Sensipent family: Difference between revisions

Revision as of 03:32, 21 October 2024

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. Two generators make 5/3, seven make harmonic 6 and nine make harmonic 10. Its ploidacot is beta-heptacot (pergen (P8, ccP5/7)) and its color name is Sepguti.

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 are all strong extensions that use the same period and 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 -1 -1], ⟨0 7 9]]

mapping generators: ~2, ~162/125

Optimal tuning (POTE): ~2 = 1200.000, ~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 -1 -1 2], ⟨0 7 9 8]]

mapping generators: ~2, ~31/24

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

Optimal tuning (CTE): ~2 = 1200.000, ~31/24 = 443.050

Badness (Dirichlet): 0.243

Sendai

See also: User:VIxen/Table_of_sensipent_intervals

Sendai is an accurate extension of (2.3.5.31) sensipent to primes 23 and 29 found by VIxen. It is named after the body of acquis designed to prevent disaster risk and improve civil protection through international cooperation and after the city in Japan of the same name where it was signed (and where an international music competition is held).

Subgroup: 2.3.5.23.29.31

Comma list: 465/464, 576/575, 621/620, 729/725

Mapping: [⟨1 -1 -1 6 -4 2], ⟨0 7 9 -4 24 8]]

Optimal ET sequence: 19, 46j, 65, 149, 363j

Optimal tuning (CTE): ~2 = 1200.000, ~31/24 = 442.989

Badness (Dirichlet): 0.283

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 -1 -1 9], ⟨0 7 9 -15]]

mapping generators: ~2, ~128/99

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

Optimal tuning (CTE): ~2 = 1200.000, ~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 -1 -1 9 10], ⟨0 7 9 -15 -16]]

mapping generators: ~2, ~22/17

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

Optimal tuning (CTE): ~2 = 1200.000, ~22/17 = 443.188

Badness (Dirichlet): 0.639

2.3.5.11.17.23

Subgroup: 2.3.5.11.17.23

Comma list: 8019/8000, 16384/16335, 256/255, 576/575

Mapping: [⟨1 -1 -1 9 10 6], ⟨0 7 9 -15 -16 -4]]

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

Optimal tuning (CTE): ~2 = 1200.000, ~22/17 = 443.185

Badness (Dirichlet): 0.555

2.3.5.11.17.23.31

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,) S23, S24, 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: 2.3.5.11.17.23.31

Comma list: 8019/8000, 16384/16335, 256/255, 576/575, 961/960

Mapping: [⟨1 -1 -1 9 10 6 2], ⟨0 7 9 -15 -16 -4 8]]

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

Optimal tuning (CTE): ~2 = 1200.000, ~22/17 = 443.183

Badness (Dirichlet): 0.490

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: [⟨1-1 -1 -2], ⟨0 7 9 13]]

mapping generators: ~2, ~9/7

Wedgie: ⟨⟨ 7 9 13 -2 1 5 ]]

Optimal tunings:

CTE: ~2 = 1200.000, ~9/7 = 443.3166
POTE: ~2 = 1200.000, ~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

Mapping: [⟨1 -1 -1 -2 0], ⟨0 7 9 13 10]]

mapping generators: ~2, ~9/7

Optimal tuning (CTE): ~2 = 1200.000, ~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 -1 -1 -2 9], ⟨0 7 9 13 -15]]

mapping generators: ~2, ~9/7

Wedgie: ⟨⟨ 7 9 13 -15 -2 1 -48 5 -66 -87 ]]

Optimal tunings:

CTE: ~2 = 1200.000, ~9/7 = 443.2987
POTE: ~2 = 1200.000, ~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 -1 -1 -2 9 0], ⟨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 = 1200.000, ~9/7 = 443.3658
POTE: ~2 = 1200.000, ~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 -1 -1 -2 9 0 10], ⟨0 7 9 13 -15 10 -16]]

Optimal tunings:

CTE: ~2 = 1200.000, ~9/7 = 443.3775
POTE: ~2 = 1200.000, ~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 -1 -1 -2 -8], ⟨0 7 9 13 31]]

mapping generators: ~2, ~9/7

Wedgie: ⟨⟨ 7 9 13 31 -2 1 25 5 41 42 ]]

Optimal tunings:

CTE: ~2 = 1200.000, ~9/7 = 443.4783
POTE: ~2 = 1200.000, ~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 -1 -1 -2 -8 0], ⟨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 = 1200.000, ~9/7 = 443.5075
POTE: ~2 = 1200.000, ~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 -1 -1 -2 -8 0 -7], ⟨0 7 9 13 31 10 30]]

Optimal tunings:

CTE: ~2 = 1200.000, ~9/7 = 443.5050
POTE: ~2 = 1200.000, ~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 -1 -1 -2 2], ⟨0 7 9 13 4]]

mapping generators: ~2, ~9/7

Wedgie: ⟨⟨ 7 9 13 4 -2 1 -18 5 -22 -34 ]]

Optimal tunings:

CTE: ~2 = 1200.000, ~9/7 = 443.1886
POTE: ~2 = 1200.000, ~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 -1 -1 -2 2 0], ⟨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 = 1200.000, ~9/7 = 443.2863
POTE: ~2 = 1200.000, ~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 -1 -1 -2 -1], ⟨0 7 9 13 12]]

mapping generators: ~2, ~9/7

Optimal tunings:

CTE: ~2 = 1200.000, ~9/7 = 443.7814
POTE: ~2 = 1200.000, ~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 -1 -1 -2 -1 0], ⟨0 7 9 13 12 11]]

Optimal tunings:

CTE: ~2 = 1200.000, ~9/7 = 443.7877
POTE: ~2 = 1200.000, ~9/7 = 443.506

Optimal ET sequence: 8d, 19e, 27

Badness: 0.023258

Bisensi

Bisensi has a 1/2-octave period. Its ploidacot is diploid delta-heptacot (pergen (P8/2, ccP5/7)).

Subgroup: 2.3.5.7.11

Comma list: 121/120, 126/125, 245/243

Mapping:

common form: [⟨2 -2 -2 -4 1], ⟨0 7 9 13 8]]

mapping generators: ~99/70, ~9/7

mingen form: [⟨2 5 7 9 9], ⟨0 -7 -9 -13 -8]]

mapping generators: ~99/70, ~11/10

Optimal tunings:

CTE: ~99/70 = 600.000, ~9/7 = 443.3688 (~11/10 = 156.6312)
POTE: ~99/70 = 600.000, ~9/7 = 443.308 (~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:

common form: [⟨2 -2 -2 -4 1 0], ⟨0 7 9 13 8 10]]

mapping generators: ~99/70, ~9/7

mingen form: [⟨2 5 7 9 9 10], ⟨0 -7 -9 -13 -8 -10]]

mapping generators: ~99/70, ~11/10

Optimal tunings:

CTE: ~55/39 = 600.000, ~9/7 = 443.4416, ~11/10 = 156.5584
POTE: ~55/39 = 600.000, ~9/7 = 443.275, ~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:

common form: [⟨2 -2 -2 -4 1 0 3], ⟨0 7 9 13 8 10 7]]

mapping generators: ~99/70, ~9/7

mingen form: [⟨2 5 7 9 9 10 10], ⟨0 -7 -9 -13 -8 -10 -7]]

mapping generators: ~99/70, ~11/10

Optimal tunings:

CTE: ~17/12 = 600.000, ~9/7 = 443.4466 (~11/10 = 156.5534)

Optimal ET sequence: 8d, …, 38df, 46

Badness: 0.0188

Hemisensi

Hemisensi splits the ~9/7 generator in two, each for ~25/22. Its ploidacot is beta-tetradecacot (pergen (P8, ccP5/14)).

Subgroup: 2.3.5.7.11

Comma list: 126/125, 243/242, 245/242

Mapping: [⟨1 -1 -1 -2 -3], ⟨0 14 18 26 35]]

mapping generators: ~2, ~25/22

Optimal tunings:

CTE: ~2 = 1200.000, ~25/22 = 221.5981
POTE: ~2 = 1200.000, ~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 -1 -1 -2 -3 0], ⟨0 14 18 26 35 20]]

Optimal tunings:

CTE: ~2 = 1200.000, ~25/22 = 221.6333
POTE: ~2 = 1200.000, ~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 -1 -1 -9], ⟨0 7 9 32]]

mapping generators: ~2, ~162/125

Wedgie: ⟨⟨ 7 9 32 -2 31 49 ]]

Optimal tuning (POTE): ~2 = 1200.000, ~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 -1 -1 15], ⟨0 7 9 -33]]

mapping generators: ~2, ~162/125

Wedgie: ⟨⟨ 7 9 -33 -2 -72 -102 ]]

Optimal tuning (POTE): ~2 = 1200.000, ~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 -1 -1 15 9], ⟨0 7 9 -33 -15]]

mapping generators: ~2, ~128/99

Optimal tuning (POTE): ~2 = 1200.000, ~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 -1 -1 15 9 17], ⟨0 7 9 -33 -15 -36]]

mapping generators: ~2, ~84/65

Optimal tuning (POTE): ~2 = 1200.000, ~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 -1 -1 15 9 17 10], ⟨0 7 9 -33 -15 -36 -16]]

mapping generators: ~2, ~22/17

Optimal tuning (POTE): ~2 = 1200.000, ~22/17 = 443.270

Optimal ET sequence: 46, 65d, 111, 268cdg, 379cddfg

Badness: 0.018105

Bison

Bison has a 1/2-octave period. Its ploidacot is diploid delta-heptacot (pergen (P8/2, ccP5/7)). Related page: Bison/Eliora's Approach.

Subgroup: 2.3.5.7

Comma list: 6144/6125, 78732/78125

Mapping:

common form: [⟨2 -2 -2 13], ⟨0 7 9 -10]]

mapping generators: ~567/400, ~162/125

mingen form: [⟨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 = 600.000, ~162/125 = 443.075 (~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:

common form: [⟨2 -2 -2 13 18], ⟨0 7 9 -10 -15]]

mapping generators: ~567/400, ~162/125

mingen form: [⟨2 5 7 3 3], ⟨0 -7 -9 10 15]]

mapping generators: ~567/400, ~35/32

Optimal tuning (POTE): ~99/70 = 600.000, ~162/125 = 443.117 (~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:

common form: [⟨2 -2 -2 13 18 17], ⟨0 7 9 -10 -15 -13]]

mapping generators: ~55/39, ~162/125

mingen form: [⟨2 5 7 3 3 4], ⟨0 -7 -9 10 15 13]]

mapping generators: ~55/39, ~35/32

Optimal tuning (POTE): ~55/39 = 600.000, ~162/125 = 443.096 (~35/32 = 156.904)

Optimal ET sequence: 46, 84, 130, 566ce, 596cef

Badness: 0.023504

Subpental

Subpental splits the generator ~14/9 in two. Its ploidacot is theta-tetradecacot (pergen (P8, c⁴P4/14)).

Subgroup: 2.3.5.7

Comma list: 3136/3125, 19683/19600

Mapping: [⟨1 6 8 17], ⟨0 -14 -18 -45]]

mapping generators: ~2, ~56/45

Wedgie: ⟨⟨ 14 18 45 -4 32 54 ]]

Optimal tuning (POTE): ~2 = 1200.000, ~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 = 1200.000, ~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 = 1200.000, ~56/45 = 378.437

Optimal ET sequence: 19, 111, 130, 241, 371ce

Badness: 0.023940

Heinz

Heinz splits the generator ~18/7 in three. Its ploidacot is theta-21-cot (pergen (P8, c⁹P5/21)). A notable tuning of heinz not shown below for those who like 19edo's representation of the 5-limit is 57edo (57 = 103 - 46).

Subgroup: 2.3.5.7

Comma list: 1029/1024, 78732/78125

Mapping: [⟨1 -8 -10 6], ⟨0 21 27 -7]]

mapping generators: ~2, ~48/35

Wedgie: ⟨⟨ 21 27 -7 -6 -70 -92 ]]

Optimal tuning (POTE): ~2 = 1200.000, ~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 -8 -10 6 3], ⟨0 21 27 -7 1]]

mapping generators: ~2, ~11/8

Optimal tuning (POTE): ~2 = 1200.000, ~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 -8 -10 6 3 11], ⟨0 21 27 -7 1 -16]]

Optimal tuning (POTE): ~2 = 1200.000, ~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 -8 -10 6 3 11 5], ⟨0 21 27 -7 1 -16 -2]]

Optimal tuning (POTE): ~2 = 1200.000, ~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 -8 -10 6 3 11 5 12], ⟨0 21 27 -7 1 -16 -2 -17]]

Optimal tuning (POTE): ~2 = 1200.000, ~11/8 = 547.614

Optimal ET sequence: 46, 103h, 149h, 252efhh

Badness: 0.019005

Trisensory

Trisensory has 1/3-octave period. Its ploidacot is triploid digamma-heptacot (pergen (P8/3, M6/21)).

Subgroup: 2.3.5.7

Comma list: 1728/1715, 78732/78125

Mapping: [⟨3 4 6 8], ⟨0 7 9 4]]

mapping generators: ~63/50, ~36/35

Wedgie: ⟨⟨ 21 27 12 -6 -40 -48 ]]

Optimal tuning (POTE): ~63/50 = 400.000, ~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 = 400.000, ~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]]

mapping generators: ~49/39, ~36/35

Optimal tuning (POTE): ~49/39 = 400.000, ~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 = 400.000, ~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 = 400.000, ~36/35 = 43.292

Optimal ET sequence: 27eg, 84e, 111

Badness: 0.018466

@@ Line 26: / Line 26: @@
 : mapping generators: ~2, ~162/125
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~162/125 = 443.058
+[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~162/125 = 443.058
 {{Optimal ET sequence|legend=1| 8, 19, 46, 65, 539, 604c, 669c, 734c, 799c, 864c, 929c }}
@@ Line 47: / Line 47: @@
 {{Optimal ET sequence|legend=1| 8, 11c, 19, 46, 65, 344, 409, 474, 539, 604c }}
-[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~31/24 = 443.050
+[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000, ~31/24 = 443.050
 Badness (Dirichlet): 0.243
@@ Line 63: / Line 63: @@
 {{Optimal ET sequence|legend=1| 19, 46j, 65, 149, 363j }}
-[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~31/24 = 442.989
+[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000, ~31/24 = 442.989
 Badness (Dirichlet): 0.283
@@ Line 80: / Line 80: @@
 {{Optimal ET sequence|legend=1| 19, 46, 65, 176, 241, 306 }}
-[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~128/99 = 443.115
+[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000, ~128/99 = 443.115
 Badness (Dirichlet): 0.728
@@ Line 97: / Line 97: @@
 {{Optimal ET sequence|legend=1| 19, 46, 65, 111, 176g }}
-[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~22/17 = 443.188
+[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000, ~22/17 = 443.188
 Badness (Dirichlet): 0.639
@@ Line 110: / Line 110: @@
 {{Optimal ET sequence|legend=1| 19, 46, 65, 111, 176g }}
-[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~22/17 = 443.185
+[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000, ~22/17 = 443.185
 Badness (Dirichlet): 0.555
@@ Line 125: / Line 125: @@
 {{Optimal ET sequence|legend=1| 19, 46, 65, 111, 176g }}
-[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~22/17 = 443.183
+[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000, ~22/17 = 443.183
 Badness (Dirichlet): 0.490
@@ Line 146: / Line 146: @@
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1\1, ~9/7 = 443.3166
+* [[CTE]]: ~2 = 1200.000, ~9/7 = 443.3166
-* [[POTE]]: ~2 = 1\1, ~9/7 = 443.383
+* [[POTE]]: ~2 = 1200.000, ~9/7 = 443.383
 [[Minimax tuning]]:
@@ Line 176: / Line 176: @@
 : mapping generators: ~2, ~9/7
-Optimal tuning (CTE): ~2 = 1\1, ~9/7 = 443.4016
+Optimal tuning (CTE): ~2 = 1200.000, ~9/7 = 443.4016
 {{Optimal ET sequence|legend=1| 19, 27, 46, 111df }}
@@ Line 192: / Line 192: @@
 Optimal tunings:
-* CTE: ~2 = 1\1, ~9/7 = 443.2987
+* CTE: ~2 = 1200.000, ~9/7 = 443.2987
-* POTE: ~2 = 1\1, ~9/7 = 443.294
+* POTE: ~2 = 1200.000, ~9/7 = 443.294
 {{Optimal ET sequence|legend=1| 19, 27, 46, 111d }}
@@ Line 209: / Line 209: @@
 Optimal tunings:
-* CTE: ~2 = 1\1, ~9/7 = 443.3658
+* CTE: ~2 = 1200.000, ~9/7 = 443.3658
-* POTE: ~2 = 1\1, ~9/7 = 443.321
+* POTE: ~2 = 1200.000, ~9/7 = 443.321
 {{Optimal ET sequence|legend=1| 19, 27, 46, 111df }}
@@ Line 224: / Line 224: @@
 Optimal tunings:
-* CTE: ~2 = 1\1, ~9/7 = 443.3775
+* CTE: ~2 = 1200.000, ~9/7 = 443.3775
-* POTE: ~2 = 1\1, ~9/7 = 443.365
+* POTE: ~2 = 1200.000, ~9/7 = 443.365
 {{Optimal ET sequence|legend=1| 19, 27, 46 }}
@@ Line 243: / Line 243: @@
 Optimal tunings:
-* CTE: ~2 = 1\1, ~9/7 = 443.4783
+* CTE: ~2 = 1200.000, ~9/7 = 443.4783
-* POTE: ~2 = 1\1, ~9/7 = 443.626
+* POTE: ~2 = 1200.000, ~9/7 = 443.626
 {{Optimal ET sequence|legend=1| 19e, 27e, 46, 119c }}
@@ Line 260: / Line 260: @@
 Optimal tunings:
-* CTE: ~2 = 1\1, ~9/7 = 443.5075
+* CTE: ~2 = 1200.000, ~9/7 = 443.5075
-* POTE: ~2 = 1\1, ~9/7 = 443.559
+* POTE: ~2 = 1200.000, ~9/7 = 443.559
 {{Optimal ET sequence|legend=1| 19e, 27e, 46 }}
@@ Line 275: / Line 275: @@
 Optimal tunings:
-* CTE: ~2 = 1\1, ~9/7 = 443.5050
+* CTE: ~2 = 1200.000, ~9/7 = 443.5050
-* POTE: ~2 = 1\1, ~9/7 = 443.551
+* POTE: ~2 = 1200.000, ~9/7 = 443.551
 {{Optimal ET sequence|legend=1| 19eg, 27eg, 46 }}
@@ Line 294: / Line 294: @@
 Optimal tunings:
-* CTE: ~2 = 1\1, ~9/7 = 443.1886
+* CTE: ~2 = 1200.000, ~9/7 = 443.1886
-* POTE: ~2 = 1\1, ~9/7 = 443.962
+* POTE: ~2 = 1200.000, ~9/7 = 443.962
 {{Optimal ET sequence|legend=1| 8d, 19, 27e }}
@@ Line 311: / Line 311: @@
 Optimal tunings:
-* CTE: ~2 = 1\1, ~9/7 = 443.2863
+* CTE: ~2 = 1200.000, ~9/7 = 443.2863
-* POTE: ~2 = 1\1, ~9/7 = 443.945
+* POTE: ~2 = 1200.000, ~9/7 = 443.945
 {{Optimal ET sequence|legend=1| 8d, 19, 27e }}
@@ Line 328: / Line 328: @@
 Optimal tunings:
-* CTE: ~2 = 1\1, ~9/7 = 443.7814
+* CTE: ~2 = 1200.000, ~9/7 = 443.7814
-* POTE: ~2 = 1\1, ~9/7 = 443.518
+* POTE: ~2 = 1200.000, ~9/7 = 443.518
 {{Optimal ET sequence|legend=1| 8d, 19e, 27 }}
@@ Line 343: / Line 343: @@
 Optimal tunings:
-* CTE: ~2 = 1\1, ~9/7 = 443.7877
+* CTE: ~2 = 1200.000, ~9/7 = 443.7877
-* POTE: ~2 = 1\1, ~9/7 = 443.506
+* POTE: ~2 = 1200.000, ~9/7 = 443.506
 {{Optimal ET sequence|legend=1| 8d, 19e, 27 }}
@@ Line 364: / Line 364: @@
 Optimal tunings:
-* CTE: ~99/70 = 1\2, ~9/7 = 443.3688 (~11/10 = 156.6312)
+* CTE: ~99/70 = 600.000, ~9/7 = 443.3688 (~11/10 = 156.6312)
-* POTE: ~99/70 = 1\2, ~9/7 = 443.308 (~11/10 = 156.692)
+* POTE: ~99/70 = 600.000, ~9/7 = 443.308 (~11/10 = 156.692)
 {{Optimal ET sequence|legend=1| 8d, …, 38d, 46 }}
@@ Line 383: / Line 383: @@
 Optimal tunings:
-* CTE: ~55/39 = 1\2, ~9/7 = 443.4416, ~11/10 = 156.5584
+* CTE: ~55/39 = 600.000, ~9/7 = 443.4416, ~11/10 = 156.5584
-* POTE: ~55/39 = 1\2, ~9/7 = 443.275, ~11/10 = 156.725
+* POTE: ~55/39 = 600.000, ~9/7 = 443.275, ~11/10 = 156.725
 {{Optimal ET sequence|legend=1| 8d, …, 38df, 46 }}
@@ Line 402: / Line 402: @@
 Optimal tunings:
-* CTE: ~17/12 = 1\2, ~9/7 = 443.4466 (~11/10 = 156.5534)
+* CTE: ~17/12 = 600.000, ~9/7 = 443.4466 (~11/10 = 156.5534)
 {{Optimal ET sequence|legend=1| 8d, …, 38df, 46 }}
@@ Line 420: / Line 420: @@
 Optimal tunings:
-* CTE: ~2 = 1\1, ~25/22 = 221.5981
+* CTE: ~2 = 1200.000, ~25/22 = 221.5981
-* POTE: ~2 = 1\1, ~25/22 = 221.605
+* POTE: ~2 = 1200.000, ~25/22 = 221.605
 {{Optimal ET sequence|legend=1| 27e, 38d, 65 }}
@@ Line 435: / Line 435: @@
 Optimal tunings:
-* CTE: ~2 = 1\1, ~25/22 = 221.6333
+* CTE: ~2 = 1200.000, ~25/22 = 221.6333
-* POTE: ~2 = 1\1, ~25/22 = 221.556
+* POTE: ~2 = 1200.000, ~25/22 = 221.556
 {{Optimal ET sequence|legend=1| 27e, 38df, 65f }}
@@ Line 453: / Line 453: @@
 {{Multival|legend=1| 7 9 32 -2 31 49 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~162/125 = 442.755
+[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~162/125 = 442.755
 {{Optimal ET sequence|legend=1| 19, 65d, 84, 103, 187, 290b }}
@@ Line 470: / Line 470: @@
 {{Multival|legend=1| 7 9 -33 -2 -72 -102 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~162/125 = 443.289
+[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~162/125 = 443.289
 {{Optimal ET sequence|legend=1| 46, 111, 157, 268cd }}
@@ Line 485: / Line 485: @@
 : mapping generators: ~2, ~128/99
-Optimal tuning (POTE): ~2 = 1\1, ~128/99 = 443.274
+Optimal tuning (POTE): ~2 = 1200.000, ~128/99 = 443.274
 {{Optimal ET sequence|legend=1| 46, 65d, 111, 268cd, 379cdd}}
@@ Line 500: / Line 500: @@
 : mapping generators: ~2, ~84/65
-Optimal tuning (POTE): ~2 = 1\1, ~84/65 = 443.270
+Optimal tuning (POTE): ~2 = 1200.000, ~84/65 = 443.270
 {{Optimal ET sequence|legend=1| 46, 65d, 111, 268cd, 379cddf}}
@@ Line 515: / Line 515: @@
 : mapping generators: ~2, ~22/17
-Optimal tuning (POTE): ~2 = 1\1, ~22/17 = 443.270
+Optimal tuning (POTE): ~2 = 1200.000, ~22/17 = 443.270
 {{Optimal ET sequence|legend=1| 46, 65d, 111, 268cdg, 379cddfg }}
@@ Line 536: / Line 536: @@
 {{Multival|legend=1| 14 18 -20 -4 -71 -97 }}
-[[Optimal tuning]] ([[POTE]]): ~567/400 = 1\2, ~162/125 = 443.075 (~35/32 = 156.925)
+[[Optimal tuning]] ([[POTE]]): ~567/400 = 600.000, ~162/125 = 443.075 (~35/32 = 156.925)
 {{Optimal ET sequence|legend=1| 8, 38, 46, 84, 130 }}
@@ Line 553: / Line 553: @@
 :: mapping generators: ~567/400, ~35/32
-Optimal tuning (POTE): ~99/70 = 1\2, ~162/125 = 443.117 (~35/32 = 156.883)
+Optimal tuning (POTE): ~99/70 = 600.000, ~162/125 = 443.117 (~35/32 = 156.883)
 {{Optimal ET sequence|legend=1| 46, 84, 130, 306, 436ce }}
@@ Line 570: / Line 570: @@
 :: mapping generators: ~55/39, ~35/32
-Optimal tuning (POTE): ~55/39 = 1\2, ~162/125 = 443.096 (~35/32 = 156.904)
+Optimal tuning (POTE): ~55/39 = 600.000, ~162/125 = 443.096 (~35/32 = 156.904)
 {{Optimal ET sequence|legend=1| 46, 84, 130, 566ce, 596cef }}
@@ Line 589: / Line 589: @@
 {{Multival|legend=1| 14 18 45 -4 32 54 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~56/45 = 378.467
+[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~56/45 = 378.467
 {{Optimal ET sequence|legend=1| 19, 111, 130, 929c, 1059c, 1189bc, 1319bc }}
@@ Line 602: / Line 602: @@
 Mapping: {{mapping| 1 6 8 17 -6 | 0 -14 -18 -45 30 }}
-Optimal tuning (POTE): ~2 = 1\1, ~56/45 = 378.440
+Optimal tuning (POTE): ~2 = 1200.000, ~56/45 = 378.440
 {{Optimal ET sequence|legend=1| 19, 111, 130, 241, 371ce, 501cde, 872cde }}
@@ Line 615: / Line 615: @@
 Mapping: {{mapping| 1 6 8 17 -6 16 | 0 -14 -18 -45 30 -39 }}
-Optimal tuning (POTE): ~2 = 1\1, ~56/45 = 378.437
+Optimal tuning (POTE): ~2 = 1200.000, ~56/45 = 378.437
 {{Optimal ET sequence|legend=1| 19, 111, 130, 241, 371ce }}
@@ Line 634: / Line 634: @@
 {{Multival|legend=1| 21 27 -7 -6 -70 -92 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~48/35 = 546.815
+[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~48/35 = 546.815
 {{Optimal ET sequence|legend=1| 46, 103, 149, 699bdd }}
@@ Line 649: / Line 649: @@
 : mapping generators: ~2, ~11/8
-Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 547.631
+Optimal tuning (POTE): ~2 = 1200.000, ~11/8 = 547.631
 {{Optimal ET sequence|legend=1| 46, 103, 149, 252e, 401bdee }}
@@ Line 662: / Line 662: @@
 {{Mapping|legend=1| 1 -8 -10 6 3 11 | 0 21 27 -7 1 -16}}
-Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 547.629
+Optimal tuning (POTE): ~2 = 1200.000, ~11/8 = 547.629
 {{Optimal ET sequence|legend=1| 46, 103, 149, 252ef, 401bdeef }}
@@ Line 675: / Line 675: @@
 {{Mapping|legend=1| 1 -8 -10 6 3 11 5 | 0 21 27 -7 1 -16 -2}}
-Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 547.635
+Optimal tuning (POTE): ~2 = 1200.000, ~11/8 = 547.635
 {{Optimal ET sequence|legend=1| 46, 103, 149, 252ef, 401bdeef }}
@@ Line 688: / Line 688: @@
 {{Mapping|legend=1| 1 -8 -10 6 3 11 5 12 | 0 21 27 -7 1 -16 -2 -17}}
-Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 547.614
+Optimal tuning (POTE): ~2 = 1200.000, ~11/8 = 547.614
 {{Optimal ET sequence|legend=1| 46, 103h, 149h, 252efhh }}
@@ Line 707: / Line 707: @@
 {{Multival|legend=1| 21 27 12 -6 -40 -48 }}
-[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~36/35 = 43.147
+[[Optimal tuning]] ([[POTE]]): ~63/50 = 400.000, ~36/35 = 43.147
 {{Optimal ET sequence|legend=1| 27, 57, 84, 111, 195d, 306d }}
@@ Line 720: / Line 720: @@
 Mapping: {{mapping| 3 4 6 8 8 | 0 7 9 4 22 }}
-Optimal tuning (POTE): ~63/50 = 1\3, ~36/35 = 43.292
+Optimal tuning (POTE): ~63/50 = 400.000, ~36/35 = 43.292
 {{Optimal ET sequence|legend=1| 27e, 84e, 111 }}
@@ Line 735: / Line 735: @@
 : mapping generators: ~49/39, ~36/35
-Optimal tuning (POTE): ~49/39 = 1\3, ~36/35 = 43.288
+Optimal tuning (POTE): ~49/39 = 400.000, ~36/35 = 43.288
 {{Optimal ET sequence|legend=1| 27e, 84e, 111 }}
@@ Line 748: / Line 748: @@
 Mapping: {{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 tuning (POTE): ~49/39 = 400.000, ~36/35 = 43.276
 {{Optimal ET sequence|legend=1| 27eg, 84e, 111 }}
@@ Line 761: / Line 761: @@
 Mapping: {{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 tuning (POTE): ~49/39 = 400.000, ~36/35 = 43.292
 {{Optimal ET sequence|legend=1| 27eg, 84e, 111 }}

Sensipent family: Difference between revisions

Revision as of 03:32, 21 October 2024

Sensipent

2.3.5.31

Sendai

Sensible

2.3.5.11.17

2.3.5.11.17.23

2.3.5.11.17.23.31

Sensi

Septimal sensi

2.3.5.7.13 subgroup (sensation)

Sensor

13-limit

17-limit

Sensus

13-limit

17-limit

Sensis

13-limit

Sensa

13-limit

Bisensi

13-limit

17-limit

Hemisensi

13-limit

Sensei

Warrior

11-limit

13-limit

17-limit

Bison

11-limit

13-limit

Subpental

11-limit

13-limit

Heinz

11-limit

13-limit

17-limit

19-limit

Trisensory

11-limit

13-limit

17-limit

19-limit

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