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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.


== Porkypine ==
== 2.3.5.11 temperaments ==
Related temperament: [[Porcupine]]
=== Twentcufo ===
: <small>''For full 11- and 13-limit extensions, see [[Hemimean clan#Undetrita]].''</small>


Subgroup: 2.3.5.11
Subgroup: 2.3.5.11


[[Comma list]]: 55/54, 100/99
[[Comma list]]: 8019/8000, 14641/14580


[[Gencom]]: [2 10/9; 55/54, 100/99]
[[Gencom]]: [2 400/363; 8019/8000 14641/14580]


[[Gencom|Gencom mapping]]: [{{val|1 2 3 0 4}}, {{val|0 -3 -5 0 -4}}]
[[Gencom|Gencom mapping]]: [{{val|1 0 -2 0 0}}, {{val|0 11 30 0 24}}]


[[Mapping|Sval mapping]]: [{{val|1 2 3 4}}, {{val|0 -3 -5 -4}}]
[[Mapping|Sval mapping]]: [{{val|1 0 -2 0}}, {{val|0 11 30 24}}]


[[Tp tuning|POL2 generator]]: ~11/10 = 164.078
[[Tp tuning|POL2 generator]]: ~400/363 = 172.8796


{{Val list|legend=1| 7, 15, 22, 37, 73cd, 95cd, 117bcd }}
{{Optimal ET sequence|legend=1| 7, 111, 118 }}


[[Tp tuning #T2 tuning|RMS error]]: 2.287 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.2393 cents


== Mohaha ==
=== Emka ===
Related temperament: [[mohajira]], [[Meantone family #Migration|migration]]
: <small>''For full 11- and 13-limit extensions, see [[Hemimean clan #Emka]] or [[Horwell temperaments #Emkay]].''</small>


Subgroup: 2.3.5.11
Subgroup: 2.3.5.11


[[Comma list]]: 81/80, 121/120
[[Comma list]]: 4000/3993, 9453125/9437184
 
[[Gencom]]: [2 11/8; 4000/3993 9453125/9437184]


[[Gencom]]: [2 11/9; 81/80 121/120]
[[Gencom|Gencom mapping]]: [{{val|1 14 6 0 3}}, {{val|0 -27 -8 0 1}}]


[[Gencom|Gencom mapping]]: [{{val|1 1 0 0 2}}, {{val|0 2 8 0 5}}]
[[Mapping|Sval mapping]]: [{{val|1 14 6 3}}, {{val|0 -27 -8 1}}]


[[Mapping|Sval mapping]]: [{{val|1 1 0 2}}, {{val|0 2 8 5}}]
[[Tp tuning|POL2 generator]]: ~11/8 = 551.7778


[[Tp tuning|POL2 generator]]: ~11/9 = 348.094
{{Optimal ET sequence|legend=1| 37, 50, 87, 137, 224 }}


{{Val list|legend=1| 7, 10, 17, 24, 31, 55, 69d, 100d, 131bd }}
[[Tp tuning #T2 tuning|RMS error]]: 0.1188 cents


[[Tp tuning #T2 tuning|RMS error]]: 1.392 cents
==== 2.3.5.11.13 ====
Subgroup: 2.3.5.11.13


; Music
[[Comma list]]: 625/624, 2200/2197, 4000/3993
[http://micro.soonlabel.com/gene_ward_smith/Others/Stiltner/mohaha10ping2.mp3 Mohaha10ping2] by [[Billy Stiltner]]


== Tetracot ==
[[Gencom]]: [2 11/8; 625/624 2200/2197 4000/3993]
: <small>''For full 11- and 13-limit extensions, see [[Tetracot family]].''</small>
 
[[Gencom|Gencom mapping]]: [{{val|1 14 6 0 3 6}}, {{val|0 -27 -8 0 1 -5}}]
 
[[Mapping|Sval mapping]]: [{{val|1 14 6 3 6}}, {{val|0 -27 -8 1 -5}}]
 
[[Tp tuning|POL2 generator]]: ~11/8 = 551.7753
 
{{Optimal ET sequence|legend=1| 37, 50, 87, 137, 224 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.1250 cents
 
=== Tremka ===
: <small>''For full 11- and 13-limit extensions, see [[Hemimean clan#Tremka]].''</small>


Subgroup: 2.3.5.11
Subgroup: 2.3.5.11


[[Comma list]]: 100/99, 243/242
[[Comma list]]: 6912/6875, 526153617/524288000


[[Gencom]]: [2 10/9; 100/99 243/242]
[[Gencom]]: [2 363/320; 6912/6875 526153617/524288000]


[[Gencom|Gencom mapping]]: [{{val|1 1 1 0 2}}, {{val|0 4 9 0 10}}]
[[Gencom|Gencom mapping]]: [{{val|1 -4 -2 0 4}}, {{val|0 31 24 0 -3}}]


[[Mapping|Sval mapping]]: [{{val|1 1 1 2}}, {{val|0 4 9 10}}]
[[Mapping|Sval mapping]]: [{{val|1 -4 -2 4}}, {{val|0 31 24 -3}}]


[[Tp tuning|POL2 generator]]: ~10/9 = 175.985
[[Tp tuning|POL2 generator]]: ~363/320 = 216.182


{{Val list|legend=1| 7, 27d, 34, 41, 75d }}
{{Optimal ET sequence|legend=1| 11b, 28bbcc, 39bc, 50, 111, 161, 272 }}


[[Tp tuning #T2 tuning|RMS error]]: 1.182 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.3717 cents


=== 2.3.5.11.13 ===
==== 2.3.5.11.13 ====
Subgroup: 2.3.5.11.13
Subgroup: 2.3.5.11.13


[[Comma list]]: 100/99, 144/143, 243/242
[[Comma list]]: 2200/2197, 3267/3250, 6656/6655


[[Gencom]]: [2 10/9; 100/99 243/242]
[[Gencom]]: [2 192/169; 2200/2197 3267/3250 6656/6655]


[[Gencom|Gencom mapping]]: [{{val|1 1 1 0 2 4}}, {{val|0 4 9 0 10 -2}}]
[[Gencom|Gencom mapping]]: [{{val|1 -4 -2 0 4 1}}, {{val|0 31 24 0 -3 15}}]


[[Mapping|Sval mapping]]: [{{val|1 1 1 2 4}}, {{val|0 4 9 10 -2}}]
[[Mapping|Sval mapping]]: [{{val|1 -4 -2 4 1}}, {{val|0 31 24 -3 15}}]


[[Tp tuning|POL2 generator]]: ~10/9 = 176.196
[[Tp tuning|POL2 generator]]: ~192/169 = 216.186


{{Val list|legend=1| 7, 27d, 34, 41, 75d }}
{{Optimal ET sequence|legend=1| 11b, 28bbccf, 39bc, 50, 111, 272, 383cf }}


[[Tp tuning #T2 tuning|RMS error]]: 1.140 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.3705 cents
 
=== Yarman ===
: <small>''For full 11- and 13-limit extensions, see [[Quartonic family]].''</small>


== Larry ==
Subgroup: 2.3.5.11
Subgroup: 2.3.5.11


[[Comma list]]: 243/242, 4000/3993
[[Comma list]]: 4000/3993, 390625000/387420489
 
[[Gencom]]: [2 100/99; 4000/3993 390625000/387420489]


Related temperaments: [[Gravity family|gravity]], [[Gravity family #Harry|harry]]
[[Gencom|Gencom mapping]]: [{{val|1 2 3 0 4}}, {{val|0 -33 -54 0 -43}}]


[[Gencom]]: [2 40/27; 243/242 4000/3993]
[[Mapping|Sval mapping]]: [{{val|1 2 3 4}}, {{val|0 -33 -54 -43}}]


[[Gencom|Gencom mapping]]: [{{val|1 5 12 0 12}}, {{val|0 -6 -17 0 -15}}]
[[Tp tuning|POL2 generator]]: ~100/99 = 15.0773


[[Mapping|Sval mapping]]: [{{val|1 5 12 12}}, {{val|0 -6 -17 -15}}]
{{Optimal ET sequence|legend=1| 79, 80, 159, 239, 398 }}


[[Tp tuning|POL2 generator]]: ~40/27 = 683.166
[[Tp tuning #T2 tuning|RMS error]]: 0.1682 cents


{{Val list|legend=1| 7, 58, 65, 137, 202 }}
==== 2.3.5.11.13 ====
Subgroup: 2.3.5.11.13


[[Tp tuning #T2 tuning|RMS error]]: 0.3025 cents
[[Comma list]]: 325/324, 4000/3993, 85683/85184


== Twentcufo ==
[[Gencom]]: [2 100/99; 325/324 4000/3993 85683/85184]
: <small>''For full 11- and 13-limit extensions, see [[Hemimean clan #Undetrita]].''</small>


Subgroup: 2.3.5.11
[[Gencom|Gencom mapping]]: [{{val|1 2 3 0 4 4}}, {{val|0 -33 -54 0 -43 -24}}]


[[Comma list]]: 8019/8000, 14641/14580
[[Mapping|Sval mapping]]: [{{val|1 2 3 4 4}}, {{val|0 -33 -54 -43 -24}}]


[[Gencom]]: [2 400/363; 8019/8000 14641/14580]
[[Tp tuning|POL2 generator]]: ~100/99 = 15.0885


[[Gencom|Gencom mapping]]: [{{val|1 0 -2 0 0}}, {{val|0 11 30 0 24}}]
{{Optimal ET sequence|legend=1| 79, 80, 159 }}


[[Mapping|Sval mapping]]: [{{val|1 0 -2 0}}, {{val|0 11 30 24}}]
[[Tp tuning #T2 tuning|RMS error]]: 0.2830 cents


[[Tp tuning|POL2 generator]]: ~400/363 = 172.8796
=== Superpine ===
{{see also|Meantone family #Superpine}}
{{see also|Meantone family #Trimean}}


{{Val list|legend=1| 7, 111, 118 }}
Subgroup: 2.3.5.11


[[Tp tuning #T2 tuning|RMS error]]: 0.2393 cents
Comma list: 81/80, 1350/1331


== Emka ==
Mapping: {{mapping| 1 2 4 5 | 0 -3 -12 -11 }}
: <small>''For full 11- and 13-limit extensions, see [[Hemimean clan #Emka]] or [[Horwell temperaments #Emkay]].''</small>


Subgroup: 2.3.5.11
Optimal tunings:  
* CTE: ~2 = 1200.000, ~11/10 = 167.712
* CWE: ~2 = 1200.000, ~11/10 = 167.882


[[Comma list]]: 4000/3993, 9453125/9437184
{{Optimal ET sequence|legend=0| 7, 29ce, 36, 43, 50 }}


[[Gencom]]: [2 11/8; 4000/3993 9453125/9437184]
==== 2.3.5.11.13 ====


[[Gencom|Gencom mapping]]: [{{val|1 14 6 0 3}}, {{val|0 -27 -8 0 1}}]
Subgroup: 2.3.5.11.13


[[Mapping|Sval mapping]]: [{{val|1 14 6 3}}, {{val|0 -27 -8 1}}]
Comma list: 81/80, 144/143, 975/968


[[Tp tuning|POL2 generator]]: ~11/8 = 551.7778
Mapping: {{mapping| 1 2 4 5 | 0 -3 -12 -11 5 }}


{{Val list|legend=1| 37, 50, 87, 137, 224 }}
Optimal tunings:
* CTE: ~2 = 1200.000, ~11/10 = 167.729
* CWE: ~2 = 1200.000, ~11/10 = 167.846


[[Tp tuning #T2 tuning|RMS error]]: 0.1188 cents
{{Optimal ET sequence|legend=0| 7, 29ce, 36, 43, 50 }}


=== 2.3.5.11.13 ===
=== Dicot ===
Subgroup: 2.3.5.11.13
{{main|Dicot family #2.3.5.11 subgroup}}


[[Comma list]]: 625/624, 2200/2197, 4000/3993
=== Mavila ===
{{main|Mavila family #2.3.5.11 subgroup}}


[[Gencom]]: [2 11/8; 625/624 2200/2197 4000/3993]
=== Tetracot ===
{{main|Tetracot family #2.3.5.11 subgroup }}


[[Gencom|Gencom mapping]]: [{{val|1 14 6 0 3 6}}, {{val|0 -27 -8 0 1 -5}}]
=== Porkypine ===
{{main|Porcupine family #2.3.5.11 subgroup (porkypine)}}


[[Mapping|Sval mapping]]: [{{val|1 14 6 3 6}}, {{val|0 -27 -8 1 -5}}]
=== Mohaha ===
{{main|Rastmic clan #Mohaha }}


[[Tp tuning|POL2 generator]]: ~11/8 = 551.7753
=== Larry ===
{{main|Gravity family #Larry}}


{{Val list|legend=1| 37, 50, 87, 137, 224 }}
=== Dequarter ===
{{main|Meantone family #Dequarter }}


[[Tp tuning #T2 tuning|RMS error]]: 0.1250 cents
=== Hypnotone ===
{{main|Meantone family #Hypnotone }}


== Majvam ==
== 2.3.5.13 temperaments ==
: <small>''For full 13- and 17-limit extensions, see [[Parkleiness temperaments #Majvamic]] or [[Cataharry temperaments #Majvamoid]].''</small>
=== 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
Subgroup: 2.3.5.13
Line 167: Line 201:
[[Tp tuning|POL2 generator]]: ~125/96 = 458.988
[[Tp tuning|POL2 generator]]: ~125/96 = 458.988


{{Val list|legend=1| 34, 149, 183, 217, 400 }}
{{Optimal ET sequence|legend=1| 34, 149, 183, 217, 400 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.1171 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.1171 cents


=== 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).
 
Subgroup: 2.3.5.13.17
Subgroup: 2.3.5.13.17


Line 184: Line 220:
[[Tp tuning|POL2 generator]]: ~125/96 = 458.991
[[Tp tuning|POL2 generator]]: ~125/96 = 458.991


{{Val list|legend=1| 34, 149, 183, 217, 400 }}
{{Optimal ET sequence|legend=1| 34, 149, 183, 217, 400 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.1129 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.1129 cents


== Photia ==
=== Tricot ===
Related temperament: [[Schismatic family|Schismic]]
{{Main| Tricot family #2.3.5.13 subgroup }}
 
=== Taylor ===
{{Main| 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 [[58edo|58]] (58h val), [[111edo|111]], and [[164edo|164]] are also possible.
 
[[Subgroup]]: 2.3.5.13
 
[[Comma list]]: 676/675, 256000/255879
 
{{Mapping|legend=2| 1 0 -6 -7 | 0 4 21 27 }}
 
[[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 }}
 
{{Optimal ET sequence|legend=1| 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: {{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|legend=0| 53, 164, 217, 270 }}


Subgroup: 2.3.5.17
Badness (Sintel): 0.190


[[Comma list]]: 256/255, 1458/1445
== 2.3.5.17 temperaments ==
=== Quintaleap ===
{{Main| Quintaleap family #Subgroup temperament }}


[[Gencom]]: [2 4/3; 256/255 1458/1445]
=== Quindromeda ===
{{Main| Quindromeda family #Subgroup temperament }}


[[Gencom|Gencom mapping]]: [{{val|1 2 -1 0 0 0 7}}, {{val|0 -1 8 0 0 0 -7}}]
=== Photia ===
{{See also| No-elevens subgroup temperaments #Garibaldia }}
{{Main| Schismatic family #Photia (2.3.5.17) }}


[[Mapping|Sval mapping]]: [{{val|1 2 -1 7}}, {{val|0 -1 8 -7}}]
=== Quintilischis ===
: ''For full 17- and 19-limit extensions, see [[Schismatic family #Quintilipyth]] or [[Schismatic family #Quintaschis]].''
{{Main| Schismatic family #Quintilischis (2.3.5.17) }}


[[Tp tuning|POL2 generator]]: ~3/2 = 701.491
=== Srutal Archagall ===
{{Main|Srutal archagall}}


{{Val list|legend=1| 12, 41, 53, 65 }}
== 2.3.5.19 temperaments ==
=== Rarity ===
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Rarity]].''


[[Tp tuning #T2 tuning|RMS error]]: 0.4842 cents
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.


=== 2.3.5.17.19 ===
[[Subgroup]]: 2.3.5.19.23
Subgroup: 2.3.5.17.19


[[Comma list]]: 171/170, 256/255, 324/323
[[Comma list]]: 1035/1024, 16875/16606, 192375/188416


[[Gencom]]: [2 4/3; 171/170 256/255 324/323]
{{Mapping|legend=2| 1 11 -10 -3 -2 | 0 -13 17 10 9 }}


[[Gencom|Gencom mapping]]: [{{val|1 2 -1 0 0 0 7 3}}, {{val|0 -1 8 0 0 0 -7 3}}]
: Sval mapping generators: ~2 = 1200.000, ~368/225 = 869.420


[[Mapping|Sval mapping]]: [{{val|1 2 -1 7 3}}, {{val|0 -1 8 -7 3}}]
[[Optimal tuning]] ([[CTE]]): ~368/225 = 869.420


[[Tp tuning|POL2 generator]]: ~3/2 = 701.470
{{Optimal ET sequence|legend=1| 11, 29, 69, 98c }}


{{Val list|legend=1| 12, 41, 53, 65 }}
[[Badness]]: 0.0656


[[Tp tuning #T2 tuning|RMS error]]: 0.5374 cents
=== Devisemi ===
: ''See also: [[No-elevens subgroup temperaments #Devisemi]]''
{{Main| Tetracot family #2.3.5.19 subgroup }}


== Nestoria ==
=== Nestoria ===
Related temperament: [[Schismatic family|Schismic]]
: ''See also: [[No-elevens subgroup temperaments #Garibaldia]] and [[No-elevens subgroup temperaments #Pontia|#Pontia]]''
{{Main| Schismatic family #Nestoria (2.3.5.19) }}


Subgroup: 2.3.5.19
=== Stützel ===
{{Main| Meantone family #Stützel (2.3.5.19) }}


[[Comma list]]: 361/360, 513/512
== 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.


[[Gencom]]: [2 4/3; 361/360 513/512]
Subgroup: 2.3.5.113


[[Gencom|Gencom mapping]]: [{{val|1 2 -1 0 0 0 0 3}}, {{val|0 -1 8 0 0 0 0 3}}]
[[Comma list]]: 226/225


[[Mapping|Sval mapping]]: [{{val|1 2 -1 3}}, {{val|0 -1 8 3}}]
{{Mapping|legend=2|1 0 0 -1|0 1 0 2|0 0 1 2}}


[[Tp tuning|POL2 generator]]: ~3/2 = 701.746
: Sval mapping generators: ~2, ~3, ~5


{{Val list|legend=1| 12, 29, 41, 53, 118, 171 }}
[[Optimal tuning]]s:
* CTE: ~3/2 = 702.449, ~5/4 = 387.373
* CWE: ~3/2 = 702.522, ~5/4 = 387.479


[[Tp tuning #T2 tuning|RMS error]]: 0.1763 cents
{{Optimal ET sequence|legend=1| 12, 22, 31, 34, 46, 53 }}


[[Category:Regular temperament theory]]
[[Category:Temperament collections]]
[[Category:Temperament collection]]
[[Category:Subgroup temperaments]]
[[Category:Subgroup temperaments]]
[[Category:Rank 2]]
[[Category:Rank 2]]
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