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 =
== 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 ====
==== 2.3.25 subgroup ====
The temperament is simplest in this subgroup, although there is a tradeoff of breaking up the simplicity of the 36/35 quartertone.
[[Subgroup]]: 2.3.25


[[Subgroup]]: 2.3.11.19
[[Comma list]]: 2048/2025


{{Mapping|legend=3| 4 5 13 18 | 0 8 5 -6 }}
{{Mapping|legend=3| 1 0 11 | 0 1 -4 }}
: mapping generators: ~2, ~3


: sval mapping generators: ~6291456/5285401, ~25289/24576
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1198.8214{{c}}, ~3/2 = 704.2059{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.9585{{c}}


[[Optimal tuning]] ([[CTE]]): ~6291456/5285401 = 1\4, ~25289/24576 = 50.257
{{Optimal ET sequence|legend=1| 12, 17, 46, 63, 80, 143* }}


[[Support]]ing [[ET]]s: {{EDOs|24, 596, 620, 644, 668, 692, 716}}, ...
<nowiki/>* Wart for 25


==== 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, 97*, 114* }}


=== Shrub ===
<nowiki/>* Wart for 25
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
Badness (Sintel): 0.223


Edo join: 17 & 12
===== 2.3.25.23.41 subgroup =====
{{See also| Reversed meantone }}


Comma list: [[2048/2025]]
Subgroup: 2.3.25.23.41


{{Mapping|legend=3| 1 1 7| 0 1 -4}}
Comma list: 82/81, 369/368, 576/575


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 705.136
{{Mapping|legend=2| 1 0 11 -5 -1 | 0 1 -4 6 4 }}


==== 2.3.23.25.41 subgroup ====
Optimal tunings:
''See also: [[Reversed meantone]]''
* WE: ~2 = 1199.3246{{c}}, ~3/2 = 705.2238{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 705.5491{{c}}


Edo join: 17 & 12
{{Optimal ET sequence|legend=0| 5, 12, 17 }}


Comma list: 2048/2025, 576/575, 82/81
Badness (Sintel): 0.284


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


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 705.264
Subgroup: 2.3.25.7


===== Sburb =====
Comma list: 50/49, 64/63
This temperament sets the [[octave reduction|octave-reduced]] 413th harmonic (413/256, 827.998{{c}}) to the diminished seventh.


Subgroup: 2.3.7.23.25.41.59
{{Mapping|legend=2| 1 0 11 6 | 0 1 -4 -2 }}


Edo join: 17 & 12
Optimal tunings:  
* WE: ~2 = 1197.6967{{c}}, ~3/2 = 705.6906{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 707.3438{{c}}


Comma list: 64/63, 225/224, 162/161, 82/81, 177/175
{{Optimal ET sequence|legend=0| 5, 12, 17, 39d }}


{{Mapping|legend=3| 1 1 4 1 7 3 10| 0 1 -2 6 -4 4 -7}}
Badness (Sintel): 0.253


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 706.387
===== 2.3.25.7.23 subgroup =====
Subgroup: 2.3.25.7.23


== 2.9.… subgroups ==
Comma list: 50/49, 64/63, 162/161
See also [[Jubilismic clan #Antikythera|antikythera]] and [[Hemimean clan #Isra|isra]].


=== Commatose ===
{{Mapping|legend=2| 1 0 11 6 -5 | 0 1 -4 -2 6 }}
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]].


[[Subgroup]]: 2.9.5.7
Optimal tunings:  
* WE: ~2 = 1197.5577{{c}}, ~3/2 = 705.3089{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 706.7010{{c}}


[[Comma list]]: {{monzo| 28 -2 -19 8 }}, {{monzo| 9 -25 23 6 }}
{{Optimal ET sequence|legend=0| 5, 12, 17 }}


{{Mapping|legend=3| 1 9 6 13 | 0 -298 -188 -521 }}
Badness (Sintel): 0.419


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~531441/524288 = 23.4765
===== 2.3.25.7.23.41 subgroup =====
Subgroup: 2.3.25.7.23.41


{{Optimal ET sequence|legend=1| 460, 869, 1329 }}
Comma list: 50/49, 64/63, 82/81, 162/161


[[Badness]]: 0.611
{{Mapping|legend=2| 1 0 11 6 -5 -1 | 0 1 -4 -2 6 4 }}


==== 2.9.5.7.11 ====
Optimal tunings:
Subgroup: 2.9.5.7.11
* WE: ~2 = 1197.9403{{c}}, ~3/2 = 705.9522{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 707.0779{{c}}


Comma list: {{monzo| -7 7 -3 2 -4 }}, {{monzo| 17 0 -13 1 3 }}, {{monzo| 11 -2 -6 7 -3 }}
{{Optimal ET sequence|legend=0| 5, 12, 17, 39d }}


Sval mapping: {{mapping| 1 9 6 13 16 | 0 -298 -188 -521 -641 }}
Badness (Sintel): 0.450


Optimal tuning (CTE): ~2 = 1\1, ~531441/524288 = 23.4767
=== Hypnosis ===
Related temperaments: [[swetismic temperaments #Hypnos|hypnos]], [[alphatricot family #Alphatrimot|alphatrimot]].  


{{Optimal ET sequence|legend=1| 460, 869e, 1329, 1789, 3118 }}
[[Subgroup]]: 2.3.7.11/5.13


Badness: 0.165
[[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 }})


==== 2.9.5.7.11.13 ====
{{Mapping|legend=3| 1 0 -3 8 0 | 0 3 11 -13 7 }}
Subgroup: 2.9.5.7.11.13
: mapping generators: ~2, ~13/9


Comma list: 123201/123200, 1016064/1015625, 2250423/2249390, 2599051/2598156
[[Optimal tuning]]s:  
* [[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}}


Sval mapping: {{mapping| 0 9 6 13 16 10 | -298 -188 -521 -641 -322 }}
{{Optimal ET sequence|legend=1| 17, 36, 125f, 161f, 197f }}


Optimal tuning (CTE): ~2 = 1\1, ~3575/3528 = 23.4767
[[Tp tuning #T2 tuning|RMS error]]: 0.5379 cents


{{Optimal ET sequence|legend=1| 460, 869e, 1329, 1789, 3118 }}
=== 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]].


Badness: 0.0564
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.


=== Daemotertiaschis ===
Darian calendar was named by [[Eliora]] in 2023 after a certain calendar layout by the same name.  
{{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
[[Subgroup]]: 2.3.35.11.19


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


{{Mapping|legend=2|1 1 11 -16 13 -18 20|0 3 -12 26 -11 30 -22}}
{{Mapping|legend=3| 4 5 18 13 18 | 0 8 15 5 -6 }}
: mapping generators: ~2240/1881, ~36/35


Optimal tuning (CTE): ~2 = 1\1, 33/20 = 867.982
[[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}}


[[Support]]ing [[ET]]s: {{Optimal ET sequence|47, 65f, 112, 159, 206, 253}}
{{Optimal ET sequence|legend=1| 24, …, 524, 548, 572, 596, 620, 644, 668 }}


=== Baldy ===
=== Hydrothermal ===
{{See also|Schismatic family #Garibaldi}}
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.
{{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


[[Subgroup]]: 2.9.5.7
[[Comma list]]: 50/49 ({{monzo| 1 0 -2 }})


[[Comma list]]: 225/224, 3125/3087
{{Mapping|legend=3| 2 0 1 | 0 1 0 }}
: mapping generators: ~7/5, ~3


{{Mapping|legend=3| 1 3 3 4 | 0 1 -4 -7 }}
[[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]] ([[POTE]]): ~2 = 1\1, ~9/8 = 204.170
{{Optimal ET sequence|legend=1| 2, 6, 8, 10, 12, 34, 46* }}


{{Optimal ET sequence|legend=1| 6, 29, 35, 41, 47 }}
<nowiki/>* Wart for 7/5


Related temperament: [[Schismatic family #Garibaldi|Garibaldi]]
=== Argentic ===
Argentic is the 2.3.7/5-subgroup temperament tempering out 5120/5103, the [[aberschisma]].


==== 2.9.5.7.13 ====
[[Subgroup]]: 2.3.7/5
{{See also|Chromatic pairs #Baldy}}


Baldy is every other step of [[garibaldi]], without the mapping of prime 11. It can be described as the 6 &amp; 35 temperament.
[[Comma list]]: 5120/5103 ({{monzo| 10 -6 -1 }})


[[Subgroup]]: 2.9.5.7.13
{{Mapping|legend=3| 1 0 10 | 0 1 -6 }}
: mapping generators: ~2, ~3


[[Comma list]]: [[225/224]], [[325/324]], [[640/637]]
[[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=3| 1 0 15 25 -28 | 0 1 -4 -7 10 }}
{{Optimal ET sequence|legend=1| 12, 29, 41, 70, 321, 391, 461, 531, 601 }}


{{Mapping|legend=5| 1 3/2 3 4 0 2 | 0 1/2 -4 -7 0 10 }}
[[Badness]] (Sintel): 0.119


: [[gencom]]: [2 9/8; 225/224 325/324 640/637]
==== Edson ====
{{See also| Chromatic pairs #Edson }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 204.090
Edson is related to [[aberschismic family #Pele|pele]] and [[schismatic family #Andromeda|andromeda]].  


{{Optimal ET sequence|legend=1| 6, 11, 17, 23, 29, 35, 41, 47, 100, 147, 488cd, 635cd }}
[[Subgroup]]: 2.3.7/5.11/5.13/5


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


Related temperament: [[Schismatic family #Garibaldi|Cassandra]]
{{Mapping|legend=3| 1 0 10 17 22 | 0 1 -6 -10 -13 }}


==== Baldanders ====
{{Mapping|legend=5| 1 0 -49/4 -9/4 19/4 39/4 | 0 1 29/4 5/4 -11/4 -23/4 }}
Baldanders results from taking every other generator of the andromeda, with mapping 11/8 to -9 whole tones.


Subgroup: 2.9.5.7.11
[[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}}


Comma list: 100/99, 225/224, 245/242
{{Optimal ET sequence|legend=1| 12, 17, 29 }}


{{Mapping|legend=3| 1 3 3 4 5 | 0 1 -4 -7 -9 }}
[[Tp tuning #T2 tuning|RMS error]]: 0.5102 cents


Optimal tuning (POTE): ~2 = 1\1, ~9/8 = 204.743
==== Haumea ====
{{See also| Chromatic pairs #Haumea }}


{{Optimal ET sequence|legend=1| 6, 23de, 29, 35, 41 }}
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]].


Related temperament: [[Schismatic family #Garibaldi|Andromeda]]
[[Subgroup]]: 2.3.7/5.11/5.13/5


===== 2.9.5.7.11.13 =====
[[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 }})
Subgroup: 2.9.5.7.11.13


Comma list: 100/99, 144/143, 225/224, 245/242
{{Mapping|legend=3| 1 0 10 -6 -1 | 0 2 -12 9 3 }}


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


Optimal tuning (POTE): ~2 = 1\1, ~9/8 = 204.414
[[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 ET sequence|legend=1| 6, 23def, 29f, 35, 41, 47 }}
{{Optimal ET sequence|legend=1| 24, 29, 111, 140, 169, 198 }}


=== Glacial ===
[[Tp tuning #T2 tuning|RMS error]]: 0.2668 cents
{{See also| Chromatic pairs #Glacial }}


[[Subgroup]]: 2.9.5.11.13
=== Historical ===
{{Distinguish|Historical temperaments}}
{{Distinguish|History (temperament)}}


[[Comma list]]: 45/44, 65/64, 81/80
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.


{{Mapping|legend=3| 1 0 -4 -6 10 | 0 1 2 3 -2 }}
[[Subgroup]]: 2.3.7/5.11/5.13/5


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


: [[gencom]]: [2 9/8; 45/44 65/64 81/80]
{{Mapping|legend=3| 1 2 0 1 2 | 0 -6 7 2 -9 }}
: mapping generators: ~2, ~21/20


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 186.151
[[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 ET sequence|legend=1| 14, 29, 101, 130, 159 }}


{{Optimal ET sequence|legend=1| 6, 13, 45be, 58bce, 71bce, 84bce }}
[[Tp tuning #T2 tuning|RMS error]]: 0.2562 cents


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


Music:
[[Subgroup]]: 2.3.49/5
* ''[[Thundersnow]]'' - [[Sevish]] (2021)


=== Mabon ===
[[Comma list]]: 2401/2400 ({{monzo| -5 -1 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| 1 1 3 | 0 2 1 }}
: mapping generators: ~2, ~49/40


Comma basis: 44957696/43046721
[[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}}


Sval mapping: [{{val|1 1 -3}}, {{val|0 3 8}}]
{{Optimal ET sequence|legend=1| 7, 17, 24, 41, 65, 106, 253, 359, 465, 1036, 1501, 1966*, 5433* }}


Optimal tuning (CTE): ~729/448 = 870.792
<nowiki/>* Wart for 49/5


{{Optimal ET sequence|legend=1|7d, 11, 18d, 29, 40, 62}}, ...
==== Tobago ====
{{See also| Chromatic pairs #Tobago }}


==== 2.9.7.11 subgroup ====
Tobago, the 10 & 14 temperament in the 2.3.11.13/5 subgroup, extends [[rastmic clan #Neutral|neutral]] and [[#Barbados|barbados]].  
Subgroup: 2.9.7.11


Comma basis: 896/891, 1331/1296
[[Subgroup]]: 2.3.11.13/5


Sval mapping: [{{val|1 1 -3 2}}, {{val|0 3 8 2}}]
[[Comma list]]: [[243/242]] ({{monzo| -1 5 -2 0 }}), [[676/675]] ({{monzo| 2 -3 0 2 }})


Optimal tuning (CTE): ~16/11 = 870.966
{{Mapping|legend=3| 2 0 -1 -2 | 0 2 5 3 }}


{{Optimal ET sequence|legend=1| 7d, 11, 40, 51, 62 }}
{{Mapping|legend=5| 2 0 -5 0 -1 -1 | 0 2 -3/2 0 5 3/2 }}
: mapping generators: ~55/39, ~26/15


=== Apparatus ===
[[Optimal tuning]]s:
[[Subgroup]]: 2.9.7.11
* [[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}}


[[Comma list]]: 41503/41472, 322102/321489
{{Optimal ET sequence|legend=1| 10, 14, 24, 82, 106, 130, 154 }}


{{Mapping|legend=3| 1 5 3 5 | 0 -19 -2 -16 }}
[[Tp tuning #T2 tuning|RMS error]]: 0.3533 cents


: mapping generators: ~2, ~77/72
==== Pakkanian hemipyth ====
[[Subgroup]]: 2.3.11.13/5.17


{{Mapping|legend=5| 1 5/2 0 3 5 | 0 -19/2 0 -2 -16 }}
[[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 }})


: [[gencom]]: [2 77/72; 41503/41472 322102/321489]
{{Mapping|legend=3| 2 0 -1 -2 5 | 0 2 5 3 2 }}


[[Optimal tuning]] ([[CTE]]): ~77/72 = 115.5685
[[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}})


{{Optimal ET sequence|legend=1| 10e, 21, 31, 52, 83, 135, 353, 488, 623 }}
{{Optimal ET sequence|legend=1| 10, 14, 24, 106, 130, 154, 178*, 202* }}


[[Badness]]: 0.00263
: <nowiki/>* wart for 13/5


=== Joan ===
=== Bridgetown ===
{{See also| Chromatic pairs #Joan }}
{{See also| Chromatic pairs #Bridgetown }}


Joan is related to [[casablanca]] as well as to [[orwell]].  
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]].  


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


[[Comma list]]: 99/98, 9317/9216
[[Comma list]]: [[352/351]] ({{monzo| 5 -3 1 -1 }}), [[676/675]] ({{monzo| 2 -3 0 2 }})


{{Mapping|legend=3| 1 0 1 3 | 0 7 4 1 }}
{{Mapping|legend=3| 1 0 -6 -1 | 0 2 9 3 }}


{{Mapping|legend=5| 1 0 0 1 3 | 0 7/2 0 4 1 }}
{{Mapping|legend=5| 1 0 7/3 0 -11/3 4/3 | 0 2 -4 0 5 -1 }}
: mapping generators: ~2, ~26/15


: [[gencom]]: [2 11/8; 99/98 9317/9216]
[[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}}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~11/8 = 542.672 cents
{{Optimal ET sequence|legend=1| 5, 19*, 24, 29, 169, 198, 227, 256, 285, 314, 343 }}


{{Optimal ET sequence|legend=1| 11, 20, 31, 42, 115bd, 157bd }}
<nowiki/>* Wart for 11/5


[[Tp tuning #T2 tuning|RMS error]]: 1.424 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.2513 cents


=== Machine ===
=== Seventeen-cot ===
Machine is every other step of [[supra]], most interesting for its scale patterns.  
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.  


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


[[Comma list]]: 64/63, 99/98
[[Comma basis]]: 43940/43923 ({{monzo| 2 -1 -4 3 }}), 225000/224939 ({{monzo| 3 2 -3 -2 }})


{{Mapping|legend=3| 1 0 6 13 | 0 1 -1 -3 }}
{{Mapping|legend=3| 1 1 1 1 | 0 17 4 11 }}
: mapping generators: ~2, ~169/165


: sval mapping generators: ~2, ~9
[[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 }}


{{Mapping|legend=5| 1 3/2 0 3 4 | 0 1/2 0 -1 -3 }}
{{Optimal ET sequence|legend=1| 29, 262, 291, 320, 349, 378, 407, 436, 465, 959, 1424, 6161* }}


: [[gencom]]: [2 8/7; 64/63 99/98]
[[Badness]] (Sintel): 0.080


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


{{Optimal ET sequence|legend=1| 5, 6, 11, 17, 28 }}
[[Subgroup]]: 2.3.11/7


[[Badness]]: 0.00233
[[Comma list]]: 256/243 ({{monzo| 8 -5 0 }})


=== Penta a.k.a. mechanism ===
{{Mapping|legend=3| 5 8 0 | 0 0 1 }}
Penta or mechanism is the 8 &amp; 11 temperament in the 2.9.7.11 subgroup.
: mapping generators: ~9/8, ~11/7


[[Subgroup]]: 2.9.7.11
[[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 }}


[[Comma list]]: 896/891, 26411/26244
{{Optimal ET sequence|legend=1| 15, 20, 35b, 55b }}


{{Mapping|legend=3| 1 0 -1 6 | 0 5 6 -4 }}
=== Pepperoni ===
{{Main| Parapyth }}
{{See also| Chromatic pairs #Pepperoni }}


: sval mapping generators: ~2, ~14/9
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.


{{Mapping|legend=5| 1 5/2 0 5 2 | 0 -5/2 0 -6 4 }}
[[Subgroup]]: 2.3.11/7.13/7


: [[gencom]]: [2 9/7; 896/891 26411/26244]
[[Comma list]]: 352/351 ({{monzo| 5 -3 1 -1 }}), 364/363 ({{monzo| 2 -1 -2 1 }})


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~14/9 = 761.3782
{{Mapping|legend=3| 1 0 7 12 | 0 1 -4 -7 }}


{{Optimal ET sequence|legend=1| 8, 11, 30, 41, 52 }}
{{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


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


[[Badness]]: 0.00439
{{Optimal ET sequence|legend=1| 12, 17, 29, 75, 104 }}


Scales: [[penta5]], [[penta8]], [[penta11]], [[penta19]]
[[Tp tuning #T2 tuning|RMS error]]: 0.3789 cents


=== Novisept ===
=== Fendo ===
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]].
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''.  


[[Subgroup]]: 2.9.7.13.17
In fendo, the fourth / tendo third serves the role of both, making the main chords of triadic harmony in fendo essentially suspended chords.  


[[Comma list]]: 729/728, 442/441, 833/832
[[5edo]] and [[13edo]] make good fendo tunings.


{{Mapping|legend=3| 1 1 1 -1 3| 0 6 5 13 3 }}
[[Subgroup]]: 2.3.13/5


[[Optimal tuning]] ([[CWE]]): ~2 = 1\1, ~9/7 = 433.836
[[Comma list]]: 40/39 ({{monzo| 3 -1 -1 }})


Badness (Dirichlet): 0.142
{{Mapping|legend=3| 1 0 3 | 0 1 -1 }}
: mapping generators: ~2, ~3


=== Demon ===
[[Optimal tuning]]s:
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.
* [[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}}


[[Subgroup]]: 2.9.11
{{Optimal ET sequence|legend=1| 1, 2, 3, 5, 17*, 22*, 27*, 32* }}


[[Comma list]]: [[1331/1296]]
<nowiki/>* Wart for 13/5


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


[[Optimal tuning]] ([[CTE]]): ~[[18/11]] = 870.060
[[Subgroup]]: 2.3.13/5


{{Optimal ET sequence|legend=1|4, 7, 11, 18, 29, 76e}}
[[Comma list]]: 676/675 ({{monzo| 2 -3 2 }})


=== Genius ===
{{Mapping|legend=3| 1 0 -1 | 0 2 3 }}
: mapping generators: ~2, ~26/15


Named after the genius in Roman religion, following the demon (daimon) in Greek mythology.
[[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}}


[[Subgroup]]: 2.9.11
{{Optimal ET sequence|legend=1| 5, 14, 19, 24, 29, 53, 82, 135 }}


[[Comma list]]: [[131769/131072]]
[[Badness]] (Smith): 0.002335


{{Mapping|legend=2|1 1 4|0 4 -1}}
; 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]


[[Optimal tuning]] ([[CTE]]): ~[[16/11]] = 650.863
=== 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.


{{Optimal ET sequence|legend=1|9, 11, 24, 59, 83, 142, 225, 367}}[-11], 592[-11], 959[-9, --11], 1326[-9, --11]
[[Subgroup]]: 2.3.17/5


=== Stacks (a.k.a. 2magic) ===
[[Comma list]]: 136/135 ({{monzo| 3 -3 1 }})
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
{{Mapping|legend=3| 1 0 -3 | 0 1 3 }}
: mapping generators: ~2, ~3


[[Comma list]]: 225/224, 245/243
[[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}}


{{Mapping|legend=3| 1 0 2 -1 | 0 5 3 6 }}
{{Optimal ET sequence|legend=1| 5, 12, 17, 46, 63, 143 }}


: sval mapping generators: ~2, ~14/9
=== 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.


{{Mapping|legend=5| 1 5/2 5/2 5 | 0 -5/2 -1/2 -6 }}
[[Subgroup]]: 2.3.19/7


: [[gencom]]: [2 9/7; 225/224 245/243]
[[Comma list]]: [[57/56]] ({{monzo| -3 1 1 }})


[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~2 = 1\1, ~14/9 = 760.704
{{Mapping|legend=3| 1 0 3 | 0 1 -1 }}
: mapping generators: ~2, ~3


{{Optimal ET sequence|legend=1| 8, 11, 30, 41, 71, 93, 112c, 134c, 175c }}
[[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}}


[[Tp tuning #T2 tuning|RMS error]]: 1.074 cents
{{Optimal ET sequence|legend=1| 5, 7, 12, 19, 31*, 50* }}


==== 2.9.15.7.11 ====
<nowiki/>* wart for 19/7
Subgroup: 2.9.15.7.11


Comma list: 100/99, 225/224, 245/243
[[Badness]] (Sintel): 0.082


Sval mapping: {{mapping| 1 0 2 -1 6 | 0 5 3 6 -4 }}
== 2.9.… subgroups ==
See also [[Jubilismic clan #Antikythera|antikythera]] and [[Hemimean clan #Isra|isra]].


{{Mapping|legend=4| 1 5/2 5/2 5 2 | 0 -5/2 -1/2 -6 4 }}
=== Commatose ===
Commatose is a 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]].


: gencom: [2 9/7; 100/99 225/224 245/243]
[[Subgroup]]: 2.9.5.7


Optimal tuning (subgroup POTE): ~2 = 1\1, ~14/9 = 761.393
[[Comma list]]: {{monzo| 28 -2 -19 8 }}, {{monzo| 9 -25 23 6 }}


Optimal ET sequence: {{Optimal ET sequence| 8, 11, 30, 41, 52, 93, 145, 342bce }}
{{Mapping|legend=3| 1 -289 -182 -508 | 0 298 188 521 }}
: mapping generators: ~2, ~1048576/531441


RMS error: 1.226 cents
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.9727{{c}}, ~1048576/531441 = 1176.4969{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~1048576/531441 = 1176.5236{{c}}


==== 2.9.15.7.11.13 ====
{{Optimal ET sequence|legend=1| 460, 869, 1329 }}
Subgroup: 2.9.15.7.11.13


Comma list: 100/99, 105/104, 144/143, 196/195
[[Badness]] (Sintel): 30.9


Sval mapping: {{mapping| 1 0 2 -1 6 -2 | 0 5 3 6 -4 9 }}
==== 2.9.5.7.11 ====
Subgroup: 2.9.5.7.11


{{Mapping|legend=4| 1 5/2 5/2 5 2 7 | 0 -5/2 -1/2 -6 4 -9 }}
Comma list: {{monzo| -7 7 -3 2 -4 }}, {{monzo| 17 0 -13 1 3 }}, {{monzo| 11 -2 -6 7 -3 }}


: gencom: [2 9/7; 100/99 105/104 144/143 196/195]
{{Mapping|legend=2| 1 -289 -182 -508 -625 | 0 298 188 521 641 }}


Optimal tuning (subgroup POTE): ~2 = 1\1, ~14/9 = 761.023
Optimal tunings:
* WE: ~2 = 1199.9833{{c}}, ~531441/524288 = 1176.5070{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~531441/524288 = 1176.5234{{c}}


Optimal ET sequence: {{Optimal ET sequence| 11, 30, 41, 153cdef, 194cdef, 235cdef }}
{{Optimal ET sequence|legend=0| 460, 869e, 1329, 1789, 3118 }}


RMS error: 1.540 cents
Badness (Sintel): 8.67


=== A-team ===
==== 2.9.5.7.11.13 ====
A-team is every other step of [[slendric]]; the 2.9.5.21.11 extension below specifically restricts [[mothra]].  
Subgroup: 2.9.5.7.11.13


[[Subgroup]]: 2.9.21
Comma list: 123201/123200, 1016064/1015625, 2250423/2249390, 2599051/2598156


[[Comma list]]: 1029/1024
{{Mapping|legend=2| 0 -289 -182 -508 -625 -312 | 0 298 188 521 641 322 }}


{{Mapping|legend=3| 1 2 4 | 0 3 1 }}
Optimal tunings:
* WE: ~2 = 1200.0000{{c}}, ~7056/3575 = 23.4767{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~7056/3575 = 23.4767{{c}}


: sval mapping generators: ~2, ~21/16
{{Optimal ET sequence|legend=0| 460, 869e, 1329, 1789, 3118 }}


{{Mapping|legend=5| 1 1 0 3 | 0 3/2 0 -1/2 }}
Badness (Sintel): 3.30


: [[gencom]]: [2 21/16; 1029/1024]
=== Daemotertiaschis ===
{{See also|Schismatic family #Tertiaschis}}


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


{{Optimal ET sequence|legend=1| 5, 13, 18, 41, 59, 77, 95 }}
Subgroup: 2.9.5.7.33.13.17


[[Tp tuning #T2 tuning|RMS error]]: 0.3202 cents
Comma list: 325/324, 375/374, 385/384, 595/594, 10985/10976


==== 2.9.5.21 ====
{{Mapping|legend=2| 1 1 11 -16 13 -18 20 | 0 3 -12 26 -11 30 -22 }}
''Lookalike temperament: [[Dual-fifth_temperaments#Dual-3_A-Team|Dual-3 A-Team]]''
: mapping generators: ~2, ~33/20


Subgroup: 2.9.5.21
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}}


[[Comma]] list: 81/80, 1029/1024
{{Optimal ET sequence|legend=1| 47, 159, 206, 253, 459* }}


Sval mapping: {{mapping| 1 2 0 4 | 0 3 6 1 }}
Badness (Sintel): 0.439


Mapping generators: ~2, ~21/16
=== Baldy ===
{{See also| Chromatic pairs #Baldy }}


Optimal ([[Lp tuning|POL2]]) generator: 464.3865
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.


{{Optimal ET sequence|legend=1| 13, 18, 31, 44 }}
[[Subgroup]]: 2.9.5.7


===== 2.9.5.21.11 =====
[[Comma list]]: 225/224, 3125/3087
Subgroup: 2.9.5.21.11


Comma list: 81/80, 99/98, 385/384
{{Mapping|legend=3| 1 0 15 25 | 0 1 -4 -7 }}
: mapping generators: ~2, ~9


Sval mapping: {{mapping| 1 2 0 4 5 | 0 3 6 1 -4 }}
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.1233{{c}}, ~9/8 = 204.1913{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~9/8 = 204.1549{{c}}


{{Mapping|legend=4| 1 1 0 3 5 | 0 3/2 6 -1/2 -4 }}
{{Optimal ET sequence|legend=1| 6, 29, 35, 41, 47, 194 }}


: gencom: [2 21/16; 81/80 99/98 385/384]
[[Badness]] (Sintel): 0.274


Optimal tuning (subgroup POTE): ~2 = 1\1, ~21/16 = 463.956
==== 2.9.5.7.13 subgroup ====
Subgroup: 2.9.5.7.13


{{Optimal ET sequence|legend=1| 5, 13, 31 }}
Comma list: 225/224, 325/324, 640/637


==== B-team ====
{{Mapping|legend=2| 1 0 15 25 -28 | 0 1 -4 -7 10 }}
B-team (23 & 41) is every other step of [[rodan]].


Subgroup: 2.9.15.21.33
{{Mapping|legend=4| 1 0 15 25 0 28 | 0 1/2 -4 -7 0 10 }}


Comma list: 245/243, 385/384, 441/440
Optimal tunings:  
* WE: ~2 = 1200.0155{{c}}, ~9/8 = 204.0931{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/8 = 204.0902{{c}}


Sval mapping: {{mapping| 1 2 0 4 7 | 0 3 10 1 -5 }}
{{Optimal ET sequence|legend=0| 6, 35, 41, 47, 100, 147 }}


Optimal tuning (subgroup POTE): ~2 = 1\1, ~21/16 = 468.918
Badness (Sintel): 0.386


{{Optimal ET sequence|legend=1| 5, 13c, 18, 23, 41, 64, 87, 151 }}
RMS error: 0.5999 cents


== 2.75.… subgroups ==
==== Baldanders ====
=== Archagall ===
Baldanders results from taking every other generator of the [[schismatic family #Andromeda|andromeda]], mapping 11/8 to -9 whole tones.
{{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.
Subgroup: 2.9.5.7.11


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


[[Subgroup]]: 2.75.85
{{Mapping|legend=2| 1 0 15 25 32 | 0 1 -4 -7 -9 }}


[[Comma list]]: [[24576/24565]] ({{monzo| 13 1 -3 }})
Optimal tunings:  
* WE: ~2 = 1200.1917{{c}}, ~9/8 = 204.7755{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/8 = 204.7197{{c}}


{{Mapping|legend=3| 1 2 5 | 0 3 1 }}
{{Optimal ET sequence|legend=0| 6, 23de, 29, 35, 41 }}
: mapping generators: ~2, ~85/32


[[Optimal tuning]]s:  
Badness (Sintel): 0.389
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.9692{{c}}, ~85/64 = 491.5853{{c}}
* [[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* }}
===== 2.9.5.7.11.13 =====
Subgroup: 2.9.5.7.11.13


==== 2.75.9/7.85 subgroup ====
Comma list: 100/99, 144/143, 225/224, 245/242
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
{{Mapping|legend=2| 1 0 15 25 32 -28 | 0 1 -4 -7 -9 10 }}


Comma list: 2025/2023 ({{monzo| 2 -2 1 0 }}), 24576/24565 ({{monzo| 13 1 0 -3 }})
Optimal tunings:  
* WE: ~2 = 1199.6385{{c}}, ~9/8 = 204.3526{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/8 = 204.4234{{c}}


{{Mapping|legend=2| 1 2 6 5 | 0 3 -4 1 }}
{{Optimal ET sequence|legend=0| 6, 35, 41, 47, 88e }}


Optimal tunings:  
Badness (Sintel): 0.653
* 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* }}
=== Glacial ===
{{See also| Chromatic pairs #Glacial }}


== 4.3.… subgroups ==
[[Subgroup]]: 2.9.5.11.13
=== Tetrahanson ===
{{Main| Tetrahanson }}


[[Subgroup]]: 4.3.5
[[Comma list]]: 45/44, 65/64, 81/80


[[Comma list]]: 15625/15552
{{Mapping|legend=3| 1 0 -4 -6 10 | 0 1 2 3 -2 }}


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


: Mapping generators: ~4, ~5/3
: [[gencom]]: [2 9/8; 45/44 65/64 81/80]


[[Optimal tuning]] ([[CTE]]): ~4 = 2\1, ~5/3 = 882.941
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 186.151


[[Support]]ing [[ET]]s: {{EDs|19, 106, 87, 68, 11, 8, 125, 49, 30, 27, 117, 46, 41b, 79|equave=4}}
{{Optimal ET sequence|legend=1| 6, 13, 45be, 58bce, 71bce, 84bce }}


=== Tetrameantone ===
[[Tp tuning #T2 tuning|RMS error]]: 2.887 cents
{{Main| Tetrameantone }}


[[Subgroup]]: 4.3.5
Music:
* ''[[Thundersnow]]'' by [[Sevish]] (2021)


[[Comma list]]: 81/80
=== Mabon ===
Derived from a [http://individual.utoronto.ca/kalendis/leap/index.htm#se calendar leap cycle built for the autumn equinox], hence the name. Defined as the 11 & 62 temperament.


{{Mapping|legend=3| 1 1 2 | 0 -1 -4 }}
Subgroup: 2.9.7


: Mapping generators: ~4, ~4/3
Comma basis: 44957696/43046721


[[Optimal tuning]] ([[POTE]]): 4 = 2400.0, ~4/3 = 503.761
Sval mapping: [{{val|1 1 -3}}, {{val|0 3 8}}]


[[Support]]ing [[ET]]s: {{EDs|5, 9, 14, 19, 24, 43, 62, 81, 100|equave=4}}
Optimal tuning (CTE): ~729/448 = 870.792


=== Tetramagic ===
{{Optimal ET sequence|legend=1|7d, 11, 18d, 29, 40, 62}}, ...


[[Subgroup]]: 4.3.5
==== 2.9.7.11 subgroup ====
Subgroup: 2.9.7.11


[[Comma list]]: 3125/3072
Comma basis: 896/891, 1331/1296


{{Mapping|legend=3| 1 0 1 | 0 5 1 }}
Sval mapping: [{{val|1 1 -3 2}}, {{val|0 3 8 2}}]


: Mapping generators: ~4, ~5/4
Optimal tuning (CTE): ~16/11 = 870.966


[[Optimal tuning]] ([[POTE]]): 4 = 2400.0, ~5/4 = 380.059
{{Optimal ET sequence|legend=1| 7d, 11, 40, 51, 62 }}


[[Support]]ing [[ET]]s: {{EDs|6, 13, 19, 25, 38, 44, 63, 82|equave=4}}
=== Apparatus ===
[[Subgroup]]: 2.9.7.11


=== Blacktetra ===
[[Comma list]]: 41503/41472, 322102/321489


[[Subgroup]]: 4.3.5
{{Mapping|legend=3| 1 5 3 5 | 0 -19 -2 -16 }}


[[Comma list]]: 256/243
: mapping generators: ~2, ~77/72


{{Mapping|legend=3| 5 4 6 | 0 0 -1 }}
{{Mapping|legend=5| 1 5/2 0 3 5 | 0 -19/2 0 -2 -16 }}


: Mapping generators: ~4, ~16/15
: [[gencom]]: [2 77/72; 41503/41472 322102/321489]


[[Optimal tuning]] ([[POTE]]): 1\5ed4 = 480.0, ~16/15 = 80.4062
[[Optimal tuning]] ([[CTE]]): ~77/72 = 115.5685


[[Support]]ing [[ET]]s: {{EDs|5, 10, 15, 20, 25, 30, 55, 85, 115|equave=4}}
{{Optimal ET sequence|legend=1| 10e, 21, 31, 52, 83, 135, 353, 488, 623 }}


== 4.6.… subgroups ==
[[Badness]]: 0.00263
=== Meanquad ===
{{Main| Meanquad }}


[[Subgroup]]: 4.6.5
=== Joan ===
{{See also| Chromatic pairs #Joan }}


[[Comma list]]: [[81/80]] = {{monzo| -4 4 -1 }}
Joan is related to [[casablanca]] as well as to [[orwell]].


{{Mapping|legend=3| 1 0 -4| 0 1 4 }}
[[Subgroup]]: 2.9.7.11


: mapping generators: ~4, ~6
[[Comma list]]: 99/98, 9317/9216


[[Optimal tuning]] (subgroup [[CTE]]): ~4 = 2\1, ~3/2 = 697.214
{{Mapping|legend=3| 1 0 1 3 | 0 7 4 1 }}


[[Support]]ing [[ET]]s: *7, *10, *11[-5], *13[+5], *17, *24, *27[+5], *31, *38, *41, *45, *52, *55, *69
{{Mapping|legend=5| 1 0 0 1 3 | 0 7/2 0 4 1 }}


<nowiki />* Wart for 4
: [[gencom]]: [2 11/8; 99/98 9317/9216]


==== 4.6.5.7 subgroup (tetrominant) ====
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~11/8 = 542.672 cents
[[Subgroup]]: 4.6.5.7


[[Comma list]]: [[36/35]] = {{monzo| 0 2 -1 -1 }}, [[64/63]] = {{monzo| 4 -2 0 -1 }}
{{Optimal ET sequence|legend=1| 11, 20, 31, 42, 115bd, 157bd }}


{{Mapping|legend=3| 1 0 -4 4 | 0 1 4 -2 }}
[[Tp tuning #T2 tuning|RMS error]]: 1.424 cents


[[Optimal tuning]] (subgroup [[CTE]]): ~4 = 2\1, ~3/2 = 699.622
=== Machine ===
Machine is every other step of [[supra]], most interesting for its scale patterns.
 
[[Subgroup]]: 2.9.7.11
 
[[Comma list]]: 64/63, 99/98


[[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]
{{Mapping|legend=3| 1 0 6 13 | 0 1 -1 -3 }}


<nowiki />* Wart for 4
: sval mapping generators: ~2, ~9


=== Fourwar ===
{{Mapping|legend=5| 1 3/2 0 3 4 | 0 1/2 0 -1 -3 }}
The 23-limit version of Fourwar was created first, as an attempt to approximate subgroup 4.6.5.7.11.13.17.19.23 as accurately as possible using 25 to 35 notes per equave. Then the lower limit versions were created by simply extrapolating the temperament downwards.


Fourwar is named after the closely related [[hemiwar]] temperament.
: [[gencom]]: [2 8/7; 64/63 99/98]


{{Todo|inline=1|cleanup}}
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~9/8 = 216.9128
* [[POTE]]: ~2 = 1\1, ~9/8 = 214.3843


<pre>
{{Optimal ET sequence|legend=1| 5, 6, 11, 17, 28 }}
Reduced Mapping
4 6 5
[ ⟨ 1 0 1 ]
⟨ 0 16 2 ] ⟩
TE Generator Tunings (cents)
⟨2399.3973, 193.8643]
TE Step Tunings (cents)
⟨25.21211, 47.81337]
TE Tuning Map (cents)
⟨2399.397, 3101.829, 2787.126]
TE Mistunings (cents)
⟨-0.603, -0.126, 0.812]
Complexity 1.369085
Adjusted Error 0.692892 cents
TE Error 0.268047 cents/octave
Unison Vector
[8, 1, -8⟩ (393216:390625)


Subsets
[[Badness]]: 0.00233
q99, q62, q37, q161, q136, q198, q25, q124, q74, q235
</pre>


==== 4.6.5.7 ====
=== Penta a.k.a. mechanism ===
<pre>
Penta or mechanism is the 8 &amp; 11 temperament in the 2.9.7.11 subgroup.  
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
[[Subgroup]]: 2.9.7.11
q99, q62, q37, q161, q136, q198, q25, q124, q74, q235
 
</pre>
[[Comma list]]: 896/891, 26411/26244
 
{{Mapping|legend=3| 1 0 -1 6 | 0 5 6 -4 }}
 
: sval mapping generators: ~2, ~14/9
 
{{Mapping|legend=5| 1 5/2 0 5 2 | 0 -5/2 0 -6 4 }}
 
: [[gencom]]: [2 9/7; 896/891 26411/26244]


==== 4.6.5.7.11 ====
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~14/9 = 761.3782
<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
{{Optimal ET sequence|legend=1| 8, 11, 30, 41, 52 }}
q37, q25, q62, q12, q74, q99, q87, q49r, q50r, q124
</pre>


==== 4.6.5.7.11.13 ====
[[Tp tuning #T2 tuning|RMS error]]: 0.4262 cents


<pre>
[[Badness]]: 0.00439
Reduced Mapping
 
4 6 5 7 11 13
Scales: [[penta5]], [[penta8]], [[penta11]], [[penta19]]
[ ⟨ 1 0 1 1 1 0 ]
 
⟨ 0 16 2 5 9 23 ] ⟩
=== Novisept ===
Novisept is generated by a one-cent-flat 9/7, such that stacking 5 of them gives you 7/4. It can be formed by doubling both generator and period of [[gizzard]].
TE Generator Tunings (cents)
 
⟨2401.2305, 193.5378]
[[Subgroup]]: 2.9.7.13.17
 
TE Step Tunings (cents)
[[Comma list]]: 729/728, 442/441, 833/832
⟨42.79107, 35.98524]
 
{{Mapping|legend=3| 1 1 1 -1 3| 0 6 5 13 3 }}
TE Tuning Map (cents)
 
⟨2401.230, 3096.606, 2788.306, 3368.920, 4143.071, 4451.371]
[[Optimal tuning]] ([[CWE]]): ~2 = 1\1, ~9/7 = 433.836
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
Badness (Dirichlet): 0.142
q25, q37f, q12f, q62, q50rf, q13rff, q49rff, q87, q74ff, q24rfff
</pre>


==== 4.6.5.7.11.13.17 ====
=== Demon ===
<pre>
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.
Reduced Mapping
 
4 6 5 7 11 13 17
[[Subgroup]]: 2.9.11
[ ⟨ 1 0 1 1 1 0 1 ]
 
⟨ 0 16 2 5 9 23 13 ] ⟩
[[Comma list]]: [[1331/1296]]
 
TE Generator Tunings (cents)
{{Mapping|legend=2|1 1 2|0 3 2}}
⟨2400.4701, 193.4599]
 
[[Optimal tuning]] ([[CTE]]): ~[[18/11]] = 870.060
TE Step Tunings (cents)
 
⟨43.39350, 35.55764]
{{Optimal ET sequence|legend=1|4, 7, 11, 18, 29, 76e}}
 
TE Tuning Map (cents)
=== Genius ===
⟨2400.470, 3095.359, 2787.390, 3367.770, 4141.609, 4449.578, 4915.449]
 
Named after the genius in Roman religion, following the demon (daimon) in Greek mythology.
TE Mistunings (cents)
 
⟨0.470, -6.596, 1.076, -1.056, -9.709, 9.050, 10.494]
[[Subgroup]]: 2.9.11
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
[[Comma list]]: [[131769/131072]]
q25, q12f, q37f, q13rffg, q50rf, q62, q49rffg, q24rfffg, q38rreffg, q74ffg
</pre>


==== 4.6.5.7.11.13.17.19 ====
{{Mapping|legend=2|1 1 4|0 4 -1}}
<pre>
 
Reduced Mapping
[[Optimal tuning]] ([[CTE]]): ~[[16/11]] = 650.863
4 6 5 7 11 13 17 19
 
[ ⟨ 1 0 1 1 1 0 1 1 ]
{{Optimal ET sequence|legend=1|9, 11, 24, 59, 83, 142, 225, 367}}[-11], 592[-11], 959[-9, --11], 1326[-9, --11]
⟨ 0 16 2 5 9 23 13 14 ] ⟩
 
=== Stacks (a.k.a. 2magic) ===
TE Generator Tunings (cents)
Stacks, the 11 &amp; 30 temperament in the 2.9.15.7.11.13 subgroup, is every other step of [[magic]].  
⟨2399.9219, 193.3952]
 
[[Subgroup]]: 2.9.15.7
TE Step Tunings (cents)
 
⟨44.14256, 35.03670]
[[Comma list]]: 225/224, 245/243
 
TE Tuning Map (cents)
{{Mapping|legend=3| 1 0 2 -1 | 0 5 3 6 }}
⟨2399.922, 3094.324, 2786.712, 3366.898, 4140.479, 4448.090, 4914.060, 5107.455]
 
: sval mapping generators: ~2, ~14/9
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
{{Mapping|legend=5| 1 5/2 5/2 5 | 0 -5/2 -1/2 -6 }}
q25, q12fh, q37f, q13rffgh, q50rf, q62, q49rffgh, q24rfffghh, q38rreffgh, q74ffgh
</pre>


==== 4.6.5.7.11.13.17.19.23 ====
: [[gencom]]: [2 9/7; 225/224 245/243]
<pre>
 
Reduced Mapping
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~2 = 1\1, ~14/9 = 760.704
4 6 5 7 11 13 17 19 23
 
[ ⟨ 1 0 1 1 1 0 1 1 0 ]
{{Optimal ET sequence|legend=1| 8, 11, 30, 41, 71, 93, 112c, 134c, 175c }}
⟨ 0 16 2 5 9 23 13 14 28 ] ⟩
 
[[Tp tuning #T2 tuning|RMS error]]: 1.074 cents
TE Generator Tunings (cents)
 
⟨2399.3286, 193.5316]
==== 2.9.15.7.11 ====
Subgroup: 2.9.15.7.11
TE Step Tunings (cents)
 
⟨37.31613, 39.63311]
Comma list: 100/99, 225/224, 245/243
 
TE Tuning Map (cents)
Sval mapping: {{mapping| 1 0 2 -1 6 | 0 5 3 6 -4 }}
⟨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
{{Mapping|legend=4| 1 5/2 5/2 5 2 | 0 -5/2 -1/2 -6 4 }}
q25i, q12fhi, q37f, q13rffghii, q62, q50rfii, q49rffghii, q24rfffghhiii, q74ffghi, q38rreffghiii
</pre>


== 4.9.… subgroups ==
: gencom: [2 9/7; 100/99 225/224 245/243]
=== Meansquared ===
[[Subgroup]]: 4.9.25


[[Comma list]]: [[6561/6400]]
Optimal tuning (subgroup POTE): ~2 = 1\1, ~14/9 = 761.393


{{Mapping|legend=3| 1 3 4 | 0 1 4 }}
Optimal ET sequence: {{Optimal ET sequence| 8, 11, 30, 41, 52, 93, 145, 342bce }}


Mapping generators: ~4, ~9/64
RMS error: 1.226 cents


[[Optimal tuning]] ([[CTE]]): ~4 = 2\1, ~9/4 = 1394.429
==== 2.9.15.7.11.13 ====
Subgroup: 2.9.15.7.11.13


[[Support]]ing [[ET]]s: 12, 7, 19, 5, 31, 26, 17[+25], 43, 9[-25], 33[-25], 45, 29[+25], 8[+25], 22[+25]
Comma list: 100/99, 105/104, 144/143, 196/195


=== Archsquared ===
Sval mapping: {{mapping| 1 0 2 -1 6 -2 | 0 5 3 6 -4 9 }}
[[Subgroup]]: 4.9.49


[[Comma list]]: 4096/3969
{{Mapping|legend=4| 1 5/2 5/2 5 2 7 | 0 -5/2 -1/2 -6 4 -9 }}


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


Mapping generators: ~4, ~9/64
Optimal tuning (subgroup POTE): ~2 = 1\1, ~14/9 = 761.023


[[Optimal tuning]] ([[CTE]]): ~9/4 = 1419.190
Optimal ET sequence: {{Optimal ET sequence| 11, 30, 41, 153cdef, 194cdef, 235cdef }}


[[Support]]ing [[ET]]s: 5, 17, 22, 12, 7, 27, 32, 8, 39[+49], 29[+49], 9[+49], 19[+49], 37, 49
RMS error: 1.540 cents


== 8.9.… subgroups ==
=== A-team ===
=== Sixscared ===
A-team is every other step of [[slendric]]; the 2.9.5.21.11 extension below specifically restricts [[mothra]].
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
[[Subgroup]]: 2.9.21


[[Comma list]]: 64/63
[[Comma list]]: 1029/1024


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


: sval mapping generators: ~8, ~9
: sval mapping generators: ~2, ~21/16


: [[gencom]]: [8 9/8; 64/63]
{{Mapping|legend=5| 1 1 0 3 | 0 3/2 0 -1/2 }}


[[Optimal tuning]] ([[CTE]]): ~9/8 = 219.1898
: [[gencom]]: [2 21/16; 1029/1024]


[[Optimal ET sequence]]: {{val| 16 17 15 }}, {{val| 33 35 31 }}, {{val| 148 … }}, {{val| 181 … }}, {{val| 214 … }}, {{val| 247 … }}
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~2 = 1\1, ~21/16 = 467.375


[[Badness]]: 0.0215 × 10<sup>-3</sup>
{{Optimal ET sequence|legend=1| 5, 13, 18, 41, 59, 77, 95 }}


= Fractional subgroup temperaments =
[[Tp tuning #T2 tuning|RMS error]]: 0.3202 cents
== 2.3.… subgroups ==
=== Hypnosis ===
Related temperaments: [[Swetismic temperaments #Hypnos|hypnos]], [[Alphatricot family #Alphatricot|alphatricot]]


[[Subgroup]]: 2.3.7.11/5.13
==== 2.9.5.21 ====
''Lookalike temperament: [[Dual-fifth_temperaments#Dual-3_A-Team|Dual-3 A-Team]]''


[[Comma list]]: 169/168, 540/539, 729/728
Subgroup: 2.9.5.21


{{Mapping|legend=3| 1 0 -3 8 0 | 0 3 11 -13 7 }}
[[Comma]] list: 81/80, 1029/1024


[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~13/9 = 633.518
Sval mapping: {{mapping| 1 2 0 4 | 0 3 6 1 }}


{{Optimal ET sequence|legend=1| 17, 36, 118f, 125f, 161f, 197f }}
Mapping generators: ~2, ~21/16


[[Tp tuning #T2 tuning|RMS error]]: 0.5379 cents
Optimal ([[Lp tuning|POL2]]) generator: 464.3865


=== Hydrothermal ===
{{Optimal ET sequence|legend=1| 13, 18, 31, 44 }}
A tuning whose distinctively sharp (but still consonant) fifth, and flat (but still consonant) octave, lend it a mysterious, heavy atmosphere. The 6-tone (hexatonic) MOS is melodically interesting and flavorful. The 18-tone MOS is a useful 'chromatic' scale for taking subsets of.


[[Subgroup]]: 2.3.7/5
===== 2.9.5.21.11 =====
Subgroup: 2.9.5.21.11


[[Comma list]]: [[50/49]]
Comma list: 81/80, 99/98, 385/384


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


[[Optimal tuning]] (inharmonic [[TE]]): ~1\2 = 590.998, ~[[10/7]]-1\2 = 128.962
{{Mapping|legend=4| 1 1 0 3 5 | 0 3/2 6 -1/2 -4 }}


[[Support]]ing [[ET]]s: {{EDOs|4, 6, 8, 10, 18, 28, 46, 64, 110}}
: gencom: [2 21/16; 81/80 99/98 385/384]


=== Argentic ===
Optimal tuning (subgroup POTE): ~2 = 1\1, ~21/16 = 463.956
Argentic is the 2.3.7/5 subgroup temperament tempering out [[5120/5103]].  


[[Subgroup]]: 2.3.7/5
{{Optimal ET sequence|legend=1| 5, 13, 31 }}


[[Comma list]]: [[5120/5103]] = {{monzo| 10 -6 -1 }}
==== B-team ====
B-team (23 & 41) is every other step of [[rodan]].


{{Mapping|legend=3| 1 0 10 | 0 1 -6 }}
Subgroup: 2.9.15.21.33
: mapping generators: ~2, ~3


[[Optimal tuning]]s:  
Comma list: 245/243, 385/384, 441/440
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1\1, ~3/2 = 702.792
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1\1, ~3/2 = 702.830


{{Optimal ET sequence|legend=1| 12, 29, 41, 70, 321, 391, 461, 531, 601 }}
Sval mapping: {{mapping| 1 2 0 4 7 | 0 3 10 1 -5 }}
<small> based on subgroup TE </small>


Badness (Sintel): 0.119
Optimal tuning (subgroup POTE): ~2 = 1\1, ~21/16 = 468.918


==== Edson ====
{{Optimal ET sequence|legend=1| 5, 13c, 18, 23, 41, 64, 87, 151 }}
{{See also| Chromatic pairs #Edson }}


Edson is related to [[pele]] and [[andromeda]].  
== 4.3.… subgroups ==
=== Tetrahanson ===
{{Main| Tetrahanson }}


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


[[Comma list]]: [[196/195]] = {{monzo| 2 -1 2 0 -1 }}, [[352/351]] = {{monzo| 5 -3 0 1 -1 }}, [[364/363]] = {{monzo| 2 -1 1 -2 1 }}
[[Comma list]]: 15625/15552


{{Mapping|legend=3| 1 0 10 17 22 | 0 1 -6 -10 -13 }}
{{Mapping|legend=3| 1 3 3 | 0 -6 -5 }}
: mapping generators: ~2, ~3


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


[[Optimal tuning]]s:
[[Optimal tuning]] ([[CTE]]): ~4 = 2\1, ~5/3 = 882.941
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1\1, ~3/2 = 703.4398
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1\1, ~3/2 = 703.414


{{Optimal ET sequence|legend=1| 12, 17, 29 }}
[[Support]]ing [[ET]]s: {{EDs|19, 106, 87, 68, 11, 8, 125, 49, 30, 27, 117, 46, 41b, 79|equave=4}}


[[Tp tuning #T2 tuning|RMS error]]: 0.5102 cents
=== Tetrameantone ===
{{Main| Tetrameantone }}


==== Haumea ====
[[Subgroup]]: 4.3.5
{{See also| Chromatic pairs #Haumea }}


Related temperaments include [[#Bridgetown|bridgetown]], [[namaka]], [[hemigari]], [[#Barbados|barbados]], and [[parizekmic]].
[[Comma list]]: 81/80


[[Subgroup]]: 2.3.7/5.11/5.13/5
{{Mapping|legend=3| 1 1 2 | 0 -1 -4 }}


[[Comma list]]: [[352/351]], [[676/675]], [[847/845]]
: Mapping generators: ~4, ~4/3


{{Mapping|legend=3| 1 0 10 -6 -1 | 0 2 -12 9 3 }}
[[Optimal tuning]] ([[POTE]]): 4 = 2400.0, ~4/3 = 503.761


{{Mapping|legend=5| 1 2 -3/4 -11/4 9/4 5/4 | 0 -2 0 12 -9 -3 }}
[[Support]]ing [[ET]]s: {{EDs|5, 9, 14, 19, 24, 43, 62, 81, 100|equave=4}}
: [[gencom]]: [2 15/13; 352/351 676/675 847/845]


[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~15/13 = 248.491
=== Tetramagic ===


{{Optimal ET sequence|legend=1| 24, 29, 111, 140, 169, 198, 565d, 763bd, 961bd }}
[[Subgroup]]: 4.3.5


[[Tp tuning #T2 tuning|RMS error]]: 0.2668 cents
[[Comma list]]: 3125/3072


=== Historical ===
{{Mapping|legend=3| 1 0 1 | 0 5 1 }}
{{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.
: Mapping generators: ~4, ~5/4


[[Subgroup]]: 2.3.7/5.11/5.13/5
[[Optimal tuning]] ([[POTE]]): 4 = 2400.0, ~5/4 = 380.059


[[Comma list]]: 364/363, 441/440, 1001/1000
[[Support]]ing [[ET]]s: {{EDs|6, 13, 19, 25, 38, 44, 63, 82|equave=4}}


{{Mapping|legend=3| 1 2 0 1 2 | 0 -6 7 2 -9 }}
=== Blacktetra ===


[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~21/20 = 83.016
[[Subgroup]]: 4.3.5


{{Optimal ET sequence|legend=1| 14, 29, 72, 101, 130, 159 }}
[[Comma list]]: 256/243


[[Tp tuning #T2 tuning|RMS error]]: 0.2562 cents
{{Mapping|legend=3| 5 4 6 | 0 0 -1 }}


=== Direct breedsmic ===
: Mapping generators: ~4, ~16/15
Related temperament: [[hemithirds]], [[newt]]


[[Subgroup]]: 2.3.49/5
[[Optimal tuning]] ([[POTE]]): 1\5ed4 = 480.0, ~16/15 = 80.4062


[[Comma list]]: 2401/2400
[[Support]]ing [[ET]]s: {{EDs|5, 10, 15, 20, 25, 30, 55, 85, 115|equave=4}}


{{Mapping|legend=3| 1 1 3 | 0 2 1 }}
== 4.6.… subgroups ==
=== Meanquad ===
{{Main| Meanquad }}


[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~49/40 = 350.966
[[Subgroup]]: 4.6.5


{{Optimal ET sequence|legend=1|7, 10, 17}}
[[Comma list]]: [[81/80]] = {{monzo| -4 4 -1 }}


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


==== Bridgetown ====
: mapping generators: ~4, ~6
{{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]].  
[[Optimal tuning]] (subgroup [[CTE]]): ~4 = 2\1, ~3/2 = 697.214


[[Subgroup]]: 2.3.11/5.13/5
[[Support]]ing [[ET]]s: *7, *10, *11[-5], *13[+5], *17, *24, *27[+5], *31, *38, *41, *45, *52, *55, *69


[[Comma list]]: [[352/351]], [[676/675]]
<nowiki />* Wart for 4


{{Mapping|legend=3| 1 0 -6 -1 | 0 2 9 3 }}
==== 4.6.5.7 subgroup (tetrominant) ====
[[Subgroup]]: 4.6.5.7


{{Mapping|legend=5| 1 2 -5/3 0 4/3 1/3 | 0 -2 4 0 -5 1 }}
[[Comma list]]: [[36/35]] = {{monzo| 0 2 -1 -1 }}, [[64/63]] = {{monzo| 4 -2 0 -1 }}
: [[gencom]]: [2 15/13; 352/351 676/675]


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


{{Optimal ET sequence|legend=1| 5, 9, 14, 19, 24, 29, 169, 198, 227, 256, 285, 314 }}
[[Optimal tuning]] (subgroup [[CTE]]): ~4 = 2\1, ~3/2 = 699.622


[[Tp tuning #T2 tuning|RMS error]]: 0.2513 cents
[[Support]]ing [[ET]]s: *7, *10, *17, *24, *27[+5], *31, *38[+7], *41, *44[+5], *55[+7], *58[+5, +7], *65[+5, +7], *75[+5, +7]


=== Blackweed ===
<nowiki />* Wart for 4
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
=== Fourwar ===
The 23-limit version of Fourwar was created first, as an attempt to approximate subgroup 4.6.5.7.11.13.17.19.23 as accurately as possible using 25 to 35 notes per equave. Then the lower limit versions were created by simply extrapolating the temperament downwards.


[[Comma list]]: {{monzo| 8 -5 }} (256/243)
Fourwar is named after the closely related [[hemiwar]] temperament.


{{Mapping|legend=3| 5 8 0 | 0 0 1 }}
{{Todo|inline=1|cleanup}}
: mapping generators: ~9/8, ~11/7


[[Optimal tuning]]s:
<pre>
* [[Tp tuning|subgroup]] [[WE]]: ~8/7 = 238.851{{c}}, ~11/7 = 782.457{{c}}
Reduced Mapping
: [[error map]]: {{val| -5.746 +8.852 -0.035 }}
4 6 5
* [[Tp tuning|subgroup]] [[CWE]]: ~8/7 = 240.000{{c}}, ~11/7 = 784.967{{c}}
[ ⟨ 1 0 1 ]
: error map: {{val| 0.000 +18.045 +2.475 }}
⟨ 0 16 2 ] ⟩
 
{{Optimal ET sequence|legend=1| 15, 20, 35b, 55b }}
TE Generator Tunings (cents)
 
⟨2399.3973, 193.8643]
=== Pepperoni ===
{{Main| Parapyth }}
TE Step Tunings (cents)
{{See also| Chromatic pairs #Pepperoni }}
⟨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)


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.
Subsets
q99, q62, q37, q161, q136, q198, q25, q124, q74, q235
</pre>


[[Subgroup]]: 2.3.11/7.13/7
==== 4.6.5.7 ====
 
<pre>
[[Comma list]]: 352/351, 364/363
Reduced Mapping
 
4 6 5 7
{{Mapping|legend=3| 1 0 7 12 | 0 1 -4 -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)


{{Mapping|legend=5| 1 1 0 -8/3 1/3 7/3 | 0 1 0 11/3 -1/3 -10/3 }}
Subsets
: [[gencom]]: [2 3/2; 352/351 364/363]
q99, q62, q37, q161, q136, q198, q25, q124, q74, q235
</pre>


[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~3/2 = 703.856
==== 4.6.5.7.11 ====
 
<pre>
{{Optimal ET sequence|legend=1| 5, 7, 12f, 17, 29, 46, 58, 75, 80, 87, 104, 121, 167, 196, 208, 271, 595b*<sup>†</sup> }}
Reduced Mapping
: <nowiki />* wart for 11/7
4 6 5 7 11
: <sup>†</sup> wart for 13/7
[ ⟨ 1 0 1 1 1 ]
 
⟨ 0 16 2 5 9 ] ⟩
[[Tp tuning #T2 tuning|RMS error]]: 0.3789 cents
 
TE Generator Tunings (cents)
=== Barbados ===
⟨2400.1097, 193.9498]
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.
 
TE Step Tunings (cents)
[[Subgroup]]: 2.3.13/5
⟨24.18752, 48.52491]
 
[[Comma list]]: 676/675 = {{monzo| 2 -3 2 }}
TE Tuning Map (cents)
 
⟨2400.110, 3103.196, 2788.009, 3369.859, 4145.658]
[[Sval]] [[mapping]]: [{{val| 1 0 -1 }}, {{val| 0 2 3 }}]
 
TE Mistunings (cents)
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~2 = 1\1, ~15/13 = 248.621
⟨0.110, 1.241, 1.696, 1.033, -5.660]
 
{{Optimal ET sequence|legend=1| 5, 9, 14, 19, 24, 29, 53, 82, 111, 140, 251, 362 }}
Complexity 1.068792
 
Adjusted Error 2.926965 cents
[[Badness]]: 0.002335
TE Error 0.846083 cents/octave
 
; Music
Unison Vectors
* [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]
[-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)


==== Tobago ====
Subsets
{{See also| Chromatic pairs #Tobago }}
q37, q25, q62, q12, q74, q99, q87, q49r, q50r, q124
</pre>


Tobago, the 10 &amp; 14 temperament in the 2.3.11.13/5 subgroup, extends [[neutral]] and [[barbados]].
==== 4.6.5.7.11.13 ====


[[Subgroup]]: 2.3.11.13/5
<pre>
 
Reduced Mapping
[[Comma list]]: [[243/242]], [[676/675]]
4 6 5 7 11 13
 
[ ⟨ 1 0 1 1 1 0 ]
{{Mapping|legend=3| 2 0 -1 -2 | 0 2 5 3 }}
⟨ 0 16 2 5 9 23 ] ⟩
 
{{Mapping|legend=5| 2 4 -2 0 9 2 | 0 -2 3/2 0 -5 -3/2 }}
TE Generator Tunings (cents)
: [[gencom]]: [55/39 15/13; 243/242 676/675]
⟨2401.2305, 193.5378]
 
[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~55/39 = 1\2, ~15/13 = 249.312
TE Step Tunings (cents)
 
⟨42.79107, 35.98524]
{{Optimal ET sequence|legend=1| 10, 14, 24, 58, 82, 130 }}
 
TE Tuning Map (cents)
[[Tp tuning #T2 tuning|RMS error]]: 0.3533 cents
⟨2401.230, 3096.606, 2788.306, 3368.920, 4143.071, 4451.371]
 
==== Pakkanian hemipyth ====
TE Mistunings (cents)
[[Subgroup]]: 2.3.11.13/5.17
⟨1.230, -5.349, 1.992, 0.094, -8.247, 10.843]
 
[[Comma list]]: 221/220, 243/242, 289/288
Complexity 1.219191
 
Adjusted Error 6.699599 cents
{{Mapping|legend=3| 2 0 -1 -2 5 | 0 2 5 3 2 }}
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)


[[Optimal tuning]]s:
Subsets
* [[Tp tuning|subgroup CTE]]: ~17/12 = 1\2, ~26/15 = 950.7656 (~15/13 = 249.2344)
q25, q37f, q12f, q62, q50rf, q13rff, q49rff, q87, q74ff, q24rfff
* [[Tp tuning|subgroup CWE]]: ~17/12 = 1\2, ~26/15 = 950.6011 (~15/13 = 249.3989)
</pre>


{{Optimal ET sequence|legend=1| 10, 14, 24, 106, 130, 154, 178*, 202* }}
==== 4.6.5.7.11.13.17 ====
: <nowiki />* wart for 13/5
<pre>
 
Reduced Mapping
=== Oceanfront ===
4 6 5 7 11 13 17
Related temperaments: [[Archytas clan #Superpyth|superpyth]], [[Archytas clan #Ultrapyth|ultrapyth]]
[ ⟨ 1 0 1 1 1 0 1 ]
 
⟨ 0 16 2 5 9 23 13 ] ⟩
[[Subgroup]]: 2.3.7.13/5
 
TE Generator Tunings (cents)
[[Comma list]]: 64/63, 91/90
⟨2400.4701, 193.4599]
 
{{Mapping|legend=3| 1 0 6 -5 | 0 1 -2 4 }}
TE Step Tunings (cents)
 
⟨43.39350, 35.55764]
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~3/2 = 713.910
 
TE Tuning Map (cents)
{{Optimal ET sequence|legend=1| 5, 22, 27, 32, 37 }}
⟨2400.470, 3095.359, 2787.390, 3367.770, 4141.609, 4449.578, 4915.449]
 
[[Tp tuning #T2 tuning|RMS error]]: 2.063 cents
TE Mistunings (cents)
 
⟨0.470, -6.596, 1.076, -1.056, -9.709, 9.050, 10.494]
Scales: [[Oceanfront scales]]
 
Complexity 1.129881
=== Seventeen-cot ===
Adjusted Error 8.082725 cents
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.
TE Error 1.977443 cents/octave
 
====2.3.13/5 subgroup====
Unison Vectors
 
[0, 1, -1, 0, 1, -1, 0⟩ (66:65)
Comma basis: {{monzo| -6 -11 17 }} (2.3.13/5)
[1, 1, 1, -1, 0, 0, -1⟩ (120:119)
 
[1, 2, 0, 0, -1, -1, 0⟩ (144:143)
edo join: 29 & 146
[-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)


{{Mapping|legend=3| 1 1 1 | 0 17 11 }}
Subsets
: mapping generators: ~2, ~2250/2197
q25, q12f, q37f, q13rffg, q50rf, q62, q49rffg, q24rfffg, q38rreffg, q74ffg
</pre>


Optimal tunings:
==== 4.6.5.7.11.13.17.19 ====
* WE: ~2 = 1200.0001354{{c}}, ~2250/2197 = 41.2914993{{c}}
<pre>
: error map: {{val| +0.0001354 +0.0006234 -0.0073197}}
Reduced Mapping
* CWE: ~2 = 2/1, ~2250/2197 = 41.2915011{{c}}
4 6 5 7 11 13 17 19
: error map: {{val| 0.0000000 +0.0005177 -0.0074359}}
[ ⟨ 1 0 1 1 1 0 1 1 ]
 
⟨ 0 16 2 5 9 23 13 14 ] ⟩
edos: 29, 465, 494, 436, 523, 407, 378, 349, 30[-3], 28[+3], 320, 291, 59[-3], 262
 
TE Generator Tunings (cents)
Badness (Sintel): 0.064
⟨2399.9219, 193.3952]
 
====2.3.11/5.13/5 subgroup====
TE Step Tunings (cents)
 
⟨44.14256, 35.03670]
Comma basis: 225000/224939, 43940/43923
 
TE Tuning Map (cents)
edo join: 29 & 146
⟨2399.922, 3094.324, 2786.712, 3366.898, 4140.479, 4448.090, 4914.060, 5107.455]
 
{{Mapping|legend=3| 1 1 1 1 | 0 17 4 11 }}
TE Mistunings (cents)
: mapping generators: ~2, ~169/165
⟨-0.078, -7.631, 0.399, -1.928, -10.839, 7.562, 9.104, 9.942]
 
Optimal tunings:  
Complexity 1.058472
* WE: ~2 = 1199.9934923{{c}}, ~169/165 = 41.2918271{{c}}
Adjusted Error 8.712222 cents
: error map: {{val| -0.0065077 -0.0004485 +0.1565720 -0.0103579}}
TE Error 2.050935 cents/octave
* CWE: ~2 = 2/1, ~169/165 = 41.2917463{{c}}
: error map: {{val| 0.0000000 +0.0046870 +0.1627569 -0.0043781}}
Unison Vectors
 
[0, 1, -1, 0, 1, -1, 0, 0⟩ (66:65)
edos: 29, 465, 494, 436, 523, 407, 378, 349, 320, 291, 30[-3], 262, 28[+3], 233
[-1, 0, 0, 1, 1, 0, 0, -1⟩ (77:76)
 
[2, 1, -1, 0, 0, 0, 0, -1⟩ (96:95)
Badness (Sintel): 0.080
[1, 1, 1, -1, 0, 0, -1, 0⟩ (120:119)
 
[0, 1, 1, 1, -1, 0, 0, -1⟩ (210:209)
=== Fiventeen ===
[0, 0, 1, -2, 1, 0, 1, -1⟩ (935:931)
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.
[2, 0, -3, 1, 0, 0, -1, 1⟩ (2128:2125)
 
[[Subgroup]]: 2.3.17/5


[[Comma list]]: 136/135 ({{monzo| 3 -3 1 }})
Subsets
q25, q12fh, q37f, q13rffgh, q50rf, q62, q49rffgh, q24rfffghh, q38rreffgh, q74ffgh
</pre>


{{Mapping|legend=3| 1 0 -3 | 0 1 3 }}
==== 4.6.5.7.11.13.17.19.23 ====
: mapping generators: ~2, ~3
<pre>
 
Reduced Mapping
[[Optimal tuning]]s:
4 6 5 7 11 13 17 19 23
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.2838{{c}}, ~3/2 = 704.4600{{c}}
[ ⟨ 1 0 1 1 1 0 1 1 0 ]
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.5286{{c}}
⟨ 0 16 2 5 9 23 13 14 28 ] ⟩
 
{{Optimal ET sequence|legend=1| 5, 12, 17, 46, 63, 143 }}
TE Generator Tunings (cents)
 
⟨2399.3286, 193.5316]
=== 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.  
TE Step Tunings (cents)
 
⟨37.31613, 39.63311]
[[Subgroup]]: 2.3.19/7
 
TE Tuning Map (cents)
[[Comma list]]: [[57/56]] ({{Monzo| -3 1 1 }})
⟨2399.329, 3096.506, 2786.392, 3366.987, 4141.113, 4451.227, 4915.240, 5108.771, 5418.885]
 
{{Mapping|legend=3| 1 0 3 | 0 1 -1 }}
TE Mistunings (cents)
: mapping generators: ~2, ~3
⟨-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)


[[Optimal tuning]]s:
Subsets
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1202.4345{{c}}, ~3/2 = 697.4314{{c}}
q25i, q12fhi, q37f, q13rffghii, q62, q50rfii, q49rffghii, q24rfffghhiii, q74ffghi, q38rreffghiii
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 697.3981{{c}}
</pre>


{{Optimal ET sequence|legend=1| 5, 7, 12, 19, 31*, 50* }}
== 4.9.… subgroups ==
=== Meansquared ===
[[Subgroup]]: 4.9.25
 
[[Comma list]]: [[6561/6400]]


<nowiki/>* wart for 19/7
{{Mapping|legend=3| 1 3 4 | 0 1 4 }}


[[Badness]] (Sintel): 0.082
Mapping generators: ~4, ~9/64


== 2.5.… subgroups ==
[[Optimal tuning]] ([[CTE]]): ~4 = 2\1, ~9/4 = 1394.429
=== 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.
[[Support]]ing [[ET]]s: 12, 7, 19, 5, 31, 26, 17[+25], 43, 9[-25], 33[-25], 45, 29[+25], 8[+25], 22[+25]


[[Subgroup]]: 2.5.7/3.11/3
=== Archsquared ===
[[Subgroup]]: 4.9.49


[[Comma list]]: [[176/175]] ({{monzo| 4 -2 -1 1 }}), [[540/539]] ({{monzo| 2 1 -2 -1 }})
[[Comma list]]: 4096/3969


{{Mapping|legend=3| 1 0 1 3 | 0 -3 1 -5 }}
{{Mapping|legend=3| 1 3 0 | 0 1 -2 }}
: 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 generators: ~4, ~9/64
: [[gencom]]: [2 7/6; 176/175 540/539]


[[Optimal tuning]]s:
[[Optimal tuning]] ([[CTE]]): ~9/4 = 1419.190
* ([[Tp tuning|subgroup]] [[CTE]]): ~2 = 1200.000, ~7/6 = 270.455
* ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1200.000, ~7/6 = 270.093


{{Optimal ET sequence|legend=1| 9, 22, 31, 40, 191c*, 231c*, 271c*, 311c* }}
[[Support]]ing [[ET]]s: 5, 17, 22, 12, 7, 27, 32, 8, 39[+49], 29[+49], 9[+49], 19[+49], 37, 49
: <nowiki/>* wart for 7/3


[[Tp tuning #T2 tuning|RMS error]]: 0.6028 cents
== 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 is named for the classic dad joke: "Why was six scared? Because seven ate nine."


==== Tridecimal guanyintet ====
[[Subgroup]]: 8.9.7
Guanyintet can extend to the 13th harmonic by the equivalences ([[12/11]])<sup>3</sup> = [[13/10]] and ([[15/14]])<sup>3</sup> = [[16/13]], therefore tempering out {S11/S12/S14/S15}. However, note that it is not supported by the 31 & 53 orwell extension dubbed "tridecimal orwell", but instead the less accurate [[winston]] (22f & 31), as orwell prefers slightly sharper tunings than guanyintet. [[40edo]] remains an excellent tuning.


[[Subgroup]]: 2.5.7/3.11/3.13
[[Comma list]]: 64/63


[[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 2 | 0 1 -1 }}


{{Mapping|legend=3| 1 0 1 3 1 | 0 -3 1 -5 12 }}
: sval mapping generators: ~8, ~9
: mapping generators: ~2, ~12/7


[[Optimal tuning]]s:  
: [[gencom]]: [8 9/8; 64/63]
* ([[Tp tuning|subgroup]] [[CTE]]): ~2 = 1200.000, ~7/6 = 270.152
* ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1200.000, ~7/6 = 270.218


{{Optimal ET sequence|legend=1| 9, 22, 31, 40, 71, 111, 151, 262c*}} <small> using subgroup TE </small>
[[Optimal tuning]] ([[CTE]]): ~9/8 = 219.1898
: <nowiki/>* wart for 7/3
 
[[Optimal ET sequence]]: {{val| 16 17 15 }}, {{val| 33 35 31 }}, {{val| 148 … }}, {{val| 181 … }}, {{val| 214 … }}, {{val| 247 … }}


Badness (Sintel): 0.329
[[Badness]]: 0.0215 × 10<sup>-3</sup>


==== Laz ====
== 2.5.… subgroups ==
{{See also | Chromatic pairs #Laz }}
=== Guanyintet ===
{{See also | Chromatic pairs #Guanyintet }}


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


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


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


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


{{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=5| 1 -4/3 3 -1/3 5/3 | 0 4/3 -3 7/3 -11/3 }}
: [[gencom]]: [2 7/6; 144/143 176/175 196/195]
: [[gencom]]: [2 7/6; 176/175 540/539]


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~12/7 = 930.598
* ([[Tp tuning|subgroup]] [[CTE]]): ~2 = 1200.000, ~7/6 = 270.455
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~12/7 = 930.700
* ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1200.000, ~7/6 = 270.093


{{Optimal ET sequence|legend=1| 9, 31, 40, 49, 156c*†, 205c*† }}
{{Optimal ET sequence|legend=1| 9, 22, 31, 40, 191c*, 231c*, 271c*, 311c* }}
: <nowiki/>* wart for 7/3
: <nowiki/>* wart for 7/3
: † wart for 11/3


[[Tp tuning #T2 tuning|RMS error]]: 0.8790 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.6028 cents


=== Kryptonite ===
==== Tridecimal guanyintet ====
{{See also| Chromatic pairs #Kryptonite }}
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.


Kryptonite is related to [[krypton]].  
[[Subgroup]]: 2.5.7/3.11/3.13


[[Subgroup]]: 2.5.7/3.11/3.13/3
[[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]]: 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 0 1 3 1 | 0 -3 1 -5 12 }}
: mapping generators: ~2, ~12/7


{{Mapping|legend=3| 1 2 1 2 2 | 0 3 2 -1 1 }}
[[Optimal tuning]]s:
: mapping generators: ~2, ~13/12
* ([[Tp tuning|subgroup]] [[CTE]]): ~2 = 1200.000, ~7/6 = 270.152
* ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1200.000, ~7/6 = 270.218


{{Mapping|legend=5| 1 -5/4 2 -1/4 3/4 3/4 | 0 -1/2 3 3/2 -3/2 1/2 }}
{{Optimal ET sequence|legend=1| 9, 22, 31, 40, 71, 111, 151, 262c*}} <small> using subgroup TE </small>
: [[gencom]]: [2 13/12; 56/55 78/77 91/90]
: <nowiki/>* wart for 7/3


[[Optimal tuning]]s:  
Badness (Sintel): 0.329
* [[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 }}
==== Laz ====
{{See also | Chromatic pairs #Laz }}


[[Tp tuning #T2 tuning|RMS error]]: 2.545 cents
Laz is related to [[avalokita]] as well as to [[winston]].  


=== Marveltri ===
[[Subgroup]]: 2.5.7/3.11/3.13/3
{{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.
[[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 }}


[[Subgroup]]: 2.5.9/7
{{Mapping|legend=3| 1 0 2 -2 6 | 0 3 -1 5 -5 }}


[[Comma list]]: 225/224 ({{monzo| -5 2 1 }})
{{Mapping|legend=5| 1 -5/4 3 -1/4 7/4 -1/4 | 0 -1/4 -3 3/4 -21/4 19/4 }}
: [[gencom]]: [2 7/6; 144/143 176/175 196/195]


{{Mapping|legend=3| 1 0 5 | 0 1 -2 }}
[[Optimal tuning]]s:
: mapping generators: ~2, ~5
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~12/7 = 930.598
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~12/7 = 930.700


{{Mapping|legend=5| 1 2 0 -1 | 0 -4/5 1 2/5 }}
{{Optimal ET sequence|legend=1| 9, 31, 40, 49, 156c*†, 205c*† }}
: [[gencom]]: [2 5; 225/224]
: <nowiki/>* wart for 7/3
: † wart for 11/3


[[Optimal tuning]]s:
[[Tp tuning #T2 tuning|RMS error]]: 0.8790 cents
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/4 = 384.208
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/4 = 383.638


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


[[Tp tuning #T2 tuning|RMS error]]: 0.4801 cents
Kryptonite is related to [[krypton]].  


==== Sulis ====
[[Subgroup]]: 2.5.7/3.11/3.13/3
Sulis is related to [[minerva]] and [[würschmidt]].  


[[Subgroup]]: 2.5.9/7.11/9
[[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]]: 99/98 ({{monzo| -1 0 2 1 }}), 176/175 ({{monzo| 4 -2 1 1 }})
{{Mapping|legend=3| 1 2 1 2 2 | 0 3 2 -1 1 }}
: mapping generators: ~2, ~13/12


{{Mapping|legend=3| 1 0 5 -9 | 0 1 -2 4 }}]
{{Mapping|legend=5| 1 -5/4 2 -1/4 3/4 3/4 | 0 -1/2 3 3/2 -3/2 1/2 }}
: [[gencom]]: [2 13/12; 56/55 78/77 91/90]


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/4 = 386.617
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~13/12 = 130.945
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/4 = 386.558
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~13/12 = 132.428


{{Optimal ET sequence|legend=1| 3, …, 22, 25, 28, 31, 59 }}
{{Optimal ET sequence|legend=1| 1, …, 8, 9 }}


[[Tp tuning #T2 tuning|RMS error]]: 1.074 cents
[[Tp tuning #T2 tuning|RMS error]]: 2.545 cents


== 2.5/3.… subgroups ==
=== Marveltri ===
=== Magicaltet ===
{{See also| Chromatic pairs #Marveltri }}
{{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.  
Marveltri, the {{nowrap| 3 & 13 }} temperament in the 2.5.9/7 subgroup, is related to [[marvel]], [[magic]], and the unnamed {{nowrap| 22 & 47 }} temperament. The tonic and the first two generator steps make a [[marvel triad]], hence the name.  


[[Subgroup]]: 2.5/3.7.11
[[Subgroup]]: 2.5.9/7


[[Comma list]]: 100/99 ({{monzo| 2 2 0 -1 }}), 385/384 ({{monzo| -7 1 1 1 }})
[[Comma list]]: 225/224 ({{monzo| -5 2 1 }})


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


{{Mapping|legend=5| 1 -1/2 1/2 2 4 | 0 1/2 -1/2 3 -2 }}
{{Mapping|legend=5| 1 2 0 -1 | 0 -4/5 1 2/5 }}
: [[gencom]]: [2 6/5; 100/99 385/384]
: [[gencom]]: [2 5; 225/224]


[[Optimal tuning]]s:
[[Optimal tuning]]s:  
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 877.343
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/4 = 384.208
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 877.351
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/4 = 383.638


{{Optimal ET sequence|legend=1| 4, 7, 11, 15, 26, 67, 93* }}
{{Optimal ET sequence|legend=1| 3, 13, 16, 19, 22, 25, 72, 97, 122, 269c* }}
: <nowiki/>* wart for 5/3
: <nowiki/>* wart for 9/7


[[Tp tuning #T2 tuning|RMS error]]: 1.206 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.4801 cents


=== Starlingtet ===
==== Sulis ====
{{See also | Chromatic pairs #Starlingtet }}
Sulis is related to [[minerva]] and [[würschmidt]].


Starlingtet, the {{nowrap| 4 & 15 }} temperament in the 2.5/3.7/3 subgroup, is related to [[starling]] as well as to [[myna]]. The tonic and the first three generator steps make a [[starling tetrad]], hence the name.
[[Subgroup]]: 2.5.9/7.11/9


[[Subgroup]]: 2.5/3.7/3
[[Comma list]]: 99/98 ({{monzo| -1 0 2 1 }}), 176/175 ({{monzo| 4 -2 1 1 }})


[[Comma list]]: [[126/125]] ({{monzo| 1 -3 1 }})
{{Mapping|legend=3| 1 0 5 -9 | 0 1 -2 4 }}]


{{Mapping|legend=3| 1 0 -1 | 0 1 3 }}
[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/4 = 386.617
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/4 = 386.558


: mapping generators: ~2, ~5/3
{{Optimal ET sequence|legend=1| 3, …, 22, 25, 28, 31, 59 }}


{{Mapping|legend=5| 1 -1 0 1 | 0 4/3 1/3 -5/3 }}
[[Tp tuning #T2 tuning|RMS error]]: 1.074 cents
: [[gencom]]: [2 6/5; 126/125]


[[Optimal tuning]]s:
== 2.5/3.… subgroups ==
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 888.759
=== Magicaltet ===
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 888.846
{{See also| Chromatic pairs #Magicaltet }}


{{Optimal ET sequence|legend=1| 4, 15, 19, 23, 27 }}
Magicaltet is related to [[keemic]], [[superkleismic]], and [[magic]]. The tonic and the first three generator steps make a [[magical seventh chord]], hence the name.


[[Tp tuning #T2 tuning|RMS error]]: 0.8398 cents
[[Subgroup]]: 2.5/3.7.11


==== Greeley ====
[[Comma list]]: 100/99 ({{monzo| 2 2 0 -1 }}), 385/384 ({{monzo| -7 1 1 1 }})
{{See also| Chromatic pairs #Greeley }}


Greeley is related to [[opossum]] as well as to [[nusecond]].
{{Mapping|legend=3| 1 0 5 2 | 0 1 -3 2 }}
: mapping generators: ~2, ~5/3


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


[[Comma list]]: 121/120 ({{monzo| -3 -1 0 2 }}), 126/125 ({{monzo| 1 -3 1 }})
[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 877.343
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 877.351


{{Mapping|legend=3| 1 1 2 2 | 0 -2 -6 -1 }}
{{Optimal ET sequence|legend=1| 4, 7, 11, 15, 26, 67, 93* }}
: <nowiki/>* wart for 5/3


{{Mapping|legend=5| 1 -5/4 -1/4 3/4 3/4 | 0 9/4 1/4 -15/4 5/4 }}
[[Tp tuning #T2 tuning|RMS error]]: 1.206 cents
: [[gencom]]: [2 11/10; 121/120 126/125]


[[Optimal tuning]]s:
=== Starlingtet ===
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~11/10 = 155.696
{{See also | Chromatic pairs #Starlingtet }}
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~11/10 = 155.776


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


[[Tp tuning #T2 tuning|RMS error]]: 1.034 cents
[[Subgroup]]: 2.5/3.7/3


==== Skateboard ====
[[Comma list]]: [[126/125]] ({{monzo| 1 -3 1 }})
{{See also| Chromatic pairs #Skateboard }}


Skateboard is related to [[thrasher]].
{{Mapping|legend=3| 1 0 -1 | 0 1 3 }}


[[Subgroup]]: 2.5/3.7/3.11.13/9
: mapping generators: ~2, ~5/3


[[Comma list]]: 56/55 ({{monzo| 3 -1 1 -1 }}), 91/90 ({{monzo| -1 -1 1 0 1 }}), 100/99 ({{monzo| 2 2 0 -1 }})
{{Mapping|legend=5| 1 -1 0 1 | 0 4/3 1/3 -5/3 }}
: [[gencom]]: [2 6/5; 126/125]


{{Mapping|legend=3| 1 0 -1 2 2 | 0 1 3 2 -2 }}
[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 888.759
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 888.846


{{Mapping|legend=5| 1 -3/7 4/7 11/7 4 -6/7 | 0 0 -1 -3 -2 2 }}
{{Optimal ET sequence|legend=1| 4, 15, 19, 23, 27 }}
: [[gencom]]: [2 6/5; 56/55 91/90 100/99]


[[Optimal tuning]]s:
[[Tp tuning #T2 tuning|RMS error]]: 0.8398 cents
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 886.158
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 886.158


{{Optimal ET sequence|legend=1| 11, 15, 19, 23, 42d, 65d }}
==== Greeley ====
{{See also| Chromatic pairs #Greeley }}


[[Tp tuning #T2 tuning|RMS error]]: 2.396 cents
Greeley is related to [[opossum]] as well as to [[nusecond]].  


=== Gariberttet ===
[[Subgroup]]: 2.5/3.7/3.11/3
Gariberttet is the 2.5/3.7/3 [[Subgroup temperament families, relationships, and genes|altergene]] of [[sirius]].


==== Gariberttet (2.5/3.7/3.13/11 subgroup) ====
[[Comma list]]: 121/120 ({{monzo| -3 -1 0 2 }}), 126/125 ({{monzo| 1 -3 1 }})
{{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]].
{{Mapping|legend=3| 1 1 2 2 | 0 -2 -6 -1 }}


[[Subgroup]]: 2.5/3.7/3.13/11
{{Mapping|legend=5| 1 -5/4 -1/4 3/4 3/4 | 0 9/4 1/4 -15/4 5/4 }}
: [[gencom]]: [2 11/10; 121/120 126/125]


[[Comma list]]: [[275/273]] ({{monzo| 0 2 -1 -1 }}), [[847/845]] ({{monzo| 0 -1 1 -2 }})
[[Optimal tuning]]s:  
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~11/10 = 155.696
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~11/10 = 155.776


{{Mapping|legend=3| 1 0 0 0 | 0 3 5 1 }}
{{Optimal ET sequence|legend=1| 8, 15, 23, 54, 77, 100, 131* }}
: <nowiki/>* wart for 11/3


{{Mapping|legend=5| 1 0 0 0 0 0 | 0 -8/3 1/3 7/3 -1/2 1/2 }}
[[Tp tuning #T2 tuning|RMS error]]: 1.034 cents
: [[gencom]]: [2 13/11; 275/273 847/845]


[[Optimal tuning]]s:
==== Skateboard ====
* [[Tp tuning|subgroup]] [[CTE]] and [[POTE]]: ~2 = 1200.000, ~13/11 = 293.679
{{See also| Chromatic pairs #Skateboard }}


{{Optimal ET sequence|legend=1| 29, 33, 37, 41, 45, 49, 78, 94, 143* }}
Skateboard is related to [[thrasher]].
: <nowiki/>* wart for 13/11


[[Tp tuning #T2 tuning|RMS error]]: 0.6914 cents
[[Subgroup]]: 2.5/3.7/3.11.13/9


==== Indium ====
[[Comma list]]: 56/55 ({{monzo| 3 -1 1 -1 }}), 91/90 ({{monzo| -1 -1 1 0 1 }}), 100/99 ({{monzo| 2 2 0 -1 }})
{{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.
{{Mapping|legend=3| 1 0 -1 2 2 | 0 1 3 2 -2 }}


[[Subgroup]]: 2.5/3.7/3.11/3
{{Mapping|legend=5| 1 -3/7 4/7 11/7 4 -6/7 | 0 0 -1 -3 -2 2 }}
: [[gencom]]: [2 6/5; 56/55 91/90 100/99]


[[Comma list]]: [[3025/3024]] ({{monzo| -4 2 -1 2 }}), [[3125/3087]] ({{monzo| 0 5 -3 }})
[[Optimal tuning]]s:  
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 886.158
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 886.158


{{Mapping|legend=3| 1 0 0 2 | 0 6 10 -1 }}
{{Optimal ET sequence|legend=1| 11, 15, 19, 23, 42d, 65d }}


{{Mapping|legend=5| 1 -1/2 -1/2 -1/2 3/2 | 0 -15/4 9/4 25/4 -19/4 }}
[[Tp tuning #T2 tuning|RMS error]]: 2.396 cents
: [[gencom]]: [2 12/11; 3025/3024 3125/3087]


[[Optimal tuning]]s:
=== Gariberttet ===
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~12/11 = 146.978
Gariberttet is the 2.5/3.7/3 [[Subgroup temperament families, relationships, and genes|altergene]] of [[sirius]].
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~12/11 = 147.010


{{Optimal ET sequence|legend=1| 8, 33, 41, 49, 204*<sup>†</sup> }}
==== Gariberttet (2.5/3.7/3.13/11 subgroup) ====
: <nowiki/>* wart for 7/3
{{See also | Chromatic pairs #Gariberttet }}
: <sup>†</sup> wart for 11/3


[[Tp tuning #T2 tuning|RMS error]]: 0.7788 cents
Gariberttet can be described as the {{nowrap| 4 & 29 }} temperament in the 2.5/3.7/3.13/11 subgroup. Extensions to the full 7-, 11-, and 13-limits include [[quasitemp]].


==== Ammon ====
[[Subgroup]]: 2.5/3.7/3.13/11
{{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.
[[Comma list]]: [[275/273]] ({{monzo| 0 2 -1 -1 }}), [[847/845]] ({{monzo| 0 -1 1 -2 }})


[[Subgroup]]: 2.5/3.7/3.11/3.13/3
{{Mapping|legend=3| 1 0 0 0 | 0 3 5 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=5| 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]


{{Mapping|legend=3| 1 3 5 3 4 | 0 -6 -10 -3 -5 }}
[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]] and [[POTE]]: ~2 = 1200.000, ~13/11 = 293.679


{{Mapping|legend=5| 1 -3 0 2 0 1 | 0 24/5 -6/5 -26/5 9/5 -1/5 }}
{{Optimal ET sequence|legend=1| 29, 33, 37, 41, 45, 49, 78, 94, 143* }}
: [[gencom]]: [2 13/10; 121/120 169/168 275/273]
: <nowiki/>* wart for 13/11


[[Optimal tuning]]s:
[[Tp tuning #T2 tuning|RMS error]]: 0.6914 cents
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~13/10 = 453.121
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~13/10 = 453.242


{{Optimal ET sequence|legend=1| 8, 29, 37, 45 }}
==== Indium ====
{{See also | Chromatic pairs #Indium }}


[[Tp tuning #T2 tuning|RMS error]]: 1.052 cents
Indium can be described as the {{nowrap| 8 & 33 }} temperament in the 2.5/3.7/3.11/3 subgroup.  


=== Sentry ===
[[Subgroup]]: 2.5/3.7/3.11/3
{{See also | Chromatic pairs #Sentry }}


Sentry, the {{nowrap| 3 & 5 }} temperament in the 2.5/3.9/7 subgroup, is related to [[sensi]].
[[Comma list]]: [[3025/3024]] ({{monzo| -4 2 -1 2 }}), [[3125/3087]] ({{monzo| 0 5 -3 }})


[[Subgroup]]: 2.5/3.9/7
{{Mapping|legend=3| 1 0 0 2 | 0 6 10 -1 }}


[[Comma list]]: [[245/243]] ({{monzo| 0 1 -2 }})
{{Mapping|legend=5| 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|legend=3| 1 0 0 | 0 2 1 }}
 
{{Mapping|legend=5| 1 0 0 0 | 0 0 2 -1 }}
: [[gencom]]: [2 9/7; 245/243]


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[Tp tuning|subgroup]] [[CTE]] and [[POTE]]: ~2 = 1200.000, ~9/7 = 440.902
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~12/11 = 146.978
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~12/11 = 147.010


{{Optimal ET sequence|legend=1| 8, 11, 19, 30, 41, 49, 52, 145*, 166<sup>†</sup>, 197*<sup>†</sup>, 215<sup>†</sup>, 264*<sup>†</sup> }}
{{Optimal ET sequence|legend=1| 8, 33, 41, 49, 204*<sup>†</sup> }}
: <nowiki/>* wart for 5/3
: <nowiki/>* wart for 7/3
: <sup>†</sup> wart for 9/7
: <sup>†</sup> wart for 11/3


[[Tp tuning #T2 tuning|RMS error]]: 0.7105 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.7788 cents


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


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


[[Subgroup]]: 2.5/3.13/9
[[Subgroup]]: 2.5/3.7/3.11/3.13/3


[[Comma list]]: [[325/324]] ({{monzo| -2 2 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 0 2 | 0 1 -2 }}
{{Mapping|legend=3| 1 3 5 3 4 | 0 -6 -10 -3 -5 }}


{{Mapping|legend=5| 1 -1/6 5/6 0 0 -1/3 | 0 -1/2 -3/2 0 0 1 }}
{{Mapping|legend=5| 1 -3 0 2 0 1 | 0 24/5 -6/5 -26/5 9/5 -1/5 }}
: [[gencom]]: [2 6/5; 325/324]
: [[gencom]]: [2 13/10; 121/120 169/168 275/273]


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 882.886
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~13/10 = 453.121
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 882.861
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~13/10 = 453.242


{{Optimal ET sequence|legend=1| 3, 4, 11, 15, 19, 34, 53, 87, 140 }}
{{Optimal ET sequence|legend=1| 8, 29, 37, 45 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.2444 cents
[[Tp tuning #T2 tuning|RMS error]]: 1.052 cents


== 2.9/5.… subgroups ==
=== Sentry ===
=== Kiribati ===
{{See also | Chromatic pairs #Sentry }}
{{See also| Chromatic pairs #Kiribati }}


Kiribati is related to [[nakika]] as well as to [[octacot]].  
Sentry, the {{nowrap| 3 & 5 }} temperament in the 2.5/3.9/7 subgroup, is related to [[sensi]].  


[[Subgroup]]: 2.9/5.7/3.11/9
[[Subgroup]]: 2.5/3.9/7


[[Comma list]]: 100/99 ({{monzo| 2 -2 0 -1 }}), 245/242 ({{monzo| -1 -1 2 -2 }})
[[Comma list]]: [[245/243]] ({{monzo| 0 1 -2 }})


{{Mapping|legend=3| 1 1 1 0 | 0 -2 3 4 }}
{{Mapping|legend=3| 1 0 0 | 0 2 1 }}
: 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 0 0 0 | 0 0 2 -1 }}
: [[gencom]]: [2 21/20; 100/99 245/242]
: [[gencom]]: [2 9/7; 245/243]


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~21/20 = 87.776
* [[Tp tuning|subgroup]] [[CTE]] and [[POTE]]: ~2 = 1200.000, ~9/7 = 440.902
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~21/20 = 87.892
 
{{Optimal ET sequence|legend=1| 8, 11, 19, 30, 41, 49, 52, 145*, 166<sup>†</sup>, 197*<sup>†</sup>, 215<sup>†</sup>, 264*<sup>†</sup> }}
: <nowiki/>* wart for 5/3
: <sup>†</sup> wart for 9/7
 
[[Tp tuning #T2 tuning|RMS error]]: 0.7105 cents


{{Optimal ET sequence|legend=1| 13, 14, 27, 41 }}
=== Marveltwintri ===
{{See also| Chromatic pairs #Marveltwintri }}
 
Marveltwintri can be described as the {{nowrap| 3 & 4 }} temperament in the 2.5/3.13/9 subgroup. The tonic and the first two generator steps make a [[marveltwin triad]], hence the name. [[Cata]] is a very natural extension of this temperament to the [[2.3.5.13 subgroup|2.3.5.13-subgroup]].
 
[[Subgroup]]: 2.5/3.13/9
 
[[Comma list]]: [[325/324]] ({{monzo| -2 2 1 }})
 
{{Mapping|legend=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 }}
: [[gencom]]: [2 6/5; 325/324]
 
[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 882.886
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 882.861
 
{{Optimal ET sequence|legend=1| 3, 4, 11, 15, 19, 34, 53, 87, 140 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.2444 cents
 
== 2.9/5.… 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 ({{monzo| 2 -2 0 -1 }}), 245/242 ({{monzo| -1 -1 2 -2 }})
 
{{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 }}
: [[gencom]]: [2 21/20; 100/99 245/242]
 
[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~21/20 = 87.776
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~21/20 = 87.892
 
{{Optimal ET sequence|legend=1| 13, 14, 27, 41 }}


[[Tp tuning #T2 tuning|RMS error]]: 1.245 cents
[[Tp tuning #T2 tuning|RMS error]]: 1.245 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]] ({{monzo| 13 1 -3 }})
{{Mapping|legend=3| 1 2 5 | 0 3 1 }}
: mapping generators: ~2, ~85/32
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.9692{{c}}, ~85/64 = 491.5853{{c}}
* [[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 tunings:
* 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* }}


== 2.7/3.… subgroups ==
== 2.7/3.… subgroups ==

Latest revision as of 09:21, 11 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.

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.

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, 143*

* Wart for 25

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, 97*, 114*

* Wart for 25

Badness (Sintel): 0.223

2.3.25.23.41 subgroup
See also: Reversed meantone

Subgroup: 2.3.25.23.41

Comma list: 82/81, 369/368, 576/575

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

Optimal tunings:

  • WE: ~2 = 1199.3246 ¢, ~3/2 = 705.2238 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 705.5491 ¢

Optimal ET sequence: 5, 12, 17

Badness (Sintel): 0.284

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

2.3.25.7.23.41 subgroup

Subgroup: 2.3.25.7.23.41

Comma list: 50/49, 64/63, 82/81, 162/161

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

Optimal tunings:

  • WE: ~2 = 1197.9403 ¢, ~3/2 = 705.9522 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 707.0779 ¢

Optimal ET sequence: 5, 12, 17, 39d

Badness (Sintel): 0.450

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.

Commatose

Commatose is a 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.

Subgroup: 2.9.5.7

Comma list: [28 -2 -19 8⟩, [9 -25 23 6⟩

Subgroup-val mapping: [⟨1 -289 -182 -508], ⟨0 298 188 521]]

mapping generators: ~2, ~1048576/531441

Optimal tunings:

  • WE: ~2 = 1199.9727 ¢, ~1048576/531441 = 1176.4969 ¢
  • CWE: ~2 = 1200.0000 ¢, ~1048576/531441 = 1176.5236 ¢

Optimal ET sequence: 460, 869, 1329

Badness (Sintel): 30.9

2.9.5.7.11

Subgroup: 2.9.5.7.11

Comma list: [-7 7 -3 2 -4⟩, [17 0 -13 1 3⟩, [11 -2 -6 7 -3⟩

Subgroup-val mapping: [⟨1 -289 -182 -508 -625], ⟨0 298 188 521 641]]

Optimal tunings:

  • WE: ~2 = 1199.9833 ¢, ~531441/524288 = 1176.5070 ¢
  • CWE: ~2 = 1200.0000 ¢, ~531441/524288 = 1176.5234 ¢

Optimal ET sequence: 460, 869e, 1329, 1789, 3118

Badness (Sintel): 8.67

2.9.5.7.11.13

Subgroup: 2.9.5.7.11.13

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

Subgroup-val mapping: [⟨0 -289 -182 -508 -625 -312], ⟨0 298 188 521 641 322]]

Optimal tunings:

  • WE: ~2 = 1200.0000 ¢, ~7056/3575 = 23.4767 ¢
  • CWE: ~2 = 1200.0000 ¢, ~7056/3575 = 23.4767 ¢

Optimal ET sequence: 460, 869e, 1329, 1789, 3118

Badness (Sintel): 3.30

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

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

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 3/2 2 0 3 4], ⟨0 1/2 2 0 3 -2]]

gencom: [2 9/8; 45/44 65/64 81/80]

Optimal tuning (POTE): ~2 = 1\1, ~9/8 = 186.151

Optimal ET sequence: 6, 13, 45be, 58bce, 71bce, 84bce

RMS error: 2.887 cents

Music:

Mabon

Derived from a calendar leap cycle built for the autumn equinox, hence the name. Defined as the 11 & 62 temperament.

Subgroup: 2.9.7

Comma basis: 44957696/43046721

Sval mapping: [⟨1 1 -3], ⟨0 3 8]]

Optimal tuning (CTE): ~729/448 = 870.792

Optimal ET sequence: 7d, 11, 18d, 29, 40, 62, ...

2.9.7.11 subgroup

Subgroup: 2.9.7.11

Comma basis: 896/891, 1331/1296

Sval mapping: [⟨1 1 -3 2], ⟨0 3 8 2]]

Optimal tuning (CTE): ~16/11 = 870.966

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

Apparatus

Subgroup: 2.9.7.11

Comma list: 41503/41472, 322102/321489

Subgroup-val mapping: [⟨1 5 3 5], ⟨0 -19 -2 -16]]

mapping generators: ~2, ~77/72

Gencom mapping: [⟨1 5/2 0 3 5], ⟨0 -19/2 0 -2 -16]]

gencom: [2 77/72; 41503/41472 322102/321489]

Optimal tuning (CTE): ~77/72 = 115.5685

Optimal ET sequence: 10e, 21, 31, 52, 83, 135, 353, 488, 623

Badness: 0.00263

Joan

See also: Chromatic pairs § Joan

Joan is related to casablanca as well as to orwell.

Subgroup: 2.9.7.11

Comma list: 99/98, 9317/9216

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

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

gencom: [2 11/8; 99/98 9317/9216]

Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 542.672 cents

Optimal ET sequence: 11, 20, 31, 42, 115bd, 157bd

RMS error: 1.424 cents

Machine

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

Subgroup: 2.9.7.11

Comma list: 64/63, 99/98

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

sval mapping generators: ~2, ~9

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

gencom: [2 8/7; 64/63 99/98]

Optimal tunings:

  • CTE: ~2 = 1\1, ~9/8 = 216.9128
  • POTE: ~2 = 1\1, ~9/8 = 214.3843

Optimal ET sequence: 5, 6, 11, 17, 28

Badness: 0.00233

Penta a.k.a. mechanism

Penta or mechanism is the 8 & 11 temperament in the 2.9.7.11 subgroup.

Subgroup: 2.9.7.11

Comma list: 896/891, 26411/26244

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

sval mapping generators: ~2, ~14/9

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

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

Optimal tuning (POTE): ~2 = 1\1, ~14/9 = 761.3782

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

RMS error: 0.4262 cents

Badness: 0.00439

Scales: penta5, penta8, penta11, penta19

Novisept

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

Subgroup: 2.9.7.13.17

Comma list: 729/728, 442/441, 833/832

Subgroup-val mapping: [⟨1 1 1 -1 3], ⟨0 6 5 13 3]]

Optimal tuning (CWE): ~2 = 1\1, ~9/7 = 433.836

Badness (Dirichlet): 0.142

Demon

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

Subgroup: 2.9.11

Comma list: 1331/1296

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

Optimal tuning (CTE): ~18/11 = 870.060

Optimal ET sequence: 4, 7, 11, 18, 29, 76e

Genius

Named after the genius in Roman religion, following the demon (daimon) in Greek mythology.

Subgroup: 2.9.11

Comma list: 131769/131072

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

Optimal tuning (CTE): ~16/11 = 650.863

Optimal ET sequence: 9, 11, 24, 59, 83, 142, 225, 367[-11], 592[-11], 959[-9, --11], 1326[-9, --11]

Stacks (a.k.a. 2magic)

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

Subgroup: 2.9.15.7

Comma list: 225/224, 245/243

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

sval mapping generators: ~2, ~14/9

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

gencom: [2 9/7; 225/224 245/243]

Optimal tuning (subgroup POTE): ~2 = 1\1, ~14/9 = 760.704

Optimal ET sequence: 8, 11, 30, 41, 71, 93, 112c, 134c, 175c

RMS error: 1.074 cents

2.9.15.7.11

Subgroup: 2.9.15.7.11

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

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

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

gencom: [2 9/7; 100/99 225/224 245/243]

Optimal tuning (subgroup POTE): ~2 = 1\1, ~14/9 = 761.393

Optimal ET sequence: 8, 11, 30, 41, 52, 93, 145, 342bce

RMS error: 1.226 cents

2.9.15.7.11.13

Subgroup: 2.9.15.7.11.13

Comma list: 100/99, 105/104, 144/143, 196/195

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

Gencom mapping: [⟨1 5/2 5/2 5 2 7], ⟨0 -5/2 -1/2 -6 4 -9]]

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

Optimal tuning (subgroup POTE): ~2 = 1\1, ~14/9 = 761.023

Optimal ET sequence: 11, 30, 41, 153cdef, 194cdef, 235cdef

RMS error: 1.540 cents

A-team

A-team is every other step of slendric; the 2.9.5.21.11 extension below specifically restricts mothra.

Subgroup: 2.9.21

Comma list: 1029/1024

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

sval mapping generators: ~2, ~21/16

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

gencom: [2 21/16; 1029/1024]

Optimal tuning (subgroup POTE): ~2 = 1\1, ~21/16 = 467.375

Optimal ET sequence: 5, 13, 18, 41, 59, 77, 95

RMS error: 0.3202 cents

2.9.5.21

Lookalike temperament: Dual-3 A-Team

Subgroup: 2.9.5.21

Comma list: 81/80, 1029/1024

Sval mapping: [⟨1 2 0 4], ⟨0 3 6 1]]

Mapping generators: ~2, ~21/16

Optimal (POL2) generator: 464.3865

Optimal ET sequence: 13, 18, 31, 44

2.9.5.21.11

Subgroup: 2.9.5.21.11

Comma list: 81/80, 99/98, 385/384

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

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

gencom: [2 21/16; 81/80 99/98 385/384]

Optimal tuning (subgroup POTE): ~2 = 1\1, ~21/16 = 463.956

Optimal ET sequence: 5, 13, 31

B-team

B-team (23 & 41) is every other step of rodan.

Subgroup: 2.9.15.21.33

Comma list: 245/243, 385/384, 441/440

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

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

Optimal ET sequence: 5, 13c, 18, 23, 41, 64, 87, 151

4.3.… subgroups

Tetrahanson

Main article: Tetrahanson

Subgroup: 4.3.5

Comma list: 15625/15552

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

Mapping generators: ~4, ~5/3

Optimal tuning (CTE): ~4 = 2\1, ~5/3 = 882.941

Supporting ETs: 19, 106, 87, 68, 11, 8, 125, 49, 30, 27, 117, 46, 41b, 79

Tetrameantone

Main article: Tetrameantone

Subgroup: 4.3.5

Comma list: 81/80

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

Mapping generators: ~4, ~4/3

Optimal tuning (POTE): 4 = 2400.0, ~4/3 = 503.761

Supporting ETs: 5, 9, 14, 19, 24, 43, 62, 81, 100

Tetramagic

Subgroup: 4.3.5

Comma list: 3125/3072

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

Mapping generators: ~4, ~5/4

Optimal tuning (POTE): 4 = 2400.0, ~5/4 = 380.059

Supporting ETs: 6, 13, 19, 25, 38, 44, 63, 82

Blacktetra

Subgroup: 4.3.5

Comma list: 256/243

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

Mapping generators: ~4, ~16/15

Optimal tuning (POTE): 1\5ed4 = 480.0, ~16/15 = 80.4062

Supporting ETs: 5, 10, 15, 20, 25, 30, 55, 85, 115

4.6.… subgroups

Meanquad

Main article: Meanquad

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 tuning (subgroup CTE): ~4 = 2\1, ~3/2 = 697.214

Supporting ETs: *7, *10, *11[-5], *13[+5], *17, *24, *27[+5], *31, *38, *41, *45, *52, *55, *69

* Wart for 4

4.6.5.7 subgroup (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 tuning (subgroup CTE): ~4 = 2\1, ~3/2 = 699.622

Supporting ETs: *7, *10, *17, *24, *27[+5], *31, *38[+7], *41, *44[+5], *55[+7], *58[+5, +7], *65[+5, +7], *75[+5, +7]

* Wart for 4

Fourwar

The 23-limit version of Fourwar was created first, as an attempt to approximate subgroup 4.6.5.7.11.13.17.19.23 as accurately as possible using 25 to 35 notes per equave. Then the lower limit versions were created by simply extrapolating the temperament downwards.

Fourwar is named after the closely related hemiwar temperament.

Todo: cleanup 
 
Reduced Mapping
4	6	5	
[ ⟨	1	0	1	]
⟨	0	16	2	] ⟩
 
TE Generator Tunings (cents)
⟨2399.3973, 193.8643]
 
TE Step Tunings (cents)
⟨25.21211, 47.81337]
 
TE Tuning Map (cents)
⟨2399.397, 3101.829, 2787.126]
 
TE Mistunings (cents)
⟨-0.603, -0.126, 0.812]
 
Complexity	1.369085
Adjusted Error	0.692892 cents
TE Error	0.268047 cents/octave
 
Unison Vector
[8, 1, -8⟩ (393216:390625)

Subsets
q99, q62, q37, q161, q136, q198, q25, q124, q74, q235

4.6.5.7

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

Subsets
q99, q62, q37, q161, q136, q198, q25, q124, q74, q235

4.6.5.7.11

Reduced Mapping
4	6	5	7	11	
[ ⟨	1	0	1	1	1	]
⟨	0	16	2	5	9	] ⟩
 
TE Generator Tunings (cents)
⟨2400.1097, 193.9498]
 
TE Step Tunings (cents)
⟨24.18752, 48.52491]
 
TE Tuning Map (cents)
⟨2400.110, 3103.196, 2788.009, 3369.859, 4145.658]
 
TE Mistunings (cents)
⟨0.110, 1.241, 1.696, 1.033, -5.660]
 
Complexity	1.068792
Adjusted Error	2.926965 cents
TE Error	0.846083 cents/octave
 
Unison Vectors
[-1, -1, -1, 0, 2⟩ (121:120)
[2, 0, -2, -1, 1⟩ (176:175)
[-3, -1, 1, 1, 1⟩ (385:384)
[-1, 0, 3, -3, 1⟩ (1375:1372)
[-2, -1, -2, 4, 0⟩ (2401:2400)
[1, 0, 1, -4, 2⟩ (2420:2401)

Subsets
q37, q25, q62, q12, q74, q99, q87, q49r, q50r, q124

4.6.5.7.11.13

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

Subsets
q25, q37f, q12f, q62, q50rf, q13rff, q49rff, q87, q74ff, q24rfff

4.6.5.7.11.13.17

Reduced Mapping
4	6	5	7	11	13	17	
[ ⟨	1	0	1	1	1	0	1	]
⟨	0	16	2	5	9	23	13	] ⟩
 
TE Generator Tunings (cents)
⟨2400.4701, 193.4599]
 
TE Step Tunings (cents)
⟨43.39350, 35.55764]
 
TE Tuning Map (cents)
⟨2400.470, 3095.359, 2787.390, 3367.770, 4141.609, 4449.578, 4915.449]
 
TE Mistunings (cents)
⟨0.470, -6.596, 1.076, -1.056, -9.709, 9.050, 10.494]
 
Complexity	1.129881
Adjusted Error	8.082725 cents
TE Error	1.977443 cents/octave
 
Unison Vectors
[0, 1, -1, 0, 1, -1, 0⟩ (66:65)
[1, 1, 1, -1, 0, 0, -1⟩ (120:119)
[1, 2, 0, 0, -1, -1, 0⟩ (144:143)
[-2, 1, 1, 1, 0, -1, 0⟩ (105:104)
[-1, 2, 2, 0, 0, -1, -1⟩ (225:221)
[-1, 1, 2, -2, 0, -1, 1⟩ (1275:1274)

Subsets
q25, q12f, q37f, q13rffg, q50rf, q62, q49rffg, q24rfffg, q38rreffg, q74ffg

4.6.5.7.11.13.17.19

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

Subsets
q25, q12fh, q37f, q13rffgh, q50rf, q62, q49rffgh, q24rfffghh, q38rreffgh, q74ffgh

4.6.5.7.11.13.17.19.23

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

Subsets
q25i, q12fhi, q37f, q13rffghii, q62, q50rfii, q49rffghii, q24rfffghhiii, q74ffghi, q38rreffghiii

4.9.… subgroups

Meansquared

Subgroup: 4.9.25

Comma list: 6561/6400

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

Mapping generators: ~4, ~9/64

Optimal tuning (CTE): ~4 = 2\1, ~9/4 = 1394.429

Supporting ETs: 12, 7, 19, 5, 31, 26, 17[+25], 43, 9[-25], 33[-25], 45, 29[+25], 8[+25], 22[+25]

Archsquared

Subgroup: 4.9.49

Comma list: 4096/3969

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

Mapping generators: ~4, ~9/64

Optimal tuning (CTE): ~9/4 = 1419.190

Supporting ETs: 5, 17, 22, 12, 7, 27, 32, 8, 39[+49], 29[+49], 9[+49], 19[+49], 37, 49

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 is named 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]]

sval mapping generators: ~8, ~9
gencom: [8 9/8; 64/63]

Optimal tuning (CTE): ~9/8 = 219.1898

Optimal ET sequence: ⟨16 17 15], ⟨33 35 31], ⟨148 …], ⟨181 …], ⟨214 …], ⟨247 …]

Badness: 0.0215 × 10-3

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 1728/1715 (S6/S7) and 540/539 (S12/S14), which imply 176/175 (S8/S10) as well as S11/S15 being tempered out. The tonic and the first three generator steps make a guanyin tetrad, hence the name.

Subgroup: 2.5.7/3.11/3

Comma list: 176/175 ([4 -2 -1 1⟩), 540/539 ([2 1 -2 -1⟩)

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

mapping generators: ~2, ~7/6

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

gencom: [2 7/6; 176/175 540/539]

Optimal tunings:

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 13th harmonic by the equivalences (12/11)3 = 13/10 and (15/14)3 = 16/13, therefore tempering out {S11/S12/S14/S15}. However, note that it is not supported by the 31 & 53 orwell extension dubbed "tridecimal orwell", but instead the less accurate winston (22f & 31), as orwell prefers slightly sharper tunings than guanyintet. 40edo remains an excellent tuning.

Subgroup: 2.5.7/3.11/3.13

Comma list: 176/175 ([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]]

mapping generators: ~2, ~12/7

Optimal tunings:

Optimal ET sequence: 9, 22, 31, 40, 71, 111, 151, 262c* using subgroup TE

* 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 -5/4 3 -1/4 7/4 -1/4], ⟨0 -1/4 -3 3/4 -21/4 19/4]]

gencom: [2 7/6; 144/143 176/175 196/195]

Optimal tunings:

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]]

mapping generators: ~2, ~13/12

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

gencom: [2 13/12; 56/55 78/77 91/90]

Optimal tunings:

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]]

mapping generators: ~2, ~5

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

gencom: [2 5; 225/224]

Optimal tunings:

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:

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]]

mapping generators: ~2, ~5/3

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

gencom: [2 6/5; 100/99 385/384]

Optimal tunings:

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]]

mapping generators: ~2, ~5/3

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

gencom: [2 6/5; 126/125]

Optimal tunings:

Optimal ET sequence: 4, 15, 19, 23, 27

RMS error: 0.8398 cents

Greeley

See also: Chromatic pairs § Greeley

Greeley is related to opossum as well as to nusecond.

Subgroup: 2.5/3.7/3.11/3

Comma list: 121/120 ([-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]]

gencom: [2 11/10; 121/120 126/125]

Optimal tunings:

Optimal ET sequence: 8, 15, 23, 54, 77, 100, 131*

* wart for 11/3

RMS error: 1.034 cents

Skateboard

See also: Chromatic pairs § Skateboard

Skateboard is related to thrasher.

Subgroup: 2.5/3.7/3.11.13/9

Comma list: 56/55 ([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 4/7 11/7 4 -6/7], ⟨0 0 -1 -3 -2 2]]

gencom: [2 6/5; 56/55 91/90 100/99]

Optimal tunings:

Optimal ET sequence: 11, 15, 19, 23, 42d, 65d

RMS error: 2.396 cents

Gariberttet

Gariberttet is the 2.5/3.7/3 altergene of sirius.

Gariberttet (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]]

gencom: [2 13/11; 275/273 847/845]

Optimal tunings:

Optimal ET sequence: 29, 33, 37, 41, 45, 49, 78, 94, 143*

* 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]]

gencom: [2 12/11; 3025/3024 3125/3087]

Optimal tunings:

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 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 3 4], ⟨0 -6 -10 -3 -5]]

Gencom mapping: [⟨1 -3 0 2 0 1], ⟨0 24/5 -6/5 -26/5 9/5 -1/5]]

gencom: [2 13/10; 121/120 169/168 275/273]

Optimal tunings:

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, is related to sensi.

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]]

gencom: [2 9/7; 245/243]

Optimal tunings:

Optimal ET sequence: 8, 11, 19, 30, 41, 49, 52, 145*, 166†, 197*†, 215†, 264*†

* wart for 5/3
† wart for 9/7

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 -1/6 5/6 0 0 -1/3], ⟨0 -1/2 -3/2 0 0 1]]

gencom: [2 6/5; 325/324]

Optimal tunings:

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]]

mapping generators: ~2, ~21/20

Gencom mapping: [⟨1 1/10 -4/5 11/10 1/5], ⟨0 -3/2 -1 3/2 1]]

gencom: [2 21/20; 100/99 245/242]

Optimal tunings:

Optimal ET sequence: 13, 14, 27, 41

RMS error: 1.245 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 1 & 4 temperament in the 2.7/3.11 subgroup, is related to orwell. The tonic and the first two generator steps make a mothwellsmic triad, hence the name.

Subgroup: 2.7/3.11

Comma list: 99/98 ([-1 -2 1⟩)

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

mapping generators: ~2, ~7/3

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

gencom: [2 7/6; 99/98]

Optimal tunings:

Optimal ET sequence: 4, 9, 13, 22, 79

RMS error: 1.064 cents

2.7/5.… subgroups

Terrain

"Terrain" redirects here. For the scale, see Terrain (scale).
See also: Chromatic pairs § Terrain

Terrain, the 6 & 21 temperament in the 2.7/5.9/5 subgroup, 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: 2.7/5.9/5

Comma list: 250047/250000

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

Gencom mapping: [⟨3 10/9 -7/9 2/9], ⟨0 -2/3 -1/3 2/3]]

gencom: [63/50 10/9; 250047/250000]

Optimal tuning (subgroup POTE): ~63/50 = 1\3, ~10/9 = 182.461

Optimal ET sequence: 6, 21, 27, 33, 105, 138, 171, 1848, 2019, 2190, 2361, 2532, 2703, 2874, 3045, 3216, 3387, 3558

RMS error: 0.00844 cents

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, 1001/1000

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

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

gencom: [2 13/10; 847/845 1001/1000]

Optimal tuning (subgroup POTE): ~2 = 1\1, ~13/10 = 454.556

Optimal ET sequence: 5, 8, 21, 29, 37, 66, 169, 235, 404c, 639c, 953bc

RMS error: 0.1613 cents

Naiadec

Subgroup: 2.7/5.11/5.13/5.17/5

Comma list: 170/169, 221/220, 847/845

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

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

gencom: [2 13/10; 170/169 221/220 847/845]

Optimal tuning (subgroup POTE): ~2 = 1\1, ~13/10 = 454.882

Optimal ET sequence: 5, 8, 21, 29, 95t, 124t

t wart for 17/5

RMS error: 0.7521 cents

2.25/7.… subgroups

Supramine

This extension approximates the 14:17:19:23:25 pentad in just six generator steps, at the cost of some accuracy. 25edo remains a strong tuning.

Subgroup: 2.17/7.19/7.23/7

Comma list: 323/322, 392/391

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

Optimal tunings:

  • Subgroup WE: ~2 = 1199.871 ¢, ~17/14 = 336.243 ¢
  • Subgroup CWE: ~2 = 1200.000 ¢, ~17/14 = 336.296 ¢

Optimal ET sequence: 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: [⟨1 -2 0 4 3], ⟨0 3 1 -2 -1]]

Optimal tunings:

  • Subgroup WE: ~2 = 1199.757 ¢, ~17/14 = 335.428 ¢
  • Subgroup CWE: ~2 = 1200.000 ¢, ~17/14 = 335.479 ¢

Optimal ET sequence: 7, 18, 25

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.

Subgroup: 2.11/5.13/5

Comma list: 2200/2197

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]]

gencom: [2 13/10; 2200/2197]

Optimal tuning (subgroup POTE): ~2 = 1\1, ~13/10 = 455.012

Optimal ET sequence: 21, 29, 153, 182, 211, 240, 269, 298, 327, 356, 385, 509, 741c, 1126c

RMS error: 0.0749 cents

Other 2-repeating subgroups

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 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 = 1200.022 ¢, ~17/14 = 335.793 ¢
  • Subgroup CWE: ~2 = 1200.000 ¢, ~17/14 = 335.785 ¢

Optimal ET sequence: 7, 18, 25

Badness (Sintel): 0.005

3/2.5/2.… subgroups

Main article: Half-prime subgroup

Hemihemi

Subgroup: 3/2.5/2.7/2

Comma list: 10976/10935

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

Optimal tuning (subgroup CTE): ~3/2 = 1\1edf, ~28/27 = 60.909

Supporting ETs: *23, *12, *11, *35, *34, *10, *13, *47, *9[+5/2], *14[-5/2], *45, *25, *21[+5/2], *8[+5/2]

Halftone

Main article: Halftone

Subgroup: 3/2.5/2.7/2

Comma list: 9604/9375

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

sval mapping generators: ~3/2, ~15/14

Optimal tuning (subgroup CTE): ~3/2 = 1\1edf, ~15/14 = 128.783

Supporting ETs: *5, *6, *7[+5/2, +7/2], *9[-5/2, --7/2], *11, *16, *17[+5/2], *23[+5/2, +7/2], *21[-7/2], *27, *28[+5/2], *38, *43[-7/2], *49

* wart for 3/2

3/2.5/2.7/2.11/2

Subgroup: 3/2.5/2.7/2.11/2

Comma list: 1232/1215, 27783/27500

Subgroup-val mapping: [⟨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

Supporting ETs: *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]

* wart for 3/2

3/2.5/2.7/2.11/2.13/2

Subgroup: 3/2.5/2.7/2.11/2.13/2

Comma list: 275/273, 1232/1215, 1323/1300

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

Optimal tuning (subgroup CTE): ~3/2 = 1\1edf, ~15/14 = 129.381

Supporting ETs: *11, *5, *16, *6, *27[-11/2]

* wart for 3/2

Semiwolf

Subgroup: 3/2.5/2.7/4

Comma list: 245/243

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

sval mapping generators: ~3/2, ~9/7

Optimal tuning (subgroup POTE): ~7/6 = 262.1728

Optimal ET sequence: 3edf, 5edf, 8edf

Semilupine

Subgroup: 3/2.5/2.7/4.11/4

Comma list: 100/99, 245/243

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

Optimal tuning (subgroup POTE): ~7/6 = 264.3771

Optimal ET sequence: 8edf, 13edf

Hemilycan

Subgroup: 3/2.5/2.7/4.11/4

Comma list: 245/243, 441/440

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

Optimal tuning (subgroup POTE): ~7/6 = 261.5939

Optimal ET sequence: 8edf, 11edf

3/2.5/4.… subgroups

Poseidon

This temperament will be subjected to renaming due to a conflict.

Subgroup: 3/2.5/4.11/8

Comma list: 121/120

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

gencom: [3/2 12/11; 121/120]

Optimal tuning (subgroup POTE): ~3/2, ~12/11 = 158.29

Optimal ET sequence: 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

Auk

Subgroup: 3/2.7.13

Comma list: 87808/85293

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

sval mapping generators: ~3/2, ~7

Optimal tuning (subgroup CTE): ~3/2 = 1\1edf, ~28/9 = 1950.859

Supporting ETs: *5, *6[+13], *7[-7, -13], *9, *11[+13], *13, *14, *17[-7, -13], *19[+13], *21[-7, -13], *22[-7], *23[+13], *25[-7, -13], *31[-7]

* wart for 3/2

Doubleton

Subgroup: 3/2.7.13

Comma list: 1352/1323

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

sval mapping generators: ~26/21, ~7

Optimal tuning (subgroup CTE): ~26/21 = 1\2edf, ~28/9 = 1971.772

Supporting ETs: *6, *10, *16, *14[-13], *8[+7], *22, *18[-13], *26, *24[-13], *28[+7], *20[+7], *36[-13], *12[+7, +13], *34[-13]

* wart for 3/2

5/2-repeating subgroups

Hyperion

Subgroup: 5/2.7.11

Comma list: [11 1 -5⟩

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

gencom: [5/2 125/88; 341796875/329832448]

Optimal tuning (subgroup POTE): ~5/2 = 1586.3137, ~125/88 = 593.6668

Supporting ETs: *5[-7], *8, *19[+7], *21[-7], *27[+7], *29[-7], *35[+7], *43[+7], *37[-7], *51[+7, +11], *45[-7], *59[+7, +11]

* wart for 5/2

Related temperament collections

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