361{{c}}
* [[CWE]]: ~4 = 2400.0000{{c}}, ~3 = 1896.6512{{c}}


[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~15/13 = 248.491
[[Optimal ET sequence]]: [[5ed4|a′5]], [[14ed4|a′14]], [[19ed4|a′19]], [[43ed4|a′43]], [[62ed4|a′62]], [[81ed4|a′81]], [[262ed4|a′262b]], [[343ed4|a′343b]]


{{Optimal ET sequence|legend=1| 24, 29, 111, 140, 169, 198, 565d, 763bd, 961bd }}
[[Badness]] (Sintel): 0.0866


[[Tp tuning #T2 tuning|RMS error]]: 0.2668 cents
=== Tetrahanson ===
{{Main| Tetrahanson }}


=== Historical ===
Named by {{u|CompactStar}} in 2023, tetrahanson is the double-octave restriction of [[kleismic family #Kleismic a.k.a. hanson|hanson]].  
{{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]]: 4.3.5


[[Subgroup]]: 2.3.7/5.11/5.13/5
[[Comma list]]: 15625/15552


[[Comma list]]: 364/363, 441/440, 1001/1000
{{Mapping|legend=3| 1 -3 -2 | 0 6 5 }}
: mapping generators: ~4, ~12/5


{{Mapping|legend=3| 1 2 0 1 2 | 0 -6 7 2 -9 }}
[[Optimal tuning]]s:
* [[WE]]: ~4 = 2400.3318{{c}}, ~12/5 = 1517.2163{{c}}
* [[CWE]]: ~4 = 2400.0000{{c}}, ~12/5 = 1517.0308{{c}}


[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~21/20 = 83.016
[[Optimal ET sequence]]: [[8ed4|a′8]], [[11ed4|a′11]], [[19ed4|a′19]], [[68ed4|a′68]], [[87ed4|a′87]], [[106ed4|a′106]], [[511ed4|a′511]], [[617ed4|a′617]], [[723ed4|a′723]], [[829ed4|a′829]], [[935ed4|a′935c]], [[1764ed4|a′1764c]]


{{Optimal ET sequence|legend=1| 14, 29, 72, 101, 130, 159 }}
[[Badness]] (Sintel): 0.155


[[Tp tuning #T2 tuning|RMS error]]: 0.2562 cents
=== Tetramagic ===
Named by [[Budjarn Lambeth]] in 2024, tetramagic is the double-octave restriction of [[magic family #Magic|magic]].  


=== Direct breedsmic ===
[[Subgroup]]: 4.3.5
Related temperament: [[hemithirds]], [[newt]]


[[Subgroup]]: 2.3.49/5
[[Comma list]]: 3125/3072


[[Comma list]]: 2401/2400
{{Mapping|legend=3| 1 0 1 | 0 5 1 }}
: mapping generators: ~4, ~5/4


{{Mapping|legend=3| 1 1 3 | 0 2 1 }}
[[Optimal tuning]]s:
* [[WE]]: ~4 = 2402.4898{{c}}, ~5/4 = 380.4527{{c}}
* [[CWE]]: ~4 = 2400.0000{{c}}, ~5/4 = 380.2194{{c}}


[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~49/40 = 350.966
[[Optimal ET sequence]]: [[6ed4|a′6]], [[13ed4|a′13]], [[19ed4|a′19]], [[63ed4|a′63]], [[82ed4|a′82]], [[101ed4|a′101]], [[120ed4|a′120]], [[221ed4|a′221c]], [[341ed4|a′341c]]


{{Optimal ET sequence|legend=1|7, 10, 17}}
[[Badness]] (Sintel): 0.459


[[Tp tuning #T2 tuning|RMS error]]: ?
=== Blacktetra ===
Named by [[Budjarn Lambeth]] in 2024, blacktetra is the double-octave restriction of [[blackwood family #Blackwood|blackwood]].


==== Bridgetown ====
[[Subgroup]]: 4.3.5
{{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]].
[[Comma list]]: 256/243


[[Subgroup]]: 2.3.11/5.13/5
{{Mapping|legend=3| 5 4 0 | 0 0 1 }}
: mapping generators: ~4, ~5


[[Comma list]]: [[352/351]], [[676/675]]
[[Optimal tuning]]s:  
 
* [[WE]]: ~4/3 = 477.7018{{c}}, ~5/4 = 397.6806{{c}} (~16/15 = 80.0212{{c}})
{{Mapping|legend=3| 1 0 -6 -1 | 0 2 9 3 }}
* [[CWE]]: ~4/3 = 480.0000{{c}}, ~5/4 = 395.1255{{c}} (~16/15 = 84.7745{{c}})


{{Mapping|legend=5| 1 2 -5/3 0 4/3 1/3 | 0 -2 4 0 -5 1 }}
[[Optimal ET sequence]]: [[5ed4|a′5]], [[20ed4|a′20]], [[25ed4|a′25]], [[30ed4|a′30]]
: [[gencom]]: [2 15/13; 352/351 676/675]


[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~15/13 = 248.399
[[Badness]] (Sintel): 0.748


{{Optimal ET sequence|legend=1| 5, 9, 14, 19, 24, 29, 169, 198, 227, 256, 285, 314 }}
== 4.6.… subgroups ==
=== Meanquad ===
Meanquad is the double-octave restriction of [[meantone family #Meantone|meantone]] to the 4.6.5 subgroup. Similarly to meantone, it has a flat [[3/2]] as generator and equates [[5/4]] and [[81/64]]. It possesses [[mos scale]]s of the forms [[3L 4s (4/1-equivalent)|3L 4s<4/1>]], [[7L 3s (4/1-equivalent)|7L 3s<4/1>]], and [[7L 10s (4/1-equivalent)|7L 10s<4/1>]], which could be viewed as equivalents to the pentic, diatonic, and chromatic scales of meantone, respectively.


[[Tp tuning #T2 tuning|RMS error]]: 0.2513 cents
[[17ed4]] is one of the smallest tunings that [[support]] this temperament and features the decatonic scale. Like [[12edo]], it can be viewed as the boundary between meanquad and [[#quarchy|quarchy]], the 4.6.7-subgroup restriction of [[archy]]. Larger options include 24ed4 (12edo), [[31ed4]] and [[55ed4]].


=== Blackweed ===
[[Subgroup]]: 4.6.5
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]]: 81/80 ({{monzo| -4 4 -1 }})


[[Comma list]]: {{monzo| 8 -5 }} (256/243)
{{Mapping|legend=3| 1 0 -4 | 0 1 4 }}
 
: mapping generators: ~4, ~6
{{Mapping|legend=3| 5 8 0 | 0 0 1 }}
: mapping generators: ~9/8, ~11/7


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[Tp tuning|subgroup]] [[WE]]: ~8/7 = 238.851{{c}}, ~11/7 = 782.457{{c}}
* [[Tp tuning|Subgroup]] [[WE]]: ~4 = 2402.7812{{c}}, ~3/2 = 697.0455{{c}}
: [[error map]]: {{val| -5.746 +8.852 -0.035 }}
* [[Tp tuning|Subgroup]] [[CWE]]: ~4 = 2400.0000{{c}}, ~3/2 = 696.6512{{c}}
* [[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 }}
[[Optimal ET sequence]]: [[7ed4|a′7]], [[17ed4|a′17]], [[24ed4|a′24]], [[31ed4|a′31]], [[100ed4|a′100]], [[131ed4|a′131]], [[293ed4|a′293*]]
: <nowiki/>* Wart for 6


=== Pepperoni ===
[[Badness]] (Sintel): 0.0146
{{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.
==== Tetrominant ====
Subgroup: 4.6.5.7


[[Subgroup]]: 2.3.11/7.13/7
Comma list: 36/35 ({{monzo| 0 2 -1 -1 }}), 64/63 ({{monzo| 4 -2 0 -1 }})


[[Comma list]]: 352/351, 364/363
{{Mapping|legend=2| 1 0 -4 4 | 0 1 4 -2 }}


{{Mapping|legend=3| 1 0 7 12 | 0 1 -4 -7 }}
Optimal tunings:
* Subgroup WE: ~4 = 2400.0000{{c}}, ~3/2 = 699.622{{c}}
* Subgroup CWE: ~4 = 2400.0000{{c}}, ~3/2 = 699.622{{c}}


{{Mapping|legend=5| 1 1 0 -8/3 1/3 7/3 | 0 1 0 11/3 -1/3 -10/3 }}
Optimal ET sequence: [[7ed4|a′7]], [[17ed4|a′17]], [[24ed4|a′24]], [[41ed4|a′41]], [[65ed4|a′65cd]]
: [[gencom]]: [2 3/2; 352/351 364/363]


[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~3/2 = 703.856
Badness (Sintel): 0.0974


{{Optimal ET sequence|legend=1| 5, 7, 12f, 17, 29, 46, 58, 75, 80, 87, 104, 121, 167, 196, 208, 271, 595b*<sup>†</sup> }}
=== Quarchy ===
: <nowiki />* wart for 11/7
[[Subgroup]]: 4.6.7
: <sup>†</sup> wart for 13/7


[[Tp tuning #T2 tuning|RMS error]]: 0.3789 cents
[[Comma list]]: 64/63 ({{monzo| 4 -2 -1 }})


=== Barbados ===
{{Mapping|legend=3| 1 0 4 | 0 1 -2 }}
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.
: mapping generators: ~4, ~6


[[Subgroup]]: 2.3.13/5
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~4 = 2393.9105{{c}}, ~3/2 = 707.5215{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~4 = 2400.0000{{c}}, ~3/2 = 709.3901{{c}}


[[Comma list]]: 676/675 = {{monzo| 2 -3 2 }}
Optimal ET sequence: [[7ed4|a′7]], [[10ed4|a′10]], [[17ed4|a′17]], [[27ed4|a′27]], [[44ed4|a′44]], [[159ed4|a′159d]], [[203ed4|a′203*dd]]
: <nowiki/>* Wart for 6


[[Sval]] [[mapping]]: [{{val| 1 0 -1 }}, {{val| 0 2 3 }}]
Badness (Sintel): 0.0134


[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~2 = 1\1, ~15/13 = 248.621
== 4.9.… subgroups ==
=== Meansquared ===
Named by {{u|CompactStar}} in 2023, meansquared tempers out the double syntonic comma, [[6561/6400]], in the 4.9.25 subgroup.  


{{Optimal ET sequence|legend=1| 5, 9, 14, 19, 24, 29, 53, 82, 111, 140, 251, 362 }}
[[Subgroup]]: 4.9.25


[[Badness]]: 0.002335
[[Comma list]]: 6561/6400


; Music
{{Mapping|legend=3| 1 0 -4 | 0 1 4 }}
* [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]
: mapping generators: ~4, ~9


==== Tobago ====
[[Optimal tuning]]s:
{{See also| Chromatic pairs #Tobago }}
* [[WE]]: ~4 = 2402.7812{{c}}, ~9/4 = 1394.0910{{c}}
* [[CWE]]: ~4 = 2400.0000{{c}}, ~9/4 = 1393.3025{{c}}


Tobago, the 10 &amp; 14 temperament in the 2.3.11.13/5 subgroup, extends [[neutral]] and [[barbados]].
[[Optimal ET sequence]]: [[5ed4|a′5]], [[7ed4|a′7]], [[12ed4|a′12]], [[19ed4|a′19]], [[31ed4|a′31]], [[50ed4|a′50]], [[81ed4|a′81]], [[131ed4|a′131b′]]


[[Subgroup]]: 2.3.11.13/5
[[Badness]] (Sintel): 0.173


[[Comma list]]: [[243/242]], [[676/675]]
=== Archsquared ===
Named by {{u|CompactStar}} in 2023, archsquared tempers out the double archytas comma, [[4096/3969]], in the 4.9.49 subgroup.


{{Mapping|legend=3| 2 0 -1 -2 | 0 2 5 3 }}
[[Subgroup]]: 4.9.49


{{Mapping|legend=5| 2 4 -2 0 9 2 | 0 -2 3/2 0 -5 -3/2 }}
[[Comma list]]: 4096/3969
: [[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=3| 2 0 -1 -2 5 | 0 2 5 3 2 }}
{{Mapping|legend=3| 1 0 6 | 0 1 -2 }}
: mapping generators: ~4, ~9


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[Tp tuning|subgroup CTE]]: ~17/12 = 1\2, ~26/15 = 950.7656 (~15/13 = 249.2344)
* [[WE]]: ~4 = 2393.9105{{c}}, ~9/4 = 1415.0430{{c}}
* [[Tp tuning|subgroup CWE]]: ~17/12 = 1\2, ~26/15 = 950.6011 (~15/13 = 249.3989)
* [[CWE]]: ~4 = 2400.0000{{c}}, ~9/4 = 1418.7801{{c}}


{{Optimal ET sequence|legend=1| 10, 14, 24, 106, 130, 154, 178*, 202* }}
[[Optimal ET sequence]]: [[5ed4|a′5]], [[12ed4|a′12]], [[17ed4|a′17]], [[22ed4|a′22]], [[137ed4|a′137b′d′d′]], [[159ed4|a′159b′d′d′d′]], [[181ed4|a′181b′b′d′d′d′]]
: <nowiki />* wart for 13/5


=== Oceanfront ===
[[Badness]] (Sintel): 0.159
Related temperaments: [[Archytas clan #Superpyth|superpyth]], [[Archytas clan #Ultrapyth|ultrapyth]]


[[Subgroup]]: 2.3.7.13/5
== 8.9.… subgroups ==
=== 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 was named by [[Budjarn Lambeth]] in 2023 for the classic dad joke: "Why was six scared? Because seven ate nine."


[[Comma list]]: 64/63, 91/90
[[Subgroup]]: 8.9.7


{{Mapping|legend=3| 1 0 6 -5 | 0 1 -2 4 }}
[[Comma list]]: 64/63


[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~3/2 = 713.910
{{Mapping|legend=3| 1 0 2 | 0 1 -1 }}
: mapping generators: ~8, ~9


{{Optimal ET sequence|legend=1| 5, 22, 27, 32, 37 }}
[[Optimal tuning]]s:  
 
* [[WE]]: ~8 = 3590.8657{{c}}, ~9/8 = 219.0878{{c}}
[[Tp tuning #T2 tuning|RMS error]]: 2.063 cents
* [[CWE]]: ~8 = 3600.0000{{c}}, ~9/8 = 218.7801{{c}}
 
Scales: [[Oceanfront scales]]
 
=== Seventeen-cot ===
Seventeen-cot is a rank-2 temperament in the 2.3.13/5 and 2.3.11/5.13/5 subgroups. It tempers out the [[Tendoartisma]] in the 2.3.13/5 subgroup. It can be generated with a ~2/1 octave and a ~2250/2197 or ~169/165 generator which is a 17th of a ~3/2 perfect fifth. It can be described as the 29 & 146 temperament in these subgroups.
 
====2.3.13/5 subgroup====
 
Comma basis: {{monzo| -6 -11 17 }} (2.3.13/5)
 
edo join: 29 & 146
 
{{Mapping|legend=3| 1 1 1 | 0 17 11 }}
: mapping generators: ~2, ~2250/2197
 
Optimal tunings:
* WE: ~2 = 1200.0001354{{c}}, ~2250/2197 = 41.2914993{{c}}
: error map: {{val| +0.0001354 +0.0006234 -0.0073197}}
* CWE: ~2 = 2/1, ~2250/2197 = 41.2915011{{c}}
: error map: {{val| 0.0000000 +0.0005177 -0.0074359}}
 
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=3| 1 1 1 1 | 0 17 4 11 }}
: mapping generators: ~2, ~169/165
 
Optimal tunings:
* WE: ~2 = 1199.9934923{{c}}, ~169/165 = 41.2918271{{c}}
: error map: {{val| -0.0065077 -0.0004485 +0.1565720 -0.0103579}}
* CWE: ~2 = 2/1, ~169/165 = 41.2917463{{c}}
: error map: {{val| 0.0000000 +0.0046870 +0.1627569 -0.0043781}}
 
edos: 29, 465, 494, 436, 523, 407, 378, 349, 320, 291, 30[-3], 262, 28[+3], 233
 
Badness (Sintel): 0.080
 
=== 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=3| 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 }}


=== Surprise ===
[[Optimal ET sequence]]: [[16ed8|a″16]], [[33ed8|a″33]], [[148ed8|a″148b′d]], [[181ed8|a″181b′d]], [[214ed8|a″214b′d]], [[247ed8|a″247b′b′d]]
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
[[Badness]] (Sintel): 0.0265
 
[[Comma list]]: [[57/56]] ({{Monzo| -3 1 1 }})
 
{{Mapping|legend=3| 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


== 2.5.… subgroups ==
== 2.5.… subgroups ==
Line 1,259: Line 1,054:
{{See also | Chromatic pairs #Guanyintet }}
{{See also | Chromatic pairs #Guanyintet }}


Guanyintet, the {{nowrap| 4 & 9 }} temperament in the 2.5.7/3.11/3 subgroup, is the main rank-2 chain of [[guanyin]] and a restriction of [[orwell]]. It is defined by tempering out [[1728/1715]] ({{S|6/S7}}) and [[540/539]] (S12/S14), which imply [[176/175]] (S8/S10) as well as S11/S15 being tempered out. The tonic and the first three generator steps make a [[guanyin tetrad]], hence the name.  
Guanyintet, the {{nowrap| 4 & 9 }} temperament in the 2.5.7/3.11/3 subgroup, is the main rank-2 chain of [[orwellismic family #Guanyin|guanyin]] and a restriction of [[semicomma family #Orwell|orwell]]. It is defined by tempering out [[176/175]] and [[540/539]], which imply [[1728/1715]] as well as [[3388/3375]] are tempered out. The tonic and the first three generator steps make a [[guanyin tetrad]], hence the name.  


[[Subgroup]]: 2.5.7/3.11/3
[[Subgroup]]: 2.5.7/3.11/3


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


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


{{Mapping|legend=5| 1 -4/3 3 -1/3 5/3 | 0 4/3 -3 7/3 -11/3 }}
{{Mapping|legend=5| 1 0 0 2 -2 | 0 -4/3 3 -7/3 11/3 }}
: [[gencom]]: [2 7/6; 176/175 540/539]
: mapping generators: ~2, ~12/7


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* ([[Tp tuning|subgroup]] [[CTE]]): ~2 = 1200.000, ~7/6 = 270.455
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.2336{{c}}, ~12/7 = 929.3128{{c}} (~7/6 = 269.9208{{c}})
* ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1200.000, ~7/6 = 270.093
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~12/7 = 929.7608{{c}} (~7/6 = 270.2392{{c}})


{{Optimal ET sequence|legend=1| 9, 22, 31, 40, 191c*, 231c*, 271c*, 311c* }}
{{Optimal ET sequence|legend=1| 9, 22, 31, 40, 191c*, 231c*, 271c*, 311c* }}
: <nowiki/>* wart for 7/3
: <nowiki/>* Wart for 7/3


[[Tp tuning #T2 tuning|RMS error]]: 0.6028 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.6028 cents


==== Tridecimal guanyintet ====
==== 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.
Guanyintet can extend to the add-13 subgroup by the equivalences ([[12/11]])<sup>3</sup>~[[13/10]] and ([[15/14]])<sup>3</sup>~[[16/13]]. However, note that it is not supported by tridecimal orwell (31 & 53), but instead the less accurate [[semicomma family #Winston|winston]] (22f & 31), as orwell prefers slightly sharper tunings than guanyintet. [[40edo]] remains an excellent tuning.


[[Subgroup]]: 2.5.7/3.11/3.13
Subgroup: 2.5.7/3.11/3.13


[[Comma list]]: [[176/175]] ({{monzo| 4 -2 -1 1 0 }}), [[540/539]] ({{monzo| 2 1 -2 -1 0 }}), [[1573/1568]] ({{monzo| -5 0 -2 2 1 }})
Comma list: 176/175 ({{monzo| 4 -2 -1 1 0 }}), 540/539 ({{monzo| 2 1 -2 -1 0 }}), 1573/1568 ({{monzo| -5 0 -2 2 1 }})


{{Mapping|legend=3| 1 0 1 3 1 | 0 -3 1 -5 12 }}
{{Mapping|legend=2| 1 0 1 3 1 | 0 3 -1 5 -12 }}
: mapping generators: ~2, ~12/7


[[Optimal tuning]]s:  
Optimal tunings:  
* ([[Tp tuning|subgroup]] [[CTE]]): ~2 = 1200.000, ~7/6 = 270.152
* Subgroup WE: ~2 = 1199.3604{{c}}, ~12/7 = 929.2866{{c}} (~7/6 = 270.0738{{c}})
* ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1200.000, ~7/6 = 270.218
* Subgroup CWE: ~2 = 1200.0000{{c}}, ~12/7 = 929.7980{{c}} (~7/6 = 270.2020{{c}})


{{Optimal ET sequence|legend=1| 9, 22, 31, 40, 71, 111, 151, 262c*}} <small> using subgroup TE </small>
{{Optimal ET sequence|legend=0| 9, 22f, 31, 40, 71, 111, 151, 262c* }}
: <nowiki/>* wart for 7/3
: <nowiki/>* Wart for 7/3


Badness (Sintel): 0.329
Badness (Sintel): 0.329
Line 1,302: Line 1,095:
{{See also | Chromatic pairs #Laz }}
{{See also | Chromatic pairs #Laz }}


Laz is related to [[avalokita]] as well as to [[winston]].  
Laz is related to [[orwellismic family #Avalokita|avalokita]] as well as to [[semicomma family #Winston|winston]].  


[[Subgroup]]: 2.5.7/3.11/3.13/3
Subgroup: 2.5.7/3.11/3.13/3


[[Comma list]]: [[144/143]] ({{monzo| 4 0 0 -1 -1 }}), [[176/175]] ({{monzo| 4 -2 -1 1 }}), [[196/195]] ({{monzo| 2 -1 2 0 -1 }}
Comma list: 144/143 ({{monzo| 4 0 0 -1 -1 }}), 176/175 ({{monzo| 4 -2 -1 1 }}), 196/195 ({{monzo| 2 -1 2 0 -1 }}


{{Mapping|legend=3| 1 0 2 -2 6 | 0 3 -1 5 -5 }}
{{Mapping|legend=2| 1 0 2 -2 6 | 0 3 -1 5 -5 }}


{{Mapping|legend=5| 1 -5/4 3 -1/4 7/4 -1/4 | 0 -1/4 -3 3/4 -21/4 19/4 }}
{{Mapping|legend=4| 1 -3/2 0 1/2 -7/2 9/2 | 0 1/4 3 -3/4 21/4 -19/4 }}
: [[gencom]]: [2 7/6; 144/143 176/175 196/195]


[[Optimal tuning]]s:  
Optimal tunings:  
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~12/7 = 930.598
* Subgroup WE: ~2 = 1198.4320{{c}}, ~12/7 = 929.4834{{c}} (~7/6 = 268.9486{{c}})
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~12/7 = 930.700
* Subgroup CWE: ~2 = 1200.0000{{c}}, ~12/7 = 930.5797{{c}} (~7/6 = 269.4203{{c}})


{{Optimal ET sequence|legend=1| 9, 31, 40, 49, 156c*†, 205c*† }}
{{Optimal ET sequence|legend=0| 9, 31, 40, 49, 156c*†, 205c*† }}
: <nowiki/>* wart for 7/3
: <nowiki/>* Wart for 7/3
: † wart for 11/3
: † Wart for 11/3


[[Tp tuning #T2 tuning|RMS error]]: 0.8790 cents
RMS error: 0.8790 cents


=== Kryptonite ===
=== Kryptonite ===
{{See also| Chromatic pairs #Kryptonite }}
{{See also| Chromatic pairs #Kryptonite }}


Kryptonite is related to [[krypton]].  
Kryptonite is related to [[sengic family #Krypton|krypton]].  


[[Subgroup]]: 2.5.7/3.11/3.13/3
Subgroup: 2.5.7/3.11/3.13/3


[[Comma list]]: 56/55 ({{monzo| 3 -1 1 -1 }}), 78/77 ({{monzo| 1 0 -1 -1 1 }}), 91/90 ({{monzo| -1 -2 1 0 1 }})
Comma list: 56/55 ({{monzo| 3 -1 1 -1 }}), 78/77 ({{monzo| 1 0 -1 -1 1 }}), 91/90 ({{monzo| -1 -2 1 0 1 }})


{{Mapping|legend=3| 1 2 1 2 2 | 0 3 2 -1 1 }}
{{Mapping|legend=2| 1 2 1 2 2 | 0 3 2 -1 1 }}
 
{{Mapping|legend=4| 1 -5/4 2 -1/4 3/4 3/4 | 0 -1/2 3 3/2 -3/2 1/2 }}
: mapping generators: ~2, ~13/12
: mapping generators: ~2, ~13/12


{{Mapping|legend=5| 1 -5/4 2 -1/4 3/4 3/4 | 0 -1/2 3 3/2 -3/2 1/2 }}
Optimal tunings:  
: [[gencom]]: [2 13/12; 56/55 78/77 91/90]
* Subgroup WE: ~2 = 1197.6484{{c}}, ~13/12 = 132.1686{{c}}
 
* Subgroup CWE: ~2 = 1200.0000{{c}}, ~13/12 = 131.8445{{c}}
[[Optimal tuning]]s:  
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~13/12 = 130.945
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~13/12 = 132.428


{{Optimal ET sequence|legend=1| 1, …, 8, 9 }}
{{Optimal ET sequence|legend=0| 1, …, 8, 9 }}


[[Tp tuning #T2 tuning|RMS error]]: 2.545 cents
RMS error: 2.545 cents


=== Marveltri ===
=== Marveltri ===
{{See also| Chromatic pairs #Marveltri }}
{{See also| Chromatic pairs #Marveltri }}


Marveltri, the {{nowrap| 3 & 13 }} temperament in the 2.5.9/7 subgroup, is related to [[marvel]], [[magic]], and the unnamed {{nowrap| 22 & 47 }} temperament. The tonic and the first two generator steps make a [[marvel triad]], hence the name.  
Marveltri, the {{nowrap| 3 & 13 }} temperament in the 2.5.9/7 subgroup, is related to [[marvel family #Marvel|marvel]], [[magic family #Magic|magic]], and the unnamed {{nowrap| 22 & 47 }} temperament. The tonic and the first two generator steps make a [[marvel triad]], hence the name.  


[[Subgroup]]: 2.5.9/7
[[Subgroup]]: 2.5.9/7
Line 1,356: Line 1,147:


{{Mapping|legend=3| 1 0 5 | 0 1 -2 }}
{{Mapping|legend=3| 1 0 5 | 0 1 -2 }}
: mapping generators: ~2, ~5


{{Mapping|legend=5| 1 2 0 -1 | 0 -4/5 1 2/5 }}
{{Mapping|legend=5| 1 2 0 -1 | 0 -4/5 1 2/5 }}
: [[gencom]]: [2 5; 225/224]
: mapping generators: ~2, ~5


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/4 = 384.208
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1200.5971{{c}}, ~5/4 = 383.8285{{c}}
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/4 = 383.638
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 383.8540{{c}}


{{Optimal ET sequence|legend=1| 3, 13, 16, 19, 22, 25, 72, 97, 122, 269c* }}
{{Optimal ET sequence|legend=1| 3, 13, 16, 19, 22, 25, 72, 97, 122, 269c* }}
: <nowiki/>* wart for 9/7
: <nowiki/>* Wart for 9/7


[[Tp tuning #T2 tuning|RMS error]]: 0.4801 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.4801 cents


==== Sulis ====
==== Sulis ====
Sulis is related to [[minerva]] and [[würschmidt]].  
Sulis is related to [[marvel family #Minerva|minerva]] and [[würschmidt family #Septimal würschmidt|würschmidt]].  


[[Subgroup]]: 2.5.9/7.11/9
Subgroup: 2.5.9/7.11/9


[[Comma list]]: 99/98 ({{monzo| -1 0 2 1 }}), 176/175 ({{monzo| 4 -2 1 1 }})
Comma list: 99/98 ({{monzo| -1 0 2 1 }}), 176/175 ({{monzo| 4 -2 1 1 }})


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


[[Optimal tuning]]s:  
Optimal tunings:  
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/4 = 386.617
* Subgroup WE: ~2 = 1200.1086{{c}}, ~5/4 = 386.5931{{c}}
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/4 = 386.558
* Subgroup CWE: ~2 = 1200.0000{{c}}, ~5/4 = 386.5785{{c}}


{{Optimal ET sequence|legend=1| 3, …, 22, 25, 28, 31, 59 }}
{{Optimal ET sequence|legend=0| 3, …, 22, 25, 28, 31, 59 }}


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


== 2.5/3.… subgroups ==
== 2.5/3.… subgroups ==
Line 1,391: Line 1,181:
{{See also| Chromatic pairs #Magicaltet }}
{{See also| Chromatic pairs #Magicaltet }}


Magicaltet is related to [[keemic]], [[superkleismic]], and [[magic]]. The tonic and the first three generator steps make a [[magical seventh chord]], hence the name.  
Magicaltet is related to [[keemic family #Keemic|keemic]], [[gamelismic clan #Superkleismic|superkleismic]], and [[magic family #Magic|magic]]. The tonic and the first three generator steps make a [[magical seventh chord]], hence the name.  


[[Subgroup]]: 2.5/3.7.11
[[Subgroup]]: 2.5/3.7.11
Line 1,398: Line 1,188:


{{Mapping|legend=3| 1 0 5 2 | 0 1 -3 2 }}
{{Mapping|legend=3| 1 0 5 2 | 0 1 -3 2 }}
{{Mapping|legend=5| 1 0 0 5 2 | 0 -1/2 1/2 -3 2 }}
: mapping generators: ~2, ~5/3
: mapping generators: ~2, ~5/3
{{Mapping|legend=5| 1 -1/2 1/2 2 4 | 0 1/2 -1/2 3 -2 }}
: [[gencom]]: [2 6/5; 100/99 385/384]


[[Optimal tuning]]s:
[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 877.343
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.9868{{c}}, ~5/3 = 877.3312{{c}}
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 877.351
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~5/3 = 877.3417{{c}}


{{Optimal ET sequence|legend=1| 4, 7, 11, 15, 26, 67, 93* }}
{{Optimal ET sequence|legend=1| 4, 7, 11, 15, 26, 67, 93* }}
: <nowiki/>* wart for 5/3
: <nowiki/>* Wart for 5/3


[[Tp tuning #T2 tuning|RMS error]]: 1.206 cents
[[Tp tuning #T2 tuning|RMS error]]: 1.206 cents
Line 1,415: Line 1,204:
{{See also | Chromatic pairs #Starlingtet }}
{{See also | Chromatic pairs #Starlingtet }}


Starlingtet, the {{nowrap| 4 & 15 }} temperament in the 2.5/3.7/3 subgroup, is related to [[starling]] as well as to [[myna]]. The tonic and the first three generator steps make a [[starling tetrad]], hence the name.  
Starlingtet, the {{nowrap| 4 & 15 }} temperament in the 2.5/3.7/3 subgroup, is related to [[starling family #Starling|starling]] as well as to [[starling temperaments #Myna|myna]]. The tonic and the first three generator steps make a [[starling tetrad]], hence the name.  


[[Subgroup]]: 2.5/3.7/3
[[Subgroup]]: 2.5/3.7/3


[[Comma list]]: [[126/125]] ({{monzo| 1 -3 1 }})
[[Comma list]]: 126/125 ({{monzo| 1 -3 1 }})


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


{{Mapping|legend=5| 1 1/3 1/3 -2/3 | 0 -4/3 -1/3 5/3 }}
: mapping generators: ~2, ~5/3
: mapping generators: ~2, ~5/3
{{Mapping|legend=5| 1 -1 0 1 | 0 4/3 1/3 -5/3 }}
: [[gencom]]: [2 6/5; 126/125]


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 888.759
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.7931{{c}}, ~5/3 = 888.6923{{c}}
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 888.846
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~5/3 = 888.8022{{c}}


{{Optimal ET sequence|legend=1| 4, 15, 19, 23, 27 }}
{{Optimal ET sequence|legend=1| 4, 15, 19, 23, 27 }}
Line 1,436: Line 1,223:
[[Tp tuning #T2 tuning|RMS error]]: 0.8398 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.8398 cents


==== Greeley ====
==== Skateboard ====
{{See also| Chromatic pairs #Greeley }}
{{See also| Chromatic pairs #Skateboard }}


Greeley is related to [[opossum]] as well as to [[nusecond]].  
Skateboard is related to [[starling family #Thrasher|thrasher]].  


[[Subgroup]]: 2.5/3.7/3.11/3
Subgroup: 2.5/3.7/3.11.13/9


[[Comma list]]: 121/120 ({{monzo| -3 -1 0 2 }}), 126/125 ({{monzo| 1 -3 1 }})
Comma list: 56/55 ({{monzo| 3 -1 1 -1 }}), 91/90 ({{monzo| -1 -1 1 0 1 }}), 100/99 ({{monzo| 2 2 0 -1 }})


{{Mapping|legend=3| 1 1 2 2 | 0 -2 -6 -1 }}
{{Mapping|legend=2| 1 0 -1 2 2 | 0 1 3 2 -2 }}


{{Mapping|legend=5| 1 -5/4 -1/4 3/4 3/4 | 0 9/4 1/4 -15/4 5/4 }}
{{Mapping|legend=4| 1 -3/7 -3/7 -10/7 2 8/7 | 0 0 1 3 2 -2 }}
: [[gencom]]: [2 11/10; 121/120 126/125]


[[Optimal tuning]]s:  
Optimal tunings:  
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~11/10 = 155.696
* Subgroup WE: ~2 = 1197.4723{{c}}, ~5/3 = 884.2917{{c}}
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~11/10 = 155.776
* Subgroup CWE: ~2 = 1200.0000{{c}}, ~5/3 = 885.6469{{c}}


{{Optimal ET sequence|legend=1| 8, 15, 23, 54, 77, 100, 131* }}
{{Optimal ET sequence|legend=0| 4, 11, 15, 19, 23, 42e }}
: <nowiki/>* wart for 11/3


[[Tp tuning #T2 tuning|RMS error]]: 1.034 cents
RMS error: 2.396 cents


==== Skateboard ====
==== Greeley ====
{{See also| Chromatic pairs #Skateboard }}
{{See also| Chromatic pairs #Greeley }}


Skateboard is related to [[thrasher]].  
Greeley is related to [[porcupine family #Opossum|opossum]] as well as to [[starling temperaments #Nusecond|nusecond]].  


[[Subgroup]]: 2.5/3.7/3.11.13/9
Subgroup: 2.5/3.7/3.11/3


[[Comma list]]: 56/55 ({{monzo| 3 -1 1 -1 }}), 91/90 ({{monzo| -1 -1 1 0 1 }}), 100/99 ({{monzo| 2 2 0 -1 }})
Comma list: 121/120 ({{monzo| -3 -1 0 2 }}), 126/125 ({{monzo| 1 -3 1 }})


{{Mapping|legend=3| 1 0 -1 2 2 | 0 1 3 2 -2 }}
{{Mapping|legend=2| 1 1 2 2 | 0 -2 -6 -1 }}


{{Mapping|legend=5| 1 -3/7 4/7 11/7 4 -6/7 | 0 0 -1 -3 -2 2 }}
{{Mapping|legend=4| 1 -5/4 -1/4 3/4 3/4 | 0 9/4 1/4 -15/4 5/4 }}
: [[gencom]]: [2 6/5; 56/55 91/90 100/99]
: mapping generators: ~2, ~11/10


[[Optimal tuning]]s:  
Optimal tunings:  
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 886.158
* Subgroup WE: ~2 = 1200.4124{{c}}, ~11/10 = 155.8292{{c}}
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 886.158
* Subgroup CWE: ~2 = 1200.0000{{c}}, ~11/10 = 155.7399{{c}}


{{Optimal ET sequence|legend=1| 11, 15, 19, 23, 42d, 65d }}
{{Optimal ET sequence|legend=0| 8, 15, 23, 54, 77, 131* }}
: <nowiki/>* Wart for 11/3


[[Tp tuning #T2 tuning|RMS error]]: 2.396 cents
RMS error: 1.034 cents


=== Gariberttet ===
=== Gariberttet ===
Gariberttet is the 2.5/3.7/3 [[Subgroup temperament families, relationships, and genes|altergene]] of [[sirius]].
Gariberttet is the 2.5/3.7/3 altergene{{idio}}{{clarify}} of [[no-twos subgroup temperaments #Sirius|sirius]], tempering out [[3125/3087]]. A notable tuning besides WE and CWE given below is [[CTE]]/[[POTE]] at 293.6785 [[cent]]s.
 
[[Subgroup]]: 2.5/3.7/3
 
[[Comma list]]: 3125/3087
 
{{Mapping|legend=3| 1 0 0 | 0 3 5 }}
: mapping generators: ~2, ~25/21


==== Gariberttet (2.5/3.7/3.13/11 subgroup) ====
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.9983{{c}}, ~25/21 = 293.7393{{c}}
: [[error map]]: {{val| -0.002 -3.141 1.826 }}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~25/21 = 293.7395{{c}}
: error map: {{val| 0.000 -3.140 1.827 }}
 
{{Optimal ET sequence|legend=0| 4, 25*, 29, 33, 37, 41, 45, 49, 94, 143, 192 }}
: <nowiki/>* Wart for 7/3
 
==== 2.5/3.7/3.13/11 subgroup ====
{{See also | Chromatic pairs #Gariberttet }}
{{See also | Chromatic pairs #Gariberttet }}


Gariberttet can be described as the {{nowrap| 4 & 29 }} temperament in the 2.5/3.7/3.13/11 subgroup. Extensions to the full 7-, 11-, and 13-limits include [[quasitemp]].
Gariberttet can be described as the {{nowrap| 4 & 29 }} temperament in the 2.5/3.7/3.13/11 subgroup. Extensions to the full 7-, 11-, and 13-limits include [[quasitemp]].


[[Subgroup]]: 2.5/3.7/3.13/11
Subgroup: 2.5/3.7/3.13/11


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


{{Mapping|legend=3| 1 0 0 0 | 0 3 5 1 }}
{{Mapping|legend=2| 1 0 0 0 | 0 3 5 1 }}


{{Mapping|legend=5| 1 0 0 0 0 0 | 0 -8/3 1/3 7/3 -1/2 1/2 }}
{{Mapping|legend=4| 1 0 0 0 0 0 | 0 -8/3 1/3 7/3 -1/2 1/2 }}
: [[gencom]]: [2 13/11; 275/273 847/845]


[[Optimal tuning]]s:  
Optimal tunings:  
* [[Tp tuning|subgroup]] [[CTE]] and [[POTE]]: ~2 = 1200.000, ~13/11 = 293.679
* Subgroup WE: ~2 = 1199.9976{{c}}, ~13/11 = 293.6780{{c}}
* Subgroup CWE: ~2 = 1200.0000{{c}}, ~13/11 = 293.6782{{c}}


{{Optimal ET sequence|legend=1| 29, 33, 37, 41, 45, 49, 78, 94, 143* }}
{{Optimal ET sequence|legend=0| 4, 25*, 29, 33, 37, 41, 45, 49, 94, 143† }}
: <nowiki/>* wart for 13/11
: <nowiki/>* Wart for 7/3
: † Wart for 13/11


[[Tp tuning #T2 tuning|RMS error]]: 0.6914 cents
RMS error: 0.6914 cents


==== Indium ====
==== Indium ====
{{See also | Chromatic pairs #Indium }}
{{See also| Chromatic pairs #Indium }}


Indium can be described as the {{nowrap| 8 & 33 }} temperament in the 2.5/3.7/3.11/3 subgroup.  
Indium can be described as the {{nowrap| 8 & 33 }} temperament in the 2.5/3.7/3.11/3 subgroup.  


[[Subgroup]]: 2.5/3.7/3.11/3
Subgroup: 2.5/3.7/3.11/3


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


{{Mapping|legend=3| 1 0 0 2 | 0 6 10 -1 }}
{{Mapping|legend=2| 1 0 0 2 | 0 6 10 -1 }}


{{Mapping|legend=5| 1 -1/2 -1/2 -1/2 3/2 | 0 -15/4 9/4 25/4 -19/4 }}
{{Mapping|legend=4| 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]
: mapping generators: ~2, ~12/11


[[Optimal tuning]]s:  
Optimal tunings:  
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~12/11 = 146.978
* Subgroup WE: ~2 = 1199.5499{{c}}, ~12/11 = 146.9551{{c}}
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~12/11 = 147.010
* Subgroup CWE: ~2 = 1200.0000{{c}}, ~12/11 = 146.9960{{c}}


{{Optimal ET sequence|legend=1| 8, 33, 41, 49, 204*<sup>†</sup> }}
{{Optimal ET sequence|legend=0| 8, 33, 41, 49, 204*† }}
: <nowiki/>* wart for 7/3
: <nowiki/>* Wart for 7/3
: <sup>†</sup> wart for 11/3
: † Wart for 11/3


[[Tp tuning #T2 tuning|RMS error]]: 0.7788 cents
RMS error: 0.7788 cents


==== Ammon ====
==== Ammon ====
{{See also| Chromatic pairs #Ammon }}
{{See also| Chromatic pairs #Ammon }}


Ammon can be described as the {{nowrap| 8 & 29 }} temperament in the 2.5/3.7/3.11/3.13/3 subgroup. It extends [[tridec]], and is related to [[ammonite]]. It is generated by a semidiminished fourth, hence the old name ''semidim'', which has been rejected since 2025 to avoid confusion with another temperament of the same name.
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|tridec]], and is related to [[porcupine family #Ammonite|ammonite]]. It is generated by ~20/13, whose octave complement, ~13/10, may be considered a semidiminished fourth, hence the old name ''semidim'', which has been rejected since 2025 to avoid confusion with another temperament of the same name.


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


[[Comma list]]: [[121/120]] ({{monzo| -3 -1 0 2 }}), [[169/168]] ({{monzo| -3 0 -1 0 2 }}), [[275/273]] ({{monzo| 0 2 -1 1 -1 }})
Comma list: 121/120 ({{monzo| -3 -1 0 2 }}), 169/168 ({{monzo| -3 0 -1 0 2 }}), 275/273 ({{monzo| 0 2 -1 1 -1 }})


{{Mapping|legend=3| 1 3 5 3 4 | 0 -6 -10 -3 -5 }}
{{Mapping|legend=2| 1 -3 -5 0 -1 | 0 6 10 3 5 }}


{{Mapping|legend=5| 1 -3 0 2 0 1 | 0 24/5 -6/5 -26/5 9/5 -1/5 }}
{{Mapping|legend=4| 1 9/5 -6/5 -16/5 9/5 4/5 | 0 -24/5 6/5 26/5 -9/5 1/5 }}
: [[gencom]]: [2 13/10; 121/120 169/168 275/273]
: mapping generators: ~2, ~20/13


[[Optimal tuning]]s:  
Optimal tunings:  
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~13/10 = 453.121
* Subgroup WE: ~2 = 1200.9125{{c}}, ~20/13 = 747.3256{{c}}
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~13/10 = 453.242
* Subgroup CWE: ~2 = 1200.0000{{c}}, ~20/13 = 746.8104{{c}}


{{Optimal ET sequence|legend=1| 8, 29, 37, 45 }}
{{Optimal ET sequence|legend=0| 8, 29, 37, 45 }}


[[Tp tuning #T2 tuning|RMS error]]: 1.052 cents
RMS error: 1.052 cents


=== Sentry ===
=== Sentry ===
{{See also | Chromatic pairs #Sentry }}
{{See also | Chromatic pairs #Sentry }}


Sentry, the {{nowrap| 3 & 5 }} temperament in the 2.5/3.9/7 subgroup, is related to [[sensi]].  
Sentry, the {{nowrap| 3 & 5 }} temperament in the 2.5/3.9/7 subgroup, tempers out [[245/243]] and shares the same generator with [[sensipent family #Sensi|sensi]]. A notable tuning besides WE and CWE given below is [[CTE]]/[[POTE]] at 440.9020 [[cent]]s.  


[[Subgroup]]: 2.5/3.9/7
[[Subgroup]]: 2.5/3.9/7


[[Comma list]]: [[245/243]] ({{monzo| 0 1 -2 }})
[[Comma list]]: 245/243 ({{monzo| 0 1 -2 }})


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


{{Mapping|legend=5| 1 0 0 0 | 0 0 2 -1 }}
{{Mapping|legend=5| 1 0 0 0 | 0 0 2 -1 }}
: [[gencom]]: [2 9/7; 245/243]
: mapping generators: ~2, ~9/7


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[Tp tuning|subgroup]] [[CTE]] and [[POTE]]: ~2 = 1200.000, ~9/7 = 440.902
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.9983{{c}}, ~9/7 = 440.9014{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~9/7 = 440.9017{{c}}


{{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> }}
{{Optimal ET sequence|legend=1| 3, 5, 8, 11, 19, 30, 41, 49 }}
: <nowiki/>* wart for 5/3
: <sup>†</sup> wart for 9/7


[[Tp tuning #T2 tuning|RMS error]]: 0.7105 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.7105 cents
Line 1,578: Line 1,380:
{{See also| Chromatic pairs #Marveltwintri }}
{{See also| Chromatic pairs #Marveltwintri }}


Marveltwintri can be described as the {{nowrap| 3 & 4 }} temperament in the 2.5/3.13/9 subgroup. The tonic and the first two generator steps make a [[marveltwin triad]], hence the name. [[Cata]] is a very natural extension of this temperament to the [[2.3.5.13 subgroup|2.3.5.13-subgroup]].
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. [[Kleismic family #2.3.5.13 subgroup (cata)|Cata]] is a very natural extension of this temperament to the [[2.3.5.13 subgroup]].


[[Subgroup]]: 2.5/3.13/9
[[Subgroup]]: 2.5/3.13/9


[[Comma list]]: [[325/324]] ({{monzo| -2 2 1 }})
[[Comma list]]: 325/324 ({{monzo| -2 2 1 }})


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


{{Mapping|legend=5| 1 -1/6 5/6 0 0 -1/3 | 0 -1/2 -3/2 0 0 1 }}
{{Mapping|legend=5| 1 -2/3 -7/3 0 0 2/3 | 0 1/2 3/2 0 0 -1 }}
: [[gencom]]: [2 6/5; 325/324]
: mapping generators: ~2, ~5/3


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 882.886
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1200.1340{{c}}, ~5/3 = 882.9597{{c}}
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 882.861
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~5/3 = 882.8731{{c}}


{{Optimal ET sequence|legend=1| 3, 4, 11, 15, 19, 34, 53, 87, 140 }}
{{Optimal ET sequence|legend=1| 3, 4, 11, 15, 19, 34, 53, 87, 140 }}
Line 1,601: Line 1,403:
{{See also| Chromatic pairs #Kiribati }}
{{See also| Chromatic pairs #Kiribati }}


Kiribati is related to [[nakika]] as well as to [[octacot]].  
Kiribati is related to [[octagar family #Nakika|nakika]] as well as to [[tetracot family #Octacot|octacot]].  


[[Subgroup]]: 2.9/5.7/3.11/9
[[Subgroup]]: 2.9/5.7/3.11/9
Line 1,608: Line 1,410:


{{Mapping|legend=3| 1 1 1 0 | 0 -2 3 4 }}
{{Mapping|legend=3| 1 1 1 0 | 0 -2 3 4 }}
: mapping generators: ~2, ~21/20


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


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~21/20 = 87.776
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.6137{{c}}, ~21/20 = 87.8636{{c}}
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~21/20 = 87.892
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 87.8385{{c}}


{{Optimal ET sequence|legend=1| 13, 14, 27, 41 }}
{{Optimal ET sequence|legend=1| 13, 14, 27, 41 }}
Line 1,621: Line 1,422:
[[Tp tuning #T2 tuning|RMS error]]: 1.245 cents
[[Tp tuning #T2 tuning|RMS error]]: 1.245 cents


== 2.7/3.… subgroups ==
=== Terrain ===
=== Mothwelltri ===
{{Redirect|Terrain|the scale|Terrain (scale)}}
{{See also| Chromatic pairs #Mothwelltri }}
{{See also| Chromatic pairs #Terrain }}
 
Terrain, the 6 & 21 temperament in the 2.9/5.9/7 subgroup, tempers out the [[landscape comma]] and is related to [[minortone family #Domain|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]]: 2.9/5.9/7
 
[[Comma list]]: 250047/250000 ({{monzo| -4 6 -3 }})
 
{{Mapping|legend=3| 3 0 -4 | 0 1 2 }}
: mapping generators: ~63/50, ~9/5
 
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~63/50 = 400.0012{{c}}, ~9/5 = 1017.5415{{c}} (~10/9 = 182.4620{{c}})
* [[Tp tuning|Subgroup]] [[CWE]]: ~63/50 = 400.0000{{c}}, ~9/5 = 1017.5388{{c}} (~10/9 = 182.4612{{c}})
 
{{Optimal ET sequence|legend=1| 6, 21, 27, 33, 105, 138, 171, 2019, 2190, 2361, 2532, 2703, 2874, 3045, 3216, 3387, 3558, 3729, 3900, 7971 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.00844 cents
 
== 2.75.… subgroups ==
=== Archagall ===
{{See also| Fifthplus }}


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.  
By tempering out the comma [[24576/24565]] in the 2.75.85 subgroup, we have three [[85/64]]'s up and one octave down as a [[75/64]] and we have two [[128/85]]'s up and one octave down as a [[17/15]] whole tone. It is because of this combination of accuracy, efficiency and mapping-wise simplicity and its corresponding explanatory power in what this comma does that the comma has been named the ''archagallisma''. The ''MVP'' stands for ''minimum viable product'', as this is the core of what the archagall logic achieves, with further extensions adding to the subgroup while avoiding significantly impacting its accuracy. This is a highly accurate temperament that could be considered to be encoding the "high-accuracy logic" of [[superpyth]] and which is inescapably related to the [[17L 5s]] scale form as it is the 17 & 22 temperament (or less accurately, the 5 & 17 temperament) in the 2.75.85 subgroup.


[[Subgroup]]: 2.7/3.11
It is perhaps worth noting that [[83edo]] is approximately an optimal tuning for getting all of [[75/64]], [[85/64]], and the interval between them, [[17/15]], close to just, in that larger tunings compromise on tone-efficiency.


[[Comma list]]: [[99/98]] ({{monzo| -1 -2 1 }})
[[Subgroup]]: 2.75.85


{{Mapping|legend=3| 1 0 1 | 0 1 2 }}
[[Comma list]]: 24576/24565 ({{monzo| 13 1 -3 }})
: mapping generators: ~2, ~7/3


{{Mapping|legend=5| 1 -1/2 0 1/2 3 | 0 -1/2 0 1/2 2 }}
{{Mapping|legend=3| 1 2 5 | 0 3 1 }}
: [[gencom]]: [2 7/6; 99/98]
: mapping generators: ~2, ~85/32


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~7/6 = 273.695
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.9692{{c}}, ~85/64 = 491.5853{{c}}
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~7/6 = 273.174
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~85/64 = 491.5794{{c}}
 
{{Optimal ET sequence|legend=1| 5, 12, 17, 22, 61, 83, 310, 393, 476, 1345, 1821, 4118…, 5939…, 7760… }}
 
==== 2.75.9/7.85 subgroup ====
A fairly natural way to extend archagall is by tempering out [[2025/2023]] ([[S-expression|S15/S17]]), which equates a stack of two [[17/15]]'s with [[9/7]] without much damage. As 9/7 was not previously in the subgroup, this does not decrease the rank of the temperament and qualifies a proper and natural extension. We can equally get the same temperament by tempering out S15/S16 instead (equating a stack of three [[16/15]]'s with [[17/14]]); however, [[16/15]] is not in the subgroup, so it is preferred to think of it as adding 2025/2023.
 
Subgroup: 2.75.9/7.85
 
Comma list: 2025/2023 ({{monzo| 2 -2 1 0 }}), 24576/24565 ({{monzo| 13 1 0 -3 }})
 
{{Mapping|legend=2| 1 2 6 5 | 0 3 -4 1 }}


{{Optimal ET sequence|legend=1| 4, 9, 13, 22, 79 }}
Optimal tunings:
* Subgroup WE: ~2 = 1200.0241{{c}}, ~85/64 = 491.3358{{c}}
* Subgroup CWE: ~2 = 1200.0000{{c}}, ~85/64 = 491.3290{{c}}


[[Tp tuning #T2 tuning|RMS error]]: 1.064 cents
{{Optimal ET sequence|legend=0| 5, 12, 17, 22, 83, 105, 127, 403, 530, 657, 784, 1441… }}


== 2.7/5.… subgroups ==
== 2.7/3.… subgroups ==
=== Terrain ===
=== Mothwelltri ===
{{Redirect|Terrain|the scale|Terrain (scale)}}
{{See also| Chromatic pairs #Mothwelltri }}
{{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.
Mothwelltri, the {{nowrap| 4 & 5 }} temperament in the 2.7/3.11 subgroup, tempers out [[99/98]] and is related to [[semicomma family #Orwell|orwell]]. The tonic and the first two generator steps make a [[mothwellsmic triad]], hence the name.  


[[Subgroup]]: 2.7/5.9/5
[[Subgroup]]: 2.7/3.11


[[Comma list]]: [[250047/250000]]
[[Comma list]]: 99/98 ({{monzo| -1 -2 1 }})


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


{{Mapping|legend=5| 3 10/9 -7/9 2/9 | 0 -2/3 -1/3 2/3 }}
{{Mapping|legend=5| 1 0 0 0 1 | 0 -1/2 0 1/2 2 }}
: [[gencom]]: [63/50 10/9; 250047/250000]
: mapping generators: ~2, ~7/3


[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~63/50 = 1\3, ~10/9 = 182.461
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1200.3175{{c}}, ~7/6 = 273.2462{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~7/6 = 273.4280{{c}}


{{Optimal ET sequence|legend=1| 6, 21, 27, 33, 105, 138, 171, 1848, 2019, 2190, 2361, 2532, 2703, 2874, 3045, 3216, 3387, 3558 }}
{{Optimal ET sequence|legend=1| 4, 9, 13, 22, 79 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.00844 cents
[[Tp tuning #T2 tuning|RMS error]]: 1.064 cents


== 2.7/5.… subgroups ==
=== Tridec ===
=== Tridec ===
{{See also| Chromatic pairs #Tridec }}
{{See also| Chromatic pairs #Tridec }}
{{See also| Non-over-1 temperament #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]].  
Tridec, the 5 & 8 temperament in the 2.7/5.11/5.13/5 subgroup, extends [[#Petrtri]].  


[[Subgroup]]: 2.7/5.11/5.13/5
[[Subgroup]]: 2.7/5.11/5.13/5


[[Comma list]]: [[847/845]], [[1001/1000]]
[[Comma list]]: 847/845 ({{monzo| 0 1 2 -2 }}), 1001/1000 ({{monzo| -3 1 1 1 }})


{{Mapping|legend=3| 1 2 0 1 | 0 -4 3 1 }}
{{Mapping|legend=3| 1 -2 3 2 | 0 4 -3 -1 }}


{{Mapping|legend=5| 1 0 -3/4 5/4 -3/4 1/4 | 0 0 0 -4 3 1 }}
{{Mapping|legend=5| 1 0 -3/4 -11/4 9/4 5/4 | 0 0 0 4 -3 -1 }}
: [[gencom]]: [2 13/10; 847/845 1001/1000]
: mapping generators: ~2, ~20/13


[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~13/10 = 454.556
[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[WE]]: ~2 = 1200.0343{{c}}, ~20/13 = 745.4658{{c}}
* [[Tp tuning|subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~20/13 = 745.4431{{c}}


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


[[Tp tuning #T2 tuning|RMS error]]: 0.1613 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.1613 cents


==== Naiadec ====
==== Naiadec ====
[[Subgroup]]: 2.7/5.11/5.13/5.17/5
Subgroup: 2.7/5.11/5.13/5.17/5


[[Comma list]]: [[170/169]], [[221/220]], [[847/845]]
Comma list: 170/169 ({{monzo| 1 0 0 -2 1 }}), 221/220 ({{monzo| -2 0 -1 1 1 }}), 847/845 ({{monzo| 0 1 2 -2 0 }})


{{Mapping|legend=3| 1 2 0 1 1 | 0 -4 3 1 2 }}
{{Mapping|legend=2| 1 -2 3 2 3 | 0 4 -3 -1 -2 }}


{{Mapping|legend=5| 1 0 -3/4 5/4 -3/4 1/4 1/4 | 0 0 0 -4 3 1 2 }}
{{Mapping|legend=4| 1 0 -3/4 -11/4 9/4 5/4 9/4 | 0 0 0 4 -3 -1 -2 }}
: [[gencom]]: [2 13/10; 170/169 221/220 847/845]
: mapping generators: ~2, ~20/13


[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~13/10 = 454.882
Optimal tunings:
* Subgroup WE: ~2 = 1200.4068{{c}}, ~20/13 = 745.3706{{c}}
* Subgroup CWE: ~2 = 1200.0000{{c}}, ~20/13 = 745.0940{{c}}


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


[[Tp tuning #T2 tuning|RMS error]]: 0.7521 cents
RMS error: 0.7521 cents


== 2.25/7.… subgroups ==
== 2.25/7.… subgroups ==
=== Supramine ===
=== 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.
Supramine extends [[#Supramin|supramin]] and 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
[[Subgroup]]: 2.25/7.17/7.19/7.23/7


Comma list: [[323/322]], [[392/391]]
[[Comma list]]: 323/322 ({{monzo| -1 0 1 1 -1 }}), 392/391 ({{monzo| 3 0 -1 0 -1 }}), 476/475 ({{monzo| 2 -1 1 -1 0 }})


Subgroup-val mapping: {{Mapping| 1 0 4 3 | 0 1 -2 -1 }}
{{Mapping|legend=3| 1 -2 0 4 3 | 0 3 1 -2 -1 }}
: mapping generators: ~2, ~17/14


Optimal tunings:
[[Optimal tuning]]s:
* Subgroup WE: ~2 = 1199.871{{c}}, ~17/14 = 336.243{{c}}
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.7558{{c}}, ~17/14 = 335.4278{{c}}
* Subgroup CWE: ~2 = 1200.000{{c}}, ~17/14 = 336.296{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~17/14 = 335.4885{{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:
{{Optimal ET sequence|legend=1| 7, 18, 25, 68, 93 }}
* 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
 
Badness (Sintel): 0.053


== 2.11/5.… subgroups ==
== 2.11/5.… subgroups ==
Line 1,744: Line 1,569:
{{See also| 5L 3s/Temperaments #Petrtri }}
{{See also| 5L 3s/Temperaments #Petrtri }}


Petrtri can be described as 3 &amp; 5 temperament in the 2.11/5.13/5 subgroup.  
Petrtri can be described as 3 & 5 temperament in the 2.11/5.13/5 subgroup and tempers out [[2200/2197]]. It restricts [[#Tridec|tridec]]. The tonic and the first two generator steps make a [[petrmic triad]], hence the name.  


[[Subgroup]]: 2.11/5.13/5
[[Subgroup]]: 2.11/5.13/5


[[Comma list]]: [[2200/2197]]
[[Comma list]]: 2200/2197 ({{monzo| 3 1 -3 }})


{{Mapping|legend=3| 1 0 1| 0 3 1 }}
{{Mapping|legend=3| 1 0 1 | 0 3 1 }}


{{Mapping|legend=5| 1 0 -1/3 0 -1/3 2/3 | 0 0 -4/3 0 5/3 -1/3 }}
{{Mapping|legend=5| 1 0 -1/3 0 -1/3 2/3 | 0 0 -4/3 0 5/3 -1/3 }}
: [[gencom]]: [2 13/10; 2200/2197]
: mapping generators: ~2, ~13/10


[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~13/10 = 455.012
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.9572{{c}}, ~13/10 = 454.9958{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~13/10 = 455.0039{{c}}


{{Optimal ET sequence|legend=1| 21, 29, 153, 182, 211, 240, 269, 298, 327, 356, 385, 509, 741c, 1126c }}
{{Optimal ET sequence|legend=1| 5, 8, 21, 29, 153, 182, 211, 240, 269, 509, 778…, 1287… }}


[[Tp tuning #T2 tuning|RMS error]]: 0.0749 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.0749 cents
Line 1,763: Line 1,590:
== Other 2-repeating subgroups ==
== Other 2-repeating subgroups ==
=== Supramin ===
=== 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]]. It is related to [[cohemimabila]].
Supramin restricts [[#Supramine|supramine]] and 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 {{u|Overthink}} in 2026 after the fact that the generator is a [[17/14]] supraminor third, two of which reach [[28/19]]. It is related to [[Mabila family #Cohemimabila|cohemimabila]].


[[Subgroup]]: 2.17/7.19/7
[[Subgroup]]: 2.17/7.19/7


[[Comma list]]: [[5491/5488]] ({{Monzo| -4 2 1 }})
[[Comma list]]: 5491/5488 ({{monzo| -4 2 1 }})


{{Mapping|legend=3| 1 0 4 | 0 1 -2 }}
{{Mapping|legend=3| 1 0 4 | 0 1 -2 }}
Line 1,773: Line 1,600:


[[Optimal tuning]]s:
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1200.022{{c}}, ~17/14 = 335.793{{c}}
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1200.0220{{c}}, ~17/14 = 335.7933{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.000{{c}}, ~17/14 = 335.785{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~17/14 = 335.7852{{c}}


{{Optimal ET sequence|legend=1| 7, 18, 25 }}
{{Optimal ET sequence|legend=1| 7, 18, 25, 118, 143, 168, 193, 218, 461, 679, 1576, 2255… }}


[[Badness]] (Sintel): 0.005
[[Badness]] (Sintel): 0.005


== 3/2.5/2.… subgroups ==
== 3/2.5/2.… subgroups ==
{{Main|Half-prime subgroup}}
{{Main| Half-prime subgroup }}


=== Hemihemi ===
=== Hemihemi ===
Named by {{u|CompactStar}} in 2023, hemihemi tempers out 10976/10935, the [[hemimage comma]], in the 3/2.5/2.7/2 subgroup.
[[Subgroup]]: 3/2.5/2.7/2
[[Subgroup]]: 3/2.5/2.7/2


[[Comma list]]: [[10976/10935]]
[[Comma list]]: 10976/10935 ({{monzo| -7 -1 3 }})


{{Mapping|legend=3| 1 2 3 | 0 3 1 }}
{{Mapping|legend=3| 1 2 3 | 0 3 1 }}
: mapping generators: ~3/2, ~28/27


[[Optimal tuning]] (subgroup [[CTE]]): ~[[3/2]] = 1\[[1edf]], ~[[28/27]] = 60.909
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~3/2 = 702.6069{{c}}, ~28/27 = 60.4665{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~3/2 = 701.9550{{c}}, ~28/27 = 60.8011{{c}}


[[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]
[[Optimal ET sequence]]: [[11edf|*11]], [[12edf|*12]], [[23edf|*23]], [[35edf|*35]], [[58edf|*58]], [[93edf|*93]], [[151edf|*151†]], [[244edf|*244†]]
: <nowiki/>* Wart for 3/2
: † Wart for 7/2


=== Halftone ===
=== Halftone ===
{{Main| Halftone }}
{{Main| Halftone }}
Named by {{u|CompactStar}} in 2023, halftone tempers out 9604/9375 in the 3/2.5/2.7/2 subgroup.


[[Subgroup]]: 3/2.5/2.7/2
[[Subgroup]]: 3/2.5/2.7/2


[[Comma list]]: 9604/9375
[[Comma list]]: 9604/9375 ({{monzo| -1 -5 4 }})


{{Mapping|legend=3| 1 3 4 | 0 -4 -5 }}
{{Mapping|legend=3| 1 3 4 | 0 -4 -5 }}
: sval mapping generators: ~3/2, ~15/14
: mapping generators: ~3/2, ~15/14


[[Optimal tuning]] (subgroup [[CTE]]): ~3/2 = 1\1edf, ~15/14 = 128.783
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~3/2 = 702.6568{{c}}, ~15/14 = 128.2849{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~3/2 = 701.9550{{c}}, ~15/14 = 128.8183{{c}}


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
[[Optimal ET sequence]]: [[5edf|*5]], [[11edf|*11]], [[27edf|*27]], [[38edf|*38]]
: <nowiki />* wart for 3/2
: <nowiki/>* Wart for 3/2


==== 3/2.5/2.7/2.11/2 ====
==== 3/2.5/2.7/2.11/2 subgroup ====
[[Subgroup]]: 3/2.5/2.7/2.11/2
Subgroup: 3/2.5/2.7/2.11/2


[[Comma list]]: 1232/1215, 27783/27500
Comma list: 1232/1215 ({{monzo| -5 -1 1 1 }}), 9604/9375 ({{monzo| -1 -5 4 0 }})


{{Mapping|legend=3| 1 3 4 4 | 0 -4 -5 1 }}
{{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
Optimal tunings:
* WE: ~3/2 = 704.8651{{c}}, ~15/14 = 130.9114{{c}}
* CWE: ~3/2 = 701.9550{{c}}, ~15/14 = 129.6376{{c}}


[[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]
Optimal ET sequence: [[5edf|*5]], [[11edf|*11]], [[16edf|*16]], [[27edf|*27…]], [[43edf|*43…]], [[70edf|*70…]]
: <nowiki />* wart for 3/2
: <nowiki/>* Wart for 3/2


==== 3/2.5/2.7/2.11/2.13/2 ====
==== 3/2.5/2.7/2.11/2.13/2 subgroup ====
[[Subgroup]]: 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
Comma list: 275/273 ({{monzo| -1 2 -1 1 -1 }}), 1232/1215 ({{monzo| -5 -1 1 1 0 }}), 1323/1300 ({{monzo| 3 -2 2 0 -1 }})


{{Mapping|legend=3| 1 3 4 4 5 | 0 -4 -5 1 -2 }}
{{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
Optimal tunings:
* WE: ~3/2 = 702.9683{{c}}, ~15/14 = 130.0424{{c}}
* CWE: ~3/2 = 701.9550{{c}}, ~15/14 = 129.6034{{c}}


[[Support]]ing [[ET]]s: *11, *5, *16, *6, *27[-11/2]
Optimal ET sequence: [[5edf|*5]], [[11edf|*11]], [[16edf|*16]], [[27edf|*27†]]
: <nowiki />* wart for 3/2
: <nowiki/>* Wart for 3/2
: † Wart for 11/2


=== Semiwolf ===
=== Semiwolf ===
Semiwolf provides an interpretation for the {{MOS scalesig|link=true|3L 2s<3/2>}} [[mos scale]]. The name ''semiwolf'' comes from two [[~]][[7/6]] generators approximating a [[27/20]] wolf fourth. This is further extended to the 11-limit in two interpretations: semilupine, where 2 major 2-mossteps (LL) equal 11/9, and hemilycan. where 1 major and 2 minor 2-mossteps (sLs) equal 11/9. Basic [[8edf]] fits both extensions.
[[Subgroup]]: 3/2.5/2.7/4
[[Subgroup]]: 3/2.5/2.7/4


[[Comma list]]: 245/243
[[Comma list]]: 245/243 ({{monzo| -5 1 2 }})


{{Mapping|legend=3| 1 1 2 | 0 2 -1 }}
{{Mapping|legend=3| 1 1 2 | 0 2 -1 }}
: mapping generators: ~3/2, ~9/7
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~3/2 = 703.7414{{c}}, ~9/7 = 440.9014{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~3/2 = 701.9550{{c}}, ~9/7 = 440.8307{{c}}
[[Optimal ET sequence]]: [[8edf|*8]], [[43edf|*43]], [[51edf|*51]], [[59edf|*59]], [[67edf|*67]], [[75edf|*75…]], [[83edf|*83…]]
: <nowiki/>* Wart for 3/2
==== Hemilycan ====
Subgroup: 3/2.5/2.7/4.11/4
Comma list: 245/243 ({{monzo| -5 1 2 0 }}), 441/440 ({{monzo| 2 -1 2 -1 }})


: sval mapping generators: ~3/2, ~9/7
{{Mapping|legend=2| 1 1 2 5 | 0 2 -1 -4 }}


[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~7/6 = 262.1728
Optimal tunings:  
* Subgroup WE: ~3/2 = 703.2870{{c}}, ~9/7 = 441.1967{{c}}
* Subgroup CWE: ~3/2 = 701.9550{{c}}, ~9/7 = 440.4501{{c}}


[[Optimal ET sequence]]: [[3edf]], [[5edf]], [[8edf]]
Optimal ET sequence: [[8edf|*8]], [[27edf|*28]], [[35edf|*35]], [[43edf|*43]], [[51edf|*51]], [[110edf|110…]]
: <nowiki/>* Wart for 3/2


==== Semilupine ====
==== Semilupine ====
[[Subgroup]]: 3/2.5/2.7/4.11/4
Subgroup: 3/2.5/2.7/4.11/4
 
Comma list: 100/99 ({{monzo| -3 2 0 -1 }}), 245/243 ({{monzo| -5 1 2 0 }})
 
{{Mapping|legend=2| 1 1 2 0 | 0 2 -1 4 }}
 
Optimal tunings:
* Subgroup WE: ~3/2 = 704.6266{{c}}, ~9/7 = 439.2433{{c}}
* Subgroup CWE: ~3/2 = 701.9550{{c}}, ~9/7 = 438.6989{{c}}


[[Comma list]]: 100/99, 245/243
Optimal ET sequence: [[5edf|*5]], [[8edf|*8]], [[45edf|*45…]], [[53edf|*53…]], [[61edf|*61…]], [[69edf|*69…]], [[77edf|*77…]]
: <nowiki/>* Wart for 3/2


{{Mapping|legend=3| 1 1 2 0 | 0 2 -1 4 }}
== 3/2.5/4.… subgroups ==
=== Ambulatory ===
Ambulatory tempers out [[686/675]] in the 3/2.5/4.7/4 subgroup and is generated by a sharp [[~]][[15/14]]. This implies a stack of two generators represents [[7/6]], and a stack of three generators represents [[5/4]].


[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~7/6 = 264.3771
Ambulatory can be considered the 3/2.5/4.7/4 analog of [[no-twos subgroup temperaments #BPS|BPS]] or [[meantone family #Meantone|meantone]], using [[5:6:7]] as its fundamental consonant chord in the place of [[3:5:7]] or of [[4:5:6]]. Ambulatory is actually similar to meantone in several ways:
* They are both of low [[badness]].{{dubious}}
* In most [[optimal tuning]]s, meantone tunes [[3/2]] flat and [[5/4]] slightly sharp, while ambulatory tunes [[7/5]] flat and [[6/5]] slightly sharp.
* Meantone[7] has [[5/4]] and [[6/5]] share a scale degree. Ambulatory[5] has [[6/5]] and [[7/6]] share a scale degree.
* Meantone[7] is a tempering of the 7-note [[AGS]](5/4, 6/5) (Ptolemy's intense diatonic) with the large and medium steps equated. Ambulatory[5] is a tempering of the 5-note AGS(6/5, 7/6) (assuming an equivalence of 3/2) with the medium and small steps equated.


[[Optimal ET sequence]]: [[8edf]], [[13edf]]
Ambulatory was first documented by {{u|CompactStar}} who originally proposed the name ''emergency'', which refers to the North American  emergency telephone number 911, because [[11edf|9\11edf]] is a possible generator. This was deemed as too dark so the name ''ambulatory''" was suggested by {{u|Frostburn}} to more subtly reference the same thing.


==== Hemilycan ====
[[Subgroup]]: 3/2.5/4.7/4
[[Subgroup]]: 3/2.5/2.7/4.11/4


[[Comma list]]: 245/243, 441/440
[[Comma list]]: 686/675 ({{monzo| -3 -2 3 }})


{{Mapping|legend=3| 1 1 2 5 | 0 2 -1 -4 }}
{{Mapping|legend=3| 1 0 1 | 0 3 2 }}]
: mapping generators: ~3/2, ~15/14


[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~7/6 = 261.5939
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~3/2 = 701.8452{{c}}, ~15/14 = 129.7976{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~3/2 = 701.9550{{c}}, ~15/14 = 129.5324{{c}}


[[Optimal ET sequence]]: [[8edf]], [[11edf]]
[[Optimal ET sequence]]: [[5edf|*5]], [[11edf|*11]], [[27edf|*27]], [[38edf|*38]]
: <nowiki/>* Wart for 3/2


== 3/2.5/4.… subgroups ==
=== Poseidon ===
=== Poseidon ===
'''This temperament will be subjected to renaming due to a conflict.'''
{{Todo|inline=1|rename|resolve temperament name conflict}}


[[Subgroup]]: 3/2.5/4.11/8
[[Subgroup]]: 3/2.5/4.11/8


[[Comma list]]: 121/120
[[Comma list]]: 121/120 ({{monzo| -1 -1 2 }})


{{Mapping|legend=3| 1 1 1 | 0 2 -1 }}]
{{Mapping|legend=3| 1 1 1 | 0 -2 -1 }}]
: mapping generators: ~3/2, ~12/11


: [[gencom]]: [3/2 12/11; 121/120]
[[Optimal tuning]]s:  
* [[Tp tuning|Subgroup]] [[WE]]: ~3/2 = 701.8452{{c}}, ~12/11 = 157.8337{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~3/2 = 701.9550{{c}}, ~12/11 = 157.8574{{c}}


[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~3/2, ~12/11 = 158.29
[[Optimal ET sequence]]: [[4edf|*4]], [[5edf|*5]], [[9edf|*9]], [[22edf|*22]], [[31edf|*31]], [[40edf|*40]]
 
: <nowiki/>* Wart for 3/2
{{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-repeating subgroups ==
== Other 3/2-repeating subgroups ==
=== Auk ===
=== Auk ===
Auk tempers out the comma [[87808/85293]] in the 3/2.7.13 subgroup, equating three instances of [[7/1]] (fifth-reduced to [[112/81]]) with [[13/1]] (fifth-reduced to 832/729). The generator when reduced is a superfourth in the neighborhood of 550 cents, representing 112/81. It possesses nontrivial mos of the families {{MOS scalesig|1L 3s<3/2>}} ("neptunian"), {{MOS scalesig|4L 1s<3/2>}}, {{MOS scalesig|5L 4s<3/2>}}, and {{MOS scalesig|9L 5s<3/2>}}. [[9edf]] is a particularly good equal temperament that [[support]]s it. It contains tempered versions of the chords 1–832/729–112/81 (fifth-reduction of 1:7:13) and 1–63/52–112/81 (fifth-reduction of 1/(13:7:1)).
[[Subgroup]]: 3/2.7.13
[[Subgroup]]: 3/2.7.13


Line 1,892: Line 1,773:


{{Mapping|legend=3| 1 0 -8 | 0 1 3 }}
{{Mapping|legend=3| 1 0 -8 | 0 1 3 }}
: mapping generators: ~3/2, ~7


: sval mapping generators: ~3/2, ~7
[[Optimal tuning]]s:  
 
* [[Tp tuning|Subgroup]] [[WE]]: ~3/2 = 706.5197{{c}}, ~112/81 = 538.8834{{c}}
[[Optimal tuning]] (subgroup [[CTE]]): ~3/2 = 1\1edf, ~28/9 = 1950.859
* [[Tp tuning|Subgroup]] [[CWE]]: ~3/2 = 701.9550{{c}}, ~112/81 = 543.8799{{c}}


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]
[[Optimal ET sequence]]: [[4edf|*4]], [[13edf|*13]], [[17edf|*17df]], [[21edf|*21df]], [[38edf|*38ddff]], [[59edf|*59ddffff]]
: <nowiki />* wart for 3/2
: <nowiki/>* Wart for 3/2


=== Doubleton ===
=== Doubleton ===
[[Subgroup]]: 3/2.7.13
[[Subgroup]]: 3/2.7.13


[[Comma list]]: 1352/1323
[[Comma list]]: 1352/1323 ({{monzo| -3 -2 2 }})


{{Mapping|legend=3| 2 0 3 | 0 1 1 }}
{{Mapping|legend=3| 2 0 3 | 0 1 1 }}
: mapping generators: ~26/21, ~7


: sval mapping generators: ~26/21, ~7
[[Optimal tuning]]s:  
* [[Tp tuning|Subgroup]] [[WE]]: ~26/21 = 352.6518{{c}}, ~112/81 = 552.6017{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~26/21 = 350.9775{{c}}, ~112/81 = 565.7161{{c}}


[[Optimal tuning]] (subgroup [[CTE]]): ~26/21 = 1\2edf, ~28/9 = 1971.772
[[Optimal ET sequence]]: [[4edf|*4]], [[10edf|*10]], [[14edf|*14f]], [[46edf|*46dfff]]
 
: <nowiki/>* Wart for 3/2
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-repeating subgroups ==
== 5/2-repeating subgroups ==
Line 1,918: Line 1,801:
[[Subgroup]]: 5/2.7.11
[[Subgroup]]: 5/2.7.11


[[Comma list]]: {{monzo| 11 1 -5 }}
[[Comma list]]: 341796875/329832448 ({{monzo| 11 1 -5 }})


{{Mapping|legend=3| 1 4 3 | 0 -5 -1 }}
{{Mapping|legend=3| 1 -1 2 | 0 5 1 }}
: mapping generators: ~5/2, ~44/25


: [[gencom]]: [5/2 125/88; 341796875/329832448]
[[Optimal tuning]]s:  
* [[Tp tuning|Subgroup]] [[WE]]: ~5/2 = 1582.2958{{c}}, ~44/25 = 990.1327{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~5/2 = 1586.3137{{c}}, ~44/25 = 991.6570{{c}}


[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~5/2 = 1586.3137, ~125/88 = 593.6668
[[Optimal ET sequence]]: [[8ed5/2|*8]], [[83ed5/2|*83…]], [[91ed5/2|*91…]], …, [[155ed5/2|*155…]], [[163ed5/2|*163…]]
: <nowiki/>* Wart for 5/2


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]
== See also ==
: <nowiki />* wart for 5/2
 
= Related temperament collections =
* [[Dual-fifth temperaments]]
* [[Dual-fifth temperaments]]
* [[Irrational-subgroup temperaments]]
* [[Irrational-subgroup temperaments]]
Line 1,935: Line 1,819:
* [[Substitute harmonic]] temperaments
* [[Substitute harmonic]] temperaments


[[Category:Subgroup temperaments| ]] <!-- main article -->
== References ==
[[Category:Temperament collections]]
<references/>
[[Category:Rank 2]]
 
{{Todo| review | cleanup }}
[[Category:Catalogs of subgroup temperaments]]
[[Category:Catalogs of rank-2 temperaments]]

Latest revision as of 10:26, 17 October 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

A subgroup temperament is a regular temperament defined on a just intonation subgroup that is not a full p-limit group.

For temperaments that omit various prime harmonics, see:

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.

This page uses an extended wart notation where squares and cubes of a prime basis element are denoted by ′ and ″, respectively, to reduce the use of temporary warts.

2.3.… subgroups

Shrub

Named by Vector in 2026, shrub is a restriction of diaschismic which omits the tritone to produce a diatonic scale. True to its name, it generates a shrubmajor[idiosyncratic term] third (~425 ¢) in quarter-comma tuning. It has an equal-temperament join of 12 & 17.

It can also admit a 41st harmonic by tempering out 82/81 (same mapping as reversed meantone). However, its tuning tends to be flat, and with the rest of the harmonics tending sharp, this will increase the overall error.

2.3.25 subgroup

Subgroup: 2.3.25

Comma list: 2048/2025

Subgroup-val mapping: [⟨1 0 11], ⟨0 1 -4]]

mapping generators: ~2, ~3

Optimal tunings:

  • WE: ~2 = 1198.8214 ¢, ~3/2 = 704.2059 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.9585 ¢

Optimal ET sequence: 12, 17, 46, 63, 80, 143c′

Badness (Sintel): 0.234

2.3.25.23 subgroup

Subgroup: 2.3.25.23

Comma list: 576/575, 736/729

Subgroup-val mapping: [⟨1 0 11 -5], ⟨0 1 -4 6]]

Optimal tunings:

  • WE: ~2 = 1198.9504 ¢, ~3/2 = 704.6585 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 705.1888 ¢

Optimal ET sequence: 12, 17, 63, 80, 97c′, 114c′

Badness (Sintel): 0.223

Sburb

Sburb is every other step of pajara, extended to the add-23 add-41 subgroup in the same way as shrub. It can also admit a 59th harmonic by setting the octave-reduced 413th harmonic (413/256, 827.998 ¢) to the diminished seventh.

Subgroup: 2.3.25.7

Comma list: 50/49, 64/63

Subgroup-val mapping: [⟨1 0 11 6], ⟨0 1 -4 -2]]

Optimal tunings:

  • WE: ~2 = 1197.6967 ¢, ~3/2 = 705.6906 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 707.3438 ¢

Optimal ET sequence: 5, 12, 17, 39d

Badness (Sintel): 0.253

2.3.25.7.23 subgroup

Subgroup: 2.3.25.7.23

Comma list: 50/49, 64/63, 162/161

Subgroup-val mapping: [⟨1 0 11 6 -5], ⟨0 1 -4 -2 6]]

Optimal tunings:

  • WE: ~2 = 1197.5577 ¢, ~3/2 = 705.3089 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 706.7010 ¢

Optimal ET sequence: 5, 12, 17

Badness (Sintel): 0.419

Hypnosis

Related temperaments: hypnos, alphatrimot.

Subgroup: 2.3.7.11/5.13

Comma list: 169/168 ([-3 -1 -1 0 2⟩), 540/539 ([2 3 -2 -1 0⟩), 729/728 ([-3 6 -1 0 -1⟩)

Subgroup-val mapping: [⟨1 0 -3 8 0], ⟨0 3 11 -13 7]]

mapping generators: ~2, ~13/9

Optimal tunings:

  • Subgroup WE: ~2 = 1200.6306 ¢, ~13/9 = 633.8505 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~13/9 = 633.5546 ¢

Optimal ET sequence: 17, 36, 125f, 161f, 197f

RMS error: 0.5379 cents

Darian calendar

Darian calendar is described as the 24 & 668 temperament in the 2.3.35.11.19 subgroup. The generator is close to 36/35. Five of them make 11/8, six of them make 32/19, and eight of them make 3/2.

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.

Darian calendar was named by Eliora in 2023 after a certain calendar layout by the same name.

Subgroup: 2.3.35.11.19

Comma list: 29282/29241 ([1 -4 0 4 -2⟩), 42875/42768 ([-4 -5 3 -1 0⟩), 885115/884736 ([-15 -3 1 3 1⟩)

Subgroup-val mapping: [⟨4 5 18 13 18], ⟨0 8 15 5 -6]]

mapping generators: ~2240/1881, ~36/35

Optimal tunings:

  • Subgroup WE: ~2240/1881 = 299.9912 ¢, ~36/35 = 50.2951 ¢
  • Subgroup CWE: ~2240/1881 = 300.0000 ¢, ~36/35 = 50.2947 ¢

Optimal ET sequence: 24, …, 524, 548, 572, 596, 620, 644, 668

Hydrothermal

Named by Budjarn Lambeth in 2024, hydrothermal tempers out 50/49, the jubilisma, in the 2.3.7/5 subgroup. A tuning whose distinctively sharp (but still consonant) fifth, and flat (but still consonant) octave, will lend it a mysterious, heavy atmosphere. The 6-tone 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 ([1 0 -2⟩)

Subgroup-val mapping: [⟨2 0 1], ⟨0 1 0]]

mapping generators: ~7/5, ~3

Optimal tunings:

  • Subgroup WE: ~2 = 599.6673 ¢, ~3/2 = 702.5906 ¢
  • Subgroup CWE: ~2 = 600.0000 ¢, ~3/2 = 702.4574 ¢

Optimal ET sequence: 2, 6, 8, 10, 12, 34, 46*

* Wart for 7/5

Argentic

Argentic is the 2.3.7/5-subgroup temperament tempering out 5120/5103, the aberschisma.

Subgroup: 2.3.7/5

Comma list: 5120/5103 ([10 -6 -1⟩)

Subgroup-val mapping: [⟨1 0 10], ⟨0 1 -6]]

mapping generators: ~2, ~3

Optimal tunings:

  • Subgroup WE: ~2 = 1199.7172 ¢, ~3/2 = 702.6636 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.8166 ¢

Optimal ET sequence: 12, 29, 41, 70, 321, 391, 461, 531, 601

Badness (Sintel): 0.119

Edson

See also: Chromatic pairs § Edson

Edson is related to pele and andromeda.

Subgroup: 2.3.7/5.11/5.13/5

Comma list: 196/195 ([2 -1 2 0 -1⟩), 352/351 ([5 -3 0 1 -1⟩), 364/363 ([2 -1 1 -2 1⟩)

Subgroup-val mapping: [⟨1 0 10 17 22], ⟨0 1 -6 -10 -13]]

Gencom mapping: [⟨1 0 -49/4 -9/4 19/4 39/4], ⟨0 1 29/4 5/4 -11/4 -23/4]]

Optimal tunings:

  • Subgroup WE: ~2 = 1199.4965 ¢, ~3/2 = 703.1192 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~3/2 = 703.4225 ¢

Optimal ET sequence: 12, 17, 29

RMS error: 0.5102 cents

Haumea

See also: Chromatic pairs § Haumea

Haumea is the add-7/5 extension of bridgetown, which is in turn the add-11/5 extension of barbados. Other related temperamnets include hemigari, parizekmic, and namaka.

Subgroup: 2.3.7/5.11/5.13/5

Comma list: 352/351 ([5 -3 0 1 -1⟩), 676/675 ([2 -3 0 0 2⟩), 847/845 ([0 0 1 2 -2⟩)

Subgroup-val mapping: [⟨1 0 10 -6 -1], ⟨0 2 -12 9 3]]

Gencom mapping: [⟨1 0 -3/4 37/4 -27/4 -7/4], ⟨0 2 0 -12 9 3]]

mapping generators: ~2, ~26/15

Optimal tunings:

  • Subgroup WE: ~2 = 1199.7072 ¢, ~26/15 = 951.2727 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~26/15 = 951.5016 ¢

Optimal ET sequence: 24, 29, 111, 140, 169, 198

RMS error: 0.2668 cents

Historical

Not to be confused with Historical temperaments.
Not to be confused with History (temperament).

Historical is essentially an analogue of miracle that splits 4/3 in six rather than 3/2. It tempers out the comma 4000/3993 (S10/S11) to set 11/10 equal to one-third of 4/3, and 676/675 (S13/S15) to equate 15/13 to one-half of 4/3, and tempers out 441/440 (S21) 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 ([2 -1 1 -2 1⟩), 441/440 ([-3 2 2 -1 0⟩, 1001/1000 ([-3 0 1 1 1⟩)

Subgroup-val mapping: [⟨1 2 0 1 2], ⟨0 -6 7 2 -9]]

mapping generators: ~2, ~21/20

Optimal tunings:

  • Subgroup WE: ~2 = 1200.0242 ¢, ~21/20 = 83.0177 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~21/20 = 83.0144 ¢

Optimal ET sequence: 14, 29, 101, 130, 159

RMS error: 0.2562 cents

Direct breedsmic

This temperament was first proposed by Royalmilktea. The name was established by Lériendil in 2024. Related temperaments: hemififths and newt.

Subgroup: 2.3.49/5

Comma list: 2401/2400 ([-5 -1 2⟩)

Subgroup-val mapping: [⟨1 1 3], ⟨0 2 1]]

mapping generators: ~2, ~49/40

Optimal tuning:

  • Subgroup WE: ~2 = 1200.0206 ¢, ~49/40 = 350.9724 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~49/40 = 350.9743 ¢

Optimal ET sequence: 7, 17, 24, 41, 65, 106, 253, 359, 465, 1036, 1501, 1966*, 5433*

* Wart for 49/5

Tobago

See also: Chromatic pairs § Tobago

Tobago, the 10 & 14 temperament in the 2.3.11.13/5 subgroup, extends neutral and barbados.

Subgroup: 2.3.11.13/5

Comma list: 243/242 ([-1 5 -2 0⟩), 676/675 ([2 -3 0 2⟩)

Subgroup-val mapping: [⟨2 0 -1 -2], ⟨0 2 5 3]]

Gencom mapping: [⟨2 0 -5 0 -1 -1], ⟨0 2 -3/2 0 5 3/2]]

mapping generators: ~55/39, ~26/15

Optimal tunings:

  • Subgroup WE: ~55/39 = 599.9845 ¢, ~26/15 = 950.6637 ¢
  • Subgroup CWE: ~55/39 = 600.0000 ¢, ~26/15 = 950.6776 ¢

Optimal ET sequence: 10, 14, 24, 82, 106, 130, 154

RMS error: 0.3533 cents

Pakkanian hemipyth

Subgroup: 2.3.11.13/5.17

Comma list: 221/220 ([-2 0 -1 1 1⟩), 243/242 ([-1 5 -2 0 0⟩), 289/288 ([-5 -2 0 0 2⟩)

Subgroup-val mapping: [⟨2 0 -1 -2 5], ⟨0 2 5 3 2]]

Optimal tunings:

  • subgroup WE: ~17/12 = 600.1781 ¢, ~26/15 = 950.7656 ¢ (~15/13 = 249.2344 ¢)
  • subgroup CWE: ~17/12 = 600.0000 ¢, ~26/15 = 950.6011 ¢ (~15/13 = 249.3989 ¢)

Optimal ET sequence: 10, 14, 24, 106, 130, 154, 178*, 202*

* Wart for 13/5

Bridgetown

See also: Chromatic pairs § Bridgetown

Bridgetown, the 5 & 24 temperament in the 2.3.11/5.13/5 subgroup, is the no-7/5 restriction of haumea and the add-11/5 extension of barbados.

Subgroup: 2.3.11/5.13/5

Comma list: 352/351 ([5 -3 1 -1⟩), 676/675 ([2 -3 0 2⟩)

Subgroup-val mapping: [⟨1 0 -6 -1], ⟨0 2 9 3]]

Gencom mapping: [⟨1 0 7/3 0 -11/3 4/3], ⟨0 2 -4 0 5 -1]]

mapping generators: ~2, ~26/15

Optimal tunings:

  • Subgroup WE: ~2 = 1199.6281 ¢, ~26/15 = 951.3060 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~26/15 = 951.5580 ¢

Optimal ET sequence: 5, 19*, 24, 29, 169, 198, 227, 256, 285, 314, 343

* Wart for 11/5

RMS error: 0.2513 cents

Seventeen-cot

Seventeen-cot may be described as the 29 & 465 temperament in the 2.3.11/5.13/5 subgroup. It tempers out the tendoartisma in the 2.3.13/5 subgroup. It can be generated with a ~2/1 octave and a ~169/165 generator which is 1/17 of a ~3/2 perfect fifth. It was named by FilterNashi in 2026.

Subgroup: 2.3.11/5.13/5

Comma basis: 43940/43923 ([2 -1 -4 3⟩), 225000/224939 ([3 2 -3 -2⟩)

Subgroup-val mapping: [⟨1 1 1 1], ⟨0 17 4 11]]

mapping generators: ~2, ~169/165

Optimal tunings:

  • Subgroup WE: ~2 = 1199.993492 ¢, ~169/165 = 41.291827 ¢
error map: ⟨-0.0065 -0.0004 +0.1566 -0.0104]
  • Subgroup CWE: ~2 = 1200.000000, ~169/165 = 41.291746 ¢
error map: ⟨0.0000 +0.0047 +0.1628 -0.0047]

Optimal ET sequence: 29, 262, 291, 320, 349, 378, 407, 436, 465, 959, 1424, 6161…

Badness (Sintel): 0.080

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 ¢ as the generator.

Subgroup: 2.3.11/7

Comma list: 256/243 ([8 -5 0⟩)

Subgroup-val mapping: [⟨5 8 0], ⟨0 0 1]]

mapping generators: ~9/8, ~11/7

Optimal tunings:

  • Subgroup WE: ~8/7 = 238.851 ¢, ~11/7 = 782.457 ¢
error map: ⟨-5.746 +8.852 -0.035]
  • Subgroup CWE: ~8/7 = 240.000 ¢, ~11/7 = 784.967 ¢
error map: ⟨0.000 +18.045 +2.475]

Optimal ET sequence: 15, 20, 35b, 55b

Pepperoni

Main article: Parapyth
See also: Chromatic pairs § Pepperoni

Pepperoni is generated by a fifth and can be described as the 5 & 12 temperament in the 2.3.11/7.13/7 subgroup. It is the single-chain retraction of parapyth. The 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 ([5 -3 1 -1⟩), 364/363 ([2 -1 -2 1⟩)

Subgroup-val mapping: [⟨1 0 7 12], ⟨0 1 -4 -7]]

Gencom mapping: [⟨1 0 0 -19/3 2/3 17/3], ⟨0 1 0 11/3 -1/3 -10/3]]

mapping generators: ~2, ~3

Optimal tunings:

  • Subgroup WE: ~2 = 1199.3706 ¢, ~11/7 = 703.4872 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~11/7 = 703.8328 ¢

Optimal ET sequence: 12, 17, 29, 75, 104

RMS error: 0.3789 cents

Fendo

Fendo tempers out 40/39 in the 2.3.13/5 subgroup. This equates 4/3 with the tendo third 13/10. Fendo can be viewed as an alternative interpretation of scales, chords, etc. commonly associated with father, as it is sometimes perceived that father's characteristics tend to suggest harmonic function more than to actually serve as a temperament. It was named by Vector in 2025 as a contraction of fourth and tendo third.

In fendo, the fourth / tendo third serves the role of both, making the main chords of triadic harmony in fendo essentially suspended chords.

5edo and 13edo make good fendo tunings.

Subgroup: 2.3.13/5

Comma list: 40/39 ([3 -1 -1⟩)

Subgroup-val mapping: [⟨1 0 3], ⟨0 1 -1]]

mapping generators: ~2, ~3

Optimal tunings:

  • Subgroup WE: ~2 = 1195.6502 ¢, ~3/2 = 709.8204 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~3/2 = 710.1529 ¢

Optimal ET sequence: 1, 2, 3, 5, 17*, 22*, 27*, 32*

* Wart for 13/5

Barbados

Barbados is the restriction of bridgetown to the 2.3.13/5 subgroup, and tempers out 676/675 alone. 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 scales 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 ([2 -3 2⟩)

Subgroup-val mapping: [⟨1 0 -1], ⟨0 2 3]]

mapping generators: ~2, ~26/15

Optimal tunings:

  • Subgroup WE: ~2 = 1199.9502 ¢, ~26/15 = 951.0713 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~26/15 = 951.0933 ¢

Optimal ET sequence: 5, 14, 19, 24, 29, 53, 82, 135

Badness (Smith): 0.002335

Music

Fiventeen

Fiventeen tempers out 136/135 ([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 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 ([3 -3 1⟩)

Subgroup-val mapping: [⟨1 0 -3], ⟨0 1 3]]

mapping generators: ~2, ~3

Optimal tunings:

  • Subgroup WE: ~2 = 1199.2838 ¢, ~3/2 = 704.4600 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.5286 ¢

Optimal ET sequence: 5, 12, 17, 46, 63, 143

Surprise

This temperament was named by Vector in 2025, as he was surprised that the temperament of 57/56 did not have a name. This is the 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 ([-3 1 1⟩)

Subgroup-val mapping: [⟨1 0 3], ⟨0 1 -1]]

mapping generators: ~2, ~3

Optimal tunings:

  • Subgroup WE: ~2 = 1202.4345 ¢, ~3/2 = 697.4314 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~3/2 = 697.3981 ¢

Optimal ET sequence: 5, 7, 12, 19, 31*, 50*

* Wart for 19/7

Badness (Sintel): 0.082

2.9.… subgroups

See also antikythera and isra.

Baldy

See also: Chromatic pairs § Baldy

Baldy is every other step of garibaldi. It is also the add-9 extension of frostburn. One of the best extension is to the 2.9.5.7.13 subgroup, mapping 13/8 to +10 whole tones, the same as the cassandra temperament.

Subgroup: 2.9.5.7

Comma list: 225/224, 3125/3087

Subgroup-val mapping: [⟨1 0 15 25], ⟨0 1 -4 -7]]

mapping generators: ~2, ~9

Optimal tunings:

  • WE: ~2 = 1200.1233 ¢, ~9/8 = 204.1913 ¢
  • CWE: ~2 = 1200.0000 ¢, ~9/8 = 204.1549 ¢

Optimal ET sequence: 6, 29, 35, 41, 47, 194

Badness (Sintel): 0.274

2.9.5.7.13 subgroup

Subgroup: 2.9.5.7.13

Comma list: 225/224, 325/324, 640/637

Subgroup-val mapping: [⟨1 0 15 25 -28], ⟨0 1 -4 -7 10]]

Gencom mapping: [⟨1 0 15 25 0 28], ⟨0 1/2 -4 -7 0 10]]

Optimal tunings:

  • WE: ~2 = 1200.0155 ¢, ~9/8 = 204.0931 ¢
  • CWE: ~2 = 1200.0000 ¢, ~9/8 = 204.0902 ¢

Optimal ET sequence: 6, 35, 41, 47, 100, 147

Badness (Sintel): 0.386

RMS error: 0.5999 cents

Baldanders

Baldanders results from taking every other generator of the andromeda, mapping 11/8 to -9 whole tones.

Subgroup: 2.9.5.7.11

Comma list: 100/99, 225/224, 245/242

Subgroup-val mapping: [⟨1 0 15 25 32], ⟨0 1 -4 -7 -9]]

Optimal tunings:

  • WE: ~2 = 1200.1917 ¢, ~9/8 = 204.7755 ¢
  • CWE: ~2 = 1200.0000 ¢, ~9/8 = 204.7197 ¢

Optimal ET sequence: 6, 23de, 29, 35, 41

Badness (Sintel): 0.389

2.9.5.7.11.13

Subgroup: 2.9.5.7.11.13

Comma list: 100/99, 144/143, 225/224, 245/242

Subgroup-val mapping: [⟨1 0 15 25 32 -28], ⟨0 1 -4 -7 -9 10]]

Optimal tunings:

  • WE: ~2 = 1199.6385 ¢, ~9/8 = 204.3526 ¢
  • CWE: ~2 = 1200.0000 ¢, ~9/8 = 204.4234 ¢

Optimal ET sequence: 6, 35, 41, 47, 88e

Badness (Sintel): 0.653

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 daemotonic 7L 4s scale of reasonable hardness (hence the name – daemo- + tertiaschis), which is notoriously difficult to approximate with simple JI or RTT methods.

Subgroup: 2.9.5.7.33.13.17

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

Subgroup-val mapping: [⟨1 1 11 -16 13 -18 20], ⟨0 3 -12 26 -11 30 -22]]

mapping generators: ~2, ~33/20

Optimal tunings:

  • Subgroup WE: ~2 = 1200.3019 ¢, 33/20 = 868.1949 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, 33/20 = 867.9773 ¢

Optimal ET sequence: 47, 159, 206, 253, 459…

Badness (Sintel): 0.439

A-team

See also: Dual-fifth temperaments § Dual-3 A-Team

A-team is every other step of mothra.

Subgroup: 2.9.5.21

Comma list: 81/80 ([-4 2 -1 0⟩), 1029/1024 ([-10 -1 0 3⟩)

Subgroup-val mapping: [⟨1 2 0 4], ⟨0 3 6 1]]

Gencom mapping: [⟨1 1 0 3], ⟨0 3/2 6 -1/2]]

mapping generators: ~2, ~21/16

Optimal tunings:

  • Subgroup WE: ~2 = 1200.9303 ¢, ~21/16 = 464.7465 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~21/16 = 464.5029 ¢

Optimal ET sequence: 13, 18, 31

Badness (Sintel): 0.177

2.9.5.21.11 subgroup

Subgroup: 2.9.5.21.11

Comma list: 81/80 ([-4 2 -1 0⟩), 99/98 ([-1 2 0 -2 1⟩), 385/384 ([-7 -1 1 1 1⟩)

Subgroup-val mapping: [⟨1 2 0 4 5], ⟨0 3 6 1 -4]]

Gencom mapping: [⟨1 1 0 3 5], ⟨0 3/2 6 -1/2 -4]]

Optimal tunings:

  • Subgroup WE: ~2 = 1201.3979 ¢, ~21/16 = 464.6020 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~21/16 = 464.1242 ¢

Optimal ET sequence: 13, 31, 44, 75b′

Badness (Sintel): 0.213

Glacial

See also: Chromatic pairs § Glacial

Subgroup: 2.9.5.11.13

Comma list: 45/44, 65/64, 81/80

Subgroup-val mapping: [⟨1 0 -4 -6 10], ⟨0 1 2 3 -2]]

Gencom mapping: [⟨1 0 -4 0 -6 10], ⟨0 1/2 2 0 3 -2]]

mapping generators: ~2, ~9

Optimal tunings:

  • WE: ~2 = 1202.6916 ¢, ~9/8 = 186.1446 ¢
  • CWE: ~2 = 1200.0000 ¢, ~9/8 = 186.1740 ¢

Optimal ET sequence: 6, 13, 45b′f

Badness (Sintel): 0.281

RMS error: 2.887 cents

Music:

Stacks (a.k.a. 2magic)

Stacks, the 11 & 30 temperament in the 2.9.15.7.11.13 subgroup, is every other step of magic.

Subgroup: 2.9.15.7

Comma list: 225/224 ([-5 0 2 -1⟩), 245/243 ([0 -3 1 2⟩)

Subgroup-val mapping: [⟨1 0 2 -1], ⟨0 5 3 6]]

Gencom mapping: [⟨1 0 2 -1], ⟨0 5/2 1/2 6]]

mapping generators: ~2, ~14/9

Optimal tunings:

  • Subgroup WE: ~2 = 1201.0786 ¢, ~14/9 = 761.3878 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~14/9 = 760.9151 ¢

Optimal ET sequence: 8, 11, 19, 30, 41, 71, 112*

* Wart for 15

Badness (Sintel): 0.0884

RMS error: 1.074 cents

2.9.15.7.11

Subgroup: 2.9.15.7.11

Comma list: 100/99 ([2 -2 2 0 -1⟩), 225/224 ([-5 0 2 -1 0⟩), 245/243 ([0 -3 1 2 0⟩)

Subgroup-val mapping: [⟨1 0 2 -1 6], ⟨0 5 3 6 -4]]

Gencom mapping: [⟨1 0 2 -1 6], ⟨0 5/2 1/2 6 -4]]

Optimal tunings:

  • Subgroup WE: ~2 = 1200.1372 ¢, ~14/9 = 761.4799 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~14/9 = 761.4017 ¢

Optimal ET sequence: 8, 11, 30, 41, 52, 93, 145

Badness (Sintel): 0.166

RMS error: 1.226 cents

2.9.15.7.11.13

Subgroup: 2.9.15.7.11.13

Comma list: 100/99 ([2 -2 2 0 -1 0⟩), 105/104 ([-3 0 1 1 0 -1⟩), 144/143 ([4 1 0 0 -1 -1⟩), 196/195 ([2 0 -1 2 0 -1⟩)

Subgroup-val mapping: [⟨1 0 2 -1 6 -2], ⟨0 5 3 6 -4 9]]

Gencom mapping: [⟨1 0 2 -1 6 -2], ⟨0 5/2 1/2 6 -4 9]]

Optimal tunings:

  • Subgroup WE: ~2 = 1200.0331 ¢, ~14/9 = 760.8754 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~14/9 = 760.8568 ¢

Optimal ET sequence: 11, 30, 41

Badness (Sintel): 0.257

RMS error: 1.540 cents

B-team

B-team may be described as 23 & 41 and is every other step of rodan.

Subgroup: 2.9.15.21

Comma list: 245/243 ([0 -4 1 2⟩), 1029/1024 ([-10 -1 0 3⟩)

Subgroup-val mapping: [⟨1 2 0 4], ⟨0 3 10 1]]

Optimal tunings:

  • Subgroup WE: ~2 = 1200.2146 ¢, ~21/16 = 468.9174 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~21/16 = 468.8518 ¢

Optimal ET sequence: 18, 23, 41, 64, 151

Badness (Sintel): 0.155

2.9.15.21.33 subgroup

Subgroup: 2.9.15.21.33

Comma list: 245/243 ([0 -4 1 2 0⟩), 385/384 ([-7 -2 1 1 1⟩), 441/440 ([-3 1 -1 2 -1⟩)

Subgroup-val mapping: [⟨1 2 0 4 7], ⟨0 3 10 1 -5]]

Optimal tunings:

  • Subgroup WE: ~2 = 1200.0553 ¢, ~21/16 = 468.9391 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~21/16 = 468.9189 ¢

Optimal ET sequence: 18, 23, 41, 64, 87, 151, 238b′

Badness (Sintel): 0.160

Mabon

Mabon may be described as the 11 & 62 temperament in the 2.9.7 subgroup. It was derived from a calendar leap cycle built for the autumn equinox, and named so by Dummy index in 2024.

Subgroup: 2.9.7

Comma basis: 44957696/43046721

Subgroup-val mapping: [⟨1 1 -3], ⟨0 3 8]]

mapping generators: ~2, ~729/448

Optimal tunings:

  • WE: ~2 = 1200.0000 ¢, ~729/448 = 870.4003 ¢
  • CWE: ~2 = 1200.0000 ¢, ~729/448 = 871.1047 ¢

Optimal ET sequence: 7d, 11, 40, 51, 62, 73b′, 157b′b′

Badness (Sintel): 4.08

2.9.7.11 subgroup

Subgroup: 2.9.7.11

Comma basis: 896/891, 1331/1296

Subgroup-val mapping: [⟨1 1 -3 2], ⟨0 3 8 2]]

Optimal tunings:

  • WE: ~2 = 1200.1182 ¢, ~18/11 = 870.9904 ¢
  • CWE: ~2 = 1200.0000 ¢, ~18/11 = 870.9234 ¢

Optimal ET sequence: 7d, 11, 40, 51, 62

Badness (Sintel): 4.08

Machine

Machine is every other step of supra, most interesting for its scale patterns.

It was named by Mike Battaglia in 2011, who spoke highly of this temperament, describing it as "stimulating, like a well-oiled machine".[1]

Subgroup: 2.9.7.11

Comma list: 64/63, 99/98

Subgroup-val mapping: [⟨1 0 6 13], ⟨0 1 -1 -3]]

Gencom mapping: [⟨1 0 0 6 13], ⟨0 1/2 0 -1 -3]]

mapping generators: ~2, ~9

Optimal tunings:

  • WE: ~2 = 1197.2650 ¢, ~9/8 = 213.8957 ¢
  • CWE: ~2 = 1200.0000 ¢, ~9/8 = 214.9961 ¢

Optimal ET sequence: 5, 6, 11, 17, 28

Badness (Sintel): 0.176

Apparatus

Apparatus is the restriction of revelation to the 2.9.7.11 subgroup, and every fourth step of quasiorwell, without prime 5.

Subgroup: 2.9.7.11

Comma list: 41503/41472, 322102/321489

Subgroup-val mapping: [⟨1 -14 1 -11], ⟨0 19 2 16]]

Gencom mapping: [⟨1 -7 0 1 -11], ⟨0 19/2 0 2 16]]

mapping generators: ~2, ~144/77

Optimal tunings:

  • WE: ~2 = 1200.0088 ¢, ~144/77 = 1084.4377 ¢
  • CWE: ~2 = 1200.0000 ¢, ~144/77 = 1084.4304 ¢

Optimal ET sequence: 10e, 21, 31, 52, 83, 135, 353, 488, 623

Badness (Sintel): 0.199

Penta a.k.a. mechanism

Penta or mechanism is the 8 & 11 temperament in the 2.9.7.11 subgroup.

Subgroup: 2.9.7.11

Comma list: 896/891, 26411/26244

Subgroup-val mapping: [⟨1 0 -1 6], ⟨0 5 6 -4]]

Gencom mapping: [⟨1 0 0 -1 6], ⟨0 5/2 0 6 -4]]

mapping generators: ~2, ~14/9

Optimal tunings:

  • WE: ~2 = 1199.4843 ¢, ~14/9 = 761.2079 ¢
  • CWE: ~2 = 1200.0000 ¢, ~14/9 = 761.4979 ¢

Optimal ET sequence: 8, 11, 30, 41, 52

Badness (Sintel): 0.331

RMS error: 0.4262 cents

Scales: penta5, penta8, penta11, penta19

Joan

See also: Chromatic pairs § Joan

Joan is the restriction of casablanca to the 2.9.7.11 subgroup, as well as every other step of orwell, without prime 5.

Subgroup: 2.9.7.11

Comma list: 99/98, 9317/9216

Subgroup-val mapping: [⟨1 0 1 3], ⟨0 7 4 1]]

Gencom mapping: [⟨1 0 0 1 3], ⟨0 7/2 0 4 1]]

mapping generators: ~2, ~11/8

Optimal tunings:

  • WE: ~2 = 1200.8640 ¢, ~11/8 = 543.0623 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/8 = 542.8446 ¢

Optimal ET sequence: 11, 20, 31, 42, 115b′d

Badness (Sintel): 0.407

RMS error: 1.424 cents

Novisept

Named by Pailiaq in 2026, novisept is generated by a one-cent-flat 9/7, such that stacking 5 of them gives 7/4. It can be formed by doubling both generator and period of gizzard.

Subgroup: 2.9.7.13.17

Comma list: 442/441, 729/728, 833/832

Subgroup-val mapping: [⟨1 1 1 -1 3], ⟨0 6 5 13 3]]

mapping generators: ~2, ~9/7

Optimal tunings:

  • WE: ~2 = 1200.3044 ¢, ~9/7 = 433.9023 ¢
  • CWE: ~2 = 1200.0000 ¢, ~9/7 = 433.8360 ¢

Optimal ET sequence: 11, 25f, 36, 47, 83, 130, 213g, 296g

Badness (Sintel): 0.142

Genius

Genius tempers out the Alpharabian comma in the 2.9.11 subgroup; it is therefore every other step of alphaxenean. It was named by Akselai in 2024 after the genius in Roman religion, following the demon (daimon) in Greek mythology.

Subgroup: 2.9.11

Comma list: 131769/131072

Subgroup-val mapping: [⟨1 1 4], ⟨0 4 -1]]

mapping generators: ~2, ~16/11

Optimal tunings:

  • WE: ~2 = 1200.3181 ¢, ~16/11 = 650.8508 ¢
  • CWE: ~2 = 1200.0000 ¢, ~16/11 = 650.7321 ¢

Optimal ET sequence: 11, 24, 59, 83, 142, 225, 367e, 592b′e, 959b′eee, 1326b′b′eeee

Badness (Sintel): 0.198

Demon

Demon is a temperament which equates three 11/9's with 16/9, or equivalently three 18/11's 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 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 the demon temperament is near the basic tuning of 7L 4s (13\18), and indeed 18edo supports this temperament.

Mabon is a strong extension of this temperament to the 2.9.7.11 subgroup.

Subgroup: 2.9.11

Comma list: 1331/1296

Subgroup-val mapping: [⟨1 1 2], ⟨0 3 2]]

mapping generators: ~2, ~18/11

Optimal tunings:

  • WE: ~2 = 1201.1151 ¢, ~18/11 = 869.4724 ¢
  • CWE: ~2 = 1200.0000 ¢, ~18/11 = 869.3196 ¢

Optimal ET sequence: 4, 7, 11, 18, 29, 76e

Badness (Sintel): 0.332

4.3.… subgroups

Tetrameantone

Main article: Tetrameantone

Named by Budjarn Lambeth in 2024, tetrameantone is the double-octave restriction of meantone.

Subgroup: 4.3.5

Comma list: 81/80

Subgroup-val mapping: [⟨1 0 -2], ⟨0 1 4]]

mapping generators: ~4, ~3

Optimal tunings:

  • WE: ~4 = 2402.7812 ¢, ~3 = 1898.4361 ¢
  • CWE: ~4 = 2400.0000 ¢, ~3 = 1896.6512 ¢

Optimal ET sequence: a′5, a′14, a′19, a′43, a′62, a′81, a′262b, a′343b

Badness (Sintel): 0.0866

Tetrahanson

Main article: Tetrahanson

Named by CompactStar in 2023, tetrahanson is the double-octave restriction of hanson.

Subgroup: 4.3.5

Comma list: 15625/15552

Subgroup-val mapping: [⟨1 -3 -2], ⟨0 6 5]]

mapping generators: ~4, ~12/5

Optimal tunings:

  • WE: ~4 = 2400.3318 ¢, ~12/5 = 1517.2163 ¢
  • CWE: ~4 = 2400.0000 ¢, ~12/5 = 1517.0308 ¢

Optimal ET sequence: a′8, a′11, a′19, a′68, a′87, a′106, a′511, a′617, a′723, a′829, a′935c, a′1764c

Badness (Sintel): 0.155

Tetramagic

Named by Budjarn Lambeth in 2024, tetramagic is the double-octave restriction of magic.

Subgroup: 4.3.5

Comma list: 3125/3072

Subgroup-val mapping: [⟨1 0 1], ⟨0 5 1]]

mapping generators: ~4, ~5/4

Optimal tunings:

  • WE: ~4 = 2402.4898 ¢, ~5/4 = 380.4527 ¢
  • CWE: ~4 = 2400.0000 ¢, ~5/4 = 380.2194 ¢

Optimal ET sequence: a′6, a′13, a′19, a′63, a′82, a′101, a′120, a′221c, a′341c

Badness (Sintel): 0.459

Blacktetra

Named by Budjarn Lambeth in 2024, blacktetra is the double-octave restriction of blackwood.

Subgroup: 4.3.5

Comma list: 256/243

Subgroup-val mapping: [⟨5 4 0], ⟨0 0 1]]

mapping generators: ~4, ~5

Optimal tunings:

  • WE: ~4/3 = 477.7018 ¢, ~5/4 = 397.6806 ¢ (~16/15 = 80.0212 ¢)
  • CWE: ~4/3 = 480.0000 ¢, ~5/4 = 395.1255 ¢ (~16/15 = 84.7745 ¢)

Optimal ET sequence: a′5, a′20, a′25, a′30

Badness (Sintel): 0.748

4.6.… subgroups

Meanquad

Meanquad is the double-octave restriction of meantone to the 4.6.5 subgroup. Similarly to meantone, it has a flat 3/2 as generator and equates 5/4 and 81/64. It possesses mos scales of the forms 3L 4s<4/1>, 7L 3s<4/1>, and 7L 10s<4/1>, which could be viewed as equivalents to the pentic, diatonic, and chromatic scales of meantone, respectively.

17ed4 is one of the smallest tunings that support this temperament and features the decatonic scale. Like 12edo, it can be viewed as the boundary between meanquad and quarchy, the 4.6.7-subgroup restriction of archy. Larger options include 24ed4 (12edo), 31ed4 and 55ed4.

Subgroup: 4.6.5

Comma list: 81/80 ([-4 4 -1⟩)

Subgroup-val mapping: [⟨1 0 -4], ⟨0 1 4]]

mapping generators: ~4, ~6

Optimal tunings:

  • Subgroup WE: ~4 = 2402.7812 ¢, ~3/2 = 697.0455 ¢
  • Subgroup CWE: ~4 = 2400.0000 ¢, ~3/2 = 696.6512 ¢

Optimal ET sequence: a′7, a′17, a′24, a′31, a′100, a′131, a′293*

* Wart for 6

Badness (Sintel): 0.0146

Tetrominant

Subgroup: 4.6.5.7

Comma list: 36/35 ([0 2 -1 -1⟩), 64/63 ([4 -2 0 -1⟩)

Subgroup-val mapping: [⟨1 0 -4 4], ⟨0 1 4 -2]]

Optimal tunings:

  • Subgroup WE: ~4 = 2400.0000 ¢, ~3/2 = 699.622 ¢
  • Subgroup CWE: ~4 = 2400.0000 ¢, ~3/2 = 699.622 ¢

Optimal ET sequence: a′7, a′17, a′24, a′41, a′65cd

Badness (Sintel): 0.0974

Quarchy

Subgroup: 4.6.7

Comma list: 64/63 ([4 -2 -1⟩)

Subgroup-val mapping: [⟨1 0 4], ⟨0 1 -2]]

mapping generators: ~4, ~6

Optimal tunings:

  • Subgroup WE: ~4 = 2393.9105 ¢, ~3/2 = 707.5215 ¢
  • Subgroup CWE: ~4 = 2400.0000 ¢, ~3/2 = 709.3901 ¢

Optimal ET sequence: a′7, a′10, a′17, a′27, a′44, a′159d, a′203*dd

* Wart for 6

Badness (Sintel): 0.0134

4.9.… subgroups

Meansquared

Named by CompactStar in 2023, meansquared tempers out the double syntonic comma, 6561/6400, in the 4.9.25 subgroup.

Subgroup: 4.9.25

Comma list: 6561/6400

Subgroup-val mapping: [⟨1 0 -4], ⟨0 1 4]]

mapping generators: ~4, ~9

Optimal tunings:

  • WE: ~4 = 2402.7812 ¢, ~9/4 = 1394.0910 ¢
  • CWE: ~4 = 2400.0000 ¢, ~9/4 = 1393.3025 ¢

Optimal ET sequence: a′5, a′7, a′12, a′19, a′31, a′50, a′81, a′131b′

Badness (Sintel): 0.173

Archsquared

Named by CompactStar in 2023, archsquared tempers out the double archytas comma, 4096/3969, in the 4.9.49 subgroup.

Subgroup: 4.9.49

Comma list: 4096/3969

Subgroup-val mapping: [⟨1 0 6], ⟨0 1 -2]]

mapping generators: ~4, ~9

Optimal tunings:

  • WE: ~4 = 2393.9105 ¢, ~9/4 = 1415.0430 ¢
  • CWE: ~4 = 2400.0000 ¢, ~9/4 = 1418.7801 ¢

Optimal ET sequence: a′5, a′12, a′17, a′22, a′137b′d′d′, a′159b′d′d′d′, a′181b′b′d′d′d′

Badness (Sintel): 0.159

8.9.… subgroups

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 was named by Budjarn Lambeth in 2023 for the classic dad joke: "Why was six scared? Because seven ate nine."

Subgroup: 8.9.7

Comma list: 64/63

Subgroup-val mapping: [⟨1 0 2], ⟨0 1 -1]]

mapping generators: ~8, ~9

Optimal tunings:

  • WE: ~8 = 3590.8657 ¢, ~9/8 = 219.0878 ¢
  • CWE: ~8 = 3600.0000 ¢, ~9/8 = 218.7801 ¢

Optimal ET sequence: a″16, a″33, a″148b′d, a″181b′d, a″214b′d, a″247b′b′d

Badness (Sintel): 0.0265

2.5.… subgroups

Guanyintet

See also: Chromatic pairs § Guanyintet

Guanyintet, the 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 176/175 and 540/539, which imply 1728/1715 as well as 3388/3375 are tempered out. The tonic and the first three generator steps make a guanyin tetrad, hence the name.

Subgroup: 2.5.7/3.11/3

Comma list: 176/175 ([4 -2 -1 1⟩), 540/539 ([2 1 -2 -1⟩)

Subgroup-val mapping: [⟨1 0 1 3], ⟨0 3 -1 5]]

Gencom mapping: [⟨1 0 0 2 -2], ⟨0 -4/3 3 -7/3 11/3]]

mapping generators: ~2, ~12/7

Optimal tunings:

  • Subgroup WE: ~2 = 1199.2336 ¢, ~12/7 = 929.3128 ¢ (~7/6 = 269.9208 ¢)
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~12/7 = 929.7608 ¢ (~7/6 = 270.2392 ¢)

Optimal ET sequence: 9, 22, 31, 40, 191c*, 231c*, 271c*, 311c*

* Wart for 7/3

RMS error: 0.6028 cents

Tridecimal guanyintet

Guanyintet can extend to the add-13 subgroup by the equivalences (12/11)3~13/10 and (15/14)3~16/13. However, note that it is not supported by tridecimal orwell (31 & 53), but instead the less accurate winston (22f & 31), as orwell prefers slightly sharper tunings than guanyintet. 40edo remains an excellent tuning.

Subgroup: 2.5.7/3.11/3.13

Comma list: 176/175 ([4 -2 -1 1 0⟩), 540/539 ([2 1 -2 -1 0⟩), 1573/1568 ([-5 0 -2 2 1⟩)

Subgroup-val mapping: [⟨1 0 1 3 1], ⟨0 3 -1 5 -12]]

Optimal tunings:

  • Subgroup WE: ~2 = 1199.3604 ¢, ~12/7 = 929.2866 ¢ (~7/6 = 270.0738 ¢)
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~12/7 = 929.7980 ¢ (~7/6 = 270.2020 ¢)

Optimal ET sequence: 9, 22f, 31, 40, 71, 111, 151, 262c*

* Wart for 7/3

Badness (Sintel): 0.329

Laz

See also: Chromatic pairs § Laz

Laz is related to avalokita as well as to winston.

Subgroup: 2.5.7/3.11/3.13/3

Comma list: 144/143 ([4 0 0 -1 -1⟩), 176/175 ([4 -2 -1 1⟩), 196/195 ([2 -1 2 0 -1⟩

Subgroup-val mapping: [⟨1 0 2 -2 6], ⟨0 3 -1 5 -5]]

Gencom mapping: [⟨1 -3/2 0 1/2 -7/2 9/2], ⟨0 1/4 3 -3/4 21/4 -19/4]]

Optimal tunings:

  • Subgroup WE: ~2 = 1198.4320 ¢, ~12/7 = 929.4834 ¢ (~7/6 = 268.9486 ¢)
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~12/7 = 930.5797 ¢ (~7/6 = 269.4203 ¢)

Optimal ET sequence: 9, 31, 40, 49, 156c*†, 205c*†

* Wart for 7/3
† Wart for 11/3

RMS error: 0.8790 cents

Kryptonite

See also: Chromatic pairs § Kryptonite

Kryptonite is related to krypton.

Subgroup: 2.5.7/3.11/3.13/3

Comma list: 56/55 ([3 -1 1 -1⟩), 78/77 ([1 0 -1 -1 1⟩), 91/90 ([-1 -2 1 0 1⟩)

Subgroup-val mapping: [⟨1 2 1 2 2], ⟨0 3 2 -1 1]]

Gencom mapping: [⟨1 -5/4 2 -1/4 3/4 3/4], ⟨0 -1/2 3 3/2 -3/2 1/2]]

mapping generators: ~2, ~13/12

Optimal tunings:

  • Subgroup WE: ~2 = 1197.6484 ¢, ~13/12 = 132.1686 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~13/12 = 131.8445 ¢

Optimal ET sequence: 1, …, 8, 9

RMS error: 2.545 cents

Marveltri

See also: Chromatic pairs § Marveltri

Marveltri, the 3 & 13 temperament in the 2.5.9/7 subgroup, is related to marvel, magic, and the unnamed 22 & 47 temperament. The tonic and the first two generator steps make a marvel triad, hence the name.

Subgroup: 2.5.9/7

Comma list: 225/224 ([-5 2 1⟩)

Subgroup-val mapping: [⟨1 0 5], ⟨0 1 -2]]

Gencom mapping: [⟨1 2 0 -1], ⟨0 -4/5 1 2/5]]

mapping generators: ~2, ~5

Optimal tunings:

  • Subgroup WE: ~2 = 1200.5971 ¢, ~5/4 = 383.8285 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~5/4 = 383.8540 ¢

Optimal ET sequence: 3, 13, 16, 19, 22, 25, 72, 97, 122, 269c*

* Wart for 9/7

RMS error: 0.4801 cents

Sulis

Sulis is related to minerva and würschmidt.

Subgroup: 2.5.9/7.11/9

Comma list: 99/98 ([-1 0 2 1⟩), 176/175 ([4 -2 1 1⟩)

Subgroup-val mapping: [⟨1 0 5 -9], ⟨0 1 -2 4]]

Optimal tunings:

  • Subgroup WE: ~2 = 1200.1086 ¢, ~5/4 = 386.5931 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~5/4 = 386.5785 ¢

Optimal ET sequence: 3, …, 22, 25, 28, 31, 59

RMS error: 1.074 cents

2.5/3.… subgroups

Magicaltet

See also: Chromatic pairs § Magicaltet

Magicaltet is related to keemic, superkleismic, and magic. The tonic and the first three generator steps make a magical seventh chord, hence the name.

Subgroup: 2.5/3.7.11

Comma list: 100/99 ([2 2 0 -1⟩), 385/384 ([-7 1 1 1⟩)

Subgroup-val mapping: [⟨1 0 5 2], ⟨0 1 -3 2]]

Gencom mapping: [⟨1 0 0 5 2], ⟨0 -1/2 1/2 -3 2]]

mapping generators: ~2, ~5/3

Optimal tunings:

  • Subgroup WE: ~2 = 1199.9868 ¢, ~5/3 = 877.3312 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~5/3 = 877.3417 ¢

Optimal ET sequence: 4, 7, 11, 15, 26, 67, 93*

* Wart for 5/3

RMS error: 1.206 cents

Starlingtet

See also: Chromatic pairs § Starlingtet

Starlingtet, the 4 & 15 temperament in the 2.5/3.7/3 subgroup, is related to starling as well as to myna. The tonic and the first three generator steps make a starling tetrad, hence the name.

Subgroup: 2.5/3.7/3

Comma list: 126/125 ([1 -3 1⟩)

Subgroup-val mapping: [⟨1 0 -1], ⟨0 1 3]]

Gencom mapping: [⟨1 1/3 1/3 -2/3], ⟨0 -4/3 -1/3 5/3]]

mapping generators: ~2, ~5/3

Optimal tunings:

  • Subgroup WE: ~2 = 1199.7931 ¢, ~5/3 = 888.6923 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~5/3 = 888.8022 ¢

Optimal ET sequence: 4, 15, 19, 23, 27

RMS error: 0.8398 cents

Skateboard

See also: Chromatic pairs § Skateboard

Skateboard is related to thrasher.

Subgroup: 2.5/3.7/3.11.13/9

Comma list: 56/55 ([3 -1 1 -1⟩), 91/90 ([-1 -1 1 0 1⟩), 100/99 ([2 2 0 -1⟩)

Subgroup-val mapping: [⟨1 0 -1 2 2], ⟨0 1 3 2 -2]]

Gencom mapping: [⟨1 -3/7 -3/7 -10/7 2 8/7], ⟨0 0 1 3 2 -2]]

Optimal tunings:

  • Subgroup WE: ~2 = 1197.4723 ¢, ~5/3 = 884.2917 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~5/3 = 885.6469 ¢

Optimal ET sequence: 4, 11, 15, 19, 23, 42e

RMS error: 2.396 cents

Greeley

See also: Chromatic pairs § Greeley

Greeley is related to opossum as well as to nusecond.

Subgroup: 2.5/3.7/3.11/3

Comma list: 121/120 ([-3 -1 0 2⟩), 126/125 ([1 -3 1⟩)

Subgroup-val mapping: [⟨1 1 2 2], ⟨0 -2 -6 -1]]

Gencom mapping: [⟨1 -5/4 -1/4 3/4 3/4], ⟨0 9/4 1/4 -15/4 5/4]]

mapping generators: ~2, ~11/10

Optimal tunings:

  • Subgroup WE: ~2 = 1200.4124 ¢, ~11/10 = 155.8292 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~11/10 = 155.7399 ¢

Optimal ET sequence: 8, 15, 23, 54, 77, 131*

* Wart for 11/3

RMS error: 1.034 cents

Gariberttet

Gariberttet is the 2.5/3.7/3 altergene[idiosyncratic term][clarification needed] of sirius, tempering out 3125/3087. A notable tuning besides WE and CWE given below is CTE/POTE at 293.6785 cents.

Subgroup: 2.5/3.7/3

Comma list: 3125/3087

Subgroup-val mapping: [⟨1 0 0], ⟨0 3 5]]

mapping generators: ~2, ~25/21

Optimal tunings:

  • Subgroup WE: ~2 = 1199.9983 ¢, ~25/21 = 293.7393 ¢
error map: ⟨-0.002 -3.141 1.826]
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~25/21 = 293.7395 ¢
error map: ⟨0.000 -3.140 1.827]

Optimal ET sequence: 4, 25*, 29, 33, 37, 41, 45, 49, 94, 143, 192

* Wart for 7/3

2.5/3.7/3.13/11 subgroup

See also: Chromatic pairs § Gariberttet

Gariberttet can be described as the 4 & 29 temperament in the 2.5/3.7/3.13/11 subgroup. Extensions to the full 7-, 11-, and 13-limits include quasitemp.

Subgroup: 2.5/3.7/3.13/11

Comma list: 275/273 ([0 2 -1 -1⟩), 847/845 ([0 -1 1 -2⟩)

Subgroup-val mapping: [⟨1 0 0 0], ⟨0 3 5 1]]

Gencom mapping: [⟨1 0 0 0 0 0], ⟨0 -8/3 1/3 7/3 -1/2 1/2]]

Optimal tunings:

  • Subgroup WE: ~2 = 1199.9976 ¢, ~13/11 = 293.6780 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~13/11 = 293.6782 ¢

Optimal ET sequence: 4, 25*, 29, 33, 37, 41, 45, 49, 94, 143†

* Wart for 7/3
† Wart for 13/11

RMS error: 0.6914 cents

Indium

See also: Chromatic pairs § Indium

Indium can be described as the 8 & 33 temperament in the 2.5/3.7/3.11/3 subgroup.

Subgroup: 2.5/3.7/3.11/3

Comma list: 3025/3024 ([-4 2 -1 2⟩), 3125/3087 ([0 5 -3⟩)

Subgroup-val mapping: [⟨1 0 0 2], ⟨0 6 10 -1]]

Gencom mapping: [⟨1 -1/2 -1/2 -1/2 3/2], ⟨0 -15/4 9/4 25/4 -19/4]]

mapping generators: ~2, ~12/11

Optimal tunings:

  • Subgroup WE: ~2 = 1199.5499 ¢, ~12/11 = 146.9551 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~12/11 = 146.9960 ¢

Optimal ET sequence: 8, 33, 41, 49, 204*†

* Wart for 7/3
† Wart for 11/3

RMS error: 0.7788 cents

Ammon

See also: Chromatic pairs § Ammon

Ammon can be described as the 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 ~20/13, whose octave complement, ~13/10, may be considered a semidiminished fourth, hence the old name semidim, which has been rejected since 2025 to avoid confusion with another temperament of the same name.

Subgroup: 2.5/3.7/3.11/3.13/3

Comma list: 121/120 ([-3 -1 0 2⟩), 169/168 ([-3 0 -1 0 2⟩), 275/273 ([0 2 -1 1 -1⟩)

Subgroup-val mapping: [⟨1 -3 -5 0 -1], ⟨0 6 10 3 5]]

Gencom mapping: [⟨1 9/5 -6/5 -16/5 9/5 4/5], ⟨0 -24/5 6/5 26/5 -9/5 1/5]]

mapping generators: ~2, ~20/13

Optimal tunings:

  • Subgroup WE: ~2 = 1200.9125 ¢, ~20/13 = 747.3256 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~20/13 = 746.8104 ¢

Optimal ET sequence: 8, 29, 37, 45

RMS error: 1.052 cents

Sentry

See also: Chromatic pairs § Sentry

Sentry, the 3 & 5 temperament in the 2.5/3.9/7 subgroup, tempers out 245/243 and shares the same generator with sensi. A notable tuning besides WE and CWE given below is CTE/POTE at 440.9020 cents.

Subgroup: 2.5/3.9/7

Comma list: 245/243 ([0 1 -2⟩)

Subgroup-val mapping: [⟨1 0 0], ⟨0 2 1]]

Gencom mapping: [⟨1 0 0 0], ⟨0 0 2 -1]]

mapping generators: ~2, ~9/7

Optimal tunings:

  • Subgroup WE: ~2 = 1199.9983 ¢, ~9/7 = 440.9014 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~9/7 = 440.9017 ¢

Optimal ET sequence: 3, 5, 8, 11, 19, 30, 41, 49

RMS error: 0.7105 cents

Marveltwintri

See also: Chromatic pairs § Marveltwintri

Marveltwintri can be described as the 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.

Subgroup: 2.5/3.13/9

Comma list: 325/324 ([-2 2 1⟩)

Subgroup-val mapping: [⟨1 0 2], ⟨0 1 -2]]

Gencom mapping: [⟨1 -2/3 -7/3 0 0 2/3], ⟨0 1/2 3/2 0 0 -1]]

mapping generators: ~2, ~5/3

Optimal tunings:

  • Subgroup WE: ~2 = 1200.1340 ¢, ~5/3 = 882.9597 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~5/3 = 882.8731 ¢

Optimal ET sequence: 3, 4, 11, 15, 19, 34, 53, 87, 140

RMS error: 0.2444 cents

2.9/5.… subgroups

Kiribati

See also: Chromatic pairs § Kiribati

Kiribati is related to nakika as well as to octacot.

Subgroup: 2.9/5.7/3.11/9

Comma list: 100/99 ([2 -2 0 -1⟩), 245/242 ([-1 -1 2 -2⟩)

Subgroup-val mapping: [⟨1 1 1 0], ⟨0 -2 3 4]]

Gencom mapping: [⟨1 1/10 -4/5 11/10 1/5], ⟨0 -3/2 -1 3/2 1]]

mapping generators: ~2, ~21/20

Optimal tunings:

  • Subgroup WE: ~2 = 1199.6137 ¢, ~21/20 = 87.8636 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~21/20 = 87.8385 ¢

Optimal ET sequence: 13, 14, 27, 41

RMS error: 1.245 cents

Terrain

"Terrain" redirects here. For the scale, see Terrain (scale).
See also: Chromatic pairs § Terrain

Terrain, the 6 & 21 temperament in the 2.9/5.9/7 subgroup, tempers out the landscape comma and is related to 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: 2.9/5.9/7

Comma list: 250047/250000 ([-4 6 -3⟩)

Subgroup-val mapping: [⟨3 0 -4], ⟨0 1 2]]

mapping generators: ~63/50, ~9/5

Optimal tunings:

  • Subgroup WE: ~63/50 = 400.0012 ¢, ~9/5 = 1017.5415 ¢ (~10/9 = 182.4620 ¢)
  • Subgroup CWE: ~63/50 = 400.0000 ¢, ~9/5 = 1017.5388 ¢ (~10/9 = 182.4612 ¢)

Optimal ET sequence: 6, 21, 27, 33, 105, 138, 171, 2019, 2190, 2361, 2532, 2703, 2874, 3045, 3216, 3387, 3558, 3729, 3900, 7971

RMS error: 0.00844 cents

2.75.… subgroups

Archagall

See also: Fifthplus

By tempering out the comma 24576/24565 in the 2.75.85 subgroup, we have three 85/64's up and one octave down as a 75/64 and we have two 128/85's up and one octave down as a 17/15 whole tone. It is because of this combination of accuracy, efficiency and mapping-wise simplicity and its corresponding explanatory power in what this comma does that the comma has been named the archagallisma. The MVP stands for minimum viable product, as this is the core of what the archagall logic achieves, with further extensions adding to the subgroup while avoiding significantly impacting its accuracy. This is a highly accurate temperament that could be considered to be encoding the "high-accuracy logic" of superpyth and which is inescapably related to the 17L 5s scale form as it is the 17 & 22 temperament (or less accurately, the 5 & 17 temperament) in the 2.75.85 subgroup.

It is perhaps worth noting that 83edo is approximately an optimal tuning for getting all of 75/64, 85/64, and the interval between them, 17/15, close to just, in that larger tunings compromise on tone-efficiency.

Subgroup: 2.75.85

Comma list: 24576/24565 ([13 1 -3⟩)

Subgroup-val mapping: [⟨1 2 5], ⟨0 3 1]]

mapping generators: ~2, ~85/32

Optimal tunings:

  • Subgroup WE: ~2 = 1199.9692 ¢, ~85/64 = 491.5853 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~85/64 = 491.5794 ¢

Optimal ET sequence: 5, 12, 17, 22, 61, 83, 310, 393, 476, 1345, 1821, 4118…, 5939…, 7760…

2.75.9/7.85 subgroup

A fairly natural way to extend archagall is by tempering out 2025/2023 (S15/S17), which equates a stack of two 17/15's with 9/7 without much damage. As 9/7 was not previously in the subgroup, this does not decrease the rank of the temperament and qualifies a proper and natural extension. We can equally get the same temperament by tempering out S15/S16 instead (equating a stack of three 16/15's with 17/14); however, 16/15 is not in the subgroup, so it is preferred to think of it as adding 2025/2023.

Subgroup: 2.75.9/7.85

Comma list: 2025/2023 ([2 -2 1 0⟩), 24576/24565 ([13 1 0 -3⟩)

Subgroup-val mapping: [⟨1 2 6 5], ⟨0 3 -4 1]]

Optimal tunings:

  • Subgroup WE: ~2 = 1200.0241 ¢, ~85/64 = 491.3358 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~85/64 = 491.3290 ¢

Optimal ET sequence: 5, 12, 17, 22, 83, 105, 127, 403, 530, 657, 784, 1441…

2.7/3.… subgroups

Mothwelltri

See also: Chromatic pairs § Mothwelltri

Mothwelltri, the 4 & 5 temperament in the 2.7/3.11 subgroup, tempers out 99/98 and is related to orwell. The tonic and the first two generator steps make a mothwellsmic triad, hence the name.

Subgroup: 2.7/3.11

Comma list: 99/98 ([-1 -2 1⟩)

Subgroup-val mapping: [⟨1 0 1], ⟨0 1 2]]

Gencom mapping: [⟨1 0 0 0 1], ⟨0 -1/2 0 1/2 2]]

mapping generators: ~2, ~7/3

Optimal tunings:

  • Subgroup WE: ~2 = 1200.3175 ¢, ~7/6 = 273.2462 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~7/6 = 273.4280 ¢

Optimal ET sequence: 4, 9, 13, 22, 79

RMS error: 1.064 cents

2.7/5.… subgroups

Tridec

See also: Chromatic pairs § Tridec
See also: Non-over-1 temperament § Tridec

Tridec, the 5 & 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 ([0 1 2 -2⟩), 1001/1000 ([-3 1 1 1⟩)

Subgroup-val mapping: [⟨1 -2 3 2], ⟨0 4 -3 -1]]

Gencom mapping: [⟨1 0 -3/4 -11/4 9/4 5/4], ⟨0 0 0 4 -3 -1]]

mapping generators: ~2, ~20/13

Optimal tunings:

  • subgroup WE: ~2 = 1200.0343 ¢, ~20/13 = 745.4658 ¢
  • subgroup CWE: ~2 = 1200.0000 ¢, ~20/13 = 745.4431 ¢

Optimal ET sequence: 5, 8, 21, 29, 37, 66

RMS error: 0.1613 cents

Naiadec

Subgroup: 2.7/5.11/5.13/5.17/5

Comma list: 170/169 ([1 0 0 -2 1⟩), 221/220 ([-2 0 -1 1 1⟩), 847/845 ([0 1 2 -2 0⟩)

Subgroup-val mapping: [⟨1 -2 3 2 3], ⟨0 4 -3 -1 -2]]

Gencom mapping: [⟨1 0 -3/4 -11/4 9/4 5/4 9/4], ⟨0 0 0 4 -3 -1 -2]]

mapping generators: ~2, ~20/13

Optimal tunings:

  • Subgroup WE: ~2 = 1200.4068 ¢, ~20/13 = 745.3706 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~20/13 = 745.0940 ¢

Optimal ET sequence: 5, 8, 21, 29, 95*, 124*

* Wart for 17/5

RMS error: 0.7521 cents

2.25/7.… subgroups

Supramine

Supramine extends supramin and 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: 2.25/7.17/7.19/7.23/7

Comma list: 323/322 ([-1 0 1 1 -1⟩), 392/391 ([3 0 -1 0 -1⟩), 476/475 ([2 -1 1 -1 0⟩)

Subgroup-val mapping: [⟨1 -2 0 4 3], ⟨0 3 1 -2 -1]]

mapping generators: ~2, ~17/14

Optimal tunings:

  • Subgroup WE: ~2 = 1199.7558 ¢, ~17/14 = 335.4278 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~17/14 = 335.4885 ¢

Optimal ET sequence: 7, 18, 25, 68, 93

Badness (Sintel): 0.053

2.11/5.… subgroups

Petrtri

See also: Chromatic pairs § Petrtri
See also: 5L 3s/Temperaments § Petrtri

Petrtri can be described as 3 & 5 temperament in the 2.11/5.13/5 subgroup and tempers out 2200/2197. It restricts tridec. The tonic and the first two generator steps make a petrmic triad, hence the name.

Subgroup: 2.11/5.13/5

Comma list: 2200/2197 ([3 1 -3⟩)

Subgroup-val mapping: [⟨1 0 1], ⟨0 3 1]]

Gencom mapping: [⟨1 0 -1/3 0 -1/3 2/3], ⟨0 0 -4/3 0 5/3 -1/3]]

mapping generators: ~2, ~13/10

Optimal tunings:

  • Subgroup WE: ~2 = 1199.9572 ¢, ~13/10 = 454.9958 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~13/10 = 455.0039 ¢

Optimal ET sequence: 5, 8, 21, 29, 153, 182, 211, 240, 269, 509, 778…, 1287…

RMS error: 0.0749 cents

Other 2-repeating subgroups

Supramin

Supramin restricts supramine and 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 Overthink in 2026 after the fact that the generator is a 17/14 supraminor third, two of which reach 28/19. It is related to cohemimabila.

Subgroup: 2.17/7.19/7

Comma list: 5491/5488 ([-4 2 1⟩)

Subgroup-val mapping: [⟨1 0 4], ⟨0 1 -2]]

mapping generators: ~2, ~17/7

Optimal tunings:

  • Subgroup WE: ~2 = 1200.0220 ¢, ~17/14 = 335.7933 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~17/14 = 335.7852 ¢

Optimal ET sequence: 7, 18, 25, 118, 143, 168, 193, 218, 461, 679, 1576, 2255…

Badness (Sintel): 0.005

3/2.5/2.… subgroups

Main article: Half-prime subgroup

Hemihemi

Named by CompactStar in 2023, hemihemi tempers out 10976/10935, the hemimage comma, in the 3/2.5/2.7/2 subgroup.

Subgroup: 3/2.5/2.7/2

Comma list: 10976/10935 ([-7 -1 3⟩)

Subgroup-val mapping: [⟨1 2 3], ⟨0 3 1]]

mapping generators: ~3/2, ~28/27

Optimal tunings:

  • Subgroup WE: ~3/2 = 702.6069 ¢, ~28/27 = 60.4665 ¢
  • Subgroup CWE: ~3/2 = 701.9550 ¢, ~28/27 = 60.8011 ¢

Optimal ET sequence: *11, *12, *23, *35, *58, *93, *151†, *244†

* Wart for 3/2
† Wart for 7/2

Halftone

Main article: Halftone

Named by CompactStar in 2023, halftone tempers out 9604/9375 in the 3/2.5/2.7/2 subgroup.

Subgroup: 3/2.5/2.7/2

Comma list: 9604/9375 ([-1 -5 4⟩)

Subgroup-val mapping: [⟨1 3 4], ⟨0 -4 -5]]

mapping generators: ~3/2, ~15/14

Optimal tunings:

  • Subgroup WE: ~3/2 = 702.6568 ¢, ~15/14 = 128.2849 ¢
  • Subgroup CWE: ~3/2 = 701.9550 ¢, ~15/14 = 128.8183 ¢

Optimal ET sequence: *5, *11, *27, *38

* Wart for 3/2

3/2.5/2.7/2.11/2 subgroup

Subgroup: 3/2.5/2.7/2.11/2

Comma list: 1232/1215 ([-5 -1 1 1⟩), 9604/9375 ([-1 -5 4 0⟩)

Subgroup-val mapping: [⟨1 3 4 4], ⟨0 -4 -5 1]]

Optimal tunings:

  • WE: ~3/2 = 704.8651 ¢, ~15/14 = 130.9114 ¢
  • CWE: ~3/2 = 701.9550 ¢, ~15/14 = 129.6376 ¢

Optimal ET sequence: *5, *11, *16, *27…, *43…, *70…

* Wart for 3/2

3/2.5/2.7/2.11/2.13/2 subgroup

Subgroup: 3/2.5/2.7/2.11/2.13/2

Comma list: 275/273 ([-1 2 -1 1 -1⟩), 1232/1215 ([-5 -1 1 1 0⟩), 1323/1300 ([3 -2 2 0 -1⟩)

Subgroup-val mapping: [⟨1 3 4 4 5], ⟨0 -4 -5 1 -2]]

Optimal tunings:

  • WE: ~3/2 = 702.9683 ¢, ~15/14 = 130.0424 ¢
  • CWE: ~3/2 = 701.9550 ¢, ~15/14 = 129.6034 ¢

Optimal ET sequence: *5, *11, *16, *27†

* Wart for 3/2
† Wart for 11/2

Semiwolf

Semiwolf provides an interpretation for the 3L 2s⟨3/2⟩ mos scale. The name semiwolf comes from two ~7/6 generators approximating a 27/20 wolf fourth. This is further extended to the 11-limit in two interpretations: semilupine, where 2 major 2-mossteps (LL) equal 11/9, and hemilycan. where 1 major and 2 minor 2-mossteps (sLs) equal 11/9. Basic 8edf fits both extensions.

Subgroup: 3/2.5/2.7/4

Comma list: 245/243 ([-5 1 2⟩)

Subgroup-val mapping: [⟨1 1 2], ⟨0 2 -1]]

mapping generators: ~3/2, ~9/7

Optimal tunings:

  • Subgroup WE: ~3/2 = 703.7414 ¢, ~9/7 = 440.9014 ¢
  • Subgroup CWE: ~3/2 = 701.9550 ¢, ~9/7 = 440.8307 ¢

Optimal ET sequence: *8, *43, *51, *59, *67, *75…, *83…

* Wart for 3/2

Hemilycan

Subgroup: 3/2.5/2.7/4.11/4

Comma list: 245/243 ([-5 1 2 0⟩), 441/440 ([2 -1 2 -1⟩)

Subgroup-val mapping: [⟨1 1 2 5], ⟨0 2 -1 -4]]

Optimal tunings:

  • Subgroup WE: ~3/2 = 703.2870 ¢, ~9/7 = 441.1967 ¢
  • Subgroup CWE: ~3/2 = 701.9550 ¢, ~9/7 = 440.4501 ¢

Optimal ET sequence: *8, *28, *35, *43, *51, 110…

* Wart for 3/2

Semilupine

Subgroup: 3/2.5/2.7/4.11/4

Comma list: 100/99 ([-3 2 0 -1⟩), 245/243 ([-5 1 2 0⟩)

Subgroup-val mapping: [⟨1 1 2 0], ⟨0 2 -1 4]]

Optimal tunings:

  • Subgroup WE: ~3/2 = 704.6266 ¢, ~9/7 = 439.2433 ¢
  • Subgroup CWE: ~3/2 = 701.9550 ¢, ~9/7 = 438.6989 ¢

Optimal ET sequence: *5, *8, *45…, *53…, *61…, *69…, *77…

* Wart for 3/2

3/2.5/4.… subgroups

Ambulatory

Ambulatory tempers out 686/675 in the 3/2.5/4.7/4 subgroup and is generated by a sharp ~15/14. This implies a stack of two generators represents 7/6, and a stack of three generators represents 5/4.

Ambulatory can be considered the 3/2.5/4.7/4 analog of BPS or meantone, using 5:6:7 as its fundamental consonant chord in the place of 3:5:7 or of 4:5:6. Ambulatory is actually similar to meantone in several ways:

  • They are both of low badness.[dubious – discuss]
  • In most optimal tunings, meantone tunes 3/2 flat and 5/4 slightly sharp, while ambulatory tunes 7/5 flat and 6/5 slightly sharp.
  • Meantone[7] has 5/4 and 6/5 share a scale degree. Ambulatory[5] has 6/5 and 7/6 share a scale degree.
  • Meantone[7] is a tempering of the 7-note AGS(5/4, 6/5) (Ptolemy's intense diatonic) with the large and medium steps equated. Ambulatory[5] is a tempering of the 5-note AGS(6/5, 7/6) (assuming an equivalence of 3/2) with the medium and small steps equated.

Ambulatory was first documented by CompactStar who originally proposed the name emergency, which refers to the North American emergency telephone number 911, because 9\11edf is a possible generator. This was deemed as too dark so the name ambulatory" was suggested by Frostburn to more subtly reference the same thing.

Subgroup: 3/2.5/4.7/4

Comma list: 686/675 ([-3 -2 3⟩)

Subgroup-val mapping: [⟨1 0 1], ⟨0 3 2]]]

mapping generators: ~3/2, ~15/14

Optimal tunings:

  • Subgroup WE: ~3/2 = 701.8452 ¢, ~15/14 = 129.7976 ¢
  • Subgroup CWE: ~3/2 = 701.9550 ¢, ~15/14 = 129.5324 ¢

Optimal ET sequence: *5, *11, *27, *38

* Wart for 3/2

Poseidon

Todo: rename, resolve temperament name conflict

Subgroup: 3/2.5/4.11/8

Comma list: 121/120 ([-1 -1 2⟩)

Subgroup-val mapping: [⟨1 1 1], ⟨0 -2 -1]]]

mapping generators: ~3/2, ~12/11

Optimal tunings:

  • Subgroup WE: ~3/2 = 701.8452 ¢, ~12/11 = 157.8337 ¢
  • Subgroup CWE: ~3/2 = 701.9550 ¢, ~12/11 = 157.8574 ¢

Optimal ET sequence: *4, *5, *9, *22, *31, *40

* Wart for 3/2

Other 3/2-repeating subgroups

Auk

Auk tempers out the comma 87808/85293 in the 3/2.7.13 subgroup, equating three instances of 7/1 (fifth-reduced to 112/81) with 13/1 (fifth-reduced to 832/729). The generator when reduced is a superfourth in the neighborhood of 550 cents, representing 112/81. It possesses nontrivial mos of the families 1L 3s⟨3/2⟩ ("neptunian"), 4L 1s⟨3/2⟩, 5L 4s⟨3/2⟩, and 9L 5s⟨3/2⟩. 9edf is a particularly good equal temperament that supports it. It contains tempered versions of the chords 1–832/729–112/81 (fifth-reduction of 1:7:13) and 1–63/52–112/81 (fifth-reduction of 1/(13:7:1)).

Subgroup: 3/2.7.13

Comma list: 87808/85293

Subgroup-val mapping: [⟨1 0 -8], ⟨0 1 3]]

mapping generators: ~3/2, ~7

Optimal tunings:

  • Subgroup WE: ~3/2 = 706.5197 ¢, ~112/81 = 538.8834 ¢
  • Subgroup CWE: ~3/2 = 701.9550 ¢, ~112/81 = 543.8799 ¢

Optimal ET sequence: *4, *13, *17df, *21df, *38ddff, *59ddffff

* Wart for 3/2

Doubleton

Subgroup: 3/2.7.13

Comma list: 1352/1323 ([-3 -2 2⟩)

Subgroup-val mapping: [⟨2 0 3], ⟨0 1 1]]

mapping generators: ~26/21, ~7

Optimal tunings:

  • Subgroup WE: ~26/21 = 352.6518 ¢, ~112/81 = 552.6017 ¢
  • Subgroup CWE: ~26/21 = 350.9775 ¢, ~112/81 = 565.7161 ¢

Optimal ET sequence: *4, *10, *14f, *46dfff

* Wart for 3/2

5/2-repeating subgroups

Hyperion

Subgroup: 5/2.7.11

Comma list: 341796875/329832448 ([11 1 -5⟩)

Subgroup-val mapping: [⟨1 -1 2], ⟨0 5 1]]

mapping generators: ~5/2, ~44/25

Optimal tunings:

  • Subgroup WE: ~5/2 = 1582.2958 ¢, ~44/25 = 990.1327 ¢
  • Subgroup CWE: ~5/2 = 1586.3137 ¢, ~44/25 = 991.6570 ¢

Optimal ET sequence: *8, *83…, *91…, …, *155…, *163…

* Wart for 5/2

See also

References

  1. ↑ Yahoo! Tuning Group | Ridiculously awesome 6&11 2.3.7.11.13 subgroup temperament
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