Sensamagic clan

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The sensamagic clan tempers out the sensamagic comma, 245/243, a triprime comma with no factors of 2, 0 -5 1 2] to be exact.

For full 7-limit extensions, we have sensi, bohpier, escaped, salsa, pycnic, cohemiripple, superthird, magus and leapweek discussed below, as well as

Tempering out 245/243 alone in the full 7-limit leads to a rank-3 temperament, sensamagic, for which 283EDO is the optimal patent val.

BPS

The BPS, for Bohlen–Pierce–Stearns, is the 3.5.7 subgroup temperament tempering out 245/243. This subgroup temperament was formerly called as lambda temperament, which was named after lambda scale.

Subgroup: 3.5.7

Comma list: 245/243

Sval mapping: [1 1 2], 0 -2 1]]

Sval mapping generators: ~3, ~9/7

POTE generator: ~9/7 = 440.4881

Optimal GPV sequence: b4, b9, b13, b56, b69, b82, b95

Sensi

Main article: Sensi
See also: Sensipent family #Sensi

Sensi tempers out 126/125, 686/675 and 4375/4374 in addition to 245/243, 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 of size 11, 19 and 27 are available. The name "sensi" is a play on the words "semi-" and "sixth."

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]]

POTE generator: ~9/7 = 443.383

Optimal GPV sequence19, 27, 46, 157d, 203cd, 249cdd, 295ccdd

Badness: 0.025622

Sensation

Subgroup: 2.3.5.7.13

Comma list: 91/90, 126/125, 169/168

Sval mapping: [1 -1 -1 -2 0], 0 7 9 13 10]]

Gencom mapping: [1 -1 -1 -2 0 0], 0 7 9 13 0 10]]

Gencom: [2 9/7; 91/90 126/125 169/168]

POTE generator: ~9/7 = 443.322

Optimal GPV sequence: 19, 27, 46, 111de, 157de

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]]

POTE generator: ~9/7 = 443.294

Optimal GPV sequence: 19, 27, 46, 111d, 157d, 268cdd

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]]

POTE generator: ~9/7 = 443.321

Optimal GPV sequence: 19, 27, 46, 111df, 157df

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]]

POTE generator: ~9/7 = 443.365

Optimal GPV sequence: 19, 27, 46, 157df, 203cdff, 249cddff

Badness: 0.022908

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]]

POTE generator: ~9/7 = 443.962

Optimal GPV sequence: 8d, 19, 27e, 73ee

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]]

POTE generator: ~9/7 = 443.945

Optimal GPV sequence: 19, 27e, 46e, 73ee

Badness: 0.020017

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]]

POTE generator: ~9/7 = 443.626

Optimal GPV sequence: 19e, 27e, 46, 119c, 165c

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]]

POTE generator: ~9/7 = 443.559

Optimal GPV sequence: 19e, 27e, 46, 165cf, 211bccf, 257bccff, 303bccdff

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]]

POTE generator: ~9/7 = 443.551

Optimal GPV sequence: 19eg, 27eg, 46

Badness: 0.016238

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]]

POTE generator: ~9/7 = 443.518

Optimal GPV sequence: 19e, 27, 46ee

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]]

POTE generator: ~9/7 = 443.506

Optimal GPV sequence: 19e, 27, 46ee

Badness: 0.023258

Hemisensi

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]]

POTE generator: ~25/22 = 221.605

Optimal GPV sequence: 27e, 38d, 65, 157de, 222cde

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 30]]

POTE generator: ~25/22 = 221.556

Optimal GPV sequence: 27e, 38df, 65f

Badness: 0.033016

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]]

POTE generator: ~11/10 = 156.692

Optimal GPV sequence: 8d, …, 38d, 46, 176dde, 222cdde, 268cddee

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]]

POTE generator: ~11/10 = 156.725

Optimal GPV sequence: 8d, …, 38df, 46

Badness: 0.026339

Bohpier

For the 5-limit version of this temperament, see High badness temperaments #Bohpier.
Main article: Bohpier

Bohpier is named after its interesting relationship with the non-octave Bohlen-Pierce equal temperament.

Subgroup: 2.3.5.7

Comma list: 245/243, 3125/3087

Mapping: [1 0 0 0], 0 13 19 23]]

Wedgie⟨⟨13 19 23 0 0 0]]

POTE generator: ~27/25 = 146.474

Minimax tuning:

  • 7-odd-limit: ~27/25 = [0 0 1/19
Eigenmonzos (unchanged intervals): 2, 5/4
  • 9-odd-limit: ~27/25 = [0 1/13
Eigenmonzos (unchanged intervals): 2, 4/3

Optimal GPV sequence41, 131, 172, 213c

Badness: 0.068237

11-limit

Subgroup: 2.3.5.7.11

Comma list: 100/99, 245/243, 1344/1331

Mapping: [1 0 0 0 2], 0 13 19 23 12]]

POTE generator: ~12/11 = 146.545

Minimax tuning:

  • 11-odd-limit: ~12/11 = [1/7 1/7 0 0 -1/14
Eigenmonzos (unchanged intervals): 2, 11/9

Optimal GPV sequence: 41, 90e, 131e

Badness: 0.033949

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 144/143, 196/195, 275/273

Mapping: [1 0 0 0 2 2], 0 13 19 23 12 14]]

POTE generator: ~12/11 = 146.603

Minimax tuning:

  • 13- and 15-odd-limit: ~12/11 = [0 0 1/19
Eigenmonzos (unchanged intervals): 2, 5/4

Optimal GPV sequence: 41, 90ef, 131ef, 221bdeff

Badness: 0.024864

Music

by Chris Vaisvil:

Triboh

Triboh is named after "Triple Bohlen-Pierce scale", which divides each step of the equal-tempered Bohlen-Pierce scale into three equal parts.

Subgroup: 2.3.5.7.11

Comma list: 245/243, 1331/1323, 3125/3087

Mapping: [1 0 0 0 0], 0 39 57 69 85]]

POTE generator: ~77/75 = 48.828

Optimal GPV sequence: 49, 123ce, 172

Badness: 0.162592

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 245/243, 275/273, 847/845, 1331/1323

Mapping: [1 0 0 0 0 0], 0 39 57 69 85 91]]

POTE generator: ~77/75 = 48.822

Optimal GPV sequence: 49f, 123ce, 172f, 295ce, 467bccef

Badness: 0.082158

Escaped

See also: Escapade family #Escaped

This temperament is also called as "sensa" because it tempers out 245/243, 352/351, and 385/384 as a sensamagic temperament. Not to be confused with 19e&27 temperament (sensi extension).

Subgroup: 2.3.5.7

Comma list: 245/243, 65625/65536

Mapping: [1 2 2 4], 0 -9 7 -26]]

Wedgie⟨⟨9 -7 26 -32 16 80]]

POTE generator: ~28/27 = 55.122

Optimal GPV sequence22, 65, 87, 196, 283

Badness: 0.088746

11-limit

Subgroup: 2.3.5.7.11

Comma list: 245/243, 385/384, 4000/3993

Mapping: [1 2 2 4 3], 0 -9 7 -26 10]]

POTE generator: ~28/27 = 55.126

Optimal GPV sequence: 22, 65, 87, 196, 283

Badness: 0.035844

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 245/243, 352/351, 385/384, 625/624

Mapping: [1 2 2 4 3 2], 0 -9 7 -26 10 37]]

POTE generator: ~28/27 = 55.138

Optimal GPV sequence: 22, 65, 87, 283

Badness: 0.031366

Salsa

See also: Schismatic family

Subgroup: 2.3.5.7

Comma list: 245/243, 32805/32768

Mapping: [1 1 7 -1], 0 2 -16 13]]

Wedgie⟨⟨2 -16 13 -30 15 75]]

POTE generator: ~128/105 = 351.049

Optimal GPV sequence17, 24, 41, 106d, 147d, 188cd, 335cd

Badness: 0.080152

11-limit

Subgroup: 2.3.5.7.11

Comma list: 243/242, 245/242, 385/384

Mapping: [1 1 7 -1 2], 0 2 -16 13 5]]

POTE generator: ~11/9 = 351.014

Optimal GPV sequence: 17, 24, 41, 106d, 147d

Badness: 0.039444

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 144/143, 243/242, 245/242

Mapping: [1 1 7 -1 2 4], 0 2 -16 13 5 -1]]

POTE generator: ~11/9 = 351.025

Optimal GPV sequence: 17, 24, 41, 106df, 147df

Badness: 0.030793

Pycnic

See also: High badness temperaments #Stump

The fifth of pycnic in size is a meantone fifth, but four of them are not used to reach 5. This has the effect of making the Pythagorean major third, nominally 81/64, very close to 5/4 in tuning, being a cent sharp of it in the POTE tuning for instance. Pycnic has MOS of size 9, 11, 13, 15, 17... which contain these alternative thirds, leading to two kinds of major triads, an official one and a nominally Pythagorean one which is actually in better tune.

Subgroup: 2.3.5.7

Comma list: 245/243, 525/512

Mapping: [1 3 -1 8], 0 -3 7 -11]]

Wedgie⟨⟨3 -7 11 -18 9 45]]

POTE generator: ~45/32 = 567.720

Optimal GPV sequence17, 19, 55c, 74cd, 93cdd

Badness: 0.073735

Cohemiripple

See also: Ripple family

Subgroup: 2.3.5.7

Comma list: 245/243, 1323/1250

Mapping: [1 -3 -5 -5], 0 10 16 17]]

Wedgie⟨⟨10 16 17 2 -1 -5]]

POTE generator: ~7/5 = 549.944

Optimal GPV sequence11cd, 13cd, 24

Badness: 0.190208

11-limit

Subgroup: 2.3.5.7.11

Comma list: 77/75, 243/242, 245/242

Mapping: [1 -3 -5 -5 -8], 0 10 16 17 25]]

POTE generator: ~7/5 = 549.945

Optimal GPV sequence: 11cdee, 13cdee, 24

Badness: 0.082716

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 66/65, 77/75, 147/143, 243/242

Mapping: [1 -3 -5 -5 -8 -5], 0 -10 -16 -17 -25 -19]]

POTE generator: ~7/5 = 549.958

Optimal GPV sequence: 11cdeef, 13cdeef, 24

Badness: 0.049933

Superthird

See also: Shibboleth family

Subgroup: 2.3.5.7

Comma list: 245/243, 78125/76832

Mapping: [1 -5 -5 -10], 0 18 20 35]]

Wedgie⟨⟨18 20 35 -10 5 25]]

POTE generator: ~9/7 = 439.076

Optimal GPV sequence11cd, 30d, 41, 317bcc, 358bcc, 399bcc

Badness: 0.139379

11-limit

Subgroup: 2.3.5.7.11

Comma list: 100/99, 245/243, 78125/76832

Mapping: [1 -5 -5 -10 2], 0 18 20 35 4]]

POTE generator: ~9/7 = 439.152

Optimal GPV sequence: 11cd, 30d, 41, 153be, 194be, 235bcee

Badness: 0.070917

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 144/143, 196/195, 1375/1352

Mapping: [1 -5 -5 -10 2 -8], 0 18 20 35 4 32]]

POTE generator: ~9/7 = 439.119

Optimal GPV sequence: 11cdf, 30df, 41

Badness: 0.052835

Superenneadecal

Superenneadecal is a cousin of enneadecal but sharper fifth is used to temper 245/243.

Subgroup: 2.3.5.7

Comma list: 245/243, 395136/390625

Mapping: [⟨19 0 14 -7], ⟨0 1 1 2]]

Optimal GPV sequence19, 76bcd, 95, 114

11-limit

Subgroup: 2.3.5.7.11

Comma list: 245/243, 2560/2541, 3773/3750

Mapping: [⟨19 0 14 -7 96], ⟨0 1 1 2 -1]]

Optimal GPV sequence19, 76bcd, 95, 114e

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 245/243, 832/825, 1001/1000

Mapping: [⟨19 0 14 -7 96 10], ⟨0 1 1 2 -1 2]]

Optimal GPV sequence19, 76bcdf, 95, 114e

Magus

For the 5-limit version of this temperament, see High badness temperaments #Magus.

Magus temperament tempers out 50331648/48828125 (salegu) in the 5-limit. This temperament can be described as 46&49 temperament, which tempers out the sensamagic and 28672/28125 (sazoquingu). Alternative extension amigo (43&46) tempers out the same 5-limit comma as the magus, but with the starling comma (126/125) rather than the sensamagic tempered out.

Subgroup: 2.3.5.7

Comma list: 245/243, 28672/28125

Mapping: [1 -2 2 -6], 0 11 1 27]]

Wedgie⟨⟨11 1 27 -24 12 60]]

POTE generator: ~5/4 = 391.465

Optimal GPV sequence46, 95, 141bc, 187bc, 328bbcc

Badness: 0.108417

11-limit

Subgroup: 2.3.5.7.11

Comma list: 176/175, 245/243, 1331/1323

Mapping: [1 -2 2 -6 -6], 0 11 1 27 29]]

POTE generator: ~5/4 = 391.503

Optimal GPV sequence: 46, 95, 141bc

Badness: 0.045108

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 176/175, 245/243, 1331/1323

Mapping: [1 -2 2 -6 -6 5], 0 11 1 27 29 -4]]

POTE generator: ~5/4 = 391.366

Optimal GPV sequence: 46, 233bcff, 279bccff

Badness: 0.043024

Leapweek

Subgroup: 2.3.5.7

Comma list: 245/243, 2097152/2066715

Mapping: [1 1 17 -6], 0 1 -25 15]]

POTE generator: ~3/2 = 704.536

Optimal GPV sequence17, 29c, 46, 109, 155, 264b, 419b

Badness: 0.140577

11-limit

Subgroup: 2.3.5.7.11

Comma list: 245/243, 385/384, 1331/1323

Mapping: [1 1 17 -6 -3], 0 1 -25 15 11]]

POTE generator: ~3/2 = 704.554

Optimal GPV sequence: 17, 29c, 46, 109, 264b, 373b, 637bbe

Badness: 0.050679

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 245/243, 352/351, 364/363

Mapping: [1 1 17 -6 -3 -1], 0 1 -25 15 11 8]]

POTE generator: ~3/2 = 704.571

Optimal GPV sequence: 17, 29c, 46, 63, 109

Badness: 0.032727

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 154/153, 169/168, 245/243, 256/255, 273/272

Mapping: [1 1 17 -6 -3 -1 -10], 0 1 -25 15 11 8 24]]

POTE generator: ~3/2 = 704.540

Optimal GPV sequence: 17g, 29cg, 46, 109, 155f, 264bfg

Badness: 0.026243

Leapweeker

Subgroup: 2.3.5.7.11.13.17

Comma list: 136/135, 169/168, 221/220, 245/243, 364/363

Mapping: [1 1 17 -6 -3 -1 17], 0 1 -25 15 11 8 -22]]

POTE generator: ~3/2 = 704.537

Optimal GPV sequence: 17, 29c, 46, 109g, 155fg, 264bfgg

Badness: 0.026774

Semiwolf

Subgroup: 3/2.7/4.5/2

Comma list: 245/243

Mapping: [1 1 3], 0 1 -2]]

POL2 generator: ~7/6 = 262.1728

Optimal GPV sequence: 3edf, 5edf, 8edf

Semilupine

Subgroup: 3/2.7/4.5/2.11/4

Comma list: 100/99, 245/243

Mapping: [1 1 3 4], 0 1 -2 -4]]

POL2 generator: ~7/6 = 264.3771

Optimal GPV sequence: 8edf, 13edf

Hemilycan

Subgroup: 3/2.7/4.5/2.11/4

Comma list: 245/243, 441/440

Mapping: [1 1 3 1], 0 1 -2 4]]

POL2 generator: ~7/6 = 261.5939

Optimal GPV sequence: 8edf, 11edf