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. 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 the lambda temperament, which was named after the lambda scale.

Subgroup: 3.5.7

Comma list: 245/243

Subgroup-val mapping: [⟨1 1 2], ⟨0 -2 1]]

sval mapping generators: ~3, ~9/7

Optimal tuning (POTE): ~3 = 1\1edt, ~9/7 = 440.4881

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

Overview to extensions

The full 7-limit extensions' relation to BPS is clearer if the mapping is normalized in terms of 3.5.7.2. In fact, the strong extensions are

• #Sensi
• #Cohemiripple
• Hedgehog, {50/49, 245/243} → Porcupine family
• Fourfives, {245/243, 235298/234375} → Fifive family

The others are weak extensions. See

• Father, {16/15, 28/27} → Father family
• Sidi, {25/24, 245/243} → Dicot family
• Godzilla, {49/48, 81/80} → Meantone family
• Superpyth, {64/63, 245/243} → Archytas clan
• Hemiaug, {128/125, 245/243} → Augmented family
• Magic, {225/224, 245/243} → Magic family
• Rodan, {245/243, 1029/1024} → Gamelismic clan
• Shrutar, {245/243, 2048/2025} → Diaschismic family
• Octacot, {245/243, 2401/2400} → Tetracot family
• Clyde, {245/243, 3136/3125} → Kleismic family
• Pental, {245/243, 16807/16384} → Pental family
• Bamity, {245/243, 64827/64000} → Amity family
• Escaped, {245/243, 65625/65536} → Escapade family

as well as bohpier, salsa, pycnic, superthird, magus and leapweek discussed below.

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 scales of 8-, 11-, 19- and 27-tones 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

⟨⟨7 9 13 -2 1 5]]

Optimal tuning (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]
• 7-odd-limit diamond monotone and tradeoff: ~9/7 = [442.179, 445.628]
• 9-odd-limit diamond monotone and tradeoff: ~9/7 = [442.105, 444.444]

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

Optimal ET sequence: 19, 27, 46, 157d, 203cd, 249cdd, 295ccdd

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 (POTE): ~2 = 1\1, ~9/7 = 443.322

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

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]]

Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 443.294

Optimal ET 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 6 8 11 -6 10], ⟨0 -7 -9 -13 15 -10]]

Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 443.321

Optimal ET 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 6 8 11 -6 10 -6], ⟨0 -7 -9 -13 15 -10 16]]

Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 443.365

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

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]]

Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 443.626

Optimal ET 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 6 8 11 23 10], ⟨0 -7 -9 -13 -31 -10]]

Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 443.559

Optimal ET 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 6 8 11 23 10 23], ⟨0 -7 -9 -13 -31 -10 -30]]

Optimal tuning (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]]

Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 443.962

Optimal ET 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 6 8 11 6 10], ⟨0 -7 -9 -13 -4 -10]]

Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 443.945

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

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 tuning (POTE): ~2 = 1\1, ~9/7 = 443.518

Optimal ET 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 6 8 11 11 11], ⟨0 -7 -9 -13 -12 -11]]

Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 443.506

Optimal ET sequence: 19e, 27, 46ee

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 tuning (POTE): ~99/70 = 1\2, ~11/10 = 156.692

Optimal ET 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]]

Optimal tuning (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 tuning (POTE): ~17/12 = 1\2, ~11/10 = 156.7158

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 tuning (POTE): ~2 = 1\1, ~25/22 = 221.605

Optimal ET 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 13 17 24 32 30], ⟨0 -14 -18 -26 -35 -30]]

Optimal tuning (POTE): ~2 = 1\1, ~25/22 = 221.556

Optimal ET sequence: 27e, 38df, 65f

Badness: 0.033016

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]]

⟨⟨10 16 17 2 -1 -5]]

Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 549.944

Optimal ET sequence: 11cd, 13cd, 24

Badness: 0.190208

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]]

Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 549.945

Optimal ET 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 7 11 12 17 14], ⟨0 -10 -16 -17 -25 -19]]

Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 549.958

Optimal ET sequence: 11cdeef, 13cdeef, 24

Badness: 0.049933

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]]

⟨⟨13 19 23 0 0 0]]

Optimal tuning (POTE): ~2 = 1\1, ~27/25 = 146.474

Minimax tuning:

• 7-odd-limit: ~27/25 = [0 0 1/19⟩

Eigenmonzo (unchanged-interval) basis: 2.5

• 9-odd-limit: ~27/25 = [0 1/13⟩

Eigenmonzo (unchanged-interval) basis: 2.3

Optimal ET sequence: 41, 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]]

Optimal tuning (POTE): ~2 = 1\1, ~12/11 = 146.545

Minimax tuning:

• 11-odd-limit: ~12/11 = [1/7 1/7 0 0 -1/14⟩

Eigenmonzo basis (unchanged-interval basis): 2.11/9

Optimal ET 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]]

Optimal tuning (POTE): ~2 = 1\1, ~12/11 = 146.603

Minimax tuning:

• 13- and 15-odd-limit: ~12/11 = [0 0 1/19⟩

Eigenmonzo (unchanged-interval) basis: 2.5

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

Badness: 0.024864

Music

by Chris Vaisvil:

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

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]]

Optimal tuning (POTE): ~2 = 1\1, ~77/75 = 48.828

Optimal ET 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]]

Optimal tuning (POTE): ~2 = 1\1, ~77/75 = 48.822

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

Badness: 0.082158

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]]

⟨⟨2 -16 13 -30 15 75]]

Optimal tuning (POTE): ~2 = 1\1, ~128/105 = 351.049

Optimal ET sequence: 17, 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]]

Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 351.014

Optimal ET 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]]

Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 351.025

Optimal ET 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]]

⟨⟨3 -7 11 -18 9 45]]

Optimal tuning (POTE): ~2 = 1\1, ~45/32 = 567.720

Optimal ET sequence: 17, 19, 55c, 74cd, 93cdd

Badness: 0.073735

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]]

⟨⟨18 20 35 -10 5 25]]

Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 439.076

Optimal ET sequence: 11cd, 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]]

Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 439.152

Optimal ET 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]]

Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 439.119

Optimal ET 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 tuning (POTE): ~392/375 = 1\19, ~3/2 = 704.166

Optimal ET sequence: 19, 76bcd, 95, 114, 133, 247b, 380bcd

Badness: 0.132311

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 tuning (POTE): ~33/32 = 1\19, ~3/2 = 705.667

Optimal ET sequence: 19, 76bcd, 95, 114e

Badness: 0.101496

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 tuning (POTE): ~33/32 = 1\19, ~3/2 = 705.801

Optimal ET sequence: 19, 76bcdf, 95, 114e

Badness: 0.053197

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). The 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]]

⟨⟨11 1 27 -24 12 60]]

Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 391.465

Optimal ET sequence: 46, 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]]

Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 391.503

Optimal ET 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]]

Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 391.366

Optimal ET sequence: 46, 233bcff, 279bccff

Badness: 0.043024

Leapweek

Not to be confused with scales produced by leap week calendars such as Symmetry454.

Subgroup: 2.3.5.7

Comma list: 245/243, 2097152/2066715

Mapping: [⟨1 0 42 -21], ⟨0 1 -25 15]]

mapping generators: ~2, ~3

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.536

Optimal ET sequence: 17, 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 0 42 -21 -14], ⟨0 1 -25 15 11]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.554

Optimal ET 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 0 42 -21 -14 -9], ⟨0 1 -25 15 11 8]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.571

Optimal ET 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 0 42 -21 -14 -9 -34], ⟨0 1 -25 15 11 8 24]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.540

Optimal ET 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 0 42 -21 -14 -9 39], ⟨0 1 -25 15 11 8 -22]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.537

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

Badness: 0.026774