# 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. 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

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

sval mapping generators: ~3, ~9/7

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

### 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, and fourfives.

The others are weak extensions. Father tempers out 16/15, splitting the generator in two. Godzilla tempers out 49/48 with a hemitwelfth period. Sidi tempers out 25/24, splitting the generator in two with a hemitwelfth period. Clyde tempers out 3136/3125 with a 1/6-twelfth period. Superpyth tempers out 64/63, splitting the generator in six. Magic tempers out 225/224 with a 1/5-twelfth period. Octacot tempers out 2401/2400, splitting the generator in five. Hemiaug tempers out 128/125. Pental tempers out 16807/16384. These split the generator in seven. Bamity tempers out 64827/64000, splitting the generator in nine. Rodan tempers out 1029/1024, splitting the generator in ten. Shrutar tempers out 2048/2025, splitting the generator in eleven. Finally, escaped tempers out 65625/65536, splitting the generator in sixteen.

Discussed elsewhere are

Considered below are bohpier, salsa, pycnic, superthird, magus and leapweek.

## 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

Wedgie⟨⟨7 9 13 -2 1 5]]

• CTE: ~2 = 1\1, ~9/7 = 443.3166
• POTE: ~2 = 1\1, ~9/7 = 443.383
eigenmonzo (unchanged-interval) basis: 2.7
eigenmonzo (unchanged-interval) basis: 2.9/5
• 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.

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 (CTE): ~2 = 1\1, ~9/7 = 443.4016

### 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 tunings:

• CTE: ~2 = 1\1, ~9/7 = 443.2987
• POTE: ~2 = 1\1, ~9/7 = 443.294

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 tunings:

• CTE: ~2 = 1\1, ~9/7 = 443.3658
• POTE: ~2 = 1\1, ~9/7 = 443.321

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 tunings:

• CTE: ~2 = 1\1, ~9/7 = 443.3775
• POTE: ~2 = 1\1, ~9/7 = 443.365

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 tunings:

• CTE: ~2 = 1\1, ~9/7 = 443.4783
• POTE: ~2 = 1\1, ~9/7 = 443.626

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 tunings:

• CTE: ~2 = 1\1, ~9/7 = 443.5075
• POTE: ~2 = 1\1, ~9/7 = 443.559

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 tunings:

• CTE: ~2 = 1\1, ~9/7 = 443.5050
• POTE: ~2 = 1\1, ~9/7 = 443.551

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 tunings:

• CTE: ~2 = 1\1, ~9/7 = 443.1886
• POTE: ~2 = 1\1, ~9/7 = 443.962

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 tunings:

• CTE: ~2 = 1\1, ~9/7 = 443.2863
• POTE: ~2 = 1\1, ~9/7 = 443.945

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 tunings:

• CTE: ~2 = 1\1, ~9/7 = 443.7814
• POTE: ~2 = 1\1, ~9/7 = 443.518

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 tunings:

• CTE: ~2 = 1\1, ~9/7 = 443.7877
• POTE: ~2 = 1\1, ~9/7 = 443.506

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 tunings:

• CTE: ~99/70 = 1\2, ~11/10 = 156.6312
• POTE: ~99/70 = 1\2, ~11/10 = 156.692

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 tunings:

• CTE: ~55/39 = 1\2, ~11/10 = 156.5584
• POTE: ~55/39 = 1\2, ~11/10 = 156.725

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 tunings:

• CTE: ~17/12 = 1\2, ~11/10 = 156.5534

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 tunings:

• CTE: ~2 = 1\1, ~25/22 = 221.5981
• POTE: ~2 = 1\1, ~25/22 = 221.605

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 tunings:

• CTE: ~2 = 1\1, ~25/22 = 221.6333
• POTE: ~2 = 1\1, ~25/22 = 221.556

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]]

Wedgie⟨⟨10 16 17 2 -1 -5]]

Optimal tuning (POTE): ~2 = 1\1, ~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 7 11 12 17], 0 -10 -16 -17 -25]]

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

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

Badness: 0.049933

## Fourfives

See also: Fifive family

Subgroup: 2.3.5.7

Comma list: 245/243, 235298/234375

Mapping[4 4 6 7], 0 5 7 9]]

mapping generators: ~25/21, ~27/25

Optimal tuning (POTE): ~25/21 = 1\4, ~27/25 = 140.754

Badness: 0.114143

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 245/243, 385/384, 235298/234375

Mapping: [4 4 6 7 19], 0 5 7 9 -11]]

Optimal tuning (POTE): ~25/21 = 1\4, ~27/25 = 140.771

Badness: 0.120165

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 245/243, 385/384, 20000/19773

Mapping: [4 4 6 7 19 12], 0 5 7 9 -11 6]]

Optimal tuning (POTE): ~25/21 = 1\4, ~13/12 = 140.760

Badness: 0.067365

### Quadrafives

Subgroup: 2.3.5.7.11

Comma list: 121/120, 245/243, 1375/1372

Mapping: [4 4 6 7 11], 0 5 7 9 6]]

Optimal tuning (POTE): ~25/21 = 1\4, ~27/25 = 140.630

Badness: 0.057268

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 196/195, 245/243, 275/273

Mapping: [4 4 6 7 11 12], 0 5 7 9 6 6]]

Optimal tuning (POTE): ~25/21 = 1\4, ~13/12 = 140.728

Badness: 0.036128

#### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 121/120, 154/153, 170/169, 196/195, 245/243

Mapping: [4 4 6 7 11 12 14], 0 5 7 9 6 6 5]]

Optimal tuning (POTE): ~25/21 = 1\4, ~13/12 = 140.718

Badness: 0.024796

## Bohpier

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

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]]

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

Eigenmonzo (unchanged-interval) basis: 2.5
Eigenmonzo (unchanged-interval) basis: 2.3

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

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

Badness: 0.024864

Music

### 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

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

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]]

Wedgie⟨⟨2 -16 13 -30 15 75]]

Optimal tuning (POTE): ~2 = 1\1, ~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]]

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

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

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]]

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

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]]

Wedgie⟨⟨18 20 35 -10 5 25]]

Optimal tuning (POTE): ~2 = 1\1, ~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]]

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

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

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

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

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

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]]

Wedgie⟨⟨11 1 27 -24 12 60]]

Optimal tuning (POTE): ~2 = 1\1, ~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]]

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

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

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

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

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

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

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

Badness: 0.026774