Sensipent family
Temperaments of the sensipent family temper out the sensipent comma, 78732/78125, also known as medium semicomma. The head of this family is sensipent i.e. the 5-limit version of sensi, generated by the naiadic interval of tempered 162/125, seven make harmonic 6 and nine make harmonic 10.
The second comma of the comma list determines which 7-limit family member we are looking at. Sensi adds 126/125. Sensei adds 225/224. Warrior adds 5120/5103. These all use the same nominal generator as sensipent.
Bison adds 6144/6125 with a semioctave period. Subpental adds 3136/3125 or 19683/19600 with a generator of ~56/45; two generator steps make the original. Trisensory adds 1728/1715 with a 1/3-octave period. Heinz adds 1029/1024 with a generator of ~48/35; three make the original. Catafourth adds 2401/2400 with a generator of ~250/189; four make the original. Finally, browser adds 16875/16807 with a generator of ~49/45; five make the original.
Temperaments discussed elsewhere include:
- Catafourth → Breedsmic temperaments (+2401/2400)
- Browser → Mirkwai clan (+16875/16807)
Considered below are sensi, sensei, warrior, bison, subpental, trisensory and heinz.
Sensipent
Subgroup: 2.3.5
Comma list: 78732/78125
Mapping: [⟨1 6 8], ⟨0 -7 -9]]
- mapping generators: ~2, ~125/81
Optimal tuning (POTE): ~2 = 1\1, 162/125 = 443.058
Optimal ET sequence: 8, 19, 46, 65, 539, 604c, 669c, 734c, 799c, 864c, 929c
Badness: 0.035220
Sensi
Sensi tempers out 245/243, 686/675 and 4375/4374 in addition to 126/125, and can be described as the 19 & 27 temperament. It has as a generator half the size of a slightly wide major sixth, which gives an interval sharp of 9/7 and flat of 13/10, both of which can be used to identify it, as 2.3.5.7.13 sensi (sensation) tempers out 91/90. 22/17, in the middle, is even closer to the generator. 46edo is an excellent sensi tuning, and mos scales of size 8, 11, 19 and 27 are available.
Septimal sensi
Subgroup: 2.3.5.7
Comma list: 126/125, 245/243
Mapping: [⟨1 6 8 11], ⟨0 -7 -9 -13]]
- mapping generators: ~2, ~14/9
Wedgie: ⟨⟨7 9 13 -2 1 5]]
- 7-odd-limit: ~9/7 = [2/13 0 0 1/13⟩
- 9-odd-limit: ~9/7 = [1/5 2/5 -1/5 0⟩
- 7-odd-limit diamond monotone: ~9/7 = [442.105, 450.000] (7\19 to 3\8)
- 9-odd-limit diamond monotone: ~9/7 = [442.105, 444.444] (7\19 to 10\27)
- 7-odd-limit diamond tradeoff: ~9/7 = [442.179, 445.628]
- 9-odd-limit diamond tradeoff: ~9/7 = [435.084, 445.628]
Algebraic generator: The real root of x5 + x4 - 4x2 + x - 1, at 443.3783 cents.
Optimal ET sequence: 19, 27, 46
Badness: 0.025622
2.3.5.7.13 subgroup (sensation)
Subgroup: 2.3.5.7.13
Comma list: 91/90, 126/125, 169/168
Sval mapping: [⟨1 6 8 11 10], ⟨0 -7 -9 -13 -10]]
Gencom mapping: [⟨1 6 8 11 0 10], ⟨0 -7 -9 -13 0 -10]]
- gencom: [2 14/9; 91/90 126/125 169/168]
Optimal tuning (CTE): ~2 = 1\1, ~9/7 = 443.4016
Optimal ET sequence: 19, 27, 46, 111df
Sensor
Subgroup: 2.3.5.7.11
Comma list: 126/125, 245/243, 385/384
Mapping: [⟨1 6 8 11 -6], ⟨0 -7 -9 -13 15]]
Wedgie: ⟨⟨7 9 13 -15 -2 1 -48 5 -66 -87]]
Optimal tunings:
- CTE: ~2 = 1\1, ~9/7 = 443.2987
- POTE: ~2 = 1\1, ~9/7 = 443.294
Optimal ET sequence: 19, 27, 46, 111d
Badness: 0.037942
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 91/90, 126/125, 169/168, 385/384
Mapping: [⟨1 6 8 11 -6 10], ⟨0 -7 -9 -13 15 -10]]
Wedgie: ⟨⟨7 9 13 -15 10 -2 1 -48 -10 5 -66 -10 -87 -20 90]]
Optimal tunings:
- CTE: ~2 = 1\1, ~9/7 = 443.3658
- POTE: ~2 = 1\1, ~9/7 = 443.321
Optimal ET sequence: 19, 27, 46, 111df
Badness: 0.025575
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 91/90, 126/125, 154/153, 169/168, 256/255
Mapping: [⟨1 6 8 11 -6 10 -6], ⟨0 -7 -9 -13 15 -10 16]]
Optimal tunings:
- CTE: ~2 = 1\1, ~9/7 = 443.3775
- POTE: ~2 = 1\1, ~9/7 = 443.365
Optimal ET sequence: 19, 27, 46
Badness: 0.022908
Sensus
Subgroup: 2.3.5.7.11
Comma list: 126/125, 176/175, 245/243
Mapping: [⟨1 6 8 11 23], ⟨0 -7 -9 -13 -31]]
Wedgie: ⟨⟨7 9 13 31 -2 1 25 5 41 42]]
Optimal tunings:
- CTE: ~2 = 1\1, ~9/7 = 443.4783
- POTE: ~2 = 1\1, ~9/7 = 443.626
Optimal ET sequence: 19e, 27e, 46, 119c
Badness: 0.029486
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 91/90, 126/125, 169/168, 352/351
Mapping: [⟨1 6 8 11 23 10], ⟨0 -7 -9 -13 -31 -10]]
Wedgie: ⟨⟨7 9 13 31 10 -2 1 25 -10 5 41 -10 42 -20 -80]]
Optimal tunings:
- CTE: ~2 = 1\1, ~9/7 = 443.5075
- POTE: ~2 = 1\1, ~9/7 = 443.559
Optimal ET sequence: 19e, 27e, 46
Badness: 0.020789
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 91/90, 126/125, 136/135, 154/153, 169/168
Mapping: [⟨1 6 8 11 23 10 23], ⟨0 -7 -9 -13 -31 -10 -30]]
Optimal tunings:
- CTE: ~2 = 1\1, ~9/7 = 443.5050
- POTE: ~2 = 1\1, ~9/7 = 443.551
Optimal ET sequence: 19eg, 27eg, 46
Badness: 0.016238
Sensis
Subgroup: 2.3.5.7.11
Comma list: 56/55, 100/99, 245/243
Mapping: [⟨1 6 8 11 6], ⟨0 -7 -9 -13 -4]]
Wedgie: ⟨⟨7 9 13 4 -2 1 -18 5 -22 -34]]
Optimal tunings:
- CTE: ~2 = 1\1, ~9/7 = 443.1886
- POTE: ~2 = 1\1, ~9/7 = 443.962
Optimal ET sequence: 8d, 19, 27e
Badness: 0.028680
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 56/55, 78/77, 91/90, 100/99
Mapping: [⟨1 6 8 11 6 10], ⟨0 -7 -9 -13 -4 -10]]
Wedgie: ⟨⟨7 9 13 4 10 -2 1 -18 -10 5 -22 -10 -34 -20 20]]
Optimal tunings:
- CTE: ~2 = 1\1, ~9/7 = 443.2863
- POTE: ~2 = 1\1, ~9/7 = 443.945
Optimal ET sequence: 8d, 19, 27e
Badness: 0.020017
Sensa
Subgroup: 2.3.5.7.11
Comma list: 55/54, 77/75, 99/98
Mapping: [⟨1 6 8 11 11], ⟨0 -7 -9 -13 -12]]
Optimal tunings:
- CTE: ~2 = 1\1, ~9/7 = 443.7814
- POTE: ~2 = 1\1, ~9/7 = 443.518
Optimal ET sequence: 8d, 19e, 27
Badness: 0.036835
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 55/54, 66/65, 77/75, 143/140
Mapping: [⟨1 6 8 11 11 11], ⟨0 -7 -9 -13 -12 -11]]
Optimal tunings:
- CTE: ~2 = 1\1, ~9/7 = 443.7877
- POTE: ~2 = 1\1, ~9/7 = 443.506
Optimal ET sequence: 8d, 19e, 27
Badness: 0.023258
Bisensi
Subgroup: 2.3.5.7.11
Comma list: 121/120, 126/125, 245/243
Mapping: [⟨2 5 7 9 9], ⟨0 -7 -9 -13 -8]]
- mapping generators: ~99/70, ~11/10
Optimal tunings:
- CTE: ~99/70 = 1\2, ~11/10 = 156.6312
- POTE: ~99/70 = 1\2, ~11/10 = 156.692
Optimal ET sequence: 8d, …, 38d, 46
Badness: 0.041723
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 91/90, 121/120, 126/125, 169/168
Mapping: [⟨2 5 7 9 9 10], ⟨0 -7 -9 -13 -8 -10]]
Optimal tunings:
- CTE: ~55/39 = 1\2, ~11/10 = 156.5584
- POTE: ~55/39 = 1\2, ~11/10 = 156.725
Optimal ET sequence: 8d, …, 38df, 46
Badness: 0.026339
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 91/90, 121/120, 126/125, 154/153, 169/168
Mapping: [⟨2 5 7 9 9 10 10], ⟨0 -7 -9 -13 -8 -10 -7]]
Optimal tunings:
- CTE: ~17/12 = 1\2, ~11/10 = 156.5534
Optimal ET sequence: 8d, …, 38df, 46
Badness: 0.0188
Hemisensi
Subgroup: 2.3.5.7.11
Comma list: 126/125, 243/242, 245/242
Mapping: [⟨1 13 17 24 32], ⟨0 -14 -18 -26 -35]]
- mapping generators: ~2, ~44/25
Optimal tunings:
- CTE: ~2 = 1\1, ~25/22 = 221.5981
- POTE: ~2 = 1\1, ~25/22 = 221.605
Optimal ET sequence: 27e, 38d, 65
Badness: 0.048714
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 91/90, 126/125, 169/168, 243/242
Mapping: [⟨1 13 17 24 32 30], ⟨0 -14 -18 -26 -35 -30]]
Optimal tunings:
- CTE: ~2 = 1\1, ~25/22 = 221.6333
- POTE: ~2 = 1\1, ~25/22 = 221.556
Optimal ET sequence: 27e, 38df, 65f
Badness: 0.033016
Sensei
Subgroup: 2.3.5.7
Comma list: 225/224, 78732/78125
Mapping: [⟨1 6 8 23], ⟨0 -7 -9 -32]]
Wedgie: ⟨⟨7 9 32 -2 31 49]]
Optimal tuning (POTE): ~2 = 1\1, ~162/125 = 442.755
Optimal ET sequence: 19, 65d, 84, 103, 187, 290b
Badness: 0.059218
Warrior
Subgroup: 2.3.5.7
Comma list: 5120/5103, 78732/78125
Mapping: [⟨1 6 8 -18], ⟨0 -7 -9 33]]
Wedgie: ⟨⟨7 9 -33 -2 -72 -102]]
Optimal tuning (POTE): ~2 = 1\1, ~162/125 = 443.289
Optimal ET sequence: 46, 111, 157, 268cd
Badness: 0.118239
11-limit
Subgroup: 2.3.5.7.11
Comma list: 176/175, 1331/1323, 5120/5103
Mapping: [⟨1 6 8 -18 -6], ⟨0 -7 -9 33 15]]
Optimal tuning (POTE): ~2 = 1\1, ~128/99 = 443.274
Optimal ET sequence: 46, 65d, 111, 268cd, 379cdd
Badness: 0.046383
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 176/175, 351/350, 847/845, 1331/1323
Mapping: [⟨1 6 8 -18 -6 -19], ⟨0 -7 -9 33 15 36]]
Optimal tuning (POTE): ~2 = 1\1, ~84/65 = 443.270
Optimal ET sequence: 46, 65d, 111, 268cd, 379cddf
Badness: 0.028735
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 176/175, 256/255, 351/350, 442/441, 715/714
Mapping: [⟨1 6 8 -18 -6 -19 -6], ⟨0 -7 -9 33 15 36 16]]
Optimal tuning (POTE): ~2 = 1\1, ~22/17 = 443.270
Optimal ET sequence: 46, 65d, 111, 268cdg, 379cddfg
Badness: 0.018105
Bison
Subgroup: 2.3.5.7
Comma list: 6144/6125, 78732/78125
Mapping: [⟨2 5 7 3], ⟨0 -7 -9 10]]
- mapping generators: ~567/400, ~35/32
Wedgie: ⟨⟨14 18 -20 -4 -71 -97]]
Optimal tuning (POTE): ~567/400 = 1\2, ~35/32 = 156.925
Optimal ET sequence: 8, 38, 46, 84, 130
Badness: 0.070375
11-limit
Subgroup: 2.3.5.7.11
Comma list: 441/440, 6144/6125, 8019/8000
Mapping: [⟨2 5 7 3 3], ⟨0 -7 -9 10 15]]
Optimal tuning (POTE): ~99/70 = 1\2, ~35/32 = 156.883
Optimal ET sequence: 46, 84, 130, 306, 436ce
Badness: 0.037132
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 351/350, 364/363, 441/440, 10985/10976
Mapping: [⟨2 5 7 3 3 4], ⟨0 -7 -9 10 15 13]]
Optimal tuning (POTE): ~55/39 = 1\2, ~35/32 = 156.904
Optimal ET sequence: 46, 84, 130, 566ce, 596cef
Badness: 0.023504
Subpental
Subgroup: 2.3.5.7
Comma list: 3136/3125, 19683/19600
Mapping: [⟨1 6 8 17], ⟨0 -14 -18 -45]]
Wedgie: ⟨⟨14 18 45 -4 32 54]]
Optimal tuning (POTE): ~2 = 1\1, ~56/45 = 378.467
Optimal ET sequence: 19, 111, 130, 929c, 1059c, 1189bc, 1319bc
Badness: 0.054303
11-limit
Subgroup: 2.3.5.7.11
Comma list: 540/539, 3136/3125, 8019/8000
Mapping: [⟨1 6 8 17 -6], ⟨0 -14 -18 -45 30]]
Optimal tuning (POTE): ~2 = 1\1, ~56/45 = 378.440
Optimal ET sequence: 19, 111, 130, 241, 371ce, 501cde, 872cde
Badness: 0.045352
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 351/350, 540/539, 676/675, 3136/3125
Mapping: [⟨1 6 8 17 -6 16], ⟨0 -14 -18 -45 30 -39]]
Optimal tuning (POTE): ~2 = 1\1, ~56/45 = 378.437
Optimal ET sequence: 19, 111, 130, 241, 371ce
Badness: 0.023940
Heinz
A notable tuning of heinz not shown below for those who like 19edo's representation of the 5-limit is 57edo (= 103 - 46).
Subgroup: 2.3.5.7
Comma list: 1029/1024, 78732/78125
Mapping: [⟨1 13 17 -1], ⟨0 -21 -27 7]]
- mapping generators: ~2, ~35/24
Wedgie: ⟨⟨21 27 -7 -6 -70 -92]]
Optimal tuning (POTE): ~2 = 1\1, ~48/35 = 546.815
Optimal ET sequence: 46, 103, 149, 699bdd
Badness: 0.115385
11-limit
Subgroup: 2.3.5.7.11
Comma list: 385/384, 441/440, 78732/78125
Mapping: [⟨1 13 17 -1 4], ⟨0 -21 -27 7 -1]]
Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 547.631
Optimal ET sequence: 46, 103, 149, 252e, 401bdee
Badness: 0.042412
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 351/350, 385/384, 441/440, 847/845
Mapping: [⟨1 13 17 -1 4 -5], ⟨0 -21 -27 7 -1 16]]
Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 547.629
Optimal ET sequence: 46, 103, 149, 252ef, 401bdeef
Badness: 0.025779
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 273/272, 351/350, 385/384, 441/440, 847/845
Mapping: [⟨1 13 17 -1 4 -5 3], ⟨0 -21 -27 7 -1 16 2]]
Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 547.635
Optimal ET sequence: 46, 103, 149, 252ef, 401bdeef
Badness: 0.018479
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 171/170, 209/208, 351/350, 385/384, 441/440, 969/968
Mapping: [⟨1 13 17 -1 4 -5 3 -5], ⟨0 -21 -27 7 -1 16 2 17]]
Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 547.614
Optimal ET sequence: 46, 103h, 149h, 252efhh
Badness: 0.019005
Trisensory
Subgroup: 2.3.5.7
Comma list: 1728/1715, 78732/78125
Mapping: [⟨3 4 6 8], ⟨0 7 9 4]]
Wedgie: ⟨⟨21 27 12 -6 -40 -48]]
Optimal tuning (POTE): ~63/50 = 1\3, ~36/35 = 43.147
Optimal ET sequence: 27, 57, 84, 111, 195d, 306d
Badness: 0.089740
11-limit
Subgroup: 2.3.5.7.11
Comma list: 176/175, 540/539, 78732/78125
Mapping: [⟨3 4 6 8 8], ⟨0 7 9 4 22]]
Optimal tuning (POTE): ~63/50 = 1\3, ~36/35 = 43.292
Optimal ET sequence: 27e, 84e, 111
Badness: 0.058413
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 176/175, 351/350, 540/539, 9295/9261
Mapping: [⟨3 4 6 8 8 11], ⟨0 7 9 4 22 1]]
Optimal tuning (POTE): ~49/39 = 1\3, ~36/35 = 43.288
Optimal ET sequence: 27e, 84e, 111
Badness: 0.034829
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 176/175, 351/350, 442/441, 540/539, 715/714
Mapping: [⟨3 4 6 8 8 11 10], ⟨0 7 9 4 22 1 21]]
Optimal tuning (POTE): ~49/39 = 1\3, ~36/35 = 43.276
Optimal ET sequence: 27eg, 84e, 111
Badness: 0.024120
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 176/175, 286/285, 324/323, 351/350, 400/399, 476/475
Mapping: [⟨3 4 6 8 8 11 10 12], ⟨0 7 9 4 22 1 21 7]]
Optimal tuning (POTE): ~49/39 = 1\3, ~36/35 = 43.292
Optimal ET sequence: 27eg, 84e, 111
Badness: 0.018466