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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 ==
: <small>''For full 11- and 13-limit extensions, see [[Dicot family]].''</small>
=== 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]]: 25/24, 45/44
[[Comma list]]: 8019/8000, 14641/14580


[[Gencom]]: [2 5/4; 25/24 45/44]
[[Gencom]]: [2 400/363; 8019/8000 14641/14580]


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


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


[[Tp tuning|POL2 generator]]: ~5/4 = 346.734
[[Tp tuning|POL2 generator]]: ~400/363 = 172.8796


{{Optimal ET sequence|legend=1| 3e, 4e, 7, 24c, 31c, 38cc, 45cce }}
{{Optimal ET sequence|legend=1| 7, 111, 118 }}


[[Tp tuning #T2 tuning|RMS error]]: 5.621 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.2393 cents


Related temperaments: [[Dicot family #Septimal dicot|dicot]], [[Dicot family #Sharp|sharp]], [[Dicot family #Dichotic|dichotic]]
=== Emka ===
 
: <small>''For full 11- and 13-limit extensions, see [[Hemimean clan #Emka]] or [[Horwell temperaments #Emkay]].''</small>
=== 2.3.5.11.13 ===
Subgroup: 2.3.5.11.13
 
[[Comma list]]: 25/24, 40/39, 45/44
 
[[Gencom]]: [2 5/4; 25/24 40/39 45/44]
 
[[Gencom|Gencom mapping]]: [{{val|1 1 2 0 2 4}}, {{val|0 2 1 0 5 -1}}]
 
[[Mapping|Sval mapping]]: [{{val|1 1 2 2 4}}, {{val|0 2 1 5 -1}}]
 
[[Tp tuning|POL2 generator]]: ~5/4 = 350.526
 
{{Optimal ET sequence|legend=1| 3e, 7, 17, 24c }}
 
[[Tp tuning #T2 tuning|RMS error]]: 5.916 cents


== Mavila ==
Subgroup: 2.3.5.11
Subgroup: 2.3.5.11


[[Comma list]]: 33/32, 45/44
[[Comma list]]: 4000/3993, 9453125/9437184


[[Gencom]]: [2 4/3; 33/32, 45/44]
[[Gencom]]: [2 11/8; 4000/3993 9453125/9437184]


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


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


[[Tp tuning|POL2 generator]]: ~4/3 = 520.212
[[Tp tuning|POL2 generator]]: ~11/8 = 551.7778


{{Optimal ET sequence|legend=1| 7, 16, 23e, 30bce }}
{{Optimal ET sequence|legend=1| 37, 50, 87, 137, 224 }}


[[Tp tuning #T2 tuning|RMS error]]: 4.705 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.1188 cents


Related temperaments: [[Pelogic family #Mavila|mavila]], [[Pelogic family #Pelogic|pelogic]], [[Pelogic family #Armodue|armodue]]
==== 2.3.5.11.13 ====
 
== Porkypine ==
The [[S-expression]]-based comma list of this temperament is {[[100/99|S10]], [[121/120|S11]]}.
 
Subgroup: 2.3.5.11
 
[[Comma list]]: 55/54, 100/99
 
[[Gencom]]: [2 10/9; 55/54 100/99]
 
[[Gencom|Gencom mapping]]: [{{val|1 2 3 0 4}}, {{val|0 -3 -5 0 -4}}]
 
[[Mapping|Sval mapping]]: [{{val|1 2 3 4}}, {{val|0 -3 -5 -4}}]
 
[[Tp tuning|POL2 generator]]: ~11/10 = 164.078
 
{{Optimal ET sequence|legend=1| 7, 15, 22, 37, 73ce, 95ce, 117bce }}
 
[[Tp tuning #T2 tuning|RMS error]]: 2.287 cents
 
Related temperament: [[Porcupine]]
 
== Mohaha ==
The [[S-expression]]-based comma list of this temperament is {[[81/80|S9 = S6/S8]], [[121/120|S11]]}.
 
Subgroup: 2.3.5.11
 
[[Comma list]]: 81/80, 121/120
 
[[Gencom]]: [2 11/9; 81/80 121/120]
 
[[Gencom|Gencom mapping]]: [{{val|1 1 0 0 2}}, {{val|0 2 8 0 5}}]
 
[[Mapping|Sval mapping]]: [{{val|1 1 0 2}}, {{val|0 2 8 5}}]
 
[[Tp tuning|POL2 generator]]: ~11/9 = 348.094
 
{{Optimal ET sequence|legend=1| 7, 17c, 24, 31, 69e, 100e, 131bee }}
 
[[Tp tuning #T2 tuning|RMS error]]: 1.392 cents
 
Related temperament: [[mohajira]], [[Meantone family #Migration|migration]]
 
; Music
[http://micro.soonlabel.com/gene_ward_smith/Others/Stiltner/mohaha10ping2.mp3 Mohaha10ping2] by [[Billy Stiltner]]
 
=== Mohoho ===
Subgroup: 2.3.5.11.13
Subgroup: 2.3.5.11.13


[[Comma list]]: 66/65, 81/80, 121/120
[[Comma list]]: 625/624, 2200/2197, 4000/3993


[[Gencom]]: [2 11/9; 66/65 81/80 121/120]
[[Gencom]]: [2 11/8; 625/624 2200/2197 4000/3993]


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


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


[[Tp tuning|POL2 generator]]: ~11/9 = 348.916
[[Tp tuning|POL2 generator]]: ~11/8 = 551.7753


{{Optimal ET sequence|legend=1| 7, 17c, 24, 31, 55, 86ef, 141ceff }}
{{Optimal ET sequence|legend=1| 37, 50, 87, 137, 224 }}


[[Tp tuning #T2 tuning|RMS error]]: 1.971 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.1250 cents
 
== 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
 
{{Optimal ET sequence|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
 
[[Comma list]]: 100/99, 144/143, 243/242
 
[[Gencom]]: [2 10/9; 100/99 243/242]
 
[[Gencom|Gencom mapping]]: [{{val|1 1 1 0 2 4}}, {{val|0 4 9 0 10 -2}}]
 
[[Mapping|Sval mapping]]: [{{val|1 1 1 2 4}}, {{val|0 4 9 10 -2}}]


[[Tp tuning|POL2 generator]]: ~10/9 = 176.196
=== Tremka ===
 
: <small>''For full 11- and 13-limit extensions, see [[Hemimean clan#Tremka]].''</small>
{{Optimal ET sequence|legend=1| 7, 27e, 34, 41, 75e }}
 
[[Tp tuning #T2 tuning|RMS error]]: 1.140 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
Line 177: Line 77:
[[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 194: Line 94:
[[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
 
{{Optimal ET sequence|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
 
{{Optimal ET sequence|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 251: Line 113:
[[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 268: Line 130:
[[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}}]
 
[[Mapping|Sval mapping]]: [{{val|1 14 6 3}}, {{val|0 -27 -8 1}}]


[[Tp tuning|POL2 generator]]: ~11/8 = 551.7778
Optimal tunings:  
* CTE: ~2 = 1200.000, ~11/10 = 167.712
* CWE: ~2 = 1200.000, ~11/10 = 167.882


{{Optimal ET sequence|legend=1| 37, 50, 87, 137, 224 }}
{{Optimal ET sequence|legend=0| 7, 29ce, 36, 43, 50 }}


[[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


[[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
 
{{Optimal ET sequence|legend=1| 37, 50, 87, 137, 224 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.1250 cents
 
== Tricot ==
: <small>''For full 13-limit extensions, see [[Tricot family]].''</small>
 
Subgroup: 2.3.5.13
 
[[Comma list]]: 2197/2187, 41067/40960
 
[[Gencom]]: [2 13/9; 2197/2187, 41067/40960]
 
[[Gencom|Gencom mapping]]: [{{val|1 0 -13 0 0 0}}, {{val|0 3 29 0 0 7}}]
 
[[Mapping|Sval mapping]]: [{{val|1 0 -13 0}}, {{val|0 3 29 7}}]
 
[[Tp tuning|POL2 generator]]: ~13/9 = 633.997


{{Optimal ET sequence|legend=1| 17c, 36c, 53 }}
Mapping: {{mapping| 1 2 4 5 | 0 -3 -12 -11 5 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.2342 cents
Optimal tunings:  
* CTE: ~2 = 1200.000, ~11/10 = 167.729
* CWE: ~2 = 1200.000, ~11/10 = 167.846


== Taylor ==
{{Optimal ET sequence|legend=0| 7, 29ce, 36, 43, 50 }}
Subgroup: 2.3.5.13


[[Comma list]]: 676/675, 32805/32768
=== Dicot ===
{{main|Dicot family #2.3.5.11 subgroup}}


[[Gencom]]: [2 15/13; 676/675 32805/32768]
=== Mavila ===
{{main|Mavila family #2.3.5.11 subgroup}}


[[Gencom|Gencom mapping]]: [{{val|1 2 -1 0 0 1}}, {{val|0 -2 16 0 0 13}}]
=== Tetracot ===
{{main|Tetracot family #2.3.5.11 subgroup }}


[[Mapping|Sval mapping]]: [{{val|1 2 -1 1}}, {{val|0 -2 16 13}}]
=== Porkypine ===
{{main|Porcupine family #2.3.5.11 subgroup (porkypine)}}


[[Tp tuning|POL2 generator]]: ~15/13 = 249.145
=== Mohaha ===
{{main|Rastmic clan #Mohaha }}


{{Optimal ET sequence|legend=1| 24, 53, 130, 183, 236 }}
=== Larry ===
{{main|Gravity family #Larry}}


[[Tp tuning #T2 tuning|RMS error]]: 0.1485 cents
=== Dequarter ===
{{main|Meantone family #Dequarter }}


Related temperament: [[Schismatic family|hemischis]]
=== 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 361: Line 205:
[[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 378: Line 224:
[[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
 
[[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
 
{{Optimal ET sequence|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
 
{{Optimal ET sequence|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
 
{{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 ==
: <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]
=== Taylor ===
{{Main| Schismatic family #Taylor (2.3.5.13) }}


[[Gencom|Gencom mapping]]: [{{val|1 2 -1 0 0 0 5}}, {{val|0 -5 40 0 0 0 -11}}]
=== Vulture ===
: ''For the 5-limit restriction and full 13-limit extension, see [[Vulture family]].''


[[Mapping|Sval mapping]]: [{{val|1 2 -1 5}}, {{val|0 -5 40 -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.


[[Tp tuning|POL2 generator]]: ~18/17 = 99.649
[[Subgroup]]: 2.3.5.13


{{Optimal ET sequence|legend=1| 12, 253, 265, 277, 289 }}
[[Comma list]]: 676/675, 256000/255879


[[Tp tuning #T2 tuning|RMS error]]: 0.0719 cents
{{Mapping|legend=2| 1 0 -6 -7 | 0 4 21 27 }}


=== 2.3.5.17.19 ===
[[Optimal tuning]]s:
Subgroup: 2.3.5.17.19
* [[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 }}


[[Comma list]]: 4624/4617, 6144/6137, 6885/6859
{{Optimal ET sequence|legend=1| 53, 164, 217, 270 }}


[[Gencom]]: [2 18/17; 4624/4617 6144/6137 6885/6859]
[[Badness]] (Sintel): 0.301


[[Gencom|Gencom mapping]]: [{{val|1 2 -1 0 0 0 5 4}}, {{val|0 -5 40 0 0 0 -11 3}}]
==== 2.3.5.13.19 ====
Subgroup: 2.3.5.13.19


[[Mapping|Sval mapping]]: [{{val|1 2 -1 5 4}}, {{val|0 -5 40 -11 3}}]
Comma list: 676/675, 1216/1215, 20007/20000


[[Tp tuning|POL2 generator]]: ~18/17 = 99.652
Sval mapping: {{mapping| 1 0 -6 -7 -12 | 0 4 21 27 41 }}


{{Optimal ET sequence|legend=1| 12, 253, 265, 277, 289 }}
Optimal tunings:
* CTE: ~2 = 1200.0000, ~320/243 = 475.5498
* CWE: ~2 = 1200.0000, ~320/243 = 475.5533


[[Tp tuning #T2 tuning|RMS error]]: 0.1636 cents
{{Optimal ET sequence|legend=0| 53, 164, 217, 270 }}


== Stützel ==
Badness (Sintel): 0.190
Subgroup: 2.3.5.19


[[Comma list]]: 81/80, 96/95
== 2.3.5.17 temperaments ==
=== Quintaleap ===
{{Main| Quintaleap family #Subgroup temperament }}


[[Gencom]]: [2 4/3; 81/80 96/95]
=== Quindromeda ===
{{Main| Quindromeda family #Subgroup temperament }}


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


[[Mapping|Sval mapping]]: [{{val|1 2 4 3}}, {{val|0 -1 -4 3}}]
=== 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 = 697.867
=== Srutal Archagall ===
{{Main|Srutal archagall}}


{{Optimal ET sequence|legend=1| 5, 7, 12, 31, 43 }}
== 2.3.5.19 temperaments ==
 
=== Rarity ===
[[Tp tuning #T2 tuning|RMS error]]: 1.378 cents
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Rarity]].''
 
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}}]
 
[[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]]''
 
The [[S-expression]]-based comma list of this temperament is {[[1216/1215|S16/S18]], [[361/360|S19]](, [[513/512|S15/S20]])}.
 
[[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]]
 
== 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.
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.
Line 600: Line 296:
{{Mapping|legend=2| 1 11 -10 -3 -2 | 0 -13 17 10 9 }}
{{Mapping|legend=2| 1 11 -10 -3 -2 | 0 -13 17 10 9 }}


: Sval mapping generators: ~1\1 = 2, ~368/225 = 869.420
: Sval mapping generators: ~2 = 1200.000, ~368/225 = 869.420


[[Optimal tuning]] ([[CTE]]): ~368/225 = 869.420
[[Optimal tuning]] ([[CTE]]): ~368/225 = 869.420
Line 608: Line 304:
[[Badness]]: 0.0656
[[Badness]]: 0.0656


== Dequarter ==
=== Devisemi ===
[[Subgroup]]: 2.3.5.11
: ''See also: [[No-elevens subgroup temperaments #Devisemi]]''
{{Main| Tetracot family #2.3.5.19 subgroup }}
 
=== Nestoria ===
: ''See also: [[No-elevens subgroup temperaments #Garibaldia]] and [[No-elevens subgroup temperaments #Pontia|#Pontia]]''
{{Main| Schismatic family #Nestoria (2.3.5.19) }}
 
=== Stützel ===
{{Main| Meantone family #Stützel (2.3.5.19) }}
 
== 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.
 
Subgroup: 2.3.5.113


[[Comma list]]: 33/32, 55/54
[[Comma list]]: 226/225


{{Mapping|legend=2| 1 0 -4 5 | 0 1 4 -1 }}
{{Mapping|legend=2|1 0 0 -1|0 1 0 2|0 0 1 2}}


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 696.0387
: Sval mapping generators: ~2, ~3, ~5


{{Optimal ET sequence|legend=1| 5, 7, 19e, 26e }}
[[Optimal tuning]]s:
* CTE: ~3/2 = 702.449, ~5/4 = 387.373
* CWE: ~3/2 = 702.522, ~5/4 = 387.479


[[Badness]]: 0.0145
{{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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