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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 ===
Line 145: Line 176:


=== Larry ===
=== Larry ===
{{main|Gravity family #2.3.5.11 subgroup (Larry)}}
{{main|Gravity family #Larry}}


=== 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 ==
=== 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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=== Tricot ===
=== Tricot ===
{{main|Tricot family #2.3.5.13 subgroup }}
{{Main| Tricot family #2.3.5.13 subgroup }}


=== Taylor ===
=== Taylor ===
{{main| Schismatic family #Taylor }}
{{Main| Schismatic family #Taylor (2.3.5.13) }}
 
== 2.3.5.17 temperaments == 
 
=== Photia ===
: <small>''See also: [[No-elevens subgroup temperaments #Garibaldia]]''</small>
 
Subgroup: 2.3.5.17
 
[[Comma list]]: 256/255, 1458/1445
 
[[Gencom]]: [2 4/3; 256/255 1458/1445]


[[Gencom|Gencom mapping]]: [{{val|1 2 -1 0 0 0 7}}, {{val|0 -1 8 0 0 0 -7}}]
=== Vulture ===
: ''For the 5-limit restriction and full 13-limit extension, see [[Vulture family]].''


[[Mapping|Sval mapping]]: [{{val|1 2 -1 7}}, {{val|0 -1 8 -7}}]
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.


[[Tp tuning|POL2 generator]]: ~3/2 = 701.491
[[Subgroup]]: 2.3.5.13


{{Optimal ET sequence|legend=1| 12, 41, 53, 65 }}
[[Comma list]]: 676/675, 256000/255879


[[Tp tuning #T2 tuning|RMS error]]: 0.4842 cents
{{Mapping|legend=2| 1 0 -6 -7 | 0 4 21 27 }}


Related temperament: [[Schismatic family|Schismic]]
[[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 }}


==== 2.3.5.17.19 ====
{{Optimal ET sequence|legend=1| 53, 164, 217, 270 }}
Subgroup: 2.3.5.17.19


[[Comma list]]: 171/170, 256/255, 324/323
[[Badness]] (Sintel): 0.301


[[Gencom]]: [2 4/3; 171/170 256/255 324/323]
==== 2.3.5.13.19 ====
Subgroup: 2.3.5.13.19


[[Gencom|Gencom mapping]]: [{{val|1 2 -1 0 0 0 7 3}}, {{val|0 -1 8 0 0 0 -7 3}}]
Comma list: 676/675, 1216/1215, 20007/20000


[[Mapping|Sval mapping]]: [{{val|1 2 -1 7 3}}, {{val|0 -1 8 -7 3}}]
Sval mapping: {{mapping| 1 0 -6 -7 -12 | 0 4 21 27 41 }}


[[Tp tuning|POL2 generator]]: ~3/2 = 701.470
Optimal tunings:
* CTE: ~2 = 1200.0000, ~320/243 = 475.5498
* CWE: ~2 = 1200.0000, ~320/243 = 475.5533


{{Optimal ET sequence|legend=1| 12, 41, 53, 65 }}
{{Optimal ET sequence|legend=0| 53, 164, 217, 270 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.5374 cents
Badness (Sintel): 0.190


== 2.3.5.17 temperaments ==
=== Quintaleap ===
=== Quintaleap ===
: <small>''For full 17- and 19-limit extensions, see [[Quintaleap family]].''</small>
{{Main| Quintaleap family #Subgroup temperament }}
 
Subgroup: 2.3.5.17
 
[[Comma list]]: 256/255, 1419857/1417176
 
[[Gencom]]: [2 18/17; 256/255 1419857/1417176]
 
[[Gencom|Gencom mapping]]: [{{val| 1 2 1 0 0 0 5 }}, {{val| 0 -5 16 0 0 0 -11 }}]
 
[[Mapping|Sval mapping]]: [{{val| 1 2 1 5 }}, {{val| 0 -5 16 -11 }}]
 
[[Tp tuning|POL2 generator]]: ~18/17 = 99.272
 
{{Optimal ET sequence|legend=1| 12, 109, 121, 133 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.3051 cents
 
==== 2.3.5.17.19 ====
Subgroup: 2.3.5.17.19
 
[[Comma list]]: 256/255, 361/360, 4624/4617
 
[[Gencom]]: [2 18/17; 256/255 361/360 4624/4617]
 
[[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 ===
=== Quindromeda ===
: <small>''For full 17- and 19-limit extensions, see [[Quindromeda family]].''</small>
{{Main| Quindromeda family #Subgroup temperament }}


Subgroup: 2.3.5.17
=== Photia ===
 
{{See also| No-elevens subgroup temperaments #Garibaldia }}
[[Comma list]]: 24576/24565, 295936/295245
{{Main| Schismatic family #Photia (2.3.5.17) }}
 
[[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 ===
=== Quintilischis ===
: <small>''For full 17- and 19-limit extensions, see [[Schismatic family #Quintilipyth]] or [[Schismatic family #Quintaschis]].''</small>
: ''For full 17- and 19-limit extensions, see [[Schismatic family #Quintilipyth]] or [[Schismatic family #Quintaschis]].''
{{Main| Schismatic family #Quintilischis (2.3.5.17) }}


Subgroup: 2.3.5.17
=== Srutal Archagall ===
 
{{Main|Srutal archagall}}
[[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


== 2.3.5.19 temperaments ==
== 2.3.5.19 temperaments ==
=== Rarity ===
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Rarity]].''


=== Stützel ===
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


[[Comma list]]: 81/80, 96/95
[[Subgroup]]: 2.3.5.19.23


[[Gencom]]: [2 4/3; 81/80 96/95]
[[Comma list]]: 1035/1024, 16875/16606, 192375/188416


[[Gencom|Gencom mapping]]: [{{val|1 2 4 0 0 0 0 3}}, {{val|0 -1 -4 0 0 0 0 3}}]
{{Mapping|legend=2| 1 11 -10 -3 -2 | 0 -13 17 10 9 }}


[[Mapping|Sval mapping]]: [{{val|1 2 4 3}}, {{val|0 -1 -4 3}}]
: Sval mapping generators: ~2 = 1200.000, ~368/225 = 869.420


[[Tp tuning|POL2 generator]]: ~3/2 = 697.867
[[Optimal tuning]] ([[CTE]]): ~368/225 = 869.420


{{Optimal ET sequence|legend=1| 5, 7, 12, 31, 43 }}
{{Optimal ET sequence|legend=1| 11, 29, 69, 98c }}


[[Tp tuning #T2 tuning|RMS error]]: 1.378 cents
[[Badness]]: 0.0656
 
Related temperament: [[Meantone family|meantone]]


=== Devisemi ===
=== Devisemi ===
: <small>''See also: [[No-elevens subgroup temperaments #Devisemi]]''</small>
: ''See also: [[No-elevens subgroup temperaments #Devisemi]]''
 
{{Main| Tetracot family #2.3.5.19 subgroup }}
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}}]
 
[[Tp tuning|POL2 generator]]: ~19/18 = 88.077
 
{{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 ===
=== 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
== 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.


{{Mapping|legend=3| 1 2 -1 0 0 0 0 3 | 0 -1 8 0 0 0 0 3 }}
Subgroup: 2.3.5.113


: [[gencom]]: [2 4/3; 361/360 513/512]
[[Comma list]]: 226/225


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 701.746
{{Mapping|legend=2|1 0 0 -1|0 1 0 2|0 0 1 2}}


{{Optimal ET sequence|legend=1| 12, 29, 41, 53, 118, 171 }}
: Sval mapping generators: ~2, ~3, ~5


[[Tp tuning #T2 tuning|RMS error]]: 0.1763 cents
[[Optimal tuning]]s:
* CTE: ~3/2 = 702.449, ~5/4 = 387.373
* CWE: ~3/2 = 702.522, ~5/4 = 387.479


Related temperament: [[Schismatic family|schismic]]
{{Optimal ET sequence|legend=1| 12, 22, 31, 34, 46, 53 }}
 
=== Rarity ===
: <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.
 
[[Subgroup]]: 2.3.5.19.23
 
[[Comma list]]: 1035/1024, 16875/16606, 192375/188416
 
{{Mapping|legend=2| 1 11 -10 -3 -2 | 0 -13 17 10 9 }}
 
: 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]]
Retrieved from "https://en.xen.wiki/w/No-sevens_subgroup_temperaments"