Subgroup temperaments: Difference between revisions

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A '''subgroup temperament''' is a regular temperament defined on a [[just intonation subgroup]] that is not a full ''p-limit'' group.  
A '''subgroup temperament''' is a regular temperament defined on a [[just intonation subgroup]] that is not a full ''p''-limit group.  


Below are some subgroups and temperaments for them. Obviously, no attempt has been made at completeness; attention is focused on subgroups containing interesting chords. The reader may also want to consult the page on [[Chromatic pairs]].
For temperaments that omit various prime harmonics, see:
* [[No-elevens subgroup temperaments]]
* [[No-sevens subgroup temperaments]]
* [[No-fives subgroup temperaments]]
* [[No-threes subgroup temperaments]]


== No-elevens subgroup ==
Below are some other miscellaneous subgroups and temperaments for them. Obviously, no attempt has been made at completeness; attention is focused on subgroups containing interesting chords. The reader may also want to consult the page on [[Chromatic pairs]].
 
=== [[Ultrapyth]] ===
: <small>''For full 13-limit extensions, see [[Archytas clan #Ultrapyth]].''</small>
 
Subgroup: 2.3.5.7.13
 
[[Comma list]]: 64/63, 91/90, 4394/4375
 
[[Gencom]]: [2 4/3; 64/63 91/90 4394/4375]
 
[[Gencom|Gencom mapping]]: [{{val|1 2 8 2 0 11}}, {{val|0 -1 -14 2 0 -18}}]
 
[[Mapping|Sval mapping]]: [{{val|1 2 8 2 11}}, {{val|0 -1 -14 2 -18}}]
 
[[Tp tuning|POL2 generator]]: ~4/3 = 486.255
 
{{Val list|legend=1| 5, 32, 37 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 2.318 cents
 
=== Sensi (aka Sensation) ===
: <small>''For full 13-limit extensions, see [[Sensipent family]] or [[Sensi extensions]].''</small>
 
Subgroup: 2.3.5.7.13
 
[[Comma list]]: 91/90, 126/125, 169/168
 
[[Gencom]]: [2 9/7; 91/90 126/125 169/168]
 
[[Gencom|Gencom mapping]]: [{{val|1 6 8 11 0 10}}, {{val|0 -7 -9 -13 0 -10}}]
 
[[Mapping|Sval mapping]]: [{{val|1 6 8 11 10}}, {{val|0 -7 -9 -13 -10}}]
 
[[Tp tuning|POL2 generator]]: ~9/7 = 443.322
 
{{Val list|legend=1| 19, 27, 46, 111de, 157de }}
 
[[Tp tuning #T2 tuning|RMS error]]: 1.321 cents
 
=== Quartonic ===
: <small>''For full 13-limit extensions, see [[Orwellismic temperaments #Quartonic]].''</small>
 
Subgroup: 2.3.5.7.13
 
[[Comma list]]: 169/168, 325/324, 640/637
 
[[Gencom]]: [2 36/35; 169/168 325/324 640/637]
 
[[Gencom|Gencom mapping]]: [{{val|1 2 3 3 0 4}}, {{val|0 -11 -18 -5 0 -8}}]
 
[[Mapping|Sval mapping]]: [{{val|1 2 3 3 4}}, {{val|0 -11 -18 -5 -8}}]
 
[[Tp tuning|POL2 generator]]: ~36/35 = 45.1821
 
{{Val list|legend=1| 26, 27, 53 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.7439 cents
 
=== [[Septidiasemi]] ===
: <small>''For full 13- and 17-limit extensions, see [[Breedsmic temperaments #Septidiasemi]].''</small>
 
Subgroup: 2.3.5.7.13
 
[[Comma list]]: 2205/2197, 2401/2400, 4096/4095
 
[[Gencom]]: [2 15/14; 2205/2197 2401/2400 4096/4095]
 
[[Gencom|Gencom mapping]]: [{{val|1 -1 6 4 0 4}}, {{val|0 26 -37 -12 0 -3}}]
 
[[Mapping|Sval mapping]]: [{{val|1 -1 6 4 4}}, {{val|0 26 -37 -12 -3}}]
 
[[Tp tuning|POL2 generator]]: ~15/14 = 119.297
 
{{Val list|legend=1| 10, 151, 161, 171 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.2002 cents
 
==== 2.3.5.7.13.17 ====
Subgroup: 2.3.5.7.13.17
 
[[Comma list]]: 833/832, 1275/1274, 2025/2023, 2205/2197
 
[[Gencom]]: [2 15/14; 833/832 1275/1274 2025/2023 2205/2197]
 
[[Gencom|Gencom mapping]]: [{{val|1 -1 6 4 0 4 2}}, {{val|0 26 -37 -12 0 -3 21}}]
 
[[Mapping|Sval mapping]]: [{{val|1 -1 6 4 4 2}}, {{val|0 26 -37 -12 -3 21}}]
 
[[Tp tuning|POL2 generator]]: ~15/14 = 119.297
 
{{Val list|legend=1| 10, 151, 161, 171 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.1867 cents
 
=== [[Pontiac]] ===
: <small>''For full 13- and 17-limit extensions, see [[Schismatic family #Pontiac]].''</small>
 
Subgroup: 2.3.5.7.13
 
[[Comma list]]: 625/624 729/728 4096/4095
 
[[Gencom]]: [2 4/3; 625/624 729/728 4096/4095]
 
[[Gencom|Gencom mapping]]: [{{val|1 2 -1 19 0 -10}}, {{val|0 -1 8 -39 0 33}}]
 
[[Mapping|Sval mapping]]: [{{val|1 2 -1 19 -10}}, {{val|0 -1 8 -39 33}}]
 
[[Tp tuning|POL2 generator]]: ~3/2 = 701.773
 
{{Val list|legend=1| 53, 118, 171, 224, 395 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.1525 cents
 
==== 2.3.5.7.13.17 ====
Subgroup: 2.3.5.7.13.17
 
[[Comma list]]: 625/624, 729/728, 1225/1224, 2880/2873
 
[[Gencom]]: [2 4/3; 625/624 729/728 1225/1224 2880/2873]
 
[[Gencom|Gencom mapping]]: [{{val|1 2 -1 19 0 -10 29}}, {{val|0 -1 8 -39 0 33 -60}}]
 
[[Mapping|Sval mapping]]: [{{val|1 2 -1 19 -10 29}}, {{val|0 -1 8 -39 33 -60}}]
 
[[Tp tuning|POL2 generator]]: ~3/2 = 701.764
 
{{Val list|legend=1| 53, 118, 171, 395, 566… }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.1696 cents
 
=== [[Mitonic]] ===
: <small>''For full 13- and 17-limit extensions, see [[Minortonic family #Mitonic]].''</small>
 
Subgroup: 2.3.5.7.13
 
[[Comma list]]: 4096/4095, 4375/4374, 13720/13689
 
[[Gencom]]: [2 10/9; 4096/4095 4375/4374 13720/13689]
 
[[Gencom|Gencom mapping]]: [{{val|1 -1 -3 6 0 11}}, {{val|0 17 35 -21 0 -48}}]
 
[[Mapping|Sval mapping]]: [{{val|1 -1 -3 6 11}}, {{val|0 17 35 -21 -48}}]
 
[[Tp tuning|POL2 generator]]: ~10/9 = 182.471
 
{{Val list|legend=1| 46, 125, 171, 388 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.1442 cents
 
==== 2.3.5.7.13.17 ====
Subgroup: 2.3.5.7.13.17
 
[[Comma list]]: 833/832, 1225/1224, 1701/1700, 4096/4095
 
[[Gencom]]: [2 10/9; 833/832 1225/1224 1701/1700 4096/4095]
 
[[Gencom|Gencom mapping]]: [{{val|1 -1 -3 6 0 11 5}}, {{val|0 17 35 -21 0 -48 -6}}]
 
[[Mapping|Sval mapping]]: [{{val|1 -1 -3 6 11 5}}, {{val|0 17 35 -21 -48 -6}}]
 
[[Tp tuning|POL2 generator]]: ~10/9 = 182.471
 
{{Val list|legend=1| 46, 125, 171, 388 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.1341 cents
 
=== [[Tertiaseptal]] ===
: <small>''For full 13- and 17-limit extensions, see [[Breedsmic temperaments #Tertiaseptal]].''</small>
 
Subgroup: 2.3.5.7.13
 
[[Comma list]]: 625/624, 2401/2400, 4096/4095
 
[[Gencom]]: [2 117/112; 625/624 2401/2400 4096/4095]
 
[[Gencom|Gencom mapping]]: [{{val|1 3 2 3 0 1}}, {{val|0 -22 5 -3 0 42}}]
 
[[Mapping|Sval mapping]]: [{{val|1 3 2 3 1}}, {{val|0 -22 5 -3 42}}]
 
[[Tp tuning|POL2 generator]]: ~117/112 = 77.173
 
{{Val list|legend=1| 31, 109, 140, 171, 311 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.1383 cents
 
==== 2.3.5.7.13.17 ====
Subgroup: 2.3.5.7.13.17
 
[[Comma list]]: 625/624, 833/832, 1225/1224, 4096/4095
 
[[Gencom]]: [2 68/65; 625/624 833/832 1225/1224 4096/4095]
 
[[Gencom|Gencom mapping]]: [{{val|1 3 2 3 0 1 1}}, {{val|0 -22 5 -3 0 42 48}}]
 
[[Mapping|Sval mapping]]: [{{val|1 3 2 3 1 1}}, {{val|0 -22 5 -3 42 48}}]
 
[[Tp tuning|POL2 generator]]: ~68/65 = 77.177
 
{{Val list|legend=1| 31, 109f, 140, 171, 311 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.1367 cents
 
=== Oquatonic ===
: <small>''For full 13-limit extensions, see [[Horwell temperaments #Oquatonic]].''</small>
 
Subgroup: 2.3.5.7.13
 
[[Comma list]]: 625/624, 4096/4095, 10985/10976
 
[[Gencom]]: [40/39 105/104; 625/624 4096/4095 10985/10976]
 
[[Gencom|Gencom mapping]]: [{{val|28 44 65 79 0 104}}, {{val|0 1 0 -1 0 -1}}]
 
[[Mapping|Sval mapping]]: [{{val|28 44 65 79 104}}, {{val|0 1 0 -1 -1}}]
 
[[Tp tuning|POL2 generator]]: ~105/104 = 16.4240
 
{{Val list|legend=1| 28, 56, 84, 140, 224, 364 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.1347 cents
 
== No-sevens subgroup ==
 
=== Porkypine ===
Related temperament: [[Porcupine]]
 
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
 
{{Val list|legend=1| 7, 15, 22, 37, 73cd, 95cd, 117bcd }}
 
[[Tp tuning #T2 tuning|RMS error]]: 2.287 cents
 
=== Mohaha ===
Related temperament: [[mohajira]], [[Meantone family #Migration|migration]]
 
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
 
{{Val list|legend=1| 7, 10, 17, 24, 31, 55, 69d, 100d, 131bd }}
 
[[Tp tuning #T2 tuning|RMS error]]: 1.392 cents
 
; 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, 27d, 34, 41, 75d }}
 
[[Tp tuning #T2 tuning|RMS error]]: 1.182 cents
 
==== 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
 
{{Val list|legend=1| 7, 27d, 34, 41, 75d }}
 
[[Tp tuning #T2 tuning|RMS error]]: 1.140 cents
 
=== Larry ===
Subgroup: 2.3.5.11
 
[[Comma list]]: 243/242, 4000/3993
 
Related temperaments: [[Gravity family|gravity]], [[Gravity family #Harry|harry]]
 
[[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
 
=== Twentcufo ===
: <small>''For full 11- and 13-limit extensions, see [[Hemimean clan #Undetrita]].''</small>
 
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
 
=== Emka ===
: <small>''For full 11- and 13-limit extensions, see [[Hemimean clan #Emka]] or [[Horwell temperaments #Emkay]].''</small>
 
Subgroup: 2.3.5.11
 
[[Comma list]]: 4000/3993, 9453125/9437184
 
[[Gencom]]: [2 11/8; 4000/3993 9453125/9437184]
 
[[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
 
{{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
 
[[Comma list]]: 625/624, 2200/2197, 4000/3993
 
[[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 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.1250 cents
 
=== Majvam ===
: <small>''For full 13- and 17-limit extensions, see [[Parkleiness temperaments #Majvamic]] or [[Cataharry temperaments #Majvamoid]].''</small>
 
Subgroup: 2.3.5.13
 
[[Comma list]]: 676/675, 127401984/126953125
 
[[Gencom]]: [2 125/96; 676/675 127401984/126953125]
 
[[Gencom|Gencom mapping]]: [{{val|1 10 5 0 0 19}}, {{val|0 -22 -7 0 0 -40}}]
 
[[Mapping|Sval mapping]]: [{{val|1 10 5 19}}, {{val|0 -22 -7 -40}}]
 
[[Tp tuning|POL2 generator]]: ~125/96 = 458.988
 
{{Val list|legend=1| 34, 149, 183, 217, 400 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.1171 cents
 
==== 2.3.5.13.17 ====
Subgroup: 2.3.5.13.17
 
[[Comma list]]: 676/675, 2601/2600, 24576/24565
 
[[Gencom]]: [2 125/96; 676/675 2601/2600 24576/24565]
 
[[Gencom|Gencom mapping]]: [{{val|1 10 5 0 0 19 6}}, {{val|0 -22 -7 0 0 -40 -5}}]
 
[[Mapping|Sval mapping]]: [{{val|1 10 5 19 6}}, {{val|0 -22 -7 -40 -5}}]
 
[[Tp tuning|POL2 generator]]: ~125/96 = 458.991
 
{{Val list|legend=1| 34, 149, 183, 217, 400 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.1129 cents
 
=== Photia ===
Related temperament: [[Schismatic family|Schismic]]
 
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
 
{{Val list|legend=1| 12, 41, 53, 65 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.4842 cents
 
==== 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
 
=== Nestoria ===
Related temperament: [[Schismatic family|Schismic]]
 
Subgroup: 2.3.5.19
 
[[Comma list]]: 361/360, 513/512
 
[[Gencom]]: [2 4/3; 361/360 513/512]
 
[[Gencom|Gencom mapping]]: [{{val|1 2 -1 0 0 0 0 3}}, {{val|0 -1 8 0 0 0 0 3}}]
 
[[Mapping|Sval mapping]]: [{{val|1 2 -1 3}}, {{val|0 -1 8 3}}]
 
[[Tp tuning|POL2 generator]]: ~3/2 = 701.746
 
{{Val list|legend=1| 12, 29, 41, 53, 118, 171 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.1763 cents
 
== No-fives subgroup ==
 
=== [[Semaphore and Godzilla|Semaphore]] ===
Subgroup: 2.3.7
 
[[Comma]]: 49/48
 
[[Gencom]]: [2 8/7; 49/48]
 
[[Gencom|Gencom mapping]]: [{{val|1 2 0 3}}, {{val|0 -2 0 -1}}]
 
[[Mapping|Sval mapping]]: [{{val|1 2 3}}, {{val|0 -2 -1}}]
 
[[Tp tuning|POL2 generator]]: ~7/6 = 250.385
 
{{Val list|legend=1| 5, 14, 19, 24, 67c, 91c }}
 
[[Tp tuning# T2 tuning|RMS error]]: 2.523 cents
 
=== Bleu ===
Subgroup: 2.3.7
 
[[Comma]]: 17496/16807
 
[[Gencom]]: [2 54/49; 17496/16807]
 
[[Gencom|Gencom mapping]]: [{{val|1 1 0 2}}, {{val|0 5 0 7}}]
 
[[Mapping|Sval mapping]]: [{{val|1 1 2}}, {{val|0 5 7}}]
 
[[Tp tuning|POL2 generator]]: ~54/49 = 139.848
 
{{Val list| 9, 17, 43, 60c }}
 
[[Tp tuning #T2 tuning|RMS error]]: 1.917 cents
 
==== 2.3.7.11 ====
Subgroup: 2.3.7.11
 
[[Comma list]]: 99/98, 864/847
 
[[Gencom]]: [2 12/11; 99/98 864/847]
 
[[Gencom|Gencom mapping]]: [{{val|1 1 0 2 3}}, {{val|0 5 0 7 4}}]
 
[[Mapping|Sval mapping]]: [{{val|1 1 2 3}}, {{val|0 5 7 4}}]
 
[[Tp tuning|POL2 generator]]: ~12/11 = 140.005
 
{{Val list| 9, 17, 43, 60c }}
 
[[Tp tuning #T2 tuning|RMS error]]: 1.829 cents
 
==== 2.3.7.11.13 ====
Subgroup: 2.3.7.11.13
 
[[Comma list]]: 78/77, 99/98, 144/143
 
[[Gencom]]: [2 12/11; 78/77 99/98 144/143]
 
[[Gencom|Gencom mapping]]: [{{val|1 1 0 2 3 3}}, {{val|0 5 0 7 4 6}}]
 
[[Mapping|Sval mapping]]: [{{val|1 1 2 3 3}}, {{val|0 5 7 4 6}}]
 
[[Tp tuning|POL2 generator]]: ~12/11 = 139.990
 
{{Val list|legend=1| 17, 43, 60c }}
 
[[Tp tuning #T2 tuning|RMS error]]: 1.752 cents
 
=== Archy ===
Archy (properly pronounced "arky", after the Greek theorist Archytas) can be thought of as "no-fives [[dominant]]" or "no-fives [[superpyth]]". The name comes from the fact that it tempers out 64/63, the Archytas comma.
 
Subgroup: 2.3.7
 
[[Comma]]: 64/63
 
[[Gencom]]: [2 3/2; 64/63]
 
[[Gencom|Gencom mapping]]: [{{val|1 1 0 4}}, {{val|0 1 0 -2}}]
 
[[Mapping|Sval mapping]]: [{{val|1 2 2}}, {{val|0 -1 2}}]
 
[[Tp tuning|POL2 generator]]: ~3/2 = 709.321
 
{{Val list|legend=1| 5, 12, 17, 22, 27, 137bc }}
 
[[Tp tuning #T2 tuning|RMS error]]: 1.856 cents
 
==== Supra ====
Subgroup: 2.3.7.11
 
[[Comma list]]: 64/63, 99/98
 
[[Gencom]]: [2 3/2; 64/63 99/98]
 
[[Gencom|Gencom mapping]]: [{{val|1 1 0 4 7}}, {{val|0 1 0 -2 -6}}]
 
[[Mapping|Sval mapping]]: [{{val|1 0 6 13}}, {{val|0 1 -2 -6}}]
 
[[Tp tuning|POL2 generator]]: ~3/2 = 707.192
 
{{Val list|legend=1| 5, 12, 17, 39c, 56c }}
 
[[Tp tuning #T2 tuning|RMS error]]: 1.977 cents
 
===== Supraphon =====
Subgroup: 2.3.7.11.13
 
[[Comma list]]: 64/63, 78/77, 99/98
 
[[Gencom]]: [2 3/2; 64/63 78/77 99/98]
 
[[Gencom|Gencom mapping]]: [{{val|1 1 0 4 7 9}}, {{val|0 1 0 -2 -6 -9}}]
 
[[Mapping|Sval mapping]]: [{{val|1 0 6 13 18}}, {{val|0 1 -2 -6 -9}}]
 
[[Tp tuning|POL2 generator]]: ~3/2 = 706.137
 
{{Val list|legend=1| 5e, 12e, 17, 22, 39c, 56c }}
 
[[Tp tuning #T2 tuning|RMS error]]: 2.095 cents
 
==== Suhajira ====
Subgroup: 2.3.7.11
 
[[Comma list]]: 64/63, 243/242
 
[[Gencom]]: [2 11/9; 64/63 243/242]
 
[[Gencom|Gencom mapping]]: [{{val|1 1 0 4 2}}, {{val|0 2 0 -4 5}}]
 
[[Mapping|Sval mapping]]: [{{val|1 1 4 2}}, {{val|0 2 -4 5}}]
 
[[Tp tuning|POL2 generator]]: ~11/9 = 353.958
 
{{Val list|legend=1| 7, 10, 17, 44d, 61cd, 78cd }}
 
[[Tp tuning #T2 tuning|RMS error]]: 1.968 cents
 
===== 2.3.7.11.13 =====
Subgroup: 2.3.7.11.13
 
[[Comma list]]: 64/63, 78/77, 144/143
 
[[Gencom]]: [2 11/9; 64/63 78/77 144/143]
 
[[Gencom|Gencom mapping]]: [{{val|1 1 0 4 2 4}}, {{val|0 2 0 -4 5 -1}}]
 
[[Mapping|Sval mapping]]: [{{val|1 1 4 2 4}}, {{val|0 2 -4 5 -1}}]
 
[[Tp tuning|POL2 generator]]: ~11/9 = 353.775
 
{{Val list|legend=1| 7, 10, 17, 44d, 61cd, 78cd }}
 
[[Tp tuning #T2 tuning|RMS error]]: 1.953 cents
 
=== Skwares ===
Related temperament: [[Meantone family #Squares|squares]]
 
Subgroup: 2.3.7
 
[[Comma]]: 19683/19208
 
[[Gencom]]: [2 9/7; 19683/19208]
 
[[Gencom|Gencom mapping]]: [{{val|1 3 6}}, {{val|0 -4 -9}}]
 
[[Mapping|Sval mapping]]: [{{val|1 3 6}}, {{val|0 -4 -9}}]
 
[[Tp tuning|POL2 generator]]: ~9/7 = 425.365
 
{{Val list|legend=1| 14, 17, 31, 48, 79, 189b, 268bc, 347bc }}
 
[[Tp tuning #T2 tuning|RMS error]]: 1.149 cents
 
==== 2.3.7.11 ====
Subgroup: 2.3.7.11
 
[[Comma list]]: 99/98, 243/242
 
[[Gencom]]: [2 9/7; 99/98 243/242]
 
[[Gencom|Gencom mapping]]: [{{val|1 3 0 6 7}}, {{val|0 -4 0 -9 -10}}]
 
[[Mapping|Sval mapping]]: [{{val|1 3 6 7}}, {{val|0 -4 -9 -10}}]
 
[[Tp tuning|POL2 generator]]: ~9/7 = 425.244
 
{{Val list|legend=1| 14, 17, 31, 48, 79, 127, 206bcd }}
 
[[Tp tuning #T2 tuning|RMS error]]: 1.099 cents
 
==== 2.3.7.11.13 ====
Subgroup: 2.3.7.11.13
 
[[Comma list]]: 78/77, 99/98, 243/242
 
[[Gencom]]: [2 9/7; 78/77, 99/98, 243/242]
 
[[Gencom|Gencom mapping]]: [{{val|1 3 0 6 7 9}}, {{val|0 -4 0 -9 -10 -15}}]
 
[[Mapping|Sval mapping]]: [{{val|1 3 6 7 9}}, {{val|0 -4 -9 -10 -15}}]
 
[[Tp tuning|POL2 generator]]: ~9/7 = 424.457
 
{{Val list|legend=1| 17, 48e, 65de, 82c, 147ce }}
 
[[Tp tuning #T2 tuning|RMS error]]: 1.769 cents
 
==== Skwairs ====
Subgroup: 2.3.7.11.13
 
[[Comma list]]: 99/98, 144/143, 243/242
 
[[Gencom]]: [2 9/7; 99/98, 144/143, 243/242]
 
[[Gencom|Gencom mapping]]: [{{val|1 3 0 6 7 3}}, {{val|0 -4 0 -9 -10 2}}]
 
[[Mapping|Sval mapping]]: [{{val|1 3 6 7 3}}, {{val|0 -4 -9 -10 2}}]
 
[[Tp tuning|POL2 generator]]: ~9/7 = 424.702
 
{{Val list|legend=1| 14, 17, 31 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 1.290 cents
=== Harrison ===
Subgroup: 2.3.7
 
[[Comma]]: [[59049/57344]]
 
[[Gencom]]: [2 3/2; 59049/57344]
 
[[Gencom|Gencom mapping]]: [{{val|1 1 0 -3}}, {{val|0 1 0 10}}]
 
[[Mapping|Sval mapping]]: [{{val|1 1 -3}}, {{val|0 1 10}}]
 
[[Tp tuning|POL2 generator]]: ~3/2 = 696.544
 
{{Val list|legend=1| 12, 19, 31, 112b, 143b, 174b}}
=== Leapfrog ===
{{see also|Gentle region}}
 
Subgroup: 2.3.7
 
[[Comma list]]: 14680064/14348907
 
[[Gencom]]: [2 3/2; 14680064/14348907]
 
[[Gencom|Gencom mapping]]: [{{val|1 1 0 -6}}, {{val|0 1 0 15}}]
 
[[Mapping|Sval mapping]]: [{{val|1 0 -21}}, {{val|0 1 15}}]
 
[[Tp tuning|POL2 generator]]: ~3/2 = 704.721 cents
 
{{Val list|legend=1| 17, 46, 63 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.6202 cents
 
==== 2.3.7.11 ====
Subgroup: 2.3.7.11
 
[[Comma list]]: 896/891, 1331/1323
 
[[Gencom]]: [2 3/2; 896/891 1331/1323]
 
[[Gencom|Gencom mapping]]: [{{val|1 1 0 -6 -3}}, {{val|0 1 0 15 11}}]
 
[[Mapping|Sval mapping]]: [{{val|1 0 -21 -14}}, {{val|0 1 15 11}}]
 
[[Tp tuning|POL2 generator]]: ~3/2 = 704.753 cents
 
{{Val list|legend=1| 17, 46, 63 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.6047 cents
 
==== 2.3.7.11.13 ====
Subgroup: 2.3.7.11.13
 
[[Comma list]]: 169/168, 352/351, 364/363
 
[[Gencom]]: [2 3/2; 169/169 352/351 364/363]
 
[[Gencom|Gencom mapping]]: [{{val|1 1 0 -6 -3 -1}}, {{val|0 1 0 15 11 8}}]
 
[[Mapping|Sval mapping]]: [{{val|1 0 -21 -14 -9}}, {{val|0 1 15 11 8}}]
 
[[Tp tuning|POL2 generator]]: ~3/2 = 704.745 cents
 
{{Val list|legend=1| 17, 46, 63 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.7541 cents
 
; Music
* Suite for Harpsichord in A Locrian, tuning: Eb-G# in [[46edo|46EDO]] by [[User:IlL|IlL]] (in progress):
** I. Prelude
** II. Allemande
** III. Courante
** [[:File:Locrian Suite Sarabande.mp3|IV. Sarabande]] ([[:File:Locrian Suite Sarabande Score.pdf|score]], [[:File:Locrian Suite Sarabande 17edo.mp3|17EDO version]])
** [[:File:Locrian Suite Menuet.mp3|V. Menuet and Trio]]
** [[:File:Locrian Suite Gavotte.mp3|VI. Gavotte I and II]]
** [[:File:Locrian Suite Gigue.mp3|VII. Gigue]]
 
=== Lee ===
Subgroup: 2.3.7
 
[[Comma]]: 177147/175616
 
[[Gencom]]: [2 81/56; 177147/175616]
 
[[Gencom|Gencom mapping]]: [{{val|1 0 0 -3}}, {{val|0 3 0 11}}]
 
[[Mapping|Sval mapping]]: [{{val|1 0 -3}}, {{val|0 3 11}}]
 
[[Tp tuning|POL2 generator]]: ~81/56 = 633.525
 
{{Val list|legend=1| 17, 36, 89, 125, 161, 358, 519b }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.3519 cents
 
=== [[Slendric]] ===
Subgroup: 2.3.7
 
[[Comma]]: 1029/1024
 
[[Gencom]]: [2 8/7; 1029/1024]
 
[[Gencom|Gencom mapping]]: [{{val|1 1 0 3}}, {{val|0 3 0 -1}}]
 
[[Mapping|Sval mapping]]: [{{val|1 1 3}}, {{val|0 3 -1}}]
 
[[Tp tuning|POL2 generator]]: ~8/7 = 233.688
 
{{Val list|legend=1| 5, 21, 26, 31, 36, 77, 113, 190 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.3202 cents
 
==== [[Gamelismic clan #Baladic|Baladic]] ====
Subgroup: 2.3.7.13
 
[[Comma list]]: 169/168, 1029/1024
 
[[Gencom]]: [91/64 8/7; 169/168 1029/1024]
 
[[Mapping|Sval mapping]]: [{{val|2 2 6 7}}, {{val|0 3 -1 1}}]
 
[[Tp tuning|POL2 generator]]: ~8/7 = 233.6044
 
{{Val list|legend=1| 10, 26, 36, 154…, 190…, 226…, 262… }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.5452 cents
 
==== 2.3.7.13.17 ====
Subgroup: 2.3.7.13.17
 
[[Comma list]]: 169/168, 273/272, 289/288
 
[[Gencom]]: [17/12 8/7; 169/168 273/272 289/288]
 
[[Mapping|Sval mapping]]: [{{val|2 2 6 7 7}}, {{val|0 3 -1 1 3}}]
 
[[Tp tuning|POL2 generator]]: ~8/7 = 233.6155
 
{{Val list|legend=1| 10, 26, 36, 154…, 190…, 226… }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.5073 cents
 
=== Hemif ===
Related temperament: [[Breedsmic temperaments #Hemififths|hemififths]], namo
 
Subgroup: 2.3.7
 
[[Comma]]: 1605632/1594323
 
[[Gencom]]: [2 2187/1792; 1605632/1594323]
 
[[Gencom|Gencom mapping]]: [{{val|1 1 0 -1}}, {{val|0 2 0 13}}]
 
[[Mapping|Sval mapping]]: [{{val|1 1 -1}}, {{val|0 2 13}}]
 
[[Tp tuning|POL2 generator]]: ~2187/1792 = 351.485
 
{{Val list|legend=1| 7, 17, 41, 58, 99 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.2344 cents
 
==== 2.3.7.11 ====
Subgroup: 2.3.7.11
 
[[Comma list]]: 243/242, 896/891
 
[[Gencom]]: [2 11/9; 243/242 896/891]
 
[[Gencom|Gencom mapping]]: [{{val|1 1 0 -1 2}}, {{val|0 2 0 13 5}}]
 
[[Mapping|Sval mapping]]: [{{val|1 1 -1 2}}, {{val|0 2 13 5}}]
 
[[Tp tuning|POL2 generator]]: ~11/9 = 351.535
 
{{Val list|legend=1| 7, 17, 41, 58, 99d }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.6108 cents
 
==== 2.3.7.11.13 ====
Subgroup: 2.3.7.11.13
 
[[Comma list]]: 144/143, 243/242, 364/363
 
[[Gencom]]: [2 11/9; 144/143 243/242 364/363]
 
[[Gencom|Gencom mapping]]: [{{val|1 1 0 -1 2 4}}, {{val|0 2 0 13 5 -1}}]
 
[[Mapping|Sval mapping]]: [{{val|1 1 -1 2 4}}, {{val|0 2 13 5 -1}}]
 
[[Tp tuning|POL2 generator]]: ~11/9 = 351.691
 
{{Val list|legend=1| 7, 10, 17, 24, 41, 58 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.7167 cents
 
=== Ennea ===
Subgroup: 2.3.7.11
 
[[Comma list]]: 41503/41472, 43923/43904
 
[[Gencom]]: [2 99/98; 41503/41472, 43923/43904]
 
[[Gencom|Gencom mapping]]: [{{val|1 14/9 0 25/9 31/9}}, {{val|0 2 0 2 1}}]
 
[[Mapping|Sval mapping]]: [{{val|9 0 11 24}}, {{val|0 2 2 1}}]
 
[[Tp tuning|POL2 generator]]: ~99/98 = 17.6258
 
{{Val list|legend=1| 54, 63, 72, 135, 342, 477, 1089, 1566 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.0383 cents
 
=== Parapyth (Rank 3) ===
{{see also| Pentacircle temperaments #Parapyth }}
 
Subgroup: 2.3.7.11
 
[[Comma list]]: 896/891
 
[[Gencom]]: [2 3/2 28/27; 896/891]
 
[[Gencom]] [[mapping]]: [{{val| 1 1 0 1 4 }}, {{val| 0 1 0 3 -1 }}, {{val| 0 0 0 1 1 }}]
 
[[Sval]] [[mapping]]: [{{val| 1 0 0 7 }}, {{val| 0 1 0 -4 }}, {{val| 0 0 1 1 }}]
 
[[Tp tuning|POL2 tuning]]: ~3 = 1903.834, ~7 = 3369.872
 
{{Val list|legend=1| 17, 36, 41, 58, 63, 104 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.4149 cents
 
==== 2.3.7.11.13 ====
Subgroup: 2.3.7.11.13
 
[[Comma list]]: 352/351, 364/363
 
The gencom below gives [[Margo Schulter]]'s favored basis
 
[[Gencom]]: [2 3/2 28/27; 352/351 364/363]
 
[[Gencom]] [[mapping]]: [{{val| 1 1 0 1 4 6 }}, {{val| 0 1 0 3 -1 -4 }}, {{val| 0 0 0 1 1 1 }}]
 
[[Sval]] [[mapping]]: [{{val| 1 0 0 7 12 }}, {{val| 0 1 0 -4 -7 }}, {{val| 0 0 1 1 1 }}]
 
[[Tp tuning|POL2 tuning]]: ~3 = 1903.856, ~7 = 3369.907
 
{{Val list|legend=1| 17, 41, 46, 58, 87, 104 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.3789 cents
 
=== Neutral ===
Neutral can be thought of as the 2.3.11 version of either [[mohajira]] or [[maqamic]], as well as suhajira and ringo. Among other things, it is the temperament optimizing the [[neutral tetrad]].
 
Subgroup: 2.3.11
 
[[Comma]]: 243/242
 
[[Gencom]]: [2 11/9; 243/242]
 
[[Gencom|Gencom mapping]]: [{{val|1 1 0 0 2}}, {{val|0 2 0 0 5}}]
 
[[Mapping|Sval mapping]]: [{{val|1 1 2}}, {{val|0 2 5}}]
 
[[Tp tuning|POL2 generator]]: ~11/9 = 350.525
 
{{Val list|legend=1| 7, 10, 17, 24, 41, 65, 89, 202, 291, 380 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.3021 cents
 
[[neutraltet7|Seven note albitonic scale]]
 
[[neutraltet10|Ten note chromatic scale]]
 
[[neutraltet17|Seventeen note mega chromatic scale]]
 
==== Namo ====
Subgroup: 2.3.11.13
 
[[Comma list]]: 144/143, 243/242
 
[[Gencom]]: [2 11/9; 144/143 243/242]
 
[[Gencom|Gencom mapping]]: [{{val|1 1 0 0 2 4}}, {{val|0 2 0 0 5 -1}}]
 
[[Mapping|Sval mapping]]: [{{val|1 1 2 4}}, {{val|0 2 5 -1}}]
 
[[Tp tuning|POL2 generator]]: ~11/9 = 351.488
 
{{Val list|legend=1| 7, 10, 17, 24, 41 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.7038 cents
 
== No-threes subgroup ==
=== Llywelyn ===
Subgroup: 2.5.7
 
[[Comma]]: 4194304/4117715
 
[[Gencom]]: [2 8/7; 4194304/4117715]
 
[[Gencom|Gencom mapping]]: [{{val|1 0 1 3}}, {{val|0 0 7 -1}}]
 
[[Mapping|Sval mapping]]: [{{val|1 1 3}}, {{val|0 7 -1}}]
 
[[Tp tuning|POL2 generator]]: ~8/7 = 226.910
 
{{Val list|legend=1| 16, 37 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.5391 cents
 
=== Didacus ===
{{See also| Hemimean clan #Didacus }}
 
Related temperaments: [[Chromatic pairs #Roulette|roulette]], [[Gamelismic clan #Hemithirds|hemithirds]]
 
Subgroup: 2.5.7
 
[[Comma]]: 3136/3125
 
[[Gencom]]: [2 28/25; 3136/3125]
 
[[Gencom|Gencom mapping]]: [{{val|1 0 2 2}}, {{val|0 0 2 5}}]
 
[[Mapping|Sval mapping]]: [{{val|1 2 2}}, {{val|0 2 5}}]
 
[[Tp tuning|POL2 generator]]: ~28/25 = 93.772
 
{{Val list|legend=1| 6, 19, 25, 31, 37, 99, 130, 161, 353 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.2138 cents
 
=== Rainy ===
 
Three generators make an [[8/7]]; five generators make a [[5/4]]. This is the no-threes version of [[tertiaseptal]].
 
Subgroup: 2.5.7
 
[[Comma]]: [[2100875/2097152]]
 
[[Gencom]]: [2 256/245; 2100875/2097152]
 
[[Gencom|Gencom mapping]]: [{{val|1 0 2 3 }}, {{val|0 0 5 -3}}]
 
[[Mapping|Sval mapping]]: [{{val|1 2 3 }}, {{val|0 5 -3}}]
 
[[Tp tuning|POL2 generator]]: ~256/245 = 77.205
 
{{Val list|legend=1| 31, 47, 78, 109, 140, 171, 202, 233 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.0586 cents
 
=== Mercy ===
{{See also| Quince clan #Mercy }}
 
Two generators make an [[8/7]]; seven generators make an [[8/5]]. Mercy can be thought of as a way to conceptualize the 2.5.7.13.17.19 subgroup of [[31edo]], and is the no-threes or elevens version of [[miracle]].
 
Subgroup: 2.5.7
 
[[Comma list]]: 823543/819200
 
[[Gencom]]: [2 2744/2560; 823543/819200]
 
[[Gencom|Gencom mapping]]: [{{val| 1 0 3 3 }}, {{val| 0 0 -7 -2 }}]
 
[[Sval]] [[mapping]]: [{{val| 1 3 3 }}, {{val| 0 -7 -2 }}]
 
[[Tp tuning|POL2 generator]]: ~343/320 = 116.291
 
{{Val list|legend=1| 10, 21, 31, 134, 165, 196, 227, 485d, 712d, 1197dd }}
 
==== 2.5.7.13 ====
Subgroup: 2.5.7.13
 
[[Comma list]]: 343/338, 640/637
 
[[Gencom]]: [2 14/13; 343/338 640/637]
 
[[Gencom|Gencom mapping]]: [{{val|1 0 3 3 4}}, {{val|0 0 -7 -2 -3}}]
 
[[Mapping|Sval mapping]]: [{{val|1 3 3 4}}, {{val|0 -7 -2 -3}}]
 
[[Tp tuning|POL2 generator]]: ~14/13 = 116.094
 
{{Val list|legend=1| 10, 21, 31}}
 
==== 2.5.7.13.17 ====
Subgroup: 2.5.7.13.17
 
[[Comma list]]: 170/169, 224/221, 640/637
 
[[Gencom]]: [2 14/13; 170/169 224/221 640/637]
 
[[Gencom|Gencom mapping]]: [{{val|1 0 3 3 4 4}}, {{val|0 0 -7 -2 -3 1}}]
 
[[Mapping|Sval mapping]]: [{{val|1 3 3 4 4}}, {{val|0 -7 -2 -3 1}}]
 
[[Tp tuning|POL2 generator]]: ~14/13 = 115.769
 
{{Val list|legend=1| 10, 21, 31}}
 
==== 2.5.7.13.17.19 ====
Subgroup: 2.5.7.13.17.19
 
[[Comma list]]: 170/169, 343/338, 640/637, 16384/16055
 
[[Gencom]]: [2 14/13; 170/169 343/338 640/637 16384/16055]
 
[[Gencom|Gencom mapping]]: [{{val|1 0 3 3 4 4 3}}, {{val|0 0 -7 -2 -3 1 13}}]
 
[[Mapping|Sval mapping]]: [{{val|1 3 3 4 4 3}}, {{val|0 -7 -2 -3 1 13}}]
 
[[Tp tuning|POL2 generator]]: ~14/13 = 115.716
 
{{Val list|legend=1| 10, 21, 31, 52f}}


== 2.9.7.11 subgroup ==
== 2.9.7.11 subgroup ==
Line 1,251: Line 151:
[[Tp tuning #T2 tuning|RMS error]]: 0.4898 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.4898 cents


== Miscellaneous subgroup temperaments ==
== Fractional subgroup temperaments ==


=== Historical ===
=== Historical ===

Revision as of 21:03, 6 November 2021

A subgroup temperament is a regular temperament defined on a just intonation subgroup that is not a full p-limit group.

For temperaments that omit various prime harmonics, see:

Below are some other miscellaneous subgroups and temperaments for them. Obviously, no attempt has been made at completeness; attention is focused on subgroups containing interesting chords. The reader may also want to consult the page on Chromatic pairs.

2.9.7.11 subgroup

Machine

Subgroup: 2.9.7.11

Commas: 64/63, 99/98

Gencom: [2 8/7; 64/63 99/98]

Gencom mapping: [⟨1 3/2 0 3 4], ⟨0 1/2 0 -1 -3]]

Sval mapping: [⟨1 0 6 13], ⟨0 1 -1 -3]]

POL2 generator: ~8/7 = 214.384

Template:Val list

RMS error: 1.977 cents

Mechanism

Subgroup: 2.9.7.11

Comma list: 896/891, 26411/26244

Gencom: [2 9/7; 896/891 26411/26244]

Gencom mapping: [⟨1 5/2 0 5 2], ⟨0 -5/2 0 -6 4]]

Sval mapping: [⟨1 5 5 2], ⟨0 -5 -6 4]]

POL2 generator: ~9/7 = 438.465

Template:Val list

RMS error: 0.4262 cents

Apparatus

Subgroup: 2.9.7.11

Comma list: 41503/41472, 322102/321489

Gencom: [2 77/72; 41503/41472 322102/321489]

Gencom mapping: [⟨1 5/2 0 3 5], ⟨0 -19/2 0 -2 -16]]

Sval mapping: [⟨1 5 3 5], ⟨0 -19 -2 -16]]

POL2 generator: ~77/72 = 115.570

Template:Val list

RMS error: 0.0673 cents

2.9.15.7 subgroup

Stacks (aka 2magic)

Subgroup: 2.9.15.7

Comma list: 225/224, 245/243

Gencom: [2 9/7; 225/224 245/243]

Gencom mapping: [⟨1 5/2 5/2 5], ⟨0 -5/2 -1/2 -6]]

Sval mapping: [⟨1 0 2 -1], ⟨0 5 3 6]]

POL2 generator: ~9/7 = 439.296

Template:Val list

RMS error: 1.074 cents

2.9.15.7.11

Subgroup: 2.9.15.7.11

Comma list: 100/99, 225/224, 245/243

Gencom: [2 9/7; 100/99 225/224 245/243]

Gencom mapping: [⟨1 5/2 5/2 5 2], ⟨0 -5/2 -1/2 -6 4]]

Sval mapping: [⟨1 0 2 -1 6], ⟨0 5 3 6 -4]]

POL2 generator: ~9/7 = 438.607

Vals: Template:Val list

RMS error: 1.226 cents

2.9.15.7.11.13

Subgroup: 2.9.15.7.11.13

Comma list: 100/99, 105/104, 144/143, 196/195

Gencom: [2 9/7; 100/99 105/104 144/143 196/195]

Gencom mapping: [⟨1 5/2 5/2 5 2 7], ⟨0 -5/2 -1/2 -6 4 -9]]

Sval mapping: [⟨1 0 2 -1 6 -2], ⟨0 5 3 6 -4 9]]

POL2 generator: ~9/7 = 438.977

Vals: Template:Val list

RMS error: 1.540 cents

2.9.21 subgroup

A-team

Subgroup: 2.9.21

Comma: 1029/1024

Gencom: [2 21/16; 1029/1024]

Gencom mapping: [⟨1 1 0 3], ⟨0 3/2 0 -1/2]]

Sval mapping: [⟨1 2 4], ⟨0 3 1]]

POL2 generator: ~21/16 = 467.375

Template:Val list

RMS error: 0.3202 cents

2.11.13.17.19 subgroup

Yamablu

Yamablu, with a generator of ~17/13, is named for it's tempering of the yama comma (209/208) and the blume comma (2057/2048), which also implies the blumeyer comma (2432/2431). The 13th Yamablu[13] scale is a linear-temperament version of Gjaeck.

Subgroup: 2.11.13.17.19

Comma list: 209/208, 2057/2048, 83521/83486

Sval mapping: [⟨1 5 1 1 0], ⟨0 -4 7 8 11]]

POL2 generator: ~17/13 = 462.9606

Template:Val list

RMS error: 0.4898 cents

Fractional subgroup temperaments

Historical

Subgroup: 2.3.7/5.11/5.13/5

Comma list: 364/363, 441/440, 1001/1000

Sval mapping: [⟨1 2 0 1 2], ⟨0 -6 7 2 -9]]

POL2 generator: ~21/20 = 83.016

Template:Val list

RMS error: 0.2562 cents

Hypnosis

Related temperament: hypnos, tricot

Subgroup: 2.3.7.11/5.13

Comma list: 169/168, 540/539, 729/728

Sval mapping: [⟨1 0 -3 8 0], ⟨0 3 11 -13 7]]

POL2 generator: ~13/9 = 633.518

Template:Val list

RMS error: 0.5379 cents

Oceanfront

Subgroup: 2.3.7.13/5

Related temperament: superpyth, ultrapyth

Comma list: 64/63, 91/90

Sval mapping: [⟨1 0 6 -5], ⟨0 1 -2 4]]

POL2 generator: ~3/2 = 713.910

Template:Val list

RMS error: 2.063 cents

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