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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]].  
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=== 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.
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{{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 ]
⟨ 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)


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


[[Subgroup]]: 4.3.5
==== 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)


[[Comma list]]: 3125/3072
Subsets
q99, q62, q37, q161, q136, q198, q25, q124, q74, q235
</pre>


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


[[Comma list]]: 256/243
Subsets
q37, q25, q62, q12, q74, q99, q87, q49r, q50r, q124
</pre>


{{Mapping|legend=2| 5 4 6 | 0 0 -1 }}
==== 4.6.5.7.11.13 ====


: Mapping generators: ~4, ~16/15
<pre>
 
Reduced Mapping
[[Optimal tuning]] ([[POTE]]): 1\5ed4 = 480.0, ~16/15 = 80.4062
4 6 5 7 11 13
 
[ ⟨ 1 0 1 1 1 0 ]
[[Support]]ing [[ET]]s: {{EDs|5, 10, 15, 20, 25, 30, 55, 85, 115|equave=4}}
⟨ 0 16 2 5 9 23 ] ⟩
 
== 4.6.5 subgroup ==
TE Generator Tunings (cents)
=== Meanquad ===
⟨2401.2305, 193.5378]
{{Main| Meanquad }}
 
TE Step Tunings (cents)
[[Subgroup]]: 4.6.5
⟨42.79107, 35.98524]
 
[[Comma list]]: [[81/80]] = {{monzo| -4 4 -1 }}
TE Tuning Map (cents)
 
⟨2401.230, 3096.606, 2788.306, 3368.920, 4143.071, 4451.371]
{{Mapping|legend=2| 1 0 -4| 0 1 4 }}
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)


: mapping generators: ~4, ~6
Subsets
q25, q37f, q12f, q62, q50rf, q13rff, q49rff, q87, q74ff, q24rfff
</pre>


[[Optimal tuning]] (subgroup [[CTE]]): ~4 = 2\1, ~3/2 = 697.214
==== 4.6.5.7.11.13.17 ====
 
<pre>
[[Support]]ing [[ET]]s: *7, *10, *11[-5], *13[+5], *17, *24, *27[+5], *31, *38, *41, *45, *52, *55, *69
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)


<nowiki />* Wart for 4
Subsets
q25, q12f, q37f, q13rffg, q50rf, q62, q49rffg, q24rfffg, q38rreffg, q74ffg
</pre>


==== 4.6.5.7 subgroup (tetrominant) ====
==== 4.6.5.7.11.13.17.19 ====
[[Subgroup]]: 4.6.5.7
<pre>
 
[[Comma list]]: [[36/35]] = {{monzo| 0 2 -1 -1 }}, [[64/63]] = {{monzo| 4 -2 0 -1 }}
 
{{Mapping|legend=2| 1 0 -4 4 | 0 1 4 -2 }}
 
[[Optimal tuning]] (subgroup [[CTE]]): ~4 = 2\1, ~3/2 = 699.622
 
[[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
Reduced Mapping
4 6 5
4 6 5 7 11 13 17 19
[ ⟨ 1 0 1 ]
[ ⟨ 1 0 1 1 1 0 1 1 ]
⟨ 0 16 2 ] ⟩
⟨ 0 16 2 5 9 23 13 14 ] ⟩
   
   
TE Generator Tunings (cents)
TE Generator Tunings (cents)
⟨2399.3973, 193.8643]
⟨2399.9219, 193.3952]
   
   
TE Step Tunings (cents)
TE Step Tunings (cents)
⟨25.21211, 47.81337]
⟨44.14256, 35.03670]
   
   
TE Tuning Map (cents)
TE Tuning Map (cents)
⟨2399.397, 3101.829, 2787.126]
⟨2399.922, 3094.324, 2786.712, 3366.898, 4140.479, 4448.090, 4914.060, 5107.455]
   
   
TE Mistunings (cents)
TE Mistunings (cents)
⟨-0.603, -0.126, 0.812]
⟨-0.078, -7.631, 0.399, -1.928, -10.839, 7.562, 9.104, 9.942]
   
   
Complexity 1.369085
Complexity 1.058472
Adjusted Error 0.692892 cents
Adjusted Error 8.712222 cents
TE Error 0.268047 cents/octave
TE Error 2.050935 cents/octave
   
   
Unison Vector
Unison Vectors
[8, 1, -8⟩ (393216:390625)
[0, 1, -1, 0, 1, -1, 0, 0⟩ (66:65)
 
[-1, 0, 0, 1, 1, 0, 0, -1⟩ (77:76)
Subsets
[2, 1, -1, 0, 0, 0, 0, -1⟩ (96:95)
q99, q62, q37, q161, q136, q198, q25, q124, q74, q235
[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>
</pre>


==== 4.6.5.7 ====
==== 4.6.5.7.11.13.17.19.23 ====
<pre>
<pre>
Reduced Mapping
Reduced Mapping
4 6 5 7
4 6 5 7 11 13 17 19 23
[ ⟨ 1 0 1 1 ]
[ ⟨ 1 0 1 1 1 0 1 1 0 ]
⟨ 0 16 2 5 ] ⟩
⟨ 0 16 2 5 9 23 13 14 28 ] ⟩
   
   
TE Generator Tunings (cents)
TE Generator Tunings (cents)
⟨2399.4195, 193.8654]
⟨2399.3286, 193.5316]
   
   
TE Step Tunings (cents)
TE Step Tunings (cents)
⟨25.23883, 47.79592]
⟨37.31613, 39.63311]
   
   
TE Tuning Map (cents)
TE Tuning Map (cents)
⟨2399.420, 3101.846, 2787.150, 3368.747]
⟨2399.329, 3096.506, 2786.392, 3366.987, 4141.113, 4451.227, 4915.240, 5108.771, 5418.885]
   
   
TE Mistunings (cents)
TE Mistunings (cents)
⟨-0.580, -0.109, 0.837, -0.079]
⟨-0.671, -5.449, 0.078, -1.839, -10.205, 10.699, 10.284, 11.258, -9.389]
   
   
Complexity 1.192044
Complexity 1.115920
Adjusted Error 0.653313 cents
Adjusted Error 9.502017 cents
TE Error 0.232715 cents/octave
TE Error 2.100561 cents/octave
   
   
Unison Vectors
Unison Vectors
[-2, -1, -2, 4⟩ (2401:2400)
[0, 1, -1, 0, 1, -1, 0, 0, 0⟩ (66:65)
[3, 0, -5, 2⟩ (3136:3125)
[1, 0, 0, -1, 0, -1, 0, 0, 1⟩ (92:91)
[5, 1, -3, -2⟩ (6144:6125)
[0, -1, 1, 0, 0, 0, 0, -1, 1⟩ (115:114)
[8, 1, -8, 0⟩ (393216:390625)
[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
Subsets
q99, q62, q37, q161, q136, q198, q25, q124, q74, q235
q25i, q12fhi, q37f, q13rffghii, q62, q50rfii, q49rffghii, q24rfffghhiii, q74ffghi, q38rreffghiii
</pre>
</pre>


==== 4.6.5.7.11 ====
== 4.9.25 subgroup ==
<pre>
=== Meansquared ===
Reduced Mapping
[[Subgroup]]: 4.9.25
4 6 5 7 11
 
[ ⟨ 1 0 1 1 1 ]
[[Comma list]]: [[6561/6400]]
⟨ 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
{{Mapping|legend=2| 1 3 4 | 0 1 4 }}
q37, q25, q62, q12, q74, q99, q87, q49r, q50r, q124
 
</pre>
Mapping generators: ~4, ~9/64
 
[[Optimal tuning]] ([[CTE]]): ~4 = 2\1, ~9/4 = 1394.429
 
[[Support]]ing [[ET]]s: 12, 7, 19, 5, 31, 26, 17[+25], 43, 9[-25], 33[-25], 45, 29[+25], 8[+25], 22[+25]
 
== 4.9.49 subgroup ==
=== Archsquared ===
[[Subgroup]]: 4.9.49


==== 4.6.5.7.11.13 ====
[[Comma list]]: 4096/3969


<pre>
{{Mapping|legend=2| 1 3 0 | 0 1 -2 }}
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
Mapping generators: ~4, ~9/64
q25, q37f, q12f, q62, q50rf, q13rff, q49rff, q87, q74ff, q24rfff
</pre>


==== 4.6.5.7.11.13.17 ====
[[Optimal tuning]] ([[CTE]]): ~[[9/8]] = 219.190
<pre>
 
Reduced Mapping
[[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
 
[ ⟨ 1 0 1 1 1 0 1 ]
== 8.9.7 subgroup ==
⟨ 0 16 2 5 9 23 13 ] ⟩
=== 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."
TE Generator Tunings (cents)
 
⟨2400.4701, 193.4599]
[[Subgroup]]: 8.9.7
 
TE Step Tunings (cents)
[[Comma list]]: 64/63
⟨43.39350, 35.55764]
 
{{Mapping|legend=2| 1 0 2 | 0 1 -1 }}
TE Tuning Map (cents)
 
⟨2400.470, 3095.359, 2787.390, 3367.770, 4141.609, 4449.578, 4915.449]
: sval mapping generators: ~8, ~9
 
TE Mistunings (cents)
: [[gencom]]: [8 9/8; 64/63]
⟨0.470, -6.596, 1.076, -1.056, -9.709, 9.050, 10.494]
 
[[Optimal tuning]] ([[CTE]]): 1\[[3ed8]] = 1600.0, ~9/8 = 219.1898
Complexity 1.129881
Adjusted Error 8.082725 cents
TE Error 1.977443 cents/octave
Unison Vectors
[0, 1, -1, 0, 1, -1, 0⟩ (66:65)
[1, 1, 1, -1, 0, 0, -1⟩ (120:119)
[1, 2, 0, 0, -1, -1, 0⟩ (144:143)
[-2, 1, 1, 1, 0, -1, 0⟩ (105:104)
[-1, 2, 2, 0, 0, -1, -1⟩ (225:221)
[-1, 1, 2, -2, 0, -1, 1⟩ (1275:1274)


Subsets
[[Optimal ET sequence]]: {{val| 16 17 15 }}, {{val| 33 35 31 }}, {{val| 148 … }}, {{val| 181 … }}, {{val| 214 … }}, {{val| 247 … }}
q25, q12f, q37f, q13rffg, q50rf, q62, q49rffg, q24rfffg, q38rreffg, q74ffg
</pre>


==== 4.6.5.7.11.13.17.19 ====
[[Badness]]: 0.0215 × 10<sup>-3</sup>
<pre>
 
Reduced Mapping
= Fractional subgroup temperaments =
4 6 5 7 11 13 17 19
== 2.5/3… subgroups ==
[ ⟨ 1 0 1 1 1 0 1 1 ]
=== Magicaltet ===
⟨ 0 16 2 5 9 23 13 14 ] ⟩
{{See also| Chromatic pairs #Magicaltet }}
 
TE Generator Tunings (cents)
Magicaltet is related to [[supermagic]], [[superkleismic]], and [[magic]]. The tonic and the first three generator steps make a [[magical seventh chord]], hence the name.  
⟨2399.9219, 193.3952]
 
[[Subgroup]]: 2.5/3.7.11
TE Step Tunings (cents)
 
⟨44.14256, 35.03670]
[[Comma list]]: 100/99 ({{monzo| 2 2 0 -1 }}), 385/384 ({{monzo| -7 1 1 1 }})
 
TE Tuning Map (cents)
{{Mapping|legend=2| 1 0 5 2 | 0 1 -3 2 }}
⟨2399.922, 3094.324, 2786.712, 3366.898, 4140.479, 4448.090, 4914.060, 5107.455]
 
: mapping generators: ~2, ~5/3
TE Mistunings (cents)
 
⟨-0.078, -7.631, 0.399, -1.928, -10.839, 7.562, 9.104, 9.942]
{{Mapping|legend=3| 1 -1/2 1/2 2 4 | 0 1/2 -1/2 3 -2 }}
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
: [[gencom]]: [2 6/5; 100/99 385/384]
q25, q12fh, q37f, q13rffgh, q50rf, q62, q49rffgh, q24rfffghh, q38rreffgh, q74ffgh
</pre>


==== 4.6.5.7.11.13.17.19.23 ====
[[Optimal tuning]]s:
<pre>
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 877.343
Reduced Mapping
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 877.351
4 6 5 7 11 13 17 19 23
 
[ ⟨ 1 0 1 1 1 0 1 1 0 ]
{{Optimal ET sequence|legend=1| 4, 7, 11, 15, 26, 67, 93* }}
⟨ 0 16 2 5 9 23 13 14 28 ] ⟩
 
<nowiki/>* Wart for 5/3
TE Generator Tunings (cents)
 
⟨2399.3286, 193.5316]
[[Tp tuning #T2 tuning|RMS error]]: 1.206 cents
 
TE Step Tunings (cents)
=== Starlingtet ===
⟨37.31613, 39.63311]
{{See also | Chromatic pairs #Starlingtet }}
 
TE Tuning Map (cents)
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.  
⟨2399.329, 3096.506, 2786.392, 3366.987, 4141.113, 4451.227, 4915.240, 5108.771, 5418.885]
 
[[Subgroup]]: 2.5/3.7/3
TE Mistunings (cents)
 
⟨-0.671, -5.449, 0.078, -1.839, -10.205, 10.699, 10.284, 11.258, -9.389]
[[Comma list]]: [[126/125]] ({{monzo| 1 -3 1 }})
 
Complexity 1.115920
{{Mapping|legend=2| 1 0 -1 | 0 1 3 }}
Adjusted Error 9.502017 cents
TE Error 2.100561 cents/octave
Unison Vectors
[0, 1, -1, 0, 1, -1, 0, 0, 0⟩ (66:65)
[1, 0, 0, -1, 0, -1, 0, 0, 1⟩ (92:91)
[0, -1, 1, 0, 0, 0, 0, -1, 1⟩ (115:114)
[1, 1, 1, -1, 0, 0, -1, 0, 0⟩ (120:119)
[2, 0, -2, -1, 1, 0, 0, 0, 0⟩ (176:175)
[-3, -1, 1, 1, 1, 0, 0, 0, 0⟩ (385:384)
[1, 0, -2, 1, 0, 0, 1, -1, 0⟩ (476:475)
[1, 0, 0, -2, 1, 0, -1, 1, 0⟩ (836:833)
[0, 0, 1, -2, 1, 0, 1, -1, 0⟩ (935:931)
[1, -1, 0, 0, 0, 0, -2, 1, 1⟩ (874:867)


Subsets
: mapping generators: ~2, ~5/3
q25i, q12fhi, q37f, q13rffghii, q62, q50rfii, q49rffghii, q24rfffghhiii, q74ffghi, q38rreffghiii
</pre>


== 4.9.25 subgroup ==
{{Mapping|legend=3| 1 -1 0 1 | 0 4/3 1/3 -5/3 }}
=== Meansquared ===
[[Subgroup]]: 4.9.25


[[Comma list]]: [[6561/6400]]
: [[gencom]]: [2 6/5; 126/125]


{{Mapping|legend=2| 1 3 4 | 0 1 4 }}
[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 888.759
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 888.846


Mapping generators: ~4, ~9/64
{{Optimal ET sequence|legend=1| 4, 15, 19, 23, 27 }}


[[Optimal tuning]] ([[CTE]]): ~4 = 2\1, ~9/4 = 1394.429
[[Tp tuning #T2 tuning|RMS error]]: 0.8398 cents


[[Support]]ing [[ET]]s: 12, 7, 19, 5, 31, 26, 17[+25], 43, 9[-25], 33[-25], 45, 29[+25], 8[+25], 22[+25]
==== Greeley ====
{{See also| Chromatic pairs #Greeley }}


== 4.9.49 subgroup ==
Greeley is related to [[opossum]] as well as to [[nusecond]].  
=== Archsquared ===
[[Subgroup]]: 4.9.49


[[Comma list]]: 4096/3969
[[Subgroup]]: 2.5/3.7/3.11/3


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


Mapping generators: ~4, ~9/64
{{Mapping|legend=2| 1 1 2 2 | 0 -2 -6 -1 }}


[[Optimal tuning]] ([[CTE]]): ~[[9/8]] = 219.190
{{Mapping|legend=3| 1 -5/4 -1/4 3/4 3/4 | 0 9/4 1/4 -15/4 5/4 }}


[[Support]]ing [[ET]]s: 5, 17, 22, 12, 7, 27, 32, 8, 39[+49], 29[+49], 9[+49], 19[+49], 37, 49
: [[gencom]]: [2 11/10; 121/120 126/125]


== 8.9.7 subgroup ==
[[Optimal tuning]]s:
=== Sixscared ===
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~11/10 = 155.696
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."
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~11/10 = 155.776


[[Subgroup]]: 8.9.7
{{Optimal ET sequence|legend=1| 8, 15, 23, 54, 77, 100, 131* }}


[[Comma list]]: 64/63
<nowiki/>* Wart for 11/3


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


: sval mapping generators: ~8, ~9
==== Skateboard ====
{{See also| Chromatic pairs #Skateboard }}


: [[gencom]]: [8 9/8; 64/63]
Skateboard is related to [[thrasher]].


[[Optimal tuning]] ([[CTE]]): 1\[[3ed8]] = 1600.0, ~9/8 = 219.1898
[[Subgroup]]: 2.5/3.7/3.11.13/9


[[Optimal ET sequence]]: {{val| 16 17 15 }}, {{val| 33 35 31 }}, {{val| 148 … }}, {{val| 181 … }}, {{val| 214 … }}, {{val| 247 … }}
[[Comma list]]: 56/55 ({{monzo| 3 -1 1 -1 }}), 91/90 ({{monzo| -1 -1 1 0 1 }}), 100/99 ({{monzo| 2 2 0 -1 }})


[[Badness]]: 0.0215 × 10<sup>-3</sup>
{{Mapping|legend=2| 1 0 -1 2 2 | 0 1 3 2 -2 }}


= Fractional subgroup temperaments =
{{Mapping|legend=3| 1 -3/7 4/7 11/7 4 -6/7 | 0 0 -1 -3 -2 2 }}
== 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.
: [[gencom]]: [2 6/5; 56/55 91/90 100/99]


[[Subgroup]]: 2.5/3.7.11
[[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


[[Comma list]]: 100/99 = {{monzo| 2 2 0 -1 }}, 385/384 = {{monzo| -7 1 1 1 }}
{{Optimal ET sequence|legend=1| 11, 15, 19, 23, 42d, 65d }}


{{Mapping|legend=2| 1 0 5 2 | 0 1 -3 2 }}
[[Tp tuning #T2 tuning|RMS error]]: 2.396 cents


: mapping generators: ~2, ~5/3
=== Gariberttet ===
Gariberttet is the 2.5/3.7/3 [[Subgroup temperament families, relationships, and genes|altergene]] of [[sirius]].


{{Mapping|legend=3| 1 -1/2 1/2 2 4 | 0 1/2 -1/2 3 -2 }}
==== Gariberttet (2.5/3.7/3.13/11 subgroup) ====
{{See also | Chromatic pairs #Gariberttet }}


: [[gencom]]: [2 6/5; 100/99 385/384]
Gariberttet can be described as the {{nowrap| 4 & 29 }} temperament in the 2.5/3.7/3.13/11 subgroup.


[[Optimal tuning]]s:
[[Subgroup]]: 2.5/3.7/3.13/11
* [[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* }}
[[Comma list]]: [[275/273]] ({{monzo| 0 2 -1 -1 }}), [[847/845]] ({{monzo| 0 -1 1 -2 }})


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


[[Tp tuning #T2 tuning|RMS error]]: 1.206 cents
{{Mapping|legend=3| 1 0 0 0 0 0 | 0 -8/3 1/3 7/3 -1/2 1/2 }}


=== Starlingtet ===
: [[gencom]]: [2 13/11; 275/273 847/845]
{{See also | Chromatic pairs #Starlingtet }}


Starlingtet, the 4 &amp; 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.  
[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]] and [[POTE]]: ~2 = 1200.000, ~13/11 = 293.679


[[Subgroup]]: 2.5/3.7/3
{{Optimal ET sequence|legend=1| 29, 33, 37, 41, 45, 49, 78, 94, 143* }}


[[Comma list]]: [[126/125]] = {{monzo| 1 -3 1 }}
<nowiki/>* Wart for 13/11


{{Mapping|legend=2| 1 0 -1 | 0 1 3 }}
[[Tp tuning #T2 tuning|RMS error]]: 0.6914 cents


: mapping generators: ~2, ~5/3
==== Indium ====
{{See also | Chromatic pairs #Indium }}


{{Mapping|legend=3| 1 -1 0 1 | 0 4/3 1/3 -5/3 }}
Indium can be described as the {{nowrap| 8 & 33 }} temperament in the 2.5/3.7/3.11/3 subgroup.


: [[gencom]]: [2 6/5; 126/125]
[[Subgroup]]: 2.5/3.7/3.11/3


[[Optimal tuning]]s:  
[[Comma list]]: [[3025/3024]] ({{monzo| -4 2 -1 2 }}), [[3125/3087]] ({{monzo| 0 5 -3 }})
* [[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 }}
{{Mapping|legend=2| 1 0 0 2 | 0 6 10 -1 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.8398 cents
{{Mapping|legend=3| 1 -1/2 -1/2 -1/2 3/2 | 0 -15/4 9/4 25/4 -19/4 }}


==== Greeley ====
: [[gencom]]: [2 12/11; 3025/3024 3125/3087]
{{See also| Chromatic pairs #Greeley }}


Greeley is related to [[opossum]] as well as to [[nusecond]].  
[[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


[[Subgroup]]: 2.5/3.7/3.11/3
{{Optimal ET sequence|legend=1| 8, 33, 41, 49, 204*<sup>†</sup> }}


[[Comma list]]: 121/120 = {{monzo| -3 -1 0 2 }}, 126/125 = {{monzo| 1 -3 1 }}
<nowiki/>* Wart for 7/3


{{Mapping|legend=2| 1 1 2 2 | 0 -2 -6 -1 }}
<sup>†</sup> Wart for 11/3


{{Mapping|legend=3| 1 -5/4 -1/4 3/4 3/4 | 0 9/4 1/4 -15/4 5/4 }}
[[Tp tuning #T2 tuning|RMS error]]: 0.7788 cents
 
==== Ammon ====
{{See also| Chromatic pairs #Ammon }}


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


[[Optimal tuning]]s:  
[[Subgroup]]: 2.5/3.7/3.11/3.13/3
* [[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* }}
[[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 }})


<nowiki />* Wart for 11/3
{{Mapping|legend=2| 1 3 5 3 4 | 0 -6 -10 -3 -5 }}


[[Tp tuning#T2 tuning|RMS error]]: 1.034 cents
{{Mapping|legend=3| 1 -3 0 2 0 1 | 0 24/5 -6/5 -26/5 9/5 -1/5 }}


==== Skateboard ====
: [[gencom]]: [2 13/10; 121/120 169/168 275/273]
{{See also| Chromatic pairs #Skateboard }}


Skateboard is related to [[thrasher]].  
[[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


[[Subgroup]]: 2.5/3.7/3.11.13/9
{{Optimal ET sequence|legend=1| 8, 29, 37, 45 }}


[[Comma list]]: 56/55 = {{monzo| 3 -1 1 -1 }}, 91/90 = {{monzo| -1 -1 1 0 1 }}, 100/99 = {{monzo| 2 2 0 -1 }}
[[Tp tuning #T2 tuning|RMS error]]: 1.052 cents


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


{{Mapping|legend=3| 1 -3/7 4/7 11/7 4 -6/7 | 0 0 -1 -3 -2 2 }}
Sentry, the {{nowrap| 3 & 5 }} temperament in the 2.5/3.9/7 subgroup, is related to [[sensi]].


: [[gencom]]: [2 6/5; 56/55 91/90 100/99]
[[Subgroup]]: 2.5/3.9/7


[[Optimal tuning]]s:  
[[Comma list]]: [[245/243]] ({{monzo| 0 1 -2 }})
* [[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 }}
{{Mapping|legend=2| 1 0 0 | 0 2 1 }}


[[Tp tuning #T2 tuning|RMS error]]: 2.396 cents
{{Mapping|legend=3| 1 0 0 0 | 0 0 2 -1 }}


=== Gariberttet ===
: [[gencom]]: [2 9/7; 245/243]
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) ====
[[Optimal tuning]]s:
{{See also | Chromatic pairs #Gariberttet }}
* [[Tp tuning|subgroup]] [[CTE]] and [[POTE]]: ~2 = 1200.000, ~9/7 = 440.902


Gariberttet can be described as the 4 &amp; 29 temperament in the 2.5/3.7/3.13/11 subgroup.
{{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> }}


[[Subgroup]]: 2.5/3.7/3.13/11
<nowiki/>* Wart for 5/3


[[Comma list]]: [[275/273]] = {{monzo| 0 2 -1 -1 }}, [[847/845]] = {{monzo| 0 -1 1 -2 }}
<sup>†</sup> Wart for 9/7


{{Mapping|legend=2| 1 0 0 0 | 0 3 5 1 }}
[[Tp tuning #T2 tuning|RMS error]]: 0.7105 cents


{{Mapping|legend=3| 1 0 0 0 0 0 | 0 -8/3 1/3 7/3 -1/2 1/2 }}
=== Marveltwintri ===
{{See also| Chromatic pairs #Marveltwintri }}


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


[[Optimal tuning]]s:  
[[Subgroup]]: 2.5/3.13/9
* [[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* }}
[[Comma list]]: [[325/324]] ({{monzo| -2 2 1 }})


<nowiki />* Wart for 13/11
{{Mapping|legend=2| 1 0 2 | 0 1 -2 }}


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


==== Indium ====
: [[gencom]]: [2 6/5; 325/324]
{{See also | Chromatic pairs #Indium }}


Indium can be described as the 8 &amp; 33 temperament in the 2.5/3.7/3.11/3 subgroup.  
[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 886.886
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 886.861


[[Subgroup]]: 2.5/3.7/3.11/3
{{Optimal ET sequence|legend=1| 3, 4, 11, 15, 19, 34, 53, 87, 140 }}


[[Comma list]]: [[3025/3024]] = {{monzo| -4 2 -1 2 }}, [[3125/3087]] = {{monzo| 0 5 -3 }}
[[Tp tuning #T2 tuning|RMS error]]: 0.2444 cents


{{Mapping|legend=2| 1 0 0 2 | 0 6 10 -1 }}
== 2.….7/3… subgroups ==
=== Guanyintet ===
{{See also | Chromatic pairs #Guanyintet }}


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


: [[gencom]]: [2 12/11; 3025/3024 3125/3087]
[[Subgroup]]: 2.5.7/3.11/3


[[Optimal tuning]]s:  
[[Comma list]]: [[176/175]] ({{monzo| 4 -2 -1 1 }}), [[540/539]] ({{monzo| 2 1 -2 -1 }})
* [[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> }}
{{Mapping|legend=2| 1 0 2 -2 | 0 3 -1 5 }}


<nowiki />* Wart for 7/3
: mapping generators: ~2, ~12/7


<sup>†</sup> Wart for 11/3
{{Mapping|legend=3| 1 -4/3 3 -1/3 5/3 | 0 4/3 -3 7/3 -11/3 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.7788 cents
: [[gencom]]: [2 7/6; 176/175 540/539]


==== Semidim ====
[[Optimal tuning]]s:
{{See also| Chromatic pairs #Semidim }}
* ([[Tp tuning|subgroup]] [[CTE]]): ~2 = 1200.000, ~12/7 = 929.545
* ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1200.000, ~12/7 = 929.907


Semidim can be described as the 8 &amp; 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.
{{Optimal ET sequence|legend=1| 9, 22, 31, 40, 191c*, 231c*, 271c*, 311c* }}


[[Subgroup]]: 2.5/3.7/3.11/3.13/3
<nowiki/>* wart for 7/3


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


{{Mapping|legend=2| 1 3 5 3 4 | 0 -6 -10 -3 -5 }}
==== Laz ====
{{See also | Chromatic pairs #Laz }}


{{Mapping|legend=3| 1 -3 0 2 0 1 | 0 24/5 -6/5 -26/5 9/5 -1/5 }}
Laz is related to [[avalokita]] as well as to [[winston]].


: [[gencom]]: [2 13/10; 121/120 169/168 275/273]
[[Subgroup]]: 2.5.7/3.11/3.13/3


[[Optimal tuning]]s:  
[[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 }}
* [[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 }}
{{Mapping|legend=2| 1 0 2 -2 6 | 0 3 -1 5 -5 }}


[[Tp tuning #T2 tuning|RMS error]]: 1.052 cents
{{Mapping|legend=3| 1 -5/4 3 -1/4 7/4 -1/4 | 0 -1/4 -3 3/4 -21/4 19/4 }}


=== Sentry ===
: [[gencom]]: [2 7/6; 144/143 176/175 196/195]
{{See also | Chromatic pairs #Sentry }}


Sentry, the 3 &amp; 5 temperament in the 2.5/3.9/7 subgroup, is related to [[sensi]].  
[[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


[[Subgroup]]: 2.5/3.9/7
{{Optimal ET sequence|legend=1| 9, 31, 40, 49, 156c*†, 205c*† }}


[[Comma list]]: [[245/243]] = {{monzo| 0 1 -2 }}
<nowiki/>* wart for 7/3


{{Mapping|legend=2| 1 0 0 | 0 2 1 }}
† wart for 11/3


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


: [[gencom]]: [2 9/7; 245/243]
=== Kryptonite ===
{{See also| Chromatic pairs #Kryptonite }}


[[Optimal tuning]]s:
Kryptonite is related to [[krypton]].  
* [[Tp tuning|subgroup]] [[CTE]] and [[POTE]]: ~2 = 1\1, ~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> }}
[[Subgroup]]: 2.5.7/3.11/3.13/3


<nowiki />* Wart for 5/3
[[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 }})


<sup>†</sup> Wart for 9/7
{{Mapping|legend=2| 1 2 1 2 2 | 0 3 2 -1 1 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.7105 cents
: mapping generators: ~2, ~13/12


=== Marveltwintri ===
{{Mapping|legend=3| 1 -5/4 2 -1/4 3/4 3/4 | 0 -1/2 3 3/2 -3/2 1/2 }}
{{See also| Chromatic pairs #Marveltwintri }}


Marveltwintri can be described as the 3 &amp; 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.
: [[gencom]]: [2 13/12; 56/55 78/77 91/90]


[[Subgroup]]: 2.5/3.13/9
[[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


[[Comma list]]: [[325/324]] = {{monzo| -2 2 1 }}
{{Optimal ET sequence|legend=1| 1, …, 8, 9 }}


{{Mapping|legend=2| 1 0 2 | 0 1 -2 }}
[[Tp tuning #T2 tuning|RMS error]]: 2.545 cents


{{Mapping|legend=3| 1 -1/6 5/6 0 0 -1/3 | 0 -1/2 -3/2 0 0 1 }}
=== Kiribati ===
{{See also| Chromatic pairs #Kiribati }}


: [[gencom]]: [2 6/5; 325/324]
Kiribati is related to [[nakika]] as well as to [[octacot]].


[[Optimal tuning]]s:  
[[Subgroup]]: 2.9/5.7/3.11/9
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 886.886
* [[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 }}
[[Comma list]]: 100/99 ({{monzo| 2 -2 0 -1 }}), 245/242 ({{monzo| -1 -1 2 -2 }})


[[Tp tuning #T2 tuning|RMS error]]: 0.2444 cents
{{Mapping|legend=2| 1 1 1 0 | 0 -2 3 4 }}


== 2.….7/3… subgroups ==
: mapping generators: ~2, ~21/20
=== 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]].
{{Mapping|legend=3| 1 1/10 -4/5 11/10 1/5 | 0 -3/2 -1 3/2 1 }}


[[Subgroup]]: 2.5.7/3.11/3
: [[gencom]]: [2 21/20; 100/99 245/242]


[[Comma list]]: [[176/175]], [[540/539]]
[[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


{{Mapping|legend=2| 1 0 2 -2 | 0 3 -1 5 }}
{{Optimal ET sequence|legend=1| 13, 14, 27, 41 }}


{{Mapping|legend=3| 1 -4/3 3 -1/3 5/3 | 0 4/3 -3 7/3 -11/3 }}
[[Tp tuning #T2 tuning|RMS error]]: 1.245 cents


: [[gencom]]: [2 7/6; 176/175 540/539]
=== Mothwelltri ===
{{See also| Chromatic pairs #Mothwelltri }}


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


{{Optimal ET sequence|legend=1| 9, 31, 40, 49, 89, 191bc, 227bc, 231bc, 271bc, 311bc, 316bcd }}
[[Subgroup]]: 2.7/3.11


[[Tp tuning #T2 tuning|RMS error]]: 0.6028 cents
[[Comma list]]: [[99/98]] ({{monzo| -1 -2 1 }})


==== Laz ====
{{Mapping|legend=2| 1 0 1 | 0 1 2 }}
{{See also | Chromatic pairs #Laz }}


Laz is related to [[georgian]] as well as to [[winston]].
: mapping generators: ~2, ~7/3


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


[[Comma list]]: [[144/143]], [[176/175]], [[196/195]]
: [[gencom]]: [2 7/6; 99/98]


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


{{Mapping|legend=3| 1 -5/4 3 -1/4 7/4 -1/4 | 0 -1/4 -3 3/4 -21/4 19/4 }}
{{Optimal ET sequence|legend=1| 4, 9, 13, 22, 79 }}


: [[gencom]]: [2 7/6; 144/143 176/175 196/195]
[[Tp tuning #T2 tuning|RMS error]]: 1.064 cents


[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~7/6 = 269.300
== 2.….9/7… subgroups ==
=== Marveltri ===
{{See also| Chromatic pairs #Marveltri }}


{{Optimal ET sequence|legend=1| 9, 31, 40, 49, 58, 156bde, 205bde }}
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.


[[Tp tuning #T2 tuning|RMS error]]: 0.8790 cents
[[Subgroup]]: 2.5.9/7


=== Kryptonite ===
[[Comma list]]: 225/224 ({{monzo| -5 2 1 }})
{{See also| Chromatic pairs #Kryptonite }}


Kryptonite is related to [[krypton]].
{{Mapping|legend=2| 1 0 5 | 0 1 -2 }}


[[Subgroup]]: 2.5.7/3.11/3.13/3
: mapping generators: ~2, ~5


[[Comma list]]: 56/55, 78/77, 91/90
{{Mapping|legend=3| 1 2 0 -1 | 0 -4/5 1 2/5 }}


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


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


: [[gencom]]: [2 13/12; 56/55 78/77 91/90]
{{Optimal ET sequence|legend=1| 3, 13, 16, 19, 22, 25, 72, 97, 122, 269c* }}


[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~13/12 = 132.428
<nowiki/>* Wart for 9/7


{{Optimal ET sequence|legend=1| 9, 63, 82bd, 91bde }}
[[Tp tuning #T2 tuning|RMS error]]: 0.4801 cents


[[Tp tuning #T2 tuning|RMS error]]: 2.545 cents
==== Sulis ====
Sulis is related to [[minerva]] and [[würschmidt]].  


=== Kiribati ===
[[Subgroup]]: 2.5.9/7.11/9
{{See also| Chromatic pairs #Kiribati }}


Kiribati is related to [[nakika]] as well as to [[octacot]].
[[Comma list]]: 99/98 ({{monzo| -1 0 2 1 }}), 176/175 ({{monzo| 4 -2 1 1 }})


[[Subgroup]]: 2.9/5.7/3.11/9
{{Mapping|legend=2| 1 0 5 -9 | 0 1 -2 4 }}]


[[Comma list]]: 100/99, 245/242
[[Optimal tuning]]s:  
 
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/4 = 386.617
{{Mapping|legend=2| 1 1 1 0 | 0 -2 3 4 }}
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/4 = 386.558
 
{{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]
 
[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~21/20 = 87.892
 
{{Optimal ET sequence|legend=1| 13, 14, 27, 41, 55, 191bd, 232bcd, 273bcd }}
 
[[Tp tuning #T2 tuning|RMS error]]: 1.245 cents
 
=== Mothwelltri ===
{{See also| Chromatic pairs #Mothwelltri }}
 
Mothwelltri, the 1 &amp; 4 temperament in the 2.7/3.11 subgroup, is related to [[orwell]].
 
[[Subgroup]]: 2.7/3.11
 
[[Comma list]]: [[99/98]]
 
{{Mapping|legend=2| 1 0 1 | 0 1 2 }}
 
{{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 tuning]] ([[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 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.4801 cents
{{Optimal ET sequence|legend=1| 3, …, 22, 25, 28, 31, 59 }}
 
==== Sulis ====
Related temperament: [[marvel family|minerva]], [[würschmidt family|würschmidt]]
 
[[Subgroup]]: 2.5.9/7.11/9
 
[[Comma list]]: 99/98 ({{monzo| -1 0 2 1 }}), 176/175 ({{monzo| 4 -2 1 1 }})
 
{{Mapping|legend=2| 1 0 5 -9 | 0 1 -2 4 }}]
 
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~2 = 1200.000, ~5/4 = 386.558
 
{{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 1,354: 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 1,754: 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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