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Sensamagic clan

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This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The sensamagic clan of temperaments 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

Main article: BPS

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]]

mapping generators: ~3, ~9/7

Optimal tunings:

  • WE: ~3 = 1903.7398 ¢, ~9/7 = 440.9014 ¢
error map: ⟨+1.785 -0.771 -2.248]
  • CWE: ~3 = 1901.9550 ¢, ~9/7 = 440.6646 ¢
error map: ⟨0.000 -3.030 -5.580]

Optimal ET sequence: b4, b9, b13, b56, b69, b82, b95, b367cdd, b462cdd

Badness (Sintel): 0.0659

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.

These temperaments are distributed into different family pages.

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. Pentacloud 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. Salsa tempers out 32805/32768, splitting the generator in fifteen. Finally, escaped tempers out 65625/65536, splitting the generator in sixteen.

Discussed elsewhere are

For no-twos extensions, see No-twos subgroup temperaments #BPS.

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

Bohpier

Main article: Bohpier
For the 5-limit version, see Miscellaneous 5-limit temperaments #Bohpier.

Bohpier tempers out 3125/3087 and may be described as the 41 & 49 temperament. It is named after its interesting relationship with the non-octave Bohlen–Pierce equal temperament.

41edo itself makes for an excellent tuning, though 90edo and 131edo are interesting alternatives. Another notable tuning is given by TE, CTE and POTE, all coinciding at 146.4741 ¢ with pure octaves since prime 2 is not involved in the comma to begin with, though its difference from WE and/or CWE (shown below) is largely unnoticeable.

Subgroup: 2.3.5.7

Comma list: 245/243, 3125/3087

Mapping: [⟨1 0 0 0], ⟨0 13 19 23]]

mapping generators: ~2, ~27/25

Optimal tunings:

  • WE: ~2 = 1199.9967 ¢, ~27/25 = 146.4737 ¢
error map: ⟨-0.003 +2.203 -3.314 +0.068]
  • CWE: ~2 = 1200.0000 ¢, ~27/25 = 146.4739 ¢
error map: ⟨0.000 +2.205 -3.310 +0.073]

Minimax tuning:

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

Optimal ET sequence: 8d, …, 41, 131, 172, 213c

Badness (Sintel): 1.73

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 tunings:

  • WE: ~2 = 1199.2309 ¢, ~12/11 = 146.4507 ¢
  • CWE: ~2 = 1200.0000 ¢, ~12/11 = 146.5009 ¢

Minimax tuning:

  • 11-odd-limit: ~12/11 = [1/7 1/7 0 0 -1/14⟩
unchanged-interval (eigenmonzo) basis: 2.11/9

Optimal ET sequence: 8d, …, 41, 90e, 131e

Badness (Sintel): 1.12

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 tunings:

  • WE: ~2 = 1198.5478 ¢, ~12/11 = 146.4252 ¢
  • CWE: ~2 = 1200.0000 ¢, ~12/11 = 146.5230 ¢

Minimax tuning:

  • 13- and 15-odd-limit: ~12/11 = [0 0 1/19⟩
unchanged-interval (eigenmonzo) basis: 2.5

Optimal ET sequence: 8d, …, 41, 90ef

Badness (Sintel): 1.03

Triboh

Triboh is named after the "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]]

mapping generators: ~2, ~77/75

Optimal tunings:

  • WE: ~2 = 1199.9966 ¢, ~77/75 = 48.8281 ¢
  • CWE: ~2 = 1200.0000 ¢, ~77/75 = 48.8282 ¢

Optimal ET sequence: 49, 123ce, 172

Badness (Sintel): 5.38

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 tunings:

  • WE: ~2 = 1199.9962 ¢, ~77/75 = 48.8219 ¢
  • CWE: ~2 = 1200.0000 ¢, ~77/75 = 48.8219 ¢

Optimal ET sequence: 49f, 123ce, 172f

Badness (Sintel): 3.39

Pycnic

For the 5-limit version, see Syntonic–kleismic equivalence continuum #Stump.

Pycnic is related to triton, but its mapping differs for the 7th harmonic. It is also related to liese, from which its mapping differs for the 5th harmonic.

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 two cents sharp of it in the CWE 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 0 6 -3], ⟨0 3 -7 11]]

mapping generators: ~2, ~64/45

Optimal tunings:

  • WE: ~2 = 1203.3437 ¢, ~64/45 = 634.0416 ¢
error map: ⟨+3.344 +0.170 -4.542 -4.400]
  • CWE: ~2 = 1200.0000 ¢, ~64/45 = 632.3502 ¢
error map: ⟨0.000 -4.904 -12.765 -12.973]

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

Badness (Sintel): 1.87

Xenia

For the 5-limit version, see Syntonic–kleismic equivalence continuum #Xenial.

Xenia is related to xenial, but its mapping differs for the 7th harmonic. It may be described as 19 & 51c or 19 & 70d, which tempers out the sensamagic and keega, 1029/1000.

Subgroup: 2.3.5.7

Comma list: 245/243, 1029/1000

Mapping: [⟨1 -6 -12 -9], ⟨0 9 17 14]]

mapping generators: ~2, ~9/5

Optimal tunings:

  • WE: ~2 = 1201.0862 ¢, ~9/5 = 1012.0503 ¢
error map: ⟨+1.086 -0.020 +5.507 -9.898]
  • CWE: ~2 = 1200.0000 ¢, ~9/5 = 1011.2199 ¢
error map: ⟨0.000 -0.976 +4.424 -11.748]

Optimal ET sequence: 19, 70d, 89d

Badness (Sintel): 2.25

Magus

For the 5-limit version, see Miscellaneous 5-limit temperaments #Magus.

Magus temperament tempers out 50331648/48828125 in the 5-limit. This temperament can be described as 46 & 49 temperament, which tempers out the sensamagic and 28672/28125. 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.

Magus has a generator of a sharp ~5/4, and ~25/16 is twice as sharp so that it makes sense to equate with 11/7 by tempering out 176/175), so that three reaches 128/125 short of the octave, where 128/125 is tuned narrow; this is significant because magus reaches 3/2 as (25/16)/(128/125)3, that is, 2 + 3 × 3 = 11 generators. Therefore, it implies that 25/24 is split into three 128/125's. Therefore, in the 5-limit, magus can be thought of as a higher-complexity and sharper analogue of würschmidt (which reaches 3/2 as (25/16)/(128/125)2 implying 25/24 is split into two 128/125's thus having a guaranteed neutral third), which itself is a higher-complexity and sharper analogue of magic (which equates 25/24 with 128/125 by flattening 5). For more details on these connections see Würschmidt comma.

Subgroup: 2.3.5.7

Comma list: 245/243, 28672/28125

Mapping: [⟨1 -2 2 -6], ⟨0 11 1 27]]

Optimal tunings:

  • WE: ~2 = 1198.7187 ¢, ~5/4 = 391.0473 ¢
error map: ⟨-1.281 +2.128 +2.171 -2.860]
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 391.4129 ¢
error map: ⟨0.000 +3.587 +5.099 -0.678]

Optimal ET sequence: 46, 95, 141bc, 187bc

Badness (Sintel): 2.74

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 tunings:

  • WE: ~2 = 1198.7144 ¢, ~5/4 = 391.0836 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 391.4506 ¢

Optimal ET sequence: 46, 95, 141bc

Badness (Sintel): 1.49

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 tunings:

  • WE: ~2 = 1199.7708 ¢, ~5/4 = 391.2912 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 391.3597 ¢

Optimal ET sequence: 3de, 43de, 46

Badness (Sintel): 1.78

Superenneadecal

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

Subgroup: 2.3.5.7

Comma list: 245/243, 395136/390625

Mapping: [⟨19 0 14 -7], ⟨0 1 1 2]]

mapping generators: ~392/375, ~3

Optimal tunings:

  • WE: ~392/375 = 63.1399 ¢, ~3/2 = 703.9652 ¢
error map: ⟨-0.343 +1.668 +1.267 -3.560]
  • CWE: ~392/375 = 63.1579 ¢, ~3/2 = 703.9028 ¢
error map: ⟨0.000 +1.948 +1.800 -3.126]

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

Badness (Sintel): 3.35

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 tunings:

  • WE: ~33/32 = 63.0966 ¢, ~3/2 = 704.9824 ¢
  • CWE: ~33/32 = 63.1579 ¢, ~3/2 = 705.3096 ¢

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

Badness (Sintel): 3.36

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 tunings:

  • WE: ~33/32 = 63.0988 ¢, ~3/2 = 705.1402 ¢
  • CWE: ~33/32 = 63.1579 ¢, ~3/2 = 705.4315 ¢

Optimal ET sequence: 19, 76bcdf, 95, 114e, 209bcef

Badness (Sintel): 2.20

Superthird

For the 5-limit version, see Miscellaneous 5-limit temperaments #Shibboleth.

Subgroup: 2.3.5.7

Comma list: 245/243, 78125/76832

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

mapping generators: ~2, ~9/7

Optimal tunings:

  • WE: ~2 = 1200.3935 ¢, ~9/7 = 439.2199 ¢
error map: ⟨+0.394 +2.035 -3.884 -0.066]
  • CWE: ~2 = 1200.0000 ¢, ~9/7 = 439.0931 ¢
error map: ⟨0.000 +1.721 -4.452 -0.568]

Optimal ET sequence: 11cd, 30d, 41

Badness (Sintel): 3.53

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 tunings:

  • WE: ~2 = 1199.5116{c}}, ~9/7 = 438.9734 ¢
  • CWE: ~2 = 1200.0000 ¢, ~9/7 = 439.1362 ¢

Optimal ET sequence: 11cd, 30d, 41, 153be

Badness (Sintel): 2.34

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 tunings:

  • WE: ~2 = 1199.2631{c}}, ~9/7 = 438.8494 ¢
  • CWE: ~2 = 1200.0000 ¢, ~9/7 = 439.0943 ¢

Optimal ET sequence: 11cdf, 30df, 41

Badness (Sintel): 2.18

Leapweek

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

Leapweek may be described as the 46 & 63 temperament, generated by a perfect fifth and being a strong extension of leapfrog. 109edo makes for an excellent tuning.

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 tunings:

  • WE: ~2 = 1199.6301 ¢, ~3/2 = 704.3191 ¢
error map: ⟨-0.370 +1.994 -0.578 -1.821]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.5387 ¢
error map: ⟨0.000 +2.584 +0.218 -0.745]

Optimal ET sequence: 17, 46, 109, 155, 264b

Badness (Sintel): 3.56

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 tunings:

  • WE: ~2 = 1199.7910 ¢, ~3/2 = 704.4312 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.5542 ¢

Optimal ET sequence: 17, 46, 109, 264b

Badness (Sintel): 1.68

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 tunings:

  • WE: ~2 = 1200.0070 ¢, ~3/2 = 704.5751 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.5709 ¢

Optimal ET sequence: 17, 46, 63, 109

Badness (Sintel): 1.35

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 tunings:

  • WE: ~2 = 1199.8670 ¢, ~3/2 = 704.4620 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.5395 ¢

Optimal ET sequence: 17g, 46, 109

Badness (Sintel): 1.34

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 tunings:

  • WE: ~2 = 1200.1737 ¢, ~3/2 = 704.6390 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.5364 ¢

Optimal ET sequence: 17, 46, 109g, 155fg

Badness (Sintel): 1.36

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