Compton family
The compton family, otherwise known as the aristoxenean family, tempers out the Pythagorean comma, 531441/524288 = [-19 12⟩, and hence the fifths form a closed 12-note circle of fifths, identical to 12edo. While the tuning of the fifth will be that of 12edo, two cents flat, the tuning of the larger primes is not so constrained, and the point of these temperaments is to improve on it.
Compton
3-limit as in 12edo; intervals of 5 are off by one generator. In the 7-limit (sometimes called waage), intervals of 7 are off by two generators. In the 11-limit, intervals of 11 are off by 3 generators. Thinking of 72edo might make this more concrete.
5-limit compton is also known as aristoxenean. It tempers out the Pythagorean comma and has a period of 1\12, so it is the 12edo circle of fifths with an independent dimension for the harmonic 5. Equivalent generators are 5/4, 6/5, 10/9, 16/15 (the secor), 45/32, 135/128 and most importantly, 81/80. In terms of equal temperaments, it is the 12 & 72 temperament, and 72edo, 84edo or 240edo make for good tunings.
Subgroup: 2.3.5
Comma list: 531441/524288
Mapping: [⟨12 19 0], ⟨0 0 1]]
- mapping generators: ~256/243, ~5
Optimal tuning (POTE): ~256/243 = 1\12, ~5/4 = 384.884 (~81/80 = 15.116)
Optimal ET sequence: 12, 48, 60, 72, 84, 156, 240, 396b, 636bbc
Badness: 0.094494
Septimal compton
Septimal compton is also known as waage. In terms of the normal list, compton adds 413343/409600 = [-14 10 -2 1⟩ to the Pythagorean comma; however, it can also be characterized by saying it adds 225/224.
In either the 5- or 7-limit, 240edo is an excellent tuning, with 81/80 coming in at 15 cents exactly. In the 12edo, the major third is sharp by 13.686 cents, and the minor third flat by 15.641 cents; adjusting these down and up by 15 cents puts them in excellent tune.
In terms of the normal comma list, we may add 8019/8000 to get to the 11-limit version of compton, which also adds 441/440. For this 72edo can be recommended as a tuning.
Subgroup: 2.3.5.7
Comma list: 225/224, 250047/250000
Mapping: [⟨12 19 0 -22], ⟨0 0 1 2]]
Optimal tuning (POTE): ~256/243 = 1\12, ~5/4 = 383.7752 (~126/125 = 16.2248)
Optimal ET sequence: 12, 48d, 60, 72, 228, 300c, 372bc, 444bc
Badness: 0.035686
11-limit
Subgroup: 2.3.5.7.11
Comma list: 225/224, 441/440, 4375/4356
Mapping: [⟨12 19 0 -22 -42], ⟨0 0 1 2 3]]
Optimal tuning (POTE): ~256/243 = 1\12, ~5/4 = 383.2660 (~100/99 = 16.7340)
Optimal ET sequence: 12, 48dee, 60e, 72
Badness: 0.022235
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 225/224, 351/350, 364/363, 441/440
Mapping: [⟨12 19 0 -22 -42 -67], ⟨0 0 1 2 3 4]]
Optimal tuning (POTE): ~256/243 = 1\12, ~5/4 = 383.9628 (~105/104 = 16.0372)
Optimal ET sequence: 12f, 48defff, 60eff, 72, 228f
Badness: 0.021852
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 221/220, 225/224, 289/288, 351/350, 441/440
Mapping: [⟨12 19 0 -22 -42 -67 49], ⟨0 0 1 2 3 4 0]]
Optimal tuning (POTE): ~18/17 = 1\12, ~5/4 = 383.7500 (~105/104 = 16.2500)
Optimal ET sequence: 12f, 60eff, 72
Badness: 0.017131
Comptone
Subgroup: 2.3.5.7.11.13
Comma list: 225/224, 325/324, 441/440, 1001/1000
Mapping: [⟨12 19 0 -22 -42 100], ⟨0 0 1 2 3 -2]]
Optimal tuning (POTE): ~256/243 = 1\12, ~5/4 = 382.6116 (~100/99 = 17.3884)
Optimal ET sequence: 12, 60e, 72, 204cdef, 276cdeff
Badness: 0.025144
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 225/224, 273/272, 289/288, 325/324, 441/440
Mapping: [⟨12 19 0 -22 -42 100 49], ⟨0 0 1 2 3 -2 0]]
Optimal tuning (POTE): ~18/17 = 1\12, ~5/4 = 382.5968 (~100/99 = 17.4032)
Optimal ET sequence: 12, 60e, 72, 204cdefg, 276cdeffgg
Badness: 0.016361
Catler
In terms of the normal comma list, catler is characterized by the addition of the schisma, 32805/32768, to the Pythagorean comma, though it can also be characterized as adding 81/80, 128/125 or 648/625. In any event, the 5-limit is exactly the same as the 5-limit of 12edo. Catler can also be characterized as the 12 & 24 temperament. 36edo or 48edo are possible tunings. Possible generators are 36/35, 21/20, 15/14, 8/7, 7/6, 9/7, 7/5, and most importantly, 64/63.
Subgroup: 2.3.5.7
Comma list: 81/80, 128/125
Mapping: [⟨12 19 28 0], ⟨0 0 0 1]]
- mapping generators: ~16/15, ~7
Optimal tuning (POTE): ~16/15 = 1\12, ~7/4 = 973.210 (~64/63 = 26.790)
Optimal ET sequence: 12, 24, 36, 48c
Badness: 0.050297
11-limit
Subgroup: 2.3.5.7.11
Comma list: 81/80, 99/98, 128/125
Mapping: [⟨12 19 28 0 -26], ⟨0 0 0 1 2]]
Optimal tuning (POTE): ~16/15 = 1\12, ~7/4 = 977.277 (~64/63 = 22.723)
Optimal ET sequence: 12, 36e, 48c, 108ccd
Badness: 0.058213
Catlat
Subgroup: 2.3.5.7.11
Comma list: 81/80, 128/125, 540/539
Mapping: [⟨12 19 28 0 109], ⟨0 0 0 1 -2]]
Optimal tuning (POTE): ~16/15 = 1\12, ~7/4 = 972.136 (~64/63 = 27.864)
Optimal ET sequence: 36, 48c, 84c
Badness: 0.081909
Catnip
Subgroup: 2.3.5.7.11
Comma list: 56/55, 81/80, 128/125
Mapping: [⟨12 19 28 0 8], ⟨0 0 0 1 1]]
Optimal tuning (POTE): ~16/15 = 1\12, ~7/4 = 967.224 (~64/63 = 32.776)
Optimal ET sequence: 12, 24, 36, 72ce
Badness: 0.034478
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 56/55, 66/65, 81/80, 105/104
Mapping: [⟨12 19 28 0 8 11], ⟨0 0 0 1 1 1]]
Optimal tuning (POTE): ~16/15 = 1\12, ~7/4 = 962.778 (~40/39 = 37.232)
Optimal ET sequence: 12f, 24, 36f, 60cf
Badness: 0.028363
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 51/50, 56/55, 66/65, 81/80, 105/104
Mapping: [⟨12 19 28 0 8 11 49], ⟨0 0 0 1 1 1 0]]
Optimal tuning (POTE): ~18/17 = 1\12, ~7/4 = 960.223 (~40/39 = 39.777)
Optimal ET sequence: 12f, 24, 36f, 60cf
Badness: 0.023246
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 51/50, 56/55, 66/65, 76/75, 81/80, 96/95
Mapping: [⟨12 19 28 0 8 11 49 51], ⟨0 0 0 1 1 1 0 0]]
Optimal tuning (POTE): ~18/17 = 1\12, ~7/4 = 959.835 (~40/39 = 40.165)
Optimal ET sequence: 12f, 24, 36f, 60cf
Badness: 0.018985
Duodecic
Subgroup: 2.3.5.7.11.13
Comma list: 56/55, 81/80, 91/90, 128/125
Mapping: [⟨12 19 28 0 8 78], ⟨0 0 0 1 1 -1]]
Optimal tuning (POTE): ~16/15 = 1\12, ~7/4 = 962.312 (~64/63 = 37.688)
Optimal ET sequence: 12, 24, 36, 60c
Badness: 0.038307
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 51/50, 56/55, 81/80, 91/90, 128/125
Mapping:[⟨12 19 28 0 8 78 49], ⟨0 0 0 1 1 -1 0]]
Optimal tuning (POTE): ~18/17 = 1\12, ~7/4 = 961.903 (~64/63 = 38.097)
Optimal ET sequence: 12, 24, 36, 60c
Badness: 0.027487
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 51/50, 56/55, 76/75, 81/80, 91/90, 96/95
Mapping: [⟨12 19 28 0 8 78 49 51], ⟨0 0 0 1 1 -1 0 0]]
Optimal tuning (POTE): ~18/17 = 1\12, ~7/4 = 961.920 (~64/63 = 38.080)
Optimal ET sequence: 12, 24, 36, 60c
Badness: 0.020939
Duodecim
- See also: Jubilismic clan #Duodecim
Subgroup: 2.3.5.7.11
Comma list: 36/35, 50/49, 64/63
Mapping: [⟨12 19 28 34 0], ⟨0 0 0 0 1]]
- mapping generators: ~16/15, ~11
Optimal tuning (POTE): ~16/15 = 1\12, ~11/8 = 565.023 (~55/54 = 34.977)
Badness: 0.030536
Hours
The hours temperament has a period of 1/24 octave and tempers out the cataharry comma (19683/19600) and the mirwomo comma (33075/32768). The name "hours" was so named for the following reasons – the period is 1/24 octave, and there are 24 hours per a day.
Subgroup: 2.3.5.7
Comma list: 19683/19600, 33075/32768
Mapping: [⟨24 38 0 123], ⟨0 0 1 -1]]
Wedgie: ⟨⟨0 24 -24 38 -38 -123]]
- mapping generators: ~36/35, ~5
Optimal tuning (POTE): ~36/35 = 1\24, ~5/4 = 384.033
Optimal ET sequence: 24, 48, 72, 312bd, 384bcdd, 456bcdd, 528bcdd, 600bccdd
Badness: 0.116091
11-limit
Subgroup: 2.3.5.7.11
Comma list: 243/242, 385/384, 9801/9800
Mapping: [⟨24 38 0 123 83], ⟨0 0 1 -1 0]]
Optimal tuning (POTE): ~36/35 = 1\24, ~5/4 = 384.054
Optimal ET sequence: 24, 48, 72, 312bd, 384bcdd, 456bcdde, 528bcdde
Badness: 0.036248
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 243/242, 351/350, 364/363, 385/384
Mapping: [⟨24 38 0 123 83 33], ⟨0 0 1 -1 0 1]]
Optimal tuning (POTE): ~36/35 = 1\24, ~5/4 = 384.652
Optimal ET sequence: 24, 48f, 72, 168df, 240dff
Badness: 0.026931
Decades
The decades temperament has a period of 1/36 octave and tempers out the gamelisma (1029/1024) and the stearnsma (118098/117649). The name "decades" was so named for the following reasons – the period is 1/36 octave, and there are 36 decades (ten days) per a year (12 months × 3 decades per a month).
Subgroup: 2.3.5.7
Comma list: 1029/1024, 118098/117649
Mapping: [⟨36 57 0 101], ⟨0 0 1 0]]
- mapping generators: ~49/48, ~5
Wedgie: ⟨⟨0 36 0 57 0 -101]]
Optimal tuning (POTE): ~49/48 = 1\36, ~5/4 = 384.764
Optimal ET sequence: 36, 72, 252, 324bd, 396bd
Badness: 0.108016
11-limit
Subgroup: 2.3.5.7.11
Comma list: 540/539, 1029/1024, 4000/3993
Mapping: [⟨36 57 0 101 41], ⟨0 0 1 0 1]]
Optimal tuning (POTE): ~49/48 = 1\36, ~5/4 = 384.150
Optimal ET sequence: 36, 72, 396bd, 468bcd, 540bcd, 612bccdd, 684bbccdd, 756bbccdd
Badness: 0.043088
Omicronbeta
Subgroup: 2.3.5.7.11.13
Comma list: 225/224, 243/242, 441/440, 4375/4356
Mapping: [⟨72 114 167 202 249 266], ⟨0 0 0 0 0 1]]
- mapping generators: ~100/99, ~13
Optimal tuning (POTE): ~100/99 = 1\72, ~13/8 = 837.814
Optimal ET sequence: 72, 144, 216c, 288cdf, 504bcdef
Badness: 0.029956