No-sevens subgroup temperaments: Difference between revisions

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This is a collection of [[subgroup temperament]]s which omit the prime harmonic of 7.  
{{Technical data page}}
This is a collection of [[subgroup temperament]]s which omit the prime harmonic of 7. Many of these have been removed from this page and placed on their appropriate family pages in an effort to include extensions other than incremental-prime-limit; sections for them remain on this page due to extensive linkage.


== 2.3.5.11 temperaments ==
== 2.3.5.11 temperaments ==
Line 128: Line 129:


[[Tp tuning #T2 tuning|RMS error]]: 0.2830 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.2830 cents
=== Superpine ===
{{see also|Meantone family #Superpine}}
{{see also|Meantone family #Trimean}}
Subgroup: 2.3.5.11
Comma list: 81/80, 1350/1331
Mapping: {{mapping| 1 2 4 5 | 0 -3 -12 -11 }}
Optimal tunings:
* CTE: ~2 = 1200.000, ~11/10 = 167.712
* CWE: ~2 = 1200.000, ~11/10 = 167.882
{{Optimal ET sequence|legend=0| 7, 29ce, 36, 43, 50 }}
==== 2.3.5.11.13 ====
Subgroup: 2.3.5.11.13
Comma list: 81/80, 144/143, 975/968
Mapping: {{mapping| 1 2 4 5 | 0 -3 -12 -11 5 }}
Optimal tunings:
* CTE: ~2 = 1200.000, ~11/10 = 167.729
* CWE: ~2 = 1200.000, ~11/10 = 167.846
{{Optimal ET sequence|legend=0| 7, 29ce, 36, 43, 50 }}


=== Dicot ===
=== Dicot ===
Line 134: Line 165:
=== Mavila ===
=== Mavila ===
{{main|Mavila family #2.3.5.11 subgroup}}
{{main|Mavila family #2.3.5.11 subgroup}}
=== Tetracot ===
{{main|Tetracot family #2.3.5.11 subgroup }}


=== Porkypine ===
=== Porkypine ===
Line 140: Line 174:
=== Mohaha ===
=== Mohaha ===
{{main|Rastmic clan #Mohaha }}
{{main|Rastmic clan #Mohaha }}
=== Tetracot ===
{{main|Tetracot family #2.3.5.11 subgroup }}


=== Larry ===
=== Larry ===
Line 148: Line 179:


=== Dequarter ===
=== Dequarter ===
{{main|Meantone family #2.3.5.11 subgroup (Dequarter) }}
{{main|Meantone family #Dequarter }}


=== Hypnotone ===
=== Hypnotone ===
{{main|Meantone family #2.3.5.11 subgroup (Hypnotone) }}
{{main|Meantone family #Hypnotone }}


== 2.3.5.13 temperaments ==
== 2.3.5.13 temperaments ==
=== Tricot ===
{{main|Tricot family #2.3.5.13 subgroup }}
=== Taylor ===
Subgroup: 2.3.5.13
[[Comma list]]: 676/675, 32805/32768
[[Gencom]]: [2 15/13; 676/675 32805/32768]
[[Gencom|Gencom mapping]]: [{{val|1 2 -1 0 0 1}}, {{val|0 -2 16 0 0 13}}]
[[Mapping|Sval mapping]]: [{{val|1 2 -1 1}}, {{val|0 -2 16 13}}]
[[Tp tuning|POL2 generator]]: ~15/13 = 249.145
{{Optimal ET sequence|legend=1| 24, 53, 130, 183, 236 }}
[[Tp tuning #T2 tuning|RMS error]]: 0.1485 cents
Related temperament: [[Schismatic family|hemischis]]
=== Majvam ===
=== Majvam ===
: <small>''For full 13- and 17-limit extensions, see [[Parkleiness temperaments #Majvamic]] or [[Cataharry temperaments #Majvamoid]].''</small>
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Majvam]].''
: ''For full 13- and 17-limit extensions, see [[Parkleiness temperaments #Majvamic]] or [[Cataharry temperaments #Majvamoid]].''


Subgroup: 2.3.5.13
Subgroup: 2.3.5.13
Line 197: Line 206:


==== 2.3.5.13.17 ====
==== 2.3.5.13.17 ====
The comma list shown here is interestingly a possible minimal S-expression-based comma list for majvam: {[[676/675|S13/S15 = S26]], [[24576/24565|S16/S17]], [[2601/2600|S51]]} (though the order of the last two commas is swapped to make the commas appear in ascending prime limit).
The comma list shown here is interestingly a possible minimal [[S-expression]]-based comma list for majvam: {[[676/675|S13/S15 = S26]], [[24576/24565|S16/S17]], [[2601/2600|S51]]} (though the order of the last two commas is swapped to make the commas appear in ascending prime limit).


Subgroup: 2.3.5.13.17
Subgroup: 2.3.5.13.17
Line 215: Line 224:
[[Tp tuning #T2 tuning|RMS error]]: 0.1129 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.1129 cents


== Photia ==
=== Tricot ===
: <small>''See also: [[No-elevens subgroup temperaments #Garibaldia]]''</small>
{{Main| Tricot family #2.3.5.13 subgroup }}


Subgroup: 2.3.5.17
=== Taylor ===
{{Main| Schismatic family #Taylor (2.3.5.13) }}


[[Comma list]]: 256/255, 1458/1445
=== Vulture ===
: ''For the 5-limit restriction and full 13-limit extension, see [[Vulture family]].''


[[Gencom]]: [2 4/3; 256/255 1458/1445]
This entry is interesting because [[vulture]] and [[buzzard]] unite to the same temperament on the 2.3.5.13.19 subgroup. It results in a surprising decrease in Dirichlet badness, and up to [[octave equivalence]] finds [[13/8]] at 27 generators and [[19/16]] at 41 generators. In this temperament, the [[schisma]] is equated with [[325/324]], [[361/360]], [[513/512]], and [[625/624]]. [[270edo]] is especially ideal, whose step size being between 361/360 and 513/512, with [[217edo]] exaggerating the comma to be slightly sharp of 361/360. Smaller edos such as [[58edo|58]] (58h val), [[111edo|111]], and [[164edo|164]] are also possible.


[[Gencom|Gencom mapping]]: [{{val|1 2 -1 0 0 0 7}}, {{val|0 -1 8 0 0 0 -7}}]
[[Subgroup]]: 2.3.5.13


[[Mapping|Sval mapping]]: [{{val|1 2 -1 7}}, {{val|0 -1 8 -7}}]
[[Comma list]]: 676/675, 256000/255879


[[Tp tuning|POL2 generator]]: ~3/2 = 701.491
{{Mapping|legend=2| 1 0 -6 -7 | 0 4 21 27 }}


{{Optimal ET sequence|legend=1| 12, 41, 53, 65 }}
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.0000, ~320/243 = 475.5502
: [[error map]]: {{val| 0.000 +0.2457 +0.2401 -0.6728 }}
* [[CWE]]: ~2 = 1200.0000, ~320/243 = 475.5543
: [[error map]]: {{val| 0.000 +0.2622 +0.3266 -0.5616 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.4842 cents
{{Optimal ET sequence|legend=1| 53, 164, 217, 270 }}


Related temperament: [[Schismatic family|Schismic]]
[[Badness]] (Sintel): 0.301


=== 2.3.5.17.19 ===
==== 2.3.5.13.19 ====
Subgroup: 2.3.5.17.19
Subgroup: 2.3.5.13.19


[[Comma list]]: 171/170, 256/255, 324/323
Comma list: 676/675, 1216/1215, 20007/20000


[[Gencom]]: [2 4/3; 171/170 256/255 324/323]
Sval mapping: {{mapping| 1 0 -6 -7 -12 | 0 4 21 27 41 }}


[[Gencom|Gencom mapping]]: [{{val|1 2 -1 0 0 0 7 3}}, {{val|0 -1 8 0 0 0 -7 3}}]
Optimal tunings:  
* CTE: ~2 = 1200.0000, ~320/243 = 475.5498
* CWE: ~2 = 1200.0000, ~320/243 = 475.5533


[[Mapping|Sval mapping]]: [{{val|1 2 -1 7 3}}, {{val|0 -1 8 -7 3}}]
{{Optimal ET sequence|legend=0| 53, 164, 217, 270 }}


[[Tp tuning|POL2 generator]]: ~3/2 = 701.470
Badness (Sintel): 0.190


{{Optimal ET sequence|legend=1| 12, 41, 53, 65 }}
== 2.3.5.17 temperaments ==
=== Quintaleap ===
{{Main| Quintaleap family #Subgroup temperament }}


[[Tp tuning #T2 tuning|RMS error]]: 0.5374 cents
=== Quindromeda ===
{{Main| Quindromeda family #Subgroup temperament }}


== Quintaleap ==
=== Photia ===
: <small>''For full 17- and 19-limit extensions, see [[Quintaleap family]].''</small>
{{See also| No-elevens subgroup temperaments #Garibaldia }}
{{Main| Schismatic family #Photia (2.3.5.17) }}


Subgroup: 2.3.5.17
=== Quintilischis ===
: ''For full 17- and 19-limit extensions, see [[Schismatic family #Quintilipyth]] or [[Schismatic family #Quintaschis]].''
{{Main| Schismatic family #Quintilischis (2.3.5.17) }}


[[Comma list]]: 256/255, 1419857/1417176
=== Srutal Archagall ===
{{Main|Srutal archagall}}


[[Gencom]]: [2 18/17; 256/255 1419857/1417176]
== 2.3.5.19 temperaments ==
=== Rarity ===
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Rarity]].''


[[Gencom|Gencom mapping]]: [{{val| 1 2 1 0 0 0 5 }}, {{val| 0 -5 16 0 0 0 -11 }}]
Rarity generator is so close to [[23/19]], and some edos supporting it have good 19th and 23rd harmonics that it is natural to prescribe a 2.3.5.19.23 extension. Since the generator is also mapped to [[368/225]], this means that [[4275/4232]] is tempered out.


[[Mapping|Sval mapping]]: [{{val| 1 2 1 5 }}, {{val| 0 -5 16 -11 }}]
[[Subgroup]]: 2.3.5.19.23


[[Tp tuning|POL2 generator]]: ~18/17 = 99.272
[[Comma list]]: 1035/1024, 16875/16606, 192375/188416


{{Optimal ET sequence|legend=1| 12, 109, 121, 133 }}
{{Mapping|legend=2| 1 11 -10 -3 -2 | 0 -13 17 10 9 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.3051 cents
: Sval mapping generators: ~2 = 1200.000, ~368/225 = 869.420


=== 2.3.5.17.19 ===
[[Optimal tuning]] ([[CTE]]): ~368/225 = 869.420
Subgroup: 2.3.5.17.19


[[Comma list]]: 256/255, 361/360, 4624/4617
{{Optimal ET sequence|legend=1| 11, 29, 69, 98c }}


[[Gencom]]: [2 18/17; 256/255 361/360 4624/4617]
[[Badness]]: 0.0656
 
[[Gencom|Gencom mapping]]: [{{val| 1 2 1 0 0 0 5 4 }}, {{val| 0 -5 16 0 0 0 -11 3 }}]
 
[[Mapping|Sval mapping]]: [{{val| 1 2 1 5 4 }}, {{val| 0 -5 16 -11 3 }}]
 
[[Tp tuning|POL2 generator]]: ~18/17 = 99.276
 
{{Optimal ET sequence|legend=1| 12, 109, 121, 133 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.3427 cents
 
== Quindromeda ==
: <small>''For full 17- and 19-limit extensions, see [[Quindromeda family]].''</small>
 
Subgroup: 2.3.5.17
 
[[Comma list]]: 24576/24565, 295936/295245
 
[[Gencom]]: [2 18/17; 24576/24565 295936/295245]
 
[[Gencom|Gencom mapping]]: [{{val| 1 2 0 0 0 0 5 }}, {{val| 0 -5 28 0 0 0 -11 }}]
 
[[Mapping|Sval mapping]]: [{{val| 1 2 0 5 }}, {{val| 0 -5 28 -11 }}]
 
[[Tp tuning|POL2 generator]]: ~18/17 = 99.524
 
{{Optimal ET sequence|legend=1| 12, 169, 181, 193, 205, 422 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.0757 cents
 
=== 2.3.5.17.19 ===
Subgroup: 2.3.5.17.19
 
[[Comma list]]: 1216/1215, 1445/1444, 6144/6137
 
[[Gencom]]: [2 18/17; 1216/1215 1445/1444 6144/6137]
 
[[Gencom|Gencom mapping]]: [{{val| 1 2 0 0 0 0 5 4 }}, {{val| 0 -5 28 0 0 0 -11 3 }}]
 
[[Mapping|Sval mapping]]: [{{val| 1 2 0 5 4 }}, {{val| 0 -5 28 -11 3 }}]
 
[[Tp tuning|POL2 generator]]: ~18/17 = 99.524
 
{{Optimal ET sequence|legend=1| 12, 169, 181, 193, 205, 422 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.0813 cents
 
== Quintilischis ==
: <small>''For full 17- and 19-limit extensions, see [[Schismatic family #Quintilipyth]] or [[Schismatic family #Quintaschis]].''</small>
 
Subgroup: 2.3.5.17
 
[[Comma list]]: 32805/32768, 1419857/1417176
 
[[Gencom]]: [2 18/17; 32805/32768 1419857/1417176]
 
[[Gencom|Gencom mapping]]: [{{val|1 2 -1 0 0 0 5}}, {{val|0 -5 40 0 0 0 -11}}]
 
[[Mapping|Sval mapping]]: [{{val|1 2 -1 5}}, {{val|0 -5 40 -11}}]
 
[[Tp tuning|POL2 generator]]: ~18/17 = 99.649
 
{{Optimal ET sequence|legend=1| 12, 253, 265, 277, 289 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.0719 cents
 
=== 2.3.5.17.19 ===
Subgroup: 2.3.5.17.19
 
[[Comma list]]: 4624/4617, 6144/6137, 6885/6859
 
[[Gencom]]: [2 18/17; 4624/4617 6144/6137 6885/6859]
 
[[Gencom|Gencom mapping]]: [{{val|1 2 -1 0 0 0 5 4}}, {{val|0 -5 40 0 0 0 -11 3}}]
 
[[Mapping|Sval mapping]]: [{{val|1 2 -1 5 4}}, {{val|0 -5 40 -11 3}}]
 
[[Tp tuning|POL2 generator]]: ~18/17 = 99.652
 
{{Optimal ET sequence|legend=1| 12, 253, 265, 277, 289 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.1636 cents
 
== Stützel ==
Subgroup: 2.3.5.19
 
[[Comma list]]: 81/80, 96/95
 
[[Gencom]]: [2 4/3; 81/80 96/95]
 
[[Gencom|Gencom mapping]]: [{{val|1 2 4 0 0 0 0 3}}, {{val|0 -1 -4 0 0 0 0 3}}]
 
[[Mapping|Sval mapping]]: [{{val|1 2 4 3}}, {{val|0 -1 -4 3}}]
 
[[Tp tuning|POL2 generator]]: ~3/2 = 697.867
 
{{Optimal ET sequence|legend=1| 5, 7, 12, 31, 43 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 1.378 cents
 
Related temperament: [[Meantone family|meantone]]
 
== Devisemi ==
: <small>''See also: [[No-elevens subgroup temperaments #Devisemi]]''</small>
 
Subgroup: 2.3.5.19
 
[[Comma list]]: 361/360, 20000/19683
 
[[Gencom]]: [2 19/18; 361/360 20000/19683]
 
[[Gencom|Gencom mapping]]: [{{val|1 1 1 0 0 0 0 3}}, {{val|0 8 18 0 0 0 0 17}}]


[[Mapping|Sval mapping]]: [{{val|1 1 1 3}}, {{val|0 8 18 17}}]
=== Devisemi ===
: ''See also: [[No-elevens subgroup temperaments #Devisemi]]''
{{Main| Tetracot family #2.3.5.19 subgroup }}


[[Tp tuning|POL2 generator]]: ~19/18 = 88.077
=== Nestoria ===
 
{{Optimal ET sequence|legend=1| 14c, 27, 41, 68, 109 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.5701 cents
 
Related temperament: [[Tetracot family|octacot]]
 
== Nestoria ==
: ''See also: [[No-elevens subgroup temperaments #Garibaldia]] and [[No-elevens subgroup temperaments #Pontia|#Pontia]]''
: ''See also: [[No-elevens subgroup temperaments #Garibaldia]] and [[No-elevens subgroup temperaments #Pontia|#Pontia]]''
{{Main| Schismatic family #Nestoria (2.3.5.19) }}


The [[S-expression]]-based comma list of this temperament is {[[1216/1215|S16/S18]], [[361/360|S19]](, [[513/512|S15/S20]])}.
=== Stützel ===
 
{{Main| Meantone family #Stützel (2.3.5.19) }}
[[Subgroup]]: 2.3.5.19
 
[[Comma list]]: 361/360, 513/512
 
{{Mapping|legend=2| 1 0 15 9 | 0 1 -8 -3 }}
 
: mapping generators: ~2, ~3
 
{{Mapping|legend=3| 1 2 -1 0 0 0 0 3 | 0 -1 8 0 0 0 0 3 }}
 
: [[gencom]]: [2 4/3; 361/360 513/512]
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 701.746
 
{{Optimal ET sequence|legend=1| 12, 29, 41, 53, 118, 171 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.1763 cents
== Higher-limit subgroup temperaments ==
=== Reversed marvel ===
As the [[Marvel|marvel temperament]] is based on the marvel comma, [[225/224]], identifying the [[7/4|harmonic seventh (7/4)]] as a stack of two [[15/8|classical major sevenths (15/8)]] [[Octave reduction|octave-reduced]] and tempering 3rd and 5th harmonics slightly flat, tempering [[226/225]] instead results in sharper 3rd and 5th harmonics, and a stack of two classical major sevenths equivalent to the 113th harmonic instead of the 7th, so it might as well be called reversed marvel.


Related temperament: [[Schismatic family|schismic]]
Subgroup: 2.3.5.113


== Rarity ==
[[Comma list]]: 226/225
: <small>''For the 5-limit version of this temperament, see [[High badness temperaments #Rarity]].''</small>


Rarity generator is so close to [[23/19]], and some edos supporting it have good 19th and 23rd harmonics that it is natural to prescribe a 2.3.5.19.23 extension. Since the generator is also mapped to [[368/225]], this means that [[4275/4232]] is tempered out.
{{Mapping|legend=2|1 0 0 -1|0 1 0 2|0 0 1 2}}


[[Subgroup]]: 2.3.5.19.23
: Sval mapping generators: ~2, ~3, ~5


[[Comma list]]: 1035/1024, 16875/16606, 192375/188416
[[Optimal tuning]]s:
* CTE: ~3/2 = 702.449, ~5/4 = 387.373
* CWE: ~3/2 = 702.522, ~5/4 = 387.479


{{Mapping|legend=2| 1 11 -10 -3 -2 | 0 -13 17 10 9 }}
{{Optimal ET sequence|legend=1| 12, 22, 31, 34, 46, 53 }}
 
: Sval mapping generators: ~1\1 = 2, ~368/225 = 869.420
 
[[Optimal tuning]] ([[CTE]]): ~368/225 = 869.420
 
{{Optimal ET sequence|legend=1| 11, 29, 69, 98c }}
 
[[Badness]]: 0.0656


[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Subgroup temperaments]]
[[Category:Subgroup temperaments]]
[[Category:Rank 2]]
[[Category:Rank 2]]

Latest revision as of 12:29, 27 November 2025

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

This is a collection of subgroup temperaments which omit the prime harmonic of 7. Many of these have been removed from this page and placed on their appropriate family pages in an effort to include extensions other than incremental-prime-limit; sections for them remain on this page due to extensive linkage.

2.3.5.11 temperaments

Twentcufo

For full 11- and 13-limit extensions, see Hemimean clan#Undetrita.

Subgroup: 2.3.5.11

Comma list: 8019/8000, 14641/14580

Gencom: [2 400/363; 8019/8000 14641/14580]

Gencom mapping: [⟨1 0 -2 0 0], ⟨0 11 30 0 24]]

Sval mapping: [⟨1 0 -2 0], ⟨0 11 30 24]]

POL2 generator: ~400/363 = 172.8796

Optimal ET sequence: 7, 111, 118

RMS error: 0.2393 cents

Emka

For full 11- and 13-limit extensions, see Hemimean clan #Emka or Horwell temperaments #Emkay.

Subgroup: 2.3.5.11

Comma list: 4000/3993, 9453125/9437184

Gencom: [2 11/8; 4000/3993 9453125/9437184]

Gencom mapping: [⟨1 14 6 0 3], ⟨0 -27 -8 0 1]]

Sval mapping: [⟨1 14 6 3], ⟨0 -27 -8 1]]

POL2 generator: ~11/8 = 551.7778

Optimal ET sequence: 37, 50, 87, 137, 224

RMS error: 0.1188 cents

2.3.5.11.13

Subgroup: 2.3.5.11.13

Comma list: 625/624, 2200/2197, 4000/3993

Gencom: [2 11/8; 625/624 2200/2197 4000/3993]

Gencom mapping: [⟨1 14 6 0 3 6], ⟨0 -27 -8 0 1 -5]]

Sval mapping: [⟨1 14 6 3 6], ⟨0 -27 -8 1 -5]]

POL2 generator: ~11/8 = 551.7753

Optimal ET sequence: 37, 50, 87, 137, 224

RMS error: 0.1250 cents

Tremka

For full 11- and 13-limit extensions, see Hemimean clan#Tremka.

Subgroup: 2.3.5.11

Comma list: 6912/6875, 526153617/524288000

Gencom: [2 363/320; 6912/6875 526153617/524288000]

Gencom mapping: [⟨1 -4 -2 0 4], ⟨0 31 24 0 -3]]

Sval mapping: [⟨1 -4 -2 4], ⟨0 31 24 -3]]

POL2 generator: ~363/320 = 216.182

Optimal ET sequence: 11b, 28bbcc, 39bc, 50, 111, 161, 272

RMS error: 0.3717 cents

2.3.5.11.13

Subgroup: 2.3.5.11.13

Comma list: 2200/2197, 3267/3250, 6656/6655

Gencom: [2 192/169; 2200/2197 3267/3250 6656/6655]

Gencom mapping: [⟨1 -4 -2 0 4 1], ⟨0 31 24 0 -3 15]]

Sval mapping: [⟨1 -4 -2 4 1], ⟨0 31 24 -3 15]]

POL2 generator: ~192/169 = 216.186

Optimal ET sequence: 11b, 28bbccf, 39bc, 50, 111, 272, 383cf

RMS error: 0.3705 cents

Yarman

For full 11- and 13-limit extensions, see Quartonic family.

Subgroup: 2.3.5.11

Comma list: 4000/3993, 390625000/387420489

Gencom: [2 100/99; 4000/3993 390625000/387420489]

Gencom mapping: [⟨1 2 3 0 4], ⟨0 -33 -54 0 -43]]

Sval mapping: [⟨1 2 3 4], ⟨0 -33 -54 -43]]

POL2 generator: ~100/99 = 15.0773

Optimal ET sequence: 79, 80, 159, 239, 398

RMS error: 0.1682 cents

2.3.5.11.13

Subgroup: 2.3.5.11.13

Comma list: 325/324, 4000/3993, 85683/85184

Gencom: [2 100/99; 325/324 4000/3993 85683/85184]

Gencom mapping: [⟨1 2 3 0 4 4], ⟨0 -33 -54 0 -43 -24]]

Sval mapping: [⟨1 2 3 4 4], ⟨0 -33 -54 -43 -24]]

POL2 generator: ~100/99 = 15.0885

Optimal ET sequence: 79, 80, 159

RMS error: 0.2830 cents

Superpine

See also: Meantone family § Superpine
See also: Meantone family § Trimean

Subgroup: 2.3.5.11

Comma list: 81/80, 1350/1331

Mapping: [⟨1 2 4 5], ⟨0 -3 -12 -11]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~11/10 = 167.712
  • CWE: ~2 = 1200.000, ~11/10 = 167.882

Optimal ET sequence: 7, 29ce, 36, 43, 50

2.3.5.11.13

Subgroup: 2.3.5.11.13

Comma list: 81/80, 144/143, 975/968

Mapping: [⟨1 2 4 5], ⟨0 -3 -12 -11 5]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~11/10 = 167.729
  • CWE: ~2 = 1200.000, ~11/10 = 167.846

Optimal ET sequence: 7, 29ce, 36, 43, 50

Dicot

Main article: Dicot family § 2.3.5.11 subgroup

Mavila

Main article: Mavila family § 2.3.5.11 subgroup

Tetracot

Main article: Tetracot family § 2.3.5.11 subgroup

Porkypine

Main article: Porcupine family § 2.3.5.11 subgroup (porkypine)

Mohaha

Main article: Rastmic clan § Mohaha

Larry

Main article: Gravity family § Larry

Dequarter

Main article: Meantone family § Dequarter

Hypnotone

Main article: Meantone family § Hypnotone

2.3.5.13 temperaments

Majvam

For the 5-limit version, see Miscellaneous 5-limit temperaments #Majvam.
For full 13- and 17-limit extensions, see Parkleiness temperaments #Majvamic or Cataharry temperaments #Majvamoid.

Subgroup: 2.3.5.13

Comma list: 676/675, 127401984/126953125

Gencom: [2 125/96; 676/675 127401984/126953125]

Gencom mapping: [⟨1 10 5 0 0 19], ⟨0 -22 -7 0 0 -40]]

Sval mapping: [⟨1 10 5 19], ⟨0 -22 -7 -40]]

POL2 generator: ~125/96 = 458.988

Optimal ET sequence: 34, 149, 183, 217, 400

RMS error: 0.1171 cents

2.3.5.13.17

The comma list shown here is interestingly a possible minimal S-expression-based comma list for majvam: {S13/S15 = S26, S16/S17, S51} (though the order of the last two commas is swapped to make the commas appear in ascending prime limit).

Subgroup: 2.3.5.13.17

Comma list: 676/675, 2601/2600, 24576/24565

Gencom: [2 125/96; 676/675 2601/2600 24576/24565]

Gencom mapping: [⟨1 10 5 0 0 19 6], ⟨0 -22 -7 0 0 -40 -5]]

Sval mapping: [⟨1 10 5 19 6], ⟨0 -22 -7 -40 -5]]

POL2 generator: ~125/96 = 458.991

Optimal ET sequence: 34, 149, 183, 217, 400

RMS error: 0.1129 cents

Tricot

Main article: Tricot family § 2.3.5.13 subgroup

Taylor

Main article: Schismatic family § Taylor (2.3.5.13)

Vulture

For the 5-limit restriction and full 13-limit extension, see Vulture family.

This entry is interesting because vulture and buzzard unite to the same temperament on the 2.3.5.13.19 subgroup. It results in a surprising decrease in Dirichlet badness, and up to octave equivalence finds 13/8 at 27 generators and 19/16 at 41 generators. In this temperament, the schisma is equated with 325/324, 361/360, 513/512, and 625/624. 270edo is especially ideal, whose step size being between 361/360 and 513/512, with 217edo exaggerating the comma to be slightly sharp of 361/360. Smaller edos such as 58 (58h val), 111, and 164 are also possible.

Subgroup: 2.3.5.13

Comma list: 676/675, 256000/255879

Subgroup-val mapping: [⟨1 0 -6 -7], ⟨0 4 21 27]]

Optimal tunings:

  • CTE: ~2 = 1200.0000, ~320/243 = 475.5502
error map: ⟨0.000 +0.2457 +0.2401 -0.6728]
  • CWE: ~2 = 1200.0000, ~320/243 = 475.5543
error map: ⟨0.000 +0.2622 +0.3266 -0.5616]

Optimal ET sequence: 53, 164, 217, 270

Badness (Sintel): 0.301

2.3.5.13.19

Subgroup: 2.3.5.13.19

Comma list: 676/675, 1216/1215, 20007/20000

Sval mapping: [⟨1 0 -6 -7 -12], ⟨0 4 21 27 41]]

Optimal tunings:

  • CTE: ~2 = 1200.0000, ~320/243 = 475.5498
  • CWE: ~2 = 1200.0000, ~320/243 = 475.5533

Optimal ET sequence: 53, 164, 217, 270

Badness (Sintel): 0.190

2.3.5.17 temperaments

Quintaleap

Main article: Quintaleap family § Subgroup temperament

Quindromeda

Main article: Quindromeda family § Subgroup temperament

Photia

See also: No-elevens subgroup temperaments § Garibaldia
Main article: Schismatic family § Photia (2.3.5.17)

Quintilischis

For full 17- and 19-limit extensions, see Schismatic family #Quintilipyth or Schismatic family #Quintaschis.
Main article: Schismatic family § Quintilischis (2.3.5.17)

Srutal Archagall

Main article: Srutal archagall

2.3.5.19 temperaments

Rarity

For the 5-limit version, see Miscellaneous 5-limit temperaments #Rarity.

Rarity generator is so close to 23/19, and some edos supporting it have good 19th and 23rd harmonics that it is natural to prescribe a 2.3.5.19.23 extension. Since the generator is also mapped to 368/225, this means that 4275/4232 is tempered out.

Subgroup: 2.3.5.19.23

Comma list: 1035/1024, 16875/16606, 192375/188416

Subgroup-val mapping: [⟨1 11 -10 -3 -2], ⟨0 -13 17 10 9]]

Sval mapping generators: ~2 = 1200.000, ~368/225 = 869.420

Optimal tuning (CTE): ~368/225 = 869.420

Optimal ET sequence: 11, 29, 69, 98c

Badness: 0.0656

Devisemi

See also: No-elevens subgroup temperaments #Devisemi
Main article: Tetracot family § 2.3.5.19 subgroup

Nestoria

See also: No-elevens subgroup temperaments #Garibaldia and #Pontia
Main article: Schismatic family § Nestoria (2.3.5.19)

Stützel

Main article: Meantone family § Stützel (2.3.5.19)

Higher-limit subgroup temperaments

Reversed marvel

As the marvel temperament is based on the marvel comma, 225/224, identifying the harmonic seventh (7/4) as a stack of two classical major sevenths (15/8) octave-reduced and tempering 3rd and 5th harmonics slightly flat, tempering 226/225 instead results in sharper 3rd and 5th harmonics, and a stack of two classical major sevenths equivalent to the 113th harmonic instead of the 7th, so it might as well be called reversed marvel.

Subgroup: 2.3.5.113

Comma list: 226/225

Subgroup-val mapping: [⟨1 0 0 -1], ⟨0 1 0 2], ⟨0 0 1 2]]

Sval mapping generators: ~2, ~3, ~5

Optimal tunings:

  • CTE: ~3/2 = 702.449, ~5/4 = 387.373
  • CWE: ~3/2 = 702.522, ~5/4 = 387.479

Optimal ET sequence: 12, 22, 31, 34, 46, 53

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