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

Main article: 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]]

Optimal tunings:

• CTE: ~2 = 1\1, ~9/7 = 443.3166
• POTE: ~2 = 1\1, ~9/7 = 443.383

Minimax tuning:

• 7-odd-limit: ~9/7 = [2/13 0 0 1/13⟩

eigenmonzo (unchanged-interval) basis: 2.7

• 9-odd-limit: ~9/7 = [1/5 2/5 -1/5 0⟩

eigenmonzo (unchanged-interval) basis: 2.9/5

Tuning ranges:

• 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