Sensamagic clan: Difference between revisions
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== Leapweek == | == Leapweek == | ||
''Not to be confused with scales produced by leap week calendars such as [[Symmetry454]].'' | |||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
Revision as of 15:42, 30 June 2022
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.
For full 7-limit extensions, we have sensi, bohpier, escaped, salsa, pycnic, cohemiripple, superthird, magus and leapweek discussed below, as well as
- Father, {16/15, 28/27} → Father family
- Sidi, {25/24, 245/243} → Dicot family
- Godzilla, {49/48, 81/80} → Meantone family
- Hedgehog, {50/49, 245/243} → Porcupine 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
- Fourfives, {245/243, 235298/234375} → Fifive family
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 as lambda temperament, which was named after lambda scale.
Subgroup: 3.5.7
Comma list: 245/243
Sval mapping: [⟨1 1 2], ⟨0 -2 1]]
Sval mapping generators: ~3, ~9/7
POTE generator: ~9/7 = 440.4881
Optimal GPV sequence: b4, b9, b13, b56, b69, b82, b95
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 of size 11, 19 and 27 are available. The name "sensi" is a play on the words "semi-" and "sixth."
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 ]]
POTE generator: ~9/7 = 443.383
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]
POTE generator: ~9/7 = 443.322
Optimal GPV sequence: Template:Val list
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]]
POTE generator: ~9/7 = 443.294
Optimal GPV sequence: Template:Val list
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]]
POTE generator: ~9/7 = 443.321
Optimal GPV sequence: Template:Val list
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]]
POTE generator: ~9/7 = 443.365
Optimal GPV sequence: Template:Val list
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]]
POTE generator: ~9/7 = 443.962
Optimal GPV sequence: Template:Val list
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]]
POTE generator: ~9/7 = 443.945
Optimal GPV sequence: Template:Val list
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]]
POTE generator: ~9/7 = 443.626
Optimal GPV sequence: Template:Val list
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]]
POTE generator: ~9/7 = 443.559
Optimal GPV sequence: Template:Val list
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]]
POTE generator: ~9/7 = 443.551
Optimal GPV sequence: Template:Val list
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]]
POTE generator: ~9/7 = 443.518
Optimal GPV sequence: Template:Val list
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]]
POTE generator: ~9/7 = 443.506
Optimal GPV sequence: Template:Val list
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]]
POTE generator: ~25/22 = 221.605
Optimal GPV sequence: Template:Val list
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]]
POTE generator: ~25/22 = 221.556
Optimal GPV sequence: Template:Val list
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]]
POTE generator: ~11/10 = 156.692
Optimal GPV sequence: Template:Val list
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]]
POTE generator: ~11/10 = 156.725
Optimal GPV sequence: Template:Val list
Badness: 0.026339
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 ]]
POTE generator: ~27/25 = 146.474
- 7-odd-limit: ~27/25 = [0 0 1/19⟩
- Eigenmonzos (unchanged intervals): 2, 5/4
- 9-odd-limit: ~27/25 = [0 1/13⟩
- Eigenmonzos (unchanged intervals): 2, 4/3
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]]
POTE generator: ~12/11 = 146.545
Minimax tuning:
- 11-odd-limit: ~12/11 = [1/7 1/7 0 0 -1/14⟩
- Eigenmonzos (unchanged intervals): 2, 11/9
Optimal GPV sequence: Template:Val list
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]]
POTE generator: ~12/11 = 146.603
Minimax tuning:
- 13- and 15-odd-limit: ~12/11 = [0 0 1/19⟩
- Eigenmonzos (unchanged intervals): 2, 5/4
Optimal GPV sequence: Template:Val list
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]]
POTE generator: ~77/75 = 48.828
Optimal GPV sequence: Template:Val list
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]]
POTE generator: ~77/75 = 48.822
Optimal GPV sequence: Template:Val list
Badness: 0.082158
Escaped
This temperament is also called as "sensa" because it tempers out 245/243, 352/351, and 385/384 as a sensamagic temperament. Not to be confused with 19e&27 temperament (sensi extension).
Subgroup: 2.3.5.7
Comma list: 245/243, 65625/65536
Mapping: [⟨1 2 2 4], ⟨0 -9 7 -26]]
Wedgie: ⟨⟨ 9 -7 26 -32 16 80 ]]
POTE generator: ~28/27 = 55.122
Badness: 0.088746
11-limit
Subgroup: 2.3.5.7.11
Comma list: 245/243, 385/384, 4000/3993
Mapping: [⟨1 2 2 4 3], ⟨0 -9 7 -26 10]]
POTE generator: ~28/27 = 55.126
Optimal GPV sequence: Template:Val list
Badness: 0.035844
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 245/243, 352/351, 385/384, 625/624
Mapping: [⟨1 2 2 4 3 2], ⟨0 -9 7 -26 10 37]]
POTE generator: ~28/27 = 55.138
Optimal GPV sequence: Template:Val list
Badness: 0.031366
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 ]]
POTE generator: ~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]]
POTE generator: ~11/9 = 351.014
Optimal GPV sequence: Template:Val list
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]]
POTE generator: ~11/9 = 351.025
Optimal GPV sequence: Template:Val list
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 ]]
POTE generator: ~45/32 = 567.720
Badness: 0.073735
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 ]]
POTE generator: ~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 -3 -5 -5 -8], ⟨0 10 16 17 25]]
POTE generator: ~7/5 = 549.945
Optimal GPV sequence: Template:Val list
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]]
POTE generator: ~7/5 = 549.958
Optimal GPV sequence: Template:Val list
Badness: 0.049933
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 ]]
POTE generator: ~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]]
POTE generator: ~9/7 = 439.152
Optimal GPV sequence: Template:Val list
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]]
POTE generator: ~9/7 = 439.119
Optimal GPV sequence: Template:Val list
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]]
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]]
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]]
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). 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 ]]
POTE generator: ~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]]
POTE generator: ~5/4 = 391.503
Optimal GPV sequence: Template:Val list
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]]
POTE generator: ~5/4 = 391.366
Optimal GPV sequence: Template:Val list
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 1 17 -6], ⟨0 1 -25 15]]
POTE generator: ~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 1 17 -6 -3], ⟨0 1 -25 15 11]]
POTE generator: ~3/2 = 704.554
Optimal GPV sequence: Template:Val list
Badness: 0.050679
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 169/168, 245/243, 352/351, 364/363
Mapping: [⟨1 1 17 -6 -3 -1], ⟨0 1 -25 15 11 8]]
POTE generator: ~3/2 = 704.571
Optimal GPV sequence: Template:Val list
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 1 17 -6 -3 -1 -10], ⟨0 1 -25 15 11 8 24]]
POTE generator: ~3/2 = 704.540
Optimal GPV sequence: Template:Val list
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 1 17 -6 -3 -1 17], ⟨0 1 -25 15 11 8 -22]]
POTE generator: ~3/2 = 704.537
Optimal GPV sequence: Template:Val list
Badness: 0.026774
Semiwolf
Subgroup: 3/2.7/4.5/2
Comma list: 245/243
Mapping: [⟨1 1 3], ⟨0 1 -2]]
POL2 generator: ~7/6 = 262.1728
Optimal GPV sequence: 3edf, 5edf, 8edf
Semilupine
Subgroup: 3/2.7/4.5/2.11/4
Comma list: 100/99, 245/243
Mapping: [⟨1 1 3 4], ⟨0 1 -2 -4]]
POL2 generator: ~7/6 = 264.3771
Optimal GPV sequence: 8edf, 13edf
Hemilycan
Subgroup: 3/2.7/4.5/2.11/4
Comma list: 245/243, 441/440
Mapping: [⟨1 1 3 1], ⟨0 1 -2 4]]
POL2 generator: ~7/6 = 261.5939