Sensamagic clan
- 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 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
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 = 1901.9550, ~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, and fourfives.
These temperaments are distributed into different family pages.
- Sensi (+126/125) → Sensipent family
- Hedgehog (+50/49) → Porcupine family
- Cohemiripple (+1323/1250) → Ripple family
- Fourfives (+235298/234375) → Fifive family
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. Finally, escaped tempers out 65625/65536, splitting the generator in sixteen.
Discussed elsewhere are
- Father (+16/15 or 28/27) → Father family
- Godzilla (+49/48 or 81/80) → Semaphoresmic clan
- Sidi (+25/24) → Dicot family
- Clyde (+3136/3125) → Kleismic family
- Superpyth (+64/63) → Archytas clan
- Magic (+225/224) → Magic family
- Octacot (+2401/2400) → Tetracot family
- Hemiaug (+128/125) → Augmented family
- Pentacloud (+16807/16384) → Quintile family
- Bamity (+64827/64000) → Amity family
- Rodan (+1029/1024) → Gamelismic clan
- Shrutar (+2048/2025) → Diaschismic family
- Escaped (+65625/65536) → Escapade family
For no-twos extensions, see No-twos subgroup temperaments #BPS.
Considered below are bohpier, salsa, pycnic, superthird, magus and leapweek.
Bohpier
- For the 5-limit version, see Miscellaneous 5-limit temperaments #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]]
Optimal tuning (POTE): ~2 = 1200.000, ~27/25 = 146.474
- 7-odd-limit: ~27/25 = [0 0 1/19⟩
- 9-odd-limit: ~27/25 = [0 1/13⟩
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 = 1200.000, ~12/11 = 146.545
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: 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 = 1200.000, ~12/11 = 146.603
Minimax tuning:
- 13- and 15-odd-limit: ~12/11 = [0 0 1/19⟩
- Unchanged-interval (eigenmonzo) basis: 2.5
Optimal ET sequence: 41, 90ef, 131ef, 221bdeff
Badness: 0.024864
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]]
Optimal tuning (POTE): ~2 = 1200.000, ~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 = 1200.000, ~77/75 = 48.822
Optimal ET sequence: 49f, 123ce, 172f, 295ce, 467bccef
Badness: 0.082158
Salsa
Subgroup: 2.3.5.7
Comma list: 245/243, 32805/32768
Mapping: [⟨1 1 7 -1], ⟨0 2 -16 13]]
Optimal tuning (POTE): ~2 = 1200.000, ~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 = 1200.000, ~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 = 1200.000, ~11/9 = 351.025
Optimal ET sequence: 17, 24, 41, 106df, 147df
Badness: 0.030793
Pycnic
- For the 5-limit version, see Miscellaneous 5-limit 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]]
Optimal tuning (POTE): ~2 = 1200.000, ~45/32 = 567.720
Optimal ET sequence: 17, 19, 55c, 74cd, 93cdd
Badness: 0.073735
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]]
Optimal tuning (POTE): ~2 = 1200.000, ~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 = 1200.000, ~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 = 1200.000, ~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 = 63.158, ~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 = 63.158, ~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 = 63.158, ~3/2 = 705.801
Optimal ET sequence: 19, 76bcdf, 95, 114e
Badness: 0.053197
Magus
- For the 5-limit version, see Miscellaneous 5-limit 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.
Magus has a generator of a sharp ~5/4 (so that ~25/16 is twice as sharp so that it makes sense to equate with 11/7 by tempering 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 tuning (POTE): ~2 = 1200.000, ~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 = 1200.000, ~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 = 1200.000, ~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 = 1200.000, ~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 = 1200.000, ~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 = 1200.000, ~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 = 1200.000, ~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 = 1200.000, ~3/2 = 704.537
Optimal ET sequence: 17, 29c, 46, 109g, 155fg, 264bfgg
Badness: 0.026774