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Subgroup temperaments: Difference between revisions

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{{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]]
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= Composite 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.
Line 439: Line 424:
{{Optimal ET sequence|legend=1| 5, 13c, 18, 23, 41, 64, 87, 151 }}
{{Optimal ET sequence|legend=1| 5, 13c, 18, 23, 41, 64, 87, 151 }}


== 2.15.55 subgroup ==
== 4.3.5 subgroup ==
=== Spog ===
=== Tetrahanson ===
{{Main| Tetrahanson }}


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]]: 4.3.5


[[Subgroup]]: 2.15.55
[[Comma list]]: 15625/15552


[[Comma list]]: [[100663296/100656875]]
{{Mapping|legend=2| 1 3 3 | 0 -6 -5 }}


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


[[Optimal tuning]] (subgroup [[CTE]]): ~55/32 = 937.655
[[Optimal tuning]] ([[CTE]]): ~4 = 2\1, ~5/3 = 882.941


{{Optimal ET sequence|legend=1|5, 9, 23, 32, 151, 183, 215, 247, 956, 1203, 1450, 3147, 4597 }}
[[Support]]ing [[ET]]s: {{EDs|19, 106, 87, 68, 11, 8, 125, 49, 30, 27, 117, 46, 41b, 79|equave=4}}


==== 2.15.55.325 ====
=== Tetrameantone ===
[[Subgroup]]: 2.15.55.325
{{Main| Tetrameantone }}


[[Comma list]]: [[4225/4224]], [[6656/6655]]
[[Subgroup]]: 4.3.5


{{Mapping|legend=2| 1 0 5 6 | 0 5 1 3 }}
[[Comma list]]: 81/80


[[Optimal tuning]] (subgroup [[CTE]]): ~55/32 = 937.647
{{Mapping|legend=2| 1 1 2 | 0 -1 -4 }}


[[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
: Mapping generators: ~4, ~4/3


==== 2.15.189.55.325 ====
[[Optimal tuning]] ([[POTE]]): 4 = 2400.0, ~4/3 = 503.761
Related temperament: [[Lehmerismic_temperaments#Lux|lux]]


[[Subgroup]]: 2.15.189.55.325
[[Support]]ing [[ET]]s: {{EDs|5, 9, 14, 19, 24, 43, 62, 81, 100|equave=4}}


[[Comma list]]: [[2080/2079]], [[3025/3024]], [[4096/4095]]
=== Tetramagic ===


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


[[Optimal tuning]] (subgroup [[CTE]]): ~55/32 = 937.677
[[Comma list]]: 3125/3072


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


==== 2.15.189.55.325.725 ====
: Mapping generators: ~4, ~5/4


[[Subgroup]]: 2.15.189.55.325.725
[[Optimal tuning]] ([[POTE]]): 4 = 2400.0, ~5/4 = 380.059


[[Comma list]]: [[1625/1624]], [[2080/2079]], [[3025/3024]], [[4096/4095]]
[[Support]]ing [[ET]]s: {{EDs|6, 13, 19, 25, 38, 44, 63, 82|equave=4}}


{{Mapping|legend=2| 1 0 6 5 6 -3 | 0 5 2 1 3 16 }}
=== Blacktetra ===


[[Optimal tuning]] (subgroup [[CTE]]): ~55/32 = 937.649
[[Subgroup]]: 4.3.5


[[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]]: 256/243


==== 2.15.189.55.325.725.279 ====
{{Mapping|legend=2| 5 4 6 | 0 0 -1 }}


Here are rational approximations to the intervals of the semiquartal scale.
: Mapping generators: ~4, ~16/15


Sharp: 12/11, 25/21, 33/26, 18/13, 31/21 ~ 65/44 ~ 96/65, 50/31 ~ 29/18, 55/32, 15/8.
[[Optimal tuning]] ([[POTE]]): 1\5ed4 = 480.0, ~16/15 = 80.4062


Flat: 16/15, 64/55, 31/25 ~ 36/29, 42/31 ~ 65/48 ~ 88/65, 13/9, 52/33, 42/25, 11/6.
[[Support]]ing [[ET]]s: {{EDs|5, 10, 15, 20, 25, 30, 55, 85, 115|equave=4}}
[[Subgroup]]: 2.15.189.55.325.725.279


[[Comma list]]: [[1625/1624]], [[2016/2015]], [[2080/2079]], [[3025/3024]], [[4096/4095]]
== 4.6.5 subgroup ==
=== Meanquad ===
{{Main| Meanquad }}


{{Mapping|legend=2| 1 0 6 5 6 -3 5 | 0 5 2 1 3 16 4 }}
[[Subgroup]]: 4.6.5


[[Optimal tuning]] (subgroup [[CTE]]): ~55/32 = 937.638
[[Comma list]]: [[81/80]] = {{monzo| -4 4 -1 }}


[[Support]]ing [[ET]]s: 9[-725], 14[+725], 23, 32, 41[-725], 55, 73[-725], 87, 105[-725], 119, 151, 183, 206[+725], 311
{{Mapping|legend=2| 1 0 -4| 0 1 4 }}


== 4.3.5 subgroup ==
: mapping generators: ~4, ~6
=== Tetrahanson ===
{{Main| Tetrahanson }}


[[Subgroup]]: 4.3.5
[[Optimal tuning]] (subgroup [[CTE]]): ~4 = 2\1, ~3/2 = 697.214


[[Comma list]]: 15625/15552
[[Support]]ing [[ET]]s: *7, *10, *11[-5], *13[+5], *17, *24, *27[+5], *31, *38, *41, *45, *52, *55, *69


{{Mapping|legend=2| 1 3 3 | 0 -6 -5 }}
<nowiki />* Wart for 4


: Mapping generators: ~4, ~5/3
==== 4.6.5.7 subgroup (tetrominant) ====
[[Subgroup]]: 4.6.5.7


[[Optimal tuning]] ([[CTE]]): ~4 = 2\1, ~5/3 = 882.941
[[Comma list]]: [[36/35]] = {{monzo| 0 2 -1 -1 }}, [[64/63]] = {{monzo| 4 -2 0 -1 }}


[[Support]]ing [[ET]]s: {{EDs|19, 106, 87, 68, 11, 8, 125, 49, 30, 27, 117, 46, 41b, 79|equave=4}}
{{Mapping|legend=2| 1 0 -4 4 | 0 1 4 -2 }}


=== Tetrameantone ===
[[Optimal tuning]] (subgroup [[CTE]]): ~4 = 2\1, ~3/2 = 699.622
{{Main| Tetrameantone }}


[[Subgroup]]: 4.3.5
[[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]


[[Comma list]]: 81/80
<nowiki />* Wart for 4


{{Mapping|legend=2| 1 1 2 | 0 -1 -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.


: Mapping generators: ~4, ~4/3
Fourwar is named after the closely related [[hemiwar]] temperament.


[[Optimal tuning]] ([[POTE]]): 4 = 2400.0, ~4/3 = 503.761
<pre>
 
Reduced Mapping
[[Support]]ing [[ET]]s: {{EDs|5, 9, 14, 19, 24, 43, 62, 81, 100|equave=4}}
4 6 5
 
[ ⟨ 1 0 1 ]
=== Tetramagic ===
⟨ 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)


[[Subgroup]]: 4.3.5
Subsets
q99, q62, q37, q161, q136, q198, q25, q124, q74, q235
</pre>


[[Comma list]]: 3125/3072
==== 4.6.5.7 ====
 
<pre>
{{Mapping|legend=2| 1 0 1 | 0 5 1 }}
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)


: Mapping generators: ~4, ~5/4
Subsets
q99, q62, q37, q161, q136, q198, q25, q124, q74, q235
</pre>


[[Optimal tuning]] ([[POTE]]): 4 = 2400.0, ~5/4 = 380.059
==== 4.6.5.7.11 ====
 
<pre>
[[Support]]ing [[ET]]s: {{EDs|6, 13, 19, 25, 38, 44, 63, 82|equave=4}}
Reduced Mapping
 
4 6 5 7 11
=== Blacktetra ===
[ ⟨ 1 0 1 1 1 ]
 
⟨ 0 16 2 5 9 ] ⟩
[[Subgroup]]: 4.3.5
 
TE Generator Tunings (cents)
[[Comma list]]: 256/243
⟨2400.1097, 193.9498]
 
{{Mapping|legend=2| 5 4 6 | 0 0 -1 }}
TE Step Tunings (cents)
 
⟨24.18752, 48.52491]
: Mapping generators: ~4, ~16/15
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)


[[Optimal tuning]] ([[POTE]]): 1\5ed4 = 480.0, ~16/15 = 80.4062
Subsets
q37, q25, q62, q12, q74, q99, q87, q49r, q50r, q124
</pre>


[[Support]]ing [[ET]]s: {{EDs|5, 10, 15, 20, 25, 30, 55, 85, 115|equave=4}}
==== 4.6.5.7.11.13 ====


== 4.6.5 subgroup ==
<pre>
=== Meanquad ===
Reduced Mapping
{{Main| Meanquad }}
4 6 5 7 11 13
 
[ ⟨ 1 0 1 1 1 0 ]
[[Subgroup]]: 4.6.5
⟨ 0 16 2 5 9 23 ] ⟩
 
[[Comma list]]: [[81/80]] = {{monzo| -4 4 -1 }}
TE Generator Tunings (cents)
 
⟨2401.2305, 193.5378]
{{Mapping|legend=2| 1 0 -4| 0 1 4 }}
 
TE Step Tunings (cents)
: mapping generators: ~4, ~6
⟨42.79107, 35.98524]
 
[[Optimal tuning]] (subgroup [[CTE]]): ~4 = 2\1, ~3/2 = 697.214
TE Tuning Map (cents)
 
⟨2401.230, 3096.606, 2788.306, 3368.920, 4143.071, 4451.371]
[[Support]]ing [[ET]]s: *7, *10, *11[-5], *13[+5], *17, *24, *27[+5], *31, *38, *41, *45, *52, *55, *69
 
TE Mistunings (cents)
<nowiki />* Wart for 4
⟨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)


==== 4.6.5.7 subgroup (tetrominant) ====
Subsets
[[Subgroup]]: 4.6.5.7
q25, q37f, q12f, q62, q50rf, q13rff, q49rff, q87, q74ff, q24rfff
</pre>


[[Comma list]]: [[36/35]] = {{monzo| 0 2 -1 -1 }}, [[64/63]] = {{monzo| 4 -2 0 -1 }}
==== 4.6.5.7.11.13.17 ====
 
<pre>
{{Mapping|legend=2| 1 0 -4 4 | 0 1 4 -2 }}
Reduced Mapping
 
4 6 5 7 11 13 17
[[Optimal tuning]] (subgroup [[CTE]]): ~4 = 2\1, ~3/2 = 699.622
[ ⟨ 1 0 1 1 1 0 1 ]
 
⟨ 0 16 2 5 9 23 13 ] ⟩
[[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 />* 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.
 
<pre>  
Reduced Mapping
4 6 5
[ ⟨ 1 0 1 ]
⟨ 0 16 2 ] ⟩
   
   
TE Generator Tunings (cents)
TE Generator Tunings (cents)
⟨2399.3973, 193.8643]
⟨2400.4701, 193.4599]
   
   
TE Step Tunings (cents)
TE Step Tunings (cents)
⟨25.21211, 47.81337]
⟨43.39350, 35.55764]
   
   
TE Tuning Map (cents)
TE Tuning Map (cents)
⟨2399.397, 3101.829, 2787.126]
⟨2400.470, 3095.359, 2787.390, 3367.770, 4141.609, 4449.578, 4915.449]
   
   
TE Mistunings (cents)
TE Mistunings (cents)
⟨-0.603, -0.126, 0.812]
⟨0.470, -6.596, 1.076, -1.056, -9.709, 9.050, 10.494]
   
   
Complexity 1.369085
Complexity 1.129881
Adjusted Error 0.692892 cents
Adjusted Error 8.082725 cents
TE Error 0.268047 cents/octave
TE Error 1.977443 cents/octave
   
   
Unison Vector
Unison Vectors
[8, 1, -8⟩ (393216:390625)
[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
Subsets
q99, q62, q37, q161, q136, q198, q25, q124, q74, q235
q25, q12f, q37f, q13rffg, q50rf, q62, q49rffg, q24rfffg, q38rreffg, q74ffg
</pre>
</pre>


==== 4.6.5.7 ====
==== 4.6.5.7.11.13.17.19 ====
<pre>
<pre>
Reduced Mapping
Reduced Mapping
4 6 5 7
4 6 5 7 11 13 17 19
[ ⟨ 1 0 1 1 ]
[ ⟨ 1 0 1 1 1 0 1 1 ]
⟨ 0 16 2 5 ] ⟩
⟨ 0 16 2 5 9 23 13 14 ] ⟩
   
   
TE Generator Tunings (cents)
TE Generator Tunings (cents)
⟨2399.4195, 193.8654]
⟨2399.9219, 193.3952]
   
   
TE Step Tunings (cents)
TE Step Tunings (cents)
⟨25.23883, 47.79592]
⟨44.14256, 35.03670]
   
   
TE Tuning Map (cents)
TE Tuning Map (cents)
⟨2399.420, 3101.846, 2787.150, 3368.747]
⟨2399.922, 3094.324, 2786.712, 3366.898, 4140.479, 4448.090, 4914.060, 5107.455]
   
   
TE Mistunings (cents)
TE Mistunings (cents)
⟨-0.580, -0.109, 0.837, -0.079]
⟨-0.078, -7.631, 0.399, -1.928, -10.839, 7.562, 9.104, 9.942]
   
   
Complexity 1.192044
Complexity 1.058472
Adjusted Error 0.653313 cents
Adjusted Error 8.712222 cents
TE Error 0.232715 cents/octave
TE Error 2.050935 cents/octave
   
   
Unison Vectors
Unison Vectors
[-2, -1, -2, 4⟩ (2401:2400)
[0, 1, -1, 0, 1, -1, 0, 0⟩ (66:65)
[3, 0, -5, 2⟩ (3136:3125)
[-1, 0, 0, 1, 1, 0, 0, -1⟩ (77:76)
[5, 1, -3, -2⟩ (6144:6125)
[2, 1, -1, 0, 0, 0, 0, -1⟩ (96:95)
[8, 1, -8, 0⟩ (393216:390625)
[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
Subsets
q99, q62, q37, q161, q136, q198, q25, q124, q74, q235
q25, q12fh, q37f, q13rffgh, q50rf, q62, q49rffgh, q24rfffghh, q38rreffgh, q74ffgh
</pre>
</pre>


==== 4.6.5.7.11 ====
==== 4.6.5.7.11.13.17.19.23 ====
<pre>
<pre>
Reduced Mapping
Reduced Mapping
4 6 5 7 11
4 6 5 7 11 13 17 19 23
[ ⟨ 1 0 1 1 1 ]
[ ⟨ 1 0 1 1 1 0 1 1 0 ]
⟨ 0 16 2 5 9 ] ⟩
⟨ 0 16 2 5 9 23 13 14 28 ] ⟩
   
   
TE Generator Tunings (cents)
TE Generator Tunings (cents)
⟨2400.1097, 193.9498]
⟨2399.3286, 193.5316]
   
   
TE Step Tunings (cents)
TE Step Tunings (cents)
⟨24.18752, 48.52491]
⟨37.31613, 39.63311]
   
   
TE Tuning Map (cents)
TE Tuning Map (cents)
⟨2400.110, 3103.196, 2788.009, 3369.859, 4145.658]
⟨2399.329, 3096.506, 2786.392, 3366.987, 4141.113, 4451.227, 4915.240, 5108.771, 5418.885]
   
   
TE Mistunings (cents)
TE Mistunings (cents)
⟨0.110, 1.241, 1.696, 1.033, -5.660]
⟨-0.671, -5.449, 0.078, -1.839, -10.205, 10.699, 10.284, 11.258, -9.389]
   
   
Complexity 1.068792
Complexity 1.115920
Adjusted Error 2.926965 cents
Adjusted Error 9.502017 cents
TE Error 0.846083 cents/octave
TE Error 2.100561 cents/octave
   
   
Unison Vectors
Unison Vectors
[-1, -1, -1, 0, 2⟩ (121:120)
[0, 1, -1, 0, 1, -1, 0, 0, 0⟩ (66:65)
[2, 0, -2, -1, 1⟩ (176:175)
[1, 0, 0, -1, 0, -1, 0, 0, 1⟩ (92:91)
[-3, -1, 1, 1, 1⟩ (385:384)
[0, -1, 1, 0, 0, 0, 0, -1, 1⟩ (115:114)
[-1, 0, 3, -3, 1⟩ (1375:1372)
[1, 1, 1, -1, 0, 0, -1, 0, 0⟩ (120:119)
[-2, -1, -2, 4, 0⟩ (2401:2400)
[2, 0, -2, -1, 1, 0, 0, 0, 0⟩ (176:175)
[1, 0, 1, -4, 2⟩ (2420:2401)
[-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
Subsets
q37, q25, q62, q12, q74, q99, q87, q49r, q50r, q124
q25i, q12fhi, q37f, q13rffghii, q62, q50rfii, q49rffghii, q24rfffghhiii, q74ffghi, q38rreffghiii
</pre>
</pre>


==== 4.6.5.7.11.13 ====
== 4.9.25 subgroup ==
=== Meansquared ===
[[Subgroup]]: 4.9.25
 
[[Comma list]]: [[6561/6400]]
 
{{Mapping|legend=2| 1 3 4 | 0 1 4 }}
 
Mapping generators: ~4, ~9/64
 
[[Optimal tuning]] ([[CTE]]): ~4 = 2\1, ~9/4 = 1394.429


<pre>
[[Support]]ing [[ET]]s: 12, 7, 19, 5, 31, 26, 17[+25], 43, 9[-25], 33[-25], 45, 29[+25], 8[+25], 22[+25]
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
== 4.9.49 subgroup ==
q25, q37f, q12f, q62, q50rf, q13rff, q49rff, q87, q74ff, q24rfff
=== Archsquared ===
</pre>
[[Subgroup]]: 4.9.49
 
[[Comma list]]: 4096/3969
 
{{Mapping|legend=2| 1 3 0 | 0 1 -2 }}
 
Mapping generators: ~4, ~9/64
 
[[Optimal tuning]] ([[CTE]]): ~[[9/8]] = 219.190
 
[[Support]]ing [[ET]]s: 5, 17, 22, 12, 7, 27, 32, 8, 39[+49], 29[+49], 9[+49], 19[+49], 37, 49


==== 4.6.5.7.11.13.17 ====
== 8.9.7 subgroup ==
<pre>
=== Sixscared ===
Reduced Mapping
Sixscared is a tuning which still maintains some consonance, while eviscerating the rules of conventional 12-tone harmony. The familiar major, minor and perfect intervals are nowhere to be found, and octaves are far and few between, so the seventh harmonic becomes the backbone of harmony. Approximating the harmonics 7, 8, 9, Sixscared is named for the classic dad joke: "Why was six scared? Because seven ate nine."
4 6 5 7 11 13 17
 
[ ⟨ 1 0 1 1 1 0 1 ]
[[Subgroup]]: 8.9.7
⟨ 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
[[Comma list]]: 64/63
q25, q12f, q37f, q13rffg, q50rf, q62, q49rffg, q24rfffg, q38rreffg, q74ffg
</pre>


==== 4.6.5.7.11.13.17.19 ====
{{Mapping|legend=2| 1 0 2 | 0 1 -1 }}
<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
: sval mapping generators: ~8, ~9
q25, q12fh, q37f, q13rffgh, q50rf, q62, q49rffgh, q24rfffghh, q38rreffgh, q74ffgh
 
</pre>
: [[gencom]]: [8 9/8; 64/63]
 
[[Optimal tuning]] ([[CTE]]): 1\[[3ed8]] = 1600.0, ~9/8 = 219.1898
 
[[Optimal ET sequence]]: {{val| 16 17 15 }}, {{val| 33 35 31 }}, {{val| 148 … }}, {{val| 181 … }}, {{val| 214 … }}, {{val| 247 … }}
 
[[Badness]]: 0.0215 × 10<sup>-3</sup>


==== 4.6.5.7.11.13.17.19.23 ====
= Fractional subgroup temperaments =
<pre>
== 2.5/3… subgroups ==
Reduced Mapping
=== Magicaltet ===
4 6 5 7 11 13 17 19 23
{{See also| Chromatic pairs #Magicaltet }}
[ ⟨ 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
Magicaltet is related to [[supermagic]], [[superkleismic]], and [[magic]]. The tonic and the first three generator steps make a [[magical seventh chord]], hence the name.
q25i, q12fhi, q37f, q13rffghii, q62, q50rfii, q49rffghii, q24rfffghhiii, q74ffghi, q38rreffghiii
</pre>


== 4.9.25 subgroup ==
[[Subgroup]]: 2.5/3.7.11
=== Meansquared ===
[[Subgroup]]: 4.9.25


[[Comma list]]: [[6561/6400]]
[[Comma list]]: 100/99 ({{monzo| 2 2 0 -1 }}), 385/384 ({{monzo| -7 1 1 1 }})


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


Mapping generators: ~4, ~9/64
: mapping generators: ~2, ~5/3


[[Optimal tuning]] ([[CTE]]): ~4 = 2\1, ~9/4 = 1394.429
{{Mapping|legend=3| 1 -1/2 1/2 2 4 | 0 1/2 -1/2 3 -2 }}


[[Support]]ing [[ET]]s: 12, 7, 19, 5, 31, 26, 17[+25], 43, 9[-25], 33[-25], 45, 29[+25], 8[+25], 22[+25]
: [[gencom]]: [2 6/5; 100/99 385/384]


== 4.9.49 subgroup ==
[[Optimal tuning]]s:
=== Archsquared ===
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 877.343
[[Subgroup]]: 4.9.49
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 877.351


[[Comma list]]: 4096/3969
{{Optimal ET sequence|legend=1| 4, 7, 11, 15, 26, 67, 93* }}


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


Mapping generators: ~4, ~9/64
[[Tp tuning #T2 tuning|RMS error]]: 1.206 cents


[[Optimal tuning]] ([[CTE]]): ~[[9/8]] = 219.190
=== Starlingtet ===
{{See also | Chromatic pairs #Starlingtet }}


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


== 8.9.7 subgroup ==
[[Comma list]]: [[126/125]] ({{monzo| 1 -3 1 }})
=== Sixscared ===
Sixscared is a tuning which still maintains some consonance, while eviscerating the rules of conventional 12-tone harmony. The familiar major, minor and perfect intervals are nowhere to be found, and octaves are far and few between, so the seventh harmonic becomes the backbone of harmony. Approximating the harmonics 7, 8, 9, Sixscared is named for the classic dad joke: "Why was six scared? Because seven ate nine."


[[Subgroup]]: 8.9.7
{{Mapping|legend=2| 1 0 -1 | 0 1 3 }}


[[Comma list]]: 64/63
: mapping generators: ~2, ~5/3


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


: sval mapping generators: ~8, ~9
: [[gencom]]: [2 6/5; 126/125]


: [[gencom]]: [8 9/8; 64/63]
[[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 tuning]] ([[CTE]]): 1\[[3ed8]] = 1600.0, ~9/8 = 219.1898
{{Optimal ET sequence|legend=1| 4, 15, 19, 23, 27 }}


[[Optimal ET sequence]]: {{val| 16 17 15 }}, {{val| 33 35 31 }}, {{val| 148 … }}, {{val| 181 … }}, {{val| 214 … }}, {{val| 247 … }}
[[Tp tuning #T2 tuning|RMS error]]: 0.8398 cents


[[Badness]]: 0.0215 × 10<sup>-3</sup>
==== Greeley ====
{{See also| Chromatic pairs #Greeley }}


= Fractional subgroup temperaments =
Greeley is related to [[opossum]] as well as to [[nusecond]].  
== 2.5/3… subgroups ==
=== Magicaltet ===
{{See also| Chromatic pairs #Magicaltet }}


Magicaltet is related to [[supermagic]], [[superkleismic]], and [[magic]]. The tonic and the first three generator steps make a [[magical seventh chord]], hence the name.  
[[Subgroup]]: 2.5/3.7/3.11/3


[[Subgroup]]: 2.5/3.7.11
[[Comma list]]: 121/120 ({{monzo| -3 -1 0 2 }}), 126/125 ({{monzo| 1 -3 1 }})


[[Comma list]]: 100/99 ({{monzo| 2 2 0 -1 }}), 385/384 ({{monzo| -7 1 1 1 }})
{{Mapping|legend=2| 1 1 2 2 | 0 -2 -6 -1 }}


{{Mapping|legend=2| 1 0 5 2 | 0 1 -3 2 }}
{{Mapping|legend=3| 1 -5/4 -1/4 3/4 3/4 | 0 9/4 1/4 -15/4 5/4 }}


: mapping generators: ~2, ~5/3
: [[gencom]]: [2 11/10; 121/120 126/125]


{{Mapping|legend=3| 1 -1/2 1/2 2 4 | 0 1/2 -1/2 3 -2 }}
[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~11/10 = 155.696
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~11/10 = 155.776


: [[gencom]]: [2 6/5; 100/99 385/384]
{{Optimal ET sequence|legend=1| 8, 15, 23, 54, 77, 100, 131* }}


[[Optimal tuning]]s:
<nowiki/>* Wart for 11/3
* [[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* }}
[[Tp tuning #T2 tuning|RMS error]]: 1.034 cents


<nowiki/>* Wart for 5/3
==== Skateboard ====
{{See also| Chromatic pairs #Skateboard }}


[[Tp tuning #T2 tuning|RMS error]]: 1.206 cents
Skateboard is related to [[thrasher]].  


=== Starlingtet ===
[[Subgroup]]: 2.5/3.7/3.11.13/9
{{See also | Chromatic pairs #Starlingtet }}


Starlingtet, the {{nowrap| 4 & 15 }} temperament in the 2.5/3.7/3 subgroup, is related to [[starling]] as well as to [[myna]]. The tonic and the first three generator steps make a [[starling tetrad]], hence the name.
[[Comma list]]: 56/55 ({{monzo| 3 -1 1 -1 }}), 91/90 ({{monzo| -1 -1 1 0 1 }}), 100/99 ({{monzo| 2 2 0 -1 }})


[[Subgroup]]: 2.5/3.7/3
{{Mapping|legend=2| 1 0 -1 2 2 | 0 1 3 2 -2 }}


[[Comma list]]: [[126/125]] ({{monzo| 1 -3 1 }})
{{Mapping|legend=3| 1 -3/7 4/7 11/7 4 -6/7 | 0 0 -1 -3 -2 2 }}


{{Mapping|legend=2| 1 0 -1 | 0 1 3 }}
: [[gencom]]: [2 6/5; 56/55 91/90 100/99]


: mapping generators: ~2, ~5/3
[[Optimal tuning]]s:  
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 886.158
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 886.158


{{Mapping|legend=3| 1 -1 0 1 | 0 4/3 1/3 -5/3 }}
{{Optimal ET sequence|legend=1| 11, 15, 19, 23, 42d, 65d }}


: [[gencom]]: [2 6/5; 126/125]
[[Tp tuning #T2 tuning|RMS error]]: 2.396 cents


[[Optimal tuning]]s:
=== Gariberttet ===
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 888.759
Gariberttet is the 2.5/3.7/3 [[Subgroup temperament families, relationships, and genes|altergene]] of [[sirius]].
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 888.846


{{Optimal ET sequence|legend=1| 4, 15, 19, 23, 27 }}
==== Gariberttet (2.5/3.7/3.13/11 subgroup) ====
{{See also | Chromatic pairs #Gariberttet }}


[[Tp tuning #T2 tuning|RMS error]]: 0.8398 cents
Gariberttet can be described as the {{nowrap| 4 & 29 }} temperament in the 2.5/3.7/3.13/11 subgroup.  


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


Greeley is related to [[opossum]] as well as to [[nusecond]].
[[Comma list]]: [[275/273]] ({{monzo| 0 2 -1 -1 }}), [[847/845]] ({{monzo| 0 -1 1 -2 }})


[[Subgroup]]: 2.5/3.7/3.11/3
{{Mapping|legend=2| 1 0 0 0 | 0 3 5 1 }}


[[Comma list]]: 121/120 ({{monzo| -3 -1 0 2 }}), 126/125 ({{monzo| 1 -3 1 }})
{{Mapping|legend=3| 1 0 0 0 0 0 | 0 -8/3 1/3 7/3 -1/2 1/2 }}


{{Mapping|legend=2| 1 1 2 2 | 0 -2 -6 -1 }}
: [[gencom]]: [2 13/11; 275/273 847/845]
 
{{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:  
[[Optimal tuning]]s:  
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~11/10 = 155.696
* [[Tp tuning|subgroup]] [[CTE]] and [[POTE]]: ~2 = 1200.000, ~13/11 = 293.679
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~11/10 = 155.776


{{Optimal ET sequence|legend=1| 8, 15, 23, 54, 77, 100, 131* }}
{{Optimal ET sequence|legend=1| 29, 33, 37, 41, 45, 49, 78, 94, 143* }}


<nowiki/>* Wart for 11/3
<nowiki/>* Wart for 13/11


[[Tp tuning #T2 tuning|RMS error]]: 1.034 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.6914 cents


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


Skateboard is related to [[thrasher]].  
Indium can be described as the {{nowrap| 8 & 33 }} temperament in the 2.5/3.7/3.11/3 subgroup.  


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


[[Comma list]]: 56/55 ({{monzo| 3 -1 1 -1 }}), 91/90 ({{monzo| -1 -1 1 0 1 }}), 100/99 ({{monzo| 2 2 0 -1 }})
[[Comma list]]: [[3025/3024]] ({{monzo| -4 2 -1 2 }}), [[3125/3087]] ({{monzo| 0 5 -3 }})


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


{{Mapping|legend=3| 1 -3/7 4/7 11/7 4 -6/7 | 0 0 -1 -3 -2 2 }}
{{Mapping|legend=3| 1 -1/2 -1/2 -1/2 3/2 | 0 -15/4 9/4 25/4 -19/4 }}


: [[gencom]]: [2 6/5; 56/55 91/90 100/99]
: [[gencom]]: [2 12/11; 3025/3024 3125/3087]


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 886.158
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~12/11 = 146.978
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 886.158
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~12/11 = 147.010


{{Optimal ET sequence|legend=1| 11, 15, 19, 23, 42d, 65d }}
{{Optimal ET sequence|legend=1| 8, 33, 41, 49, 204*<sup>†</sup> }}


[[Tp tuning #T2 tuning|RMS error]]: 2.396 cents
<nowiki/>* Wart for 7/3


=== Gariberttet ===
<sup>†</sup> Wart for 11/3
Gariberttet is the 2.5/3.7/3 [[Subgroup Temperament Families, Relationships, and Genes|altergene]] of [[sirius]].


==== Gariberttet (2.5/3.7/3.13/11 subgroup) ====
[[Tp tuning #T2 tuning|RMS error]]: 0.7788 cents
{{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.
==== Ammon ====
{{See also| Chromatic pairs #Ammon }}


[[Subgroup]]: 2.5/3.7/3.13/11
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]]: [[275/273]] ({{monzo| 0 2 -1 -1 }}), [[847/845]] ({{monzo| 0 -1 1 -2 }})
[[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 0 0 0 | 0 3 5 1 }}
{{Mapping|legend=2| 1 3 5 3 4 | 0 -6 -10 -3 -5 }}


{{Mapping|legend=3| 1 0 0 0 0 0 | 0 -8/3 1/3 7/3 -1/2 1/2 }}
{{Mapping|legend=3| 1 -3 0 2 0 1 | 0 24/5 -6/5 -26/5 9/5 -1/5 }}


: [[gencom]]: [2 13/11; 275/273 847/845]
: [[gencom]]: [2 13/10; 121/120 169/168 275/273]


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[Tp tuning|subgroup]] [[CTE]] and [[POTE]]: ~2 = 1200.000, ~13/11 = 293.679
* [[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| 29, 33, 37, 41, 45, 49, 78, 94, 143* }}
{{Optimal ET sequence|legend=1| 8, 29, 37, 45 }}


<nowiki/>* Wart for 13/11
[[Tp tuning #T2 tuning|RMS error]]: 1.052 cents


[[Tp tuning #T2 tuning|RMS error]]: 0.6914 cents
=== Sentry ===
{{See also | Chromatic pairs #Sentry }}


==== Indium ====
Sentry, the {{nowrap| 3 & 5 }} temperament in the 2.5/3.9/7 subgroup, is related to [[sensi]].
{{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]]: 2.5/3.9/7


[[Subgroup]]: 2.5/3.7/3.11/3
[[Comma list]]: [[245/243]] ({{monzo| 0 1 -2 }})


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


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


{{Mapping|legend=3| 1 -1/2 -1/2 -1/2 3/2 | 0 -15/4 9/4 25/4 -19/4 }}
: [[gencom]]: [2 9/7; 245/243]
 
: [[gencom]]: [2 12/11; 3025/3024 3125/3087]


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


{{Optimal ET sequence|legend=1| 8, 33, 41, 49, 204*<sup>†</sup> }}
{{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 7/3
<nowiki/>* Wart for 5/3


<sup>†</sup> Wart for 11/3
<sup>†</sup> Wart for 9/7


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


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


Semidim 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 name.  
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.  


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


[[Comma list]]: [[121/120]] ({{monzo| -3 -1 0 2 }}), [[169/168]] ({{monzo| -3 0 -1 0 2 }}), [[275/273]] ({{monzo| 0 2 -1 1 -1 }})
[[Comma list]]: [[325/324]] ({{monzo| -2 2 1 }})


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


{{Mapping|legend=3| 1 -3 0 2 0 1 | 0 24/5 -6/5 -26/5 9/5 -1/5 }}
{{Mapping|legend=3| 1 -1/6 5/6 0 0 -1/3 | 0 -1/2 -3/2 0 0 1 }}


: [[gencom]]: [2 13/10; 121/120 169/168 275/273]
: [[gencom]]: [2 6/5; 325/324]


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~13/10 = 453.121
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 886.886
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~13/10 = 453.242
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 886.861
 
{{Optimal ET sequence|legend=1| 3, 4, 11, 15, 19, 34, 53, 87, 140 }}


{{Optimal ET sequence|legend=1| 8, 29, 37, 45 }}
[[Tp tuning #T2 tuning|RMS error]]: 0.2444 cents


[[Tp tuning #T2 tuning|RMS error]]: 1.052 cents
== 2.….7/3… subgroups ==
=== Guanyintet ===
{{See also | Chromatic pairs #Guanyintet }}


=== Sentry ===
Guanyintet, the {{nowrap| 4 & 9 }} temperament in the 2.5.7/3.11/3 subgroup, is related to [[guanyin]] as well as to [[orwell]]. The tonic and the first three generator steps make a [[guanyin tetrad]], hence the name.
{{See also | Chromatic pairs #Sentry }}


Sentry, the {{nowrap| 3 & 5 }} temperament in the 2.5/3.9/7 subgroup, is related to [[sensi]].
[[Subgroup]]: 2.5.7/3.11/3


[[Subgroup]]: 2.5/3.9/7
[[Comma list]]: [[176/175]] ({{monzo| 4 -2 -1 1 }}), [[540/539]] ({{monzo| 2 1 -2 -1 }})


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


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


{{Mapping|legend=3| 1 0 0 0 | 0 0 2 -1 }}
{{Mapping|legend=3| 1 -4/3 3 -1/3 5/3 | 0 4/3 -3 7/3 -11/3 }}


: [[gencom]]: [2 9/7; 245/243]
: [[gencom]]: [2 7/6; 176/175 540/539]


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[Tp tuning|subgroup]] [[CTE]] and [[POTE]]: ~2 = 1200.000, ~9/7 = 440.902
* ([[Tp tuning|subgroup]] [[CTE]]): ~2 = 1200.000, ~12/7 = 929.545
* ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1200.000, ~12/7 = 929.907


{{Optimal ET sequence|legend=1| 8, 11, 19, 30, 41, 49, 52, 145*, 166<sup>†</sup>, 197*<sup>†</sup>, 215<sup>†</sup>, 264*<sup>†</sup> }}
{{Optimal ET sequence|legend=1| 9, 22, 31, 40, 191c*, 231c*, 271c*, 311c* }}


<nowiki/>* Wart for 5/3
<nowiki/>* wart for 7/3


<sup>†</sup> Wart for 9/7
[[Tp tuning #T2 tuning|RMS error]]: 0.6028 cents


[[Tp tuning #T2 tuning|RMS error]]: 0.7105 cents
==== Laz ====
{{See also | Chromatic pairs #Laz }}


=== Marveltwintri ===
Laz is related to [[avalokita]] as well as to [[winston]].
{{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.
[[Subgroup]]: 2.5.7/3.11/3.13/3


[[Subgroup]]: 2.5/3.13/9
[[Comma list]]: [[144/143]] ({{monzo| 4 0 0 -1 -1 }}), [[176/175]] ({{monzo| 4 -2 -1 1 }}), [[196/195]] ({{monzo| 2 -1 2 0 -1 }}


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


{{Mapping|legend=2| 1 0 2 | 0 1 -2 }}
{{Mapping|legend=3| 1 -5/4 3 -1/4 7/4 -1/4 | 0 -1/4 -3 3/4 -21/4 19/4 }}


{{Mapping|legend=3| 1 -1/6 5/6 0 0 -1/3 | 0 -1/2 -3/2 0 0 1 }}
: [[gencom]]: [2 7/6; 144/143 176/175 196/195]
 
: [[gencom]]: [2 6/5; 325/324]


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 886.886
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~12/7 = 930.598
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 886.861
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~12/7 = 930.700


{{Optimal ET sequence|legend=1| 3, 4, 11, 15, 19, 34, 53, 87, 140 }}
{{Optimal ET sequence|legend=1| 9, 31, 40, 49, 156c*†, 205c*† }}


[[Tp tuning #T2 tuning|RMS error]]: 0.2444 cents
<nowiki/>* wart for 7/3


== 2.….7/3… subgroups ==
† wart for 11/3
=== Guanyintet ===
{{See also | Chromatic pairs #Guanyintet }}


Guanyintet, the 4 &amp; 9 temperament in the 2.5.7/3.11/3 subgroup, is related to [[guanyin]] as well as to [[orwell]].  
[[Tp tuning #T2 tuning|RMS error]]: 0.8790 cents


[[Subgroup]]: 2.5.7/3.11/3
=== Kryptonite ===
{{See also| Chromatic pairs #Kryptonite }}


[[Comma list]]: [[176/175]], [[540/539]]
Kryptonite is related to [[krypton]].


{{Mapping|legend=2| 1 0 2 -2 | 0 3 -1 5 }}
[[Subgroup]]: 2.5.7/3.11/3.13/3


{{Mapping|legend=3| 1 -4/3 3 -1/3 5/3 | 0 4/3 -3 7/3 -11/3 }}
[[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 }})


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


[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~7/6 = 270.093
: mapping generators: ~2, ~13/12


{{Optimal ET sequence|legend=1| 9, 31, 40, 49, 89, 191bc, 227bc, 231bc, 271bc, 311bc, 316bcd }}
{{Mapping|legend=3| 1 -5/4 2 -1/4 3/4 3/4 | 0 -1/2 3 3/2 -3/2 1/2 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.6028 cents
: [[gencom]]: [2 13/12; 56/55 78/77 91/90]


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


Laz is related to [[georgian]] as well as to [[winston]].
{{Optimal ET sequence|legend=1| 1, …, 8, 9 }}


[[Subgroup]]: 2.5.7/3.11/3.13/3
[[Tp tuning #T2 tuning|RMS error]]: 2.545 cents


[[Comma list]]: [[144/143]], [[176/175]], [[196/195]]
=== Kiribati ===
{{See also| Chromatic pairs #Kiribati }}


{{Mapping|legend=2| 1 0 2 -2 6 | 0 3 -1 5 -5 }}
Kiribati is related to [[nakika]] as well as to [[octacot]].


{{Mapping|legend=3| 1 -5/4 3 -1/4 7/4 -1/4 | 0 -1/4 -3 3/4 -21/4 19/4 }}
[[Subgroup]]: 2.9/5.7/3.11/9


: [[gencom]]: [2 7/6; 144/143 176/175 196/195]
[[Comma list]]: 100/99 ({{monzo| 2 -2 0 -1 }}), 245/242 ({{monzo| -1 -1 2 -2 }})


[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~7/6 = 269.300
{{Mapping|legend=2| 1 1 1 0 | 0 -2 3 4 }}


{{Optimal ET sequence|legend=1| 9, 31, 40, 49, 58, 156bde, 205bde }}
: mapping generators: ~2, ~21/20


[[Tp tuning #T2 tuning|RMS error]]: 0.8790 cents
{{Mapping|legend=3| 1 1/10 -4/5 11/10 1/5 | 0 -3/2 -1 3/2 1 }}


=== Kryptonite ===
: [[gencom]]: [2 21/20; 100/99 245/242]
{{See also| Chromatic pairs #Kryptonite }}


Kryptonite is related to [[krypton]].  
[[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


[[Subgroup]]: 2.5.7/3.11/3.13/3
{{Optimal ET sequence|legend=1| 13, 14, 27, 41 }}


[[Comma list]]: 56/55, 78/77, 91/90
[[Tp tuning #T2 tuning|RMS error]]: 1.245 cents


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


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


: [[gencom]]: [2 13/12; 56/55 78/77 91/90]
[[Subgroup]]: 2.7/3.11


[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~13/12 = 132.428
[[Comma list]]: [[99/98]] ({{monzo| -1 -2 1 }})


{{Optimal ET sequence|legend=1| 9, 63, 82bd, 91bde }}
{{Mapping|legend=2| 1 0 1 | 0 1 2 }}


[[Tp tuning #T2 tuning|RMS error]]: 2.545 cents
: mapping generators: ~2, ~7/3


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


Kiribati is related to [[nakika]] as well as to [[octacot]].
: [[gencom]]: [2 7/6; 99/98]


[[Subgroup]]: 2.9/5.7/3.11/9
[[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


[[Comma list]]: 100/99, 245/242
{{Optimal ET sequence|legend=1| 4, 9, 13, 22, 79 }}


{{Mapping|legend=2| 1 1 1 0 | 0 -2 3 4 }}
[[Tp tuning #T2 tuning|RMS error]]: 1.064 cents


{{Mapping|legend=3| 1 1/10 -4/5 11/10 1/5 | 0 -3/2 -1 3/2 1 }}
== 2.….9/7… subgroups ==
=== Marveltri ===
{{See also| Chromatic pairs #Marveltri }}


: [[gencom]]: [2 21/20; 100/99 245/242]
Marveltri, the {{nowrap| 3 & 13 }} temperament in the 2.5.9/7 subgroup, is related to [[marvel]], [[magic]], and the unnamed {{nowrap| 22 & 47 }} temperament. The tonic and the first two generator steps make a [[marvel triad]], hence the name.


[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~21/20 = 87.892
[[Subgroup]]: 2.5.9/7


{{Optimal ET sequence|legend=1| 13, 14, 27, 41, 55, 191bd, 232bcd, 273bcd }}
[[Comma list]]: 225/224 ({{monzo| -5 2 1 }})


[[Tp tuning #T2 tuning|RMS error]]: 1.245 cents
{{Mapping|legend=2| 1 0 5 | 0 1 -2 }}


=== Mothwelltri ===
: mapping generators: ~2, ~5
{{See also| Chromatic pairs #Mothwelltri }}


Mothwelltri, the 1 &amp; 4 temperament in the 2.7/3.11 subgroup, is related to [[orwell]].
{{Mapping|legend=3| 1 2 0 -1 | 0 -4/5 1 2/5 }}


[[Subgroup]]: 2.7/3.11
: [[gencom]]: [2 5; 225/224]


[[Comma list]]: [[99/98]]
[[Optimal tuning]]s:  
 
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/4 = 384.208
{{Mapping|legend=2| 1 0 1 | 0 1 2 }}
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/4 = 383.638
 
{{Mapping|legend=3| 1 -1/2 0 1/2 3 | 0 -1/2 0 1/2 2 }}
 
: [[gencom]]: [2 7/6; 99/98]
 
[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~7/6 = 273.174
 
{{Optimal ET sequence|legend=1| 9, 22, 40, 49c, 58c, 67c, 76c, 79, 101b, 123bc }}
 
[[Tp tuning #T2 tuning|RMS error]]: 1.064 cents
 
== 2.….9/7… subgroups ==
=== Marveltri ===
{{See also| Chromatic pairs #Marveltri }}
 
Marveltri, the {{nowrap| 3 & 13 }} temperament in the 2.5.9/7 subgroup, is related to [[marvel]], [[magic]], and the unnamed {{nowrap| 22 & 47 }} temperament. The tonic and the first two generator steps make a [[marvel triad]], hence the name.
 
[[Subgroup]]: 2.5.9/7
 
[[Comma list]]: 225/224 ({{monzo| -5 2 1 }})
 
{{Mapping|legend=2| 1 0 5 | 0 1 -2 }}
 
: mapping generators: ~2, ~5
 
{{Mapping|legend=3| 1 2 0 -1 | 0 -4/5 1 2/5 }}


: [[gencom]]: [2 5; 225/224]
{{Optimal ET sequence|legend=1| 3, 13, 16, 19, 22, 25, 72, 97, 122, 269c* }}
 
[[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, 13, 16, 19, 22, 25, 72, 97, 122 }}
<nowiki/>* Wart for 9/7


[[Tp tuning #T2 tuning|RMS error]]: 0.4801 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.4801 cents
Line 1,267: Line 1,209:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* ([[Tp tuning|subgroup]] [[CTE]]): ~2 = 1200.000, ~5/4 = 386.617
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/4 = 386.617
* ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1200.000, ~5/4 = 386.558
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/4 = 386.558


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


[[Tp tuning #T2 tuning|RMS error]]: 1.074 cents
[[Tp tuning #T2 tuning|RMS error]]: 1.074 cents
== 2.….15/11… subgroups ==
=== Poggers ===
Related temperaments: [[Stearnsmic_clan#Pogo|pogo]], [[Stearnsmic_clan#Supers|supers]]
[[Subgroup]]: 2.9.7.15/11.13
[[Comma list]]: [[540/539]], [[1716/1715]], [[2080/2079]]
{{Mapping|legend=2| 1 1 1 -1 -1 | 0 6 5 4 13 }}
[[Optimal tuning]] (subgroup [[CTE]]): ~9/7 = 433.888
[[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


== 2.….7/5… subgroups ==
== 2.….7/5… subgroups ==
Line 1,357: Line 1,284:
{{distinguish|History (temperament)}}, which is the rank-3 version of this temperament in the full 13-limit.
{{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]].
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 1,757: Line 1,684:
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[[Category:Temperament collections]][[Category:Subgroup]]
[[Category:Temperament collections]]
[[Category:Pages with mostly numerical content]][[Category:Subgroup]]
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