Xenharmonic Wiki

Subgroup temperaments: Difference between revisions

← Older edit
VisualWikitext
BudjarnLambeth (talk | contribs)
autopatrolled, emailconfirmed
21,263 edits
m →Oceanfront: Add link to "oceanfront scales"
Hkm (talk | contribs)
autopatrolled
989 edits
 
(97 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]]
* [[No-fives subgroup temperaments]]
* [[No-fives subgroup temperaments]]
* [[No-threes subgroup temperaments]]
* [[No-threes subgroup temperaments]]
* For no-twos, see [[Catalog of 3.5.7 subgroup rank two temperaments]] and [[No-twos subgroup temperaments]].
* [[No-twos subgroup temperaments]] (additionally, [[Catalog of 3.5.7 subgroup rank two temperaments]]).


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}}
 
Baldy is every other step of [[garibaldi]], without the mapping of prime 11. It can be described as the 6 & 35 temperament.
 
[[Subgroup]]: 2.9.5.7.13
 
[[Comma list]]: [[225/224]], [[325/324]], [[640/637]]
 
{{Mapping|legend=2| 1 0 15 25 -28 | 0 1 -4 -7 10 }}


Comma list: 225/224, 325/324, 640/637
{{Mapping|legend=3| 1 3/2 3 4 0 2 | 0 1/2 -4 -7 0 10 }}


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


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


{{Optimal ET sequence|legend=1| 6, 29f, 35, 41, 47 }}
{{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]
{{Mapping|legend=2| 1 1 1 -1 3| 0 6 5 13 3 }}
 
[[Optimal tuning]] ([[CTE]]): ~77/72 = 115.5685


{{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 484: Line 538:


== 4.6.5 subgroup ==
== 4.6.5 subgroup ==
=== Meanquadi ===
=== Meanquad ===
{{Main| Meanquad }}
{{Main| Meanquad }}


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 ===
The 23-limit version of Fourwar was created first, as an attempt to approximate subgroup 4.6.5.7.11.13.17.19.23 as accurately as possible using 25 to 35 notes per equave. Then the lower limit versions were created by simply extrapolating the temperament downwards.
 
Fourwar is named after the closely related [[hemiwar]] temperament.
 
{{Todo|inline=1|cleanup}}
 
<pre>
Reduced Mapping
4 6 5
[ ⟨ 1 0 1 ]
⟨ 0 16 2 ] ⟩
TE Generator Tunings (cents)
⟨2399.3973, 193.8643]
TE Step Tunings (cents)
⟨25.21211, 47.81337]
TE Tuning Map (cents)
⟨2399.397, 3101.829, 2787.126]
TE Mistunings (cents)
⟨-0.603, -0.126, 0.812]
Complexity 1.369085
Adjusted Error 0.692892 cents
TE Error 0.268047 cents/octave
Unison Vector
[8, 1, -8⟩ (393216:390625)
 
Subsets
q99, q62, q37, q161, q136, q198, q25, q124, q74, q235
</pre>
 
==== 4.6.5.7 ====
<pre>
Reduced Mapping
4 6 5 7
[ ⟨ 1 0 1 1 ]
⟨ 0 16 2 5 ] ⟩
TE Generator Tunings (cents)
⟨2399.4195, 193.8654]
TE Step Tunings (cents)
⟨25.23883, 47.79592]
TE Tuning Map (cents)
⟨2399.420, 3101.846, 2787.150, 3368.747]
TE Mistunings (cents)
⟨-0.580, -0.109, 0.837, -0.079]
Complexity 1.192044
Adjusted Error 0.653313 cents
TE Error 0.232715 cents/octave
Unison Vectors
[-2, -1, -2, 4⟩ (2401:2400)
[3, 0, -5, 2⟩ (3136:3125)
[5, 1, -3, -2⟩ (6144:6125)
[8, 1, -8, 0⟩ (393216:390625)
 
Subsets
q99, q62, q37, q161, q136, q198, q25, q124, q74, q235
</pre>
 
==== 4.6.5.7.11 ====
<pre>
Reduced Mapping
4 6 5 7 11
[ ⟨ 1 0 1 1 1 ]
⟨ 0 16 2 5 9 ] ⟩
TE Generator Tunings (cents)
⟨2400.1097, 193.9498]
TE Step Tunings (cents)
⟨24.18752, 48.52491]
TE Tuning Map (cents)
⟨2400.110, 3103.196, 2788.009, 3369.859, 4145.658]
TE Mistunings (cents)
⟨0.110, 1.241, 1.696, 1.033, -5.660]
Complexity 1.068792
Adjusted Error 2.926965 cents
TE Error 0.846083 cents/octave
Unison Vectors
[-1, -1, -1, 0, 2⟩ (121:120)
[2, 0, -2, -1, 1⟩ (176:175)
[-3, -1, 1, 1, 1⟩ (385:384)
[-1, 0, 3, -3, 1⟩ (1375:1372)
[-2, -1, -2, 4, 0⟩ (2401:2400)
[1, 0, 1, -4, 2⟩ (2420:2401)
 
Subsets
q37, q25, q62, q12, q74, q99, q87, q49r, q50r, q124
</pre>
 
==== 4.6.5.7.11.13 ====
 
<pre>
Reduced Mapping
4 6 5 7 11 13
[ ⟨ 1 0 1 1 1 0 ]
⟨ 0 16 2 5 9 23 ] ⟩
TE Generator Tunings (cents)
⟨2401.2305, 193.5378]
TE Step Tunings (cents)
⟨42.79107, 35.98524]
TE Tuning Map (cents)
⟨2401.230, 3096.606, 2788.306, 3368.920, 4143.071, 4451.371]
TE Mistunings (cents)
⟨1.230, -5.349, 1.992, 0.094, -8.247, 10.843]
Complexity 1.219191
Adjusted Error 6.699599 cents
TE Error 1.810487 cents/octave
Unison Vectors
[0, 1, -1, 0, 1, -1⟩ (66:65)
[-1, -1, -1, 0, 2, 0⟩ (121:120)
[1, 2, 0, 0, -1, -1⟩ (144:143)
[2, 0, -2, -1, 1, 0⟩ (176:175)
[-2, 1, 1, 1, 0, -1⟩ (105:104)
[-3, -1, 1, 1, 1, 0⟩ (385:384)
[-3, 0, 0, 1, 2, -1⟩ (847:832)
[1, 3, -1, 0, 0, -2⟩ (864:845)
[-1, 0, 3, -3, 1, 0⟩ (1375:1372)
 
Subsets
q25, q37f, q12f, q62, q50rf, q13rff, q49rff, q87, q74ff, q24rfff
</pre>
 
==== 4.6.5.7.11.13.17 ====
<pre>
Reduced Mapping
4 6 5 7 11 13 17
[ ⟨ 1 0 1 1 1 0 1 ]
⟨ 0 16 2 5 9 23 13 ] ⟩
TE Generator Tunings (cents)
⟨2400.4701, 193.4599]
TE Step Tunings (cents)
⟨43.39350, 35.55764]
TE Tuning Map (cents)
⟨2400.470, 3095.359, 2787.390, 3367.770, 4141.609, 4449.578, 4915.449]
TE Mistunings (cents)
⟨0.470, -6.596, 1.076, -1.056, -9.709, 9.050, 10.494]
Complexity 1.129881
Adjusted Error 8.082725 cents
TE Error 1.977443 cents/octave
Unison Vectors
[0, 1, -1, 0, 1, -1, 0⟩ (66:65)
[1, 1, 1, -1, 0, 0, -1⟩ (120:119)
[1, 2, 0, 0, -1, -1, 0⟩ (144:143)
[-2, 1, 1, 1, 0, -1, 0⟩ (105:104)
[-1, 2, 2, 0, 0, -1, -1⟩ (225:221)
[-1, 1, 2, -2, 0, -1, 1⟩ (1275:1274)
 
Subsets
q25, q12f, q37f, q13rffg, q50rf, q62, q49rffg, q24rfffg, q38rreffg, q74ffg
</pre>
 
==== 4.6.5.7.11.13.17.19 ====
<pre>
Reduced Mapping
4 6 5 7 11 13 17 19
[ ⟨ 1 0 1 1 1 0 1 1 ]
⟨ 0 16 2 5 9 23 13 14 ] ⟩
TE Generator Tunings (cents)
⟨2399.9219, 193.3952]
TE Step Tunings (cents)
⟨44.14256, 35.03670]
TE Tuning Map (cents)
⟨2399.922, 3094.324, 2786.712, 3366.898, 4140.479, 4448.090, 4914.060, 5107.455]
TE Mistunings (cents)
⟨-0.078, -7.631, 0.399, -1.928, -10.839, 7.562, 9.104, 9.942]
Complexity 1.058472
Adjusted Error 8.712222 cents
TE Error 2.050935 cents/octave
Unison Vectors
[0, 1, -1, 0, 1, -1, 0, 0⟩ (66:65)
[-1, 0, 0, 1, 1, 0, 0, -1⟩ (77:76)
[2, 1, -1, 0, 0, 0, 0, -1⟩ (96:95)
[1, 1, 1, -1, 0, 0, -1, 0⟩ (120:119)
[0, 1, 1, 1, -1, 0, 0, -1⟩ (210:209)
[0, 0, 1, -2, 1, 0, 1, -1⟩ (935:931)
[2, 0, -3, 1, 0, 0, -1, 1⟩ (2128:2125)
 
Subsets
q25, q12fh, q37f, q13rffgh, q50rf, q62, q49rffgh, q24rfffghh, q38rreffgh, q74ffgh
</pre>
 
==== 4.6.5.7.11.13.17.19.23 ====
<pre>
Reduced Mapping
4 6 5 7 11 13 17 19 23
[ ⟨ 1 0 1 1 1 0 1 1 0 ]
⟨ 0 16 2 5 9 23 13 14 28 ] ⟩
TE Generator Tunings (cents)
⟨2399.3286, 193.5316]
TE Step Tunings (cents)
⟨37.31613, 39.63311]
TE Tuning Map (cents)
⟨2399.329, 3096.506, 2786.392, 3366.987, 4141.113, 4451.227, 4915.240, 5108.771, 5418.885]
TE Mistunings (cents)
⟨-0.671, -5.449, 0.078, -1.839, -10.205, 10.699, 10.284, 11.258, -9.389]
Complexity 1.115920
Adjusted Error 9.502017 cents
TE Error 2.100561 cents/octave
Unison Vectors
[0, 1, -1, 0, 1, -1, 0, 0, 0⟩ (66:65)
[1, 0, 0, -1, 0, -1, 0, 0, 1⟩ (92:91)
[0, -1, 1, 0, 0, 0, 0, -1, 1⟩ (115:114)
[1, 1, 1, -1, 0, 0, -1, 0, 0⟩ (120:119)
[2, 0, -2, -1, 1, 0, 0, 0, 0⟩ (176:175)
[-3, -1, 1, 1, 1, 0, 0, 0, 0⟩ (385:384)
[1, 0, -2, 1, 0, 0, 1, -1, 0⟩ (476:475)
[1, 0, 0, -2, 1, 0, -1, 1, 0⟩ (836:833)
[0, 0, 1, -2, 1, 0, 1, -1, 0⟩ (935:931)
[1, -1, 0, 0, 0, 0, -2, 1, 1⟩ (874:867)
 
Subsets
q25i, q12fhi, q37f, q13rffghii, q62, q50rfii, q49rffghii, q24rfffghhiii, q74ffghi, q38rreffghiii
</pre>


== 4.9.25 subgroup ==
== 4.9.25 subgroup ==
Line 538: 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 556: 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 563: Line 870:


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


== 3/2.5/2… subgroup ==
[[Tp tuning #T2 tuning|RMS error]]: 0.8790 cents
{{Main|Half-prime subgroup}}
 
=== Kryptonite ===
{{See also| Chromatic pairs #Kryptonite }}
 
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 }})


=== Hemihemi ===
{{Mapping|legend=2| 1 1 1 0 | 0 -2 3 4 }}
[[Subgroup]]: 3/2.5/2.7/2
: mapping generators: ~2, ~21/20


[[Comma list]]: [[10976/10935]]
{{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]


{{Mapping|legend=2| 1 2 3 | 0 3 1  }}
[[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 tuning]] (subgroup [[CTE]]): ~[[3/2]] = 1\[[1edf]], ~[[28/27]] = 60.909
{{Optimal ET sequence|legend=1| 13, 14, 27, 41 }}


[[Support]]ing [[ET]]s: *23, *12, *11, *35, *34, *10, *13, *47, *9[+5/2], *14[-5/2], *45, *25, *21[+5/2], *8[+5/2]
[[Tp tuning #T2 tuning|RMS error]]: 1.245 cents


=== Halftone ===
=== Mothwelltri ===
{{Main| Halftone }}
{{See also| Chromatic pairs #Mothwelltri }}


[[Subgroup]]: 3/2.5/2.7/2
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.  


[[Comma list]]: 9604/9375
[[Subgroup]]: 2.7/3.11


{{Mapping|legend=2| 1 3 4 | 0 -4 -5 }}
[[Comma list]]: [[99/98]] ({{monzo| -1 -2 1 }})


: sval mapping generators: ~3/2, ~15/14
{{Mapping|legend=2| 1 0 1 | 0 1 2 }}
: mapping generators: ~2, ~7/3


[[Optimal tuning]] (subgroup [[CTE]]): ~3/2 = 1\1edf, ~15/14 = 128.783
{{Mapping|legend=3| 1 -1/2 0 1/2 3 | 0 -1/2 0 1/2 2 }}
: [[gencom]]: [2 7/6; 99/98]


Supporting ETs: *5, *6, *7[+5/2, +7/2], *9[-5/2, --7/2], *11, *16, *17[+5/2], *23[+5/2, +7/2], *21[-7/2], *27, *28[+5/2], *38, *43[-7/2], *49
[[Optimal tuning]]s:  
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~7/6 = 273.695
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~7/6 = 273.174


<nowiki>*</nowiki> wart for 3/2
{{Optimal ET sequence|legend=1| 4, 9, 13, 22, 79 }}


==== 3/2.5/2.7/2.11/2 ====
[[Tp tuning #T2 tuning|RMS error]]: 1.064 cents
[[Subgroup]]: 3/2.5/2.7/2.11/2


[[Comma list]]: 1232/1215, 27783/27500
== 2.….9/7.… subgroups ==
=== Marveltri ===
{{See also| Chromatic pairs #Marveltri }}


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


: sval mapping generators: ~3/2, ~15/14
[[Subgroup]]: 2.5.9/7


[[Optimal tuning]] (subgroup [[CTE]]): ~3/2 = 1\1edf, ~15/14 = 129.186
[[Comma list]]: 225/224 ({{monzo| -5 2 1 }})


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


<nowiki>*</nowiki> wart for 3/2
{{Mapping|legend=3| 1 2 0 -1 | 0 -4/5 1 2/5 }}
: [[gencom]]: [2 5; 225/224]


==== 3/2.5/2.7/2.11/2.13/2 ====
[[Optimal tuning]]s:
[[Subgroup]]: 3/2.5/2.7/2.11/2.13/2
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/4 = 384.208
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/4 = 383.638


[[Comma list]]: 275/273, 1232/1215, 1323/1300
{{Optimal ET sequence|legend=1| 3, 13, 16, 19, 22, 25, 72, 97, 122, 269c* }}
: <nowiki/>* wart for 9/7


{{Mapping|legend=2| 1 3 4 4 5 | 0 -4 -5 1 -2 }}
[[Tp tuning #T2 tuning|RMS error]]: 0.4801 cents


[[Optimal tuning]] (subgroup [[CTE]]): ~3/2 = 1\1edf, ~15/14 = 129.381
==== Sulis ====
Sulis is related to [[minerva]] and [[würschmidt]].  


[[Support]]ing [[ET]]s: *11, *5, *16, *6, *27[-11/2]
[[Subgroup]]: 2.5.9/7.11/9


<nowiki>*</nowiki> wart for 3/2
[[Comma list]]: 99/98 ({{monzo| -1 0 2 1 }}), 176/175 ({{monzo| 4 -2 1 1 }})


=== Semiwolf ===
{{Mapping|legend=2| 1 0 5 -9 | 0 1 -2 4 }}]
[[Subgroup]]: 3/2.5/2.7/4


[[Comma list]]: 245/243
[[Optimal tuning]]s:  
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/4 = 386.617
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/4 = 386.558


{{Mapping|legend=2| 1 1 2 | 0 2 -1 }}
{{Optimal ET sequence|legend=1| 3, …, 22, 25, 28, 31, 59 }}


: sval mapping generators: ~3/2, ~9/7
[[Tp tuning #T2 tuning|RMS error]]: 1.074 cents


[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~7/6 = 262.1728
== 2.….7/5.… subgroups ==
=== 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.


[[Optimal ET sequence]]: [[3edf]], [[5edf]], [[8edf]]
[[Subgroup]]: 2.3.7/5


==== Semilupine ====
[[Comma list]]: [[50/49]]
[[Subgroup]]: 3/2.5/2.7/4.11/4


[[Comma list]]: 100/99, 245/243
{{Mapping|legend=2| 2 3 1 | 0 1 0 }}


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


[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~7/6 = 264.3771
[[Support]]ing [[ET]]s: {{EDOs|4, 6, 8, 10, 18, 28, 46, 64, 110}}


[[Optimal ET sequence]]: [[8edf]], [[13edf]]
=== Argentic ===
Argentic is the 2.3.7/5 subgroup temperament tempering out [[5120/5103]].


==== Hemilycan ====
[[Subgroup]]: 2.3.7/5
[[Subgroup]]: 3/2.5/2.7/4.11/4


[[Comma list]]: 245/243, 441/440
[[Comma list]]: [[5120/5103]] = {{monzo| 10 -6 -1 }}


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


[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~7/6 = 261.5939
[[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]]: [[8edf]], [[11edf]]
{{Optimal ET sequence|legend=1| 12, 29, 41, 70, 321, 391, 461, 531, 601 }}
<small> based on subgroup TE </small>


<!-- name conflict
Badness (Sintel): 0.119
== 3/2.5/4 subgroup ==
=== Poseidon ===
[[Subgroup]]: 3/2.5/4.11/8


[[Comma list]]: 121/120
==== Edson (2.3.7/5.11/5.13/5 subgroup) ====
{{See also| Chromatic pairs #Edson }}


{{Mapping|legend=2| 1 1 1 | 0 2 -1 }}]
Edson is related to [[pele]] and [[andromeda]].


: [[gencom]]: [3/2 12/11; 121/120]
[[Subgroup]]: 2.3.7/5.11/5.13/5


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


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


== 5/3.7/3… subgroup ==
{{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]


=== Greeley ===
[[Optimal tuning]]s:
Related temperaments: [[Porcupine family #Opossum|Opossum]], [[Starling temperaments #Nusecond|Nusecond]]
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1\1, ~3/2 = 703.4398
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1\1, ~3/2 = 703.414


[[Subgroup]]: 2.5/3.7/3.11/3
{{Optimal ET sequence|legend=1| 12, 17, 29 }}


[[Comma list]]: 121/120, 126/125
[[Tp tuning #T2 tuning|RMS error]]: 0.5102 cents


{{Mapping|legend=2| 1 1 2 2 | 0 -2 -6 -1 }}
==== Haumea ====
{{See also| Chromatic pairs #Haumea }}


{{Mapping|legend=3| 1 -5/4 -1/4 3/4 3/4 | 0 9/4 1/4 -15/4 5/4 }}
Related temperaments include [[#Bridgetown|bridgetown]], [[namaka]], [[hemigari]], [[#Barbados|barbados]], and [[parizekmic]].


: [[gencom]]: [2 11/10; 121/120 126/125]
[[Subgroup]]: 2.3.7/5.11/5.13/5


[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~11/10 = 155.776
[[Comma list]]: [[352/351]], [[676/675]], [[847/845]]


{{Optimal ET sequence|legend=1| 8, 15, 23, 54, 77, 100, 131†, 208*† }}
{{Mapping|legend=2| 1 0 10 -6 -1 | 0 2 -12 9 3 }}


<nowiki>*</nowiki> wart for 5/3
{{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]


<nowiki>†</nowiki> wart for 11/3
[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~15/13 = 248.491


[[Tp tuning#T2 tuning|RMS error]]: 1.034 cents
{{Optimal ET sequence|legend=1| 24, 29, 111, 140, 169, 198, 565d, 763bd, 961bd }}


== 7/5.11/5… subgroup ==
[[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 715: Line 1,353:
[[Tp tuning #T2 tuning|RMS error]]: 0.2562 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.2562 cents


== 11/7.13/7… subgroup ==
=== Terrain ===
{{Redirect|Terrain|the scale|Terrain (scale)}}
{{See also| Chromatic pairs #Terrain }}


=== Pepperoni ===
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.
{{Main| Parapyth }}


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


[[Subgroup]]: 2.3.11/7.13/7
[[Comma list]]: [[250047/250000]]


[[Comma list]]: 352/351, 364/363
{{Mapping|legend=2| 3 1 3 | 0 1 -1 }}


{{Mapping|legend=2| 1 0 7 12 | 0 1 -4 -7 }}
{{Mapping|legend=3| 3 10/9 -7/9 2/9 | 0 -2/3 -1/3 2/3 }}
: [[gencom]]: [63/50 10/9; 250047/250000]


{{Mapping|legend=3| 1 1 0 -8/3 1/3 7/3 | 0 1 0 11/3 -1/3 -10/3 }}
[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~63/50 = 1\3, ~10/9 = 182.461


: [[gencom]]: [2 3/2; 352/351 364/363]
{{Optimal ET sequence|legend=1| 6, 21, 27, 33, 105, 138, 171, 1848, 2019, 2190, 2361, 2532, 2703, 2874, 3045, 3216, 3387, 3558 }}


[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~3/2 = 703.856
[[Tp tuning #T2 tuning|RMS error]]: 0.00844 cents


{{Optimal ET sequence|legend=1| 5, 7, 12, 17, 29, 46, 58, 75, 80, 87, 104, 121, 167, 196, 208, 271, 595b*† }}
=== Tridec ===
{{See also| Chromatic pairs #Tridec }}
{{See also| Non-over-1 temperament #Tridec }}


<nowiki>*</nowiki> wart for 11/7
Tridec, the 5 &amp; 8 temperament in the 2.7/5.11/5.13/5 subgroup, extends [[#Petrtri]].


<nowiki>†</nowiki> wart for 13/7
[[Subgroup]]: 2.7/5.11/5.13/5


[[Tp tuning #T2 tuning|RMS error]]: 0.3789 cents
[[Comma list]]: [[847/845]], [[1001/1000]]


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


=== Auk ===
{{Mapping|legend=3| 1 0 -3/4 5/4 -3/4 1/4 | 0 0 0 -4 3 1 }}
[[Subgroup]]: 3/2.7.13
: [[gencom]]: [2 13/10; 847/845 1001/1000]


[[Comma list]]: 87808/85293
[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~13/10 = 454.556


{{Mapping|legend=2| 1 0 -8 | 0 1 3 }}
{{Optimal ET sequence|legend=1| 5, 8, 21, 29, 37, 66, 169, 235, 404c, 639c, 953bc }}


: sval mapping generators: ~3/2, ~7
[[Tp tuning #T2 tuning|RMS error]]: 0.1613 cents


[[Optimal tuning]] (subgroup [[CTE]]): ~3/2 = 1\1edf, ~28/9 = 1950.859
==== Naiadec ====
[[Subgroup]]: 2.7/5.11/5.13/5.17/5


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


<nowiki>*</nowiki> wart for 3/2
{{Mapping|legend=2| 1 2 0 1 1 | 0 -4 3 1 2 }}


=== Doubleton ===
{{Mapping|legend=3| 1 0 -3/4 5/4 -3/4 1/4 1/4 | 0 0 0 -4 3 1 2 }}
[[Subgroup]]: 3/2.7.13
: [[gencom]]: [2 13/10; 170/169 221/220 847/845]


[[Comma list]]: 1352/1323
[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~13/10 = 454.882


{{Mapping|legend=2| 2 0 3 | 0 1 1 }}
{{Optimal ET sequence|legend=1| 5, 8, 21, 29, 95<sup>t</sup>, 124<sup>t</sup> }}
: <sup>t</sup> wart for 17/5


: sval mapping generators: ~26/21, ~7
[[Tp tuning #T2 tuning|RMS error]]: 0.7521 cents


[[Optimal tuning]] (subgroup [[CTE]]): ~26/21 = 1\2edf, ~28/9 = 1971.772
== 2.….11/5.… subgroups ==
=== Petrtri ===
{{See also| Chromatic pairs #Petrtri }}
{{See also| 5L 3s/Temperaments #Petrtri }}


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]
Petrtri can be described as 3 &amp; 5 temperament in the 2.11/5.13/5 subgroup.


<nowiki>*</nowiki> wart for 3/2
[[Subgroup]]: 2.11/5.13/5


== Other 5/2 subgroups ==
[[Comma list]]: [[2200/2197]]


=== Hyperion ===
{{Mapping|legend=2| 1 0 1| 0 3 1 }}
[[Subgroup]]: 5/2.7.11


[[Comma list]]: {{monzo| 11 1 -5 }}
{{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]


{{Mapping|legend=2| 1 4 3 | 0 -5 -1 }}
[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~13/10 = 455.012


: [[gencom]]: [5/2 125/88; 341796875/329832448]
{{Optimal ET sequence|legend=1| 21, 29, 153, 182, 211, 240, 269, 298, 327, 356, 385, 509, 741c, 1126c }}


[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~5/2 = 1586.3137, ~125/88 = 593.6668
[[Tp tuning #T2 tuning|RMS error]]: 0.0749 cents


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


<nowiki>*</nowiki> wart for 5/2
Bridgetown, the 5 &amp; 24 temperament in the 2.3.11/5.13/5 subgroup, is related to [[#Haumea|haumea]] and [[#Barbados|barbados]].


== Other 7/5 subgroups ==
[[Subgroup]]: 2.3.11/5.13/5


=== Hydrothermal ===
[[Comma list]]: [[352/351]], [[676/675]]
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=2| 1 0 -6 -1 | 0 2 9 3 }}


[[Comma list]]: [[50/49]]
{{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]


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


[[Optimal tuning]] (inharmonic [[TE]]): ~1\2 = 590.998, ~[[10/7]]-1\2 = 128.962
{{Optimal ET sequence|legend=1| 5, 9, 14, 19, 24, 29, 169, 198, 227, 256, 285, 314 }}


[[Support]]ing [[ET]]s: {{EDOs|4, 6, 8, 10, 18, 28, 46, 64, 110}}
[[Tp tuning #T2 tuning|RMS error]]: 0.2513 cents
 
== Other 11/5 subgroups ==


=== 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 823: Line 1,469:
[[Tp tuning #T2 tuning|RMS error]]: 0.5379 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.5379 cents


== Other 13/5 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 ===
{{Main| Parapyth }}
{{See also| Chromatic pairs #Pepperoni }}
 
Pepperoni is generated by a fifth and can be described as the 5 &amp; 12 temperament in the 2.3.11/7.13/7 subgroup. It is the single-chain retraction of [[parapyth]]. The [[Peppermint-24|Pepper fifth]], which is (40200 + 600 sqrt(5))/59 = 704.096 cents, is a good pepperoni generator, hence the name.
 
[[Subgroup]]: 2.3.11/7.13/7
 
[[Comma list]]: 352/351, 364/363
 
{{Mapping|legend=2| 1 0 7 12 | 0 1 -4 -7 }}
 
{{Mapping|legend=3| 1 1 0 -8/3 1/3 7/3 | 0 1 0 11/3 -1/3 -10/3 }}
: [[gencom]]: [2 3/2; 352/351 364/363]
 
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~3/2 = 703.856
 
{{Optimal ET sequence|legend=1| 5, 7, 12f, 17, 29, 46, 58, 75, 80, 87, 104, 121, 167, 196, 208, 271, 595b*<sup>†</sup> }}
: <nowiki />* wart for 11/7
: <sup>†</sup> wart for 13/7
 
[[Tp tuning #T2 tuning|RMS error]]: 0.3789 cents
 
== 2.….13/5.… subgroups ==
=== Barbados ===
=== 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 837: 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 869: 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
== Other 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 885: 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]]: ?
-->


== Other 9/7 subgroups ==
== 2.….17/5.… subgroups ==
=== Fiventeen ===
Fiventeen tempers out [[136/135]] ({{monzo| 3 -3 1 }}) in 2.3.17/5. It equates [[17/15]] with [[9/8]], so it implies a [[supersoft]] [[pentic]] [[pentad]] of [[~]]30:34:40:45:51. [[17edo]] makes a good tuning especially for its size, which gives a [[supersoft]] pentic scale corresponding approximately to a just [[20/17]] tuning, although [[80edo]] might be preferred for an approximately just [[51/40]] to optimize plausibility slightly more, and [[97edo]] (= 80 + 17) and  [[114edo]] (= 97 + 17) do even better in striking a balance between 80edo's more stable tuning and that having 20/17 more accurate (as in 17edo) is useful because of the more convincing suggestion of the two 15:17:20 chords present in the fiventeen pentad. The same is true of the related rank-3 temperament diatic, for which the [[optimal ET sequence]] is much more characteristic of optimized tunings, finding [[34edo]], then [[80edo]], then [[114edo]] (= 34 + 80) and even [[194edo|194bc-edo]] (= 80 + 114), though because of its focus on primes 5 and 17 it misses 97edo as a tuning, and slightly less optimized though still interesting [[63edo]] and [[143edo]] (= 63 + 80) tunings are found in the optimal ET sequence for fiventeen.
 
[[Subgroup]]: 2.3.17/5
 
[[Comma list]]: 136/135 ({{monzo| 3 -3 1 }})
 
{{Mapping|legend=2| 1 0 -3 | 0 1 3 }}
: mapping generators: ~2, ~3
 
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.2838{{c}}, ~3/2 = 704.4600{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.5286{{c}}
 
{{Optimal ET sequence|legend=1| 5, 12, 17, 46, 63, 143 }}
 
== 2.….19/7.… subgroups ==
=== Surprise ===
This temperament was named by [[User:VectorGraphics|Vector]] in 2025, as he was surprised that the temperament of [[57/56]] did not have a name. This is the [[rank-2 temperament|rank-2]] version of the temperament; Vector surmises that the name ''hendrix'' would be more thoughtfully given to the [[rank-3]] version.
 
[[Subgroup]]: 2.3.19/7
 
[[Comma list]]: [[57/56]] ({{Monzo| -3 1 1 }})
 
{{Mapping|legend=2| 1 0 3 | 0 1 -1 }}
: mapping generators: ~2, ~3
 
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1202.4345{{c}}, ~3/2 = 697.4314{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 697.3981{{c}}
 
{{Optimal ET sequence|legend=1| 5, 7, 12, 19, 31*, 50* }}
 
<nowiki/>* wart for 19/7
 
[[Badness]] (Sintel): 0.082
 
=== Supramin ===
This is a remarkable low-complexity microtemperament that contains the 14:17:19 triad within just four generator steps. An excellent tuning is [[25edo]], which provides an accurate yet tone-efficient tuning of this temperament. It was named by [[User:Overthink|Overthink]] in 2026 after the fact that the generator is a [[17/14]] supraminor third, two of which reach [[28/19]].
 
[[Subgroup]]: 2.17/7.19/7
 
[[Comma list]]: [[5491/5488]] ({{Monzo| -4 2 1 }})
 
{{Mapping|legend=2| 1 0 4 | 0 1 -2 }}
: mapping generators: ~2, ~17/7
 
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1200.022{{c}}, ~17/14 = 335.793{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.000{{c}}, ~17/14 = 335.785{{c}}
 
{{Optimal ET sequence|legend=1| 7, 18, 25 }}
 
[[Badness]] (Sintel): 0.005
 
==== Supramine ====
This extension approximates the 14:17:19:23:25 pentad in just six generator steps, at the cost of some accuracy. 25edo remains a strong tuning.
 
Subgroup: 2.17/7.19/7.23/7
 
Comma list: [[323/322]], [[392/391]]
 
Subgroup-val mapping: {{Mapping| 1 0 4 3 | 0 1 -2 -1 }}
 
Optimal tunings:
* Subgroup WE: ~2 = 1199.871{{c}}, ~17/14 = 336.243{{c}}
* Subgroup CWE: ~2 = 1200.000{{c}}, ~17/14 = 336.296{{c}}
 
{{Optimal ET sequence|legend=0| 7, 18, 25 }}
 
Badness (Sintel): 0.029
 
==== 2.25/7.17/7.19/7.23/7 subgroup ====
 
Subgroup: 2.25/7.17/7.19/7.23/7
 
Comma list: [[323/322]], [[392/391]], [[476/475]]
 
Subgroup-val mapping: {{Mapping| 1 -2 0 4 3 | 0 3 1 -2 -1 }}
 
Optimal tunings:
* Subgroup WE: ~2 = 1199.757{{c}}, ~17/14 = 335.428{{c}}
* Subgroup CWE: ~2 = 1200.000{{c}}, ~17/14 = 335.479{{c}}
 
{{Optimal ET sequence|legend=0| 7, 18, 25 }}
 
Badness (Sintel): 0.053
 
== 3/2.5/2.… subgroups ==
{{Main|Half-prime subgroup}}
 
=== Hemihemi ===
[[Subgroup]]: 3/2.5/2.7/2
 
[[Comma list]]: [[10976/10935]]
 
{{Mapping|legend=2| 1 2 3 | 0 3 1  }}
 
[[Optimal tuning]] (subgroup [[CTE]]): ~[[3/2]] = 1\[[1edf]], ~[[28/27]] = 60.909
 
[[Support]]ing [[ET]]s: *23, *12, *11, *35, *34, *10, *13, *47, *9[+5/2], *14[-5/2], *45, *25, *21[+5/2], *8[+5/2]
 
=== Halftone ===
{{Main| Halftone }}
 
[[Subgroup]]: 3/2.5/2.7/2
 
[[Comma list]]: 9604/9375
 
{{Mapping|legend=2| 1 3 4 | 0 -4 -5 }}
: sval mapping generators: ~3/2, ~15/14
 
[[Optimal tuning]] (subgroup [[CTE]]): ~3/2 = 1\1edf, ~15/14 = 128.783
 
Supporting ETs: *5, *6, *7[+5/2, +7/2], *9[-5/2, --7/2], *11, *16, *17[+5/2], *23[+5/2, +7/2], *21[-7/2], *27, *28[+5/2], *38, *43[-7/2], *49
: <nowiki />* wart for 3/2
 
==== 3/2.5/2.7/2.11/2 ====
[[Subgroup]]: 3/2.5/2.7/2.11/2
 
[[Comma list]]: 1232/1215, 27783/27500
 
{{Mapping|legend=2| 1 3 4 4 | 0 -4 -5 1 }}
: sval mapping generators: ~3/2, ~15/14
 
[[Optimal tuning]] (subgroup [[CTE]]): ~3/2 = 1\1edf, ~15/14 = 129.186
 
[[Support]]ing [[ET]]s: *11, *5, *16, *6, *27[-11/2], *21[-7/2], *38[-11/2], *43[-7/2, -11/2], *59[-7/2, -11/2], *70[-7/2, -11/2], *75[--7/2, -11/2]
: <nowiki />* wart for 3/2
 
==== 3/2.5/2.7/2.11/2.13/2 ====
[[Subgroup]]: 3/2.5/2.7/2.11/2.13/2
 
[[Comma list]]: 275/273, 1232/1215, 1323/1300
 
{{Mapping|legend=2| 1 3 4 4 5 | 0 -4 -5 1 -2 }}
 
[[Optimal tuning]] (subgroup [[CTE]]): ~3/2 = 1\1edf, ~15/14 = 129.381
 
[[Support]]ing [[ET]]s: *11, *5, *16, *6, *27[-11/2]
: <nowiki />* wart for 3/2
 
=== Semiwolf ===
[[Subgroup]]: 3/2.5/2.7/4
 
[[Comma list]]: 245/243
 
{{Mapping|legend=2| 1 1 2 | 0 2 -1 }}
 
: sval mapping generators: ~3/2, ~9/7
 
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~7/6 = 262.1728
 
[[Optimal ET sequence]]: [[3edf]], [[5edf]], [[8edf]]
 
==== Semilupine ====
[[Subgroup]]: 3/2.5/2.7/4.11/4
 
[[Comma list]]: 100/99, 245/243
 
{{Mapping|legend=2| 1 1 2 0 | 0 2 -1 4 }}
 
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~7/6 = 264.3771
 
[[Optimal ET sequence]]: [[8edf]], [[13edf]]
 
==== Hemilycan ====
[[Subgroup]]: 3/2.5/2.7/4.11/4
 
[[Comma list]]: 245/243, 441/440
 
{{Mapping|legend=2| 1 1 2 5 | 0 2 -1 -4 }}
 
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~7/6 = 261.5939


=== Marveltri ===
[[Optimal ET sequence]]: [[8edf]], [[11edf]]
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.
 
== 3/2.5/4.… subgroups ==
=== Poseidon ===
'''This temperament will be subjected to renaming due to a conflict.'''
 
[[Subgroup]]: 3/2.5/4.11/8


[[Subgroup]]: 2.5.9/7
[[Comma list]]: 121/120


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


{{Mapping|legend=2| 1 2 1 | 0 1 -2 }}
: [[gencom]]: [3/2 12/11; 121/120]


{{Mapping|legend=3| 1 2/5 2 -1/5 | 0 -4/5 1 2/5 }}
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~3/2, ~12/11 = 158.29


: [[gencom]]: [2 5/4; 225/224]
{{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 tuning]] ([[Tp tuning|subgroup POTE]]): ~5/4 = 383.638
== Other 3/2-equave subgroups ==
=== Auk ===
[[Subgroup]]: 3/2.7.13


{{Optimal ET sequence|legend=1| 12, 13, 16, 19, 22, 25, 47, 69, 72, 97, 122, 269c*, 660c* }}
[[Comma list]]: 87808/85293


<nowiki>*</nowiki> wart for 9/7
{{Mapping|legend=2| 1 0 -8 | 0 1 3 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.4801 cents
: sval mapping generators: ~3/2, ~7


==== Sulis ====
[[Optimal tuning]] (subgroup [[CTE]]): ~3/2 = 1\1edf, ~28/9 = 1950.859
Related temperament: [[Marvel family|minerva]], [[Würschmidt family|würschmidt]]


[[Subgroup]]: 2.5.9/7.11/9
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


[[Comma list]]: 99/98, 176/175
=== Doubleton ===
[[Subgroup]]: 3/2.7.13


{{Mapping|legend=2| 1 2 1 -1 | 0 1 -2 4 }}]
[[Comma list]]: 1352/1323


[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~5/4 = 386.558
{{Mapping|legend=2| 2 0 3 | 0 1 1 }}


{{Optimal ET sequence|legend=1| 3, …, 22, 25, 28, 31, 59 }}
: sval mapping generators: ~26/21, ~7


[[Tp tuning #T2 tuning|RMS error]]: 1.074 cents
[[Optimal tuning]] (subgroup [[CTE]]): ~26/21 = 1\2edf, ~28/9 = 1971.772


== Other 15/11 subgroups ==
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


=== Poggers ===
== 5/2-equave subgroups ==
Related temperaments: [[Stearnsmic_clan#Pogo|pogo]], [[Stearnsmic_clan#Supers|supers]]
=== Hyperion ===
[[Subgroup]]: 5/2.7.11


[[Subgroup]]: 2.9.7.15/11.13
[[Comma list]]: {{monzo| 11 1 -5 }}


[[Comma list]]: [[540/539]], [[1716/1715]], [[2080/2079]]
{{Mapping|legend=2| 1 4 3 | 0 -5 -1 }}


{{Mapping|legend=2| 1 1 1 -1 -1 | 0 6 5 4 13 }}
: [[gencom]]: [5/2 125/88; 341796875/329832448]


[[Optimal tuning]] (subgroup [[CTE]]): ~9/7 = 433.888
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~5/2 = 1586.3137, ~125/88 = 593.6668


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


= Related temperament collections =
= Related temperament collections =
Line 948: 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"