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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.


[[Subgroup]]: 2.9.5.7
[[Subgroup]]: 2.9.5.7
Line 152: Line 137:


{{Optimal ET sequence|legend=1| 6, 23def, 29f, 35, 41, 47 }}
{{Optimal ET sequence|legend=1| 6, 23def, 29f, 35, 41, 47 }}
== 2.3.25 subgroup ==
=== Shrub ===
This is a restriction of diaschismic which omits the tritone to produce a diatonic scale. True to its name, it generates a [[shrubmajor]] third (~425c) in quarter-comma tuning.
Subgroup: 2.3.25
Edo join: 17 & 12
Comma list: [[2048/2025]]
{{Mapping|legend=2| 1 1 7| 0 1 -4}}
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 705.136
==== 2.3.23.25.41 subgroup ====
''See also: [[Reversed meantone]]''
Edo join: 17 & 12
Comma list: 2048/2025, 576/575, 82/81
{{Mapping|legend=2| 1 1 1 7 3| 0 1 6 -4 4}}
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 705.264
===== Sburb =====
This temperament sets the [[octave reduction|octave-reduced]] 413th harmonic (413/256, 827.998{{c}}) to the diminished seventh.
Subgroup: 2.3.7.23.25.41.59
Edo join: 17 & 12
Comma list: 64/63, 225/224, 162/161, 82/81, 177/175
{{Mapping|legend=2| 1 1 4 1 7 3 10| 0 1 -2 6 -4 4 -7}}
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 706.387


== 2.9.5.11 subgroup ==
== 2.9.5.11 subgroup ==
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Scales: [[penta5]], [[penta8]], [[penta11]], [[penta19]]
Scales: [[penta5]], [[penta8]], [[penta11]], [[penta19]]
== 2.9.7.13.17 subgroup ==
=== Novisept ===
Novisept is generated by a one-cent-flat 9/7, such that stacking 5 of them gives you 7/4. It can be formed by doubling both generator and period of [[gizzard]].
[[Subgroup]]: 2.9.7.13.17
[[Comma list]]: 729/728, 442/441, 833/832
{{Mapping|legend=2| 1 1 1 -1 3| 0 6 5 13 3 }}
[[Optimal tuning]] ([[CWE]]): ~2 = 1\1, ~9/7 = 433.836
Badness (Dirichlet): 0.142


== 2.9.11 subgroup ==
== 2.9.11 subgroup ==
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{{Optimal ET sequence|legend=1| 13, 18, 31, 44 }}
{{Optimal ET sequence|legend=1| 13, 18, 31, 44 }}


==== 2.9.5.21.11 ====
===== 2.9.5.21.11 =====
Subgroup: 2.9.5.21.11
Subgroup: 2.9.5.21.11


Line 437: Line 476:
Optimal tuning (subgroup POTE): ~2 = 1\1, ~21/16 = 468.918
Optimal tuning (subgroup POTE): ~2 = 1\1, ~21/16 = 468.918


{{Optimal ET sequence|legend=1| 5, 13, 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
{{Todo|inline=1|cleanup}}


[[Support]]ing [[ET]]s: {{EDs|5, 9, 14, 19, 24, 43, 62, 81, 100|equave=4}}
<pre>
 
Reduced Mapping
=== Tetramagic ===
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)


[[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>
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|legend=2| 1 0 1 | 0 5 1 }}
Subsets
q99, q62, q37, q161, q136, q198, q25, q124, q74, q235
</pre>


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


: Mapping generators: ~4, ~16/15
Subsets
q37, q25, q62, q12, q74, q99, q87, q49r, q50r, q124
</pre>


[[Optimal tuning]] ([[POTE]]): 1\5ed4 = 480.0, ~16/15 = 80.4062
==== 4.6.5.7.11.13 ====


[[Support]]ing [[ET]]s: {{EDs|5, 10, 15, 20, 25, 30, 55, 85, 115|equave=4}}
<pre>
 
Reduced Mapping
== 4.6.5 subgroup ==
4 6 5 7 11 13
=== Meanquad ===
[ ⟨ 1 0 1 1 1 0 ]
{{Main| Meanquad }}
⟨ 0 16 2 5 9 23 ] ⟩
 
[[Subgroup]]: 4.6.5
TE Generator Tunings (cents)
 
⟨2401.2305, 193.5378]
[[Comma list]]: [[81/80]] = {{monzo| -4 4 -1 }}
 
TE Step Tunings (cents)
{{Mapping|legend=2| 1 0 -4| 0 1 4 }}
⟨42.79107, 35.98524]
 
: mapping generators: ~4, ~6
TE Tuning Map (cents)
 
⟨2401.230, 3096.606, 2788.306, 3368.920, 4143.071, 4451.371]
[[Optimal tuning]] (subgroup [[CTE]]): ~4 = 2\1, ~3/2 = 697.214
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)


[[Support]]ing [[ET]]s: *7, *10, *11[-5], *13[+5], *17, *24, *27[+5], *31, *38, *41, *45, *52, *55, *69
Subsets
q25, q37f, q12f, q62, q50rf, q13rff, q49rff, q87, q74ff, q24rfff
</pre>


<nowiki />* Wart for 4
==== 4.6.5.7.11.13.17 ====
 
<pre>
==== 4.6.5.7 subgroup (tetrominant) ====
Reduced Mapping
[[Subgroup]]: 4.6.5.7
4 6 5 7 11 13 17
 
[ ⟨ 1 0 1 1 1 0 1 ]
[[Comma list]]: [[36/35]] = {{monzo| 0 2 -1 -1 }}, [[64/63]] = {{monzo| 4 -2 0 -1 }}
⟨ 0 16 2 5 9 23 13 ] ⟩
 
{{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
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]]


<pre>
{{Mapping|legend=2| 1 3 4 | 0 1 4 }}
Reduced Mapping
 
4 6 5 7 11 13
Mapping generators: ~4, ~9/64
[ ⟨ 1 0 1 1 1 0 ]
 
⟨ 0 16 2 5 9 23 ] ⟩
[[Optimal tuning]] ([[CTE]]): ~4 = 2\1, ~9/4 = 1394.429
 
TE Generator Tunings (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]
⟨2401.2305, 193.5378]
 
== 4.9.49 subgroup ==
TE Step Tunings (cents)
=== Archsquared ===
⟨42.79107, 35.98524]
[[Subgroup]]: 4.9.49
 
TE Tuning Map (cents)
[[Comma list]]: 4096/3969
⟨2401.230, 3096.606, 2788.306, 3368.920, 4143.071, 4451.371]
 
{{Mapping|legend=2| 1 3 0 | 0 1 -2 }}
TE Mistunings (cents)
 
⟨1.230, -5.349, 1.992, 0.094, -8.247, 10.843]
Mapping generators: ~4, ~9/64
 
Complexity 1.219191
[[Optimal tuning]] ([[CTE]]): ~9/4 = 1419.190
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
[[Support]]ing [[ET]]s: 5, 17, 22, 12, 7, 27, 32, 8, 39[+49], 29[+49], 9[+49], 19[+49], 37, 49
q25, q37f, q12f, q62, q50rf, q13rff, q49rff, q87, q74ff, q24rfff
</pre>


==== 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 ] ⟩
 
[[Comma list]]: 64/63
TE Generator Tunings (cents)
 
⟨2400.4701, 193.4599]
{{Mapping|legend=2| 1 0 2 | 0 1 -1 }}
 
TE Step Tunings (cents)
: sval mapping generators: ~8, ~9
⟨43.39350, 35.55764]
 
: [[gencom]]: [8 9/8; 64/63]
TE Tuning Map (cents)
 
⟨2400.470, 3095.359, 2787.390, 3367.770, 4141.609, 4449.578, 4915.449]
[[Optimal tuning]] ([[CTE]]): ~9/8 = 219.1898
 
TE Mistunings (cents)
[[Optimal ET sequence]]: {{val| 16 17 15 }}, {{val| 33 35 31 }}, {{val| 148 … }}, {{val| 181 … }}, {{val| 214 … }}, {{val| 247 … }}
⟨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
[[Badness]]: 0.0215 × 10<sup>-3</sup>
q25, q12f, q37f, q13rffg, q50rf, q62, q49rffg, q24rfffg, q38rreffg, q74ffg
</pre>


==== 4.6.5.7.11.13.17.19 ====
= Fractional subgroup temperaments =
<pre>
== 2.5/3.… subgroups ==
Reduced Mapping
=== Magicaltet ===
4 6 5 7 11 13 17 19
{{See also| Chromatic pairs #Magicaltet }}
[ ⟨ 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
Magicaltet is related to [[keemic]], [[superkleismic]], and [[magic]]. The tonic and the first three generator steps make a [[magical seventh chord]], hence the name.
q25, q12fh, q37f, q13rffgh, q50rf, q62, q49rffgh, q24rfffghh, q38rreffgh, q74ffgh
 
</pre>
[[Subgroup]]: 2.5/3.7.11
 
[[Comma list]]: 100/99 ({{monzo| 2 2 0 -1 }}), 385/384 ({{monzo| -7 1 1 1 }})
 
{{Mapping|legend=2| 1 0 5 2 | 0 1 -3 2 }}
: mapping generators: ~2, ~5/3
 
{{Mapping|legend=3| 1 -1/2 1/2 2 4 | 0 1/2 -1/2 3 -2 }}
: [[gencom]]: [2 6/5; 100/99 385/384]


==== 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 ]
⟨ 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
{{Optimal ET sequence|legend=1| 4, 7, 11, 15, 26, 67, 93* }}
q25i, q12fhi, q37f, q13rffghii, q62, q50rfii, q49rffghii, q24rfffghhiii, q74ffghi, q38rreffghiii
: <nowiki/>* wart for 5/3
</pre>


== 4.9.25 subgroup ==
[[Tp tuning #T2 tuning|RMS error]]: 1.206 cents
=== Meansquared ===
[[Subgroup]]: 4.9.25


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


{{Mapping|legend=2| 1 3 4 | 0 1 4 }}
Starlingtet, the {{nowrap| 4 & 15 }} temperament in the 2.5/3.7/3 subgroup, is related to [[starling]] as well as to [[myna]]. The tonic and the first three generator steps make a [[starling tetrad]], hence the name.


Mapping generators: ~4, ~9/64
[[Subgroup]]: 2.5/3.7/3


[[Optimal tuning]] ([[CTE]]): ~4 = 2\1, ~9/4 = 1394.429
[[Comma list]]: [[126/125]] ({{monzo| 1 -3 1 }})


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


== 4.9.49 subgroup ==
: mapping generators: ~2, ~5/3
=== Archsquared ===
[[Subgroup]]: 4.9.49


[[Comma list]]: 4096/3969
{{Mapping|legend=3| 1 -1 0 1 | 0 4/3 1/3 -5/3 }}
: [[gencom]]: [2 6/5; 126/125]


{{Mapping|legend=2| 1 3 0 | 0 1 -2 }}
[[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]]): ~[[9/8]] = 219.190
[[Tp tuning #T2 tuning|RMS error]]: 0.8398 cents


[[Support]]ing [[ET]]s: 5, 17, 22, 12, 7, 27, 32, 8, 39[+49], 29[+49], 9[+49], 19[+49], 37, 49
==== Greeley ====
{{See also| Chromatic pairs #Greeley }}


== 8.9.7 subgroup ==
Greeley is related to [[opossum]] as well as to [[nusecond]].  
=== 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
[[Subgroup]]: 2.5/3.7/3.11/3


[[Comma list]]: 64/63
[[Comma list]]: 121/120 ({{monzo| -3 -1 0 2 }}), 126/125 ({{monzo| 1 -3 1 }})


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


: sval mapping generators: ~8, ~9
{{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]


: [[gencom]]: [8 9/8; 64/63]
[[Optimal tuning]]s:  
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~11/10 = 155.696
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~11/10 = 155.776


[[Optimal tuning]] ([[CTE]]): 1\[[3ed8]] = 1600.0, ~9/8 = 219.1898
{{Optimal ET sequence|legend=1| 8, 15, 23, 54, 77, 100, 131* }}
: <nowiki/>* wart for 11/3


[[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]]: 1.034 cents


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


= Fractional subgroup temperaments =
Skateboard is related to [[thrasher]].  
== 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]]: 2.5/3.7/3.11.13/9


[[Subgroup]]: 2.5/3.7.11
[[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]]: 100/99 = {{monzo| 2 2 0 -1 }}, 385/384 = {{monzo| -7 1 1 1 }}
{{Mapping|legend=2| 1 0 -1 2 2 | 0 1 3 2 -2 }}


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


: 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 -1/2 1/2 2 4 | 0 1/2 -1/2 3 -2 }}
{{Optimal ET sequence|legend=1| 11, 15, 19, 23, 42d, 65d }}


: [[gencom]]: [2 6/5; 100/99 385/384]
[[Tp tuning #T2 tuning|RMS error]]: 2.396 cents


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


{{Optimal ET sequence|legend=1| 4, 7, 11, 15, 26, 67, 93* }}
==== Gariberttet (2.5/3.7/3.13/11 subgroup) ====
{{See also | Chromatic pairs #Gariberttet }}


<nowiki />* Wart for 5/3
Gariberttet can be described as the {{nowrap| 4 & 29 }} temperament in the 2.5/3.7/3.13/11 subgroup. Extensions to the full 7-, 11-, and 13-limits include [[quasitemp]].


[[Tp tuning #T2 tuning|RMS error]]: 1.206 cents
[[Subgroup]]: 2.5/3.7/3.13/11


=== Starlingtet ===
[[Comma list]]: [[275/273]] ({{monzo| 0 2 -1 -1 }}), [[847/845]] ({{monzo| 0 -1 1 -2 }})
{{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.
{{Mapping|legend=2| 1 0 0 0 | 0 3 5 1 }}


[[Subgroup]]: 2.5/3.7/3
{{Mapping|legend=3| 1 0 0 0 0 0 | 0 -8/3 1/3 7/3 -1/2 1/2 }}
 
: [[gencom]]: [2 13/11; 275/273 847/845]
[[Comma list]]: [[126/125]] = {{monzo| 1 -3 1 }}
 
{{Mapping|legend=2| 1 0 -1 | 0 1 3 }}
 
: mapping generators: ~2, ~5/3
 
{{Mapping|legend=3| 1 -1 0 1 | 0 4/3 1/3 -5/3 }}
 
: [[gencom]]: [2 6/5; 126/125]


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 888.759
* [[Tp tuning|subgroup]] [[CTE]] and [[POTE]]: ~2 = 1200.000, ~13/11 = 293.679
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 888.846


{{Optimal ET sequence|legend=1| 4, 15, 19, 23, 27 }}
{{Optimal ET sequence|legend=1| 29, 33, 37, 41, 45, 49, 78, 94, 143* }}
: <nowiki/>* wart for 13/11


[[Tp tuning #T2 tuning|RMS error]]: 0.8398 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.6914 cents


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


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


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


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


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


{{Mapping|legend=3| 1 -5/4 -1/4 3/4 3/4 | 0 9/4 1/4 -15/4 5/4 }}
{{Mapping|legend=3| 1 -1/2 -1/2 -1/2 3/2 | 0 -15/4 9/4 25/4 -19/4 }}
 
: [[gencom]]: [2 12/11; 3025/3024 3125/3087]
: [[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]]: ~2 = 1200.000, ~12/11 = 146.978
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~11/10 = 155.776
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~12/11 = 147.010


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


<nowiki />* Wart for 11/3
[[Tp tuning #T2 tuning|RMS error]]: 0.7788 cents


[[Tp tuning#T2 tuning|RMS error]]: 1.034 cents
==== Ammon ====
{{See also| Chromatic pairs #Ammon }}


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


Skateboard is related to [[thrasher]].  
[[Subgroup]]: 2.5/3.7/3.11/3.13/3


[[Subgroup]]: 2.5/3.7/3.11.13/9
[[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]]: 56/55 = {{monzo| 3 -1 1 -1 }}, 91/90 = {{monzo| -1 -1 1 0 1 }}, 100/99 = {{monzo| 2 2 0 -1 }}
{{Mapping|legend=2| 1 3 5 3 4 | 0 -6 -10 -3 -5 }}


{{Mapping|legend=2| 1 0 -1 2 2 | 0 1 3 2 -2 }}
{{Mapping|legend=3| 1 -3 0 2 0 1 | 0 24/5 -6/5 -26/5 9/5 -1/5 }}
: [[gencom]]: [2 13/10; 121/120 169/168 275/273]


{{Mapping|legend=3| 1 -3/7 4/7 11/7 4 -6/7 | 0 0 -1 -3 -2 2 }}
[[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


: [[gencom]]: [2 6/5; 56/55 91/90 100/99]
{{Optimal ET sequence|legend=1| 8, 29, 37, 45 }}


[[Optimal tuning]]s:
[[Tp tuning #T2 tuning|RMS error]]: 1.052 cents
* [[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 }}
=== Sentry ===
{{See also | Chromatic pairs #Sentry }}


[[Tp tuning #T2 tuning|RMS error]]: 2.396 cents
Sentry, the {{nowrap| 3 & 5 }} temperament in the 2.5/3.9/7 subgroup, is related to [[sensi]].  


=== Gariberttet ===
[[Subgroup]]: 2.5/3.9/7
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) ====
[[Comma list]]: [[245/243]] ({{monzo| 0 1 -2 }})
{{See also | Chromatic pairs #Gariberttet }}


Gariberttet can be described as the 4 &amp; 29 temperament in the 2.5/3.7/3.13/11 subgroup.
{{Mapping|legend=2| 1 0 0 | 0 2 1 }}


[[Subgroup]]: 2.5/3.7/3.13/11
{{Mapping|legend=3| 1 0 0 0 | 0 0 2 -1 }}
: [[gencom]]: [2 9/7; 245/243]


[[Comma list]]: [[275/273]] = {{monzo| 0 2 -1 -1 }}, [[847/845]] = {{monzo| 0 -1 1 -2 }}
[[Optimal tuning]]s:  
* [[Tp tuning|subgroup]] [[CTE]] and [[POTE]]: ~2 = 1200.000, ~9/7 = 440.902


{{Mapping|legend=2| 1 0 0 0 | 0 3 5 1 }}
{{Optimal ET sequence|legend=1| 8, 11, 19, 30, 41, 49, 52, 145*, 166<sup>†</sup>, 197*<sup>†</sup>, 215<sup>†</sup>, 264*<sup>†</sup> }}
: <nowiki/>* wart for 5/3
: <sup>†</sup> wart for 9/7


{{Mapping|legend=3| 1 0 0 0 0 0 | 0 -8/3 1/3 7/3 -1/2 1/2 }}
[[Tp tuning #T2 tuning|RMS error]]: 0.7105 cents


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


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


<nowiki />* Wart for 13/11
[[Comma list]]: [[325/324]] ({{monzo| -2 2 1 }})


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


==== Indium ====
{{Mapping|legend=3| 1 -1/6 5/6 0 0 -1/3 | 0 -1/2 -3/2 0 0 1 }}
{{See also | Chromatic pairs #Indium }}
: [[gencom]]: [2 6/5; 325/324]


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 = 882.886
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 882.861


[[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 the main rank-2 chain of [[guanyin]] and a restriction of [[orwell]]. It is defined by tempering out [[1728/1715]] ({{S|6/S7}}) and [[540/539]] (S12/S14), which imply [[176/175]] (S8/S10) as well as S11/S15 being tempered out. The tonic and the first three generator steps make a [[guanyin tetrad]], hence the name.


: [[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 1 3 | 0 -3 1 -5 }}
: mapping generators: ~2, ~7/6


<nowiki />* Wart for 7/3
{{Mapping|legend=3| 1 -4/3 3 -1/3 5/3 | 0 4/3 -3 7/3 -11/3 }}
: [[gencom]]: [2 7/6; 176/175 540/539]


<sup>†</sup> Wart for 11/3
[[Optimal tuning]]s:
* ([[Tp tuning|subgroup]] [[CTE]]): ~2 = 1200.000, ~7/6 = 270.455
* ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1200.000, ~7/6 = 270.093


[[Tp tuning #T2 tuning|RMS error]]: 0.7788 cents
{{Optimal ET sequence|legend=1| 9, 22, 31, 40, 191c*, 231c*, 271c*, 311c* }}
: <nowiki/>* wart for 7/3


==== Semidim ====
[[Tp tuning #T2 tuning|RMS error]]: 0.6028 cents
{{See also| Chromatic pairs #Semidim }}


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


[[Subgroup]]: 2.5/3.7/3.11/3.13/3
[[Subgroup]]: 2.5.7/3.11/3.13


[[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]]: [[176/175]] ({{monzo| 4 -2 -1 1 0 }}), [[540/539]] ({{monzo| 2 1 -2 -1 0 }}), [[1573/1568]] ({{monzo| -5 0 -2 2 1 }})


{{Mapping|legend=2| 1 3 5 3 4 | 0 -6 -10 -3 -5 }}
{{Mapping|legend=2| 1 0 1 3 1 | 0 -3 1 -5 12 }}
: mapping generators: ~2, ~12/7


{{Mapping|legend=3| 1 -3 0 2 0 1 | 0 24/5 -6/5 -26/5 9/5 -1/5 }}
[[Optimal tuning]]s:
* ([[Tp tuning|subgroup]] [[CTE]]): ~2 = 1200.000, ~7/6 = 270.152
* ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1200.000, ~7/6 = 270.218


: [[gencom]]: [2 13/10; 121/120 169/168 275/273]
{{Optimal ET sequence|legend=1| 9, 22, 31, 40, 71, 111, 151, 262c*}} <small> using subgroup TE </small>
: <nowiki/>* wart for 7/3


[[Optimal tuning]]s:  
Badness (Sintel): 0.329
* [[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 }}
==== Laz ====
{{See also | Chromatic pairs #Laz }}


[[Tp tuning #T2 tuning|RMS error]]: 1.052 cents
Laz is related to [[avalokita]] as well as to [[winston]].  


=== Sentry ===
[[Subgroup]]: 2.5.7/3.11/3.13/3
{{See also | Chromatic pairs #Sentry }}


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


[[Subgroup]]: 2.5/3.9/7
{{Mapping|legend=2| 1 0 2 -2 6 | 0 3 -1 5 -5 }}


[[Comma list]]: [[245/243]] = {{monzo| 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 }}
: [[gencom]]: [2 7/6; 144/143 176/175 196/195]


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


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


: [[gencom]]: [2 9/7; 245/243]
[[Tp tuning #T2 tuning|RMS error]]: 0.8790 cents


[[Optimal tuning]]s:
=== Kryptonite ===
* [[Tp tuning|subgroup]] [[CTE]] and [[POTE]]: ~2 = 1\1, ~9/7 = 440.902
{{See also| Chromatic pairs #Kryptonite }}


{{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> }}
Kryptonite is related to [[krypton]].


<nowiki />* Wart for 5/3
[[Subgroup]]: 2.5.7/3.11/3.13/3


<sup>†</sup> Wart for 9/7
[[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 }})


[[Tp tuning #T2 tuning|RMS error]]: 0.7105 cents
{{Mapping|legend=2| 1 2 1 2 2 | 0 3 2 -1 1 }}
: 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 }}
: [[gencom]]: [2 13/12; 56/55 78/77 91/90]


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.  
[[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


[[Subgroup]]: 2.5/3.13/9
{{Optimal ET sequence|legend=1| 1, …, 8, 9 }}


[[Comma list]]: [[325/324]] = {{monzo| -2 2 1 }}
[[Tp tuning #T2 tuning|RMS error]]: 2.545 cents


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


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


: [[gencom]]: [2 6/5; 325/324]
[[Subgroup]]: 2.9/5.7/3.11/9


[[Optimal tuning]]s:  
[[Comma list]]: 100/99 ({{monzo| 2 -2 0 -1 }}), 245/242 ({{monzo| -1 -1 2 -2 }})
* [[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 }}
{{Mapping|legend=2| 1 1 1 0 | 0 -2 3 4 }}
: mapping generators: ~2, ~21/20


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


== 2.….7/3… subgroups ==
[[Optimal tuning]]s:
=== Guanyintet ===
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~21/20 = 87.776
{{See also | Chromatic pairs #Guanyintet }}
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~21/20 = 87.892


Guanyintet, the 4 &amp; 9 temperament in the 2.5.7/3.11/3 subgroup, is related to [[guanyin]] as well as to [[orwell]].
{{Optimal ET sequence|legend=1| 13, 14, 27, 41 }}


[[Subgroup]]: 2.5.7/3.11/3
[[Tp tuning #T2 tuning|RMS error]]: 1.245 cents


[[Comma list]]: [[176/175]], [[540/539]]
=== Mothwelltri ===
{{See also| Chromatic pairs #Mothwelltri }}


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


{{Mapping|legend=3| 1 -4/3 3 -1/3 5/3 | 0 4/3 -3 7/3 -11/3 }}
[[Subgroup]]: 2.7/3.11


: [[gencom]]: [2 7/6; 176/175 540/539]
[[Comma list]]: [[99/98]] ({{monzo| -1 -2 1 }})


[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~7/6 = 270.093
{{Mapping|legend=2| 1 0 1 | 0 1 2 }}
: mapping generators: ~2, ~7/3


{{Optimal ET sequence|legend=1| 9, 31, 40, 49, 89, 191bc, 227bc, 231bc, 271bc, 311bc, 316bcd }}
{{Mapping|legend=3| 1 -1/2 0 1/2 3 | 0 -1/2 0 1/2 2 }}
: [[gencom]]: [2 7/6; 99/98]


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


==== Laz ====
{{Optimal ET sequence|legend=1| 4, 9, 13, 22, 79 }}
{{See also | Chromatic pairs #Laz }}


Laz is related to [[georgian]] as well as to [[winston]].  
[[Tp tuning #T2 tuning|RMS error]]: 1.064 cents


[[Subgroup]]: 2.5.7/3.11/3.13/3
== 2.….9/7.… subgroups ==
=== Marveltri ===
{{See also| Chromatic pairs #Marveltri }}


[[Comma list]]: [[144/143]], [[176/175]], [[196/195]]
Marveltri, the {{nowrap| 3 & 13 }} temperament in the 2.5.9/7 subgroup, is related to [[marvel]], [[magic]], and the unnamed {{nowrap| 22 & 47 }} temperament. The tonic and the first two generator steps make a [[marvel triad]], hence the name.


{{Mapping|legend=2| 1 0 2 -2 6 | 0 3 -1 5 -5 }}
[[Subgroup]]: 2.5.9/7


{{Mapping|legend=3| 1 -5/4 3 -1/4 7/4 -1/4 | 0 -1/4 -3 3/4 -21/4 19/4 }}
[[Comma list]]: 225/224 ({{monzo| -5 2 1 }})


: [[gencom]]: [2 7/6; 144/143 176/175 196/195]
{{Mapping|legend=2| 1 0 5 | 0 1 -2 }}
: mapping generators: ~2, ~5


[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~7/6 = 269.300
{{Mapping|legend=3| 1 2 0 -1 | 0 -4/5 1 2/5 }}
: [[gencom]]: [2 5; 225/224]


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


[[Tp tuning #T2 tuning|RMS error]]: 0.8790 cents
{{Optimal ET sequence|legend=1| 3, 13, 16, 19, 22, 25, 72, 97, 122, 269c* }}
: <nowiki/>* wart for 9/7


=== Kryptonite ===
[[Tp tuning #T2 tuning|RMS error]]: 0.4801 cents
{{See also| Chromatic pairs #Kryptonite }}


Kryptonite is related to [[krypton]].  
==== Sulis ====
Sulis is related to [[minerva]] and [[würschmidt]].  


[[Subgroup]]: 2.5.7/3.11/3.13/3
[[Subgroup]]: 2.5.9/7.11/9


[[Comma list]]: 56/55, 78/77, 91/90
[[Comma list]]: 99/98 ({{monzo| -1 0 2 1 }}), 176/175 ({{monzo| 4 -2 1 1 }})


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


{{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 = 386.617
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/4 = 386.558


: [[gencom]]: [2 13/12; 56/55 78/77 91/90]
{{Optimal ET sequence|legend=1| 3, …, 22, 25, 28, 31, 59 }}


[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~13/12 = 132.428
[[Tp tuning #T2 tuning|RMS error]]: 1.074 cents


{{Optimal ET sequence|legend=1| 9, 63, 82bd, 91bde }}
== 2.….7/5.… subgroups ==
=== Hydrothermal ===
A tuning whose distinctively sharp (but still consonant) fifth, and flat (but still consonant) octave, lend it a mysterious, heavy atmosphere. The 6-tone (hexatonic) MOS is melodically interesting and flavorful. The 18-tone MOS is a useful 'chromatic' scale for taking subsets of.


[[Tp tuning #T2 tuning|RMS error]]: 2.545 cents
[[Subgroup]]: 2.3.7/5


=== Kiribati ===
[[Comma list]]: [[50/49]]
{{See also| Chromatic pairs #Kiribati }}


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


[[Subgroup]]: 2.9/5.7/3.11/9
[[Optimal tuning]] (inharmonic [[TE]]): ~1\2 = 590.998, ~[[10/7]]-1\2 = 128.962


[[Comma list]]: 100/99, 245/242
[[Support]]ing [[ET]]s: {{EDOs|4, 6, 8, 10, 18, 28, 46, 64, 110}}


{{Mapping|legend=2| 1 1 1 0 | 0 -2 3 4 }}
=== Argentic ===
Argentic is the 2.3.7/5 subgroup temperament tempering out [[5120/5103]].


{{Mapping|legend=3| 1 1/10 -4/5 11/10 1/5 | 0 -3/2 -1 3/2 1 }}
[[Subgroup]]: 2.3.7/5


: [[gencom]]: [2 21/20; 100/99 245/242]
[[Comma list]]: [[5120/5103]] = {{monzo| 10 -6 -1 }}


[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~21/20 = 87.892
{{Mapping|legend=2| 1 0 10 | 0 1 -6 }}
: mapping generators: ~2, ~3


{{Optimal ET sequence|legend=1| 13, 14, 27, 41, 55, 191bd, 232bcd, 273bcd }}
[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1\1, ~3/2 = 702.792
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1\1, ~3/2 = 702.830


[[Tp tuning #T2 tuning|RMS error]]: 1.245 cents
{{Optimal ET sequence|legend=1| 12, 29, 41, 70, 321, 391, 461, 531, 601 }}
<small> based on subgroup TE </small>


=== Mothwelltri ===
Badness (Sintel): 0.119
{{See also| Chromatic pairs #Mothwelltri }}


Mothwelltri, the 1 &amp; 4 temperament in the 2.7/3.11 subgroup, is related to [[orwell]].
==== Edson (2.3.7/5.11/5.13/5 subgroup) ====
{{See also| Chromatic pairs #Edson }}


[[Subgroup]]: 2.7/3.11
Edson is related to [[pele]] and [[andromeda]].  


[[Comma list]]: [[99/98]]
[[Subgroup]]: 2.3.7/5.11/5.13/5


{{Mapping|legend=2| 1 0 1 | 0 1 2 }}
[[Comma list]]: [[196/195]] = {{monzo| 2 -1 2 0 -1 }}, [[352/351]] = {{monzo| 5 -3 0 1 -1 }}, [[364/363]] = {{monzo| 2 -1 1 -2 1 }}


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


: [[gencom]]: [2 7/6; 99/98]
{{Mapping|legend=3| 1 1 -5 -1 2 4 | 0 1 29/4 5/4 -11/4 -23/4 }}
: [[gencom]]: [2 3/2; 196/195, 352/351, 364/363]


[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~7/6 = 273.174
[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1\1, ~3/2 = 703.4398
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1\1, ~3/2 = 703.414


{{Optimal ET sequence|legend=1| 9, 22, 40, 49c, 58c, 67c, 76c, 79, 101b, 123bc }}
{{Optimal ET sequence|legend=1| 12, 17, 29 }}


[[Tp tuning #T2 tuning|RMS error]]: 1.064 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.5102 cents


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


=== Marveltri ===
Related temperaments include [[#Bridgetown|bridgetown]], [[namaka]], [[hemigari]], [[#Barbados|barbados]], and [[parizekmic]].
{{See also| Chromatic pairs #Marveltri }}


Marveltri, the 3 &amp; 13 temperament in the 2.5.9/7 subgroup, is related to [[marvel]], [[magic]], and the unnamed 22 &amp; 47 temperament.  
[[Subgroup]]: 2.3.7/5.11/5.13/5


[[Subgroup]]: 2.5.9/7
[[Comma list]]: [[352/351]], [[676/675]], [[847/845]]


[[Comma list]]: 225/224
{{Mapping|legend=2| 1 0 10 -6 -1 | 0 2 -12 9 3 }}


{{Mapping|legend=2| 1 2 1 | 0 1 -2 }}
{{Mapping|legend=3| 1 2 -3/4 -11/4 9/4 5/4 | 0 -2 0 12 -9 -3 }}
: [[gencom]]: [2 15/13; 352/351 676/675 847/845]


{{Mapping|legend=3| 1 2/5 2 -1/5 | 0 -4/5 1 2/5 }}
[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~15/13 = 248.491


: [[gencom]]: [2 5/4; 225/224]
{{Optimal ET sequence|legend=1| 24, 29, 111, 140, 169, 198, 565d, 763bd, 961bd }}


[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~5/4 = 383.638
[[Tp tuning #T2 tuning|RMS error]]: 0.2668 cents


{{Optimal ET sequence|legend=1| 12, 13, 16, 19, 22, 25, 47, 69, 72, 97, 122, 269c*, 660c* }}
=== Historical ===
{{distinguish|Historical temperaments}}
{{distinguish|History (temperament)}}, which is the rank-3 version of this temperament in the full 13-limit.


<nowiki />* Wart for 9/7
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.


[[Tp tuning #T2 tuning|RMS error]]: 0.4801 cents
[[Subgroup]]: 2.3.7/5.11/5.13/5


==== Sulis ====
[[Comma list]]: 364/363, 441/440, 1001/1000
Related temperament: [[Marvel family|minerva]], [[Würschmidt family|würschmidt]]


[[Subgroup]]: 2.5.9/7.11/7
{{Mapping|legend=2| 1 2 0 1 2 | 0 -6 7 2 -9 }}


[[Comma list]]: 99/98, 176/175
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~21/20 = 83.016


{{Mapping|legend=2| 1 2 1 0 | 0 1 -2 2 }}]
{{Optimal ET sequence|legend=1| 14, 29, 72, 101, 130, 159 }}


[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~5/4 = 386.558
[[Tp tuning #T2 tuning|RMS error]]: 0.2562 cents


{{Optimal ET sequence|legend=1| 3, …, 22, 25, 28, 31, 59 }}
=== Terrain ===
{{Redirect|Terrain|the scale|Terrain (scale)}}
{{See also| Chromatic pairs #Terrain }}


[[Tp tuning #T2 tuning|RMS error]]: 1.074 cents
Terrain, the 6 &amp; 21 temperament in the 2.7/5.9/5 subgroup, is related to [[domain (temperament)|domain]]. It is a remarkable temperament, in that while its complexity is low, it has no discernible error. The 1–7/5–9/5 and 1–9/7–9/5 chords are characteristic.


== 2.….15/11… subgroups ==
[[Subgroup]]: 2.7/5.9/5


=== Poggers ===
[[Comma list]]: [[250047/250000]]
Related temperaments: [[Stearnsmic_clan#Pogo|pogo]], [[Stearnsmic_clan#Supers|supers]]


[[Subgroup]]: 2.9.7.15/11.13
{{Mapping|legend=2| 3 1 3 | 0 1 -1 }}


[[Comma list]]: [[540/539]], [[1716/1715]], [[2080/2079]]
{{Mapping|legend=3| 3 10/9 -7/9 2/9 | 0 -2/3 -1/3 2/3 }}
: [[gencom]]: [63/50 10/9; 250047/250000]


{{Mapping|legend=2| 1 1 1 -1 -1 | 0 6 5 4 13 }}
[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~63/50 = 1\3, ~10/9 = 182.461


[[Optimal tuning]] (subgroup [[CTE]]): ~9/7 = 433.888
{{Optimal ET sequence|legend=1| 6, 21, 27, 33, 105, 138, 171, 1848, 2019, 2190, 2361, 2532, 2703, 2874, 3045, 3216, 3387, 3558 }}


[[Support]]ing [[ET]]s: 8[+9, +7, +13], 11, 14[-13], 19[+9, +7, ++13], 25[-13], 36, 47, 58, 61[-13], 69[+13], 80[+13], 83, 91[+9, +7, +13], 105
[[Tp tuning #T2 tuning|RMS error]]: 0.00844 cents


== 2.….7/5… subgroups ==
=== Tridec ===
{{See also| Chromatic pairs #Tridec }}
{{See also| Non-over-1 temperament #Tridec }}


=== Hydrothermal ===
Tridec, the 5 &amp; 8 temperament in the 2.7/5.11/5.13/5 subgroup, extends [[#Petrtri]].  
A tuning whose distinctively sharp (but still consonant) fifth, and flat (but still consonant) octave, lend it a mysterious, heavy atmosphere. The 6-tone (hexatonic) MOS is melodically interesting and flavorful. The 18-tone MOS is a useful 'chromatic' scale for taking subsets of.


[[Subgroup]]: 2.3.7/5
[[Subgroup]]: 2.7/5.11/5.13/5


[[Comma list]]: [[50/49]]
[[Comma list]]: [[847/845]], [[1001/1000]]


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


[[Optimal tuning]] (inharmonic [[TE]]): ~1\2 = 590.998, ~[[10/7]]-1\2 = 128.962
{{Mapping|legend=3| 1 0 -3/4 5/4 -3/4 1/4 | 0 0 0 -4 3 1 }}
: [[gencom]]: [2 13/10; 847/845 1001/1000]


[[Support]]ing [[ET]]s: {{EDOs|4, 6, 8, 10, 18, 28, 46, 64, 110}}
[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~13/10 = 454.556


=== Edson ===
{{Optimal ET sequence|legend=1| 5, 8, 21, 29, 37, 66, 169, 235, 404c, 639c, 953bc }}
Edson is the 2.3.7/5 subgroup temperament tempering out [[5120/5103]].


==== Edson (2.3.7/5.11/5.13/5 subgroup) ====
[[Tp tuning #T2 tuning|RMS error]]: 0.1613 cents
{{See also| Chromatic pairs #Edson }}


Edson is related to [[pele]] and [[andromeda]].  
==== Naiadec ====
[[Subgroup]]: 2.7/5.11/5.13/5.17/5


[[Subgroup]]: 2.3.7/5.11/5.13/5
[[Comma list]]: [[170/169]], [[221/220]], [[847/845]]


[[Comma list]]: [[196/195]] = {{monzo| 2 -1 2 0 -1 }}, [[352/351]] = {{monzo| 5 -3 0 1 -1 }}, [[364/363]] = {{monzo| 2 -1 1 -2 1 }}
{{Mapping|legend=2| 1 2 0 1 1 | 0 -4 3 1 2 }}


{{Mapping|legend=2| 1 0 10 17 22 | 0 1 -6 -10 -13 }}
{{Mapping|legend=3| 1 0 -3/4 5/4 -3/4 1/4 1/4 | 0 0 0 -4 3 1 2 }}
: [[gencom]]: [2 13/10; 170/169 221/220 847/845]


: mapping generators: ~2, ~3
[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~13/10 = 454.882


{{Mapping|legend=3| 1 1 -5 -1 2 4 | 0 1 29/4 5/4 -11/4 -23/4 }}
{{Optimal ET sequence|legend=1| 5, 8, 21, 29, 95<sup>t</sup>, 124<sup>t</sup> }}
: <sup>t</sup> wart for 17/5


: [[gencom]]: [2 3/2; 196/195, 352/351, 364/363]
[[Tp tuning #T2 tuning|RMS error]]: 0.7521 cents


[[Optimal tuning]]s:
== 2.….11/5.… subgroups ==
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1\1, ~3/2 = 703.4398
=== Petrtri ===
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1\1, ~3/2 = 703.414
{{See also| Chromatic pairs #Petrtri }}
{{See also| 5L 3s/Temperaments #Petrtri }}


{{Optimal ET sequence|legend=1| 12, 17, 29 }}
Petrtri can be described as 3 &amp; 5 temperament in the 2.11/5.13/5 subgroup.


[[Tp tuning #T2 tuning|RMS error]]: 0.5102 cents
[[Subgroup]]: 2.11/5.13/5


==== Haumea ====
[[Comma list]]: [[2200/2197]]
{{See also| Chromatic pairs #Haumea }}


Related temperaments include [[#Bridgetown|bridgetown]], [[namaka]], [[hemigari]], [[#Barbados|barbados]], and [[parizekmic]].
{{Mapping|legend=2| 1 0 1| 0 3 1 }}


[[Subgroup]]: 2.3.7/5.11/5.13/5
{{Mapping|legend=3| 1 0 -1/3 0 -1/3 2/3 | 0 0 -4/3 0 5/3 -1/3 }}
: [[gencom]]: [2 13/10; 2200/2197]


[[Comma list]]: [[352/351]], [[676/675]], [[847/845]]
[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~13/10 = 455.012


{{Mapping|legend=2| 1 0 10 -6 -1 | 0 2 -12 9 3 }}
{{Optimal ET sequence|legend=1| 21, 29, 153, 182, 211, 240, 269, 298, 327, 356, 385, 509, 741c, 1126c }}


{{Mapping|legend=3| 1 2 -3/4 -11/4 9/4 5/4 | 0 -2 0 12 -9 -3 }}
[[Tp tuning #T2 tuning|RMS error]]: 0.0749 cents


: [[gencom]]: [2 15/13; 352/351 676/675 847/845]
==== Bridgetown ====
{{See also| Chromatic pairs #Bridgetown }}


[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~15/13 = 248.491
Bridgetown, the 5 &amp; 24 temperament in the 2.3.11/5.13/5 subgroup, is related to [[#Haumea|haumea]] and [[#Barbados|barbados]].  


{{Optimal ET sequence|legend=1| 24, 29, 111, 140, 169, 198, 565d, 763bd, 961bd }}
[[Subgroup]]: 2.3.11/5.13/5


[[Tp tuning #T2 tuning|RMS error]]: 0.2668 cents
[[Comma list]]: [[352/351]], [[676/675]]


=== Historical ===
{{Mapping|legend=2| 1 0 -6 -1 | 0 2 9 3 }}
{{distinguish|Historical temperaments}}
{{distinguish|History (temperament)}}, which is the rank-3 version of this temperament in the full 13-limit.


Historical is essentially an analogue of [[miracle]] that splits [[4/3]] in six rather than [[3/2]]. It tempers out the comma S10/S11 = [[4000/3993]] to set [[11/10]] equal to one-third of 4/3, and S13/S15 = [[676/675]] to equate [[15/13]] to one-half of 4/3, and tempers out S21 = [[441/440]] to split 11/10 into two instances of [[22/21]]~[[21/20]].
{{Mapping|legend=3| 1 2 -5/3 0 4/3 1/3 | 0 -2 4 0 -5 1 }}
: [[gencom]]: [2 15/13; 352/351 676/675]


[[Subgroup]]: 2.3.7/5.11/5.13/5
[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~15/13 = 248.399


[[Comma list]]: 364/363, 441/440, 1001/1000
{{Optimal ET sequence|legend=1| 5, 9, 14, 19, 24, 29, 169, 198, 227, 256, 285, 314 }}


{{Mapping|legend=2| 1 2 0 1 2 | 0 -6 7 2 -9 }}
[[Tp tuning #T2 tuning|RMS error]]: 0.2513 cents


[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~21/20 = 83.016
=== Hypnosis ===
Related temperaments: [[Swetismic temperaments #Hypnos|hypnos]], [[Alphatricot family #Alphatricot|alphatricot]]


{{Optimal ET sequence|legend=1| 14, 29, 72, 101, 130, 159 }}
[[Subgroup]]: 2.3.7.11/5.13


[[Tp tuning #T2 tuning|RMS error]]: 0.2562 cents
[[Comma list]]: 169/168, 540/539, 729/728


=== Terrain ===
{{Mapping|legend=2| 1 0 -3 8 0 | 0 3 11 -13 7 }}
{{Redirect|Terrain|the scale|Terrain (scale)}}
{{See also| Chromatic pairs #Terrain }}


Terrain, the 6 &amp; 21 temperament in the 2.7/5.9/5 subgroup, is related to [[domain (temperament)|domain]]. It is a remarkable temperament, in that while its complexity is low, it has no discernible error. The 1–7/5–9/5 and 1–9/7–9/5 chords are characteristic.
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~13/9 = 633.518


[[Subgroup]]: 2.7/5.9/5
{{Optimal ET sequence|legend=1| 17, 36, 118f, 125f, 161f, 197f }}


[[Comma list]]: [[250047/250000]]
[[Tp tuning #T2 tuning|RMS error]]: 0.5379 cents


{{Mapping|legend=2| 3 1 3 | 0 1 -1 }}
=== Trisect ===
Trisect divides every Pythagorean interval into three, and is the much more accurate subgroup restriction of [[Augmented family #Trisected|trisected]].


{{Mapping|legend=3| 3 10/9 -7/9 2/9 | 0 -2/3 -1/3 2/3 }}
Extending this temperament to the full [[11-limit|11-]], [[13-limit|13-]], or [[17-limit]] through [[portent]] or [[landscape]] results in the [[weak extension]] known as [[tritikleismic]].


: [[gencom]]: [63/50 10/9; 250047/250000]
[[Subgroup]]: 2.3.7.11/5


[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~63/50 = 1\3, ~10/9 = 182.461
[[Comma list]]: 1029/1024, 4000/3993


{{Optimal ET sequence|legend=1| 6, 21, 27, 33, 105, 138, 171, 1848, 2019, 2190, 2361, 2532, 2703, 2874, 3045, 3216, 3387, 3558 }}
{{Mapping|legend=2| 3 0 10 5 | 0 3 -1 -1 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.00844 cents
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~44/35 = 1\3, ~13/9 = 633.742


=== Tridec ===
{{Optimal ET sequence|legend=1| 15, 21, 36, 123, 159, 195, 231 }}
{{See also| Chromatic pairs #Tridec }}
{{See also| Non-over-1 temperament #Tridec }}


Tridec, the 5 &amp; 8 temperament in the 2.7/5.11/5.13/5 subgroup, extends [[#Petrtri]].
[[Tp tuning #T2 tuning|RMS error]]: ???


[[Subgroup]]: 2.7/5.11/5.13/5
==== 2.3.7.11/5.13 subgroup ====
[[Subgroup]]: 2.3.7.11/5.13


[[Comma list]]: [[847/845]], [[1001/1000]]
[[Comma list]]: 1029/1024, 1575/1573, 2080/2079


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


{{Mapping|legend=3| 1 0 -3/4 5/4 -3/4 1/4 | 0 0 0 -4 3 1 }}
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~44/35 = 1\3, ~13/9 = 633.918


: [[gencom]]: [2 13/10; 847/845 1001/1000]
{{Optimal ET sequence|legend=1| 15, 21f, 36, 87, 123, 159 }}


[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~13/10 = 454.556
[[Tp tuning #T2 tuning|RMS error]]: ???


{{Optimal ET sequence|legend=1| 5, 8, 21, 29, 37, 66, 169, 235, 404c, 639c, 953bc }}
==== 2.3.7.11/5.13.17 subgroup ====
[[Subgroup]]: 2.3.7.11/5.13.17


[[Tp tuning #T2 tuning|RMS error]]: 0.1613 cents
[[Comma list]]: 273/272, 833/832, 1575/1573, 2080/2079


== 2.….11/5… subgroups ==
{{Mapping|legend=2| 3 0 10 5 0 -2 | 0 3 -1 -1 7 9 }}


=== Petrtri ===
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~34/27 = 1\3, ~13/9 = 633.820
{{See also| Chromatic pairs #Petrtri }}
{{See also| 5L 3s/Temperaments #Petrtri }}


Petrtri can be described as 3 &amp; 5 temperament in the 2.11/5.13/5 subgroup.
{{Optimal ET sequence|legend=1| 15, 21fg, 36, 123, 159 }}


[[Subgroup]]: 2.11/5.13/5
[[Tp tuning #T2 tuning|RMS error]]: ???


[[Comma list]]: [[2200/2197]]
===== Trisector =====
[[Subgroup]]: 2.3.7.11/5.13.17.19


{{Mapping|legend=2| 1 0 1| 0 3 1 }}
[[Comma list]]: 210/209, 273/272, 286/285, 595/594, 2080/2079


{{Mapping|legend=3| 1 0 -1/3 0 -1/3 2/3 | 0 0 -4/3 0 5/3 -1/3 }}
{{Mapping|legend=2| 3 0 10 5 0 -2 8 | 0 3 -1 -1 7 9 3 }}


: [[gencom]]: [2 13/10; 2200/2197]
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~34/27 = 1\3, ~13/9 = 633.894


[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~13/10 = 455.012
{{Optimal ET sequence|legend=1| 15, 21fg, 36, 123h, 159h }}


{{Optimal ET sequence|legend=1| 21, 29, 153, 182, 211, 240, 269, 298, 327, 356, 385, 509, 741c, 1126c }}
[[Tp tuning #T2 tuning|RMS error]]: ???


[[Tp tuning #T2 tuning|RMS error]]: 0.0749 cents
===== 2.3.7.11/5.13.17.19.23 subgroup =====
[[Subgroup]]: 2.3.7.11/5.13.17.19.23


==== Bridgetown ====
[[Comma list]]: 210/209, 231/230, 273/272, 286/285, 595/594, 2080/2079
{{See also| Chromatic pairs #Bridgetown }}


Bridgetown, the 5 &amp; 24 temperament in the 2.3.11/5.13/5 subgroup, is related to [[#Haumea|haumea]] and [[#Barbados|barbados]].
{{Mapping|legend=2| 3 0 10 5 0 -2 8 12 | 0 3 -1 -1 7 9 3 1 }}


[[Subgroup]]: 2.3.11/5.13/5
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~34/27 = 1\3, ~13/9 = 634.038


[[Comma list]]: [[352/351]], [[676/675]]
{{Optimal ET sequence|legend=1| 15g, 21fg, 36, 87, 123hi }}


{{Mapping|legend=2| 1 0 -6 -1 | 0 2 9 3 }}
[[Tp tuning #T2 tuning|RMS error]]: ???


{{Mapping|legend=3| 1 2 -5/3 0 4/3 1/3 | 0 -2 4 0 -5 1 }}
===== 2.3.7.11/5.13.17.19.23.29 subgroup =====
[[Subgroup]]: 2.3.7.11/5.13.17.19.23.29


: [[gencom]]: [2 15/13; 352/351 676/675]
[[Comma list]]: 210/209, 231/230, 273/272, 286/285, 320/319, 595/594, 2080/2079


[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~15/13 = 248.399
{{Mapping|legend=2| 3 0 10 5 0 -2 8 12 13 | 0 3 -1 -1 7 9 3 1 1 }}


{{Optimal ET sequence|legend=1| 5, 9, 14, 19, 24, 29, 169, 198, 227, 256, 285, 314 }}
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~29/23 = 1\3, ~13/9 = 634.102


[[Tp tuning #T2 tuning|RMS error]]: 0.2513 cents
{{Optimal ET sequence|legend=1| 15g, 21fg, 36, 87, 123hi }}


=== Hypnosis ===
[[Tp tuning #T2 tuning|RMS error]]: ???
Related temperaments: [[Swetismic temperaments #Hypnos|hypnos]], [[Hemifamity temperaments #Tricot|tricot]]


[[Subgroup]]: 2.3.7.11/5.13
== 2.….11/7.… subgroups ==
 
[[Comma list]]: 169/168, 540/539, 729/728
 
{{Mapping|legend=2| 1 0 -3 8 0 | 0 3 11 -13 7 }}
 
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~13/9 = 633.518
 
{{Optimal ET sequence|legend=1| 17, 36, 118f, 125f, 161f, 197f }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.5379 cents
 
== 2.….11/7… subgroups ==
=== Pepperoni ===
=== Pepperoni ===
{{Main| Parapyth }}
{{Main| Parapyth }}
Line 1,486: Line 1,565:


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


Line 1,492: Line 1,570:


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


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


== 2.….13/5… subgroups ==
== 2.….13/5.… subgroups ==
=== Barbados ===
=== Barbados ===
The [[minimax tuning]] for this makes the generator the cube root of 20/13, or 248.5953 cents. Edos which may be used for it are [[24edo]], [[29edo]], [[53edo]] and [[111edo]], with [[mos scale]]s of size 5, 9, 14, 19, 24 and 29 making for a good variety of scales.
The [[minimax tuning]] for this makes the generator the cube root of 20/13, or 248.5953 cents. Edos which may be used for it are [[24edo]], [[29edo]], [[53edo]] and [[111edo]], with [[mos scale]]s of size 5, 9, 14, 19, 24 and 29 making for a good variety of scales.
Line 1,530: Line 1,606:


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


Line 1,540: Line 1,615:


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


Line 1,552: Line 1,626:


{{Optimal ET sequence|legend=1| 10, 14, 24, 106, 130, 154, 178*, 202* }}
{{Optimal ET sequence|legend=1| 10, 14, 24, 106, 130, 154, 178*, 202* }}
: <nowiki />* wart for 13/5
=== Oceanfront ===
Related temperaments: [[Archytas clan #Superpyth|superpyth]], [[Archytas clan #Ultrapyth|ultrapyth]]
[[Subgroup]]: 2.3.7.13/5
[[Comma list]]: 64/63, 91/90
{{Mapping|legend=2| 1 0 6 -5 | 0 1 -2 4 }}
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~3/2 = 713.910
{{Optimal ET sequence|legend=1| 5, 22, 27, 32, 37 }}
[[Tp tuning #T2 tuning|RMS error]]: 2.063 cents
Scales: [[Oceanfront scales]]
== 2.….49/5.… subgroups ==
=== Direct breedsmic ===
Related temperament: [[hemithirds]], [[newt]]
[[Subgroup]]: 2.3.49/5
[[Comma list]]: 2401/2400
{{Mapping|legend=2| 1 1 3 | 0 2 1 }}
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~49/40 = 350.966
{{Optimal ET sequence|legend=1|7, 10, 17}}
[[Tp tuning #T2 tuning|RMS error]]: ?
== 2.….17/5.… subgroups ==
=== Fiventeen ===
Fiventeen tempers out [[136/135]] ({{monzo| 3 -3 1 }}) in 2.3.17/5. It equates [[17/15]] with [[9/8]], so it implies a [[supersoft]] [[pentic]] [[pentad]] of [[~]]30:34:40:45:51. [[17edo]] makes a good tuning especially for its size, which gives a [[supersoft]] pentic scale corresponding approximately to a just [[20/17]] tuning, although [[80edo]] might be preferred for an approximately just [[51/40]] to optimize plausibility slightly more, and [[97edo]] (= 80 + 17) and  [[114edo]] (= 97 + 17) do even better in striking a balance between 80edo's more stable tuning and that having 20/17 more accurate (as in 17edo) is useful because of the more convincing suggestion of the two 15:17:20 chords present in the fiventeen pentad. The same is true of the related rank-3 temperament diatic, for which the [[optimal ET sequence]] is much more characteristic of optimized tunings, finding [[34edo]], then [[80edo]], then [[114edo]] (= 34 + 80) and even [[194edo|194bc-edo]] (= 80 + 114), though because of its focus on primes 5 and 17 it misses 97edo as a tuning, and slightly less optimized though still interesting [[63edo]] and [[143edo]] (= 63 + 80) tunings are found in the optimal ET sequence for fiventeen.
[[Subgroup]]: 2.3.17/5
[[Comma list]]: 136/135 ({{monzo| 3 -3 1 }})
{{Mapping|legend=2| 1 0 -3 | 0 1 3 }}
: mapping generators: ~2, ~3
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.2838{{c}}, ~3/2 = 704.4600{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.5286{{c}}
{{Optimal ET sequence|legend=1| 5, 12, 17, 46, 63, 143 }}
== 2.….19/7.… subgroups ==
=== Surprise ===
This temperament was named by [[User:VectorGraphics|Vector]] in 2025, as he was surprised that the temperament of [[57/56]] did not have a name. This is the [[rank-2 temperament|rank-2]] version of the temperament; Vector surmises that the name ''hendrix'' would be more thoughtfully given to the [[rank-3]] version.
[[Subgroup]]: 2.3.19/7
[[Comma list]]: [[57/56]] ({{Monzo| -3 1 1 }})
{{Mapping|legend=2| 1 0 3 | 0 1 -1 }}
: mapping generators: ~2, ~3
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1202.4345{{c}}, ~3/2 = 697.4314{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 697.3981{{c}}
{{Optimal ET sequence|legend=1| 5, 7, 12, 19, 31*, 50* }}
<nowiki/>* wart for 19/7
[[Badness]] (Sintel): 0.082
=== Supramin ===
This is a remarkable low-complexity microtemperament that contains the 14:17:19 triad within just four generator steps. An excellent tuning is [[25edo]], which provides an accurate yet tone-efficient tuning of this temperament. It was named by [[User:Overthink|Overthink]] in 2026 after the fact that the generator is a [[17/14]] supraminor third, two of which reach [[28/19]].
[[Subgroup]]: 2.17/7.19/7
[[Comma list]]: [[5491/5488]] ({{Monzo| -4 2 1 }})
{{Mapping|legend=2| 1 0 4 | 0 1 -2 }}
: mapping generators: ~2, ~17/7
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1200.022{{c}}, ~17/14 = 335.793{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.000{{c}}, ~17/14 = 335.785{{c}}


<nowiki />* Wart for 13/5
{{Optimal ET sequence|legend=1| 7, 18, 25 }}


=== Oceanfront ===
[[Badness]] (Sintel): 0.005
Related temperaments: [[Archytas clan #Superpyth|superpyth]], [[Archytas clan #Ultrapyth|ultrapyth]]


[[Subgroup]]: 2.3.7.13/5
==== Supramine ====
This extension approximates the 14:17:19:23:25 pentad in just six generator steps, at the cost of some accuracy. 25edo remains a strong tuning.


[[Comma list]]: 64/63, 91/90
Subgroup: 2.17/7.19/7.23/7


{{Mapping|legend=2| 1 0 6 -5 | 0 1 -2 4 }}
Comma list: [[323/322]], [[392/391]]


[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~3/2 = 713.910
Subgroup-val mapping: {{Mapping| 1 0 4 3 | 0 1 -2 -1 }}


{{Optimal ET sequence|legend=1| 5, 22, 27, 32, 37 }}
Optimal tunings:
* Subgroup WE: ~2 = 1199.871{{c}}, ~17/14 = 336.243{{c}}
* Subgroup CWE: ~2 = 1200.000{{c}}, ~17/14 = 336.296{{c}}


[[Tp tuning #T2 tuning|RMS error]]: 2.063 cents
{{Optimal ET sequence|legend=0| 7, 18, 25 }}


Scales: [[Oceanfront scales]]
Badness (Sintel): 0.029


== 2.….49/5… subgroups ==
==== 2.25/7.17/7.19/7.23/7 subgroup ====
=== Direct breedsmic ===
Related temperament: [[hemithirds]], [[newt]]


[[Subgroup]]: 2.3.49/5
Subgroup: 2.25/7.17/7.19/7.23/7


[[Comma list]]: 2401/2400
Comma list: [[323/322]], [[392/391]], [[476/475]]


{{Mapping|legend=2| 1 1 3 | 0 2 1 }}
Subgroup-val mapping: {{Mapping| 1 -2 0 4 3 | 0 3 1 -2 -1 }}


[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~49/40 = 350.966
Optimal tunings:
 
* Subgroup WE: ~2 = 1199.757{{c}}, ~17/14 = 335.428{{c}}
{{Optimal ET sequence|legend=1|?}}
* Subgroup CWE: ~2 = 1200.000{{c}}, ~17/14 = 335.479{{c}}


[[Tp tuning #T2 tuning|RMS error]]: ?
{{Optimal ET sequence|legend=0| 7, 18, 25 }}


Badness (Sintel): 0.053


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


Line 1,611: Line 1,772:


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


Line 1,617: Line 1,777:


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


==== 3/2.5/2.7/2.11/2 ====
==== 3/2.5/2.7/2.11/2 ====
Line 1,626: Line 1,785:


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


Line 1,632: Line 1,790:


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


==== 3/2.5/2.7/2.11/2.13/2 ====
==== 3/2.5/2.7/2.11/2.13/2 ====
Line 1,645: Line 1,802:


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


=== Semiwolf ===
=== Semiwolf ===
Line 1,683: Line 1,839:
[[Optimal ET sequence]]: [[8edf]], [[11edf]]
[[Optimal ET sequence]]: [[8edf]], [[11edf]]


== 3/2.5/4… subgroups ==
== 3/2.5/4.… subgroups ==
=== Poseidon ===
=== Poseidon ===
'''This temperament will be subjected to renaming due to a conflict.'''
'''This temperament will be subjected to renaming due to a conflict.'''
Line 1,712: Line 1,868:


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


=== Doubleton ===
=== Doubleton ===
Line 1,727: Line 1,882:


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


== 5/2-equave subgroups ==
== 5/2-equave subgroups ==
=== Hyperion ===
=== Hyperion ===
[[Subgroup]]: 5/2.7.11
[[Subgroup]]: 5/2.7.11
Line 1,744: Line 1,897:


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


= Related temperament collections =
= Related temperament collections =
Line 1,752: Line 1,904:
* [[Substitute harmonic]] temperaments
* [[Substitute harmonic]] temperaments


<!-- main article -->
[[Category:Subgroup temperaments| ]] <!-- main article -->
 
[[Category:Temperament collections]]
[[Category:Temperament collections]][[Category:Subgroup]]
{{Todo| review | cleanup }}
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