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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 ===
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=== 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 ===
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=== 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
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==== 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
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[[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]]
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