Compton family

The compton 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

Compton 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

Catcall

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)

Optimal ET sequence: 12, 24d

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