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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.


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


[[Comma list]]: 33/32, 45/44
[[Gencom]]: [2 4/3; 33/32, 45/44]
[[Gencom|Gencom mapping]]: [{{val|1 2 1 0 3}}, {{val|0 -1 3 0 1}}]
[[Mapping|Sval mapping]]: [{{val|1 2 1 3}}, {{val|0 -1 3 1}}]
[[Tp tuning|POL2 generator]]: ~4/3 = 520.212
{{Val list|legend=1| 7, 16, 23e, 30bce }}
[[Tp tuning #T2 tuning|RMS error]]: 4.705 cents
Related temperaments: [[Pelogic family #Mavila|mavila]], [[Pelogic family #Pelogic|pelogic]], [[Pelogic family #Armodue|armodue]]
== Porkypine ==
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, 73ce, 95ce, 117bce }}
{{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


Related temperament: [[Porcupine]]
=== Emka ===
: <small>''For full 11- and 13-limit extensions, see [[Hemimean clan #Emka]] or [[Horwell temperaments #Emkay]].''</small>


== Mohaha ==
Subgroup: 2.3.5.11
Subgroup: 2.3.5.11


[[Comma list]]: 81/80, 121/120
[[Comma list]]: 4000/3993, 9453125/9437184


[[Gencom]]: [2 11/9; 81/80 121/120]
[[Gencom]]: [2 11/8; 4000/3993 9453125/9437184]


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


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


[[Tp tuning|POL2 generator]]: ~11/9 = 348.094
[[Tp tuning|POL2 generator]]: ~11/8 = 551.7778


{{Val list|legend=1| 7, 10, 17, 24, 31, 55, 69e, 100e, 131be }}
{{Optimal ET sequence|legend=1| 37, 50, 87, 137, 224 }}


[[Tp tuning #T2 tuning|RMS error]]: 1.392 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.1188 cents


Related temperament: [[mohajira]], [[Meantone family #Migration|migration]]
==== 2.3.5.11.13 ====
 
; Music
[http://micro.soonlabel.com/gene_ward_smith/Others/Stiltner/mohaha10ping2.mp3 Mohaha10ping2] by [[Billy Stiltner]]
 
== Tetracot ==
: <small>''For full 11- and 13-limit extensions, see [[Tetracot family]].''</small>
 
Subgroup: 2.3.5.11
 
[[Comma list]]: 100/99, 243/242
 
[[Gencom]]: [2 10/9; 100/99 243/242]
 
[[Gencom|Gencom mapping]]: [{{val|1 1 1 0 2}}, {{val|0 4 9 0 10}}]
 
[[Mapping|Sval mapping]]: [{{val|1 1 1 2}}, {{val|0 4 9 10}}]
 
[[Tp tuning|POL2 generator]]: ~10/9 = 175.985
 
{{Val list|legend=1| 7, 27e, 34, 41, 75e }}
 
[[Tp tuning #T2 tuning|RMS error]]: 1.182 cents
 
Related temperaments: [[Tetracot family #Monkey|monkey]], [[Tetracot family #Bunya|bunya]], [[Tetracot family #Modus|modus]], [[Tetracot family #Wollemia|wollemia]]
 
=== 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]]: 625/624, 2200/2197, 4000/3993


[[Gencom]]: [2 10/9; 100/99 243/242]
[[Gencom]]: [2 11/8; 625/624 2200/2197 4000/3993]


[[Gencom|Gencom mapping]]: [{{val|1 1 1 0 2 4}}, {{val|0 4 9 0 10 -2}}]
[[Gencom|Gencom mapping]]: [{{val|1 14 6 0 3 6}}, {{val|0 -27 -8 0 1 -5}}]


[[Mapping|Sval mapping]]: [{{val|1 1 1 2 4}}, {{val|0 4 9 10 -2}}]
[[Mapping|Sval mapping]]: [{{val|1 14 6 3 6}}, {{val|0 -27 -8 1 -5}}]


[[Tp tuning|POL2 generator]]: ~10/9 = 176.196
[[Tp tuning|POL2 generator]]: ~11/8 = 551.7753


{{Val list|legend=1| 7, 27e, 34, 41, 75e }}
{{Optimal ET sequence|legend=1| 37, 50, 87, 137, 224 }}


[[Tp tuning #T2 tuning|RMS error]]: 1.140 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.1250 cents


== Tremka ==
=== Tremka ===
: <small>''For full 11- and 13-limit extensions, see [[Hemimean clan #Tremka]].''</small>
: <small>''For full 11- and 13-limit extensions, see [[Hemimean clan#Tremka]].''</small>


Subgroup: 2.3.5.11
Subgroup: 2.3.5.11
Line 114: Line 73:
[[Tp tuning|POL2 generator]]: ~363/320 = 216.182
[[Tp tuning|POL2 generator]]: ~363/320 = 216.182


{{Val list|legend=1| 11b, 28bbcc, 39bc, 50, 111, 161, 272 }}
{{Optimal ET sequence|legend=1| 11b, 28bbcc, 39bc, 50, 111, 161, 272 }}


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


Line 131: Line 90:
[[Tp tuning|POL2 generator]]: ~192/169 = 216.186
[[Tp tuning|POL2 generator]]: ~192/169 = 216.186


{{Val list|legend=1| 11b, 28bbccf, 39bc, 50, 111, 272, 383cf }}
{{Optimal ET sequence|legend=1| 11b, 28bbccf, 39bc, 50, 111, 272, 383cf }}


[[Tp tuning #T2 tuning|RMS error]]: 0.3705 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.3705 cents


== Larry ==
=== Yarman ===
Subgroup: 2.3.5.11
: <small>''For full 11- and 13-limit extensions, see [[Quartonic family]].''</small>
 
[[Comma list]]: 243/242, 4000/3993
 
[[Gencom]]: [2 40/27; 243/242 4000/3993]
 
[[Gencom|Gencom mapping]]: [{{val|1 5 12 0 12}}, {{val|0 -6 -17 0 -15}}]
 
[[Mapping|Sval mapping]]: [{{val|1 5 12 12}}, {{val|0 -6 -17 -15}}]
 
[[Tp tuning|POL2 generator]]: ~40/27 = 683.166
 
{{Val list|legend=1| 7, 58, 65, 137, 202 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.3025 cents
 
Related temperaments: [[Gravity family|gravity]], [[Gravity family #Harry|harry]]
 
== Twentcufo ==
Subgroup: 2.3.5.11
 
[[Comma list]]: 8019/8000, 14641/14580
 
[[Gencom]]: [2 400/363; 8019/8000 14641/14580]
 
[[Gencom|Gencom mapping]]: [{{val|1 0 -2 0 0}}, {{val|0 11 30 0 24}}]
 
[[Mapping|Sval mapping]]: [{{val|1 0 -2 0}}, {{val|0 11 30 24}}]
 
[[Tp tuning|POL2 generator]]: ~400/363 = 172.8796
 
{{Val list|legend=1| 7, 111, 118 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.2393 cents
 
Related temperament: [[Hemimean clan #Undetrita|undetrita]]
 
== Yarman ==
: <small>''For full 11- and 13-limit extensions, see [[Turkish maqam music temperaments]].''</small>


Subgroup: 2.3.5.11
Subgroup: 2.3.5.11
Line 188: Line 109:
[[Tp tuning|POL2 generator]]: ~100/99 = 15.0773
[[Tp tuning|POL2 generator]]: ~100/99 = 15.0773


{{Val list|legend=1| 79, 80, 159, 239, 398 }}
{{Optimal ET sequence|legend=1| 79, 80, 159, 239, 398 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.1682 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.1682 cents


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


Line 205: Line 126:
[[Tp tuning|POL2 generator]]: ~100/99 = 15.0885
[[Tp tuning|POL2 generator]]: ~100/99 = 15.0885


{{Val list|legend=1| 79, 80, 159 }}
{{Optimal ET sequence|legend=1| 79, 80, 159 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.2830 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.2830 cents


== Emka ==
=== Superpine ===
: <small>''For full 11- and 13-limit extensions, see [[Hemimean clan #Emka]] or [[Horwell temperaments #Emkay]].''</small>
{{see also|Meantone family #Superpine}}
{{see also|Meantone family #Trimean}}


Subgroup: 2.3.5.11
Subgroup: 2.3.5.11


[[Comma list]]: 4000/3993, 9453125/9437184
Comma list: 81/80, 1350/1331


[[Gencom]]: [2 11/8; 4000/3993 9453125/9437184]
Mapping: {{mapping| 1 2 4 5 | 0 -3 -12 -11 }}


[[Gencom|Gencom mapping]]: [{{val|1 14 6 0 3}}, {{val|0 -27 -8 0 1}}]
Optimal tunings:  
* CTE: ~2 = 1200.000, ~11/10 = 167.712
* CWE: ~2 = 1200.000, ~11/10 = 167.882


[[Mapping|Sval mapping]]: [{{val|1 14 6 3}}, {{val|0 -27 -8 1}}]
{{Optimal ET sequence|legend=0| 7, 29ce, 36, 43, 50 }}


[[Tp tuning|POL2 generator]]: ~11/8 = 551.7778
==== 2.3.5.11.13 ====
 
{{Val list|legend=1| 37, 50, 87, 137, 224 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.1188 cents


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


[[Comma list]]: 625/624, 2200/2197, 4000/3993
Comma list: 81/80, 144/143, 975/968
 
[[Gencom]]: [2 11/8; 625/624 2200/2197 4000/3993]
 
[[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


{{Val list|legend=1| 37, 50, 87, 137, 224 }}
Mapping: {{mapping| 1 2 4 5 | 0 -3 -12 -11 5 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.1250 cents
Optimal tunings:  
* CTE: ~2 = 1200.000, ~11/10 = 167.729
* CWE: ~2 = 1200.000, ~11/10 = 167.846


== Tricot ==
{{Optimal ET sequence|legend=0| 7, 29ce, 36, 43, 50 }}
: <small>''For full 13-limit extensions, see [[Tricot family]].''</small>


Subgroup: 2.3.5.13
=== Dicot ===
{{main|Dicot family #2.3.5.11 subgroup}}


[[Comma list]]: 2197/2187, 41067/40960
=== Mavila ===
{{main|Mavila family #2.3.5.11 subgroup}}


[[Gencom]]: [2 13/9; 2197/2187, 41067/40960]
=== Tetracot ===
{{main|Tetracot family #2.3.5.11 subgroup }}


[[Gencom|Gencom mapping]]: [{{val|1 0 -13 0 0 0}}, {{val|0 3 29 0 0 7}}]
=== Porkypine ===
{{main|Porcupine family #2.3.5.11 subgroup (porkypine)}}


[[Mapping|Sval mapping]]: [{{val|1 0 -13 0}}, {{val|0 3 29 7}}]
=== Mohaha ===
{{main|Rastmic clan #Mohaha }}


[[Tp tuning|POL2 generator]]: ~13/9 = 633.997
=== Larry ===
{{main|Gravity family #Larry}}


{{Val list|legend=1| 17c, 36c, 53 }}
=== Dequarter ===
{{main|Meantone family #Dequarter }}


[[Tp tuning #T2 tuning|RMS error]]: 0.2342 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 279: 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 296: 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 ===
Subgroup: 2.3.5.17
{{Main| Tricot family #2.3.5.13 subgroup }}
 
[[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}}]
 
[[Mapping|Sval mapping]]: [{{val|1 2 -1 7}}, {{val|0 -1 8 -7}}]
 
[[Tp tuning|POL2 generator]]: ~3/2 = 701.491
 
{{Val list|legend=1| 12, 41, 53, 65 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.4842 cents
 
Related temperament: [[Schismatic family|Schismic]]
 
=== 2.3.5.17.19 ===
Subgroup: 2.3.5.17.19
 
[[Comma list]]: 171/170, 256/255, 324/323
 
[[Gencom]]: [2 4/3; 171/170 256/255 324/323]
 
[[Gencom|Gencom mapping]]: [{{val|1 2 -1 0 0 0 7 3}}, {{val|0 -1 8 0 0 0 -7 3}}]
 
[[Mapping|Sval mapping]]: [{{val|1 2 -1 7 3}}, {{val|0 -1 8 -7 3}}]
 
[[Tp tuning|POL2 generator]]: ~3/2 = 701.470
 
{{Val list|legend=1| 12, 41, 53, 65 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.5374 cents
 
== Quintaleap ==
: <small>''For full 17- and 19-limit extensions, see [[Quintaleap family]].''</small>
 
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
 
{{Val list|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
 
{{Val list|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
 
{{Val list|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 }}]
=== Taylor ===
{{Main| Schismatic family #Taylor (2.3.5.13) }}


[[Tp tuning|POL2 generator]]: ~18/17 = 99.524
=== Vulture ===
: ''For the 5-limit restriction and full 13-limit extension, see [[Vulture family]].''


{{Val list|legend=1| 12, 169, 181, 193, 205, 422 }}
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 #T2 tuning|RMS error]]: 0.0813 cents
[[Subgroup]]: 2.3.5.13


== Quintilischis ==
[[Comma list]]: 676/675, 256000/255879
: <small>''For full 17- and 19-limit extensions, see [[Schismatic family #Quintilipyth]] or [[Schismatic family #Quintaschis]].''</small>


Subgroup: 2.3.5.17
{{Mapping|legend=2| 1 0 -6 -7 | 0 4 21 27 }}


[[Comma list]]: 32805/32768, 1419857/1417176
[[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 }}


[[Gencom]]: [2 18/17; 32805/32768 1419857/1417176]
{{Optimal ET sequence|legend=1| 53, 164, 217, 270 }}


[[Gencom|Gencom mapping]]: [{{val|1 2 -1 0 0 0 5}}, {{val|0 -5 40 0 0 0 -11}}]
[[Badness]] (Sintel): 0.301


[[Mapping|Sval mapping]]: [{{val|1 2 -1 5}}, {{val|0 -5 40 -11}}]
==== 2.3.5.13.19 ====
Subgroup: 2.3.5.13.19


[[Tp tuning|POL2 generator]]: ~18/17 = 99.649
Comma list: 676/675, 1216/1215, 20007/20000


{{Val list|legend=1| 12, 253, 265, 277, 289 }}
Sval mapping: {{mapping| 1 0 -6 -7 -12 | 0 4 21 27 41 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.0719 cents
Optimal tunings:  
* CTE: ~2 = 1200.0000, ~320/243 = 475.5498
* CWE: ~2 = 1200.0000, ~320/243 = 475.5533


=== 2.3.5.17.19 ===
{{Optimal ET sequence|legend=0| 53, 164, 217, 270 }}
Subgroup: 2.3.5.17.19


[[Comma list]]: 4624/4617, 6144/6137, 6885/6859
Badness (Sintel): 0.190


[[Gencom]]: [2 18/17; 4624/4617 6144/6137 6885/6859]
== 2.3.5.17 temperaments ==
=== Quintaleap ===
{{Main| Quintaleap family #Subgroup temperament }}


[[Gencom|Gencom mapping]]: [{{val|1 2 -1 0 0 0 5 4}}, {{val|0 -5 40 0 0 0 -11 3}}]
=== Quindromeda ===
{{Main| Quindromeda family #Subgroup temperament }}


[[Mapping|Sval mapping]]: [{{val|1 2 -1 5 4}}, {{val|0 -5 40 -11 3}}]
=== Photia ===
{{See also| No-elevens subgroup temperaments #Garibaldia }}
{{Main| Schismatic family #Photia (2.3.5.17) }}


[[Tp tuning|POL2 generator]]: ~18/17 = 99.652
=== Quintilischis ===
: ''For full 17- and 19-limit extensions, see [[Schismatic family #Quintilipyth]] or [[Schismatic family #Quintaschis]].''
{{Main| Schismatic family #Quintilischis (2.3.5.17) }}


{{Val list|legend=1| 12, 253, 265, 277, 289 }}
=== Srutal Archagall ===
{{Main|Srutal archagall}}


[[Tp tuning #T2 tuning|RMS error]]: 0.1636 cents
== 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


{{Val list|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 ===
: ''See also: [[No-elevens subgroup temperaments #Devisemi]]''
{{Main| Tetracot family #2.3.5.19 subgroup }}


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


[[Comma list]]: 361/360, 513/512
=== Stützel ===
{{Main| Meantone family #Stützel (2.3.5.19) }}


[[Gencom]]: [2 4/3; 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|Gencom mapping]]: [{{val|1 2 -1 0 0 0 0 3}}, {{val|0 -1 8 0 0 0 0 3}}]
Subgroup: 2.3.5.113


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


[[Tp tuning|POL2 generator]]: ~3/2 = 701.746
{{Mapping|legend=2|1 0 0 -1|0 1 0 2|0 0 1 2}}


{{Val list|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 }}


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