Compton family: Difference between revisions

The '''compton family''', otherwise known as the '''aristoxenean family''', of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[Pythagorean comma]] ([[ratio]]: 531441/524288, {{monzo|legend=1| -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{{cent}} flat of [[just]], these temperaments aim to add tunings for higher primes which are more in tune than in 12edo.

{{Main| 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 temperament]]s, compton is the {{nowrap| 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]].  

{{Mapping|legend=1| 12 19 0 | 0 0 1 }}

: mapping generators: ~256/243, ~5

[[Optimal tuning]]s:

* [[WE]]: ~256/243 = 100.0513{{c}}, ~5/4 = 385.0800{{c}} (~81/80 = 15.1253{{c}})

: [[error map]]: {{val| +0.616 -0.980 -0.001 }}

* [[CWE]]: ~256/243 = 100.0000{{c}}, ~5/4 = 385.3590{{c}} (~81/80 = 14.6410{{c}})

[[Badness]] (Sintel): 2.22

 

In terms of the [[normal forms #Normal forms for commas|normal comma list]], septimal compton adds 413343/409600 ({{monzo| -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.  

 

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.  

{{Mapping|legend=1| 12 19 0 -22 | 0 0 1 2 }}

[[Optimal tuning]]s:

* [[WE]]: ~256/243 = 100.0579{{c}}, ~5/4 = 383.9974{{c}} (~126/125 = 16.2342{{c}})

* [[CWE]]: ~256/243 = 100.0000{{c}}, ~5/4 = 384.1429{{c}} (~126/125 = 15.8571{{c}})

{{Optimal ET sequence|legend=1| 12, …, 60, 72, 228, 300c, 372bc, 444bc }}

[[Badness]] (Sintel): 0.903

Mapping: {{mapping| 12 19 0 -22 -42 | 0 0 1 2 3 }}

Optimal tunings:

* WE: ~35/33 = 100.0633{{c}}, ~5/4 = 383.5087{{c}} (~100/99 = 16.7446{{c}})

* CWE: ~35/33 = 100.0000{{c}}, ~5/4 = 383.5958{{c}} (~100/99 = 16.4042{{c}})

{{Optimal ET sequence|legend=0| 12, …, 60e, 72 }}

Badness (Sintel): 0.735

Mapping: {{mapping| 12 19 0 -22 -42 -67 | 0 0 1 2 3 4 }}

Optimal tunings:

* WE: ~35/33 = 100.0508{{c}}, ~5/4 = 384.1577{{c}} (~100/99 = 16.0454{{c}})

* CWE: ~35/33 = 100.0000{{c}}, ~5/4 = 384.1782{{c}} (~100/99 = 15.8218{{c}})

{{Optimal ET sequence|legend=0| 12f, …, 60eff, 72, 228f }}

Badness (Sintel): 0.903

Mapping: {{mapping| 12 19 0 -22 -42 -67 49 | 0 0 1 2 3 4 0 }}

Optimal tunings:

* WE: ~18/17 = 100.0658{{c}}, ~5/4 = 384.0024{{c}} (~100/99 = 16.2607{{c}})

* CWE: ~18/17 = 100.0000{{c}}, ~5/4 = 383.9647{{c}} (~100/99 = 16.0353{{c}})

{{Optimal ET sequence|legend=0| 12f, 60eff, 72 }}

Badness (Sintel): 0.873

Mapping: {{mapping| 12 19 0 -22 -42 100 | 0 0 1 2 3 -2 }}

Optimal tunings:

* WE: ~35/33 = 100.0926{{c}}, ~5/4 = 382.9660{{c}} (~100/99 = 17.4045{{c}})

* CWE: ~35/33 = 100.0000{{c}}, ~5/4 = 382.7748{{c}} (~100/99 = 17.2252{{c}})

{{Optimal ET sequence|legend=0| 12, 60e, 72, 204cdef, 276cdeff }}

Badness (Sintel): 1.04

Mapping: {{mapping| 12 19 0 -22 -42 100 49 | 0 0 1 2 3 -2 0 }}

Optimal tunings:

* WE: ~18/17 = 100.0941{{c}}, ~5/4 = 382.9567{{c}} (~100/99 = 17.4796{{c}})

* CWE: ~18/17 = 100.0000{{c}}, ~5/4 = 382.7381{{c}} (~100/99 = 17.2619{{c}})

{{Optimal ET sequence|legend=0| 12, 60e, 72, 204cdefg, 276cdeffgg }}

Badness (Sintel): 0.833

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 {{nowrap| 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]].   

{{Mapping|legend=1| 12 19 28 0 | 0 0 0 1 }}

: mapping generators: ~16/15, ~7

[[Optimal tuning]]s:

* [[WE]]: ~16/15 = 99.8680{{c}}, ~7/4 = 971.9257{{c}} (~64/63 = 26.7545{{c}})

: [[error map]]: {{val| -1.584 -4.463 +9.991 -0.068 }}

* [[CWE]]: ~16/15 = 100.0000{{c}}, ~7/4 = 972.0971{{c}} (~64/63 = 27.9029{{c}})

{{Optimal ET sequence|legend=1| 12, 24, 36, 48c, 84c }}

[[Badness]] (Sintel): 1.27

Mapping: {{mapping| 12 19 28 0 -26 | 0 0 0 1 2 }}

Optimal tunings:

* WE: ~16/15 = 99.8542{{c}}, ~7/4 = 975.8519{{c}} (~64/63 = 22.6896{{c}})

* CWE: ~16/15 = 100.0000{{c}}, ~7/4 = 976.4125{{c}} (~64/63 = 23.5875{{c}})

{{Optimal ET sequence|legend=0| 12, 36e, 48c }}

Badness (Sintel): 1.92

Mapping: {{mapping| 12 19 28 0 109 | 0 0 0 1 -2 }}

Optimal tunings:

* WE: ~16/15 = 99.8791{{c}}, ~7/4 = 970.9614{{c}} (~64/63 = 27.8300{{c}})

* CWE: ~16/15 = 100.0000{{c}}, ~7/4 = 972.2549{{c}} (~64/63 = 27.7451{{c}})

{{Optimal ET sequence|legend=0| 12e, 36, 48c, 84c }}

Badness (Sintel): 2.71

=== Catnip ===

Mapping: {{mapping| 12 19 28 0 8 | 0 0 0 1 1 }}

Optimal tunings:

* WE: ~16/15 = 99.8519{{c}}, ~7/4 = 965.7912{{c}} (~64/63 = 32.7275{{c}})

* CWE: ~16/15 = 100.0000{{c}}, ~7/4 = 965.8666{{c}} (~64/63 = 34.1334{{c}})

{{Optimal ET sequence|legend=0| 12, 24, 36 }}

Badness (Sintel): 1.14

Mapping: {{mapping| 12 19 28 0 8 11 | 0 0 0 1 1 1 }}

Optimal tunings:

* WE: ~16/15 = 99.8308{{c}}, ~7/4 = 961.1391{{c}} (~40/39 = 37.1694{{c}})

* CWE: ~16/15 = 100.0000{{c}}, ~7/4 = 961.1435{{c}} (~40/39 = 38.8565{{c}})

{{Optimal ET sequence|legend=0| 12f, 24, 36f }}

Badness (Sintel): 1.18

Mapping: {{mapping| 12 19 28 0 8 11 49 | 0 0 0 1 1 1 0 }}

Optimal tunings:

* WE: ~18/17 = 99.8958{{c}}, ~7/4 = 959.2226{{c}} (~40/39 = 39.7354{{c}})

* CWE: ~18/17 = 100.0000{{c}}, ~7/4 = 959.4216{{c}} (~40/39 = 40.5784{{c}})

{{Optimal ET sequence|legend=0| 12f, 24, 36f }}

Badness (Sintel): 1.18

Mapping: {{mapping| 12 19 28 0 8 11 49 51 | 0 0 0 1 1 1 0 0 }}

Optimal tunings:

* WE: ~18/17 = 99.9058{{c}}, ~7/4 = 958.9307{{c}} (~40/39 = 40.1270{{c}})

* CWE: ~18/17 = 100.0000{{c}}, ~7/4 = 959.2303{{c}} (~40/39 = 40.7697{{c}})

{{Optimal ET sequence|legend=0| 12f, 24, 36f }}

Badness (Sintel): 1.15

Mapping: {{mapping| 12 19 28 0 8 78 | 0 0 0 1 1 -1 }}

Optimal tunings:

* WE: ~18/17 = 99.9301{{c}}, ~7/4 = 961.6396{{c}} (~64/63 = 37.6617{{c}})

* CWE: ~18/17 = 100.0000{{c}}, ~7/4 = 962.1413{{c}} (~64/63 = 37.8587{{c}})

{{Optimal ET sequence|legend=0| 12, 24, 36 }}

Badness (Sintel): 1.58

Mapping:{{mapping| 12 19 28 0 8 78 49 | 0 0 0 1 1 -1 0 }}

Optimal tunings:

* WE: ~18/17 = 99.9556{{c}}, ~7/4 = 961.4763{{c}} (~64/63 = 38.0796{{c}})

* CWE: ~18/17 = 100.0000{{c}}, ~7/4 = 961.8075{{c}} (~64/63 = 38.1925{{c}})

{{Optimal ET sequence|legend=0| 12, 24, 36, 60c }}

Badness (Sintel): 1.40

Mapping: {{mapping| 12 19 28 0 8 78 49 51 | 0 0 0 1 1 -1 0 0 }}

Optimal tunings:

* WE: ~18/17 = 99.9545{{c}}, ~7/4 = 961.4829{{c}} (~64/63 = 38.0624{{c}})

* CWE: ~18/17 = 100.0000{{c}}, ~7/4 = 961.8354{{c}} (~64/63 = 38.1646{{c}})

{{Optimal ET sequence|legend=0| 12, 24, 36, 60c }}

Badness (Sintel): 1.27

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

{{Mapping|legend=1| 12 19 28 34 0 | 0 0 0 0 1 }}

: mapping genereators: ~16/15, ~11

[[Optimal tuning]]s:

* [[WE]]: ~16/15 = 99.6643{{c}}, ~11/8 = 563.1257{{c}} (~55/54 = 34.8599{{c}})

: [[error map]]: {{val| -4.029 -8.334 +4.285 +19.759 -0.279 }}

* [[CWE]]: ~16/15 = 100.0000{{c}}, ~11/8 = 562.2258{{c}} (~55/54 = 37.7742{{c}})

{{Optimal ET sequence|legend=1| 12, 24d, 36d }}

[[Badness]] (Sintel): 1.01

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.

{{Mapping|legend=1| 24 38 0 123 | 0 0 1 -1 }}

* [[WE]]: ~36/35 = 50.0337{{c}}, ~5/4 = 384.2919{{c}} (~81/80 = 15.9775{{c}})

: [[error map]]: {{val| +0.808 -0.675 -0.406 -0.592 }}

* [[CWE]]: ~36/35 = 50.0000{{c}}, ~5/4 = 384.0719{{c}} (~81/80 = 15.9281{{c}})

[[Badness]] (Sintel): 2.94

Mapping: {{mapping| 24 38 0 123 83 | 0 0 1 -1 0 }}

Optimal tunings:

* WE: ~36/35 = 50.0301{{c}}, ~5/4 = 384.2848{{c}} (~121/120 = 15.9559{{c}})

* CWE: ~36/35 = 50.0000{{c}}, ~5/4 = 384.0825{{c}} (~121/120 = 15.9175{{c}})

{{Optimal ET sequence|legend=0| 24, 48, 72, 312bd, 384bcdd, 456bcdde, 528bcdde }}

Badness (Sintel): 1.20

Mapping: {{mapping| 24 38 0 123 83 33 | 0 0 1 -1 0 1 }}

Optimal tunings:

* WE: ~36/35 = 50.0358{{c}}, ~5/4 = 384.9267{{c}} (~121/120 = 15.3594{{c}})

* CWE: ~36/35 = 50.0000{{c}}, ~5/4 = 384.7662{{c}} (~121/120 = 15.2338{{c}})

{{Optimal ET sequence|legend=0| 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.

{{Mapping|legend=1| 36 57 0 101 | 0 0 1 0 }}

[[Optimal tuning]]s:

* [[WE]]: ~49/48 = 33.3519{{c}}, ~5/4 = 384.9781{{c}} (~81/80 = 15.2442{{c}})

: [[error map]]: {{val| +0.667 -0.899 -0.002 -0.288 }}

* [[CWE]]: ~49/48 = 33.3333{{c}}, ~5/4 = 385.1512{{c}} (~81/80 = 14.8488{{c}})

: error map: {{val| 0.000 -1.955 -1.162 -2.159 }}

[[Badness]] (Sintel): 2.73

Mapping: {{mapping| 36 57 0 101 41 | 0 0 1 0 1 }}

Optimal tunings:

* WE: ~49/48 = 33.3504{{c}}, ~5/4 = 384.3474{{c}} (~81/80 = 15.8576{{c}})

* CWE: ~49/48 = 33.333{{c}}, ~5/4 = 384.5541{{c}} (~81/80 = 15.4459{{c}})

{{Optimal ET sequence|legend=0| 36, 72, 396bd }}

Badness (Sintel): 1.42

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

{{Mapping|legend=1| 72 114 167 202 249 0 | 0 0 0 0 0 1 }}

: mapping generators: ~100/99, ~13

[[Optimal tuning]]s:

* [[WE]]: ~100/99 = 16.6768{{c}}, ~13/8 = 838.3259{{c}} (~364/363 = 4.4838{{c}})

: [[error map]]: {{val| +0.733 -0.795 -1.281 -0.104 +1.216 -0.004 }}

* [[CWE]]: ~100/99 = 16.6667{{c}}, ~13/8 = 838.2660{{c}} (~364/363 = 4.9326{{c}})

: error map: {{val| 0.000 -1.955 -2.980 -2.159 -1.318 -2.262 }}

{{Optimal ET sequence|legend=1| 72, 144, 216c, 288cdf }}

[[Badness]] (Sintel): 1.24