No-fives subgroup temperaments
This is a collection of subgroup temperaments which omit the prime harmonic of 5.
Temperaments with a 2.3.7 gene
Semaphore
See Semaphoresmic clan #Semaphore.
Bleu
Bleu can be described as the 9 & 17 temperament in the no-5 13-limit.
Subgroup: 2.3.7
Comma list: 17496/16807
Sval mapping: [⟨1 1 2], ⟨0 5 7]]
Gencom mapping: [⟨1 1 0 2], ⟨0 5 0 7]]
- gencom: [2 54/49; 17496/16807]
Optimal tuning (POTE): ~2 = 1\1, ~54/49 = 139.848
Optimal ET sequence: 9, 17, 43, 60d
RMS error: 1.917 cents
2.3.7.11 subgroup
Subgroup: 2.3.7.11
Comma list: 99/98, 864/847
Sval mapping: [⟨1 1 2 3], ⟨0 5 7 4]]
Gencom mapping: [⟨1 1 0 2 3], ⟨0 5 0 7 4]]
- gencom: [2 12/11; 99/98 864/847]
Optimal tuning]] (POTE): ~2 = 1\1, ~12/11 = 140.005
Optimal ET sequence: 9, 17, 43, 60d
RMS error: 1.829 cents
2.3.7.11.13 subgroup
Subgroup: 2.3.7.11.13
Comma list: 78/77, 99/98, 144/143
Sval mapping: [⟨1 1 2 3 3], ⟨0 5 7 4 6]]
Gencom mapping: [⟨1 1 0 2 3 3], ⟨0 5 0 7 4 6]]
- gencom: [2 13/12; 78/77 99/98 144/143]
Optimal tuning (POTE): ~2 = 1\1, ~13/12 = 139.990
Optimal ET sequence: 17, 43, 60d
RMS error: 1.752 cents
- Music
- On a Well Worn Riff (2011) by Chris Vaisvil – blog | play – in Bleu[17]
Archy
See Archytas clan #Archy.
Supra
See Archytas clan #Supra.
Supraphon
Suhajira
Flutterpyth
Subgroup: 2.3.7.11.13.19
Comma list: 64/63, 209/208, 343/342, 364/363
Mapping: [⟨1 1 4 10 15 9], ⟨0 -1 -2 -11 -19 -8]]
Optimal tuning (CTE): ~3/2 = 713.459
Restricted to 2.3.7.11.13, this temperament is a no-5 restriction of 13-limit Ultrapyth. This temperament was created to yield blackdye tunings where aberrisma-altered 3-limit thirds become tempered 13/11~19/16 and 14/11.
Skwares
Skwares is the no-5 restriction of squares.
Subgroup: 2.3.7
Comma list: 19683/19208
Sval mapping: [⟨1 3 6], ⟨0 -4 -9]]
Gencom mapping: [⟨1 3 0 6], ⟨0 -4 0 -9]]
- gencom: [2 9/7; 19683/19208]
Optimal tuning (POTE): ~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
Sval mapping: [⟨1 3 6 7], ⟨0 -4 -9 -10]]
Gencom mapping: [⟨1 3 0 6 7], ⟨0 -4 0 -9 -10]]
- gencom: [2 9/7; 99/98 243/242]
Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 425.244
Optimal ET sequence: 5, 8, 11, 14, 17, 31, 48, 79, 127, 206bcd
RMS error: 1.099 cents
2.3.7.11.13
Subgroup: 2.3.7.11.13
Comma list: 78/77, 99/98, 243/242
Sval mapping: [⟨1 3 6 7 9], ⟨0 -4 -9 -10 -15]]
Gencom mapping: [⟨1 3 0 6 7 9], ⟨0 -4 0 -9 -10 -15]]
- gencom: [2 9/7; 78/77, 99/98, 243/242]
Optimal tuning (POTE): ~2 = 1\1, ~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
Sval mapping: [⟨1 3 6 7 3], ⟨0 -4 -9 -10 2]]
Gencom mapping: [⟨1 3 0 6 7 3], ⟨0 -4 0 -9 -10 2]]
- gencom: [2 9/7; 99/98, 144/143, 243/242]
Optimal tuning (POTE): ~2 = 1\1, ~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
Sval mapping: [⟨2 2 3 4 5], ⟨0 4 9 10 12]]
- gencom: [209/147 21/19; 99/98 243/242 363/361]
Optimal tuning (POTE): ~2 = 1\1, ~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: septimal meantone
Leapfrog
In regular 13-limit leapday, the mapping for prime 5 is very complex at +21 generator steps. Furthermore, adding prime 5 to rank-3 parapythic is arguably against the original vision of it as a 2.3.7.11.13-subgroup temperament, so avoiding prime 5 may be preferred for this reason also. This results in no-5's leapday, or leapfrog, which as aforementioned is much lower in badness, but it also allows more tunings to be used: a notable patent val tuning not appearing in the optimal ET sequence is 80edo, which is approximately the just-13's tuning (as 10edo is used as a consistent circle of ~16/13's therein), with 13/8 still tuned slightly flat so qualifying a reasonable tuning for the 2.3.13 subgroup (as evidenced by appearing in the sequence for tetris). In other words, the only reason 80edo was "disqualified" from leapday is that the mapping for prime 5 constrains the tuning range which is naturally more flexible as a no-5's 13-limit temperament, which is also a sign of leapfrog being very efficient.
Other related temperaments include leapweek and srutal.
Subgroup: 2.3.7
Comma list: 14680064/14348907
Sval mapping: [⟨1 0 -21], ⟨0 1 15]]
Gencom mapping: [⟨1 1 0 -6], ⟨0 1 0 15]]
- gencom: [2 3/2; 14680064/14348907]
- POTE: ~2 = 1\1, ~3/2 = 704.721
Optimal ET sequence: 17, 46, 63
RMS error: 0.6202 cents
2.3.7.11
Subgroup: 2.3.7.11
Comma list: 896/891, 1331/1323
Sval mapping: [⟨1 0 -21 -14], ⟨0 1 15 11]]
Gencom mapping: [⟨1 1 0 -6 -3], ⟨0 1 0 15 11]]
- gencom: [2 3/2; 896/891 1331/1323]
- POTE: ~3/2 = 704.753
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
Sval mapping: [⟨1 0 -21 -14 -9], ⟨0 1 15 11 8]]
Gencom mapping: [⟨1 1 0 -6 -3 -1], ⟨0 1 0 15 11 8]]
- gencom: [2 3/2; 169/169 352/351 364/363]
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
Sval mapping: [⟨1 0 -21 -14 -9 -5], ⟨0 1 15 11 8 6]]
Gencom mapping: [⟨1 1 0 -6 -3 -1 0 0 1], ⟨0 1 0 15 11 8 0 0 6]]
- gencom: [2 3/2; 169/169 208/207 352/351 364/363]
- POTE: ~2 = 1\1, ~3/2 = 704.729
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
Sval mapping: [⟨1 0 -21 -14 -9 -5 -38], ⟨0 1 15 11 8 6 27]]
Gencom mapping: [⟨1 1 0 -6 -3 -1 0 0 1 -11], ⟨0 1 0 15 11 8 0 0 6 27]]
- gencom: [2 3/2; 169/169 208/207 352/351 364/363]
- POTE: ~2 = 1\1, ~3/2 = 704.729
Optimal ET sequence: 17, 46, 63
- Music
- Suite for Harpsichord in A Locrian, tuning: Eb-G# in 46edo by Inthar (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
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
Related temperaments: hemififths, namo
Subgroup: 2.3.7
Comma list: 1605632/1594323
Sval mapping: [⟨1 1 -1], ⟨0 2 13]]
Gencom mapping: [⟨1 1 0 -1], ⟨0 2 0 13]]
- gencom: [2 2187/1792; 1605632/1594323]
Optimal tuning (POTE): ~2 = 1\1, ~2187/1792 = 351.485
Optimal ET sequence: 7, 17, 41, 58, 99
RMS error: 0.2344 cents
2.3.7.11
Subgroup: 2.3.7.11
Comma list: 243/242, 896/891
Sval mapping: [⟨1 1 -1 2], ⟨0 2 13 5]]
Gencom mapping: [⟨1 1 0 -1 2], ⟨0 2 0 13 5]]
- gencom: [2 11/9; 243/242 896/891]
Optimal tuning (POTE): ~2 = 1\1, ~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
Sval mapping: [⟨1 1 -1 2 4], ⟨0 2 13 5 -1]]
Gencom mapping: [⟨1 1 0 -1 2 4], ⟨0 2 0 13 5 -1]]
- gencom: [2 11/9; 144/143 243/242 364/363]
Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 351.691
Optimal ET sequence: 7, 10, 17, 24, 41, 58
RMS error: 0.7167 cents
Heartful
Related temperaments: bunya
Subgroup: 2.3.7.11.19
Comma list: 243/242, 896/891, 1083/1078
Sval mapping: [⟨1 1 -1 2 0], ⟨0 4 26 10 29]]
- gencom: [2 21/19; 243/242 896/891 1083/1078]
Optimal tuning (POTE): ~2 = 1\1, ~21/19 = 175.804
Optimal ET sequence: 34dh, 41, 116e, 157e
RMS error: 0.5360 cents
Hearts
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
Slendrismic
In slendrismic, the period (1\5) is given a very accurate interpretation of 147/128 = (3/2)/(8/7)2 = 8/7 * 1029/1024 = S7/S8, which is a significant interval as it is the "harmonic 5edostep" in that it's a rooted (/2^n) interval that approximates 1\5 very well. The generator is 1029/1024, the difference between 8/7 and 147/128 and therefore between 3/2 and (8/7)3. The temperament is named for the very "slender" generator as well as as a pun on "slendric" (which it shouldn't be confused with). One can consider this as a microtemperament counterpart to cloudy, which equates them.
Subgroup: 2.3.7
Comma list: 68719476736/68641485507
Mapping: [⟨5 8 14], ⟨0 -2 1]]
POL2 generator: ~1029/1024 = 8.9906
Optimal ET sequence: 130, 135, 265, 400, 935, 1335, 1735
RMS error: 0.0212 cents
Related temperaments: hemipental
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
Purpleheart
Subgroup: 2.3.7
Comma list: 2187/2048
Mapping: [⟨7 11 0], ⟨0 0 1]]
- mapping generators: ~9/8, ~7
Optimal tuning (CTE): ~9/8 = 1\7, ~7/4 = 968.826 (~64/63 = 59.746)
Optimal ET sequence: 7, 14, 35, 49bd
Badness: 0.0875
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)
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
Heartland (rank 3)
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
Temperaments with a 2.3.11 gene
Io
Io is a very low-complexity temperament which tempers out the undecimal quartertone 33/32. This equates very different intervals (for example, the generator itself represents both 3/2 and 16/11), and as such some consider it to be an exotemperament. It has an extremely wide generator range, but the most accurate tunings are generally inside the range of flattone temperament.
The name Io was coined by CompactStar in 2024 based on the color name ilo, prior to which it could only be termed as "undecimal temperament" with 33/32 being known as the undecimal comma.
Subgroup: 2.3.11
Comma list: 33/32
Sval mapping: [⟨1 0 5], ⟨0 1 -1]]
- mapping generators: ~2, ~3
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 692.713
Optimal ET sequence: 2, 5, 7, 12e
Badness: 0.185
Paralimmal
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
RMS error: 1.237 cents
Neutral
Namo
Huxley
Huxley, the 4 & 13 temperament in the 2.3.11.13 subgroup, extends lovecraft. Specifically it tunes the ~13/8 to exactly half of ~8/3.
Subgroup: 2.3.11.13
Comma list: 512/507, 1352/1331
Sval mapping: [⟨1 3 3 3], ⟨0 -6 2 3]]
- mapping generators: ~2, ~13/11
Optimal ET sequence: 4, 13, 17
Badness: 0.0263
Aerophore
Subgroup: 2.3.11.19
Comma list: 363/361, 729/704
Sval mapping: [⟨1 0 -6 -6], ⟨0 2 12 13]]
Optimal tuning (POTE): ~2 = 1\1, ~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
Sval mapping: [⟨1 0 2 -6 -6], ⟨0 2 1 12 13]]
Optimal tuning (POTE): ~2 = 1\1, ~7/4 = 944.667
Optimal ET sequence: 9eehh, 14, 33d, 47deh
Temperaments with a 2.3.13 gene
Superflat
Superflat 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
RMS error: 1.591 cents
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
RMS error: 4.367 cents
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
RMS error: 0.2054 cents
Temperaments with a higher-limit gene
Semitonic
Subgroup: 2.3.17
Sval mapping: [⟨2 0 5], ⟨0 1 1]]
- sval mapping generators: ~17/12, ~3
- gencom: [17/12 3; 289/288]
Optimal tuning (CTE): ~17/12 = 1\2, ~3/2 = 702.3472 (~17/16 = 102.3472)
Optimal ET sequence: 12, 58, 70, 82, 94, 106, 118, 224g
RMS error: 0.2247 cents
Gigapyth
Subgroup: 2.3.85
Comma list: 2.3.85 ⟨-40 1 6]
Sval mapping: [⟨1 4 6], ⟨0 -6 1]]
- mapping generators: ~2, ~85/64
Optimal tuning (CTE): ~2 = 1\1, ~85/64 = 483.034
Supporting ETs: 5, 47, 52, 57, 62, 67, 72, 77*, 82*, 87*, 92*, 139*, 149*, 159*
*Wart for 85
2.3.7.85 subgroup
Subgroup: 2.3.7.85
Comma list: 1029/1024, 7225/7203
Sval mapping: [⟨1 4 2 6], ⟨0 -6 2 1]]
- mapping generators: ~2, ~85/64
Optimal tuning (CTE): ~2 = 1\1, ~85/64 = 483.031
Supporting ETs: 5, 47, 52, 57, 62, 67, 72, 77*, 82*, 87*, 92*, 139*, 149*, 159*
*Wart for 85
Dog
The dog temperament is based by 2L 5s or 7L 2s scale that makes 81/76 vanish, so 19/16 is a major third. It can be viewed as a 2.3.19 subgroup analogue of mavila.
Subgroup: 2.3.19
Comma list: 81/76
Gencom: [2 4/3; 81/76]
Sval mapping: [⟨1 2 6], ⟨0 -1 -4]]
POL2 generator: ~4/3 = 521.403
Optimal ET sequence: 5h, 7, 16, 23
RMS error: 4.943 cents
Boethian
Boethian is a diatonic-based temperament that makes 513/512 vanish, so 19/16 is a minor third. It can be viewed as a 2.3.19 subgroup analogue of schismic temperament.
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
Lipsett
Lipsett temperament is a pleasantly melodic little temperament with a highly useable 5-tone and 9-tone mos. It is audibly similar to semaphore temperament, so it could be thought of as semaphore but for the 23rd harmonic instead of the 7th. It is named for Arthur Lipsett, the director off the Canadian short film ’21-87’. Leia’s prison cell in Star Wars is numbered ‘2187’, as a nod to the influence the film had on George Lucas.
Subgroup: 2.3.23
Sval mapping: [⟨1 0 -1], ⟨0 2 7]]
Optimal tuning (CTE): ~2 = 1\1, ~46/27 = 948.526
Optimal ET sequence: 5, 14, 19, 43, 62i, 81i
Badness (Smith): 8.998 × 10-3
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
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
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