Xenharmonic Wiki

Subgroup temperaments: Difference between revisions

← Older edit
VisualWikitext
Lériendil (talk | contribs)
8,044 edits
mNo edit summary
Hkm (talk | contribs)
autopatrolled
989 edits
 
(90 intermediate revisions by 13 users not shown)
Line 1: Line 1:
{{Technical data page}}
A '''subgroup temperament''' is a regular temperament defined on a [[just intonation subgroup]] that is not a full ''p''-limit group.  
A '''subgroup temperament''' is a regular temperament defined on a [[just intonation subgroup]] that is not a full ''p''-limit group.  


For temperaments that omit various prime harmonics, see:  
For temperaments that omit various prime harmonics, see:  
* [[No-thirteens subgroup temperaments]]
* [[No-elevens subgroup temperaments]]
* [[No-elevens subgroup temperaments]]
* [[No-sevens subgroup temperaments]]
* [[No-sevens subgroup temperaments]]
Line 10: Line 12:
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]].


= Integer subgroup temperaments =
= Composite subgroup temperaments =
== 2.3.35 subgroup ==
=== Shaka ===
{{See also|Kalismic temperaments}}
 
Two commas that split 2/1 in half, corresponding to convergents to sqrt(2), are the [[1682/1681|''sha''ftesburisma]] [[Square superparticular|S29]]/S41 and the [[9801/9800|''ka''lisma S99]], prompting to temper out {S29, S41, S99}, approximating /29 and /41 [[Primodality|primodal]] chords well.
 
Subgroup: 2.3.35.11.29.41
 
Comma list: 841/840, 1189/1188, 1681/1680
 
{{Mapping|legend=2|2 2 6 5 7 8|0 1 1 -1 1 1|0 0 2 2 1 1}}
 
Optimal tuning (CTE): ~41/29 = 1\2, ~3/2 = 702.031, ~41/24 = 926.693
 
[[Support]]ing [[ET]]s: {{EDOs|22, 26, 36, 48, 70, 96, 106, 118, 140, 154, 176, 188, 224, 272, 294, 342}}
 
Scale: [[Shaka10]]
 
== 2.9.5.7 subgroup ==
== 2.9.5.7 subgroup ==
See also [[Jubilismic clan #Antikythera|antikythera]] and [[Hemimean clan #Isra|isra]].  
See also [[Jubilismic clan #Antikythera|antikythera]] and [[Hemimean clan #Isra|isra]].  
Line 89: Line 73:
=== Baldy ===
=== Baldy ===
{{See also|Schismatic family #Garibaldi}}
{{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.
Baldy results from taking every other generator of the [[garibaldi]] temperament. One of the best extension is 2.9.5.7.13 subgroup with mapping 13/8 to +10 whole tones, as well as the cassandra temperament.


[[Subgroup]]: 2.9.5.7
[[Subgroup]]: 2.9.5.7
Line 105: Line 90:


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


Comma list: 225/224, 325/324, 640/637
Baldy is every other step of [[garibaldi]], without the mapping of prime 11. It can be described as the 6 & 35 temperament.


{{Mapping|legend=2| 1 3 3 4 2 | 0 1 -4 -7 10 }}
[[Subgroup]]: 2.9.5.7.13


Optimal tuning (POTE): ~2 = 1\1, ~9/8 = 204.090
[[Comma list]]: [[225/224]], [[325/324]], [[640/637]]


{{Optimal ET sequence|legend=1| 6, 29f, 35, 41, 47 }}
{{Mapping|legend=2| 1 0 15 25 -28 | 0 1 -4 -7 10 }}
 
{{Mapping|legend=3| 1 3/2 3 4 0 2 | 0 1/2 -4 -7 0 10 }}
 
: [[gencom]]: [2 9/8; 225/224 325/324 640/637]
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 204.090
 
{{Optimal ET sequence|legend=1| 6, 11, 17, 23, 29, 35, 41, 47, 100, 147, 488cd, 635cd }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.5999 cents


Related temperament: [[Schismatic family #Garibaldi|Cassandra]]
Related temperament: [[Schismatic family #Garibaldi|Cassandra]]
Line 142: Line 137:


{{Optimal ET sequence|legend=1| 6, 23def, 29f, 35, 41, 47 }}
{{Optimal ET sequence|legend=1| 6, 23def, 29f, 35, 41, 47 }}
== 2.3.25 subgroup ==
=== Shrub ===
This is a restriction of diaschismic which omits the tritone to produce a diatonic scale. True to its name, it generates a [[shrubmajor]] third (~425c) in quarter-comma tuning.
Subgroup: 2.3.25
Edo join: 17 & 12
Comma list: [[2048/2025]]
{{Mapping|legend=2| 1 1 7| 0 1 -4}}
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 705.136
==== 2.3.23.25.41 subgroup ====
''See also: [[Reversed meantone]]''
Edo join: 17 & 12
Comma list: 2048/2025, 576/575, 82/81
{{Mapping|legend=2| 1 1 1 7 3| 0 1 6 -4 4}}
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 705.264
===== Sburb =====
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
Edo join: 17 & 12
Comma list: 64/63, 225/224, 162/161, 82/81, 177/175
{{Mapping|legend=2| 1 1 4 1 7 3 10| 0 1 -2 6 -4 4 -7}}
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 706.387
== 2.9.5.11 subgroup ==
=== Glacial ===
{{See also| Chromatic pairs #Glacial }}
[[Subgroup]]: 2.9.5.11.13
[[Comma list]]: 45/44, 65/64, 81/80
{{Mapping|legend=2| 1 0 -4 -6 10 | 0 1 2 3 -2 }}
{{Mapping|legend=3| 1 3/2 2 0 3 4 | 0 1/2 2 0 3 -2 }}
: [[gencom]]: [2 9/8; 45/44 65/64 81/80]
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 186.151
{{Optimal ET sequence|legend=1| 6, 13, 45be, 58bce, 71bce, 84bce }}
[[Tp tuning #T2 tuning|RMS error]]: 2.887 cents
Music:
* ''[[Thundersnow]]'' - [[Sevish]] (2021)


== 2.9.7 subgroup ==
== 2.9.7 subgroup ==
Line 169: Line 226:


== 2.9.7.11 subgroup ==
== 2.9.7.11 subgroup ==
=== Apparatus ===
[[Subgroup]]: 2.9.7.11
[[Comma list]]: 41503/41472, 322102/321489
{{Mapping|legend=2| 1 5 3 5 | 0 -19 -2 -16 }}
: mapping generators: ~2, ~77/72
{{Mapping|legend=3| 1 5/2 0 3 5 | 0 -19/2 0 -2 -16 }}
: [[gencom]]: [2 77/72; 41503/41472 322102/321489]
[[Optimal tuning]] ([[CTE]]): ~77/72 = 115.5685
{{Optimal ET sequence|legend=1| 10e, 21, 31, 52, 83, 135, 353, 488, 623 }}
[[Badness]]: 0.00263
=== Joan ===
{{See also| Chromatic pairs #Joan }}
Joan is related to [[casablanca]] as well as to [[orwell]].
[[Subgroup]]: 2.9.7.11
[[Comma list]]: 99/98, 9317/9216
{{Mapping|legend=2| 1 0 1 3 | 0 7 4 1 }}
{{Mapping|legend=3| 1 0 0 1 3 | 0 7/2 0 4 1 }}
: [[gencom]]: [2 11/8; 99/98 9317/9216]
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~11/8 = 542.672 cents
{{Optimal ET sequence|legend=1| 11, 20, 31, 42, 115bd, 157bd }}
[[Tp tuning #T2 tuning|RMS error]]: 1.424 cents
=== Machine ===
=== Machine ===
Machine is every other step of [[supra]], most interesting for its scale patterns.  
Machine is every other step of [[supra]], most interesting for its scale patterns.  
Line 215: Line 312:
[[Badness]]: 0.00439
[[Badness]]: 0.00439


=== Apparatus ===
Scales: [[penta5]], [[penta8]], [[penta11]], [[penta19]]
[[Subgroup]]: 2.9.7.11


[[Comma list]]: 41503/41472, 322102/321489
== 2.9.7.13.17 subgroup ==


{{Mapping|legend=2| 1 5 3 5 | 0 -19 -2 -16 }}
=== Novisept ===
Novisept is generated by a one-cent-flat 9/7, such that stacking 5 of them gives you 7/4. It can be formed by doubling both generator and period of [[gizzard]].


: mapping generators: ~2, ~77/72
[[Subgroup]]: 2.9.7.13.17


{{Mapping|legend=3| 1 5/2 0 3 5 | 0 -19/2 0 -2 -16 }}
[[Comma list]]: 729/728, 442/441, 833/832
 
: [[gencom]]: [2 77/72; 41503/41472 322102/321489]


[[Optimal tuning]] ([[CTE]]): ~77/72 = 115.5685
{{Mapping|legend=2| 1 1 1 -1 3| 0 6 5 13 3 }}


{{Optimal ET sequence|legend=1| 10e, 21, 31, 52, 83, 135, 353, 488, 623 }}
[[Optimal tuning]] ([[CWE]]): ~2 = 1\1, ~9/7 = 433.836


[[Badness]]: 0.00263
Badness (Dirichlet): 0.142


== 2.9.11 subgroup ==
== 2.9.11 subgroup ==
Line 320: Line 415:
== 2.9.21 subgroup ==
== 2.9.21 subgroup ==
=== A-team ===
=== A-team ===
A-team is every other step of [[mothra]].  
A-team is every other step of [[slendric]]; the 2.9.5.21.11 extension below specifically restricts [[mothra]].  


[[Subgroup]]: 2.9.21
[[Subgroup]]: 2.9.21
Line 340: Line 435:
[[Tp tuning #T2 tuning|RMS error]]: 0.3202 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.3202 cents


==== 2.9.5.21.11 ====
==== 2.9.5.21 ====
''Lookalike temperament: [[Dual-fifth_temperaments#Dual-3_A-Team|Dual-3 A-Team]]''
 
Subgroup: 2.9.5.21
 
[[Comma]] list: 81/80, 1029/1024
 
Sval mapping: {{mapping| 1 2 0 4 | 0 3 6 1 }}
 
Mapping generators: ~2, ~21/16
 
Optimal ([[Lp tuning|POL2]]) generator: 464.3865
 
{{Optimal ET sequence|legend=1| 13, 18, 31, 44 }}
 
===== 2.9.5.21.11 =====
Subgroup: 2.9.5.21.11
Subgroup: 2.9.5.21.11


Line 355: Line 465:
{{Optimal ET sequence|legend=1| 5, 13, 31 }}
{{Optimal ET sequence|legend=1| 5, 13, 31 }}


== 2.15.55 subgroup ==
==== B-team ====
=== Spog ===
B-team (23 & 41) is every other step of [[rodan]].
 
This temperament produces [[Slendro_clan#Superpelog|superpelog]]-like [[5L 4s|semiquartal]] scales while being more accurate ([[Subgroup temperaments#2.15.189.55.325.725.279|see]] rational approximations to their intervals).
 
[[Subgroup]]: 2.15.55
 
[[Comma list]]: [[100663296/100656875]]
 
{{Mapping|legend=2| 1 0 5 | 0 5 1 }}
 
[[Optimal tuning]] (subgroup [[CTE]]): ~55/32 = 937.655
 
{{Optimal ET sequence|legend=1|5, 9, 23, 32, 151, 183, 215, 247, 956, 1203, 1450, 3147, 4597 }}
 
==== 2.15.55.325 ====
[[Subgroup]]: 2.15.55.325
 
[[Comma list]]: [[4225/4224]], [[6656/6655]]
 
{{Mapping|legend=2| 1 0 5 6 | 0 5 1 3 }}
 
[[Optimal tuning]] (subgroup [[CTE]]): ~55/32 = 937.647
 
[[Support]]ing [[ET]]s: 5, 9, 13[-15], 14, 23, 32, 37, 41, 50, 55, 64, 73, 78, 87, 96, 101, 105, 119, 128, 151, 183, 206, 311
 
==== 2.15.189.55.325 ====
Related temperament: [[Lehmerismic_temperaments#Lux|lux]]
 
[[Subgroup]]: 2.15.189.55.325
 
[[Comma list]]: [[2080/2079]], [[3025/3024]], [[4096/4095]]
 
{{Mapping|legend=2| 1 0 6 5 6 | 0 5 2 1 3 }}
 
[[Optimal tuning]] (subgroup [[CTE]]): ~55/32 = 937.677
 
[[Support]]ing [[ET]]s: 5, 9, 14, 23, 32, 37, 41, 50, 55, 64, 73, 78, 87, 96, 101, 105, 119, 128, 151, 183, 206, 311
 
==== 2.15.189.55.325.725 ====
 
[[Subgroup]]: 2.15.189.55.325.725
 
[[Comma list]]: [[1625/1624]], [[2080/2079]], [[3025/3024]], [[4096/4095]]
 
{{Mapping|legend=2| 1 0 6 5 6 -3 | 0 5 2 1 3 16 }}


[[Optimal tuning]] (subgroup [[CTE]]): ~55/32 = 937.649
Subgroup: 2.9.15.21.33


[[Support]]ing [[ET]]s: 9[-725], 14[+725], 23, 32, 41[-725], 55, 73[-725], 87, 105[-725], 119,  142[+725], 151, 183, 206[+725], 311
Comma list: 245/243, 385/384, 441/440


==== 2.15.189.55.325.725.279 ====
Sval mapping: {{mapping| 1 2 0 4 7 | 0 3 10 1 -5 }}


Here are rational approximations to the intervals of the semiquartal scale.
Optimal tuning (subgroup POTE): ~2 = 1\1, ~21/16 = 468.918


Sharp: 12/11, 25/21, 33/26, 18/13, 31/21 ~ 65/44 ~ 96/65, 50/31 ~ 29/18, 55/32, 15/8.
{{Optimal ET sequence|legend=1| 5, 13c, 18, 23, 41, 64, 87, 151 }}
 
Flat: 16/15, 64/55, 31/25 ~ 36/29, 42/31 ~ 65/48 ~ 88/65, 13/9, 52/33, 42/25, 11/6.
[[Subgroup]]: 2.15.189.55.325.725.279
 
[[Comma list]]: [[1625/1624]], [[2016/2015]], [[2080/2079]], [[3025/3024]], [[4096/4095]]
 
{{Mapping|legend=2| 1 0 6 5 6 -3 5 | 0 5 2 1 3 16 4 }}
 
[[Optimal tuning]] (subgroup [[CTE]]): ~55/32 = 937.638
 
[[Support]]ing [[ET]]s: 9[-725], 14[+725], 23, 32, 41[-725], 55, 73[-725], 87, 105[-725], 119, 151, 183, 206[+725], 311


== 4.3.5 subgroup ==
== 4.3.5 subgroup ==
Line 499: Line 553:
[[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: *7, *10, *11[-5], *13[+5], *17, *24, *27[+5], *31, *38, *41, *45, *52, *55, *69


<nowiki>*</nowiki> wart for 4
<nowiki />* Wart for 4


==== 4.6.5.7 subgroup (tetrominant) ====
==== 4.6.5.7 subgroup (tetrominant) ====
Line 512: Line 566:
[[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]
[[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]


<nowiki>*</nowiki> wart for 4
<nowiki />* Wart for 4


=== Fourwar ===
=== Fourwar ===
Line 518: Line 572:


Fourwar is named after the closely related [[hemiwar]] temperament.
Fourwar is named after the closely related [[hemiwar]] temperament.
{{Todo|inline=1|cleanup}}


<pre>  
<pre>  
Line 789: Line 845:
Mapping generators: ~4, ~9/64
Mapping generators: ~4, ~9/64


[[Optimal tuning]] ([[CTE]]): ~[[9/8]] = 219.190
[[Optimal tuning]] ([[CTE]]): ~9/4 = 1419.190


[[Support]]ing [[ET]]s: 5, 17, 22, 12, 7, 27, 32, 8, 39[+49], 29[+49], 9[+49], 19[+49], 37, 49
[[Support]]ing [[ET]]s: 5, 17, 22, 12, 7, 27, 32, 8, 39[+49], 29[+49], 9[+49], 19[+49], 37, 49
Line 807: Line 863:
: [[gencom]]: [8 9/8; 64/63]
: [[gencom]]: [8 9/8; 64/63]


[[Optimal tuning]] ([[CTE]]): 1\[[3ed8]] = 1600.0, ~9/8 = 219.1898
[[Optimal tuning]] ([[CTE]]): ~9/8 = 219.1898


[[Optimal ET sequence]]: {{val| 16 17 15 }}, {{val| 33 35 31 }}, {{val| 148 … }}, {{val| 181 … }}, {{val| 214 … }}, {{val| 247 … }}
[[Optimal ET sequence]]: {{val| 16 17 15 }}, {{val| 33 35 31 }}, {{val| 148 … }}, {{val| 181 … }}, {{val| 214 … }}, {{val| 247 … }}
Line 814: Line 870:


= Fractional subgroup temperaments =
= Fractional subgroup temperaments =
== 2.5/3… subgroups ==
== 2.5/3.… subgroups ==
=== Magicaltet ===
{{See also| Chromatic pairs #Magicaltet }}
 
Magicaltet is related to [[keemic]], [[superkleismic]], and [[magic]]. The tonic and the first three generator steps make a [[magical seventh chord]], hence the name.
 
[[Subgroup]]: 2.5/3.7.11
 
[[Comma list]]: 100/99 ({{monzo| 2 2 0 -1 }}), 385/384 ({{monzo| -7 1 1 1 }})
 
{{Mapping|legend=2| 1 0 5 2 | 0 1 -3 2 }}
: mapping generators: ~2, ~5/3


=== Greeley ===
{{Mapping|legend=3| 1 -1/2 1/2 2 4 | 0 1/2 -1/2 3 -2 }}
Related temperaments: [[Porcupine family #Opossum|Opossum]], [[Starling temperaments #Nusecond|Nusecond]]
: [[gencom]]: [2 6/5; 100/99 385/384]
 
[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 877.343
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 877.351
 
{{Optimal ET sequence|legend=1| 4, 7, 11, 15, 26, 67, 93* }}
: <nowiki/>* wart for 5/3
 
[[Tp tuning #T2 tuning|RMS error]]: 1.206 cents
 
=== Starlingtet ===
{{See also | Chromatic pairs #Starlingtet }}
 
Starlingtet, the {{nowrap| 4 & 15 }} temperament in the 2.5/3.7/3 subgroup, is related to [[starling]] as well as to [[myna]]. The tonic and the first three generator steps make a [[starling tetrad]], hence the name.
 
[[Subgroup]]: 2.5/3.7/3
 
[[Comma list]]: [[126/125]] ({{monzo| 1 -3 1 }})
 
{{Mapping|legend=2| 1 0 -1 | 0 1 3 }}
 
: mapping generators: ~2, ~5/3
 
{{Mapping|legend=3| 1 -1 0 1 | 0 4/3 1/3 -5/3 }}
: [[gencom]]: [2 6/5; 126/125]
 
[[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
 
{{Optimal ET sequence|legend=1| 4, 15, 19, 23, 27 }}
 
[[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, 126/125
[[Comma list]]: 121/120 ({{monzo| -3 -1 0 2 }}), 126/125 ({{monzo| 1 -3 1 }})


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


{{Mapping|legend=3| 1 -5/4 -1/4 3/4 3/4 | 0 9/4 1/4 -15/4 5/4 }}
{{Mapping|legend=3| 1 -5/4 -1/4 3/4 3/4 | 0 9/4 1/4 -15/4 5/4 }}
: [[gencom]]: [2 11/10; 121/120 126/125]
[[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
{{Optimal ET sequence|legend=1| 8, 15, 23, 54, 77, 100, 131* }}
: <nowiki/>* wart for 11/3
[[Tp tuning #T2 tuning|RMS error]]: 1.034 cents
==== Skateboard ====
{{See also| Chromatic pairs #Skateboard }}
Skateboard is related to [[thrasher]].
[[Subgroup]]: 2.5/3.7/3.11.13/9
[[Comma list]]: 56/55 ({{monzo| 3 -1 1 -1 }}), 91/90 ({{monzo| -1 -1 1 0 1 }}), 100/99 ({{monzo| 2 2 0 -1 }})
{{Mapping|legend=2| 1 0 -1 2 2 | 0 1 3 2 -2 }}
{{Mapping|legend=3| 1 -3/7 4/7 11/7 4 -6/7 | 0 0 -1 -3 -2 2 }}
: [[gencom]]: [2 6/5; 56/55 91/90 100/99]
[[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
{{Optimal ET sequence|legend=1| 11, 15, 19, 23, 42d, 65d }}
[[Tp tuning #T2 tuning|RMS error]]: 2.396 cents
=== Gariberttet ===
Gariberttet is the 2.5/3.7/3 [[Subgroup temperament families, relationships, and genes|altergene]] of [[sirius]].
==== Gariberttet (2.5/3.7/3.13/11 subgroup) ====
{{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]]: 2.5/3.7/3.13/11
[[Comma list]]: [[275/273]] ({{monzo| 0 2 -1 -1 }}), [[847/845]] ({{monzo| 0 -1 1 -2 }})
{{Mapping|legend=2| 1 0 0 0 | 0 3 5 1 }}
{{Mapping|legend=3| 1 0 0 0 0 0 | 0 -8/3 1/3 7/3 -1/2 1/2 }}
: [[gencom]]: [2 13/11; 275/273 847/845]
[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]] and [[POTE]]: ~2 = 1200.000, ~13/11 = 293.679
{{Optimal ET sequence|legend=1| 29, 33, 37, 41, 45, 49, 78, 94, 143* }}
: <nowiki/>* wart for 13/11
[[Tp tuning #T2 tuning|RMS error]]: 0.6914 cents
==== Indium ====
{{See also | Chromatic pairs #Indium }}
Indium can be described as the {{nowrap| 8 & 33 }} temperament in the 2.5/3.7/3.11/3 subgroup.
[[Subgroup]]: 2.5/3.7/3.11/3
[[Comma list]]: [[3025/3024]] ({{monzo| -4 2 -1 2 }}), [[3125/3087]] ({{monzo| 0 5 -3 }})
{{Mapping|legend=2| 1 0 0 2 | 0 6 10 -1 }}
{{Mapping|legend=3| 1 -1/2 -1/2 -1/2 3/2 | 0 -15/4 9/4 25/4 -19/4 }}
: [[gencom]]: [2 12/11; 3025/3024 3125/3087]
[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~12/11 = 146.978
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~12/11 = 147.010
{{Optimal ET sequence|legend=1| 8, 33, 41, 49, 204*<sup>†</sup> }}
: <nowiki/>* wart for 7/3
: <sup>†</sup> wart for 11/3
[[Tp tuning #T2 tuning|RMS error]]: 0.7788 cents
==== Ammon ====
{{See also| Chromatic pairs #Ammon }}
Ammon can be described as the {{nowrap| 8 & 29 }} temperament in the 2.5/3.7/3.11/3.13/3 subgroup. It extends [[tridec]], and is related to [[ammonite]]. It is generated by a semidiminished fourth, hence the old name ''semidim'', which has been rejected since 2025 to avoid confusion with another temperament of the same name.
[[Subgroup]]: 2.5/3.7/3.11/3.13/3
[[Comma list]]: [[121/120]] ({{monzo| -3 -1 0 2 }}), [[169/168]] ({{monzo| -3 0 -1 0 2 }}), [[275/273]] ({{monzo| 0 2 -1 1 -1 }})
{{Mapping|legend=2| 1 3 5 3 4 | 0 -6 -10 -3 -5 }}
{{Mapping|legend=3| 1 -3 0 2 0 1 | 0 24/5 -6/5 -26/5 9/5 -1/5 }}
: [[gencom]]: [2 13/10; 121/120 169/168 275/273]
[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~13/10 = 453.121
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~13/10 = 453.242
{{Optimal ET sequence|legend=1| 8, 29, 37, 45 }}
[[Tp tuning #T2 tuning|RMS error]]: 1.052 cents
=== Sentry ===
{{See also | Chromatic pairs #Sentry }}
Sentry, the {{nowrap| 3 & 5 }} temperament in the 2.5/3.9/7 subgroup, is related to [[sensi]].
[[Subgroup]]: 2.5/3.9/7
[[Comma list]]: [[245/243]] ({{monzo| 0 1 -2 }})
{{Mapping|legend=2| 1 0 0 | 0 2 1 }}
{{Mapping|legend=3| 1 0 0 0 | 0 0 2 -1 }}
: [[gencom]]: [2 9/7; 245/243]
[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]] and [[POTE]]: ~2 = 1200.000, ~9/7 = 440.902
{{Optimal ET sequence|legend=1| 8, 11, 19, 30, 41, 49, 52, 145*, 166<sup>†</sup>, 197*<sup>†</sup>, 215<sup>†</sup>, 264*<sup>†</sup> }}
: <nowiki/>* wart for 5/3
: <sup>†</sup> wart for 9/7
[[Tp tuning #T2 tuning|RMS error]]: 0.7105 cents
=== Marveltwintri ===
{{See also| Chromatic pairs #Marveltwintri }}
Marveltwintri can be described as the {{nowrap| 3 & 4 }} temperament in the 2.5/3.13/9 subgroup. The tonic and the first two generator steps make a [[marveltwin triad]], hence the name. [[Cata]] is a very natural extension of this temperament to the [[2.3.5.13 subgroup|2.3.5.13-subgroup]].
[[Subgroup]]: 2.5/3.13/9
[[Comma list]]: [[325/324]] ({{monzo| -2 2 1 }})
{{Mapping|legend=2| 1 0 2 | 0 1 -2 }}
{{Mapping|legend=3| 1 -1/6 5/6 0 0 -1/3 | 0 -1/2 -3/2 0 0 1 }}
: [[gencom]]: [2 6/5; 325/324]
[[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.….7/3.… subgroups ==
=== Guanyintet ===
{{See also | Chromatic pairs #Guanyintet }}
Guanyintet, the {{nowrap| 4 & 9 }} temperament in the 2.5.7/3.11/3 subgroup, is the main rank-2 chain of [[guanyin]] and a restriction of [[orwell]]. It is defined by tempering out [[1728/1715]] ({{S|6/S7}}) and [[540/539]] (S12/S14), which imply [[176/175]] (S8/S10) as well as S11/S15 being tempered out. The tonic and the first three generator steps make a [[guanyin tetrad]], hence the name.
[[Subgroup]]: 2.5.7/3.11/3
[[Comma list]]: [[176/175]] ({{monzo| 4 -2 -1 1 }}), [[540/539]] ({{monzo| 2 1 -2 -1 }})
{{Mapping|legend=2| 1 0 1 3 | 0 -3 1 -5 }}
: mapping generators: ~2, ~7/6
{{Mapping|legend=3| 1 -4/3 3 -1/3 5/3 | 0 4/3 -3 7/3 -11/3 }}
: [[gencom]]: [2 7/6; 176/175 540/539]
[[Optimal tuning]]s:
* ([[Tp tuning|subgroup]] [[CTE]]): ~2 = 1200.000, ~7/6 = 270.455
* ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1200.000, ~7/6 = 270.093
{{Optimal ET sequence|legend=1| 9, 22, 31, 40, 191c*, 231c*, 271c*, 311c* }}
: <nowiki/>* wart for 7/3
[[Tp tuning #T2 tuning|RMS error]]: 0.6028 cents
==== 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.
[[Subgroup]]: 2.5.7/3.11/3.13
[[Comma list]]: [[176/175]] ({{monzo| 4 -2 -1 1 0 }}), [[540/539]] ({{monzo| 2 1 -2 -1 0 }}), [[1573/1568]] ({{monzo| -5 0 -2 2 1 }})
{{Mapping|legend=2| 1 0 1 3 1 | 0 -3 1 -5 12 }}
: mapping generators: ~2, ~12/7
[[Optimal tuning]]s:
* ([[Tp tuning|subgroup]] [[CTE]]): ~2 = 1200.000, ~7/6 = 270.152
* ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1200.000, ~7/6 = 270.218
{{Optimal ET sequence|legend=1| 9, 22, 31, 40, 71, 111, 151, 262c*}} <small> using subgroup TE </small>
: <nowiki/>* wart for 7/3
Badness (Sintel): 0.329
==== Laz ====
{{See also | Chromatic pairs #Laz }}
Laz is related to [[avalokita]] as well as to [[winston]].
[[Subgroup]]: 2.5.7/3.11/3.13/3
[[Comma list]]: [[144/143]] ({{monzo| 4 0 0 -1 -1 }}), [[176/175]] ({{monzo| 4 -2 -1 1 }}), [[196/195]] ({{monzo| 2 -1 2 0 -1 }}
{{Mapping|legend=2| 1 0 2 -2 6 | 0 3 -1 5 -5 }}
{{Mapping|legend=3| 1 -5/4 3 -1/4 7/4 -1/4 | 0 -1/4 -3 3/4 -21/4 19/4 }}
: [[gencom]]: [2 7/6; 144/143 176/175 196/195]
[[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
{{Optimal ET sequence|legend=1| 9, 31, 40, 49, 156c*†, 205c*† }}
: <nowiki/>* wart for 7/3
: † wart for 11/3
[[Tp tuning #T2 tuning|RMS error]]: 0.8790 cents
=== Kryptonite ===
{{See also| Chromatic pairs #Kryptonite }}


: [[gencom]]: [2 11/10; 121/120 126/125]
Kryptonite is related to [[krypton]].
 
[[Subgroup]]: 2.5.7/3.11/3.13/3
 
[[Comma list]]: 56/55 ({{monzo| 3 -1 1 -1 }}), 78/77 ({{monzo| 1 0 -1 -1 1 }}), 91/90 ({{monzo| -1 -2 1 0 1 }})
 
{{Mapping|legend=2| 1 2 1 2 2 | 0 3 2 -1 1 }}
: mapping generators: ~2, ~13/12
 
{{Mapping|legend=3| 1 -5/4 2 -1/4 3/4 3/4 | 0 -1/2 3 3/2 -3/2 1/2 }}
: [[gencom]]: [2 13/12; 56/55 78/77 91/90]
 
[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~13/12 = 130.945
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~13/12 = 132.428
 
{{Optimal ET sequence|legend=1| 1, …, 8, 9 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 2.545 cents
 
=== 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 }})


[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~11/10 = 155.776
{{Mapping|legend=2| 1 1 1 0 | 0 -2 3 4 }}
: mapping generators: ~2, ~21/20


{{Optimal ET sequence|legend=1| 8, 15, 23, 54, 77, 100, 131†, 208*† }}
{{Mapping|legend=3| 1 1/10 -4/5 11/10 1/5 | 0 -3/2 -1 3/2 1 }}
: [[gencom]]: [2 21/20; 100/99 245/242]


<nowiki>*</nowiki> wart for 5/3
[[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


<nowiki>†</nowiki> wart for 11/3
{{Optimal ET sequence|legend=1| 13, 14, 27, 41 }}


[[Tp tuning#T2 tuning|RMS error]]: 1.034 cents
[[Tp tuning #T2 tuning|RMS error]]: 1.245 cents


== 2.….9/7… subgroups ==
=== Mothwelltri ===
{{See also| Chromatic pairs #Mothwelltri }}


=== Marveltri ===
Mothwelltri, the {{nowrap| 1 & 4 }} temperament in the 2.7/3.11 subgroup, is related to [[orwell]]. The tonic and the first two generator steps make a [[mothwellsmic triad]], hence the name.  
Marveltri, the 3 &amp; 13 temperament in the 2.5.9/7 subgroup, is related to marvel, magic, and the unnamed 22 &amp; 47 temperament.  


[[Subgroup]]: 2.5.9/7
[[Subgroup]]: 2.7/3.11


[[Comma list]]: 225/224
[[Comma list]]: [[99/98]] ({{monzo| -1 -2 1 }})


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


{{Mapping|legend=3| 1 2/5 2 -1/5 | 0 -4/5 1 2/5 }}
{{Mapping|legend=3| 1 -1/2 0 1/2 3 | 0 -1/2 0 1/2 2 }}
: [[gencom]]: [2 7/6; 99/98]


: [[gencom]]: [2 5/4; 225/224]
[[Optimal tuning]]s:  
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~7/6 = 273.695
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~7/6 = 273.174


[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~5/4 = 383.638
{{Optimal ET sequence|legend=1| 4, 9, 13, 22, 79 }}


{{Optimal ET sequence|legend=1| 12, 13, 16, 19, 22, 25, 47, 69, 72, 97, 122, 269c*, 660c* }}
[[Tp tuning #T2 tuning|RMS error]]: 1.064 cents


<nowiki>*</nowiki> wart for 9/7
== 2.….9/7.… subgroups ==
=== Marveltri ===
{{See also| Chromatic pairs #Marveltri }}


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


==== Sulis ====
[[Subgroup]]: 2.5.9/7
Related temperament: [[Marvel family|minerva]], [[Würschmidt family|würschmidt]]


[[Subgroup]]: 2.5.9/7.11/7
[[Comma list]]: 225/224 ({{monzo| -5 2 1 }})


[[Comma list]]: 99/98, 176/175
{{Mapping|legend=2| 1 0 5 | 0 1 -2 }}
: mapping generators: ~2, ~5


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


[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~5/4 = 386.558
[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/4 = 384.208
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/4 = 383.638


{{Optimal ET sequence|legend=1| 3, …, 22, 25, 28, 31, 59 }}
{{Optimal ET sequence|legend=1| 3, 13, 16, 19, 22, 25, 72, 97, 122, 269c* }}
: <nowiki/>* wart for 9/7


[[Tp tuning #T2 tuning|RMS error]]: 1.074 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.4801 cents


== 2.….15/11… subgroups ==
==== Sulis ====
Sulis is related to [[minerva]] and [[würschmidt]].


=== Poggers ===
[[Subgroup]]: 2.5.9/7.11/9
Related temperaments: [[Stearnsmic_clan#Pogo|pogo]], [[Stearnsmic_clan#Supers|supers]]


[[Subgroup]]: 2.9.7.15/11.13
[[Comma list]]: 99/98 ({{monzo| -1 0 2 1 }}), 176/175 ({{monzo| 4 -2 1 1 }})


[[Comma list]]: [[540/539]], [[1716/1715]], [[2080/2079]]
{{Mapping|legend=2| 1 0 5 -9 | 0 1 -2 4 }}]


{{Mapping|legend=2| 1 1 1 -1 -1 | 0 6 5 4 13 }}
[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/4 = 386.617
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/4 = 386.558


[[Optimal tuning]] (subgroup [[CTE]]): ~9/7 = 433.888
{{Optimal ET sequence|legend=1| 3, …, 22, 25, 28, 31, 59 }}


[[Support]]ing [[ET]]s: 8[+9, +7, +13], 11, 14[-13], 19[+9, +7, ++13], 25[-13], 36, 47, 58, 61[-13], 69[+13], 80[+13], 83, 91[+9, +7, +13], 105
[[Tp tuning #T2 tuning|RMS error]]: 1.074 cents


== 2.….7/5… subgroups ==
== 2.….7/5.… subgroups ==
=== Hydrothermal ===
=== Hydrothermal ===
A tuning whose distinctively sharp (but still consonant) fifth, and flat (but still consonant) octave, lend it a mysterious, heavy atmosphere. The 6-tone (hexatonic) MOS is melodically interesting and flavorful. The 18-tone MOS is a useful 'chromatic' scale for taking subsets of.
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.
Line 905: Line 1,272:


[[Support]]ing [[ET]]s: {{EDOs|4, 6, 8, 10, 18, 28, 46, 64, 110}}
[[Support]]ing [[ET]]s: {{EDOs|4, 6, 8, 10, 18, 28, 46, 64, 110}}
=== Argentic ===
Argentic is the 2.3.7/5 subgroup temperament tempering out [[5120/5103]].
[[Subgroup]]: 2.3.7/5
[[Comma list]]: [[5120/5103]] = {{monzo| 10 -6 -1 }}
{{Mapping|legend=2| 1 0 10 | 0 1 -6 }}
: mapping generators: ~2, ~3
[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1\1, ~3/2 = 702.792
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1\1, ~3/2 = 702.830
{{Optimal ET sequence|legend=1| 12, 29, 41, 70, 321, 391, 461, 531, 601 }}
<small> based on subgroup TE </small>
Badness (Sintel): 0.119
==== Edson (2.3.7/5.11/5.13/5 subgroup) ====
{{See also| Chromatic pairs #Edson }}
Edson is related to [[pele]] and [[andromeda]].
[[Subgroup]]: 2.3.7/5.11/5.13/5
[[Comma list]]: [[196/195]] = {{monzo| 2 -1 2 0 -1 }}, [[352/351]] = {{monzo| 5 -3 0 1 -1 }}, [[364/363]] = {{monzo| 2 -1 1 -2 1 }}
{{Mapping|legend=2| 1 0 10 17 22 | 0 1 -6 -10 -13 }}
: mapping generators: ~2, ~3
{{Mapping|legend=3| 1 1 -5 -1 2 4 | 0 1 29/4 5/4 -11/4 -23/4 }}
: [[gencom]]: [2 3/2; 196/195, 352/351, 364/363]
[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1\1, ~3/2 = 703.4398
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1\1, ~3/2 = 703.414
{{Optimal ET sequence|legend=1| 12, 17, 29 }}
[[Tp tuning #T2 tuning|RMS error]]: 0.5102 cents
==== Haumea ====
{{See also| Chromatic pairs #Haumea }}
Related temperaments include [[#Bridgetown|bridgetown]], [[namaka]], [[hemigari]], [[#Barbados|barbados]], and [[parizekmic]].
[[Subgroup]]: 2.3.7/5.11/5.13/5
[[Comma list]]: [[352/351]], [[676/675]], [[847/845]]
{{Mapping|legend=2| 1 0 10 -6 -1 | 0 2 -12 9 3 }}
{{Mapping|legend=3| 1 2 -3/4 -11/4 9/4 5/4 | 0 -2 0 12 -9 -3 }}
: [[gencom]]: [2 15/13; 352/351 676/675 847/845]
[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~15/13 = 248.491
{{Optimal ET sequence|legend=1| 24, 29, 111, 140, 169, 198, 565d, 763bd, 961bd }}
[[Tp tuning #T2 tuning|RMS error]]: 0.2668 cents


=== Historical ===
=== Historical ===
{{distinguish|Historical temperaments}}
{{distinguish|Historical temperaments}}
{{distinguish|History (temperament)}}, which is the rank-3 version of this temperament in the full 13-limit.
Historical is essentially an analogue of [[miracle]] that splits [[4/3]] in six rather than [[3/2]]. It tempers out the comma S10/S11 = [[4000/3993]] to set [[11/10]] equal to one-third of 4/3, and S13/S15 = [[676/675]] to equate [[15/13]] to one-half of 4/3, and tempers out S21 = [[441/440]] to split 11/10 into two instances of [[22/21]]~[[21/20]]. [[Sextilifourths]] adds the [[schismic]] mapping of prime 5 (reached by eight fourths) to complete the 13-limit.
[[Subgroup]]: 2.3.7/5.11/5.13/5
[[Subgroup]]: 2.3.7/5.11/5.13/5


Line 920: Line 1,353:
[[Tp tuning #T2 tuning|RMS error]]: 0.2562 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.2562 cents


== 2.….11/5… subgroups ==
=== Terrain ===
{{Redirect|Terrain|the scale|Terrain (scale)}}
{{See also| Chromatic pairs #Terrain }}
 
Terrain, the 6 &amp; 21 temperament in the 2.7/5.9/5 subgroup, is related to [[domain (temperament)|domain]]. It is a remarkable temperament, in that while its complexity is low, it has no discernible error. The 1–7/5–9/5 and 1–9/7–9/5 chords are characteristic.
 
[[Subgroup]]: 2.7/5.9/5
 
[[Comma list]]: [[250047/250000]]
 
{{Mapping|legend=2| 3 1 3 | 0 1 -1 }}
 
{{Mapping|legend=3| 3 10/9 -7/9 2/9 | 0 -2/3 -1/3 2/3 }}
: [[gencom]]: [63/50 10/9; 250047/250000]
 
[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~63/50 = 1\3, ~10/9 = 182.461
 
{{Optimal ET sequence|legend=1| 6, 21, 27, 33, 105, 138, 171, 1848, 2019, 2190, 2361, 2532, 2703, 2874, 3045, 3216, 3387, 3558 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.00844 cents
 
=== Tridec ===
{{See also| Chromatic pairs #Tridec }}
{{See also| Non-over-1 temperament #Tridec }}
 
Tridec, the 5 &amp; 8 temperament in the 2.7/5.11/5.13/5 subgroup, extends [[#Petrtri]].
 
[[Subgroup]]: 2.7/5.11/5.13/5
 
[[Comma list]]: [[847/845]], [[1001/1000]]
 
{{Mapping|legend=2| 1 2 0 1 | 0 -4 3 1 }}
 
{{Mapping|legend=3| 1 0 -3/4 5/4 -3/4 1/4 | 0 0 0 -4 3 1 }}
: [[gencom]]: [2 13/10; 847/845 1001/1000]
 
[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~13/10 = 454.556
 
{{Optimal ET sequence|legend=1| 5, 8, 21, 29, 37, 66, 169, 235, 404c, 639c, 953bc }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.1613 cents
 
==== Naiadec ====
[[Subgroup]]: 2.7/5.11/5.13/5.17/5
 
[[Comma list]]: [[170/169]], [[221/220]], [[847/845]]
 
{{Mapping|legend=2| 1 2 0 1 1 | 0 -4 3 1 2 }}
 
{{Mapping|legend=3| 1 0 -3/4 5/4 -3/4 1/4 1/4 | 0 0 0 -4 3 1 2 }}
: [[gencom]]: [2 13/10; 170/169 221/220 847/845]
 
[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~13/10 = 454.882
 
{{Optimal ET sequence|legend=1| 5, 8, 21, 29, 95<sup>t</sup>, 124<sup>t</sup> }}
: <sup>t</sup> wart for 17/5
 
[[Tp tuning #T2 tuning|RMS error]]: 0.7521 cents
 
== 2.….11/5.… subgroups ==
=== Petrtri ===
{{See also| Chromatic pairs #Petrtri }}
{{See also| 5L 3s/Temperaments #Petrtri }}
 
Petrtri can be described as 3 &amp; 5 temperament in the 2.11/5.13/5 subgroup.
 
[[Subgroup]]: 2.11/5.13/5
 
[[Comma list]]: [[2200/2197]]
 
{{Mapping|legend=2| 1 0 1| 0 3 1 }}
 
{{Mapping|legend=3| 1 0 -1/3 0 -1/3 2/3 | 0 0 -4/3 0 5/3 -1/3 }}
: [[gencom]]: [2 13/10; 2200/2197]
 
[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~13/10 = 455.012
 
{{Optimal ET sequence|legend=1| 21, 29, 153, 182, 211, 240, 269, 298, 327, 356, 385, 509, 741c, 1126c }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.0749 cents
 
==== Bridgetown ====
{{See also| Chromatic pairs #Bridgetown }}
 
Bridgetown, the 5 &amp; 24 temperament in the 2.3.11/5.13/5 subgroup, is related to [[#Haumea|haumea]] and [[#Barbados|barbados]].
 
[[Subgroup]]: 2.3.11/5.13/5
 
[[Comma list]]: [[352/351]], [[676/675]]
 
{{Mapping|legend=2| 1 0 -6 -1 | 0 2 9 3 }}
 
{{Mapping|legend=3| 1 2 -5/3 0 4/3 1/3 | 0 -2 4 0 -5 1 }}
: [[gencom]]: [2 15/13; 352/351 676/675]
 
[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~15/13 = 248.399
 
{{Optimal ET sequence|legend=1| 5, 9, 14, 19, 24, 29, 169, 198, 227, 256, 285, 314 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.2513 cents
 
=== Hypnosis ===
=== Hypnosis ===
Related temperaments: [[Swetismic temperaments #Hypnos|hypnos]], [[Hemifamity temperaments #Tricot|tricot]]
Related temperaments: [[Swetismic temperaments #Hypnos|hypnos]], [[Alphatricot family #Alphatricot|alphatricot]]


[[Subgroup]]: 2.3.7.11/5.13
[[Subgroup]]: 2.3.7.11/5.13
Line 936: Line 1,469:
[[Tp tuning #T2 tuning|RMS error]]: 0.5379 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.5379 cents


== 2.….11/7… subgroups ==
=== Trisect ===
Trisect divides every Pythagorean interval into three, and is the much more accurate subgroup restriction of [[Augmented family #Trisected|trisected]].
 
Extending this temperament to the full [[11-limit|11-]], [[13-limit|13-]], or [[17-limit]] through [[portent]] or [[landscape]] results in the [[weak extension]] known as [[tritikleismic]].
 
[[Subgroup]]: 2.3.7.11/5
 
[[Comma list]]: 1029/1024, 4000/3993
 
{{Mapping|legend=2| 3 0 10 5 | 0 3 -1 -1 }}
 
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~44/35 = 1\3, ~13/9 = 633.742
 
{{Optimal ET sequence|legend=1| 15, 21, 36, 123, 159, 195, 231 }}
 
[[Tp tuning #T2 tuning|RMS error]]: ???
 
==== 2.3.7.11/5.13 subgroup ====
[[Subgroup]]: 2.3.7.11/5.13
 
[[Comma list]]: 1029/1024, 1575/1573, 2080/2079
 
{{Mapping|legend=2| 3 0 10 5 0 | 0 3 -1 -1 7 }}
 
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~44/35 = 1\3, ~13/9 = 633.918
 
{{Optimal ET sequence|legend=1| 15, 21f, 36, 87, 123, 159 }}
 
[[Tp tuning #T2 tuning|RMS error]]: ???
 
==== 2.3.7.11/5.13.17 subgroup ====
[[Subgroup]]: 2.3.7.11/5.13.17
 
[[Comma list]]: 273/272, 833/832, 1575/1573, 2080/2079
 
{{Mapping|legend=2| 3 0 10 5 0 -2 | 0 3 -1 -1 7 9 }}
 
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~34/27 = 1\3, ~13/9 = 633.820
 
{{Optimal ET sequence|legend=1| 15, 21fg, 36, 123, 159 }}
 
[[Tp tuning #T2 tuning|RMS error]]: ???
 
===== Trisector =====
[[Subgroup]]: 2.3.7.11/5.13.17.19
 
[[Comma list]]: 210/209, 273/272, 286/285, 595/594, 2080/2079
 
{{Mapping|legend=2| 3 0 10 5 0 -2 8 | 0 3 -1 -1 7 9 3 }}
 
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~34/27 = 1\3, ~13/9 = 633.894
 
{{Optimal ET sequence|legend=1| 15, 21fg, 36, 123h, 159h }}
 
[[Tp tuning #T2 tuning|RMS error]]: ???
 
===== 2.3.7.11/5.13.17.19.23 subgroup =====
[[Subgroup]]: 2.3.7.11/5.13.17.19.23
 
[[Comma list]]: 210/209, 231/230, 273/272, 286/285, 595/594, 2080/2079
 
{{Mapping|legend=2| 3 0 10 5 0 -2 8 12 | 0 3 -1 -1 7 9 3 1 }}
 
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~34/27 = 1\3, ~13/9 = 634.038
 
{{Optimal ET sequence|legend=1| 15g, 21fg, 36, 87, 123hi }}
 
[[Tp tuning #T2 tuning|RMS error]]: ???
 
===== 2.3.7.11/5.13.17.19.23.29 subgroup =====
[[Subgroup]]: 2.3.7.11/5.13.17.19.23.29
 
[[Comma list]]: 210/209, 231/230, 273/272, 286/285, 320/319, 595/594, 2080/2079
 
{{Mapping|legend=2| 3 0 10 5 0 -2 8 12 13 | 0 3 -1 -1 7 9 3 1 1 }}
 
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~29/23 = 1\3, ~13/9 = 634.102
 
{{Optimal ET sequence|legend=1| 15g, 21fg, 36, 87, 123hi }}
 
[[Tp tuning #T2 tuning|RMS error]]: ???
 
== 2.….11/7.… subgroups ==
=== Pepperoni ===
=== Pepperoni ===
{{Main| Parapyth }}
{{Main| Parapyth }}
{{See also| Chromatic pairs #Pepperoni }}


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


[[Subgroup]]: 2.3.11/7.13/7
[[Subgroup]]: 2.3.11/7.13/7
Line 949: Line 1,565:


{{Mapping|legend=3| 1 1 0 -8/3 1/3 7/3 | 0 1 0 11/3 -1/3 -10/3 }}
{{Mapping|legend=3| 1 1 0 -8/3 1/3 7/3 | 0 1 0 11/3 -1/3 -10/3 }}
: [[gencom]]: [2 3/2; 352/351 364/363]
: [[gencom]]: [2 3/2; 352/351 364/363]


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


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


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


== 2.….13/5… subgroups ==
== 2.….13/5.… subgroups ==
=== Barbados ===
=== Barbados ===
The [[minimax tuning]] for this makes the generator the cube root of 20/13, or 248.5953 cents. Edos which may be used for it are [[24edo]], [[29edo]], [[53edo]] and [[111edo]], with [[mos scale]]s of size 5, 9, 14, 19, 24 and 29 making for a good variety of scales.
[[Subgroup]]: 2.3.13/5
[[Subgroup]]: 2.3.13/5


Line 976: Line 1,591:
[[Badness]]: 0.002335
[[Badness]]: 0.002335


<nowiki>*</nowiki> wart for 3/2
; Music
* [http://micro.soonlabel.com/gene_ward_smith/Others/Sevish/Sevish%20-%20Desert%20Island%20Rain.mp3 ''Desert Island Rain''] in 313edo tuned Barbados[9], by [https://soundcloud.com/sevish/desert-island-rain Sevish]


=== Oceanfront ===
==== Tobago ====
Related temperaments: [[Archytas clan #Superpyth|superpyth]], [[Archytas clan #Ultrapyth|ultrapyth]]
{{See also| Chromatic pairs #Tobago }}


[[Subgroup]]: 2.3.7.13/5
Tobago, the 10 &amp; 14 temperament in the 2.3.11.13/5 subgroup, extends [[neutral]] and [[barbados]].


[[Comma list]]: 64/63, 91/90
[[Subgroup]]: 2.3.11.13/5


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


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


{{Optimal ET sequence|legend=1| 5, 22, 27, 32, 37 }}
{{Mapping|legend=3| 2 4 -2 0 9 2 | 0 -2 3/2 0 -5 -3/2 }}
: [[gencom]]: [55/39 15/13; 243/242 676/675]


[[Tp tuning #T2 tuning|RMS error]]: 2.063 cents
[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~55/39 = 1\2, ~15/13 = 249.312


Scales: [[Oceanfront scales]]
{{Optimal ET sequence|legend=1| 10, 14, 24, 58, 82, 130 }}


=== Pakkanian hemipyth ===
[[Tp tuning #T2 tuning|RMS error]]: 0.3533 cents


==== Pakkanian hemipyth ====
[[Subgroup]]: 2.3.11.13/5.17  
[[Subgroup]]: 2.3.11.13/5.17  


Line 1,008: Line 1,626:


{{Optimal ET sequence|legend=1| 10, 14, 24, 106, 130, 154, 178*, 202* }}
{{Optimal ET sequence|legend=1| 10, 14, 24, 106, 130, 154, 178*, 202* }}
: <nowiki />* wart for 13/5
=== Oceanfront ===
Related temperaments: [[Archytas clan #Superpyth|superpyth]], [[Archytas clan #Ultrapyth|ultrapyth]]


<nowiki>*</nowiki> wart for 13/5
[[Subgroup]]: 2.3.7.13/5


<!-- name conflict
[[Comma list]]: 64/63, 91/90
== 2.….49/5… subgroups ==
 
=== Breedsmic ===
{{Mapping|legend=2| 1 0 6 -5 | 0 1 -2 4 }}
 
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~3/2 = 713.910
 
{{Optimal ET sequence|legend=1| 5, 22, 27, 32, 37 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 2.063 cents
 
Scales: [[Oceanfront scales]]
 
== 2.….49/5.… subgroups ==
=== Direct breedsmic ===
Related temperament: [[hemithirds]], [[newt]]
Related temperament: [[hemithirds]], [[newt]]


Line 1,024: Line 1,657:
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~49/40 = 350.966
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~49/40 = 350.966


{{Optimal ET sequence|legend=1|?}}
{{Optimal ET sequence|legend=1|7, 10, 17}}


[[Tp tuning #T2 tuning|RMS error]]: ?
[[Tp tuning #T2 tuning|RMS error]]: ?
-->


== 2.….17/5.… subgroups ==
=== Fiventeen ===
Fiventeen tempers out [[136/135]] ({{monzo| 3 -3 1 }}) in 2.3.17/5. It equates [[17/15]] with [[9/8]], so it implies a [[supersoft]] [[pentic]] [[pentad]] of [[~]]30:34:40:45:51. [[17edo]] makes a good tuning especially for its size, which gives a [[supersoft]] pentic scale corresponding approximately to a just [[20/17]] tuning, although [[80edo]] might be preferred for an approximately just [[51/40]] to optimize plausibility slightly more, and [[97edo]] (= 80 + 17) and  [[114edo]] (= 97 + 17) do even better in striking a balance between 80edo's more stable tuning and that having 20/17 more accurate (as in 17edo) is useful because of the more convincing suggestion of the two 15:17:20 chords present in the fiventeen pentad. The same is true of the related rank-3 temperament diatic, for which the [[optimal ET sequence]] is much more characteristic of optimized tunings, finding [[34edo]], then [[80edo]], then [[114edo]] (= 34 + 80) and even [[194edo|194bc-edo]] (= 80 + 114), though because of its focus on primes 5 and 17 it misses 97edo as a tuning, and slightly less optimized though still interesting [[63edo]] and [[143edo]] (= 63 + 80) tunings are found in the optimal ET sequence for fiventeen.
[[Subgroup]]: 2.3.17/5
[[Comma list]]: 136/135 ({{monzo| 3 -3 1 }})
{{Mapping|legend=2| 1 0 -3 | 0 1 3 }}
: mapping generators: ~2, ~3
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.2838{{c}}, ~3/2 = 704.4600{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.5286{{c}}
{{Optimal ET sequence|legend=1| 5, 12, 17, 46, 63, 143 }}
== 2.….19/7.… subgroups ==
=== Surprise ===
This temperament was named by [[User:VectorGraphics|Vector]] in 2025, as he was surprised that the temperament of [[57/56]] did not have a name. This is the [[rank-2 temperament|rank-2]] version of the temperament; Vector surmises that the name ''hendrix'' would be more thoughtfully given to the [[rank-3]] version.
[[Subgroup]]: 2.3.19/7
[[Comma list]]: [[57/56]] ({{Monzo| -3 1 1 }})
{{Mapping|legend=2| 1 0 3 | 0 1 -1 }}
: mapping generators: ~2, ~3
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1202.4345{{c}}, ~3/2 = 697.4314{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 697.3981{{c}}
{{Optimal ET sequence|legend=1| 5, 7, 12, 19, 31*, 50* }}
<nowiki/>* wart for 19/7
[[Badness]] (Sintel): 0.082
=== Supramin ===
This is a remarkable low-complexity microtemperament that contains the 14:17:19 triad within just four generator steps. An excellent tuning is [[25edo]], which provides an accurate yet tone-efficient tuning of this temperament. It was named by [[User:Overthink|Overthink]] in 2026 after the fact that the generator is a [[17/14]] supraminor third, two of which reach [[28/19]].
[[Subgroup]]: 2.17/7.19/7
[[Comma list]]: [[5491/5488]] ({{Monzo| -4 2 1 }})
{{Mapping|legend=2| 1 0 4 | 0 1 -2 }}
: mapping generators: ~2, ~17/7
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1200.022{{c}}, ~17/14 = 335.793{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.000{{c}}, ~17/14 = 335.785{{c}}
{{Optimal ET sequence|legend=1| 7, 18, 25 }}
[[Badness]] (Sintel): 0.005
==== Supramine ====
This extension approximates the 14:17:19:23:25 pentad in just six generator steps, at the cost of some accuracy. 25edo remains a strong tuning.
Subgroup: 2.17/7.19/7.23/7
Comma list: [[323/322]], [[392/391]]
Subgroup-val mapping: {{Mapping| 1 0 4 3 | 0 1 -2 -1 }}
Optimal tunings:
* Subgroup WE: ~2 = 1199.871{{c}}, ~17/14 = 336.243{{c}}
* Subgroup CWE: ~2 = 1200.000{{c}}, ~17/14 = 336.296{{c}}
{{Optimal ET sequence|legend=0| 7, 18, 25 }}
Badness (Sintel): 0.029
==== 2.25/7.17/7.19/7.23/7 subgroup ====
Subgroup: 2.25/7.17/7.19/7.23/7
Comma list: [[323/322]], [[392/391]], [[476/475]]
Subgroup-val mapping: {{Mapping| 1 -2 0 4 3 | 0 3 1 -2 -1 }}
Optimal tunings:
* Subgroup WE: ~2 = 1199.757{{c}}, ~17/14 = 335.428{{c}}
* Subgroup CWE: ~2 = 1200.000{{c}}, ~17/14 = 335.479{{c}}
{{Optimal ET sequence|legend=0| 7, 18, 25 }}
Badness (Sintel): 0.053


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


Line 1,052: Line 1,772:


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


Line 1,058: Line 1,777:


Supporting ETs: *5, *6, *7[+5/2, +7/2], *9[-5/2, --7/2], *11, *16, *17[+5/2], *23[+5/2, +7/2], *21[-7/2], *27, *28[+5/2], *38, *43[-7/2], *49
Supporting ETs: *5, *6, *7[+5/2, +7/2], *9[-5/2, --7/2], *11, *16, *17[+5/2], *23[+5/2, +7/2], *21[-7/2], *27, *28[+5/2], *38, *43[-7/2], *49
 
: <nowiki />* wart for 3/2
<nowiki>*</nowiki> wart for 3/2


==== 3/2.5/2.7/2.11/2 ====
==== 3/2.5/2.7/2.11/2 ====
Line 1,067: Line 1,785:


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


Line 1,073: Line 1,790:


[[Support]]ing [[ET]]s: *11, *5, *16, *6, *27[-11/2], *21[-7/2], *38[-11/2], *43[-7/2, -11/2], *59[-7/2, -11/2], *70[-7/2, -11/2], *75[--7/2, -11/2]
[[Support]]ing [[ET]]s: *11, *5, *16, *6, *27[-11/2], *21[-7/2], *38[-11/2], *43[-7/2, -11/2], *59[-7/2, -11/2], *70[-7/2, -11/2], *75[--7/2, -11/2]
 
: <nowiki />* wart for 3/2
<nowiki>*</nowiki> wart for 3/2


==== 3/2.5/2.7/2.11/2.13/2 ====
==== 3/2.5/2.7/2.11/2.13/2 ====
Line 1,086: Line 1,802:


[[Support]]ing [[ET]]s: *11, *5, *16, *6, *27[-11/2]
[[Support]]ing [[ET]]s: *11, *5, *16, *6, *27[-11/2]
 
: <nowiki />* wart for 3/2
<nowiki>*</nowiki> wart for 3/2


=== Semiwolf ===
=== Semiwolf ===
Line 1,124: Line 1,839:
[[Optimal ET sequence]]: [[8edf]], [[11edf]]
[[Optimal ET sequence]]: [[8edf]], [[11edf]]


<!-- name conflict
== 3/2.5/4.… subgroups ==
== 3/2.5/4… subgroups ==
=== Poseidon ===
=== Poseidon ===
'''This temperament will be subjected to renaming due to a conflict.'''
[[Subgroup]]: 3/2.5/4.11/8
[[Subgroup]]: 3/2.5/4.11/8


Line 1,135: Line 1,851:
: [[gencom]]: [3/2 12/11; 121/120]
: [[gencom]]: [3/2 12/11; 121/120]


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


{{Optimal ET sequence|legend=1|9, 5, 13, 22, 14, 31, 17, 6[+5/4], 23, 40, 35, 21[-5/4], 19[+5/4], 49}}
{{Optimal ET sequence|legend=1|9, 5, 13, 22, 14, 31, 17, 6[+5/4], 23, 40, 35, 21[-5/4], 19[+5/4], 49}}
-->


== Other 3/2-equave subgroups ==
== Other 3/2-equave subgroups ==
Line 1,153: Line 1,868:


Supporting ETs: *5, *6[+13], *7[-7, -13], *9, *11[+13], *13, *14, *17[-7, -13], *19[+13], *21[-7, -13], *22[-7], *23[+13], *25[-7, -13], *31[-7]
Supporting ETs: *5, *6[+13], *7[-7, -13], *9, *11[+13], *13, *14, *17[-7, -13], *19[+13], *21[-7, -13], *22[-7], *23[+13], *25[-7, -13], *31[-7]
 
: <nowiki />* wart for 3/2
<nowiki>*</nowiki> wart for 3/2


=== Doubleton ===
=== Doubleton ===
Line 1,168: Line 1,882:


Supporting ETs: *6, *10, *16, *14[-13], *8[+7], *22, *18[-13], *26, *24[-13], *28[+7], *20[+7], *36[-13], *12[+7, +13], *34[-13]
Supporting ETs: *6, *10, *16, *14[-13], *8[+7], *22, *18[-13], *26, *24[-13], *28[+7], *20[+7], *36[-13], *12[+7, +13], *34[-13]
 
: <nowiki />* wart for 3/2
<nowiki>*</nowiki> wart for 3/2


== 5/2-equave subgroups ==
== 5/2-equave subgroups ==
=== Hyperion ===
=== Hyperion ===
[[Subgroup]]: 5/2.7.11
[[Subgroup]]: 5/2.7.11
Line 1,185: Line 1,897:


Supporting ETs: *5[-7], *8, *19[+7], *21[-7], *27[+7], *29[-7], *35[+7], *43[+7], *37[-7], *51[+7, +11], *45[-7], *59[+7, +11]
Supporting ETs: *5[-7], *8, *19[+7], *21[-7], *27[+7], *29[-7], *35[+7], *43[+7], *37[-7], *51[+7, +11], *45[-7], *59[+7, +11]
 
: <nowiki />* wart for 5/2
<nowiki>*</nowiki> wart for 5/2


= Related temperament collections =
= Related temperament collections =
Line 1,193: Line 1,904:
* [[Substitute harmonic]] temperaments
* [[Substitute harmonic]] temperaments


<!-- main article -->
[[Category:Subgroup temperaments| ]] <!-- main article -->
 
[[Category:Temperament collections]]
[[Category:Temperament collections]][[Category:Subgroup]]
{{Todo| review | cleanup }}
Retrieved from "https://en.xen.wiki/w/Subgroup_temperaments"