No-elevens subgroup temperaments
This is a collection of subgroup temperaments which omit the prime harmonic of 11.
Ultrapyth
- For full 13-limit extensions, see Archytas clan #Ultrapyth.
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 mapping: [⟨1 2 8 2 0 11], ⟨0 -1 -14 2 0 -18]]
Sval mapping: [⟨1 2 8 2 11], ⟨0 -1 -14 2 -18]]
POL2 generator: ~4/3 = 486.255
Optimal ET sequence: 5, 32, 37
RMS error: 2.318 cents
Meanpop
- For full 13-limit extensions, see Meantone family #Septimal meantone.
Subgroup: 2.3.5.7.13
Comma list: 81/80, 105/104, 126/125
Gencom: [2 4/3; 81/80 105/104 126/125]
Gencom mapping: [⟨1 2 4 7 0 10], ⟨0 -1 -4 -10 0 -15]]
Sval mapping: [⟨1 2 4 7 10], ⟨0 -1 -4 -10 -15]]
POL2 generator: ~3/2 = 695.988
Optimal ET sequence: 12f, 19, 31, 50
RMS error: 1.421 cents
Sensi (aka sensation)
- For full 13-limit extensions, see Sensipent family or Sensi extensions.
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 mapping: [⟨1 6 8 11 0 10], ⟨0 -7 -9 -13 0 -10]]
Sval mapping: [⟨1 6 8 11 10], ⟨0 -7 -9 -13 -10]]
POL2 generator: ~9/7 = 443.322
Optimal ET sequence: 19, 27, 46, 111de, 157de
RMS error: 1.321 cents
Decic
- For full 13- and 17-limit extensions, see Cloudy clan #Decic.
Subgroup: 2.3.5.7.13
Comma list: 105/104, 196/195, 8281/8192
Gencom: [2 49/48; 105/104 196/195 8281/8192]
Gencom mapping: [⟨10 16 23 28 0 37], ⟨0 -1 1 0 0 0]]
Sval mapping: [⟨10 16 23 28 37], ⟨0 -1 1 0 0]]
POL2 generator: ~49/48 = 21.1859
Optimal ET sequence: 10, 30b, 40, 50, 60, 110d, 170cdd
RMS error: 1.122 cents
2.3.5.7.13.17
Subgroup: 2.3.5.7.13.17
Comma list: 105/104, 170/169, 196/195, 289/288
Gencom: [2 49/48; 105/104 170/169 196/195 289/288]
Gencom mapping: [⟨10 16 23 28 0 37 41], ⟨0 -1 1 0 0 0 -1]]
Sval mapping: [⟨10 16 23 28 37 41], ⟨0 -1 1 0 0 -1]]
POL2 generator: ~49/48 = 21.1836
Optimal ET sequence: 10, 30bg, 40, 50, 60, 110dg, 170cddg
RMS error: 1.038 cents
Quartonic
- For full 13-limit extensions, see Orwellismic temperaments #Quartonic.
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 mapping: [⟨1 2 3 3 0 4], ⟨0 -11 -18 -5 0 -8]]
Sval mapping: [⟨1 2 3 3 4], ⟨0 -11 -18 -5 -8]]
POL2 generator: ~36/35 = 45.1821
Optimal ET sequence: 26, 27, 53
RMS error: 0.7439 cents
Unicorn
- For full 13-limit extensions, see Unicorn family.
Subgroup: 2.3.5.7.13
Comma list: 126/125, 196/195, 676/675
Gencom: [2 28/27; 126/125, 196/195, 676/675]
Gencom mapping: [⟨1 2 3 4 0 5], ⟨0 -8 -13 -23 0 -25]]
Sval mapping: [⟨1 2 3 4 5], ⟨0 -8 -13 -23 -25]]
POL2 generator: ~28/27 = 62.294
Optimal ET sequence: 19, 39df, 58, 77
RMS error: 0.7092 cents
Septidiasemi
- For full 13- and 17-limit extensions, see Breedsmic temperaments #Septidiasemi.
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 mapping: [⟨1 -1 6 4 0 4], ⟨0 26 -37 -12 0 -3]]
Sval mapping: [⟨1 -1 6 4 4], ⟨0 26 -37 -12 -3]]
POL2 generator: ~15/14 = 119.297
Optimal ET sequence: 10, 151, 161, 171
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 mapping: [⟨1 -1 6 4 0 4 2], ⟨0 26 -37 -12 0 -3 21]]
Sval mapping: [⟨1 -1 6 4 4 2], ⟨0 26 -37 -12 -3 21]]
POL2 generator: ~15/14 = 119.297
Optimal ET sequence: 10, 151, 161, 171
RMS error: 0.1867 cents
Pontiac
- For full 13- and 17-limit extensions, see Schismatic family #Pontiac.
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 mapping: [⟨1 2 -1 19 0 -10], ⟨0 -1 8 -39 0 33]]
Sval mapping: [⟨1 2 -1 19 -10], ⟨0 -1 8 -39 33]]
POL2 generator: ~3/2 = 701.773
Optimal ET sequence: 53, 118, 171, 224, 395
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 mapping: [⟨1 2 -1 19 0 -10 29], ⟨0 -1 8 -39 0 33 -60]]
Sval mapping: [⟨1 2 -1 19 -10 29], ⟨0 -1 8 -39 33 -60]]
POL2 generator: ~3/2 = 701.764
Optimal ET sequence: 53, 118, 171, 395, 566…
RMS error: 0.1696 cents
Mitonic
- For full 13- and 17-limit extensions, see Minortonic family #Mitonic.
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 mapping: [⟨1 -1 -3 6 0 11], ⟨0 17 35 -21 0 -48]]
Sval mapping: [⟨1 -1 -3 6 11], ⟨0 17 35 -21 -48]]
POL2 generator: ~10/9 = 182.471
Optimal ET sequence: 46, 125, 171, 388
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 mapping: [⟨1 -1 -3 6 0 11 5], ⟨0 17 35 -21 0 -48 -6]]
Sval mapping: [⟨1 -1 -3 6 11 5], ⟨0 17 35 -21 -48 -6]]
POL2 generator: ~10/9 = 182.471
Optimal ET sequence: 46, 125, 171, 388
RMS error: 0.1341 cents
Tertiaseptal
- For full 13- and 17-limit extensions, see Breedsmic temperaments #Tertiaseptal.
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 mapping: [⟨1 3 2 3 0 1], ⟨0 -22 5 -3 0 42]]
Sval mapping: [⟨1 3 2 3 1], ⟨0 -22 5 -3 42]]
POL2 generator: ~117/112 = 77.173
Optimal ET sequence: 31, 109, 140, 171, 311
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 mapping: [⟨1 3 2 3 0 1 1], ⟨0 -22 5 -3 0 42 48]]
Sval mapping: [⟨1 3 2 3 1 1], ⟨0 -22 5 -3 42 48]]
POL2 generator: ~68/65 = 77.177
Optimal ET sequence: 31, 109f, 140, 171, 311
RMS error: 0.1367 cents
Oquatonic
- For full 13-limit extensions, see Horwell temperaments #Oquatonic.
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 mapping: [⟨28 44 65 79 0 104], ⟨0 1 0 -1 0 -1]]
Sval mapping: [⟨28 44 65 79 104], ⟨0 1 0 -1 -1]]
POL2 generator: ~105/104 = 16.4240
Optimal ET sequence: 28, 56, 84, 140, 224, 364
RMS error: 0.1347 cents
Garibaldia
- See also: No-sevens subgroup temperaments #Photia and #Nestoria
Subgroup: 2.3.5.7.17
Comma list: 120/119, 225/224, 1458/1445
Gencom: [2 4/3; 120/119 225/224 1458/1445]
Gencom mapping: [⟨1 2 -1 -3 0 0 7], ⟨0 -1 8 14 0 0 -7]]
Sval mapping: [⟨1 2 -1 -3 7], ⟨0 -1 8 14 -7]]
POL2 generator: ~3/2 = 701.862
Optimal ET sequence: 12, 29, 41, 53
RMS error: 0.8796 cents
Related temperament: Garibaldi
2.3.5.7.17.19
Subgroup: 2.3.5.7.17.19
Comma list: 120/119, 171/170, 190/189, 225/224
Gencom: [2 4/3; 120/119 171/170 190/189 225/224]
Gencom mapping: [⟨1 2 -1 -3 0 0 7 3], ⟨0 -1 8 14 0 0 -7 3]]
Sval mapping: [⟨1 2 -1 -3 7 3], ⟨0 -1 8 14 -7 3]]
POL2 generator: ~3/2 = 701.880
Optimal ET sequence: 12, 29, 41, 53
RMS error: 0.9338 cents
Devisemi
Subgroup: 2.3.5.7.19
Comma list: 190/189, 245/243, 361/360
Gencom: [2 19/18; 190/189 245/243 361/360]
Gencom mapping: [⟨1 1 1 2 0 0 0 3], ⟨0 8 18 11 0 0 0 17]]
Sval mapping: [⟨1 1 1 2 3], ⟨0 8 18 11 17]]
POL2 generator: ~19/18 = 88.075
Optimal ET sequence: 14c, 27, 41, 68, 109
RMS error: 0.5780 cents
Related temperament: octacot
Pontia
Subgroup: 2.3.5.7.19
Comma list: 361/360, 513/512, 4375/4374
Gencom: [2 4/3; 361/360 513/512 4375/4374]
Gencom mapping: [⟨1 2 -1 19 0 0 0 3], ⟨0 -1 8 -39 0 0 0 3]]
Sval mapping: [⟨1 2 -1 19 3], ⟨0 -1 8 -39 3]]
POL2 generator: ~3/2 = 701.748
Optimal ET sequence: 53, 118, 171
RMS error: 0.1763 cents
Related temperament: Pontiac
Cotoneum
- For full 19-limit extensions, see Hemimage temperaments #Cotoneum.
Subgroup: 2.3.5.7.19
Comma list: 343/342, 1216/1215, 48013/48000
Gencom: [2 4/3; 343/342 1216/1215 48013/48000]
Gencom mapping: [⟨1 2 -18 -3 0 0 0 -14], ⟨0 -1 49 14 0 0 0 44]]
Sval mapping: [⟨1 2 -18 -3 -14], ⟨0 -1 49 14 44]]
POL2 generator: ~3/2 = 702.319
Optimal ET sequence: 41, 176, 217
RMS error: 0.1852 cents
Beet
Being describable as mynucumic archy biome, this temperament has the logic common to 13-limit beatles, ringo and beetle; in other words, they all only differ by the mapping of 11. The mapping of 11 provided by ares is seen as the canonic extension of beatles to the 11-limit. If you want a good patent val tuning, 27edo is about as good as you're going to get if you want the major third to have a good chance at being able to function contextually as both a ~9/7 and ~13/10, assuming you're targeting the no-11's 13-odd-limit.
The S-expression-based comma list of this temperament is {S8, S13, S14}.
Subgroup: 2.3.5.7.13
Comma list: 64/63, 686/675, 512/507
Mapping: [⟨1 1 5 4 4], ⟨0 2 -9 -4 -1]]
Optimal tuning (CTE): 356.645
All patent val EDO tunings: 10, 27, 37