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
Sval 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
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 -1 -1 -2], ⟨0 7 9 13]]
Mapping generators: ~2, ~9/7
Wedgie: ⟨⟨ 7 9 13 -2 1 5 ]]
Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 443.383
Optimal ET sequence: 19, 27, 46, 157d, 203cd, 249cdd, 295ccdd
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]
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 -1 -1 -2 9], ⟨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 -1 -1 -2 9 0], ⟨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 -1 -1 -2 9 0 10], ⟨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
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]]
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 -1 -1 -2 2 0], ⟨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
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]]
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 -1 -1 -2 -8 0], ⟨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 -1 -1 -2 -8 0 -7], ⟨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
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]]
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 -1 -1 -2 -1 0], ⟨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
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]]
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 -1 -1 -2 -3 0], ⟨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
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]]
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): ~99/70 = 1\2, ~11/10 = 156.725
Optimal ET sequence: 8d, …, 38df, 46
Badness: 0.026339
Cohemiripple
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 ]]
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 -3 -5 -5 -8], ⟨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 -3 -5 -5 -8 -5], ⟨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.
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]]
Wedgie: ⟨⟨ 13 19 23 0 0 0 ]]
Optimal tuning (POTE): ~2 = 1\1, ~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 = 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 basis (unchanged-interval basis): 2.5
Optimal ET sequence: 41, 90ef, 131ef, 221bdeff
Badness: 0.024864
- Music
by Chris Vaisvil:
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
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
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
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
Optimal ET sequence: 17, 19, 55c, 74cd, 93cdd
Badness: 0.073735
Superthird
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
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]]
Wedgie: ⟨⟨ 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