Compton family

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 compton family, otherwise known as the aristoxenean family, of temperaments tempers out the Pythagorean comma (ratio: 531441/524288, monzo: [-19 12⟩, and hence the fifths form a closed 12-note circle of fifths, identical to 12edo. While the tuning of the fifth is fixed to 7 steps of 12edo, about 2 ¢ flat of just, these temperaments aim to add tunings for higher primes which are more in tune than in 12edo.

Compton

Main article: Compton

5-limit 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, compton is the 12 & 72 temperament; its ploidacot is dodecaploid acot. 72edo, 84edo or 240edo make for good tunings.

This temperament is documented as aristoxenean in Tonalsoft Encyclopedia.

Subgroup: 2.3.5

Comma list: 531441/524288

Mapping: [⟨12 19 0], ⟨0 0 1]]

mapping generators: ~256/243, ~5

Optimal tunings:

• WE: ~256/243 = 100.0513 ¢, ~5/4 = 385.0800 ¢ (~81/80 = 15.1253 ¢)

error map: ⟨+0.616 -0.980 -0.001]

• CWE: ~256/243 = 100.0000 ¢, ~5/4 = 385.3590 ¢ (~81/80 = 14.6410 ¢)

error map: ⟨0.000 -1.955 -0.955]

Optimal ET sequence: 12, 48, 60, 72, 84, 156, 240, 396b, 636bbc

Badness (Sintel): 2.22

Septimal compton

Main article: Compton

In terms of the normal comma list, septimal compton adds 413343/409600 ([-14 10 -2 1⟩) to the Pythagorean comma; however, it can also be characterized by saying it adds 225/224. Other important commas of this temperament are 250047/250000, the landscape comma, which sets 63/50 to 1/3 of an octave, and 390625/388962, the dimcomp comma, which sets 25/21 to 1/4 of an octave.

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.

Septimal compton is catalogued as waage in Graham Breed's temperament finder.

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 and has a natural extension to the 13-limit. In 13-limit compton, intervals of 5, 7, 11, and 13 are off by one, two, three, and four generators, respectively. For these, 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 tunings:

• WE: ~256/243 = 100.0579 ¢, ~5/4 = 383.9974 ¢ (~126/125 = 16.2342 ¢)

error map: ⟨+0.695 -0.855 -0.927 +0.674]

• CWE: ~256/243 = 100.0000 ¢, ~5/4 = 384.1429 ¢ (~126/125 = 15.8571 ¢)

error map: ⟨0.000 -1.955 -2.171 -0.540]

Optimal ET sequence: 12, …, 60, 72, 228, 300c, 372bc, 444bc

Badness (Sintel): 0.903

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 tunings:

• WE: ~35/33 = 100.0633 ¢, ~5/4 = 383.5087 ¢ (~100/99 = 16.7446 ¢)
• CWE: ~35/33 = 100.0000 ¢, ~5/4 = 383.5958 ¢ (~100/99 = 16.4042 ¢)

Optimal ET sequence: 12, …, 60e, 72

Badness (Sintel): 0.735

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 tunings:

• WE: ~35/33 = 100.0508 ¢, ~5/4 = 384.1577 ¢ (~100/99 = 16.0454 ¢)
• CWE: ~35/33 = 100.0000 ¢, ~5/4 = 384.1782 ¢ (~100/99 = 15.8218 ¢)

Optimal ET sequence: 12f, …, 60eff, 72, 228f

Badness (Sintel): 0.903

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 tunings:

• WE: ~18/17 = 100.0658 ¢, ~5/4 = 384.0024 ¢ (~100/99 = 16.2607 ¢)
• CWE: ~18/17 = 100.0000 ¢, ~5/4 = 383.9647 ¢ (~100/99 = 16.0353 ¢)

Optimal ET sequence: 12f, 60eff, 72

Badness (Sintel): 0.873

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 tunings:

• WE: ~35/33 = 100.0926 ¢, ~5/4 = 382.9660 ¢ (~100/99 = 17.4045 ¢)
• CWE: ~35/33 = 100.0000 ¢, ~5/4 = 382.7748 ¢ (~100/99 = 17.2252 ¢)

Optimal ET sequence: 12, 60e, 72, 204cdef, 276cdeff

Badness (Sintel): 1.04

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 tunings:

• WE: ~18/17 = 100.0941 ¢, ~5/4 = 382.9567 ¢ (~100/99 = 17.4796 ¢)
• CWE: ~18/17 = 100.0000 ¢, ~5/4 = 382.7381 ¢ (~100/99 = 17.2619 ¢)

Optimal ET sequence: 12, 60e, 72, 204cdefg, 276cdeffgg

Badness (Sintel): 0.833

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 tunings:

• WE: ~16/15 = 99.8680 ¢, ~7/4 = 971.9257 ¢ (~64/63 = 26.7545 ¢)

error map: ⟨-1.584 -4.463 +9.991 -0.068]

• CWE: ~16/15 = 100.0000 ¢, ~7/4 = 972.0971 ¢ (~64/63 = 27.9029 ¢)

error map: ⟨0.000 -1.955 +13.686 +3.271]

Optimal ET sequence: 12, 24, 36, 48c, 84c

Badness (Sintel): 1.27

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 tunings:

• WE: ~16/15 = 99.8542 ¢, ~7/4 = 975.8519 ¢ (~64/63 = 22.6896 ¢)
• CWE: ~16/15 = 100.0000 ¢, ~7/4 = 976.4125 ¢ (~64/63 = 23.5875 ¢)

Optimal ET sequence: 12, 36e, 48c

Badness (Sintel): 1.92

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 tunings:

• WE: ~16/15 = 99.8791 ¢, ~7/4 = 970.9614 ¢ (~64/63 = 27.8300 ¢)
• CWE: ~16/15 = 100.0000 ¢, ~7/4 = 972.2549 ¢ (~64/63 = 27.7451 ¢)

Optimal ET sequence: 12e, 36, 48c, 84c

Badness (Sintel): 2.71

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 tunings:

• WE: ~16/15 = 99.8519 ¢, ~7/4 = 965.7912 ¢ (~64/63 = 32.7275 ¢)
• CWE: ~16/15 = 100.0000 ¢, ~7/4 = 965.8666 ¢ (~64/63 = 34.1334 ¢)

Optimal ET sequence: 12, 24, 36

Badness (Sintel): 1.14

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 tunings:

• WE: ~16/15 = 99.8308 ¢, ~7/4 = 961.1391 ¢ (~40/39 = 37.1694 ¢)
• CWE: ~16/15 = 100.0000 ¢, ~7/4 = 961.1435 ¢ (~40/39 = 38.8565 ¢)

Optimal ET sequence: 12f, 24, 36f

Badness (Sintel): 1.18

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 tunings:

• WE: ~18/17 = 99.8958 ¢, ~7/4 = 959.2226 ¢ (~40/39 = 39.7354 ¢)
• CWE: ~18/17 = 100.0000 ¢, ~7/4 = 959.4216 ¢ (~40/39 = 40.5784 ¢)

Optimal ET sequence: 12f, 24, 36f

Badness (Sintel): 1.18

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 tunings:

• WE: ~18/17 = 99.9058 ¢, ~7/4 = 958.9307 ¢ (~40/39 = 40.1270 ¢)
• CWE: ~18/17 = 100.0000 ¢, ~7/4 = 959.2303 ¢ (~40/39 = 40.7697 ¢)

Optimal ET sequence: 12f, 24, 36f

Badness (Sintel): 1.15

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 tunings:

• WE: ~18/17 = 99.9301 ¢, ~7/4 = 961.6396 ¢ (~64/63 = 37.6617 ¢)
• CWE: ~18/17 = 100.0000 ¢, ~7/4 = 962.1413 ¢ (~64/63 = 37.8587 ¢)

Optimal ET sequence: 12, 24, 36

Badness (Sintel): 1.58

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 tunings:

• WE: ~18/17 = 99.9556 ¢, ~7/4 = 961.4763 ¢ (~64/63 = 38.0796 ¢)
• CWE: ~18/17 = 100.0000 ¢, ~7/4 = 961.8075 ¢ (~64/63 = 38.1925 ¢)

Optimal ET sequence: 12, 24, 36, 60c

Badness (Sintel): 1.40

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 tunings:

• WE: ~18/17 = 99.9545 ¢, ~7/4 = 961.4829 ¢ (~64/63 = 38.0624 ¢)
• CWE: ~18/17 = 100.0000 ¢, ~7/4 = 961.8354 ¢ (~64/63 = 38.1646 ¢)

Optimal ET sequence: 12, 24, 36, 60c

Badness (Sintel): 1.27

Duodecim

Duodecim uses exactly the same mapping as the 7-limit of 12edo, only correcting its poor approximation of prime 11.

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 genereators: ~16/15, ~11

Optimal tunings:

• WE: ~16/15 = 99.6643 ¢, ~11/8 = 563.1257 ¢ (~55/54 = 34.8599 ¢)

error map: ⟨-4.029 -8.334 +4.285 +19.759 -0.279]

• CWE: ~16/15 = 100.0000 ¢, ~11/8 = 562.2258 ¢ (~55/54 = 37.7742 ¢)

error map: ⟨0.000 -1.955 +13.686 +31.174 +10.908]

Optimal ET sequence: 12, 24d, 36d

Badness (Sintel): 1.01

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). Its ploidacot is 24-ploid acot. The name hours was given for the reason that the period is 1/24 octave and there are 24 hours per day.

Subgroup: 2.3.5.7

Comma list: 19683/19600, 33075/32768

Mapping: [⟨24 38 0 123], ⟨0 0 1 -1]]

mapping generators: ~36/35, ~5

Optimal tunings:

• WE: ~36/35 = 50.0337 ¢, ~5/4 = 384.2919 ¢ (~81/80 = 15.9775 ¢)

error map: ⟨+0.808 -0.675 -0.406 -0.592]

• CWE: ~36/35 = 50.0000 ¢, ~5/4 = 384.0719 ¢ (~81/80 = 15.9281 ¢)

error map: ⟨0.000 -1.955 -2.242 -2.898]

Optimal ET sequence: 24, 48, 72, 312bd, 384bcdd, 456bcdd, 528bcdd, 600bccdd

Badness (Sintel): 2.94

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 tunings:

• WE: ~36/35 = 50.0301 ¢, ~5/4 = 384.2848 ¢ (~121/120 = 15.9559 ¢)
• CWE: ~36/35 = 50.0000 ¢, ~5/4 = 384.0825 ¢ (~121/120 = 15.9175 ¢)

Optimal ET sequence: 24, 48, 72, 312bd, 384bcdd, 456bcdde, 528bcdde

Badness (Sintel): 1.20

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 tunings:

• WE: ~36/35 = 50.0358 ¢, ~5/4 = 384.9267 ¢ (~121/120 = 15.3594 ¢)
• CWE: ~36/35 = 50.0000 ¢, ~5/4 = 384.7662 ¢ (~121/120 = 15.2338 ¢)

Optimal ET sequence: 24, 48f, 72, 168df, 240dff

Badness (Sintel): 1.11

Gamelstearn

The gamelstearn temperament has a period of 1/36 octave and tempers out the gamelisma (1029/1024) and the stearnsma (118098/117649). Its ploidacot is 36-ploid acot.

It used to be called decades, but was renamed in 2025 after the above two commas because the old name was deemed too confusing.

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

Optimal tunings:

• WE: ~49/48 = 33.3519 ¢, ~5/4 = 384.9781 ¢ (~81/80 = 15.2442 ¢)

error map: ⟨+0.667 -0.899 -0.002 -0.288]

• CWE: ~49/48 = 33.3333 ¢, ~5/4 = 385.1512 ¢ (~81/80 = 14.8488 ¢)

error map: ⟨0.000 -1.955 -1.162 -2.159]

Optimal ET sequence: 36, 72, 252, 324bd, 396bd

Badness (Sintel): 2.73

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 tunings:

• WE: ~49/48 = 33.3504 ¢, ~5/4 = 384.3474 ¢ (~81/80 = 15.8576 ¢)
• CWE: ~49/48 = 33.333 ¢, ~5/4 = 384.5541 ¢ (~81/80 = 15.4459 ¢)

Optimal ET sequence: 36, 72, 396bd

Badness (Sintel): 1.42

Omicronbeta

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 243/242, 385/384, 4000/3993

Mapping: [⟨72 114 167 202 249 0], ⟨0 0 0 0 0 1]]

mapping generators: ~100/99, ~13

Optimal tunings:

• WE: ~100/99 = 16.6768 ¢, ~13/8 = 838.3259 ¢ (~364/363 = 4.4838 ¢)

error map: ⟨+0.733 -0.795 -1.281 -0.104 +1.216 -0.004]

• CWE: ~100/99 = 16.6667 ¢, ~13/8 = 838.2660 ¢ (~364/363 = 4.9326 ¢)

error map: ⟨0.000 -1.955 -2.980 -2.159 -1.318 -2.262]

Optimal ET sequence: 72, 144, 216c, 288cdf

Badness (Sintel): 1.24