Sensamagic clan

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.

Lambda

Subgroup: 3.5.7

Comma list: 245/243

Sval mapping: [⟨1 1 2], ⟨0 -2 1]]

Sval mapping generators: ~3, ~9/7

Gencom mapping: [⟨0 1 1 2], ⟨0 0 -2 1]]

POTE generator: ~9/7 = 440.4881

Vals: b4, b9, b13, b56, b69, b82, b95

Extensions

For full 7-limit extensions, we have sensi, bohpier, sensa/escaped, salsa, pycnic, cohemiripple, superthird, magus and leapweek discussed below, as well as father, sidi, godzilla, hedgehog, superpyth, hemiaug, magic, rodan, octacot, shrutar, and clyde discussed elsewhere.

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.

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."

7-limit

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 ]]

POTE generator: ~9/7 = 443.383

Minimax tuning:

• 7-odd-limit

[[1 0 0 0⟩, [1/13 0 0 7/13⟩, [5/13 0 0 9/13⟩, [0 0 0 1⟩]

Eigenmonzos: 2, 7

• 9-odd-limit

[[1 0 0 0⟩, [2/5 14/5 -7/5 0⟩, [4/5 18/5 -9/5 0⟩, [3/5 26/5 -13/5 0⟩]

Eigenmonzos: 2, 9/5

Algebraic generator: The real root of x⁵ + x⁴ - 4x² + x - 1, at 443.3783 cents.

Badness: 0.0256

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 9/7; 91/90 126/125 169/168]

POTE generator: ~9/7 = 443.322

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]]

POTE generator: ~9/7 = 443.294

Badness: 0.0379

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]]

POTE generator: ~9/7 = 443.321

Badness: 0.0256

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]]

POTE generator: ~9/7 = 443.962

Badness: 0.0287

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]]

POTE generator: ~9/7 = 443.945

Badness: 0.0200

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]]

POTE generator: ~9/7 = 443.626

Badness: 0.0295

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]]

POTE generator: ~9/7 = 443.559

Badness: 0.0208

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]]

POTE generator: ~9/7 = 443.518

Badness: 0.0368

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 10], ⟨0 -7 -9 -13 -12 -11]]

POTE generator: ~9/7 = 443.506

Badness: 0.0233

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]]

POTE generator: ~25/22 = 221.605

Badness: 0.0487

Bohpier

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

5-limit

Subgroup: 2.3.5

Comma list: 1220703125/1162261467

Mapping: [⟨1 0 0], ⟨0 13 19]]

POTE generator: ~27/25 = 146.476

Badness: 0.8605

7-limit

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

Badness: 0.0682

11-limit

Subgroup: 2.3.5.7.11

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

POTE generator: ~12/11 = 146.545

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

Badness: 0.0339

13-limit

Subgroup: 2.3.5.7.11.13

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

POTE generator: ~12/11 = 146.603

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

Badness: 0.0249

Music

by Chris Vaisvil:

• bophier-1.mp3
• bophier-12equal-six-octaves.mp3

Sensa aka escaped

See also: Escapade family § Escaped

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

Badness: 0.0887

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

Badness: 0.0358

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

Badness: 0.0317

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

Badness: 0.08015

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

Badness: 0.0394

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

Badness: 0.0310

Pycnic

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

Badness: 0.0737

Cohemiripple

See also: Ripple family

Subgroup: 2.3.5.7

Comma list: 245/243, 1323/1250

Mapping: [⟨1 7 11 12], ⟨0 -10 -16 -17]]

Wedgie: ⟨⟨ 10 16 17 2 -1 -5 ]]

POTE generator: ~7/5 = 549.944

Badness: 0.1902

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 7 11 12 17], ⟨0 -10 -16 -17 -25]]

POTE generator: ~7/5 = 549.945

Badness: 0.0827

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 7 11 12 17 14], ⟨0 -10 -16 -17 -25 -19]]

POTE generator: ~7/5 = 549.958

Badness: 0.0499

Superthird

Subgroup: 2.3.5.7

Comma list: 245/243, 78125/76832

Mapping: [⟨1 13 15 25], ⟨0 -18 -20 -35]]

Wedgie: ⟨⟨ 18 20 35 -10 5 25 ]]

POTE generator: ~9/7 = 439.076

Badness: 0.1394

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 13 15 25 6], ⟨0 -18 -20 -35 -4]]

POTE generator: ~9/7 = 439.152

Badness: 0.0709

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 13 15 25 6 24], ⟨0 -18 -20 -35 -4 -32]]

POTE generator: ~9/7 = 439.119

Badness: 0.0528

Magus

5-limit

Subgroup: 2.3.5

Comma list: 50331648/48828125

Mapping: [⟨1 9 3], ⟨0 -11 -1]]

POTE generator: ~5/4 = 391.225

Badness: 0.3602

7-limit

Subgroup: 2.3.5.7

Comma list: 245/243, 28672/28125

Mapping: [⟨1 9 3 21], ⟨0 -11 -1 -27]]

Wedgie: ⟨⟨ 11 1 27 -24 12 60 ]]

POTE generator: ~5/4 = 391.465

Badness: 0.1084

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 9 3 21 23], ⟨0 -11 -1 -27 -29]]

POTE generator: ~5/4 = 391.503

Badness: 0.0451

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 9 3 21 23 1], ⟨0 -11 -1 -27 -29 4]]

POTE generator: ~5/4 = 391.366

Badness: 0.0430

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

Badness: 0.14058

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

Badness: 0.0507

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

Badness: 0.0327