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

Badness (Dirichlet): 0.826

2.3.5.31

Fascinatingly, essentially the only simple and accurate extension that preserves the occurrence of sensipent's tempered 5-limit structure in such large edos as 539 is the one to prime 31 by interpreting the generator accurately as 40/31~31/24 by tempering S31 = 961/960, so that the large 31-limit quarter-tones 32/31 and 31/30 are equated, as sensipent splits 16/15 into two equal parts. For a less sparse subgroup present in smaller edo tunings like 111edo at the cost of slight accuracy, see the extension to the 2.3.5.11.17.31 subgroup #Sensible.

Subgroup: 2.3.5.31

Comma list: 78732/78125, 961/960

Mapping: [⟨1 6 8 10], ⟨0 -7 -9 -8]]

Optimal ET sequence: 8, 11c, 19, 46, 65, 344, 409, 474, 539, 604c

Optimal tuning (CTE): 2 = 1\1, ~31/28 = 443.050

Badness (Dirichlet): 0.243

Sensible

An extension of sensipent to prime 11 of dubious canonicity (but significantly higher accuracy than sensi) interprets the generator as 165/128~128/99 by tempering S9/S10 so that 11/8 is reached as (10/9)³. This extension is very strong as supported by the optimal ET sequence going very far and as supported by another observation: that it is equivalent to tempering the semiporwellisma which is equal to S31 * S32² (thus forming the S-expression-based comma list). The equivalence of the aforementioned lopsided comma also implies that this temperament equates (33/32)² with 16/15 as well as that a natural extension to prime 31 exists through {S31, S32}, which we will see is very accurate, but this itself suggests that an extension to prime 17 is reasonably accurate through tempering S33 so that a slightly sharp ~22/17 is equated with the generator.

Subgroup: 2.3.5.11

Comma list: 8019/8000, 16384/16335

Mapping: [⟨1 6 8 -6], ⟨0 -7 -9 15]]

Optimal ET sequence: 19, 46, 65, 176, 241, 306

Optimal tuning (CTE): 2 = 1\1, ~128/99 = 443.115

Badness (Dirichlet): 0.728

2.3.5.11.17

The aforementioned extension to prime 17 through tempering S33 is equivalent to the one by tempering S16 = 256/255 = (22/17)/(165/128).

Subgroup: 2.3.5.11.17

Comma list: 8019/8000, 16384/16335, 256/255

Mapping: [⟨1 6 8 -6 -6], ⟨0 -7 -9 15 16]]

Optimal ET sequence: 19, 46, 65, 111, 176g

Optimal tuning (CTE): 2 = 1\1, ~22/17 = 443.188

Badness (Dirichlet): 0.639

2.3.5.11.17.31

Sensible uses the accurate mapping of prime 31 in sensipent, so that the sensible generator serves many roles in subgroup harmony without the need for interpreting it as high-damage ~9/7 or ~13/10 intervals. Its S-expression-based comma list is {(S16,) S9/S10, S31, S32, S33} implying also tempering 496/495 = S31 * S32 and 528/527 = S32 * S33 as well as 16337/16335 = S31/S33 = (34/30)/(33/31)² = (17/15)/(33/31)². A notable patent val tuning not appearing in the optimal ET sequence is 157edo.

Subgroup: 2.3.5.11.17.31

Comma list: 8019/8000, 16384/16335, 256/255, 961/960

Mapping: [⟨1 6 8 -6 -6 10], ⟨0 -7 -9 15 16 -8]]

Optimal ET sequence: 19, 46, 65, 111, 176g, 287cg

Optimal tuning (CTE): 2 = 1\1, ~31/24 = 443.185

Badness (Dirichlet): 0.519

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