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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]]
Line 11: Line 13:


= Composite subgroup temperaments =
= Composite subgroup temperaments =
== 2.3.35 subgroup ==
=== Shaka ===
{{See also|Kalismic temperaments}}
Two commas that split 2/1 in half, corresponding to convergents to sqrt(2), are the [[1682/1681|''sha''ftesburisma]] [[Square superparticular|S29]]/S41 and the [[9801/9800|''ka''lisma S99]], prompting to temper out {S29, S41, S99}, approximating /29 and /41 [[Primodality|primodal]] chords well.
Subgroup: 2.3.35.11.29.41
Comma list: 841/840, 1189/1188, 1681/1680
{{Mapping|legend=2|2 2 6 5 7 8|0 1 1 -1 1 1|0 0 2 2 1 1}}
Optimal tuning (CTE): ~41/29 = 1\2, ~3/2 = 702.031, ~41/24 = 926.693
[[Support]]ing [[ET]]s: {{EDOs|22, 26, 36, 48, 70, 96, 106, 118, 140, 154, 176, 188, 224, 272, 294, 342}}
Scale: [[Shaka10]]
== 2.9.5.7 subgroup ==
== 2.9.5.7 subgroup ==
See also [[Jubilismic clan #Antikythera|antikythera]] and [[Hemimean clan #Isra|isra]].  
See also [[Jubilismic clan #Antikythera|antikythera]] and [[Hemimean clan #Isra|isra]].  
Line 89: Line 73:
=== Baldy ===
=== Baldy ===
{{See also|Schismatic family #Garibaldi}}
{{See also|Schismatic family #Garibaldi}}
{{See also|No-threes subgroup temperaments #Frostburn}}


Baldy results from taking every other generator of the [[garibaldi temperament]]. One of the best extension is 2.9.5.7.13 subgroup with mapping 13/8 to +10 whole tones, as well as the cassandra temperament.
Baldy results from taking every other generator of the [[garibaldi]] temperament. One of the best extension is 2.9.5.7.13 subgroup with mapping 13/8 to +10 whole tones, as well as the cassandra temperament.


[[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 ==
Line 289: Line 313:


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 ==
Line 396: Line 435:
[[Tp tuning #T2 tuning|RMS error]]: 0.3202 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.3202 cents


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


Line 412: Line 466:


==== B-team ====
==== B-team ====
B-team is every other step of [[rodan]].
B-team (23 & 41) is every other step of [[rodan]].


Subgroup: 2.9.15.21.33
Subgroup: 2.9.15.21.33
Line 422: 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
 
[[Optimal tuning]] ([[POTE]]): 1\5ed4 = 480.0, ~16/15 = 80.4062


Sharp: 12/11, 25/21, 33/26, 18/13, 31/21 ~ 65/44 ~ 96/65, 50/31 ~ 29/18, 55/32, 15/8.
[[Support]]ing [[ET]]s: {{EDs|5, 10, 15, 20, 25, 30, 55, 85, 115|equave=4}}


Flat: 16/15, 64/55, 31/25 ~ 36/29, 42/31 ~ 65/48 ~ 88/65, 13/9, 52/33, 42/25, 11/6.
== 4.6.5 subgroup ==
=== Meanquad ===
[[Subgroup]]: 2.15.189.55.325.725.279
{{Main| Meanquad }}


[[Comma list]]: [[1625/1624]], [[2016/2015]], [[2080/2079]], [[3025/3024]], [[4096/4095]]
[[Subgroup]]: 4.6.5


{{Mapping|legend=2| 1 0 6 5 6 -3 5 | 0 5 2 1 3 16 4 }}
[[Comma list]]: [[81/80]] = {{monzo| -4 4 -1 }}


[[Optimal tuning]] (subgroup [[CTE]]): ~55/32 = 937.638
{{Mapping|legend=2| 1 0 -4| 0 1 4 }}


[[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 generators: ~4, ~6


== 4.3.5 subgroup ==
[[Optimal tuning]] (subgroup [[CTE]]): ~4 = 2\1, ~3/2 = 697.214
=== Tetrahanson ===
{{Main| Tetrahanson }}


[[Subgroup]]: 4.3.5
[[Support]]ing [[ET]]s: *7, *10, *11[-5], *13[+5], *17, *24, *27[+5], *31, *38, *41, *45, *52, *55, *69


[[Comma list]]: 15625/15552
<nowiki />* Wart for 4


{{Mapping|legend=2| 1 3 3 | 0 -6 -5 }}
==== 4.6.5.7 subgroup (tetrominant) ====
[[Subgroup]]: 4.6.5.7


: Mapping generators: ~4, ~5/3
[[Comma list]]: [[36/35]] = {{monzo| 0 2 -1 -1 }}, [[64/63]] = {{monzo| 4 -2 0 -1 }}


[[Optimal tuning]] ([[CTE]]): ~4 = 2\1, ~5/3 = 882.941
{{Mapping|legend=2| 1 0 -4 4 | 0 1 4 -2 }}


[[Support]]ing [[ET]]s: {{EDs|19, 106, 87, 68, 11, 8, 125, 49, 30, 27, 117, 46, 41b, 79|equave=4}}
[[Optimal tuning]] (subgroup [[CTE]]): ~4 = 2\1, ~3/2 = 699.622


=== Tetrameantone ===
[[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]
{{Main| Tetrameantone }}


[[Subgroup]]: 4.3.5
<nowiki />* Wart for 4


[[Comma list]]: 81/80
=== 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|legend=2| 1 1 2 | 0 -1 -4 }}
Fourwar is named after the closely related [[hemiwar]] temperament.


: Mapping generators: ~4, ~4/3
{{Todo|inline=1|cleanup}}


[[Optimal tuning]] ([[POTE]]): 4 = 2400.0, ~4/3 = 503.761
<pre>
Reduced Mapping
4 6 5
[ ⟨ 1 0 1 ]
⟨ 0 16 2 ] ⟩
TE Generator Tunings (cents)
⟨2399.3973, 193.8643]
TE Step Tunings (cents)
⟨25.21211, 47.81337]
TE Tuning Map (cents)
⟨2399.397, 3101.829, 2787.126]
TE Mistunings (cents)
⟨-0.603, -0.126, 0.812]
Complexity 1.369085
Adjusted Error 0.692892 cents
TE Error 0.268047 cents/octave
Unison Vector
[8, 1, -8⟩ (393216:390625)


[[Support]]ing [[ET]]s: {{EDs|5, 9, 14, 19, 24, 43, 62, 81, 100|equave=4}}
Subsets
q99, q62, q37, q161, q136, q198, q25, q124, q74, q235
</pre>


=== Tetramagic ===
==== 4.6.5.7 ====
 
<pre>
[[Subgroup]]: 4.3.5
Reduced Mapping
 
4 6 5 7
[[Comma list]]: 3125/3072
[ ⟨ 1 0 1 1 ]
 
⟨ 0 16 2 5 ] ⟩
{{Mapping|legend=2| 1 0 1 | 0 5 1 }}
 
TE Generator Tunings (cents)
: Mapping generators: ~4, ~5/4
⟨2399.4195, 193.8654]
 
[[Optimal tuning]] ([[POTE]]): 4 = 2400.0, ~5/4 = 380.059
TE Step Tunings (cents)
 
⟨25.23883, 47.79592]
[[Support]]ing [[ET]]s: {{EDs|6, 13, 19, 25, 38, 44, 63, 82|equave=4}}
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)


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


[[Subgroup]]: 4.3.5
==== 4.6.5.7.11 ====
 
<pre>
[[Comma list]]: 256/243
Reduced Mapping
4 6 5 7 11
[ ⟨ 1 0 1 1 1 ]
⟨ 0 16 2 5 9 ] ⟩
TE Generator Tunings (cents)
⟨2400.1097, 193.9498]
TE Step Tunings (cents)
⟨24.18752, 48.52491]
TE Tuning Map (cents)
⟨2400.110, 3103.196, 2788.009, 3369.859, 4145.658]
TE Mistunings (cents)
⟨0.110, 1.241, 1.696, 1.033, -5.660]
Complexity 1.068792
Adjusted Error 2.926965 cents
TE Error 0.846083 cents/octave
Unison Vectors
[-1, -1, -1, 0, 2⟩ (121:120)
[2, 0, -2, -1, 1⟩ (176:175)
[-3, -1, 1, 1, 1⟩ (385:384)
[-1, 0, 3, -3, 1⟩ (1375:1372)
[-2, -1, -2, 4, 0⟩ (2401:2400)
[1, 0, 1, -4, 2⟩ (2420:2401)


{{Mapping|legend=2| 5 4 6 | 0 0 -1 }}
Subsets
q37, q25, q62, q12, q74, q99, q87, q49r, q50r, q124
</pre>


: Mapping generators: ~4, ~16/15
==== 4.6.5.7.11.13 ====


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


{{Mapping|legend=2| 1 0 -4| 0 1 4 }}
Subsets
q25, q37f, q12f, q62, q50rf, q13rff, q49rff, q87, q74ff, q24rfff
</pre>


: mapping generators: ~4, ~6
==== 4.6.5.7.11.13.17 ====
 
<pre>
[[Optimal tuning]] (subgroup [[CTE]]): ~4 = 2\1, ~3/2 = 697.214
 
[[Support]]ing [[ET]]s: *7, *10, *11[-5], *13[+5], *17, *24, *27[+5], *31, *38, *41, *45, *52, *55, *69
 
<nowiki />* Wart for 4
 
==== 4.6.5.7 subgroup (tetrominant) ====
[[Subgroup]]: 4.6.5.7
 
[[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
[ ⟨ 1 0 1 ]
[ ⟨ 1 0 1 1 1 0 1 ]
⟨ 0 16 2 ] ⟩
⟨ 0 16 2 5 9 23 13 ] ⟩
   
   
TE Generator Tunings (cents)
TE Generator Tunings (cents)
⟨2399.3973, 193.8643]
⟨2400.4701, 193.4599]
   
   
TE Step Tunings (cents)
TE Step Tunings (cents)
⟨25.21211, 47.81337]
⟨43.39350, 35.55764]
   
   
TE Tuning Map (cents)
TE Tuning Map (cents)
⟨2399.397, 3101.829, 2787.126]
⟨2400.470, 3095.359, 2787.390, 3367.770, 4141.609, 4449.578, 4915.449]
   
   
TE Mistunings (cents)
TE Mistunings (cents)
⟨-0.603, -0.126, 0.812]
⟨0.470, -6.596, 1.076, -1.056, -9.709, 9.050, 10.494]
   
   
Complexity 1.369085
Complexity 1.129881
Adjusted Error 0.692892 cents
Adjusted Error 8.082725 cents
TE Error 0.268047 cents/octave
TE Error 1.977443 cents/octave
   
   
Unison Vector
Unison Vectors
[8, 1, -8⟩ (393216:390625)
[0, 1, -1, 0, 1, -1, 0⟩ (66:65)
 
[1, 1, 1, -1, 0, 0, -1⟩ (120:119)
[1, 2, 0, 0, -1, -1, 0⟩ (144:143)
[-2, 1, 1, 1, 0, -1, 0⟩ (105:104)
[-1, 2, 2, 0, 0, -1, -1⟩ (225:221)
[-1, 1, 2, -2, 0, -1, 1⟩ (1275:1274)
 
Subsets
Subsets
q99, q62, q37, q161, q136, q198, q25, q124, q74, q235
q25, q12f, q37f, q13rffg, q50rf, q62, q49rffg, q24rfffg, q38rreffg, q74ffg
</pre>
</pre>


==== 4.6.5.7 ====
==== 4.6.5.7.11.13.17.19 ====
<pre>
<pre>
Reduced Mapping
Reduced Mapping
4 6 5 7
4 6 5 7 11 13 17 19
[ ⟨ 1 0 1 1 ]
[ ⟨ 1 0 1 1 1 0 1 1 ]
⟨ 0 16 2 5 ] ⟩
⟨ 0 16 2 5 9 23 13 14 ] ⟩
   
   
TE Generator Tunings (cents)
TE Generator Tunings (cents)
⟨2399.4195, 193.8654]
⟨2399.9219, 193.3952]
   
   
TE Step Tunings (cents)
TE Step Tunings (cents)
⟨25.23883, 47.79592]
⟨44.14256, 35.03670]
   
   
TE Tuning Map (cents)
TE Tuning Map (cents)
⟨2399.420, 3101.846, 2787.150, 3368.747]
⟨2399.922, 3094.324, 2786.712, 3366.898, 4140.479, 4448.090, 4914.060, 5107.455]
   
   
TE Mistunings (cents)
TE Mistunings (cents)
⟨-0.580, -0.109, 0.837, -0.079]
⟨-0.078, -7.631, 0.399, -1.928, -10.839, 7.562, 9.104, 9.942]
   
   
Complexity 1.192044
Complexity 1.058472
Adjusted Error 0.653313 cents
Adjusted Error 8.712222 cents
TE Error 0.232715 cents/octave
TE Error 2.050935 cents/octave
   
   
Unison Vectors
Unison Vectors
[-2, -1, -2, 4⟩ (2401:2400)
[0, 1, -1, 0, 1, -1, 0, 0⟩ (66:65)
[3, 0, -5, 2⟩ (3136:3125)
[-1, 0, 0, 1, 1, 0, 0, -1⟩ (77:76)
[5, 1, -3, -2⟩ (6144:6125)
[2, 1, -1, 0, 0, 0, 0, -1⟩ (96:95)
[8, 1, -8, 0⟩ (393216:390625)
[1, 1, 1, -1, 0, 0, -1, 0⟩ (120:119)
[0, 1, 1, 1, -1, 0, 0, -1⟩ (210:209)
[0, 0, 1, -2, 1, 0, 1, -1⟩ (935:931)
[2, 0, -3, 1, 0, 0, -1, 1⟩ (2128:2125)


Subsets
Subsets
q99, q62, q37, q161, q136, q198, q25, q124, q74, q235
q25, q12fh, q37f, q13rffgh, q50rf, q62, q49rffgh, q24rfffghh, q38rreffgh, q74ffgh
</pre>
</pre>


==== 4.6.5.7.11 ====
==== 4.6.5.7.11.13.17.19.23 ====
<pre>
<pre>
Reduced Mapping
Reduced Mapping
4 6 5 7 11
4 6 5 7 11 13 17 19 23
[ ⟨ 1 0 1 1 1 ]
[ ⟨ 1 0 1 1 1 0 1 1 0 ]
⟨ 0 16 2 5 9 ] ⟩
⟨ 0 16 2 5 9 23 13 14 28 ] ⟩
   
   
TE Generator Tunings (cents)
TE Generator Tunings (cents)
⟨2400.1097, 193.9498]
⟨2399.3286, 193.5316]
   
   
TE Step Tunings (cents)
TE Step Tunings (cents)
⟨24.18752, 48.52491]
⟨37.31613, 39.63311]
   
   
TE Tuning Map (cents)
TE Tuning Map (cents)
⟨2400.110, 3103.196, 2788.009, 3369.859, 4145.658]
⟨2399.329, 3096.506, 2786.392, 3366.987, 4141.113, 4451.227, 4915.240, 5108.771, 5418.885]
   
   
TE Mistunings (cents)
TE Mistunings (cents)
⟨0.110, 1.241, 1.696, 1.033, -5.660]
⟨-0.671, -5.449, 0.078, -1.839, -10.205, 10.699, 10.284, 11.258, -9.389]
   
   
Complexity 1.068792
Complexity 1.115920
Adjusted Error 2.926965 cents
Adjusted Error 9.502017 cents
TE Error 0.846083 cents/octave
TE Error 2.100561 cents/octave
   
   
Unison Vectors
Unison Vectors
[-1, -1, -1, 0, 2⟩ (121:120)
[0, 1, -1, 0, 1, -1, 0, 0, 0⟩ (66:65)
[2, 0, -2, -1, 1⟩ (176:175)
[1, 0, 0, -1, 0, -1, 0, 0, 1⟩ (92:91)
[-3, -1, 1, 1, 1⟩ (385:384)
[0, -1, 1, 0, 0, 0, 0, -1, 1⟩ (115:114)
[-1, 0, 3, -3, 1⟩ (1375:1372)
[1, 1, 1, -1, 0, 0, -1, 0, 0⟩ (120:119)
[-2, -1, -2, 4, 0⟩ (2401:2400)
[2, 0, -2, -1, 1, 0, 0, 0, 0⟩ (176:175)
[1, 0, 1, -4, 2⟩ (2420:2401)
[-3, -1, 1, 1, 1, 0, 0, 0, 0⟩ (385:384)
[1, 0, -2, 1, 0, 0, 1, -1, 0⟩ (476:475)
[1, 0, 0, -2, 1, 0, -1, 1, 0⟩ (836:833)
[0, 0, 1, -2, 1, 0, 1, -1, 0⟩ (935:931)
[1, -1, 0, 0, 0, 0, -2, 1, 1⟩ (874:867)


Subsets
Subsets
q37, q25, q62, q12, q74, q99, q87, q49r, q50r, q124
q25i, q12fhi, q37f, q13rffghii, q62, q50rfii, q49rffghii, q24rfffghhiii, q74ffghi, q38rreffghiii
</pre>
</pre>


==== 4.6.5.7.11.13 ====
== 4.9.25 subgroup ==
=== Meansquared ===
[[Subgroup]]: 4.9.25
 
[[Comma list]]: [[6561/6400]]
 
{{Mapping|legend=2| 1 3 4 | 0 1 4 }}
 
Mapping generators: ~4, ~9/64
 
[[Optimal tuning]] ([[CTE]]): ~4 = 2\1, ~9/4 = 1394.429
 
[[Support]]ing [[ET]]s: 12, 7, 19, 5, 31, 26, 17[+25], 43, 9[-25], 33[-25], 45, 29[+25], 8[+25], 22[+25]


<pre>
== 4.9.49 subgroup ==
Reduced Mapping
=== Archsquared ===
4 6 5 7 11 13
[[Subgroup]]: 4.9.49
[ ⟨ 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
[[Comma list]]: 4096/3969
q25, q37f, q12f, q62, q50rf, q13rff, q49rff, q87, q74ff, q24rfff
</pre>


==== 4.6.5.7.11.13.17 ====
{{Mapping|legend=2| 1 3 0 | 0 1 -2 }}
<pre>
 
Reduced Mapping
Mapping generators: ~4, ~9/64
4 6 5 7 11 13 17
 
[ ⟨ 1 0 1 1 1 0 1 ]
[[Optimal tuning]] ([[CTE]]): ~9/4 = 1419.190
⟨ 0 16 2 5 9 23 13 ] ⟩
 
[[Support]]ing [[ET]]s: 5, 17, 22, 12, 7, 27, 32, 8, 39[+49], 29[+49], 9[+49], 19[+49], 37, 49
TE Generator Tunings (cents)
 
⟨2400.4701, 193.4599]
== 8.9.7 subgroup ==
=== Sixscared ===
TE Step Tunings (cents)
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."
⟨43.39350, 35.55764]
 
[[Subgroup]]: 8.9.7
TE Tuning Map (cents)
 
⟨2400.470, 3095.359, 2787.390, 3367.770, 4141.609, 4449.578, 4915.449]
[[Comma list]]: 64/63
 
TE Mistunings (cents)
{{Mapping|legend=2| 1 0 2 | 0 1 -1 }}
⟨0.470, -6.596, 1.076, -1.056, -9.709, 9.050, 10.494]
 
: sval mapping generators: ~8, ~9
Complexity 1.129881
 
Adjusted Error 8.082725 cents
: [[gencom]]: [8 9/8; 64/63]
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 tuning]] ([[CTE]]): ~9/8 = 219.1898
q25, q12f, q37f, q13rffg, q50rf, q62, q49rffg, q24rfffg, q38rreffg, q74ffg
</pre>


==== 4.6.5.7.11.13.17.19 ====
[[Optimal ET sequence]]: {{val| 16 17 15 }}, {{val| 33 35 31 }}, {{val| 148 … }}, {{val| 181 … }}, {{val| 214 … }}, {{val| 247 … }}
<pre>
 
Reduced Mapping
[[Badness]]: 0.0215 × 10<sup>-3</sup>
4 6 5 7 11 13 17 19
 
[ ⟨ 1 0 1 1 1 0 1 1 ]
= Fractional subgroup temperaments =
⟨ 0 16 2 5 9 23 13 14 ] ⟩
== 2.5/3.… subgroups ==
=== Magicaltet ===
TE Generator Tunings (cents)
{{See also| Chromatic pairs #Magicaltet }}
⟨2399.9219, 193.3952]
 
Magicaltet is related to [[keemic]], [[superkleismic]], and [[magic]]. The tonic and the first three generator steps make a [[magical seventh chord]], hence the name.  
TE Step Tunings (cents)
 
⟨44.14256, 35.03670]
[[Subgroup]]: 2.5/3.7.11
 
TE Tuning Map (cents)
[[Comma list]]: 100/99 ({{monzo| 2 2 0 -1 }}), 385/384 ({{monzo| -7 1 1 1 }})
⟨2399.922, 3094.324, 2786.712, 3366.898, 4140.479, 4448.090, 4914.060, 5107.455]
 
{{Mapping|legend=2| 1 0 5 2 | 0 1 -3 2 }}
TE Mistunings (cents)
: mapping generators: ~2, ~5/3
⟨-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
{{Mapping|legend=3| 1 -1/2 1/2 2 4 | 0 1/2 -1/2 3 -2 }}
q25, q12fh, q37f, q13rffgh, q50rf, q62, q49rffgh, q24rfffghh, q38rreffgh, q74ffgh
: [[gencom]]: [2 6/5; 100/99 385/384]
</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)
[[Tp tuning #T2 tuning|RMS error]]: 1.206 cents
⟨2399.3286, 193.5316]
 
=== Starlingtet ===
TE Step Tunings (cents)
{{See also | Chromatic pairs #Starlingtet }}
⟨37.31613, 39.63311]
 
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.  
TE Tuning Map (cents)
 
⟨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)
[[Comma list]]: [[126/125]] ({{monzo| 1 -3 1 }})
⟨-0.671, -5.449, 0.078, -1.839, -10.205, 10.699, 10.284, 11.258, -9.389]
 
{{Mapping|legend=2| 1 0 -1 | 0 1 3 }}
Complexity 1.115920
 
Adjusted Error 9.502017 cents
: mapping generators: ~2, ~5/3
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|legend=3| 1 -1 0 1 | 0 4/3 1/3 -5/3 }}
q25i, q12fhi, q37f, q13rffghii, q62, q50rfii, q49rffghii, q24rfffghhiii, q74ffghi, q38rreffghiii
: [[gencom]]: [2 6/5; 126/125]
</pre>


== 4.9.25 subgroup ==
[[Optimal tuning]]s:
=== Meansquared ===
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 888.759
[[Subgroup]]: 4.9.25
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 888.846


[[Comma list]]: [[6561/6400]]
{{Optimal ET sequence|legend=1| 4, 15, 19, 23, 27 }}


{{Mapping|legend=2| 1 3 4 | 0 1 4 }}
[[Tp tuning #T2 tuning|RMS error]]: 0.8398 cents


Mapping generators: ~4, ~9/64
==== Greeley ====
{{See also| Chromatic pairs #Greeley }}


[[Optimal tuning]] ([[CTE]]): ~4 = 2\1, ~9/4 = 1394.429
Greeley is related to [[opossum]] as well as to [[nusecond]].  


[[Support]]ing [[ET]]s: 12, 7, 19, 5, 31, 26, 17[+25], 43, 9[-25], 33[-25], 45, 29[+25], 8[+25], 22[+25]
[[Subgroup]]: 2.5/3.7/3.11/3


== 4.9.49 subgroup ==
[[Comma list]]: 121/120 ({{monzo| -3 -1 0 2 }}), 126/125 ({{monzo| 1 -3 1 }})
=== Archsquared ===
[[Subgroup]]: 4.9.49


[[Comma list]]: 4096/3969
{{Mapping|legend=2| 1 1 2 2 | 0 -2 -6 -1 }}


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


Mapping generators: ~4, ~9/64
[[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]]): ~[[9/8]] = 219.190
{{Optimal ET sequence|legend=1| 8, 15, 23, 54, 77, 100, 131* }}
: <nowiki/>* wart for 11/3


[[Support]]ing [[ET]]s: 5, 17, 22, 12, 7, 27, 32, 8, 39[+49], 29[+49], 9[+49], 19[+49], 37, 49
[[Tp tuning #T2 tuning|RMS error]]: 1.034 cents


== 8.9.7 subgroup ==
==== Skateboard ====
=== Sixscared ===
{{See also| Chromatic pairs #Skateboard }}
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
Skateboard is related to [[thrasher]].  


[[Comma list]]: 64/63
[[Subgroup]]: 2.5/3.7/3.11.13/9


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


: sval mapping generators: ~8, ~9
{{Mapping|legend=2| 1 0 -1 2 2 | 0 1 3 2 -2 }}


: [[gencom]]: [8 9/8; 64/63]
{{Mapping|legend=3| 1 -3/7 4/7 11/7 4 -6/7 | 0 0 -1 -3 -2 2 }}
: [[gencom]]: [2 6/5; 56/55 91/90 100/99]


[[Optimal tuning]] ([[CTE]]): 1\[[3ed8]] = 1600.0, ~9/8 = 219.1898
[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 886.158
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 886.158


[[Optimal ET sequence]]: {{val| 16 17 15 }}, {{val| 33 35 31 }}, {{val| 148 … }}, {{val| 181 … }}, {{val| 214 … }}, {{val| 247 … }}
{{Optimal ET sequence|legend=1| 11, 15, 19, 23, 42d, 65d }}


[[Badness]]: 0.0215 × 10<sup>-3</sup>
[[Tp tuning #T2 tuning|RMS error]]: 2.396 cents


= Fractional subgroup temperaments =
=== Gariberttet ===
== 2.5/3… subgroups ==
Gariberttet is the 2.5/3.7/3 [[Subgroup temperament families, relationships, and genes|altergene]] of [[sirius]].
=== 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.  
==== Gariberttet (2.5/3.7/3.13/11 subgroup) ====
{{See also | Chromatic pairs #Gariberttet }}


[[Subgroup]]: 2.5/3.7.11
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]].


[[Comma list]]: 100/99 = {{monzo| 2 2 0 -1 }}, 385/384 = {{monzo| -7 1 1 1 }}
[[Subgroup]]: 2.5/3.7/3.13/11


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


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


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


: [[gencom]]: [2 6/5; 100/99 385/384]
[[Optimal tuning]]s:  
* [[Tp tuning|subgroup]] [[CTE]] and [[POTE]]: ~2 = 1200.000, ~13/11 = 293.679


[[Optimal tuning]]s:
{{Optimal ET sequence|legend=1| 29, 33, 37, 41, 45, 49, 78, 94, 143* }}
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 877.343
: <nowiki/>* wart for 13/11
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 877.351


{{Optimal ET sequence|legend=1| 4, 7, 11, 15, 26, 67, 93* }}
[[Tp tuning #T2 tuning|RMS error]]: 0.6914 cents


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


[[Tp tuning #T2 tuning|RMS error]]: 1.206 cents
Indium can be described as the {{nowrap| 8 & 33 }} temperament in the 2.5/3.7/3.11/3 subgroup.  


=== Starlingtet ===
[[Subgroup]]: 2.5/3.7/3.11/3
{{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.
[[Comma list]]: [[3025/3024]] ({{monzo| -4 2 -1 2 }}), [[3125/3087]] ({{monzo| 0 5 -3 }})


[[Subgroup]]: 2.5/3.7/3
{{Mapping|legend=2| 1 0 0 2 | 0 6 10 -1 }}


[[Comma list]]: [[126/125]] = {{monzo| 1 -3 1 }}
{{Mapping|legend=3| 1 -1/2 -1/2 -1/2 3/2 | 0 -15/4 9/4 25/4 -19/4 }}
: [[gencom]]: [2 12/11; 3025/3024 3125/3087]


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


: mapping generators: ~2, ~5/3
{{Optimal ET sequence|legend=1| 8, 33, 41, 49, 204*<sup>†</sup> }}
: <nowiki/>* wart for 7/3
: <sup>†</sup> wart for 11/3


{{Mapping|legend=3| 1 -1 0 1 | 0 4/3 1/3 -5/3 }}
[[Tp tuning #T2 tuning|RMS error]]: 0.7788 cents


: [[gencom]]: [2 6/5; 126/125]
==== Ammon ====
{{See also| Chromatic pairs #Ammon }}


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


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


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


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


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


[[Comma list]]: 121/120 = {{monzo| -3 -1 0 2 }}, 126/125 = {{monzo| 1 -3 1 }}
{{Optimal ET sequence|legend=1| 8, 29, 37, 45 }}


{{Mapping|legend=2| 1 1 2 2 | 0 -2 -6 -1 }}
[[Tp tuning #T2 tuning|RMS error]]: 1.052 cents


{{Mapping|legend=3| 1 -5/4 -1/4 3/4 3/4 | 0 9/4 1/4 -15/4 5/4 }}
=== Sentry ===
{{See also | Chromatic pairs #Sentry }}


: [[gencom]]: [2 11/10; 121/120 126/125]
Sentry, the {{nowrap| 3 & 5 }} temperament in the 2.5/3.9/7 subgroup, is related to [[sensi]].


[[Optimal tuning]]s:  
[[Subgroup]]: 2.5/3.9/7
* [[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]]: [[245/243]] ({{monzo| 0 1 -2 }})


<nowiki />* Wart for 11/3
{{Mapping|legend=2| 1 0 0 | 0 2 1 }}


[[Tp tuning#T2 tuning|RMS error]]: 1.034 cents
{{Mapping|legend=3| 1 0 0 0 | 0 0 2 -1 }}
: [[gencom]]: [2 9/7; 245/243]


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


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


[[Subgroup]]: 2.5/3.7/3.11.13/9
[[Tp tuning #T2 tuning|RMS error]]: 0.7105 cents


[[Comma list]]: 56/55 = {{monzo| 3 -1 1 -1 }}, 91/90 = {{monzo| -1 -1 1 0 1 }}, 100/99 = {{monzo| 2 2 0 -1 }}
=== Marveltwintri ===
{{See also| Chromatic pairs #Marveltwintri }}


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


{{Mapping|legend=3| 1 -3/7 4/7 11/7 4 -6/7 | 0 0 -1 -3 -2 2 }}
[[Subgroup]]: 2.5/3.13/9


: [[gencom]]: [2 6/5; 56/55 91/90 100/99]
[[Comma list]]: [[325/324]] ({{monzo| -2 2 1 }})


[[Optimal tuning]]s:
{{Mapping|legend=2| 1 0 2 | 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=3| 1 -1/6 5/6 0 0 -1/3 | 0 -1/2 -3/2 0 0 1 }}
: [[gencom]]: [2 6/5; 325/324]


[[Tp tuning #T2 tuning|RMS error]]: 2.396 cents
[[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


=== Gariberttet ===
{{Optimal ET sequence|legend=1| 3, 4, 11, 15, 19, 34, 53, 87, 140 }}
Gariberttet is the 2.5/3.7/3 [[Subgroup Temperament Families, Relationships, and Genes|altergene]] of [[sirius]].


==== Gariberttet (2.5/3.7/3.13/11 subgroup) ====
[[Tp tuning #T2 tuning|RMS error]]: 0.2444 cents
{{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.
== 2.….7/3.… subgroups ==
=== Guanyintet ===
{{See also | Chromatic pairs #Guanyintet }}


[[Subgroup]]: 2.5/3.7/3.13/11
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.


[[Comma list]]: [[275/273]] = {{monzo| 0 2 -1 -1 }}, [[847/845]] = {{monzo| 0 -1 1 -2 }}
[[Subgroup]]: 2.5.7/3.11/3


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


{{Mapping|legend=3| 1 0 0 0 0 0 | 0 -8/3 1/3 7/3 -1/2 1/2 }}
{{Mapping|legend=2| 1 0 1 3 | 0 -3 1 -5 }}
: mapping generators: ~2, ~7/6


: [[gencom]]: [2 13/11; 275/273 847/845]
{{Mapping|legend=3| 1 -4/3 3 -1/3 5/3 | 0 4/3 -3 7/3 -11/3 }}
: [[gencom]]: [2 7/6; 176/175 540/539]


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[Tp tuning|subgroup]] [[CTE]] and [[POTE]]: ~2 = 1200.000, ~13/11 = 293.679
* ([[Tp tuning|subgroup]] [[CTE]]): ~2 = 1200.000, ~7/6 = 270.455
* ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1200.000, ~7/6 = 270.093


{{Optimal ET sequence|legend=1| 29, 33, 37, 41, 45, 49, 78, 94, 143* }}
{{Optimal ET sequence|legend=1| 9, 22, 31, 40, 191c*, 231c*, 271c*, 311c* }}
: <nowiki/>* wart for 7/3


<nowiki />* Wart for 13/11
[[Tp tuning #T2 tuning|RMS error]]: 0.6028 cents


[[Tp tuning #T2 tuning|RMS error]]: 0.6914 cents
==== Tridecimal guanyintet ====
Guanyintet can extend to the 13th harmonic by the equivalences ([[12/11]])<sup>3</sup> = [[13/10]] and ([[15/14]])<sup>3</sup> = [[16/13]], therefore tempering out {S11/S12/S14/S15}. However, note that it is not supported by the 31 & 53 orwell extension dubbed "tridecimal orwell", but instead the less accurate [[winston]] (22f & 31), as orwell prefers slightly sharper tunings than guanyintet. [[40edo]] remains an excellent tuning.


==== Indium ====
[[Subgroup]]: 2.5.7/3.11/3.13
{{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.
[[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 }})


[[Subgroup]]: 2.5/3.7/3.11/3
{{Mapping|legend=2| 1 0 1 3 1 | 0 -3 1 -5 12 }}
: mapping generators: ~2, ~12/7


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


{{Mapping|legend=2| 1 0 0 2 | 0 6 10 -1 }}
{{Optimal ET sequence|legend=1| 9, 22, 31, 40, 71, 111, 151, 262c*}} <small> using subgroup TE </small>
: <nowiki/>* wart for 7/3


{{Mapping|legend=3| 1 -1/2 -1/2 -1/2 3/2 | 0 -15/4 9/4 25/4 -19/4 }}
Badness (Sintel): 0.329


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


[[Optimal tuning]]s:
Laz is related to [[avalokita]] as well as to [[winston]].  
* [[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> }}
[[Subgroup]]: 2.5.7/3.11/3.13/3


<nowiki />* Wart for 7/3
[[Comma list]]: [[144/143]] ({{monzo| 4 0 0 -1 -1 }}), [[176/175]] ({{monzo| 4 -2 -1 1 }}), [[196/195]] ({{monzo| 2 -1 2 0 -1 }}


<sup>†</sup> Wart for 11/3
{{Mapping|legend=2| 1 0 2 -2 6 | 0 3 -1 5 -5 }}


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


==== Semidim ====
[[Optimal tuning]]s:
{{See also| Chromatic pairs #Semidim }}
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~12/7 = 930.598
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~12/7 = 930.700


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, 31, 40, 49, 156c*†, 205c*† }}
: <nowiki/>* wart for 7/3
: † wart for 11/3


[[Subgroup]]: 2.5/3.7/3.11/3.13/3
[[Tp tuning #T2 tuning|RMS error]]: 0.8790 cents


[[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 }}
=== Kryptonite ===
{{See also| Chromatic pairs #Kryptonite }}


{{Mapping|legend=2| 1 3 5 3 4 | 0 -6 -10 -3 -5 }}
Kryptonite is related to [[krypton]].


{{Mapping|legend=3| 1 -3 0 2 0 1 | 0 24/5 -6/5 -26/5 9/5 -1/5 }}
[[Subgroup]]: 2.5.7/3.11/3.13/3
 
[[Comma list]]: 56/55 ({{monzo| 3 -1 1 -1 }}), 78/77 ({{monzo| 1 0 -1 -1 1 }}), 91/90 ({{monzo| -1 -2 1 0 1 }})
 
{{Mapping|legend=2| 1 2 1 2 2 | 0 3 2 -1 1 }}
: mapping generators: ~2, ~13/12


: [[gencom]]: [2 13/10; 121/120 169/168 275/273]
{{Mapping|legend=3| 1 -5/4 2 -1/4 3/4 3/4 | 0 -1/2 3 3/2 -3/2 1/2 }}
: [[gencom]]: [2 13/12; 56/55 78/77 91/90]


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~13/10 = 453.121
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~13/12 = 130.945
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~13/10 = 453.242
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~13/12 = 132.428


{{Optimal ET sequence|legend=1| 8, 29, 37, 45 }}
{{Optimal ET sequence|legend=1| 1, …, 8, 9 }}


[[Tp tuning #T2 tuning|RMS error]]: 1.052 cents
[[Tp tuning #T2 tuning|RMS error]]: 2.545 cents


=== Sentry ===
=== Kiribati ===
{{See also | Chromatic pairs #Sentry }}
{{See also| Chromatic pairs #Kiribati }}


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


[[Subgroup]]: 2.5/3.9/7
[[Subgroup]]: 2.9/5.7/3.11/9


[[Comma list]]: [[245/243]] = {{monzo| 0 1 -2 }}
[[Comma list]]: 100/99 ({{monzo| 2 -2 0 -1 }}), 245/242 ({{monzo| -1 -1 2 -2 }})


{{Mapping|legend=2| 1 0 0 | 0 2 1 }}
{{Mapping|legend=2| 1 1 1 0 | 0 -2 3 4 }}
: mapping generators: ~2, ~21/20


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


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[Tp tuning|subgroup]] [[CTE]] and [[POTE]]: ~2 = 1\1, ~9/7 = 440.902
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~21/20 = 87.776
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~21/20 = 87.892


{{Optimal ET sequence|legend=1| 8, 11, 19, 30, 41, 49, 52, 145*, 166<sup>†</sup>, 197*<sup>†</sup>, 215<sup>†</sup>, 264*<sup>†</sup> }}
{{Optimal ET sequence|legend=1| 13, 14, 27, 41 }}


<nowiki />* Wart for 5/3
[[Tp tuning #T2 tuning|RMS error]]: 1.245 cents


<sup>†</sup> Wart for 9/7
=== Mothwelltri ===
{{See also| Chromatic pairs #Mothwelltri }}


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


=== Marveltwintri ===
[[Subgroup]]: 2.7/3.11
{{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.
[[Comma list]]: [[99/98]] ({{monzo| -1 -2 1 }})


[[Subgroup]]: 2.5/3.13/9
{{Mapping|legend=2| 1 0 1 | 0 1 2 }}
: mapping generators: ~2, ~7/3


[[Comma list]]: [[325/324]] = {{monzo| -2 2 1 }}
{{Mapping|legend=3| 1 -1/2 0 1/2 3 | 0 -1/2 0 1/2 2 }}
: [[gencom]]: [2 7/6; 99/98]


{{Mapping|legend=2| 1 0 2 | 0 1 -2 }}
[[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 -1/6 5/6 0 0 -1/3 | 0 -1/2 -3/2 0 0 1 }}
{{Optimal ET sequence|legend=1| 4, 9, 13, 22, 79 }}


: [[gencom]]: [2 6/5; 325/324]
[[Tp tuning #T2 tuning|RMS error]]: 1.064 cents


[[Optimal tuning]]s:
== 2.….9/7.… subgroups ==
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 886.886
=== Marveltri ===
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 886.861
{{See also| Chromatic pairs #Marveltri }}


{{Optimal ET sequence|legend=1| 3, 4, 11, 15, 19, 34, 53, 87, 140 }}
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]]: 0.2444 cents
[[Subgroup]]: 2.5.9/7


== 2.….7/3… subgroups ==
[[Comma list]]: 225/224 ({{monzo| -5 2 1 }})
=== 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=2| 1 0 5 | 0 1 -2 }}
: mapping generators: ~2, ~5


[[Subgroup]]: 2.5.7/3.11/3
{{Mapping|legend=3| 1 2 0 -1 | 0 -4/5 1 2/5 }}
: [[gencom]]: [2 5; 225/224]


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


{{Mapping|legend=2| 1 0 2 -2 | 0 3 -1 5 }}
{{Optimal ET sequence|legend=1| 3, 13, 16, 19, 22, 25, 72, 97, 122, 269c* }}
: <nowiki/>* wart for 9/7


{{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]]: 0.4801 cents


: [[gencom]]: [2 7/6; 176/175 540/539]
==== Sulis ====
Sulis is related to [[minerva]] and [[würschmidt]].


[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~7/6 = 270.093
[[Subgroup]]: 2.5.9/7.11/9


{{Optimal ET sequence|legend=1| 9, 31, 40, 49, 89, 191bc, 227bc, 231bc, 271bc, 311bc, 316bcd }}
[[Comma list]]: 99/98 ({{monzo| -1 0 2 1 }}), 176/175 ({{monzo| 4 -2 1 1 }})


[[Tp tuning #T2 tuning|RMS error]]: 0.6028 cents
{{Mapping|legend=2| 1 0 5 -9 | 0 1 -2 4 }}]


==== Laz ====
[[Optimal tuning]]s:
{{See also | Chromatic pairs #Laz }}
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/4 = 386.617
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/4 = 386.558


Laz is related to [[georgian]] as well as to [[winston]].
{{Optimal ET sequence|legend=1| 3, …, 22, 25, 28, 31, 59 }}


[[Subgroup]]: 2.5.7/3.11/3.13/3
[[Tp tuning #T2 tuning|RMS error]]: 1.074 cents


[[Comma list]]: [[144/143]], [[176/175]], [[196/195]]
== 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.


{{Mapping|legend=2| 1 0 2 -2 6 | 0 3 -1 5 -5 }}
[[Subgroup]]: 2.3.7/5


{{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]]: [[50/49]]


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


[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~7/6 = 269.300
[[Optimal tuning]] (inharmonic [[TE]]): ~1\2 = 590.998, ~[[10/7]]-1\2 = 128.962


{{Optimal ET sequence|legend=1| 9, 31, 40, 49, 58, 156bde, 205bde }}
[[Support]]ing [[ET]]s: {{EDOs|4, 6, 8, 10, 18, 28, 46, 64, 110}}


[[Tp tuning #T2 tuning|RMS error]]: 0.8790 cents
=== Argentic ===
Argentic is the 2.3.7/5 subgroup temperament tempering out [[5120/5103]].  


=== Kryptonite ===
[[Subgroup]]: 2.3.7/5
{{See also| Chromatic pairs #Kryptonite }}


Kryptonite is related to [[krypton]].
[[Comma list]]: [[5120/5103]] = {{monzo| 10 -6 -1 }}


[[Subgroup]]: 2.5.7/3.11/3.13/3
{{Mapping|legend=2| 1 0 10 | 0 1 -6 }}
: mapping generators: ~2, ~3


[[Comma list]]: 56/55, 78/77, 91/90
[[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


{{Mapping|legend=2| 1 2 1 2 2 | 0 -3 -2 1 -1 }}
{{Optimal ET sequence|legend=1| 12, 29, 41, 70, 321, 391, 461, 531, 601 }}
<small> based on subgroup TE </small>


{{Mapping|legend=3| 1 -5/4 2 -1/4 3/4 3/4 | 0 -1/2 3 3/2 -3/2 1/2 }}
Badness (Sintel): 0.119


: [[gencom]]: [2 13/12; 56/55 78/77 91/90]
==== Edson (2.3.7/5.11/5.13/5 subgroup) ====
{{See also| Chromatic pairs #Edson }}


[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~13/12 = 132.428
Edson is related to [[pele]] and [[andromeda]].  


{{Optimal ET sequence|legend=1| 9, 63, 82bd, 91bde }}
[[Subgroup]]: 2.3.7/5.11/5.13/5


[[Tp tuning #T2 tuning|RMS error]]: 2.545 cents
[[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 }}


=== Kiribati ===
{{Mapping|legend=2| 1 0 10 17 22 | 0 1 -6 -10 -13 }}
{{See also| Chromatic pairs #Kiribati }}
: mapping generators: ~2, ~3


Kiribati is related to [[nakika]] as well as to [[octacot]].
{{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]


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


[[Comma list]]: 100/99, 245/242
{{Optimal ET sequence|legend=1| 12, 17, 29 }}


{{Mapping|legend=2| 1 1 1 0 | 0 -2 3 4 }}
[[Tp tuning #T2 tuning|RMS error]]: 0.5102 cents


{{Mapping|legend=3| 1 1/10 -4/5 11/10 1/5 | 0 -3/2 -1 3/2 1 }}
==== Haumea ====
{{See also| Chromatic pairs #Haumea }}


: [[gencom]]: [2 21/20; 100/99 245/242]
Related temperaments include [[#Bridgetown|bridgetown]], [[namaka]], [[hemigari]], [[#Barbados|barbados]], and [[parizekmic]].


[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~21/20 = 87.892
[[Subgroup]]: 2.3.7/5.11/5.13/5


{{Optimal ET sequence|legend=1| 13, 14, 27, 41, 55, 191bd, 232bcd, 273bcd }}
[[Comma list]]: [[352/351]], [[676/675]], [[847/845]]


[[Tp tuning #T2 tuning|RMS error]]: 1.245 cents
{{Mapping|legend=2| 1 0 10 -6 -1 | 0 2 -12 9 3 }}


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


Mothwelltri, the 1 &amp; 4 temperament in the 2.7/3.11 subgroup, is related to [[orwell]].  
[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~15/13 = 248.491


[[Subgroup]]: 2.7/3.11
{{Optimal ET sequence|legend=1| 24, 29, 111, 140, 169, 198, 565d, 763bd, 961bd }}


[[Comma list]]: [[99/98]]
[[Tp tuning #T2 tuning|RMS error]]: 0.2668 cents


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


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


: [[gencom]]: [2 7/6; 99/98]
[[Subgroup]]: 2.3.7/5.11/5.13/5


[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~7/6 = 273.174
[[Comma list]]: 364/363, 441/440, 1001/1000


{{Optimal ET sequence|legend=1| 9, 22, 40, 49c, 58c, 67c, 76c, 79, 101b, 123bc }}
{{Mapping|legend=2| 1 2 0 1 2 | 0 -6 7 2 -9 }}


[[Tp tuning #T2 tuning|RMS error]]: 1.064 cents
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~21/20 = 83.016


== 2.….9/7… subgroups ==
{{Optimal ET sequence|legend=1| 14, 29, 72, 101, 130, 159 }}


=== Marveltri ===
[[Tp tuning #T2 tuning|RMS error]]: 0.2562 cents
{{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.
=== Terrain ===
{{Redirect|Terrain|the scale|Terrain (scale)}}
{{See also| Chromatic pairs #Terrain }}


[[Subgroup]]: 2.5.9/7
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.


[[Comma list]]: 225/224
[[Subgroup]]: 2.7/5.9/5


{{Mapping|legend=2| 1 2 1 | 0 1 -2 }}
[[Comma list]]: [[250047/250000]]


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


: [[gencom]]: [2 5/4; 225/224]
{{Mapping|legend=3| 3 10/9 -7/9 2/9 | 0 -2/3 -1/3 2/3 }}
: [[gencom]]: [63/50 10/9; 250047/250000]


[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~5/4 = 383.638
[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~63/50 = 1\3, ~10/9 = 182.461


{{Optimal ET sequence|legend=1| 12, 13, 16, 19, 22, 25, 47, 69, 72, 97, 122, 269c*, 660c* }}
{{Optimal ET sequence|legend=1| 6, 21, 27, 33, 105, 138, 171, 1848, 2019, 2190, 2361, 2532, 2703, 2874, 3045, 3216, 3387, 3558 }}


<nowiki />* Wart for 9/7
[[Tp tuning #T2 tuning|RMS error]]: 0.00844 cents


[[Tp tuning #T2 tuning|RMS error]]: 0.4801 cents
=== Tridec ===
{{See also| Chromatic pairs #Tridec }}
{{See also| Non-over-1 temperament #Tridec }}


==== Sulis ====
Tridec, the 5 &amp; 8 temperament in the 2.7/5.11/5.13/5 subgroup, extends [[#Petrtri]].
Related temperament: [[Marvel family|minerva]], [[Würschmidt family|würschmidt]]


[[Subgroup]]: 2.5.9/7.11/7
[[Subgroup]]: 2.7/5.11/5.13/5


[[Comma list]]: 99/98, 176/175
[[Comma list]]: [[847/845]], [[1001/1000]]


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


[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~5/4 = 386.558
{{Mapping|legend=3| 1 0 -3/4 5/4 -3/4 1/4 | 0 0 0 -4 3 1 }}
: [[gencom]]: [2 13/10; 847/845 1001/1000]


{{Optimal ET sequence|legend=1| 3, …, 22, 25, 28, 31, 59 }}
[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~13/10 = 454.556


[[Tp tuning #T2 tuning|RMS error]]: 1.074 cents
{{Optimal ET sequence|legend=1| 5, 8, 21, 29, 37, 66, 169, 235, 404c, 639c, 953bc }}


== 2.….15/11… subgroups ==
[[Tp tuning #T2 tuning|RMS error]]: 0.1613 cents


=== Poggers ===
==== Naiadec ====
Related temperaments: [[Stearnsmic_clan#Pogo|pogo]], [[Stearnsmic_clan#Supers|supers]]
[[Subgroup]]: 2.7/5.11/5.13/5.17/5


[[Subgroup]]: 2.9.7.15/11.13
[[Comma list]]: [[170/169]], [[221/220]], [[847/845]]


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


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


[[Optimal tuning]] (subgroup [[CTE]]): ~9/7 = 433.888
[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~13/10 = 454.882


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


== 2.….7/5… subgroups ==
[[Tp tuning #T2 tuning|RMS error]]: 0.7521 cents


=== Hydrothermal ===
== 2.….11/5.… subgroups ==
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.
=== Petrtri ===
{{See also| Chromatic pairs #Petrtri }}
{{See also| 5L 3s/Temperaments #Petrtri }}


[[Subgroup]]: 2.3.7/5
Petrtri can be described as 3 &amp; 5 temperament in the 2.11/5.13/5 subgroup.


[[Comma list]]: [[50/49]]
[[Subgroup]]: 2.11/5.13/5


{{Mapping|legend=2| 2 3 1 | 0 1 0 }}
[[Comma list]]: [[2200/2197]]


[[Optimal tuning]] (inharmonic [[TE]]): ~1\2 = 590.998, ~[[10/7]]-1\2 = 128.962
{{Mapping|legend=2| 1 0 1| 0 3 1 }}


[[Support]]ing [[ET]]s: {{EDOs|4, 6, 8, 10, 18, 28, 46, 64, 110}}
{{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]


=== Edson ===
[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~13/10 = 455.012
Edson is the 2.3.7/5 subgroup temperament tempering out [[5120/5103]].


==== Edson (2.3.7/5.11/5.13/5 subgroup) ====
{{Optimal ET sequence|legend=1| 21, 29, 153, 182, 211, 240, 269, 298, 327, 356, 385, 509, 741c, 1126c }}
{{See also| Chromatic pairs #Edson }}


Edson is related to [[pele]] and [[andromeda]].  
[[Tp tuning #T2 tuning|RMS error]]: 0.0749 cents


[[Subgroup]]: 2.3.7/5.11/5.13/5
==== Bridgetown ====
{{See also| Chromatic pairs #Bridgetown }}


[[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 }}
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| 1 0 10 17 22 | 0 1 -6 -10 -13 }}
[[Subgroup]]: 2.3.11/5.13/5


: mapping generators: ~2, ~3
[[Comma list]]: [[352/351]], [[676/675]]


{{Mapping|legend=3| 1 1 -5 -1 2 4 | 0 1 29/4 5/4 -11/4 -23/4 }}
{{Mapping|legend=2| 1 0 -6 -1 | 0 2 9 3 }}


: [[gencom]]: [2 3/2; 196/195, 352/351, 364/363]
{{Mapping|legend=3| 1 2 -5/3 0 4/3 1/3 | 0 -2 4 0 -5 1 }}
: [[gencom]]: [2 15/13; 352/351 676/675]


[[Optimal tuning]]s:
[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~15/13 = 248.399
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1\1, ~3/2 = 703.4398
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1\1, ~3/2 = 703.414


{{Optimal ET sequence|legend=1| 12, 17, 29 }}
{{Optimal ET sequence|legend=1| 5, 9, 14, 19, 24, 29, 169, 198, 227, 256, 285, 314 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.5102 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.2513 cents


==== Haumea ====
=== Hypnosis ===
{{See also| Chromatic pairs #Haumea }}
Related temperaments: [[Swetismic temperaments #Hypnos|hypnos]], [[Alphatricot family #Alphatricot|alphatricot]]


Related temperaments include [[#Bridgetown|bridgetown]], [[namaka]], [[hemigari]], [[#Barbados|barbados]], and [[parizekmic]].  
[[Subgroup]]: 2.3.7.11/5.13


[[Subgroup]]: 2.3.7/5.11/5.13/5
[[Comma list]]: 169/168, 540/539, 729/728


[[Comma list]]: [[352/351]], [[676/675]], [[847/845]]
{{Mapping|legend=2| 1 0 -3 8 0 | 0 3 11 -13 7 }}


{{Mapping|legend=2| 1 0 10 -6 -1 | 0 2 -12 9 3 }}
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~13/9 = 633.518


{{Mapping|legend=3| 1 2 -3/4 -11/4 9/4 5/4 | 0 -2 0 12 -9 -3 }}
{{Optimal ET sequence|legend=1| 17, 36, 118f, 125f, 161f, 197f }}


: [[gencom]]: [2 15/13; 352/351 676/675 847/845]
[[Tp tuning #T2 tuning|RMS error]]: 0.5379 cents


[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~15/13 = 248.491
=== Trisect ===
Trisect divides every Pythagorean interval into three, and is the much more accurate subgroup restriction of [[Augmented family #Trisected|trisected]].


{{Optimal ET sequence|legend=1| 24, 29, 111, 140, 169, 198, 565d, 763bd, 961bd }}
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]].


[[Tp tuning #T2 tuning|RMS error]]: 0.2668 cents
[[Subgroup]]: 2.3.7.11/5


=== Historical ===
[[Comma list]]: 1029/1024, 4000/3993
{{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=2| 3 0 10 5 | 0 3 -1 -1 }}


[[Subgroup]]: 2.3.7/5.11/5.13/5
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~44/35 = 1\3, ~13/9 = 633.742


[[Comma list]]: 364/363, 441/440, 1001/1000
{{Optimal ET sequence|legend=1| 15, 21, 36, 123, 159, 195, 231 }}


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


[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~21/20 = 83.016
==== 2.3.7.11/5.13 subgroup ====
[[Subgroup]]: 2.3.7.11/5.13


{{Optimal ET sequence|legend=1| 14, 29, 72, 101, 130, 159 }}
[[Comma list]]: 1029/1024, 1575/1573, 2080/2079


[[Tp tuning #T2 tuning|RMS error]]: 0.2562 cents
{{Mapping|legend=2| 3 0 10 5 0 | 0 3 -1 -1 7 }}


=== Terrain ===
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~44/35 = 1\3, ~13/9 = 633.918
{{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 ET sequence|legend=1| 15, 21f, 36, 87, 123, 159 }}


[[Subgroup]]: 2.7/5.9/5
[[Tp tuning #T2 tuning|RMS error]]: ???


[[Comma list]]: [[250047/250000]]
==== 2.3.7.11/5.13.17 subgroup ====
[[Subgroup]]: 2.3.7.11/5.13.17


{{Mapping|legend=2| 3 1 3 | 0 1 -1 }}
[[Comma list]]: 273/272, 833/832, 1575/1573, 2080/2079


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


: [[gencom]]: [63/50 10/9; 250047/250000]
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~34/27 = 1\3, ~13/9 = 633.820


[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~63/50 = 1\3, ~10/9 = 182.461
{{Optimal ET sequence|legend=1| 15, 21fg, 36, 123, 159 }}


{{Optimal ET sequence|legend=1| 6, 21, 27, 33, 105, 138, 171, 1848, 2019, 2190, 2361, 2532, 2703, 2874, 3045, 3216, 3387, 3558 }}
[[Tp tuning #T2 tuning|RMS error]]: ???


[[Tp tuning #T2 tuning|RMS error]]: 0.00844 cents
===== Trisector =====
[[Subgroup]]: 2.3.7.11/5.13.17.19


=== Tridec ===
[[Comma list]]: 210/209, 273/272, 286/285, 595/594, 2080/2079
{{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]].
{{Mapping|legend=2| 3 0 10 5 0 -2 8 | 0 3 -1 -1 7 9 3 }}


[[Subgroup]]: 2.7/5.11/5.13/5
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~34/27 = 1\3, ~13/9 = 633.894


[[Comma list]]: [[847/845]], [[1001/1000]]
{{Optimal ET sequence|legend=1| 15, 21fg, 36, 123h, 159h }}


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


{{Mapping|legend=3| 1 0 -3/4 5/4 -3/4 1/4 | 0 0 0 -4 3 1 }}
===== 2.3.7.11/5.13.17.19.23 subgroup =====
[[Subgroup]]: 2.3.7.11/5.13.17.19.23


: [[gencom]]: [2 13/10; 847/845 1001/1000]
[[Comma list]]: 210/209, 231/230, 273/272, 286/285, 595/594, 2080/2079


[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~13/10 = 454.556
{{Mapping|legend=2| 3 0 10 5 0 -2 8 12 | 0 3 -1 -1 7 9 3 1 }}


{{Optimal ET sequence|legend=1| 5, 8, 21, 29, 37, 66, 169, 235, 404c, 639c, 953bc }}
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~34/27 = 1\3, ~13/9 = 634.038


[[Tp tuning #T2 tuning|RMS error]]: 0.1613 cents
{{Optimal ET sequence|legend=1| 15g, 21fg, 36, 87, 123hi }}


== 2.….11/5… subgroups ==
[[Tp tuning #T2 tuning|RMS error]]: ???


=== Petrtri ===
===== 2.3.7.11/5.13.17.19.23.29 subgroup =====
{{See also| Chromatic pairs #Petrtri }}
[[Subgroup]]: 2.3.7.11/5.13.17.19.23.29
{{See also| 5L 3s/Temperaments #Petrtri }}


Petrtri can be described as 3 &amp; 5 temperament in the 2.11/5.13/5 subgroup.
[[Comma list]]: 210/209, 231/230, 273/272, 286/285, 320/319, 595/594, 2080/2079


[[Subgroup]]: 2.11/5.13/5
{{Mapping|legend=2| 3 0 10 5 0 -2 8 12 13 | 0 3 -1 -1 7 9 3 1 1 }}


[[Comma list]]: [[2200/2197]]
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~29/23 = 1\3, ~13/9 = 634.102


{{Mapping|legend=2| 1 0 1| 0 3 1 }}
{{Optimal ET sequence|legend=1| 15g, 21fg, 36, 87, 123hi }}


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


: [[gencom]]: [2 13/10; 2200/2197]
== 2.….11/7.… subgroups ==
=== Pepperoni ===
{{Main| Parapyth }}
{{See also| Chromatic pairs #Pepperoni }}


[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~13/10 = 455.012
Pepperoni is generated by a fifth and can be described as the 5 &amp; 12 temperament in the 2.3.11/7.13/7 subgroup. It is the single-chain retraction of [[parapyth]]. The [[Peppermint-24|Pepper fifth]], which is (40200 + 600 sqrt(5))/59 = 704.096 cents, is a good pepperoni generator, hence the name.


{{Optimal ET sequence|legend=1| 21, 29, 153, 182, 211, 240, 269, 298, 327, 356, 385, 509, 741c, 1126c }}
[[Subgroup]]: 2.3.11/7.13/7


[[Tp tuning #T2 tuning|RMS error]]: 0.0749 cents
[[Comma list]]: 352/351, 364/363


==== Bridgetown ====
{{Mapping|legend=2| 1 0 7 12 | 0 1 -4 -7 }}
{{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=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]


[[Subgroup]]: 2.3.11/5.13/5
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~3/2 = 703.856


[[Comma list]]: [[352/351]], [[676/675]]
{{Optimal ET sequence|legend=1| 5, 7, 12f, 17, 29, 46, 58, 75, 80, 87, 104, 121, 167, 196, 208, 271, 595b*<sup>†</sup> }}
: <nowiki />* wart for 11/7
: <sup>†</sup> wart for 13/7


{{Mapping|legend=2| 1 0 -6 -1 | 0 2 9 3 }}
[[Tp tuning #T2 tuning|RMS error]]: 0.3789 cents


{{Mapping|legend=3| 1 2 -5/3 0 4/3 1/3 | 0 -2 4 0 -5 1 }}
== 2.….13/5.… subgroups ==
=== 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.


: [[gencom]]: [2 15/13; 352/351 676/675]
[[Subgroup]]: 2.3.13/5


[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~15/13 = 248.399
[[Comma list]]: 676/675 = {{monzo| 2 -3 2 }}


{{Optimal ET sequence|legend=1| 5, 9, 14, 19, 24, 29, 169, 198, 227, 256, 285, 314 }}
[[Sval]] [[mapping]]: [{{val| 1 0 -1 }}, {{val| 0 2 3 }}]


[[Tp tuning #T2 tuning|RMS error]]: 0.2513 cents
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~2 = 1\1, ~15/13 = 248.621


=== Hypnosis ===
{{Optimal ET sequence|legend=1| 5, 9, 14, 19, 24, 29, 53, 82, 111, 140, 251, 362 }}
Related temperaments: [[Swetismic temperaments #Hypnos|hypnos]], [[Hemifamity temperaments #Tricot|tricot]]


[[Subgroup]]: 2.3.7.11/5.13
[[Badness]]: 0.002335


[[Comma list]]: 169/168, 540/539, 729/728
; Music
* [http://micro.soonlabel.com/gene_ward_smith/Others/Sevish/Sevish%20-%20Desert%20Island%20Rain.mp3 ''Desert Island Rain''] in 313edo tuned Barbados[9], by [https://soundcloud.com/sevish/desert-island-rain Sevish]


{{Mapping|legend=2| 1 0 -3 8 0 | 0 3 11 -13 7 }}
==== Tobago ====
{{See also| Chromatic pairs #Tobago }}


[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~13/9 = 633.518
Tobago, the 10 &amp; 14 temperament in the 2.3.11.13/5 subgroup, extends [[neutral]] and [[barbados]].  


{{Optimal ET sequence|legend=1| 17, 36, 118f, 125f, 161f, 197f }}
[[Subgroup]]: 2.3.11.13/5


[[Tp tuning #T2 tuning|RMS error]]: 0.5379 cents
[[Comma list]]: [[243/242]], [[676/675]]


== 2.….11/7… subgroups ==
{{Mapping|legend=2| 2 0 -1 -2 | 0 2 5 3 }}
=== Pepperoni ===
{{Main| Parapyth }}
{{See also| Chromatic pairs #Pepperoni }}


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


[[Subgroup]]: 2.3.11/7.13/7
[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~55/39 = 1\2, ~15/13 = 249.312


[[Comma list]]: 352/351, 364/363
{{Optimal ET sequence|legend=1| 10, 14, 24, 58, 82, 130 }}


{{Mapping|legend=2| 1 0 7 12 | 0 1 -4 -7 }}
[[Tp tuning #T2 tuning|RMS error]]: 0.3533 cents


{{Mapping|legend=3| 1 1 0 -8/3 1/3 7/3 | 0 1 0 11/3 -1/3 -10/3 }}
==== Pakkanian hemipyth ====
[[Subgroup]]: 2.3.11.13/5.17


: [[gencom]]: [2 3/2; 352/351 364/363]
[[Comma list]]: 221/220, 243/242, 289/288


[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~3/2 = 703.856
{{Mapping|legend=2| 2 0 -1 -2 5 | 0 2 5 3 2 }}


{{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 tuning]]s:
* [[Tp tuning|subgroup CTE]]: ~17/12 = 1\2, ~26/15 = 950.7656 (~15/13 = 249.2344)
* [[Tp tuning|subgroup CWE]]: ~17/12 = 1\2, ~26/15 = 950.6011 (~15/13 = 249.3989)


<nowiki />* Wart for 11/7
{{Optimal ET sequence|legend=1| 10, 14, 24, 106, 130, 154, 178*, 202* }}
: <nowiki />* wart for 13/5


<sup>†</sup> Wart for 13/7
=== Oceanfront ===
Related temperaments: [[Archytas clan #Superpyth|superpyth]], [[Archytas clan #Ultrapyth|ultrapyth]]


[[Tp tuning #T2 tuning|RMS error]]: 0.3789 cents
[[Subgroup]]: 2.3.7.13/5


== 2.….13/5… subgroups ==
[[Comma list]]: 64/63, 91/90
=== Barbados ===
The [[minimax tuning]] for this makes the generator the cube root of 20/13, or 248.5953 cents. Edos which may be used for it are [[24edo]], [[29edo]], [[53edo]] and [[111edo]], with [[mos scale]]s of size 5, 9, 14, 19, 24 and 29 making for a good variety of scales.


[[Subgroup]]: 2.3.13/5
{{Mapping|legend=2| 1 0 6 -5 | 0 1 -2 4 }}


[[Comma list]]: 676/675 = {{monzo| 2 -3 2 }}
[[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


[[Sval]] [[mapping]]: [{{val| 1 0 -1 }}, {{val| 0 2 3 }}]
{{Optimal ET sequence|legend=1|7, 10, 17}}


[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~2 = 1\1, ~15/13 = 248.621
[[Tp tuning #T2 tuning|RMS error]]: ?


{{Optimal ET sequence|legend=1| 5, 9, 14, 19, 24, 29, 53, 82, 111, 140, 251, 362 }}
== 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.


[[Badness]]: 0.002335
[[Subgroup]]: 2.3.17/5


; Music
[[Comma list]]: 136/135 ({{monzo| 3 -3 1 }})
* [http://micro.soonlabel.com/gene_ward_smith/Others/Sevish/Sevish%20-%20Desert%20Island%20Rain.mp3 ''Desert Island Rain''] in 313edo tuned Barbados[9], by [https://soundcloud.com/sevish/desert-island-rain Sevish]


==== Tobago ====
{{Mapping|legend=2| 1 0 -3 | 0 1 3 }}
{{See also| Chromatic pairs #Tobago }}
: mapping generators: ~2, ~3


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


[[Subgroup]]: 2.3.11.13/5
{{Optimal ET sequence|legend=1| 5, 12, 17, 46, 63, 143 }}


[[Comma list]]: [[243/242]], [[676/675]]
== 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.


{{Mapping|legend=2| 2 0 -1 -2 | 0 2 5 3 }}
[[Subgroup]]: 2.3.19/7


{{Mapping|legend=3| 2 4 -2 0 9 2 | 0 -2 3/2 0 -5 -3/2 }}
[[Comma list]]: [[57/56]] ({{Monzo| -3 1 1 }})


: [[gencom]]: [55/39 15/13; 243/242 676/675]
{{Mapping|legend=2| 1 0 3 | 0 1 -1 }}
: mapping generators: ~2, ~3


[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~55/39 = 1\2, ~15/13 = 249.312
[[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| 10, 14, 24, 58, 82, 130 }}
{{Optimal ET sequence|legend=1| 5, 7, 12, 19, 31*, 50* }}


[[Tp tuning #T2 tuning|RMS error]]: 0.3533 cents
<nowiki/>* wart for 19/7


==== Pakkanian hemipyth ====
[[Badness]] (Sintel): 0.082


[[Subgroup]]: 2.3.11.13/5.17
=== 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]].


[[Comma list]]: 221/220, 243/242, 289/288
[[Subgroup]]: 2.17/7.19/7


{{Mapping|legend=2| 2 0 -1 -2 5 | 0 2 5 3 2 }}
[[Comma list]]: [[5491/5488]] ({{Monzo| -4 2 1 }})


[[Optimal tuning]]s:
{{Mapping|legend=2| 1 0 4 | 0 1 -2 }}
* [[Tp tuning|subgroup CTE]]: ~17/12 = 1\2, ~26/15 = 950.7656 (~15/13 = 249.2344)
: mapping generators: ~2, ~17/7
* [[Tp tuning|subgroup CWE]]: ~17/12 = 1\2, ~26/15 = 950.6011 (~15/13 = 249.3989)


{{Optimal ET sequence|legend=1| 10, 14, 24, 106, 130, 154, 178*, 202* }}
[[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 1,596: 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 1,602: 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 1,611: 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 1,617: 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 1,630: 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 1,668: 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 1,697: 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 1,712: 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 1,729: 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 1,737: 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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