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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.


== Dicot ==
== 2.3.5.11 temperaments ==
{{main|Dicot family #2.3.5.11 subgroup}}
=== Twentcufo ===
 
: <small>''For full 11- and 13-limit extensions, see [[Hemimean clan#Undetrita]].''</small>
== Mavila ==
{{main|Mavila family #2.3.5.11 subgroup}}
 
== Porkypine ==
{{main|Porcupine family #2.3.5.11 subgroup (porkypine)}}
 
== Mohaha ==
{{main|Rastmic clan #Mohaha }}
 
== Tetracot ==
{{main|Tetracot family #2.3.5.11 subgroup }}
 
== Tremka ==
{{main|Hemimean clan #2.3.5.11 subgroup }}
 
== Larry ==
{{main|Gravity family #Larry}}
 
== Twentcufo ==
{{main|Hemimean clan #Twentcufo}}
 
== Yarman ==
: <small>''For full 11- and 13-limit extensions, see [[Quartonic family]].''</small>


Subgroup: 2.3.5.11
Subgroup: 2.3.5.11


[[Comma list]]: 4000/3993, 390625000/387420489
[[Comma list]]: 8019/8000, 14641/14580


[[Gencom]]: [2 100/99; 4000/3993 390625000/387420489]
[[Gencom]]: [2 400/363; 8019/8000 14641/14580]


[[Gencom|Gencom mapping]]: [{{val|1 2 3 0 4}}, {{val|0 -33 -54 0 -43}}]
[[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 -33 -54 -43}}]
[[Mapping|Sval mapping]]: [{{val|1 0 -2 0}}, {{val|0 11 30 24}}]


[[Tp tuning|POL2 generator]]: ~100/99 = 15.0773
[[Tp tuning|POL2 generator]]: ~400/363 = 172.8796


{{Optimal ET sequence|legend=1| 79, 80, 159, 239, 398 }}
{{Optimal ET sequence|legend=1| 7, 111, 118 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.1682 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.2393 cents


=== 2.3.5.11.13 ===
=== Emka ===
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|Gencom mapping]]: [{{val|1 2 3 0 4 4}}, {{val|0 -33 -54 0 -43 -24}}]
 
[[Mapping|Sval mapping]]: [{{val|1 2 3 4 4}}, {{val|0 -33 -54 -43 -24}}]
 
[[Tp tuning|POL2 generator]]: ~100/99 = 15.0885
 
{{Optimal ET sequence|legend=1| 79, 80, 159 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.2830 cents
 
== Emka ==
: <small>''For full 11- and 13-limit extensions, see [[Hemimean clan #Emka]] or [[Horwell temperaments #Emkay]].''</small>
: <small>''For full 11- and 13-limit extensions, see [[Hemimean clan #Emka]] or [[Horwell temperaments #Emkay]].''</small>


Line 80: Line 41:
[[Tp tuning #T2 tuning|RMS error]]: 0.1188 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.1188 cents


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


Line 97: Line 58:
[[Tp tuning #T2 tuning|RMS error]]: 0.1250 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.1250 cents


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


Subgroup: 2.3.5.13
Subgroup: 2.3.5.11


[[Comma list]]: 2197/2187, 41067/40960
[[Comma list]]: 6912/6875, 526153617/524288000


[[Gencom]]: [2 13/9; 2197/2187, 41067/40960]
[[Gencom]]: [2 363/320; 6912/6875 526153617/524288000]


[[Gencom|Gencom mapping]]: [{{val|1 0 -13 0 0 0}}, {{val|0 3 29 0 0 7}}]
[[Gencom|Gencom mapping]]: [{{val|1 -4 -2 0 4}}, {{val|0 31 24 0 -3}}]


[[Mapping|Sval mapping]]: [{{val|1 0 -13 0}}, {{val|0 3 29 7}}]
[[Mapping|Sval mapping]]: [{{val|1 -4 -2 4}}, {{val|0 31 24 -3}}]


[[Tp tuning|POL2 generator]]: ~13/9 = 633.997
[[Tp tuning|POL2 generator]]: ~363/320 = 216.182


{{Optimal ET sequence|legend=1| 17c, 36c, 53 }}
{{Optimal ET sequence|legend=1| 11b, 28bbcc, 39bc, 50, 111, 161, 272 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.2342 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.3717 cents


== Taylor ==
==== 2.3.5.11.13 ====
Subgroup: 2.3.5.13
Subgroup: 2.3.5.11.13


[[Comma list]]: 676/675, 32805/32768
[[Comma list]]: 2200/2197, 3267/3250, 6656/6655


[[Gencom]]: [2 15/13; 676/675 32805/32768]
[[Gencom]]: [2 192/169; 2200/2197 3267/3250 6656/6655]


[[Gencom|Gencom mapping]]: [{{val|1 2 -1 0 0 1}}, {{val|0 -2 16 0 0 13}}]
[[Gencom|Gencom mapping]]: [{{val|1 -4 -2 0 4 1}}, {{val|0 31 24 0 -3 15}}]


[[Mapping|Sval mapping]]: [{{val|1 2 -1 1}}, {{val|0 -2 16 13}}]
[[Mapping|Sval mapping]]: [{{val|1 -4 -2 4 1}}, {{val|0 31 24 -3 15}}]


[[Tp tuning|POL2 generator]]: ~15/13 = 249.145
[[Tp tuning|POL2 generator]]: ~192/169 = 216.186


{{Optimal ET sequence|legend=1| 24, 53, 130, 183, 236 }}
{{Optimal ET sequence|legend=1| 11b, 28bbccf, 39bc, 50, 111, 272, 383cf }}


[[Tp tuning #T2 tuning|RMS error]]: 0.1485 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.3705 cents


Related temperament: [[Schismatic family|hemischis]]
=== Yarman ===
: <small>''For full 11- and 13-limit extensions, see [[Quartonic family]].''</small>


== Majvam ==
Subgroup: 2.3.5.11
: <small>''For full 13- and 17-limit extensions, see [[Parkleiness temperaments #Majvamic]] or [[Cataharry temperaments #Majvamoid]].''</small>


Subgroup: 2.3.5.13
[[Comma list]]: 4000/3993, 390625000/387420489


[[Comma list]]: 676/675, 127401984/126953125
[[Gencom]]: [2 100/99; 4000/3993 390625000/387420489]


[[Gencom]]: [2 125/96; 676/675 127401984/126953125]
[[Gencom|Gencom mapping]]: [{{val|1 2 3 0 4}}, {{val|0 -33 -54 0 -43}}]


[[Gencom|Gencom mapping]]: [{{val|1 10 5 0 0 19}}, {{val|0 -22 -7 0 0 -40}}]
[[Mapping|Sval mapping]]: [{{val|1 2 3 4}}, {{val|0 -33 -54 -43}}]


[[Mapping|Sval mapping]]: [{{val|1 10 5 19}}, {{val|0 -22 -7 -40}}]
[[Tp tuning|POL2 generator]]: ~100/99 = 15.0773


[[Tp tuning|POL2 generator]]: ~125/96 = 458.988
{{Optimal ET sequence|legend=1| 79, 80, 159, 239, 398 }}


{{Optimal ET sequence|legend=1| 34, 149, 183, 217, 400 }}
[[Tp tuning #T2 tuning|RMS error]]: 0.1682 cents


[[Tp tuning #T2 tuning|RMS error]]: 0.1171 cents
==== 2.3.5.11.13 ====
Subgroup: 2.3.5.11.13


=== 2.3.5.13.17 ===
[[Comma list]]: 325/324, 4000/3993, 85683/85184
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
[[Gencom]]: [2 100/99; 325/324 4000/3993 85683/85184]


[[Comma list]]: 676/675, 2601/2600, 24576/24565
[[Gencom|Gencom mapping]]: [{{val|1 2 3 0 4 4}}, {{val|0 -33 -54 0 -43 -24}}]


[[Gencom]]: [2 125/96; 676/675 2601/2600 24576/24565]
[[Mapping|Sval mapping]]: [{{val|1 2 3 4 4}}, {{val|0 -33 -54 -43 -24}}]


[[Gencom|Gencom mapping]]: [{{val|1 10 5 0 0 19 6}}, {{val|0 -22 -7 0 0 -40 -5}}]
[[Tp tuning|POL2 generator]]: ~100/99 = 15.0885


[[Mapping|Sval mapping]]: [{{val|1 10 5 19 6}}, {{val|0 -22 -7 -40 -5}}]
{{Optimal ET sequence|legend=1| 79, 80, 159 }}


[[Tp tuning|POL2 generator]]: ~125/96 = 458.991
[[Tp tuning #T2 tuning|RMS error]]: 0.2830 cents


{{Optimal ET sequence|legend=1| 34, 149, 183, 217, 400 }}
=== Superpine ===
{{see also|Meantone family #Superpine}}
{{see also|Meantone family #Trimean}}


[[Tp tuning #T2 tuning|RMS error]]: 0.1129 cents
Subgroup: 2.3.5.11


== Photia ==
Comma list: 81/80, 1350/1331
: <small>''See also: [[No-elevens subgroup temperaments #Garibaldia]]''</small>


Subgroup: 2.3.5.17
Mapping: {{mapping| 1 2 4 5 | 0 -3 -12 -11 }}


[[Comma list]]: 256/255, 1458/1445
Optimal tunings:  
* CTE: ~2 = 1200.000, ~11/10 = 167.712
* CWE: ~2 = 1200.000, ~11/10 = 167.882


[[Gencom]]: [2 4/3; 256/255 1458/1445]
{{Optimal ET sequence|legend=0| 7, 29ce, 36, 43, 50 }}


[[Gencom|Gencom mapping]]: [{{val|1 2 -1 0 0 0 7}}, {{val|0 -1 8 0 0 0 -7}}]
==== 2.3.5.11.13 ====


[[Mapping|Sval mapping]]: [{{val|1 2 -1 7}}, {{val|0 -1 8 -7}}]
Subgroup: 2.3.5.11.13


[[Tp tuning|POL2 generator]]: ~3/2 = 701.491
Comma list: 81/80, 144/143, 975/968


{{Optimal ET sequence|legend=1| 12, 41, 53, 65 }}
Mapping: {{mapping| 1 2 4 5 | 0 -3 -12 -11 5 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.4842 cents
Optimal tunings:  
* CTE: ~2 = 1200.000, ~11/10 = 167.729
* CWE: ~2 = 1200.000, ~11/10 = 167.846


Related temperament: [[Schismatic family|Schismic]]
{{Optimal ET sequence|legend=0| 7, 29ce, 36, 43, 50 }}


=== 2.3.5.17.19 ===
=== Dicot ===
Subgroup: 2.3.5.17.19
{{main|Dicot family #2.3.5.11 subgroup}}


[[Comma list]]: 171/170, 256/255, 324/323
=== Mavila ===
{{main|Mavila family #2.3.5.11 subgroup}}


[[Gencom]]: [2 4/3; 171/170 256/255 324/323]
=== Tetracot ===
{{main|Tetracot family #2.3.5.11 subgroup }}


[[Gencom|Gencom mapping]]: [{{val|1 2 -1 0 0 0 7 3}}, {{val|0 -1 8 0 0 0 -7 3}}]
=== Porkypine ===
{{main|Porcupine family #2.3.5.11 subgroup (porkypine)}}


[[Mapping|Sval mapping]]: [{{val|1 2 -1 7 3}}, {{val|0 -1 8 -7 3}}]
=== Mohaha ===
{{main|Rastmic clan #Mohaha }}


[[Tp tuning|POL2 generator]]: ~3/2 = 701.470
=== Larry ===
{{main|Gravity family #Larry}}


{{Optimal ET sequence|legend=1| 12, 41, 53, 65 }}
=== Dequarter ===
{{main|Meantone family #Dequarter }}


[[Tp tuning #T2 tuning|RMS error]]: 0.5374 cents
=== Hypnotone ===
{{main|Meantone family #Hypnotone }}


== Quintaleap ==
== 2.3.5.13 temperaments ==
: <small>''For full 17- and 19-limit extensions, see [[Quintaleap family]].''</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: 2.3.5.17
Subgroup: 2.3.5.13


[[Comma list]]: 256/255, 1419857/1417176
[[Comma list]]: 676/675, 127401984/126953125


[[Gencom]]: [2 18/17; 256/255 1419857/1417176]
[[Gencom]]: [2 125/96; 676/675 127401984/126953125]


[[Gencom|Gencom mapping]]: [{{val| 1 2 1 0 0 0 5 }}, {{val| 0 -5 16 0 0 0 -11 }}]
[[Gencom|Gencom mapping]]: [{{val|1 10 5 0 0 19}}, {{val|0 -22 -7 0 0 -40}}]


[[Mapping|Sval mapping]]: [{{val| 1 2 1 5 }}, {{val| 0 -5 16 -11 }}]
[[Mapping|Sval mapping]]: [{{val|1 10 5 19}}, {{val|0 -22 -7 -40}}]


[[Tp tuning|POL2 generator]]: ~18/17 = 99.272
[[Tp tuning|POL2 generator]]: ~125/96 = 458.988


{{Optimal ET sequence|legend=1| 12, 109, 121, 133 }}
{{Optimal ET sequence|legend=1| 34, 149, 183, 217, 400 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.3051 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.1171 cents


=== 2.3.5.17.19 ===
==== 2.3.5.13.17 ====
Subgroup: 2.3.5.17.19
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).


[[Comma list]]: 256/255, 361/360, 4624/4617
Subgroup: 2.3.5.13.17


[[Gencom]]: [2 18/17; 256/255 361/360 4624/4617]
[[Comma list]]: 676/675, 2601/2600, 24576/24565


[[Gencom|Gencom mapping]]: [{{val| 1 2 1 0 0 0 5 4 }}, {{val| 0 -5 16 0 0 0 -11 3 }}]
[[Gencom]]: [2 125/96; 676/675 2601/2600 24576/24565]


[[Mapping|Sval mapping]]: [{{val| 1 2 1 5 4 }}, {{val| 0 -5 16 -11 3 }}]
[[Gencom|Gencom mapping]]: [{{val|1 10 5 0 0 19 6}}, {{val|0 -22 -7 0 0 -40 -5}}]


[[Tp tuning|POL2 generator]]: ~18/17 = 99.276
[[Mapping|Sval mapping]]: [{{val|1 10 5 19 6}}, {{val|0 -22 -7 -40 -5}}]


{{Optimal ET sequence|legend=1| 12, 109, 121, 133 }}
[[Tp tuning|POL2 generator]]: ~125/96 = 458.991


[[Tp tuning #T2 tuning|RMS error]]: 0.3427 cents
{{Optimal ET sequence|legend=1| 34, 149, 183, 217, 400 }}


== Quindromeda ==
[[Tp tuning #T2 tuning|RMS error]]: 0.1129 cents
: <small>''For full 17- and 19-limit extensions, see [[Quindromeda family]].''</small>


Subgroup: 2.3.5.17
=== Tricot ===
{{Main| Tricot family #2.3.5.13 subgroup }}


[[Comma list]]: 24576/24565, 295936/295245
=== Taylor ===
{{Main| Schismatic family #Taylor (2.3.5.13) }}


[[Gencom]]: [2 18/17; 24576/24565 295936/295245]
=== Vulture ===
: ''For the 5-limit restriction and full 13-limit extension, see [[Vulture family]].''


[[Gencom|Gencom mapping]]: [{{val| 1 2 0 0 0 0 5 }}, {{val| 0 -5 28 0 0 0 -11 }}]
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.


[[Mapping|Sval mapping]]: [{{val| 1 2 0 5 }}, {{val| 0 -5 28 -11 }}]
[[Subgroup]]: 2.3.5.13


[[Tp tuning|POL2 generator]]: ~18/17 = 99.524
[[Comma list]]: 676/675, 256000/255879


{{Optimal ET sequence|legend=1| 12, 169, 181, 193, 205, 422 }}
{{Mapping|legend=2| 1 0 -6 -7 | 0 4 21 27 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.0757 cents
[[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]]: 1216/1215, 1445/1444, 6144/6137
[[Badness]] (Sintel): 0.301


[[Gencom]]: [2 18/17; 1216/1215 1445/1444 6144/6137]
==== 2.3.5.13.19 ====
Subgroup: 2.3.5.13.19


[[Gencom|Gencom mapping]]: [{{val| 1 2 0 0 0 0 5 4 }}, {{val| 0 -5 28 0 0 0 -11 3 }}]
Comma list: 676/675, 1216/1215, 20007/20000


[[Mapping|Sval mapping]]: [{{val| 1 2 0 5 4 }}, {{val| 0 -5 28 -11 3 }}]
Sval mapping: {{mapping| 1 0 -6 -7 -12 | 0 4 21 27 41 }}


[[Tp tuning|POL2 generator]]: ~18/17 = 99.524
Optimal tunings:  
* CTE: ~2 = 1200.0000, ~320/243 = 475.5498
* CWE: ~2 = 1200.0000, ~320/243 = 475.5533


{{Optimal ET sequence|legend=1| 12, 169, 181, 193, 205, 422 }}
{{Optimal ET sequence|legend=0| 53, 164, 217, 270 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.0813 cents
Badness (Sintel): 0.190


== Quintilischis ==
== 2.3.5.17 temperaments ==
: <small>''For full 17- and 19-limit extensions, see [[Schismatic family #Quintilipyth]] or [[Schismatic family #Quintaschis]].''</small>
=== Quintaleap ===
{{Main| Quintaleap family #Subgroup temperament }}


Subgroup: 2.3.5.17
=== Quindromeda ===
{{Main| Quindromeda family #Subgroup temperament }}


[[Comma list]]: 32805/32768, 1419857/1417176
=== Photia ===
{{See also| No-elevens subgroup temperaments #Garibaldia }}
{{Main| Schismatic family #Photia (2.3.5.17) }}


[[Gencom]]: [2 18/17; 32805/32768 1419857/1417176]
=== Quintilischis ===
: ''For full 17- and 19-limit extensions, see [[Schismatic family #Quintilipyth]] or [[Schismatic family #Quintaschis]].''
{{Main| Schismatic family #Quintilischis (2.3.5.17) }}


[[Gencom|Gencom mapping]]: [{{val|1 2 -1 0 0 0 5}}, {{val|0 -5 40 0 0 0 -11}}]
=== Srutal Archagall ===
{{Main|Srutal archagall}}


[[Mapping|Sval mapping]]: [{{val|1 2 -1 5}}, {{val|0 -5 40 -11}}]
== 2.3.5.19 temperaments ==
=== Rarity ===
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Rarity]].''


[[Tp tuning|POL2 generator]]: ~18/17 = 99.649
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.


{{Optimal ET sequence|legend=1| 12, 253, 265, 277, 289 }}
[[Subgroup]]: 2.3.5.19.23


[[Tp tuning #T2 tuning|RMS error]]: 0.0719 cents
[[Comma list]]: 1035/1024, 16875/16606, 192375/188416


=== 2.3.5.17.19 ===
{{Mapping|legend=2| 1 11 -10 -3 -2 | 0 -13 17 10 9 }}
Subgroup: 2.3.5.17.19


[[Comma list]]: 4624/4617, 6144/6137, 6885/6859
: Sval mapping generators: ~2 = 1200.000, ~368/225 = 869.420


[[Gencom]]: [2 18/17; 4624/4617 6144/6137 6885/6859]
[[Optimal tuning]] ([[CTE]]): ~368/225 = 869.420


[[Gencom|Gencom mapping]]: [{{val|1 2 -1 0 0 0 5 4}}, {{val|0 -5 40 0 0 0 -11 3}}]
{{Optimal ET sequence|legend=1| 11, 29, 69, 98c }}


[[Mapping|Sval mapping]]: [{{val|1 2 -1 5 4}}, {{val|0 -5 40 -11 3}}]
[[Badness]]: 0.0656


[[Tp tuning|POL2 generator]]: ~18/17 = 99.652
=== Devisemi ===
: ''See also: [[No-elevens subgroup temperaments #Devisemi]]''
{{Main| Tetracot family #2.3.5.19 subgroup }}


{{Optimal ET sequence|legend=1| 12, 253, 265, 277, 289 }}
=== Nestoria ===
 
[[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]]
 
== Dequarter ==
[[Subgroup]]: 2.3.5.11
 
[[Comma list]]: 33/32, 55/54
 
{{Mapping|legend=2| 1 0 -4 5 | 0 1 4 -1 }}
 
: sval mapping generators: ~2, ~3
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 696.0387
 
{{Optimal ET sequence|legend=1| 5, 7, 19e, 26e }}
 
[[Badness]]: 0.0145
 
Related temperament: [[Meantone family|meantone]]
 
=== Dreamtone ===
Subgroup: 2.3.5.11.13
 
Comma list: 33/32, 55/54, 975/968
 
Sval mapping: {{mapping| 1 0 -4 5 21 | 0 1 4 -1 -11 }}
 
: sval mapping generators: ~2, ~3
 
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 689.6993
 
Optimal ET sequence: {{Optimal ET sequence| 7, 19eff, 26eff, 33ceeff, 40ceeff }}
 
Badness: 0.0353
 
== Hypnotone ==
 
[[Subgroup]]: 2.3.5.11
 
[[Comma list]]: 45/44, 81/80
 
{{Mapping|legend=2| 1 0 -4 -6 | 0 1 4 6 }}
 
: sval mapping generators: ~2, ~3
 
[[Optimal tuning]] ([[CTE]]): ~2/1 = 1\1, ~3/2 = 694.6998
 
{{Optimal ET sequence|legend=1| 7, 12, 19, 26, 45 }}
 
[[Badness]]: 0.0104
 
Related temperament: [[flattone]]. (Hypnotone is no-sevens flattone.)
 
=== 2.3.5.11.13 subgroup ===
Subgroup: 2.3.5.11.13
 
Comma list: 45/44, 65/64, 81/80
 
Sval mapping: {{mapping| 1 0 -4 -6 10 | 0 1 4 6 -4 }}
 
: sval mapping generators: ~2, ~3
 
Optimal tuning (CTE): ~2/1 = 1\1, ~3/2 = 693.9513
 
Optimal ET sequence: {{Optimal ET sequence| 7, 12, 19, 26, 45f }}
 
Badness: 0.0141
 
== 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}}]
 
[[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 ==
: ''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


Related temperament: [[Schismatic family|schismic]]
== 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.


== Rarity ==
Subgroup: 2.3.5.113
: <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]]: 226/225


[[Comma list]]: 1035/1024, 16875/16606, 192375/188416
{{Mapping|legend=2|1 0 0 -1|0 1 0 2|0 0 1 2}}


{{Mapping|legend=2| 1 11 -10 -3 -2 | 0 -13 17 10 9 }}
: Sval mapping generators: ~2, ~3, ~5


: Sval mapping generators: ~1\1 = 2, ~368/225 = 869.420
[[Optimal tuning]]s:
* CTE: ~3/2 = 702.449, ~5/4 = 387.373
* CWE: ~3/2 = 702.522, ~5/4 = 387.479


[[Optimal tuning]] ([[CTE]]): ~368/225 = 869.420
{{Optimal ET sequence|legend=1| 12, 22, 31, 34, 46, 53 }}
 
{{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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