No-fives subgroup temperaments
This is a collection of subgroup temperaments which omit the prime harmonic of 5.
Semaphore
Subgroup: 2.3.7
Comma: 49/48
Gencom: [2 8/7; 49/48]
Gencom mapping: [⟨1 2 0 3], ⟨0 -2 0 -1]]
Sval mapping: [⟨1 2 3], ⟨0 -2 -1]]
POL2 generator: ~7/6 = 250.385
Optimal ET sequence: 5, 14, 19, 24, 67d, 91d
RMS error: 2.523 cents
Bleu
Subgroup: 2.3.7
Comma: 17496/16807
Gencom: [2 54/49; 17496/16807]
Gencom mapping: [⟨1 1 0 2], ⟨0 5 0 7]]
Sval mapping: [⟨1 1 2], ⟨0 5 7]]
POL2 generator: ~54/49 = 139.848
Optimal ET sequence: 9, 17, 43, 60d
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 mapping: [⟨1 1 0 2 3], ⟨0 5 0 7 4]]
Sval mapping: [⟨1 1 2 3], ⟨0 5 7 4]]
POL2 generator: ~12/11 = 140.005
Optimal ET sequence: 9, 17, 43, 60d
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 mapping: [⟨1 1 0 2 3 3], ⟨0 5 0 7 4 6]]
Sval mapping: [⟨1 1 2 3 3], ⟨0 5 7 4 6]]
POL2 generator: ~12/11 = 139.990
Optimal ET sequence: 17, 43, 60d
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 mapping: [⟨1 1 0 4], ⟨0 1 0 -2]]
Sval mapping: [⟨1 2 2], ⟨0 -1 2]]
POL2 generator: ~3/2 = 709.321
Optimal ET sequence: 5, 12, 17, 22, 27, 137bd
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 mapping: [⟨1 1 0 4 7], ⟨0 1 0 -2 -6]]
Sval mapping: [⟨1 0 6 13], ⟨0 1 -2 -6]]
POL2 generator: ~3/2 = 707.192
Optimal ET sequence: 5, 12, 17, 39d, 56d
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 mapping: [⟨1 1 0 4 7 9], ⟨0 1 0 -2 -6 -9]]
Sval mapping: [⟨1 0 6 13 18], ⟨0 1 -2 -6 -9]]
POL2 generator: ~3/2 = 706.137
Optimal ET sequence: 5f, 12f, 17, 22, 39d, 56d
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 mapping: [⟨1 1 0 4 2], ⟨0 2 0 -4 5]]
Sval mapping: [⟨1 1 4 2], ⟨0 2 -4 5]]
POL2 generator: ~11/9 = 353.958
Optimal ET sequence: 7, 10, 17, 44e, 61de, 78de
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 mapping: [⟨1 1 0 4 2 4], ⟨0 2 0 -4 5 -1]]
Sval mapping: [⟨1 1 4 2 4], ⟨0 2 -4 5 -1]]
POL2 generator: ~11/9 = 353.775
Optimal ET sequence: 7, 10, 17, 44e, 61de, 78de
RMS error: 1.953 cents
Skwares
Subgroup: 2.3.7
Comma: 19683/19208
Gencom: [2 9/7; 19683/19208]
Gencom mapping: [⟨1 3 6], ⟨0 -4 -9]]
Sval mapping: [⟨1 3 6], ⟨0 -4 -9]]
POL2 generator: ~9/7 = 425.365
Optimal ET sequence: 14, 17, 31, 48, 79, 189b, 268bd, 347bd
RMS error: 1.149 cents
Related temperament: squares
2.3.7.11
Subgroup: 2.3.7.11
Comma list: 99/98, 243/242
Gencom: [2 9/7; 99/98 243/242]
Gencom mapping: [⟨1 3 0 6 7], ⟨0 -4 0 -9 -10]]
Sval mapping: [⟨1 3 6 7], ⟨0 -4 -9 -10]]
POL2 generator: ~9/7 = 425.244
Optimal ET sequence: 14, 17, 31, 48, 79, 127, 206bde
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 mapping: [⟨1 3 0 6 7 9], ⟨0 -4 0 -9 -10 -15]]
Sval mapping: [⟨1 3 6 7 9], ⟨0 -4 -9 -10 -15]]
POL2 generator: ~9/7 = 424.457
Optimal ET sequence: 17, 48f, 65ef, 82d, 147df
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 mapping: [⟨1 3 0 6 7 3], ⟨0 -4 0 -9 -10 2]]
Sval mapping: [⟨1 3 6 7 3], ⟨0 -4 -9 -10 2]]
POL2 generator: ~9/7 = 424.702
Optimal ET sequence: 14, 17, 31
RMS error: 1.290 cents
Byhearted
- For the full 19-limit version of this temperament, see Tetracot family #Byhearted.
Subgroup: 2.3.7.11.19
Comma list: 99/98, 243/242, 363/361
Gencom: [209/147 21/19; 99/98 243/242 363/361]
Sval mapping: [⟨2 2 3 4 5], ⟨0 4 9 10 12]]
POL2 generator: ~21/19 = 174.735
Optimal ET sequence: 14, 34dh, 48, 110e, 158e
RMS error: 0.8727 cents
Harrison
Subgroup: 2.3.7
Gencom: [2 3/2; 59049/57344]
Gencom mapping: [⟨1 1 0 -3], ⟨0 1 0 10]]
Sval mapping: [⟨1 1 -3], ⟨0 1 10]]
POL2 generator: ~3/2 = 696.544
Optimal ET sequence: 12, 19, 31, 112b, 143b, 174b
RMS error: 1.226 cents
Related temperament: meantone
Leapfrog
- See also: Gentle region
Subgroup: 2.3.7
Comma list: 14680064/14348907
Gencom: [2 3/2; 14680064/14348907]
Gencom mapping: [⟨1 1 0 -6], ⟨0 1 0 15]]
Sval mapping: [⟨1 0 -21], ⟨0 1 15]]
POL2 generator: ~3/2 = 704.721 cents
Optimal ET sequence: 17, 46, 63
RMS error: 0.6202 cents
Related temperaments: leapday, leapweek, srutal
2.3.7.11
Subgroup: 2.3.7.11
Comma list: 896/891, 1331/1323
Gencom: [2 3/2; 896/891 1331/1323]
Gencom mapping: [⟨1 1 0 -6 -3], ⟨0 1 0 15 11]]
Sval mapping: [⟨1 0 -21 -14], ⟨0 1 15 11]]
POL2 generator: ~3/2 = 704.753 cents
Optimal ET sequence: 17, 46, 63
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 mapping: [⟨1 1 0 -6 -3 -1], ⟨0 1 0 15 11 8]]
Sval mapping: [⟨1 0 -21 -14 -9], ⟨0 1 15 11 8]]
POL2 generator: ~3/2 = 704.745 cents
Optimal ET sequence: 17, 46, 63
RMS error: 0.7541 cents
Skidoo
Subgroup: 2.3.7.11.13.23
Comma list: 169/168, 208/207, 352/351, 364/363
Gencom: [2 3/2; 169/169 208/207 352/351 364/363]
Gencom mapping: [⟨1 1 0 -6 -3 -1 0 0 1], ⟨0 1 0 15 11 8 0 0 6]]
Sval mapping: [⟨1 0 -21 -14 -9 -5], ⟨0 1 15 11 8 6]]
POL2 generator: ~3/2 = 704.729 cents
Optimal ET sequence: 17, 46, 63
RMS error: 0.6265 cents
2.3.7.11.13.23.29
Subgroup: 2.3.7.11.13.23.29
Comma list: 169/168, 208/207, 232/231, 352/351, 364/363
Gencom: [2 3/2; 169/169 208/207 352/351 364/363]
Gencom mapping: [⟨1 1 0 -6 -3 -1 0 0 1 -11], ⟨0 1 0 15 11 8 0 0 6 27]]
Sval mapping: [⟨1 0 -21 -14 -9 -5 -38], ⟨0 1 15 11 8 6 27]]
POL2 generator: ~3/2 = 704.729 cents
Optimal ET sequence: 17, 46, 63
- Music
- Suite for Harpsichord in A Locrian, tuning: Eb-G# in 46EDO by IlL (in progress):
- I. Prelude
- II. Allemande
- III. Courante
- IV. Sarabande (score, 17EDO version)
- V. Menuet and Trio
- VI. Gavotte I and II
- VII. Gigue
Doublehearted
- See also: Heartland
Subgroup: 2.3.7
Comma list: 5764801/5668704
Gencom: [2 343/324; 5764801/5668704]
Sval mapping: [⟨1 1 2], ⟨0 8 11]]
POL2 generator: ~343/324 = 87.8304
Optimal ET sequence: 14, 27, 41
RMS error: 0.5041 cents
Related temperaments: octacot
2.3.7.11
Subgroup: 2.3.7.11
Comma list: 243/242, 2401/2376
Gencom: [2 22/21; 243/242 2401/2376]
Sval mapping: [⟨1 1 2 2], ⟨0 8 11 20]]
POL2 generator: ~22/21 = 87.6512
Optimal ET sequence: 14, 27e, 41, 96d, 137d, 178d
RMS error: 0.7147 cents
Related temperaments: octacot
2.3.7.11.19
Subgroup: 2.3.7.11.19
Comma list: 133/132, 243/242, 343/342
Gencom: [2 19/18; 133/132 243/242 343/342]
Sval mapping: [⟨1 1 2 2 3], ⟨0 8 11 20 17]]
POL2 generator: ~19/18 = 87.6684
Optimal ET sequence: 14, 27e, 41
RMS error: 0.7065 cents
Related temperaments: octacot
Magi
Subgroup: 2.3.7
Comma list: 537824/531441
Gencom: [2 243/196; 537824/531441]
Sval mapping: [⟨1 0 -1], ⟨0 5 12]]
POL2 generator: ~243/196 = 380.661
Optimal ET sequence: 19, 22, 41, 104, 145, 186
RMS error: 0.4277 cents
2.3.7.11
Subgroup: 2.3.7.11
Comma list: 896/891, 26411/26244
Gencom: [2 96/77; 896/891 26411/26244]
Sval mapping: [⟨1 0 -1 6], ⟨0 5 12 -8]]
POL2 generator: ~96/77 = 380.768
Optimal ET sequence: 19, 22, 41, 63, 104
RMS error: 0.4262 cents
Balthazar
Subgroup: 2.3.7.11.13
Comma list: 169/168, 896/891, 26411/26244
Gencom: [2 143/128; 169/168 896/891 26411/26244]
Sval mapping: [⟨1 0 -1 6 1], ⟨0 10 24 -16 17]]
POL2 generator: ~143/128 = 190.407
Optimal ET sequence: 19, 44, 63, 145f
RMS error: 0.6937 cents
Caspar
Subgroup: 2.3.7.11.13
Comma list: 144/143, 343/338, 729/728
Gencom: [2 26/21; 144/143 343/338 729/728]
Sval mapping: [⟨1 0 -1 6 -2], ⟨0 5 12 -8 18]]
POL2 generator: ~26/21 = 380.531
Optimal ET sequence: 19, 22f, 41
RMS error: 1.032 cents
Melchior
Subgroup: 2.3.7.11.13
Comma list: 352/351, 364/363, 26411/26244
Gencom: [2 96/77; 352/351 364/363 26411/26244]
Sval mapping: [⟨1 0 -1 6 11], ⟨0 5 12 -8 -23]]
POL2 generator: ~96/77 = 380.766
Optimal ET sequence: 19f, 22, 41, 63, 104
RMS error: 0.3891 cents
Hogwarts
Subgroup: 2.3.7.29
Comma list: 784/783, 5887/5832
Gencom: [2 36/29; 784/783 5887/5832]
Sval mapping: [⟨1 0 -1 2], ⟨0 5 12 9]]
POL2 generator: ~36/29 = 380.618
Optimal ET sequence: 19, 22, 41, 145, 186j, 227j
Twenothology
Subgroup: 2.3.7.11.13.29
Comma list: 144/143, 232/231, 343/338, 729/728
Sval mapping: [⟨1 0 -1 6 -2 2], ⟨0 5 12 -8 18 9]]
POL2 generator: ~26/21 = 380.526
Optimal ET sequence: 19, 22f, 41
Lee
Subgroup: 2.3.7
Comma: 177147/175616
Gencom: [2 81/56; 177147/175616]
Gencom mapping: [⟨1 0 0 -3], ⟨0 3 0 11]]
Sval mapping: [⟨1 0 -3], ⟨0 3 11]]
POL2 generator: ~81/56 = 633.525
Optimal ET sequence: 17, 36, 89, 125, 161, 358, 519b
RMS error: 0.3519 cents
Slendric
Subgroup: 2.3.7
Comma: 1029/1024
Gencom: [2 8/7; 1029/1024]
Gencom mapping: [⟨1 1 0 3], ⟨0 3 0 -1]]
Sval mapping: [⟨1 1 3], ⟨0 3 -1]]
POL2 generator: ~8/7 = 233.688
Optimal ET sequence: 5, 21, 26, 31, 36, 77, 113, 190
RMS error: 0.3202 cents
Baladic
Subgroup: 2.3.7.13
Comma list: 169/168, 1029/1024
Gencom: [91/64 8/7; 169/168 1029/1024]
Sval mapping: [⟨2 2 6 7], ⟨0 3 -1 1]]
POL2 generator: ~8/7 = 233.6044
Optimal ET sequence: 10, 26, 36, 154…, 190…, 226…, 262…
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]
Sval mapping: [⟨2 2 6 7 7], ⟨0 3 -1 1 3]]
POL2 generator: ~8/7 = 233.6155
Optimal ET sequence: 10, 26, 36, 154…, 190…, 226…
RMS error: 0.5073 cents
Hemif
Subgroup: 2.3.7
Comma: 1605632/1594323
Gencom: [2 2187/1792; 1605632/1594323]
Gencom mapping: [⟨1 1 0 -1], ⟨0 2 0 13]]
Sval mapping: [⟨1 1 -1], ⟨0 2 13]]
POL2 generator: ~2187/1792 = 351.485
Optimal ET sequence: 7, 17, 41, 58, 99
RMS error: 0.2344 cents
Related temperaments: hemififths, namo
2.3.7.11
Subgroup: 2.3.7.11
Comma list: 243/242, 896/891
Gencom: [2 11/9; 243/242 896/891]
Gencom mapping: [⟨1 1 0 -1 2], ⟨0 2 0 13 5]]
Sval mapping: [⟨1 1 -1 2], ⟨0 2 13 5]]
POL2 generator: ~11/9 = 351.535
Optimal ET sequence: 7, 17, 41, 58, 99e
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 mapping: [⟨1 1 0 -1 2 4], ⟨0 2 0 13 5 -1]]
Sval mapping: [⟨1 1 -1 2 4], ⟨0 2 13 5 -1]]
POL2 generator: ~11/9 = 351.691
Optimal ET sequence: 7, 10, 17, 24, 41, 58
RMS error: 0.7167 cents
Heartful
- See also: Heartland
Subgroup: 2.3.7.11.19
Comma list: 243/242, 896/891, 1083/1078
Gencom: [2 21/19; 243/242 896/891 1083/1078]
Sval mapping: [⟨1 1 -1 2 0], ⟨0 4 26 10 29]]
POL2 generator: ~21/19 = 175.804
Optimal ET sequence: 34dh, 41, 116e, 157e
RMS error: 0.5360 cents
Related temperaments: bunya
Hearts
- See also: Heartland
Subgroup: 2.3.7
Comma list: 34451725707/34359738368 (trila-quadzo comma)
Gencom: [2 567/512; 34451725707/34359738368]
Sval mapping: [⟨1 1 5], ⟨0 4 -15]]
POL2 generator: ~567/512 = 175.433
Optimal ET sequence: 7, 27d, 34, 41, 89, 130, 171
RMS error: 0.0529 cents
Related temperaments: monkey, sesquiquartififths
2.3.7.11
Subgroup: 2.3.7.11
Comma list: 243/242, 65536/65219
Gencom: [2 256/231; 243/242 65536/65219]
Sval mapping: [⟨1 1 5 2], ⟨0 4 -15 10]]
POL2 generator: ~256/231 = 175.369
Optimal ET sequence: 7, 27de, 34, 41, 89, 130
RMS error: 0.3224 cents
Related temperaments: monkey, sesquart
2.3.7.11.19
Subgroup: 2.3.7.11.19
Comma list: 243/242, 513/512, 1083/1078
Gencom: [2 21/19; 243/242 513/512 1083/1078]
Sval mapping: [⟨1 1 5 2 6], ⟨0 4 -15 10 -12]]
POL2 generator: ~21/19 = 175.341
Optimal ET sequence: 7, 27deh, 34, 41, 89, 130, 219
RMS error: 0.3121 cents
Related temperaments: monkey, sesquart
Subgroup: 2.3.7
Comma list: 282429536481/281974669312
Mapping: [⟨1 1 0], ⟨0 5 24]]
POL2 generator: ~243/224 = 140.366
Optimal ET sequence: 17, 60, 77, 94, 171, 265, 436
RMS error: 0.0296 cents
Related temperaments: tsaharuk, quanic
2.3.7.11
Subgroup: 2.3.7.11
Comma list: 1331/1323, 19712/19683
Mapping: [⟨1 1 0 1], ⟨0 5 24 21]]
POL2 generator: ~88/81 = 140.407
Optimal ET sequence: 17, 60e, 77, 94, 359e, 453ee, 547ee, 641ee
RMS error: 0.3778 cents
Related temperaments: tsaharuk, quanic
2.3.7.11.13
Subgroup: 2.3.7.11.13
Comma list: 352/351, 729/728, 1331/1323
Mapping: [⟨1 1 0 1 3], ⟨0 5 24 21 6]]
POL2 generator: ~13/12 = 140.437
Optimal ET sequence: 17, 60e, 77, 94
RMS error: 0.4044 cents
Related temperaments: tsaharuk, quanic
Hectosaros leap week
Defined as the 320 & 1803 temperament, in the 2.3.7.13.17.19 on the basis of the fact that 1803 tropical years make up almost exactly 100 saros cycles.
Subgroup: 2.3.7
Comma list: [-50 -746 439⟩
Mapping: [⟨1 313 532], ⟨0 -439 -746]]
Optimal tuning (CTE): ~[17 343 143⟩ = 851.248
Optimal ET sequence: 320, 1163bdd, 1483bd, 1803, 2123, 4566, 6689
RMS error: 0.0164 cents
2.3.7.13 subgroup
Subgroup: 2.3.7.13
Comma list: [-42 -2 -5 16⟩, [10 -46 29 -5⟩
Mapping: [⟨1 313 532 208], ⟨0 -439 -746 -288]]
Optimal tuning (CTE): ~1235079060111/755603996672 = 851.248
Optimal ET sequence: 320, 1163bdd, 1483bd, 1803, 2123, 4566, 6689
2.3.7.13.17 subgroup
Subgroup: 2.3.7.13.17
Comma list: 39337984/39328497, [0 -14 7 4 -3⟩, [-18 -24 14 -1 5⟩
Mapping: [⟨1 313 532 208 58], ⟨0 -439 -746 -288 -76]]
Optimal tuning (CTE): ~6144/3757 = 851.248
Optimal ET sequence: 320, 1483bd, 1803, 2123
2.3.7.13.17.19 subgroup
Subgroup: 2.3.7.13.17.19
Comma list: 10081799/10077696, 39337984/39328497, 10754912/10744731, 480024727/480020256
Mapping: [⟨1 313 532 208 58 432], ⟨0 -439 -746 -288 -76 -603]]
Optimal tuning (CTE): ~6144/3757 = 851.248
Optimal ET sequence: 320, 1483bd, 1803, 2123
Ennea
Subgroup: 2.3.7.11
Comma list: 41503/41472, 43923/43904
Gencom: [2 99/98; 41503/41472, 43923/43904]
Gencom mapping: [⟨1 14/9 0 25/9 31/9], ⟨0 2 0 2 1]]
Sval mapping: [⟨9 0 11 24], ⟨0 2 2 1]]
POL2 generator: ~99/98 = 17.6258
Optimal ET sequence: 54, 63, 72, 135, 342, 477, 1089, 1566
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: [⟨1 1 0 1 4], ⟨0 1 0 3 -1], ⟨0 0 0 1 1]]
Sval mapping: [⟨1 0 0 7], ⟨0 1 0 -4], ⟨0 0 1 1]]
POL2 tuning: ~3 = 1903.834, ~7 = 3369.872
Optimal ET sequence: 17, 36, 41, 58, 63, 104
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: [⟨1 1 0 1 4 6], ⟨0 1 0 3 -1 -4], ⟨0 0 0 1 1 1]]
Sval mapping: [⟨1 0 0 7 12], ⟨0 1 0 -4 -7], ⟨0 0 1 1 1]]
POL2 tuning: ~3 = 1903.856, ~7 = 3369.907
Optimal ET sequence: 17, 41, 46, 58, 87, 104
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 mapping: [⟨1 1 0 0 2], ⟨0 2 0 0 5]]
Sval mapping: [⟨1 1 2], ⟨0 2 5]]
POL2 generator: ~11/9 = 350.525
Optimal ET sequence: 7, 10, 17, 24, 41, 65, 89, 202, 291, 380
RMS error: 0.3021 cents
- Scales
Namo
Subgroup: 2.3.11.13
Comma list: 144/143, 243/242
Gencom: [2 11/9; 144/143 243/242]
Gencom mapping: [⟨1 1 0 0 2 4], ⟨0 2 0 0 5 -1]]
Sval mapping: [⟨1 1 2 4], ⟨0 2 5 -1]]
POL2 generator: ~11/9 = 351.488
Optimal ET sequence: 7, 10, 17, 24, 41
RMS error: 0.7038 cents
Heartland (rank 3)
- Main article: Heartland
Heartland, with a generator of ~21/19, is named for its tempering of the heartlandisma, 3971/3969. Aside from the heartlandisma, the heartland temperament tempers out 243/242 (rastma) and 1083/1078 (bihendrixma), and slices the fifth in four (the number of chambers of the heart).
Subgroup: 2.3.7.11.19
Comma list: 243/242, 1083/1078
Gencom: [2 21/19 7; 243/242 1083/1078]
Sval mapping: [⟨1 1 0 2 1], ⟨0 4 0 10 3], ⟨0 0 1 0 1]]
POL2 generator: ~21/19 = 175.2713, ~7 = 3369.3784
Optimal ET sequence: 7, 14, 27e, 34dh, 41, 89, 130
RMS error: 0.3066 cents
Reversed meantone
- Main article: Reversed meantone
Subgroup: 2.3.41
Comma list: 82/81
Gencom: [2 4/3; 82/81]
Sval mapping: [⟨1 2 7], ⟨0 -1 -4]]
POL2 generator: ~4/3 = 494.509
Optimal ET sequence: 5, 12, 17
2.3.7.41 subgroup
Subgroup: 2.3.7.41
Comma list: 64/63, 82/81
Gencom: [2 4/3; 64/63 82/81]
Sval mapping: [⟨1 2 2 7], ⟨0 -1 2 -4]]
POTE generator: ~4/3 = 490.0323
TOP generators: ~2 = 1197.2342, ~4/3 = 488.9029
Optimal ET sequence: 5, 12, 17, 22, 49
2.3.7.11.41 subgroup
Subgroup: 2.3.7.11.41
Comma list: 64/63, 82/81, 99/98
Gencom: [2 4/3; 64/63 82/81 99/98]
Sval mapping: [⟨1 2 2 1 7], ⟨0 -1 2 6 -4]]
POTE generator: ~4/3 = 492.1787
TOP generators: ~2 = 1197.9683, ~4/3 = 491.3454
Optimal ET sequence: 5, 12, 17, 22, 39d
Sematology
This temperament tempers out 4107/4096 and thus equates 2 37/32's with 4/3.
Subgroup: 2.3.37
Comma list: 4107/4096
Gencom: [2 37/32; 4107/4096]
Mapping: [⟨1 1 5], ⟨0 -2 1]]
POTE generator: ~37/32 = 249.075
Optimal ET sequence: 5, 14, 19, 24, 53, 77, 130
2.3.7.37 subgroup
Subgroup: 2.3.7.37
Comma list: 4107/4096, 259/256
Gencom: [2 37/32; 4107/4096 259/256]
Mapping: [⟨1 1 1 5], ⟨0 -2 -1 1]]
POTE generator: ~37/32 = 247.782
Optimal ET sequence: 5, 14, 19, 24, 53d
2.3.5.37 subgroup
It is difficult to extend sematology to include 5, due the 5th harmonic being quite high-complexity.
Subgroup: 2.3.5.37
Comma list: 4107/4096, 17592186044416/17562397269605
Gencom: [2 37/32; 4107/4096 17592186044416/17562397269605]
Mapping: [⟨1 1 4 5], ⟨0 -2 -8 1]]
POTE generator: ~37/32 = 251.393
Optimal ET sequence: 5, 14c, 19, 43, 62
2.3.5.7.37 subgroup
Subgroup: 2.3.5.7.37
Comma list: 4107/4096, 17592186044416/17562397269605, 259/256
Gencom: [2 37/32; 4107/4096 17592186044416/17562397269605 259/256]
Mapping: [⟨1 1 4 1 5], ⟨0 -2 -8 -1 1]]
POTE generator: ~37/32 = 251.204
Optimal ET sequence: 5, 14c, 19
Reversed mavila
Subgroup: 2.3.37
Comma list: 81/74
Gencom: [2 4/3; 81/74]
Mapping: [⟨1 1 0], ⟨0 -1 12]]
POTE generator: ~4/3 = 521.397
Optimal ET sequence: 5l, 7l, 9, 16l
Aerophore
Subgroup: 2.3.11.19
Comma list: 363/361, 729/704
Mapping: [⟨1 0 -6 -6], ⟨0 2 12 13]]
POTE generator: ~19/11 = 945.4
Optimal ET sequence: 9eehh, 14, 19, 33
Semaerophore
Subgroup: 2.3.7.11.19
Comma list: 49/48, 77/76, 729/704
Mapping: [⟨1 0 2 -6 -6], ⟨0 2 1 12 13]]
POTE generator: ~7/4 = 944.667
Optimal ET sequence: 9eehh, 14, 33d, 47deh
Superflat aka tridecimal
Superflat temperament, or alternatively, tridecimal temperament, is a diatonic-based temperament that makes 1053/1024 vanish, so 13/8 is a minor sixth, and 16/13 is a major third. The more accurate tunings for this temperament are generated by a fifth at least as flat as those of flattone, although often even flatter (such as 40edo's fifth). Superflat can be viewed as a 2.3.13 subgroup analogue of meantone and archy. Superflat diatonic scales have a character somewhere between neutral third scales (or mosh) and meantone diatonic scales.
Subgroup: 2.3.13
Sval mapping: [⟨1 1 6], ⟨0 1 -4]]
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 692.939
Optimal ET sequence: 5f, 7, 12, 19, 45f, 64f, 147bfff
2.3.11.13
Subgroup: 2.3.11.13
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 692.247
Optimal ET sequence: 7, 19, 26, 59b
Ultraflat
Ultraflat is the much more inaccurate cousin of superflat, with even flatter fifths. 27/26 is tempered out rather than 1053/1024, so 13/8 is a major sixth. These temperamenets intersect in 7edo, where major sixths and minor sixths are not distinguished.
Subgroup: 2.3.13
Sval mapping: [⟨1 1 2], ⟨0 1 3]]
Optimal tuning (CTE): ~2 = 1/1, ~3/2 = 688.391
Boethian
Subgroup: 2.3.19
Mapping: [⟨1 0 9], ⟨0 1 - 3]]
- mapping generators: ~2, ~3
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.3288
Optimal ET sequence: 5, 7, 12, 41, 53, 65, 77, 219, 296
Badness: 0.000374
Porpoise
Subgroup: 2.3.29
Comma list: 24576/24389
Mapping: [⟨1 2 5], ⟨0 3 -1]]
CTE generator: ~32/29 = 166.067
Optimal ET sequence: 7, 22, 29, 94, 123, 152j, 275jj, 427jjj
Paralimmic
Subgroup: 2.3.11
Sval mapping: [⟨1 0 4], ⟨0 3 -1]]
Optimal tuning (CTE): ~2 = 1/1, ~16/11 = 634.320
Optimal ET sequence: 11b, 13, 15, 17
Threedic
Subgroup: 2.3.13
Sval mapping: [⟨1 0 0], ⟨0 3 7]]
Optimal tuning (CTE): ~2 = 1/1, ~13/9 = 634.173
Optimal ET sequence: 11bff, 13f, 15, 17, 36, 53, 70, 123, 193, 316, 755f