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.

BPS (Lambda)

The BPS, for Bohlen–Pierce–Stearns, is the 3.5.7 subgroup temperament tempering out 245/243.

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, escaped, salsa, pycnic, cohemiripple, superthird, magus and leapweek discussed below, as well as

• father, {16/15, 28/27} → Father family #Father
• sidi, {25/24, 245/243} → Dicot family #Sidi
• godzilla, {49/48, 81/80} → Meantone family #Godzilla
• hedgehog, {50/49, 245/243} → Porcupine family #Hedgehog
• superpyth, {64/63, 245/243} → Archytas clan #Superpyth
• hemiaug, {128/125, 245/243} → Augmented family #Hemiaug
• magic, {225/224, 245/243} → Magic family #Magic
• rodan, {245/243, 1029/1024} → Gamelismic clan #Rodan
• shrutar, {245/243, 2048/2025} → Diaschismic family #Shrutar
• octacot, {245/243, 2401/2400} → Tetracot family #Octacot
• clyde, {245/243, 3136/3125} → Kleismic family #Clyde
• bamity, {245/243, 64827/64000} → Amity family #Bamity

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."

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

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 (unchanged intervals): 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 (unchanged intervals): 2, 9/5

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

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

Vals: Template:Val list

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

Vals: Template:Val list

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

Vals: Template:Val list

Badness: 0.025575

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

Vals: Template:Val list

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

Vals: Template:Val list

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

Vals: Template:Val list

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

Vals: Template:Val list

Badness: 0.020789

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

Vals: Template:Val list

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

Vals: Template:Val list

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

Vals: Template:Val list

Badness: 0.048714

Bohpier

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

Subgroup: 2.3.5

Comma list: 1220703125/1162261467

Mapping: [⟨1 0 0], ⟨0 13 19]]

POTE generator: ~27/25 = 146.476

Minimax tuning:

• 5-odd-limit: ~27/25 = [0 0 1/19⟩

Eigenmonzos (unchanged intervals): 2, 5/4

Tuning ranges:

• 5-odd-limit diamond monotone: ~27/25 = [144.000, 150.000] (3\25 to 1\8)
• 5-odd-limit diamond tradeoff: ~27/25 = [146.304, 147.393]
• 5-odd-limit diamond monotone and tradeoff: ~27/25 = [146.304, 147.393]

Badness: 0.860534

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

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

Tuning ranges:

• 7-odd-limit diamond monotone: ~27/25 = [145.455, 150.000] (4\33 to 1\8)
• 9-odd-limit diamond monotone: ~27/25 = [145.455, 146.939] (4\33 to 6\49)
• 7-odd-limit diamond tradeoff: ~27/25 = [145.628, 147.393]
• 9-odd-limit diamond tradeoff: ~27/25 = [145.028, 147.393]
• 7-odd-limit diamond monotone and tradeoff: ~27/25 = [145.628, 147.393]
• 9-odd-limit diamond monotone and tradeoff: ~27/25 = [145.455, 146.939]

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

Tuning ranges:

• 11-odd-limit diamond monotone: ~12/11 = [145.455, 146.939] (4\33 to 6\49)
• 11-odd-limit diamond tradeoff: ~12/11 = [145.028, 150.637]
• 11-odd-limit diamond monotone and tradeoff: ~12/11 = [145.455, 146.939]

Vals: Template:Val list

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

Tuning ranges:

• 13-odd-limit diamond monotone: ~12/11 = [145.455, 146.939] (4\33 to 6\49)
• 15-odd-limit diamond monotone: ~12/11 = [146.341, 146.939] (5\41 to 6\49)
• 13- and 15-odd-limit diamond tradeoff: ~12/11 = [138.573, 150.637]
• 13-odd-limit diamond monotone and tradeoff: ~12/11 = [145.455, 146.939]
• 15-odd-limit diamond monotone and tradeoff: ~12/11 = [146.341, 146.939]

Vals: Template:Val list

Badness: 0.024864

Music

by Chris Vaisvil:

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

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

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

Vals: Template:Val list

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

Vals: Template:Val list

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

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

Vals: Template:Val list

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

Vals: Template:Val list

Badness: 0.030793

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: 0.073735

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

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

Vals: Template:Val list

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

Vals: Template:Val list

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

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

Vals: Template:Val list

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

Vals: Template:Val list

Badness: 0.052835

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

Comma list: 50331648/48828125

Mapping: [⟨1 -2 2], ⟨0 11 1]]

POTE generator: ~5/4 = 391.225

Badness: 0.360162

7-limit

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

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

Vals: Template:Val list

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

Vals: Template:Val list

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

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

Vals: Template:Val list

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

Vals: Template:Val list

Badness: 0.032727

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

Vals: 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

Vals: 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

Vals: 8edf, 11edf