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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 ==
== 2.3.35 subgroup ==
=== Shrub ===
=== Darian calendar ===
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.  
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]].


==== 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* }}


== 2.9.5.7 subgroup ==
<nowiki/>* Wart for 25
See also [[Jubilismic clan #Antikythera|antikythera]] and [[Hemimean clan #Isra|isra]].


=== Commatose ===
Badness (Sintel): 0.223
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
===== 2.3.25.23.41 subgroup =====
{{See also| Reversed meantone }}


[[Comma list]]: {{monzo| 28 -2 -19 8 }}, {{monzo| 9 -25 23 6 }}
Subgroup: 2.3.25.23.41


{{Mapping|legend=3| 1 9 6 13 | 0 -298 -188 -521 }}
Comma list: 82/81, 369/368, 576/575


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~531441/524288 = 23.4765
{{Mapping|legend=2| 1 0 11 -5 -1 | 0 1 -4 6 4 }}


{{Optimal ET sequence|legend=1| 460, 869, 1329 }}
Optimal tunings:
* WE: ~2 = 1199.3246{{c}}, ~3/2 = 705.2238{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 705.5491{{c}}


[[Badness]]: 0.611
{{Optimal ET sequence|legend=0| 5, 12, 17 }}


==== 2.9.5.7.11 ====
Badness (Sintel): 0.284
Subgroup: 2.9.5.7.11


Comma list: {{monzo| -7 7 -3 2 -4 }}, {{monzo| 17 0 -13 1 3 }}, {{monzo| 11 -2 -6 7 -3 }}
==== 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.


Sval mapping: {{mapping| 1 9 6 13 16 | 0 -298 -188 -521 -641 }}
Subgroup: 2.3.25.7


Optimal tuning (CTE): ~2 = 1\1, ~531441/524288 = 23.4767
Comma list: 50/49, 64/63


{{Optimal ET sequence|legend=1| 460, 869e, 1329, 1789, 3118 }}
{{Mapping|legend=2| 1 0 11 6 | 0 1 -4 -2 }}


Badness: 0.165
Optimal tunings:  
* WE: ~2 = 1197.6967{{c}}, ~3/2 = 705.6906{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 707.3438{{c}}


==== 2.9.5.7.11.13 ====
{{Optimal ET sequence|legend=0| 5, 12, 17, 39d }}
Subgroup: 2.9.5.7.11.13


Comma list: 123201/123200, 1016064/1015625, 2250423/2249390, 2599051/2598156
Badness (Sintel): 0.253


Sval mapping: {{mapping| 0 9 6 13 16 10 | -298 -188 -521 -641 -322 }}
===== 2.3.25.7.23 subgroup =====
Subgroup: 2.3.25.7.23


Optimal tuning (CTE): ~2 = 1\1, ~3575/3528 = 23.4767
Comma list: 50/49, 64/63, 162/161


{{Optimal ET sequence|legend=1| 460, 869e, 1329, 1789, 3118 }}
{{Mapping|legend=2| 1 0 11 6 -5 | 0 1 -4 -2 6 }}


Badness: 0.0564
Optimal tunings:  
* WE: ~2 = 1197.5577{{c}}, ~3/2 = 705.3089{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 706.7010{{c}}


=== Daemotertiaschis ===
{{Optimal ET sequence|legend=0| 5, 12, 17 }}
{{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
Badness (Sintel): 0.419


Comma list: 325/324, 375/374, 385/384, 595/594, 10985/10976
===== 2.3.25.7.23.41 subgroup =====
Subgroup: 2.3.25.7.23.41


{{Mapping|legend=2|1 1 11 -16 13 -18 20|0 3 -12 26 -11 30 -22}}
Comma list: 50/49, 64/63, 82/81, 162/161


Optimal tuning (CTE): ~2 = 1\1, 33/20 = 867.982
{{Mapping|legend=2| 1 0 11 6 -5 -1 | 0 1 -4 -2 6 4 }}


[[Support]]ing [[ET]]s: {{Optimal ET sequence|47, 65f, 112, 159, 206, 253}}
Optimal tunings:
* WE: ~2 = 1197.9403{{c}}, ~3/2 = 705.9522{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 707.0779{{c}}


=== Baldy ===
{{Optimal ET sequence|legend=0| 5, 12, 17, 39d }}
{{See also|Schismatic family #Garibaldi}}
{{See also|No-threes subgroup temperaments #Frostburn}}


Baldy results from taking every other generator of the [[garibaldi]] temperament. One of the best extension is 2.9.5.7.13 subgroup with mapping 13/8 to +10 whole tones, as well as the cassandra temperament.
Badness (Sintel): 0.450


[[Subgroup]]: 2.9.5.7
=== Hypnosis ===
Related temperaments: [[swetismic temperaments #Hypnos|hypnos]], [[alphatricot family #Alphatrimot|alphatrimot]].  


[[Comma list]]: 225/224, 3125/3087
[[Subgroup]]: 2.3.7.11/5.13


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


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 204.170
{{Mapping|legend=3| 1 0 -3 8 0 | 0 3 11 -13 7 }}
: mapping generators: ~2, ~13/9


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


Related temperament: [[Schismatic family #Garibaldi|Garibaldi]]
{{Optimal ET sequence|legend=1| 17, 36, 125f, 161f, 197f }}


==== 2.9.5.7.13 ====
[[Tp tuning #T2 tuning|RMS error]]: 0.5379 cents
{{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.  
=== 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]].


[[Subgroup]]: 2.9.5.7.13
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.


[[Comma list]]: [[225/224]], [[325/324]], [[640/637]]
Darian calendar was named by [[Eliora]] in 2023 after a certain calendar layout by the same name.


{{Mapping|legend=3| 1 0 15 25 -28 | 0 1 -4 -7 10 }}
[[Subgroup]]: 2.3.35.11.19


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


: [[gencom]]: [2 9/8; 225/224 325/324 640/637]
{{Mapping|legend=3| 4 5 18 13 18 | 0 8 15 5 -6 }}
: mapping generators: ~2240/1881, ~36/35


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 204.090
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2240/1881 = 299.9912{{c}}, ~36/35 = 50.2951{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2240/1881 = 300.0000{{c}}, ~36/35 = 50.2947{{c}}


{{Optimal ET sequence|legend=1| 6, 11, 17, 23, 29, 35, 41, 47, 100, 147, 488cd, 635cd }}
{{Optimal ET sequence|legend=1| 24, …, 524, 548, 572, 596, 620, 644, 668 }}


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


Related temperament: [[Schismatic family #Garibaldi|Cassandra]]
[[Subgroup]]: 2.3.7/5


==== Baldanders ====
[[Comma list]]: 50/49 ({{monzo| 1 0 -2 }})
Baldanders results from taking every other generator of the andromeda, with mapping 11/8 to -9 whole tones.


Subgroup: 2.9.5.7.11
{{Mapping|legend=3| 2 0 1 | 0 1 0 }}
: mapping generators: ~7/5, ~3


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


{{Mapping|legend=3| 1 3 3 4 5 | 0 1 -4 -7 -9 }}
{{Optimal ET sequence|legend=1| 2, 6, 8, 10, 12, 34, 46* }}


Optimal tuning (POTE): ~2 = 1\1, ~9/8 = 204.743
<nowiki/>* Wart for 7/5


{{Optimal ET sequence|legend=1| 6, 23de, 29, 35, 41 }}
=== Argentic ===
Argentic is the 2.3.7/5-subgroup temperament tempering out 5120/5103, the [[aberschisma]].


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


===== 2.9.5.7.11.13 =====
[[Comma list]]: 5120/5103 ({{monzo| 10 -6 -1 }})
Subgroup: 2.9.5.7.11.13


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


{{Mapping|legend=3| 1 3 3 4 5 2 | 0 1 -4 -7 -9 10 }}
[[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}}


Optimal tuning (POTE): ~2 = 1\1, ~9/8 = 204.414
{{Optimal ET sequence|legend=1| 12, 29, 41, 70, 321, 391, 461, 531, 601 }}


{{Optimal ET sequence|legend=1| 6, 23def, 29f, 35, 41, 47 }}
[[Badness]] (Sintel): 0.119


== 2.3.25 subgroup ==
==== Edson ====
{{See also| Chromatic pairs #Edson }}


=== Shrub ===
Edson is related to [[aberschismic family #Pele|pele]] and [[schismatic family #Andromeda|andromeda]].  
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
[[Subgroup]]: 2.3.7/5.11/5.13/5


Edo join: 17 & 12
[[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: [[2048/2025]]
{{Mapping|legend=3| 1 0 10 17 22 | 0 1 -6 -10 -13 }}


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


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 705.136
[[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}}


==== 2.3.23.25.41 subgroup ====
{{Optimal ET sequence|legend=1| 12, 17, 29 }}
''See also: [[Reversed meantone]]''


Edo join: 17 & 12
[[Tp tuning #T2 tuning|RMS error]]: 0.5102 cents


Comma list: 2048/2025, 576/575, 82/81
==== Haumea ====
{{See also| Chromatic pairs #Haumea }}


{{Mapping|legend=3| 1 1 1 7 3| 0 1 6 -4 4}}
Related temperaments include [[#Bridgetown|bridgetown]], [[aberschismic family #Namaka|namaka]], [[schismatic family #Hemigari|hemigari]], [[#Barbados|barbados]], and [[the Archipelago #Parizekmic|parizekmic]].


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 705.264
[[Subgroup]]: 2.3.7/5.11/5.13/5


===== Sburb =====
[[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 }})
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=3| 1 0 10 -6 -1 | 0 2 -12 9 3 }}


Edo join: 17 & 12
{{Mapping|legend=5| 1 0 -3/4 37/4 -27/4 -7/4 | 0 2 0 -12 9 3 }}
: mapping generators: ~2, ~26/15


Comma list: 64/63, 225/224, 162/161, 82/81, 177/175
[[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}}


{{Mapping|legend=3| 1 1 4 1 7 3 10| 0 1 -2 6 -4 4 -7}}
{{Optimal ET sequence|legend=1| 24, 29, 111, 140, 169, 198 }}


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 706.387
[[Tp tuning #T2 tuning|RMS error]]: 0.2668 cents


== 2.9.5.11 subgroup ==
=== Historical ===
=== Glacial ===
{{Distinguish|Historical temperaments}}
{{See also| Chromatic pairs #Glacial }}
{{Distinguish|History (temperament)}}


[[Subgroup]]: 2.9.5.11.13
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.


[[Comma list]]: 45/44, 65/64, 81/80
[[Subgroup]]: 2.3.7/5.11/5.13/5


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


{{Mapping|legend=5| 1 3/2 2 0 3 4 | 0 1/2 2 0 3 -2 }}
{{Mapping|legend=3| 1 2 0 1 2 | 0 -6 7 2 -9 }}
: mapping generators: ~2, ~21/20


: [[gencom]]: [2 9/8; 45/44 65/64 81/80]
[[Optimal tuning]]s:  
* [[Tp tuning|subgroup]] [[WE]]: ~2 = 1200.0242{{c}}, ~21/20 = 83.0177{{c}}
* [[Tp tuning|subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 83.0144{{c}}


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


== 2.9.7 subgroup ==
[[Comma list]]: 2401/2400 ({{monzo| -5 -1 2 }})
=== 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.


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}}, ...
=== Bridgetown ===
{{See also| Chromatic pairs #Bridgetown }}


==== 2.9.7.11 subgroup ====
Bridgetown, the 5 & 24 temperament in the 2.3.11/5.13/5 subgroup, is related to [[#Haumea|haumea]] and [[#Barbados|barbados]].  
Subgroup: 2.9.7.11


Comma basis: 896/891, 1331/1296
[[Subgroup]]: 2.3.11/5.13/5


Sval mapping: [{{val|1 1 -3 2}}, {{val|0 3 8 2}}]
[[Comma list]]: [[352/351]] ({{monzo| 5 -3 1 -1 }}), [[676/675]] ({{monzo| 2 -3 0 2 }})


Optimal tuning (CTE): ~16/11 = 870.966
{{Mapping|legend=3| 1 0 -6 -1 | 0 2 9 3 }}


{{Optimal ET sequence|legend=1| 7d, 11, 40, 51, 62 }}
{{Mapping|legend=5| 1 0 7/3 0 -11/3 4/3 | 0 2 -4 0 5 -1 }}
: mapping generators: ~2, ~26/15


== 2.9.7.11 subgroup ==
[[Optimal tuning]]s:
=== Apparatus ===
* [[Tp tuning|subgroup]] [[WE]]: ~2 = 1199.6281{{c}}, ~26/15 = 951.3060{{c}}
[[Subgroup]]: 2.9.7.11
* [[Tp tuning|subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~26/15 = 951.5580{{c}}


[[Comma list]]: 41503/41472, 322102/321489
{{Optimal ET sequence|legend=1| 5, 19*, 24, 29, 169, 198, 227, 256, 285, 314, 343 }}


{{Mapping|legend=3| 1 5 3 5 | 0 -19 -2 -16 }}
<nowiki/>* Wart for 11/5


: mapping generators: ~2, ~77/72
[[Tp tuning #T2 tuning|RMS error]]: 0.2513 cents


{{Mapping|legend=5| 1 5/2 0 3 5 | 0 -19/2 0 -2 -16 }}
=== Blackweed ===
Blackweed is a [[restriction]] of undecimal [[blackwood]] as it tempers out [[256/243]] alike but in the 2.3.11/7 subgroup. [[20edo]] is close to the optimum, which has 4\20 as the period and 420{{c}} as the generator.


: [[gencom]]: [2 77/72; 41503/41472 322102/321489]
[[Subgroup]]: 2.3.11/7


[[Optimal tuning]] ([[CTE]]): ~77/72 = 115.5685
[[Comma list]]: 256/243 ({{monzo| 8 -5 }})


{{Optimal ET sequence|legend=1| 10e, 21, 31, 52, 83, 135, 353, 488, 623 }}
{{Mapping|legend=3| 5 8 0 | 0 0 1 }}
: mapping generators: ~9/8, ~11/7


[[Badness]]: 0.00263
[[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 }}


=== Joan ===
{{Optimal ET sequence|legend=1| 15, 20, 35b, 55b }}
{{See also| Chromatic pairs #Joan }}


Joan is related to [[casablanca]] as well as to [[orwell]].
=== Pepperoni ===
{{Main| Parapyth }}
{{See also| Chromatic pairs #Pepperoni }}


[[Subgroup]]: 2.9.7.11
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.


[[Comma list]]: 99/98, 9317/9216
[[Subgroup]]: 2.3.11/7.13/7


{{Mapping|legend=3| 1 0 1 3 | 0 7 4 1 }}
[[Comma list]]: 352/351 ({{monzo| 5 -3 1 -1 }}), 364/363 ({{monzo| 2 -1 -2 1 }})


{{Mapping|legend=5| 1 0 0 1 3 | 0 7/2 0 4 1 }}
{{Mapping|legend=3| 1 0 7 12 | 0 1 -4 -7 }}


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


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


{{Optimal ET sequence|legend=1| 11, 20, 31, 42, 115bd, 157bd }}
{{Optimal ET sequence|legend=1| 12, 17, 29, 75, 104 }}


[[Tp tuning #T2 tuning|RMS error]]: 1.424 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.3789 cents


=== Machine ===
=== Barbados ===
Machine is every other step of [[supra]], most interesting for its scale patterns.  
The [[minimax tuning]] for this makes the generator the cube root of 20/13, or 248.5953 cents. Edos which may be used for it are [[24edo]], [[29edo]], [[53edo]] and [[111edo]], with [[mos scale]]s of size 5, 9, 14, 19, 24 and 29 making for a good variety of scales.


[[Subgroup]]: 2.9.7.11
[[Subgroup]]: 2.3.13/5


[[Comma list]]: 64/63, 99/98
[[Comma list]]: 676/675 = {{monzo| 2 -3 2 }}


{{Mapping|legend=3| 1 0 6 13 | 0 1 -1 -3 }}
[[Sval]] [[mapping]]: [{{val| 1 0 -1 }}, {{val| 0 2 3 }}]


: sval mapping generators: ~2, ~9
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~2 = 1\1, ~15/13 = 248.621


{{Mapping|legend=5| 1 3/2 0 3 4 | 0 1/2 0 -1 -3 }}
{{Optimal ET sequence|legend=1| 5, 9, 14, 19, 24, 29, 53, 82, 111, 140, 251, 362 }}


: [[gencom]]: [2 8/7; 64/63 99/98]
[[Badness]]: 0.002335


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


{{Optimal ET sequence|legend=1| 5, 6, 11, 17, 28 }}
==== Tobago ====
{{See also| Chromatic pairs #Tobago }}


[[Badness]]: 0.00233
Tobago, the 10 &amp; 14 temperament in the 2.3.11.13/5 subgroup, extends [[neutral]] and [[barbados]].  


=== Penta a.k.a. mechanism ===
[[Subgroup]]: 2.3.11.13/5
Penta or mechanism is the 8 &amp; 11 temperament in the 2.9.7.11 subgroup.  


[[Subgroup]]: 2.9.7.11
[[Comma list]]: [[243/242]], [[676/675]]


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


{{Mapping|legend=3| 1 0 -1 6 | 0 5 6 -4 }}
{{Mapping|legend=5| 2 4 -2 0 9 2 | 0 -2 3/2 0 -5 -3/2 }}
: [[gencom]]: [55/39 15/13; 243/242 676/675]


: sval mapping generators: ~2, ~14/9
[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~55/39 = 1\2, ~15/13 = 249.312


{{Mapping|legend=5| 1 5/2 0 5 2 | 0 -5/2 0 -6 4 }}
{{Optimal ET sequence|legend=1| 10, 14, 24, 58, 82, 130 }}


: [[gencom]]: [2 9/7; 896/891 26411/26244]
[[Tp tuning #T2 tuning|RMS error]]: 0.3533 cents


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~14/9 = 761.3782
==== Pakkanian hemipyth ====
[[Subgroup]]: 2.3.11.13/5.17


{{Optimal ET sequence|legend=1| 8, 11, 30, 41, 52 }}
[[Comma list]]: 221/220, 243/242, 289/288


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


[[Badness]]: 0.00439
[[Optimal tuning]]s:  
* [[Tp tuning|subgroup CTE]]: ~17/12 = 1\2, ~26/15 = 950.7656 (~15/13 = 249.2344)
* [[Tp tuning|subgroup CWE]]: ~17/12 = 1\2, ~26/15 = 950.6011 (~15/13 = 249.3989)


Scales: [[penta5]], [[penta8]], [[penta11]], [[penta19]]
{{Optimal ET sequence|legend=1| 10, 14, 24, 106, 130, 154, 178*, 202* }}
: <nowiki />* wart for 13/5


== 2.9.7.13.17 subgroup ==
=== Oceanfront ===
Related temperaments: [[Archytas clan #Superpyth|superpyth]], [[Archytas clan #Ultrapyth|ultrapyth]]


=== Novisept ===
[[Subgroup]]: 2.3.7.13/5
Novisept is generated by a one-cent-flat 9/7, such that stacking 5 of them gives you 7/4. It can be formed by doubling both generator and period of [[gizzard]].


[[Subgroup]]: 2.9.7.13.17
[[Comma list]]: 64/63, 91/90


[[Comma list]]: 729/728, 442/441, 833/832
{{Mapping|legend=3| 1 0 6 -5 | 0 1 -2 4 }}


{{Mapping|legend=3| 1 1 1 -1 3| 0 6 5 13 3 }}
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~3/2 = 713.910


[[Optimal tuning]] ([[CWE]]): ~2 = 1\1, ~9/7 = 433.836
{{Optimal ET sequence|legend=1| 5, 22, 27, 32, 37 }}


Badness (Dirichlet): 0.142
[[Tp tuning #T2 tuning|RMS error]]: 2.063 cents


== 2.9.11 subgroup ==
Scales: [[Oceanfront scales]]
=== Demon ===
Demon is a temperament which equates 3 [[11/9]] with [[16/9]], or equivalently 3 [[18/11]] with [[9/8]], tempering out [[1331/1296]]. This results in [[11/9]] being tuned flat to a supraminor third, and [[27/22]] being tuned sharp to a submajor third. It was discovered by [[User:CompactStar|CompactStar]] while searching for temperaments assosciated with the [[7L 4s]] ("daemotonic") MOS, known for its lack of representation of simple temperaments. The optimal tuning for demon temperament is near the basic tuning of 7L 4s (13\18), and indeed [[18edo]] supports demon temperament.


[[Subgroup]]: 2.9.11
=== Seventeen-cot ===
Seventeen-cot is a rank-2 temperament in the 2.3.13/5 and 2.3.11/5.13/5 subgroups. It tempers out the [[Tendoartisma]] in the 2.3.13/5 subgroup. It can be generated with a ~2/1 octave and a ~2250/2197 or ~169/165 generator which is a 17th of a ~3/2 perfect fifth. It can be described as the 29 & 146 temperament in these subgroups.


[[Comma list]]: [[1331/1296]]
====2.3.13/5 subgroup====


{{Mapping|legend=2|1 1 2|0 3 2}}
Comma basis: {{monzo| -6 -11 17 }} (2.3.13/5)


[[Optimal tuning]] ([[CTE]]): ~[[18/11]] = 870.060
edo join: 29 & 146


{{Optimal ET sequence|legend=1|4, 7, 11, 18, 29, 76e}}
{{Mapping|legend=3| 1 1 1 | 0 17 11 }}
: mapping generators: ~2, ~2250/2197


=== Genius ===
Optimal tunings:
* WE: ~2 = 1200.0001354{{c}}, ~2250/2197 = 41.2914993{{c}}
: error map: {{val| +0.0001354 +0.0006234 -0.0073197}}
* CWE: ~2 = 2/1, ~2250/2197 = 41.2915011{{c}}
: error map: {{val| 0.0000000 +0.0005177 -0.0074359}}


Named after the genius in Roman religion, following the demon (daimon) in Greek mythology.
edos: 29, 465, 494, 436, 523, 407, 378, 349, 30[-3], 28[+3], 320, 291, 59[-3], 262


[[Subgroup]]: 2.9.11
Badness (Sintel): 0.064


[[Comma list]]: [[131769/131072]]
====2.3.11/5.13/5 subgroup====


{{Mapping|legend=2|1 1 4|0 4 -1}}
Comma basis: 225000/224939, 43940/43923


[[Optimal tuning]] ([[CTE]]): ~[[16/11]] = 650.863
edo join: 29 & 146


{{Optimal ET sequence|legend=1|9, 11, 24, 59, 83, 142, 225, 367}}[-11], 592[-11], 959[-9, --11], 1326[-9, --11]
{{Mapping|legend=3| 1 1 1 1 | 0 17 4 11 }}
: mapping generators: ~2, ~169/165


== 2.9.15.7 subgroup ==
Optimal tunings:
=== Stacks (a.k.a. 2magic) ===
* WE: ~2 = 1199.9934923{{c}}, ~169/165 = 41.2918271{{c}}
Stacks, the 11 &amp; 30 temperament in the 2.9.15.7.11.13 subgroup, is every other step of [[magic]].  
: error map: {{val| -0.0065077 -0.0004485 +0.1565720 -0.0103579}}
* CWE: ~2 = 2/1, ~169/165 = 41.2917463{{c}}
: error map: {{val| 0.0000000 +0.0046870 +0.1627569 -0.0043781}}


[[Subgroup]]: 2.9.15.7
edos: 29, 465, 494, 436, 523, 407, 378, 349, 320, 291, 30[-3], 262, 28[+3], 233


[[Comma list]]: 225/224, 245/243
Badness (Sintel): 0.080


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


: sval mapping generators: ~2, ~14/9
[[Subgroup]]: 2.3.17/5


{{Mapping|legend=5| 1 5/2 5/2 5 | 0 -5/2 -1/2 -6 }}
[[Comma list]]: 136/135 ({{monzo| 3 -3 1 }})


: [[gencom]]: [2 9/7; 225/224 245/243]
{{Mapping|legend=3| 1 0 -3 | 0 1 3 }}
: mapping generators: ~2, ~3


[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~2 = 1\1, ~14/9 = 760.704
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.2838{{c}}, ~3/2 = 704.4600{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.5286{{c}}


{{Optimal ET sequence|legend=1| 8, 11, 30, 41, 71, 93, 112c, 134c, 175c }}
{{Optimal ET sequence|legend=1| 5, 12, 17, 46, 63, 143 }}


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


==== 2.9.15.7.11 ====
[[Subgroup]]: 2.3.19/7
Subgroup: 2.9.15.7.11


Comma list: 100/99, 225/224, 245/243
[[Comma list]]: [[57/56]] ({{Monzo| -3 1 1 }})


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


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


: gencom: [2 9/7; 100/99 225/224 245/243]
{{Optimal ET sequence|legend=1| 5, 7, 12, 19, 31*, 50* }}


Optimal tuning (subgroup POTE): ~2 = 1\1, ~14/9 = 761.393
<nowiki/>* wart for 19/7


Optimal ET sequence: {{Optimal ET sequence| 8, 11, 30, 41, 52, 93, 145, 342bce }}
[[Badness]] (Sintel): 0.082


RMS error: 1.226 cents
== 2.9.… subgroups ==
See also [[Jubilismic clan #Antikythera|antikythera]] and [[Hemimean clan #Isra|isra]].  


==== 2.9.15.7.11.13 ====
=== Commatose ===
Subgroup: 2.9.15.7.11.13
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]].


Comma list: 100/99, 105/104, 144/143, 196/195
[[Subgroup]]: 2.9.5.7


Sval mapping: {{mapping| 1 0 2 -1 6 -2 | 0 5 3 6 -4 9 }}
[[Comma list]]: {{monzo| 28 -2 -19 8 }}, {{monzo| 9 -25 23 6 }}


{{Mapping|legend=4| 1 5/2 5/2 5 2 7 | 0 -5/2 -1/2 -6 4 -9 }}
{{Mapping|legend=3| 1 9 6 13 | 0 -298 -188 -521 }}


: gencom: [2 9/7; 100/99 105/104 144/143 196/195]
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~531441/524288 = 23.4765


Optimal tuning (subgroup POTE): ~2 = 1\1, ~14/9 = 761.023
{{Optimal ET sequence|legend=1| 460, 869, 1329 }}


Optimal ET sequence: {{Optimal ET sequence| 11, 30, 41, 153cdef, 194cdef, 235cdef }}
[[Badness]]: 0.611


RMS error: 1.540 cents
==== 2.9.5.7.11 ====
Subgroup: 2.9.5.7.11


== 2.9.21 subgroup ==
Comma list: {{monzo| -7 7 -3 2 -4 }}, {{monzo| 17 0 -13 1 3 }}, {{monzo| 11 -2 -6 7 -3 }}
=== 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
Sval mapping: {{mapping| 1 9 6 13 16 | 0 -298 -188 -521 -641 }}


[[Comma list]]: 1029/1024
Optimal tuning (CTE): ~2 = 1\1, ~531441/524288 = 23.4767


{{Mapping|legend=3| 1 2 4 | 0 3 1 }}
{{Optimal ET sequence|legend=1| 460, 869e, 1329, 1789, 3118 }}


: sval mapping generators: ~2, ~21/16
Badness: 0.165


{{Mapping|legend=5| 1 1 0 3 | 0 3/2 0 -1/2 }}
==== 2.9.5.7.11.13 ====
Subgroup: 2.9.5.7.11.13


: [[gencom]]: [2 21/16; 1029/1024]
Comma list: 123201/123200, 1016064/1015625, 2250423/2249390, 2599051/2598156


[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~2 = 1\1, ~21/16 = 467.375
Sval mapping: {{mapping| 0 9 6 13 16 10 | -298 -188 -521 -641 -322 }}


{{Optimal ET sequence|legend=1| 5, 13, 18, 41, 59, 77, 95 }}
Optimal tuning (CTE): ~2 = 1\1, ~3575/3528 = 23.4767


[[Tp tuning #T2 tuning|RMS error]]: 0.3202 cents
{{Optimal ET sequence|legend=1| 460, 869e, 1329, 1789, 3118 }}


==== 2.9.5.21 ====
Badness: 0.0564
''Lookalike temperament: [[Dual-fifth_temperaments#Dual-3_A-Team|Dual-3 A-Team]]''


Subgroup: 2.9.5.21
=== Daemotertiaschis ===
{{See also|Schismatic family#Tertiaschis}}
Daemotertiaschis is produced by taking every other generator of tertiaschis, and the subgroup is chosen so it tempers out exactly the same commas. It is notable due to offering a [[7L 4s|daemotonic 7L 4s]] scale of reasonable hardness (hence the name ― daemo- + tertiaschis), which is notoriously difficult to approximate with simple JI or RTT methods.


[[Comma]] list: 81/80, 1029/1024
Subgroup: 2.9.5.7.33.13.17


Sval mapping: {{mapping| 1 2 0 4 | 0 3 6 1 }}
Comma list: 325/324, 375/374, 385/384, 595/594, 10985/10976


Mapping generators: ~2, ~21/16
{{Mapping|legend=2|1 1 11 -16 13 -18 20|0 3 -12 26 -11 30 -22}}


Optimal ([[Lp tuning|POL2]]) generator: 464.3865
Optimal tuning (CTE): ~2 = 1\1, 33/20 = 867.982


{{Optimal ET sequence|legend=1| 13, 18, 31, 44 }}
[[Support]]ing [[ET]]s: {{Optimal ET sequence|47, 65f, 112, 159, 206, 253}}


===== 2.9.5.21.11 =====
=== Baldy ===
Subgroup: 2.9.5.21.11
{{See also|Schismatic family #Garibaldi}}
{{See also|No-threes subgroup temperaments #Frostburn}}


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


Sval mapping: {{mapping| 1 2 0 4 5 | 0 3 6 1 -4 }}
[[Subgroup]]: 2.9.5.7


{{Mapping|legend=4| 1 1 0 3 5 | 0 3/2 6 -1/2 -4 }}
[[Comma list]]: 225/224, 3125/3087


: gencom: [2 21/16; 81/80 99/98 385/384]
{{Mapping|legend=3| 1 3 3 4 | 0 1 -4 -7 }}


Optimal tuning (subgroup POTE): ~2 = 1\1, ~21/16 = 463.956
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 204.170


{{Optimal ET sequence|legend=1| 5, 13, 31 }}
{{Optimal ET sequence|legend=1| 6, 29, 35, 41, 47 }}


==== B-team ====
Related temperament: [[Schismatic family #Garibaldi|Garibaldi]]
B-team (23 & 41) is every other step of [[rodan]].


Subgroup: 2.9.15.21.33
==== 2.9.5.7.13 ====
{{See also|Chromatic pairs #Baldy}}


Comma list: 245/243, 385/384, 441/440
Baldy is every other step of [[garibaldi]], without the mapping of prime 11. It can be described as the 6 &amp; 35 temperament.


Sval mapping: {{mapping| 1 2 0 4 7 | 0 3 10 1 -5 }}
[[Subgroup]]: 2.9.5.7.13


Optimal tuning (subgroup POTE): ~2 = 1\1, ~21/16 = 468.918
[[Comma list]]: [[225/224]], [[325/324]], [[640/637]]


{{Optimal ET sequence|legend=1| 5, 13c, 18, 23, 41, 64, 87, 151 }}
{{Mapping|legend=3| 1 0 15 25 -28 | 0 1 -4 -7 10 }}


== 2.75.85 subgroup ==
{{Mapping|legend=5| 1 3/2 3 4 0 2 | 0 1/2 -4 -7 0 10 }}
=== 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.
: [[gencom]]: [2 9/8; 225/224 325/324 640/637]


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.
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 204.090


[[Subgroup]]: 2.75.85
{{Optimal ET sequence|legend=1| 6, 11, 17, 23, 29, 35, 41, 47, 100, 147, 488cd, 635cd }}


[[Comma list]]: [[24576/24565]] ({{monzo| 13 1 -3 }})
[[Tp tuning #T2 tuning|RMS error]]: 0.5999 cents


{{Mapping|legend=3| 1 2 5 | 0 3 1 }}
Related temperament: [[Schismatic family #Garibaldi|Cassandra]]
: mapping generators: ~2, ~85/32


[[Optimal tuning]]s:
==== Baldanders ====
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.9692{{c}}, ~85/64 = 491.5853{{c}}
Baldanders results from taking every other generator of the andromeda, with mapping 11/8 to -9 whole tones.
* [[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* }}
Subgroup: 2.9.5.7.11


==== 2.75.9/7.85 subgroup ====
Comma list: 100/99, 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=3| 1 3 3 4 5 | 0 1 -4 -7 -9 }}


Comma list: 2025/2023 ({{monzo| 2 -2 1 0 }}), 24576/24565 ({{monzo| 13 1 0 -3 }})
Optimal tuning (POTE): ~2 = 1\1, ~9/8 = 204.743


{{Mapping|legend=2| 1 2 6 5 | 0 3 -4 1 }}
{{Optimal ET sequence|legend=1| 6, 23de, 29, 35, 41 }}


Optimal tunings:  
Related temperament: [[Schismatic family #Garibaldi|Andromeda]]
* 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* }}
===== 2.9.5.7.11.13 =====
Subgroup: 2.9.5.7.11.13


== 4.3.5 subgroup ==
Comma list: 100/99, 144/143, 225/224, 245/242
=== Tetrahanson ===
{{Main| Tetrahanson }}


[[Subgroup]]: 4.3.5
{{Mapping|legend=3| 1 3 3 4 5 2 | 0 1 -4 -7 -9 10 }}


[[Comma list]]: 15625/15552
Optimal tuning (POTE): ~2 = 1\1, ~9/8 = 204.414


{{Mapping|legend=3| 1 3 3 | 0 -6 -5 }}
{{Optimal ET sequence|legend=1| 6, 23def, 29f, 35, 41, 47 }}


: Mapping generators: ~4, ~5/3
=== Glacial ===
{{See also| Chromatic pairs #Glacial }}


[[Optimal tuning]] ([[CTE]]): ~4 = 2\1, ~5/3 = 882.941
[[Subgroup]]: 2.9.5.11.13


[[Support]]ing [[ET]]s: {{EDs|19, 106, 87, 68, 11, 8, 125, 49, 30, 27, 117, 46, 41b, 79|equave=4}}
[[Comma list]]: 45/44, 65/64, 81/80


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


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


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


{{Mapping|legend=3| 1 1 2 | 0 -1 -4 }}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 186.151


: Mapping generators: ~4, ~4/3
{{Optimal ET sequence|legend=1| 6, 13, 45be, 58bce, 71bce, 84bce }}


[[Optimal tuning]] ([[POTE]]): 4 = 2400.0, ~4/3 = 503.761
[[Tp tuning #T2 tuning|RMS error]]: 2.887 cents


[[Support]]ing [[ET]]s: {{EDs|5, 9, 14, 19, 24, 43, 62, 81, 100|equave=4}}
Music:
* ''[[Thundersnow]]'' - [[Sevish]] (2021)


=== Tetramagic ===
=== 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.


[[Subgroup]]: 4.3.5
Subgroup: 2.9.7


[[Comma list]]: 3125/3072
Comma basis: 44957696/43046721


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


: Mapping generators: ~4, ~5/4
Optimal tuning (CTE): ~729/448 = 870.792


[[Optimal tuning]] ([[POTE]]): 4 = 2400.0, ~5/4 = 380.059
{{Optimal ET sequence|legend=1|7d, 11, 18d, 29, 40, 62}}, ...


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


=== Blacktetra ===
Comma basis: 896/891, 1331/1296


[[Subgroup]]: 4.3.5
Sval mapping: [{{val|1 1 -3 2}}, {{val|0 3 8 2}}]


[[Comma list]]: 256/243
Optimal tuning (CTE): ~16/11 = 870.966


{{Mapping|legend=3| 5 4 6 | 0 0 -1 }}
{{Optimal ET sequence|legend=1| 7d, 11, 40, 51, 62 }}


: Mapping generators: ~4, ~16/15
=== Apparatus ===
[[Subgroup]]: 2.9.7.11


[[Optimal tuning]] ([[POTE]]): 1\5ed4 = 480.0, ~16/15 = 80.4062
[[Comma list]]: 41503/41472, 322102/321489


[[Support]]ing [[ET]]s: {{EDs|5, 10, 15, 20, 25, 30, 55, 85, 115|equave=4}}
{{Mapping|legend=3| 1 5 3 5 | 0 -19 -2 -16 }}


== 4.6.5 subgroup ==
: mapping generators: ~2, ~77/72
=== Meanquad ===
{{Main| Meanquad }}


[[Subgroup]]: 4.6.5
{{Mapping|legend=5| 1 5/2 0 3 5 | 0 -19/2 0 -2 -16 }}


[[Comma list]]: [[81/80]] = {{monzo| -4 4 -1 }}
: [[gencom]]: [2 77/72; 41503/41472 322102/321489]


{{Mapping|legend=3| 1 0 -4| 0 1 4 }}
[[Optimal tuning]] ([[CTE]]): ~77/72 = 115.5685


: mapping generators: ~4, ~6
{{Optimal ET sequence|legend=1| 10e, 21, 31, 52, 83, 135, 353, 488, 623 }}


[[Optimal tuning]] (subgroup [[CTE]]): ~4 = 2\1, ~3/2 = 697.214
[[Badness]]: 0.00263


[[Support]]ing [[ET]]s: *7, *10, *11[-5], *13[+5], *17, *24, *27[+5], *31, *38, *41, *45, *52, *55, *69
=== Joan ===
{{See also| Chromatic pairs #Joan }}


<nowiki />* Wart for 4
Joan is related to [[casablanca]] as well as to [[orwell]].


==== 4.6.5.7 subgroup (tetrominant) ====
[[Subgroup]]: 2.9.7.11
[[Subgroup]]: 4.6.5.7


[[Comma list]]: [[36/35]] = {{monzo| 0 2 -1 -1 }}, [[64/63]] = {{monzo| 4 -2 0 -1 }}
[[Comma list]]: 99/98, 9317/9216


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


[[Optimal tuning]] (subgroup [[CTE]]): ~4 = 2\1, ~3/2 = 699.622
{{Mapping|legend=5| 1 0 0 1 3 | 0 7/2 0 4 1 }}


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


<nowiki />* Wart for 4
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~11/8 = 542.672 cents


=== Fourwar ===
{{Optimal ET sequence|legend=1| 11, 20, 31, 42, 115bd, 157bd }}
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.
[[Tp tuning #T2 tuning|RMS error]]: 1.424 cents
 
=== Machine ===
Machine is every other step of [[supra]], most interesting for its scale patterns.
 
[[Subgroup]]: 2.9.7.11
 
[[Comma list]]: 64/63, 99/98
 
{{Mapping|legend=3| 1 0 6 13 | 0 1 -1 -3 }}
 
: sval mapping generators: ~2, ~9
 
{{Mapping|legend=5| 1 3/2 0 3 4 | 0 1/2 0 -1 -3 }}


{{Todo|inline=1|cleanup}}
: [[gencom]]: [2 8/7; 64/63 99/98]


<pre>
[[Optimal tuning]]s:
Reduced Mapping
* [[CTE]]: ~2 = 1\1, ~9/8 = 216.9128
4 6 5
* [[POTE]]: ~2 = 1\1, ~9/8 = 214.3843
[ ⟨ 1 0 1 ]
 
⟨ 0 16 2 ] ⟩
{{Optimal ET sequence|legend=1| 5, 6, 11, 17, 28 }}
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]
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~14/9 = 761.3782


==== 4.6.5.7.11 ====
{{Optimal ET sequence|legend=1| 8, 11, 30, 41, 52 }}
<pre>
 
Reduced Mapping
[[Tp tuning #T2 tuning|RMS error]]: 0.4262 cents
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
[[Badness]]: 0.00439
q37, q25, q62, q12, q74, q99, q87, q49r, q50r, q124
</pre>


==== 4.6.5.7.11.13 ====
Scales: [[penta5]], [[penta8]], [[penta11]], [[penta19]]


<pre>
=== Novisept ===
Reduced Mapping
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]].
4 6 5 7 11 13
 
[ ⟨ 1 0 1 1 1 0 ]
[[Subgroup]]: 2.9.7.13.17
⟨ 0 16 2 5 9 23 ] ⟩
 
[[Comma list]]: 729/728, 442/441, 833/832
TE Generator Tunings (cents)
 
⟨2401.2305, 193.5378]
{{Mapping|legend=3| 1 1 1 -1 3| 0 6 5 13 3 }}
 
TE Step Tunings (cents)
[[Optimal tuning]] ([[CWE]]): ~2 = 1\1, ~9/7 = 433.836
⟨42.79107, 35.98524]
 
Badness (Dirichlet): 0.142
TE Tuning Map (cents)
 
⟨2401.230, 3096.606, 2788.306, 3368.920, 4143.071, 4451.371]
=== Demon ===
Demon is a temperament which equates 3 [[11/9]] with [[16/9]], or equivalently 3 [[18/11]] with [[9/8]], tempering out [[1331/1296]]. This results in [[11/9]] being tuned flat to a supraminor third, and [[27/22]] being tuned sharp to a submajor third. It was discovered by [[User:CompactStar|CompactStar]] while searching for temperaments assosciated with the [[7L 4s]] ("daemotonic") MOS, known for its lack of representation of simple temperaments. The optimal tuning for demon temperament is near the basic tuning of 7L 4s (13\18), and indeed [[18edo]] supports demon temperament.
TE Mistunings (cents)
 
⟨1.230, -5.349, 1.992, 0.094, -8.247, 10.843]
[[Subgroup]]: 2.9.11
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
[[Comma list]]: [[1331/1296]]
q25, q37f, q12f, q62, q50rf, q13rff, q49rff, q87, q74ff, q24rfff
</pre>


==== 4.6.5.7.11.13.17 ====
{{Mapping|legend=2|1 1 2|0 3 2}}
<pre>
 
Reduced Mapping
[[Optimal tuning]] ([[CTE]]): ~[[18/11]] = 870.060
4 6 5 7 11 13 17
 
[ ⟨ 1 0 1 1 1 0 1 ]
{{Optimal ET sequence|legend=1|4, 7, 11, 18, 29, 76e}}
⟨ 0 16 2 5 9 23 13 ] ⟩
 
=== Genius ===
TE Generator Tunings (cents)
 
⟨2400.4701, 193.4599]
Named after the genius in Roman religion, following the demon (daimon) in Greek mythology.
 
TE Step Tunings (cents)
[[Subgroup]]: 2.9.11
⟨43.39350, 35.55764]
 
[[Comma list]]: [[131769/131072]]
TE Tuning Map (cents)
 
⟨2400.470, 3095.359, 2787.390, 3367.770, 4141.609, 4449.578, 4915.449]
{{Mapping|legend=2|1 1 4|0 4 -1}}
 
TE Mistunings (cents)
[[Optimal tuning]] ([[CTE]]): ~[[16/11]] = 650.863
⟨0.470, -6.596, 1.076, -1.056, -9.709, 9.050, 10.494]
Complexity 1.129881
Adjusted Error 8.082725 cents
TE Error 1.977443 cents/octave
Unison Vectors
[0, 1, -1, 0, 1, -1, 0⟩ (66:65)
[1, 1, 1, -1, 0, 0, -1⟩ (120:119)
[1, 2, 0, 0, -1, -1, 0⟩ (144:143)
[-2, 1, 1, 1, 0, -1, 0⟩ (105:104)
[-1, 2, 2, 0, 0, -1, -1⟩ (225:221)
[-1, 1, 2, -2, 0, -1, 1⟩ (1275:1274)


Subsets
{{Optimal ET sequence|legend=1|9, 11, 24, 59, 83, 142, 225, 367}}[-11], 592[-11], 959[-9, --11], 1326[-9, --11]
q25, q12f, q37f, q13rffg, q50rf, q62, q49rffg, q24rfffg, q38rreffg, q74ffg
</pre>


==== 4.6.5.7.11.13.17.19 ====
=== Stacks (a.k.a. 2magic) ===
<pre>
Stacks, the 11 &amp; 30 temperament in the 2.9.15.7.11.13 subgroup, is every other step of [[magic]].  
Reduced Mapping
 
4 6 5 7 11 13 17 19
[[Subgroup]]: 2.9.15.7
[ ⟨ 1 0 1 1 1 0 1 1 ]
 
⟨ 0 16 2 5 9 23 13 14 ] ⟩
[[Comma list]]: 225/224, 245/243
 
TE Generator Tunings (cents)
{{Mapping|legend=3| 1 0 2 -1 | 0 5 3 6 }}
⟨2399.9219, 193.3952]
 
: sval mapping generators: ~2, ~14/9
TE Step Tunings (cents)
 
⟨44.14256, 35.03670]
{{Mapping|legend=5| 1 5/2 5/2 5 | 0 -5/2 -1/2 -6 }}
 
TE Tuning Map (cents)
: [[gencom]]: [2 9/7; 225/224 245/243]
⟨2399.922, 3094.324, 2786.712, 3366.898, 4140.479, 4448.090, 4914.060, 5107.455]
 
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~2 = 1\1, ~14/9 = 760.704
TE Mistunings (cents)
 
⟨-0.078, -7.631, 0.399, -1.928, -10.839, 7.562, 9.104, 9.942]
{{Optimal ET sequence|legend=1| 8, 11, 30, 41, 71, 93, 112c, 134c, 175c }}
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
[[Tp tuning #T2 tuning|RMS error]]: 1.074 cents
q25, q12fh, q37f, q13rffgh, q50rf, q62, q49rffgh, q24rfffghh, q38rreffgh, q74ffgh
</pre>


==== 4.6.5.7.11.13.17.19.23 ====
==== 2.9.15.7.11 ====
<pre>
Subgroup: 2.9.15.7.11
Reduced Mapping
 
4 6 5 7 11 13 17 19 23
Comma list: 100/99, 225/224, 245/243
[ ⟨ 1 0 1 1 1 0 1 1 0 ]
 
⟨ 0 16 2 5 9 23 13 14 28 ] ⟩
Sval mapping: {{mapping| 1 0 2 -1 6 | 0 5 3 6 -4 }}
 
TE Generator Tunings (cents)
{{Mapping|legend=4| 1 5/2 5/2 5 2 | 0 -5/2 -1/2 -6 4 }}
⟨2399.3286, 193.5316]
 
: gencom: [2 9/7; 100/99 225/224 245/243]
TE Step Tunings (cents)
 
⟨37.31613, 39.63311]
Optimal tuning (subgroup POTE): ~2 = 1\1, ~14/9 = 761.393
 
TE Tuning Map (cents)
Optimal ET sequence: {{Optimal ET sequence| 8, 11, 30, 41, 52, 93, 145, 342bce }}
⟨2399.329, 3096.506, 2786.392, 3366.987, 4141.113, 4451.227, 4915.240, 5108.771, 5418.885]
 
RMS error: 1.226 cents
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
==== 2.9.15.7.11.13 ====
q25i, q12fhi, q37f, q13rffghii, q62, q50rfii, q49rffghii, q24rfffghhiii, q74ffghi, q38rreffghiii
Subgroup: 2.9.15.7.11.13
</pre>


== 4.9.25 subgroup ==
Comma list: 100/99, 105/104, 144/143, 196/195
=== Meansquared ===
[[Subgroup]]: 4.9.25


[[Comma list]]: [[6561/6400]]
Sval mapping: {{mapping| 1 0 2 -1 6 -2 | 0 5 3 6 -4 9 }}


{{Mapping|legend=3| 1 3 4 | 0 1 4 }}
{{Mapping|legend=4| 1 5/2 5/2 5 2 7 | 0 -5/2 -1/2 -6 4 -9 }}


Mapping generators: ~4, ~9/64
: gencom: [2 9/7; 100/99 105/104 144/143 196/195]


[[Optimal tuning]] ([[CTE]]): ~4 = 2\1, ~9/4 = 1394.429
Optimal tuning (subgroup POTE): ~2 = 1\1, ~14/9 = 761.023


[[Support]]ing [[ET]]s: 12, 7, 19, 5, 31, 26, 17[+25], 43, 9[-25], 33[-25], 45, 29[+25], 8[+25], 22[+25]
Optimal ET sequence: {{Optimal ET sequence| 11, 30, 41, 153cdef, 194cdef, 235cdef }}


== 4.9.49 subgroup ==
RMS error: 1.540 cents
=== Archsquared ===
[[Subgroup]]: 4.9.49


[[Comma list]]: 4096/3969
=== A-team ===
A-team is every other step of [[slendric]]; the 2.9.5.21.11 extension below specifically restricts [[mothra]].


{{Mapping|legend=3| 1 3 0 | 0 1 -2 }}
[[Subgroup]]: 2.9.21


Mapping generators: ~4, ~9/64
[[Comma list]]: 1029/1024


[[Optimal tuning]] ([[CTE]]): ~9/4 = 1419.190
{{Mapping|legend=3| 1 2 4 | 0 3 1 }}


[[Support]]ing [[ET]]s: 5, 17, 22, 12, 7, 27, 32, 8, 39[+49], 29[+49], 9[+49], 19[+49], 37, 49
: sval mapping generators: ~2, ~21/16


== 8.9.7 subgroup ==
{{Mapping|legend=5| 1 1 0 3 | 0 3/2 0 -1/2 }}
=== 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
: [[gencom]]: [2 21/16; 1029/1024]


[[Comma list]]: 64/63
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~2 = 1\1, ~21/16 = 467.375


{{Mapping|legend=3| 1 0 2 | 0 1 -1 }}
{{Optimal ET sequence|legend=1| 5, 13, 18, 41, 59, 77, 95 }}


: sval mapping generators: ~8, ~9
[[Tp tuning #T2 tuning|RMS error]]: 0.3202 cents


: [[gencom]]: [8 9/8; 64/63]
==== 2.9.5.21 ====
''Lookalike temperament: [[Dual-fifth_temperaments#Dual-3_A-Team|Dual-3 A-Team]]''


[[Optimal tuning]] ([[CTE]]): ~9/8 = 219.1898
Subgroup: 2.9.5.21


[[Optimal ET sequence]]: {{val| 16 17 15 }}, {{val| 33 35 31 }}, {{val| 148 … }}, {{val| 181 … }}, {{val| 214 … }}, {{val| 247 … }}
[[Comma]] list: 81/80, 1029/1024


[[Badness]]: 0.0215 × 10<sup>-3</sup>
Sval mapping: {{mapping| 1 2 0 4 | 0 3 6 1 }}


= Fractional subgroup temperaments =
Mapping generators: ~2, ~21/16
== 2.3.… subgroups ==


=== Trisect ===
Optimal ([[Lp tuning|POL2]]) generator: 464.3865
Trisect divides every Pythagorean interval into three, and is the much more accurate subgroup restriction of [[Augmented family #Trisected|trisected]].


Extending this temperament to the full [[11-limit|11-]], [[13-limit|13-]], or [[17-limit]] through [[portent]] or [[landscape]] results in the [[weak extension]] known as [[tritikleismic]].
{{Optimal ET sequence|legend=1| 13, 18, 31, 44 }}


[[Subgroup]]: 2.3.7.11/5
===== 2.9.5.21.11 =====
Subgroup: 2.9.5.21.11


[[Comma list]]: 1029/1024, 4000/3993
Comma list: 81/80, 99/98, 385/384


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


[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~44/35 = 1\3, ~13/9 = 633.742
{{Mapping|legend=4| 1 1 0 3 5 | 0 3/2 6 -1/2 -4 }}


{{Optimal ET sequence|legend=1| 15, 21, 36, 123, 159, 195, 231 }}
: gencom: [2 21/16; 81/80 99/98 385/384]


[[Tp tuning #T2 tuning|RMS error]]: ???
Optimal tuning (subgroup POTE): ~2 = 1\1, ~21/16 = 463.956


==== 2.3.7.11/5.13 subgroup ====
{{Optimal ET sequence|legend=1| 5, 13, 31 }}
[[Subgroup]]: 2.3.7.11/5.13


[[Comma list]]: 1029/1024, 1575/1573, 2080/2079
==== B-team ====
B-team (23 & 41) is every other step of [[rodan]].


{{Mapping|legend=3| 3 0 10 5 0 | 0 3 -1 -1 7 }}
Subgroup: 2.9.15.21.33


[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~44/35 = 1\3, ~13/9 = 633.918
Comma list: 245/243, 385/384, 441/440


{{Optimal ET sequence|legend=1| 15, 21f, 36, 87, 123, 159 }}
Sval mapping: {{mapping| 1 2 0 4 7 | 0 3 10 1 -5 }}


[[Tp tuning #T2 tuning|RMS error]]: ???
Optimal tuning (subgroup POTE): ~2 = 1\1, ~21/16 = 468.918


==== 2.3.7.11/5.13.17 subgroup ====
{{Optimal ET sequence|legend=1| 5, 13c, 18, 23, 41, 64, 87, 151 }}
[[Subgroup]]: 2.3.7.11/5.13.17


[[Comma list]]: 273/272, 833/832, 1575/1573, 2080/2079
== 4.3.… subgroups ==
=== Tetrahanson ===
{{Main| Tetrahanson }}


{{Mapping|legend=3| 3 0 10 5 0 -2 | 0 3 -1 -1 7 9 }}
[[Subgroup]]: 4.3.5


[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~34/27 = 1\3, ~13/9 = 633.820
[[Comma list]]: 15625/15552


{{Optimal ET sequence|legend=1| 15, 21fg, 36, 123, 159 }}
{{Mapping|legend=3| 1 3 3 | 0 -6 -5 }}


[[Tp tuning #T2 tuning|RMS error]]: ???
: Mapping generators: ~4, ~5/3


===== Trisector =====
[[Optimal tuning]] ([[CTE]]): ~4 = 2\1, ~5/3 = 882.941
[[Subgroup]]: 2.3.7.11/5.13.17.19


[[Comma list]]: 210/209, 273/272, 286/285, 595/594, 2080/2079
[[Support]]ing [[ET]]s: {{EDs|19, 106, 87, 68, 11, 8, 125, 49, 30, 27, 117, 46, 41b, 79|equave=4}}


{{Mapping|legend=3| 3 0 10 5 0 -2 8 | 0 3 -1 -1 7 9 3 }}
=== Tetrameantone ===
{{Main| Tetrameantone }}


[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~34/27 = 1\3, ~13/9 = 633.894
[[Subgroup]]: 4.3.5


{{Optimal ET sequence|legend=1| 15, 21fg, 36, 123h, 159h }}
[[Comma list]]: 81/80


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


===== 2.3.7.11/5.13.17.19.23 subgroup =====
: Mapping generators: ~4, ~4/3
[[Subgroup]]: 2.3.7.11/5.13.17.19.23


[[Comma list]]: 210/209, 231/230, 273/272, 286/285, 595/594, 2080/2079
[[Optimal tuning]] ([[POTE]]): 4 = 2400.0, ~4/3 = 503.761


{{Mapping|legend=3| 3 0 10 5 0 -2 8 12 | 0 3 -1 -1 7 9 3 1 }}
[[Support]]ing [[ET]]s: {{EDs|5, 9, 14, 19, 24, 43, 62, 81, 100|equave=4}}


[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~34/27 = 1\3, ~13/9 = 634.038
=== Tetramagic ===


{{Optimal ET sequence|legend=1| 15g, 21fg, 36, 87, 123hi }}
[[Subgroup]]: 4.3.5


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


===== 2.3.7.11/5.13.17.19.23.29 subgroup =====
{{Mapping|legend=3| 1 0 1 | 0 5 1 }}
[[Subgroup]]: 2.3.7.11/5.13.17.19.23.29


[[Comma list]]: 210/209, 231/230, 273/272, 286/285, 320/319, 595/594, 2080/2079
: Mapping generators: ~4, ~5/4


{{Mapping|legend=3| 3 0 10 5 0 -2 8 12 13 | 0 3 -1 -1 7 9 3 1 1 }}
[[Optimal tuning]] ([[POTE]]): 4 = 2400.0, ~5/4 = 380.059


[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~29/23 = 1\3, ~13/9 = 634.102
[[Support]]ing [[ET]]s: {{EDs|6, 13, 19, 25, 38, 44, 63, 82|equave=4}}


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


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


=== Hypnosis ===
[[Comma list]]: 256/243
Related temperaments: [[Swetismic temperaments #Hypnos|hypnos]], [[Alphatricot family #Alphatricot|alphatricot]]


[[Subgroup]]: 2.3.7.11/5.13
{{Mapping|legend=3| 5 4 6 | 0 0 -1 }}


[[Comma list]]: 169/168, 540/539, 729/728
: Mapping generators: ~4, ~16/15


{{Mapping|legend=3| 1 0 -3 8 0 | 0 3 11 -13 7 }}
[[Optimal tuning]] ([[POTE]]): 1\5ed4 = 480.0, ~16/15 = 80.4062


[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~13/9 = 633.518
[[Support]]ing [[ET]]s: {{EDs|5, 10, 15, 20, 25, 30, 55, 85, 115|equave=4}}


{{Optimal ET sequence|legend=1| 17, 36, 118f, 125f, 161f, 197f }}
== 4.6.… subgroups ==
=== Meanquad ===
{{Main| Meanquad }}


[[Tp tuning #T2 tuning|RMS error]]: 0.5379 cents
[[Subgroup]]: 4.6.5


=== Hydrothermal ===
[[Comma list]]: [[81/80]] = {{monzo| -4 4 -1 }}
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
{{Mapping|legend=3| 1 0 -4| 0 1 4 }}


[[Comma list]]: [[50/49]]
: mapping generators: ~4, ~6


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


[[Optimal tuning]] (inharmonic [[TE]]): ~1\2 = 590.998, ~[[10/7]]-1\2 = 128.962
[[Support]]ing [[ET]]s: *7, *10, *11[-5], *13[+5], *17, *24, *27[+5], *31, *38, *41, *45, *52, *55, *69


[[Support]]ing [[ET]]s: {{EDOs|4, 6, 8, 10, 18, 28, 46, 64, 110}}
<nowiki />* Wart for 4


=== Argentic ===
==== 4.6.5.7 subgroup (tetrominant) ====
Argentic is the 2.3.7/5 subgroup temperament tempering out [[5120/5103]].  
[[Subgroup]]: 4.6.5.7


[[Subgroup]]: 2.3.7/5
[[Comma list]]: [[36/35]] = {{monzo| 0 2 -1 -1 }}, [[64/63]] = {{monzo| 4 -2 0 -1 }}


[[Comma list]]: [[5120/5103]] = {{monzo| 10 -6 -1 }}
{{Mapping|legend=3| 1 0 -4 4 | 0 1 4 -2 }}


{{Mapping|legend=3| 1 0 10 | 0 1 -6 }}
[[Optimal tuning]] (subgroup [[CTE]]): ~4 = 2\1, ~3/2 = 699.622
: mapping generators: ~2, ~3


[[Optimal tuning]]s:  
[[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]
* [[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 }}
<nowiki />* Wart for 4
<small> based on subgroup TE </small>


Badness (Sintel): 0.119
=== 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.


==== Edson ====
Fourwar is named after the closely related [[hemiwar]] temperament.
{{See also| Chromatic pairs #Edson }}


Edson is related to [[pele]] and [[andromeda]].
{{Todo|inline=1|cleanup}}


[[Subgroup]]: 2.3.7/5.11/5.13/5
<pre>
 
Reduced Mapping
[[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 }}
4 6 5
 
[ ⟨ 1 0 1 ]
{{Mapping|legend=3| 1 0 10 17 22 | 0 1 -6 -10 -13 }}
⟨ 0 16 2 ] ⟩
: mapping generators: ~2, ~3
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)


{{Mapping|legend=5| 1 1 -5 -1 2 4 | 0 1 29/4 5/4 -11/4 -23/4 }}
Subsets
: [[gencom]]: [2 3/2; 196/195, 352/351, 364/363]
q99, q62, q37, q161, q136, q198, q25, q124, q74, q235
</pre>


[[Optimal tuning]]s:
==== 4.6.5.7 ====
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1\1, ~3/2 = 703.4398
<pre>
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1\1, ~3/2 = 703.414
Reduced Mapping
 
4 6 5 7
{{Optimal ET sequence|legend=1| 12, 17, 29 }}
[ ⟨ 1 0 1 1 ]
⟨ 0 16 2 5 ] ⟩
TE Generator Tunings (cents)
⟨2399.4195, 193.8654]
TE Step Tunings (cents)
⟨25.23883, 47.79592]
TE Tuning Map (cents)
⟨2399.420, 3101.846, 2787.150, 3368.747]
TE Mistunings (cents)
⟨-0.580, -0.109, 0.837, -0.079]
Complexity 1.192044
Adjusted Error 0.653313 cents
TE Error 0.232715 cents/octave
Unison Vectors
[-2, -1, -2, 4⟩ (2401:2400)
[3, 0, -5, 2⟩ (3136:3125)
[5, 1, -3, -2⟩ (6144:6125)
[8, 1, -8, 0⟩ (393216:390625)


[[Tp tuning #T2 tuning|RMS error]]: 0.5102 cents
Subsets
q99, q62, q37, q161, q136, q198, q25, q124, q74, q235
</pre>


==== Haumea ====
==== 4.6.5.7.11 ====
{{See also| Chromatic pairs #Haumea }}
<pre>
 
Reduced Mapping
Related temperaments include [[#Bridgetown|bridgetown]], [[namaka]], [[hemigari]], [[#Barbados|barbados]], and [[parizekmic]].
4 6 5 7 11
 
[ ⟨ 1 0 1 1 1 ]
[[Subgroup]]: 2.3.7/5.11/5.13/5
⟨ 0 16 2 5 9 ] ⟩
 
[[Comma list]]: [[352/351]], [[676/675]], [[847/845]]
TE Generator Tunings (cents)
 
⟨2400.1097, 193.9498]
{{Mapping|legend=3| 1 0 10 -6 -1 | 0 2 -12 9 3 }}
 
TE Step Tunings (cents)
{{Mapping|legend=5| 1 2 -3/4 -11/4 9/4 5/4 | 0 -2 0 12 -9 -3 }}
⟨24.18752, 48.52491]
: [[gencom]]: [2 15/13; 352/351 676/675 847/845]
 
TE Tuning Map (cents)
[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~15/13 = 248.491
⟨2400.110, 3103.196, 2788.009, 3369.859, 4145.658]
 
{{Optimal ET sequence|legend=1| 24, 29, 111, 140, 169, 198, 565d, 763bd, 961bd }}
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)


[[Tp tuning #T2 tuning|RMS error]]: 0.2668 cents
Subsets
q37, q25, q62, q12, q74, q99, q87, q49r, q50r, q124
</pre>


=== Historical ===
==== 4.6.5.7.11.13 ====
{{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.
<pre>
 
Reduced Mapping
[[Subgroup]]: 2.3.7/5.11/5.13/5
4 6 5 7 11 13
 
[ ⟨ 1 0 1 1 1 0 ]
[[Comma list]]: 364/363, 441/440, 1001/1000
⟨ 0 16 2 5 9 23 ] ⟩
 
{{Mapping|legend=3| 1 2 0 1 2 | 0 -6 7 2 -9 }}
TE Generator Tunings (cents)
 
⟨2401.2305, 193.5378]
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~21/20 = 83.016
 
TE Step Tunings (cents)
{{Optimal ET sequence|legend=1| 14, 29, 72, 101, 130, 159 }}
⟨42.79107, 35.98524]
 
[[Tp tuning #T2 tuning|RMS error]]: 0.2562 cents
TE Tuning Map (cents)
 
⟨2401.230, 3096.606, 2788.306, 3368.920, 4143.071, 4451.371]
=== Direct breedsmic ===
Related temperament: [[hemithirds]], [[newt]]
TE Mistunings (cents)
 
⟨1.230, -5.349, 1.992, 0.094, -8.247, 10.843]
[[Subgroup]]: 2.3.49/5
 
Complexity 1.219191
[[Comma list]]: 2401/2400
Adjusted Error 6.699599 cents
TE Error 1.810487 cents/octave
Unison Vectors
[0, 1, -1, 0, 1, -1⟩ (66:65)
[-1, -1, -1, 0, 2, 0⟩ (121:120)
[1, 2, 0, 0, -1, -1⟩ (144:143)
[2, 0, -2, -1, 1, 0⟩ (176:175)
[-2, 1, 1, 1, 0, -1⟩ (105:104)
[-3, -1, 1, 1, 1, 0⟩ (385:384)
[-3, 0, 0, 1, 2, -1⟩ (847:832)
[1, 3, -1, 0, 0, -2⟩ (864:845)
[-1, 0, 3, -3, 1, 0⟩ (1375:1372)


{{Mapping|legend=3| 1 1 3 | 0 2 1 }}
Subsets
q25, q37f, q12f, q62, q50rf, q13rff, q49rff, q87, q74ff, q24rfff
</pre>


[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~49/40 = 350.966
==== 4.6.5.7.11.13.17 ====
 
<pre>
{{Optimal ET sequence|legend=1|7, 10, 17}}
Reduced Mapping
 
4 6 5 7 11 13 17
[[Tp tuning #T2 tuning|RMS error]]: ?
[ ⟨ 1 0 1 1 1 0 1 ]
 
⟨ 0 16 2 5 9 23 13 ] ⟩
==== Bridgetown ====
{{See also| Chromatic pairs #Bridgetown }}
TE Generator Tunings (cents)
 
⟨2400.4701, 193.4599]
Bridgetown, the 5 &amp; 24 temperament in the 2.3.11/5.13/5 subgroup, is related to [[#Haumea|haumea]] and [[#Barbados|barbados]].  
 
TE Step Tunings (cents)
[[Subgroup]]: 2.3.11/5.13/5
⟨43.39350, 35.55764]
 
[[Comma list]]: [[352/351]], [[676/675]]
TE Tuning Map (cents)
 
⟨2400.470, 3095.359, 2787.390, 3367.770, 4141.609, 4449.578, 4915.449]
{{Mapping|legend=3| 1 0 -6 -1 | 0 2 9 3 }}
 
TE Mistunings (cents)
{{Mapping|legend=5| 1 2 -5/3 0 4/3 1/3 | 0 -2 4 0 -5 1 }}
⟨0.470, -6.596, 1.076, -1.056, -9.709, 9.050, 10.494]
: [[gencom]]: [2 15/13; 352/351 676/675]
 
Complexity 1.129881
[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~15/13 = 248.399
Adjusted Error 8.082725 cents
 
TE Error 1.977443 cents/octave
{{Optimal ET sequence|legend=1| 5, 9, 14, 19, 24, 29, 169, 198, 227, 256, 285, 314 }}
 
Unison Vectors
[[Tp tuning #T2 tuning|RMS error]]: 0.2513 cents
[0, 1, -1, 0, 1, -1, 0⟩ (66:65)
 
[1, 1, 1, -1, 0, 0, -1⟩ (120:119)
=== Blackweed ===
[1, 2, 0, 0, -1, -1, 0⟩ (144:143)
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.
[-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)


[[Subgroup]]: 2.3.11/7
Subsets
q25, q12f, q37f, q13rffg, q50rf, q62, q49rffg, q24rfffg, q38rreffg, q74ffg
</pre>


[[Comma list]]: {{monzo| 8 -5 }} (256/243)
==== 4.6.5.7.11.13.17.19 ====
 
<pre>
{{Mapping|legend=3| 5 8 0 | 0 0 1 }}
Reduced Mapping
: mapping generators: ~9/8, ~11/7
4 6 5 7 11 13 17 19
 
[ ⟨ 1 0 1 1 1 0 1 1 ]
[[Optimal tuning]]s:
⟨ 0 16 2 5 9 23 13 14 ] ⟩
* [[Tp tuning|subgroup]] [[WE]]: ~8/7 = 238.851{{c}}, ~11/7 = 782.457{{c}}
: [[error map]]: {{val| -5.746 +8.852 -0.035 }}
TE Generator Tunings (cents)
* [[Tp tuning|subgroup]] [[CWE]]: ~8/7 = 240.000{{c}}, ~11/7 = 784.967{{c}}
⟨2399.9219, 193.3952]
: error map: {{val| 0.000 +18.045 +2.475 }}
 
TE Step Tunings (cents)
{{Optimal ET sequence|legend=1| 15, 20, 35b, 55b }}
⟨44.14256, 35.03670]
 
=== Pepperoni ===
TE Tuning Map (cents)
{{Main| Parapyth }}
⟨2399.922, 3094.324, 2786.712, 3366.898, 4140.479, 4448.090, 4914.060, 5107.455]
{{See also| Chromatic pairs #Pepperoni }}
 
TE Mistunings (cents)
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.
⟨-0.078, -7.631, 0.399, -1.928, -10.839, 7.562, 9.104, 9.942]
 
[[Subgroup]]: 2.3.11/7.13/7
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)


[[Comma list]]: 352/351, 364/363
Subsets
q25, q12fh, q37f, q13rffgh, q50rf, q62, q49rffgh, q24rfffghh, q38rreffgh, q74ffgh
</pre>


{{Mapping|legend=3| 1 0 7 12 | 0 1 -4 -7 }}
==== 4.6.5.7.11.13.17.19.23 ====
 
<pre>
{{Mapping|legend=5| 1 1 0 -8/3 1/3 7/3 | 0 1 0 11/3 -1/3 -10/3 }}
Reduced Mapping
: [[gencom]]: [2 3/2; 352/351 364/363]
4 6 5 7 11 13 17 19 23
 
[ ⟨ 1 0 1 1 1 0 1 1 0 ]
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~3/2 = 703.856
⟨ 0 16 2 5 9 23 13 14 28 ] ⟩
 
{{Optimal ET sequence|legend=1| 5, 7, 12f, 17, 29, 46, 58, 75, 80, 87, 104, 121, 167, 196, 208, 271, 595b*<sup>†</sup> }}
TE Generator Tunings (cents)
: <nowiki />* wart for 11/7
⟨2399.3286, 193.5316]
: <sup>†</sup> wart for 13/7
 
TE Step Tunings (cents)
[[Tp tuning #T2 tuning|RMS error]]: 0.3789 cents
⟨37.31613, 39.63311]
 
=== Barbados ===
TE Tuning Map (cents)
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.
⟨2399.329, 3096.506, 2786.392, 3366.987, 4141.113, 4451.227, 4915.240, 5108.771, 5418.885]
 
[[Subgroup]]: 2.3.13/5
TE Mistunings (cents)
 
⟨-0.671, -5.449, 0.078, -1.839, -10.205, 10.699, 10.284, 11.258, -9.389]
[[Comma list]]: 676/675 = {{monzo| 2 -3 2 }}
 
Complexity 1.115920
[[Sval]] [[mapping]]: [{{val| 1 0 -1 }}, {{val| 0 2 3 }}]
Adjusted Error 9.502017 cents
 
TE Error 2.100561 cents/octave
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~2 = 1\1, ~15/13 = 248.621
 
Unison Vectors
{{Optimal ET sequence|legend=1| 5, 9, 14, 19, 24, 29, 53, 82, 111, 140, 251, 362 }}
[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)


[[Badness]]: 0.002335
Subsets
q25i, q12fhi, q37f, q13rffghii, q62, q50rfii, q49rffghii, q24rfffghhiii, q74ffghi, q38rreffghiii
</pre>


; Music
== 4.9.… subgroups ==
* [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]
=== Meansquared ===
[[Subgroup]]: 4.9.25


==== Tobago ====
[[Comma list]]: [[6561/6400]]
{{See also| Chromatic pairs #Tobago }}


Tobago, the 10 &amp; 14 temperament in the 2.3.11.13/5 subgroup, extends [[neutral]] and [[barbados]].
{{Mapping|legend=3| 1 3 4 | 0 1 4 }}


[[Subgroup]]: 2.3.11.13/5
Mapping generators: ~4, ~9/64


[[Comma list]]: [[243/242]], [[676/675]]
[[Optimal tuning]] ([[CTE]]): ~4 = 2\1, ~9/4 = 1394.429


{{Mapping|legend=3| 2 0 -1 -2 | 0 2 5 3 }}
[[Support]]ing [[ET]]s: 12, 7, 19, 5, 31, 26, 17[+25], 43, 9[-25], 33[-25], 45, 29[+25], 8[+25], 22[+25]


{{Mapping|legend=5| 2 4 -2 0 9 2 | 0 -2 3/2 0 -5 -3/2 }}
=== Archsquared ===
: [[gencom]]: [55/39 15/13; 243/242 676/675]
[[Subgroup]]: 4.9.49


[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~55/39 = 1\2, ~15/13 = 249.312
[[Comma list]]: 4096/3969


{{Optimal ET sequence|legend=1| 10, 14, 24, 58, 82, 130 }}
{{Mapping|legend=3| 1 3 0 | 0 1 -2 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.3533 cents
Mapping generators: ~4, ~9/64


==== Pakkanian hemipyth ====
[[Optimal tuning]] ([[CTE]]): ~9/4 = 1419.190
[[Subgroup]]: 2.3.11.13/5.17


[[Comma list]]: 221/220, 243/242, 289/288
[[Support]]ing [[ET]]s: 5, 17, 22, 12, 7, 27, 32, 8, 39[+49], 29[+49], 9[+49], 19[+49], 37, 49


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


[[Optimal tuning]]s:  
[[Subgroup]]: 8.9.7
* [[Tp tuning|subgroup CTE]]: ~17/12 = 1\2, ~26/15 = 950.7656 (~15/13 = 249.2344)
* [[Tp tuning|subgroup CWE]]: ~17/12 = 1\2, ~26/15 = 950.6011 (~15/13 = 249.3989)


{{Optimal ET sequence|legend=1| 10, 14, 24, 106, 130, 154, 178*, 202* }}
[[Comma list]]: 64/63
: <nowiki />* wart for 13/5


=== Oceanfront ===
{{Mapping|legend=3| 1 0 2 | 0 1 -1 }}
Related temperaments: [[Archytas clan #Superpyth|superpyth]], [[Archytas clan #Ultrapyth|ultrapyth]]


[[Subgroup]]: 2.3.7.13/5
: sval mapping generators: ~8, ~9


[[Comma list]]: 64/63, 91/90
: [[gencom]]: [8 9/8; 64/63]


{{Mapping|legend=3| 1 0 6 -5 | 0 1 -2 4 }}
[[Optimal tuning]] ([[CTE]]): ~9/8 = 219.1898


[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~3/2 = 713.910
[[Optimal ET sequence]]: {{val| 16 17 15 }}, {{val| 33 35 31 }}, {{val| 148 … }}, {{val| 181 … }}, {{val| 214 … }}, {{val| 247 … }}


{{Optimal ET sequence|legend=1| 5, 22, 27, 32, 37 }}
[[Badness]]: 0.0215 × 10<sup>-3</sup>


[[Tp tuning #T2 tuning|RMS error]]: 2.063 cents
== 2.5.… subgroups ==
=== Guanyintet ===
{{See also | Chromatic pairs #Guanyintet }}


Scales: [[Oceanfront scales]]
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.


=== Seventeen-cot ===
[[Subgroup]]: 2.5.7/3.11/3
Seventeen-cot is a rank-2 temperament in the 2.3.13/5 and 2.3.11/5.13/5 subgroups. It tempers out the [[Tendoartisma]] in the 2.3.13/5 subgroup. It can be generated with a ~2/1 octave and a ~2250/2197 or ~169/165 generator which is a 17th of a ~3/2 perfect fifth. It can be described as the 29 & 146 temperament in these subgroups.


====2.3.13/5 subgroup====
[[Comma list]]: [[176/175]] ({{monzo| 4 -2 -1 1 }}), [[540/539]] ({{monzo| 2 1 -2 -1 }})


Comma basis: {{monzo| -6 -11 17 }} (2.3.13/5)
{{Mapping|legend=3| 1 0 1 3 | 0 -3 1 -5 }}
: mapping generators: ~2, ~7/6


edo join: 29 & 146
{{Mapping|legend=5| 1 -4/3 3 -1/3 5/3 | 0 4/3 -3 7/3 -11/3 }}
: [[gencom]]: [2 7/6; 176/175 540/539]


{{Mapping|legend=3| 1 1 1 | 0 17 11 }}
[[Optimal tuning]]s:
: mapping generators: ~2, ~2250/2197
* ([[Tp tuning|subgroup]] [[CTE]]): ~2 = 1200.000, ~7/6 = 270.455
* ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1200.000, ~7/6 = 270.093


Optimal tunings:
{{Optimal ET sequence|legend=1| 9, 22, 31, 40, 191c*, 231c*, 271c*, 311c* }}
* WE: ~2 = 1200.0001354{{c}}, ~2250/2197 = 41.2914993{{c}}
: <nowiki/>* wart for 7/3
: error map: {{val| +0.0001354 +0.0006234 -0.0073197}}
* CWE: ~2 = 2/1, ~2250/2197 = 41.2915011{{c}}
: error map: {{val| 0.0000000 +0.0005177 -0.0074359}}


edos: 29, 465, 494, 436, 523, 407, 378, 349, 30[-3], 28[+3], 320, 291, 59[-3], 262
[[Tp tuning #T2 tuning|RMS error]]: 0.6028 cents


Badness (Sintel): 0.064
==== Tridecimal guanyintet ====
Guanyintet can extend to the 13th harmonic by the equivalences ([[12/11]])<sup>3</sup> = [[13/10]] and ([[15/14]])<sup>3</sup> = [[16/13]], therefore tempering out {S11/S12/S14/S15}. However, note that it is not supported by the 31 & 53 orwell extension dubbed "tridecimal orwell", but instead the less accurate [[winston]] (22f & 31), as orwell prefers slightly sharper tunings than guanyintet. [[40edo]] remains an excellent tuning.


====2.3.11/5.13/5 subgroup====
[[Subgroup]]: 2.5.7/3.11/3.13


Comma basis: 225000/224939, 43940/43923
[[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 }})


edo join: 29 & 146
{{Mapping|legend=3| 1 0 1 3 1 | 0 -3 1 -5 12 }}
: mapping generators: ~2, ~12/7


{{Mapping|legend=3| 1 1 1 1 | 0 17 4 11 }}
[[Optimal tuning]]s:
: mapping generators: ~2, ~169/165
* ([[Tp tuning|subgroup]] [[CTE]]): ~2 = 1200.000, ~7/6 = 270.152
* ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1200.000, ~7/6 = 270.218


Optimal tunings:
{{Optimal ET sequence|legend=1| 9, 22, 31, 40, 71, 111, 151, 262c*}} <small> using subgroup TE </small>
* WE: ~2 = 1199.9934923{{c}}, ~169/165 = 41.2918271{{c}}
: <nowiki/>* wart for 7/3
: error map: {{val| -0.0065077 -0.0004485 +0.1565720 -0.0103579}}
* CWE: ~2 = 2/1, ~169/165 = 41.2917463{{c}}
: error map: {{val| 0.0000000 +0.0046870 +0.1627569 -0.0043781}}


edos: 29, 465, 494, 436, 523, 407, 378, 349, 320, 291, 30[-3], 262, 28[+3], 233
Badness (Sintel): 0.329


Badness (Sintel): 0.080
==== Laz ====
{{See also | Chromatic pairs #Laz }}


=== Fiventeen ===
Laz is related to [[avalokita]] as well as to [[winston]].  
Fiventeen tempers out [[136/135]] ({{monzo| 3 -3 1 }}) in 2.3.17/5. It equates [[17/15]] with [[9/8]], so it implies a [[supersoft]] [[pentic]] [[pentad]] of [[~]]30:34:40:45:51. [[17edo]] makes a good tuning especially for its size, which gives a [[supersoft]] pentic scale corresponding approximately to a just [[20/17]] tuning, although [[80edo]] might be preferred for an approximately just [[51/40]] to optimize plausibility slightly more, and [[97edo]] (= 80 + 17) and  [[114edo]] (= 97 + 17) do even better in striking a balance between 80edo's more stable tuning and that having 20/17 more accurate (as in 17edo) is useful because of the more convincing suggestion of the two 15:17:20 chords present in the fiventeen pentad. The same is true of the related rank-3 temperament diatic, for which the [[optimal ET sequence]] is much more characteristic of optimized tunings, finding [[34edo]], then [[80edo]], then [[114edo]] (= 34 + 80) and even [[194edo|194bc-edo]] (= 80 + 114), though because of its focus on primes 5 and 17 it misses 97edo as a tuning, and slightly less optimized though still interesting [[63edo]] and [[143edo]] (= 63 + 80) tunings are found in the optimal ET sequence for fiventeen.


[[Subgroup]]: 2.3.17/5
[[Subgroup]]: 2.5.7/3.11/3.13/3


[[Comma list]]: 136/135 ({{monzo| 3 -3 1 }})
[[Comma list]]: [[144/143]] ({{monzo| 4 0 0 -1 -1 }}), [[176/175]] ({{monzo| 4 -2 -1 1 }}), [[196/195]] ({{monzo| 2 -1 2 0 -1 }}


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


[[Optimal tuning]]s:
{{Mapping|legend=5| 1 -5/4 3 -1/4 7/4 -1/4 | 0 -1/4 -3 3/4 -21/4 19/4 }}
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.2838{{c}}, ~3/2 = 704.4600{{c}}
: [[gencom]]: [2 7/6; 144/143 176/175 196/195]
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.5286{{c}}


{{Optimal ET sequence|legend=1| 5, 12, 17, 46, 63, 143 }}
[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~12/7 = 930.598
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~12/7 = 930.700


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


[[Subgroup]]: 2.3.19/7
[[Tp tuning #T2 tuning|RMS error]]: 0.8790 cents


[[Comma list]]: [[57/56]] ({{Monzo| -3 1 1 }})
=== Kryptonite ===
{{See also| Chromatic pairs #Kryptonite }}


{{Mapping|legend=3| 1 0 3 | 0 1 -1 }}
Kryptonite is related to [[krypton]].
: mapping generators: ~2, ~3


[[Optimal tuning]]s:
[[Subgroup]]: 2.5.7/3.11/3.13/3
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1202.4345{{c}}, ~3/2 = 697.4314{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 697.3981{{c}}


{{Optimal ET sequence|legend=1| 5, 7, 12, 19, 31*, 50* }}
[[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 }})


<nowiki/>* wart for 19/7
{{Mapping|legend=3| 1 2 1 2 2 | 0 3 2 -1 1 }}
: mapping generators: ~2, ~13/12


[[Badness]] (Sintel): 0.082
{{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]


== 2.5.… subgroups ==
[[Optimal tuning]]s:
=== Guanyintet ===
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~13/12 = 130.945
{{See also | Chromatic pairs #Guanyintet }}
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~13/12 = 132.428


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


[[Subgroup]]: 2.5.7/3.11/3
[[Tp tuning #T2 tuning|RMS error]]: 2.545 cents


[[Comma list]]: [[176/175]] ({{monzo| 4 -2 -1 1 }}), [[540/539]] ({{monzo| 2 1 -2 -1 }})
=== Marveltri ===
{{See also| Chromatic pairs #Marveltri }}


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


{{Mapping|legend=5| 1 -4/3 3 -1/3 5/3 | 0 4/3 -3 7/3 -11/3 }}
[[Subgroup]]: 2.5.9/7
: [[gencom]]: [2 7/6; 176/175 540/539]


[[Optimal tuning]]s:  
[[Comma list]]: 225/224 ({{monzo| -5 2 1 }})
* ([[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* }}
{{Mapping|legend=3| 1 0 5 | 0 1 -2 }}
: <nowiki/>* wart for 7/3
: mapping generators: ~2, ~5


[[Tp tuning #T2 tuning|RMS error]]: 0.6028 cents
{{Mapping|legend=5| 1 2 0 -1 | 0 -4/5 1 2/5 }}
: [[gencom]]: [2 5; 225/224]


==== Tridecimal guanyintet ====
[[Optimal tuning]]s:
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.
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/4 = 384.208
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/4 = 383.638


[[Subgroup]]: 2.5.7/3.11/3.13
{{Optimal ET sequence|legend=1| 3, 13, 16, 19, 22, 25, 72, 97, 122, 269c* }}
: <nowiki/>* wart for 9/7


[[Comma list]]: [[176/175]] ({{monzo| 4 -2 -1 1 0 }}), [[540/539]] ({{monzo| 2 1 -2 -1 0 }}), [[1573/1568]] ({{monzo| -5 0 -2 2 1 }})
[[Tp tuning #T2 tuning|RMS error]]: 0.4801 cents


{{Mapping|legend=3| 1 0 1 3 1 | 0 -3 1 -5 12 }}
==== Sulis ====
: mapping generators: ~2, ~12/7
Sulis is related to [[minerva]] and [[würschmidt]].


[[Optimal tuning]]s:  
[[Subgroup]]: 2.5.9/7.11/9
* ([[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>
[[Comma list]]: 99/98 ({{monzo| -1 0 2 1 }}), 176/175 ({{monzo| 4 -2 1 1 }})
: <nowiki/>* wart for 7/3


Badness (Sintel): 0.329
{{Mapping|legend=3| 1 0 5 -9 | 0 1 -2 4 }}]


==== Laz ====
[[Optimal tuning]]s:
{{See also | Chromatic pairs #Laz }}
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/4 = 386.617
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/4 = 386.558


Laz is related to [[avalokita]] as well as to [[winston]].
{{Optimal ET sequence|legend=1| 3, …, 22, 25, 28, 31, 59 }}


[[Subgroup]]: 2.5.7/3.11/3.13/3
[[Tp tuning #T2 tuning|RMS error]]: 1.074 cents


[[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 }}
== 2.5/3.… subgroups ==
=== Magicaltet ===
{{See also| Chromatic pairs #Magicaltet }}


{{Mapping|legend=3| 1 0 2 -2 6 | 0 3 -1 5 -5 }}
Magicaltet is related to [[keemic]], [[superkleismic]], and [[magic]]. The tonic and the first three generator steps make a [[magical seventh chord]], hence the name.


{{Mapping|legend=5| 1 -5/4 3 -1/4 7/4 -1/4 | 0 -1/4 -3 3/4 -21/4 19/4 }}
[[Subgroup]]: 2.5/3.7.11
: [[gencom]]: [2 7/6; 144/143 176/175 196/195]


[[Optimal tuning]]s:  
[[Comma list]]: 100/99 ({{monzo| 2 2 0 -1 }}), 385/384 ({{monzo| -7 1 1 1 }})
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~12/7 = 930.598
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~12/7 = 930.700


{{Optimal ET sequence|legend=1| 9, 31, 40, 49, 156c*†, 205c*† }}
{{Mapping|legend=3| 1 0 5 2 | 0 1 -3 2 }}
: <nowiki/>* wart for 7/3
: mapping generators: ~2, ~5/3
: † wart for 11/3


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


=== Kryptonite ===
[[Optimal tuning]]s:
{{See also| Chromatic pairs #Kryptonite }}
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 877.343
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 877.351


Kryptonite is related to [[krypton]].
{{Optimal ET sequence|legend=1| 4, 7, 11, 15, 26, 67, 93* }}
: <nowiki/>* wart for 5/3


[[Subgroup]]: 2.5.7/3.11/3.13/3
[[Tp tuning #T2 tuning|RMS error]]: 1.206 cents


[[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 }})
=== Starlingtet ===
{{See also | Chromatic pairs #Starlingtet }}


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


{{Mapping|legend=5| 1 -5/4 2 -1/4 3/4 3/4 | 0 -1/2 3 3/2 -3/2 1/2 }}
[[Subgroup]]: 2.5/3.7/3
: [[gencom]]: [2 13/12; 56/55 78/77 91/90]


[[Optimal tuning]]s:  
[[Comma list]]: [[126/125]] ({{monzo| 1 -3 1 }})
* [[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 }}
{{Mapping|legend=3| 1 0 -1 | 0 1 3 }}


[[Tp tuning #T2 tuning|RMS error]]: 2.545 cents
: mapping generators: ~2, ~5/3


=== Marveltri ===
{{Mapping|legend=5| 1 -1 0 1 | 0 4/3 1/3 -5/3 }}
{{See also| Chromatic pairs #Marveltri }}
: [[gencom]]: [2 6/5; 126/125]


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.  
[[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


[[Subgroup]]: 2.5.9/7
{{Optimal ET sequence|legend=1| 4, 15, 19, 23, 27 }}


[[Comma list]]: 225/224 ({{monzo| -5 2 1 }})
[[Tp tuning #T2 tuning|RMS error]]: 0.8398 cents


{{Mapping|legend=3| 1 0 5 | 0 1 -2 }}
==== Greeley ====
: mapping generators: ~2, ~5
{{See also| Chromatic pairs #Greeley }}


{{Mapping|legend=5| 1 2 0 -1 | 0 -4/5 1 2/5 }}
Greeley is related to [[opossum]] as well as to [[nusecond]].
: [[gencom]]: [2 5; 225/224]


[[Optimal tuning]]s:  
[[Subgroup]]: 2.5/3.7/3.11/3
* [[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* }}
[[Comma list]]: 121/120 ({{monzo| -3 -1 0 2 }}), 126/125 ({{monzo| 1 -3 1 }})
: <nowiki/>* wart for 9/7


[[Tp tuning #T2 tuning|RMS error]]: 0.4801 cents
{{Mapping|legend=3| 1 1 2 2 | 0 -2 -6 -1 }}


==== Sulis ====
{{Mapping|legend=5| 1 -5/4 -1/4 3/4 3/4 | 0 9/4 1/4 -15/4 5/4 }}
Sulis is related to [[minerva]] and [[würschmidt]].
: [[gencom]]: [2 11/10; 121/120 126/125]


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


[[Comma list]]: 99/98 ({{monzo| -1 0 2 1 }}), 176/175 ({{monzo| 4 -2 1 1 }})
{{Optimal ET sequence|legend=1| 8, 15, 23, 54, 77, 100, 131* }}
: <nowiki/>* wart for 11/3


{{Mapping|legend=3| 1 0 5 -9 | 0 1 -2 4 }}]
[[Tp tuning #T2 tuning|RMS error]]: 1.034 cents


[[Optimal tuning]]s:
==== Skateboard ====
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/4 = 386.617
{{See also| Chromatic pairs #Skateboard }}
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/4 = 386.558


{{Optimal ET sequence|legend=1| 3, …, 22, 25, 28, 31, 59 }}
Skateboard is related to [[thrasher]].


[[Tp tuning #T2 tuning|RMS error]]: 1.074 cents
[[Subgroup]]: 2.5/3.7/3.11.13/9


== 2.5/3.… subgroups ==
[[Comma list]]: 56/55 ({{monzo| 3 -1 1 -1 }}), 91/90 ({{monzo| -1 -1 1 0 1 }}), 100/99 ({{monzo| 2 2 0 -1 }})
=== 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.
{{Mapping|legend=3| 1 0 -1 2 2 | 0 1 3 2 -2 }}


[[Subgroup]]: 2.5/3.7.11
{{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]]: 100/99 ({{monzo| 2 2 0 -1 }}), 385/384 ({{monzo| -7 1 1 1 }})
[[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 5 2 | 0 1 -3 2 }}
{{Optimal ET sequence|legend=1| 11, 15, 19, 23, 42d, 65d }}
: mapping generators: ~2, ~5/3


{{Mapping|legend=5| 1 -1/2 1/2 2 4 | 0 1/2 -1/2 3 -2 }}
[[Tp tuning #T2 tuning|RMS error]]: 2.396 cents
: [[gencom]]: [2 6/5; 100/99 385/384]


[[Optimal tuning]]s:
=== Gariberttet ===
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 877.343
Gariberttet is the 2.5/3.7/3 [[Subgroup temperament families, relationships, and genes|altergene]] of [[sirius]].
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 877.351


{{Optimal ET sequence|legend=1| 4, 7, 11, 15, 26, 67, 93* }}
==== Gariberttet (2.5/3.7/3.13/11 subgroup) ====
: <nowiki/>* wart for 5/3
{{See also | Chromatic pairs #Gariberttet }}


[[Tp tuning #T2 tuning|RMS error]]: 1.206 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]].


=== Starlingtet ===
[[Subgroup]]: 2.5/3.7/3.13/11
{{See also | Chromatic pairs #Starlingtet }}


Starlingtet, the {{nowrap| 4 & 15 }} temperament in the 2.5/3.7/3 subgroup, is related to [[starling]] as well as to [[myna]]. The tonic and the first three generator steps make a [[starling tetrad]], hence the name.
[[Comma list]]: [[275/273]] ({{monzo| 0 2 -1 -1 }}), [[847/845]] ({{monzo| 0 -1 1 -2 }})


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


[[Comma list]]: [[126/125]] ({{monzo| 1 -3 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 0 -1 | 0 1 3 }}
[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]] and [[POTE]]: ~2 = 1200.000, ~13/11 = 293.679


: mapping generators: ~2, ~5/3
{{Optimal ET sequence|legend=1| 29, 33, 37, 41, 45, 49, 78, 94, 143* }}
: <nowiki/>* wart for 13/11


{{Mapping|legend=5| 1 -1 0 1 | 0 4/3 1/3 -5/3 }}
[[Tp tuning #T2 tuning|RMS error]]: 0.6914 cents
: [[gencom]]: [2 6/5; 126/125]


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


{{Optimal ET sequence|legend=1| 4, 15, 19, 23, 27 }}
Indium can be described as the {{nowrap| 8 & 33 }} temperament in the 2.5/3.7/3.11/3 subgroup.  
 
[[Tp tuning #T2 tuning|RMS error]]: 0.8398 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
[[Subgroup]]: 2.5/3.7/3.11/3


[[Comma list]]: 121/120 ({{monzo| -3 -1 0 2 }}), 126/125 ({{monzo| 1 -3 1 }})
[[Comma list]]: [[3025/3024]] ({{monzo| -4 2 -1 2 }}), [[3125/3087]] ({{monzo| 0 5 -3 }})


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


{{Mapping|legend=5| 1 -5/4 -1/4 3/4 3/4 | 0 9/4 1/4 -15/4 5/4 }}
{{Mapping|legend=5| 1 -1/2 -1/2 -1/2 3/2 | 0 -15/4 9/4 25/4 -19/4 }}
: [[gencom]]: [2 11/10; 121/120 126/125]
: [[gencom]]: [2 12/11; 3025/3024 3125/3087]


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~11/10 = 155.696
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~12/11 = 146.978
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~11/10 = 155.776
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~12/11 = 147.010


{{Optimal ET sequence|legend=1| 8, 15, 23, 54, 77, 100, 131* }}
{{Optimal ET sequence|legend=1| 8, 33, 41, 49, 204*<sup>†</sup> }}
: <nowiki/>* wart for 11/3
: <nowiki/>* wart for 7/3
: <sup>†</sup> wart for 11/3


[[Tp tuning #T2 tuning|RMS error]]: 1.034 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.7788 cents


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


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


[[Subgroup]]: 2.5/3.7/3.11.13/9
[[Subgroup]]: 2.5/3.7/3.11/3.13/3


[[Comma list]]: 56/55 ({{monzo| 3 -1 1 -1 }}), 91/90 ({{monzo| -1 -1 1 0 1 }}), 100/99 ({{monzo| 2 2 0 -1 }})
[[Comma list]]: [[121/120]] ({{monzo| -3 -1 0 2 }}), [[169/168]] ({{monzo| -3 0 -1 0 2 }}), [[275/273]] ({{monzo| 0 2 -1 1 -1 }})


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


{{Mapping|legend=5| 1 -3/7 4/7 11/7 4 -6/7 | 0 0 -1 -3 -2 2 }}
{{Mapping|legend=5| 1 -3 0 2 0 1 | 0 24/5 -6/5 -26/5 9/5 -1/5 }}
: [[gencom]]: [2 6/5; 56/55 91/90 100/99]
: [[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 = 886.158
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~13/10 = 453.121
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 886.158
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~13/10 = 453.242


{{Optimal ET sequence|legend=1| 11, 15, 19, 23, 42d, 65d }}
{{Optimal ET sequence|legend=1| 8, 29, 37, 45 }}


[[Tp tuning #T2 tuning|RMS error]]: 2.396 cents
[[Tp tuning #T2 tuning|RMS error]]: 1.052 cents


=== Gariberttet ===
=== Sentry ===
Gariberttet is the 2.5/3.7/3 [[Subgroup temperament families, relationships, and genes|altergene]] of [[sirius]].
{{See also | Chromatic pairs #Sentry }}


==== Gariberttet (2.5/3.7/3.13/11 subgroup) ====
Sentry, the {{nowrap| 3 & 5 }} temperament in the 2.5/3.9/7 subgroup, is related to [[sensi]].  
{{See also | Chromatic pairs #Gariberttet }}


Gariberttet can be described as the {{nowrap| 4 & 29 }} temperament in the 2.5/3.7/3.13/11 subgroup. Extensions to the full 7-, 11-, and 13-limits include [[quasitemp]].
[[Subgroup]]: 2.5/3.9/7


[[Subgroup]]: 2.5/3.7/3.13/11
[[Comma list]]: [[245/243]] ({{monzo| 0 1 -2 }})


[[Comma list]]: [[275/273]] ({{monzo| 0 2 -1 -1 }}), [[847/845]] ({{monzo| 0 -1 1 -2 }})
{{Mapping|legend=3| 1 0 0 | 0 2 1 }}


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


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[Tp tuning|subgroup]] [[CTE]] and [[POTE]]: ~2 = 1200.000, ~13/11 = 293.679
* [[Tp tuning|subgroup]] [[CTE]] and [[POTE]]: ~2 = 1200.000, ~9/7 = 440.902


{{Optimal ET sequence|legend=1| 29, 33, 37, 41, 45, 49, 78, 94, 143* }}
{{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 13/11
: <nowiki/>* wart for 5/3
: <sup>†</sup> wart for 9/7


[[Tp tuning #T2 tuning|RMS error]]: 0.6914 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.7105 cents


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


Indium can be described as the {{nowrap| 8 & 33 }} temperament in the 2.5/3.7/3.11/3 subgroup.  
Marveltwintri can be described as the {{nowrap| 3 & 4 }} temperament in the 2.5/3.13/9 subgroup. The tonic and the first two generator steps make a [[marveltwin triad]], hence the name. [[Cata]] is a very natural extension of this temperament to the [[2.3.5.13 subgroup|2.3.5.13-subgroup]].


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


[[Comma list]]: [[3025/3024]] ({{monzo| -4 2 -1 2 }}), [[3125/3087]] ({{monzo| 0 5 -3 }})
[[Comma list]]: [[325/324]] ({{monzo| -2 2 1 }})


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


{{Mapping|legend=5| 1 -1/2 -1/2 -1/2 3/2 | 0 -15/4 9/4 25/4 -19/4 }}
{{Mapping|legend=5| 1 -1/6 5/6 0 0 -1/3 | 0 -1/2 -3/2 0 0 1 }}
: [[gencom]]: [2 12/11; 3025/3024 3125/3087]
: [[gencom]]: [2 6/5; 325/324]


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~12/11 = 146.978
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 882.886
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~12/11 = 147.010
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 882.861


{{Optimal ET sequence|legend=1| 8, 33, 41, 49, 204*<sup>†</sup> }}
{{Optimal ET sequence|legend=1| 3, 4, 11, 15, 19, 34, 53, 87, 140 }}
: <nowiki/>* wart for 7/3
: <sup>†</sup> wart for 11/3


[[Tp tuning #T2 tuning|RMS error]]: 0.7788 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.2444 cents


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


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.
Kiribati is related to [[nakika]] as well as to [[octacot]].  


[[Subgroup]]: 2.5/3.7/3.11/3.13/3
[[Subgroup]]: 2.9/5.7/3.11/9


[[Comma list]]: [[121/120]] ({{monzo| -3 -1 0 2 }}), [[169/168]] ({{monzo| -3 0 -1 0 2 }}), [[275/273]] ({{monzo| 0 2 -1 1 -1 }})
[[Comma list]]: 100/99 ({{monzo| 2 -2 0 -1 }}), 245/242 ({{monzo| -1 -1 2 -2 }})


{{Mapping|legend=3| 1 3 5 3 4 | 0 -6 -10 -3 -5 }}
{{Mapping|legend=3| 1 1 1 0 | 0 -2 3 4 }}
: mapping generators: ~2, ~21/20


{{Mapping|legend=5| 1 -3 0 2 0 1 | 0 24/5 -6/5 -26/5 9/5 -1/5 }}
{{Mapping|legend=5| 1 1/10 -4/5 11/10 1/5 | 0 -3/2 -1 3/2 1 }}
: [[gencom]]: [2 13/10; 121/120 169/168 275/273]
: [[gencom]]: [2 21/20; 100/99 245/242]


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~13/10 = 453.121
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~21/20 = 87.776
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~13/10 = 453.242
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~21/20 = 87.892


{{Optimal ET sequence|legend=1| 8, 29, 37, 45 }}
{{Optimal ET sequence|legend=1| 13, 14, 27, 41 }}


[[Tp tuning #T2 tuning|RMS error]]: 1.052 cents
[[Tp tuning #T2 tuning|RMS error]]: 1.245 cents


=== Sentry ===
== 2.75.… subgroups ==
{{See also | Chromatic pairs #Sentry }}
=== Archagall ===
{{See also| Fifthplus }}


Sentry, the {{nowrap| 3 & 5 }} temperament in the 2.5/3.9/7 subgroup, is related to [[sensi]].  
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]]: 2.5/3.9/7
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]]: [[245/243]] ({{monzo| 0 1 -2 }})
[[Subgroup]]: 2.75.85


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


{{Mapping|legend=5| 1 0 0 0 | 0 0 2 -1 }}
{{Mapping|legend=3| 1 2 5 | 0 3 1 }}
: [[gencom]]: [2 9/7; 245/243]
: mapping generators: ~2, ~85/32


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[Tp tuning|subgroup]] [[CTE]] and [[POTE]]: ~2 = 1200.000, ~9/7 = 440.902
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.9692{{c}}, ~85/64 = 491.5853{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~85/64 = 491.5794{{c}}


{{Optimal ET sequence|legend=1| 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| 5, 12, 17, 22, 61, 83, 310, 393, 476, 1345, 1821, 4118*, 5939*, 7760* }}
: <nowiki/>* wart for 5/3
: <sup>†</sup> wart for 9/7


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


=== Marveltwintri ===
Subgroup: 2.75.9/7.85
{{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]].
Comma list: 2025/2023 ({{monzo| 2 -2 1 0 }}), 24576/24565 ({{monzo| 13 1 0 -3 }})


[[Subgroup]]: 2.5/3.13/9
{{Mapping|legend=2| 1 2 6 5 | 0 3 -4 1 }}


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


{{Mapping|legend=3| 1 0 2 | 0 1 -2 }}
{{Optimal ET sequence|legend=0| 5, 12, 17, 22, 83, 105, 127, 403, 530, 657, 784, 1441* }}
 
{{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


== 2.7/3.… subgroups ==
== 2.7/3.… subgroups ==
Retrieved from "https://en.xen.wiki/w/Subgroup_temperaments"