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.

== 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]].  

[[Subgroup]]: 2.3.5

[[Comma list]]: 531441/524288

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

: mapping generators: ~256/243, ~5

: [[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}})

: error map: {{val| 0.000 -1.955 -0.955 }}

{{Optimal ET sequence|legend=1| 12, 48, 60, 72, 84, 156, 240, 396b, 636bbc }}

[[Badness]] (Sintel): 2.22

{{Main| Compton }}

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.

 

 

 

 

 

 

 

 

[[Badness]] (Sintel): 0.903

 

{{Mapping|legend=0| 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|legend=0| 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|legend=0| 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|legend=0| 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|legend=0| 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

== 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 {{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]].   

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 81/80, 128/125

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

 

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

: error map: {{val| 0.000 -1.955 +13.686 +3.271 }}

 

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

 

 

{{Mapping|legend=0| 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|legend=0| 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

 

{{Mapping|legend=0| 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

 

 

Optimal tunings:  

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

 

 

 

 

 

 

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

Badness (Sintel): 1.18

 

 

 

 

 

Badness (Sintel): 1.15

 

 

Subgroup: 2.3.5.7.11.13

{{Mapping|legend=0| 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

 

 

 

 

 

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

 

 

{{Todo|inline=1|complete section|unify precision|comment= Sort the comma basis. Normalize the mapping. Unify the cent values in optimal tunings to 4 decimal places. Add the optimal ET sequence. <br>Add the data for the 11-, 13-, and 17-limit versions. }}

 

Comma list: 81/80, 91/90, 51/50, 128/125, 76/75, 121/120

 

 

* WE: ~33/32 = 49.994{{c}}, ~7/4 = 965.340{{c}}

* CWE: ~33/32 = 50.000{{c}}, ~7/4 = 965.449{{c}}

Badness (Sintel): 2.383

{{Distinguish|Duodecic}}

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

: mapping genereators: ~16/15, ~11

: [[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}})

: error map: {{val| 0.000 -1.955 +13.686 +31.174 +10.908 }}

{{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.

[[Comma list]]: 19683/19600, 33075/32768

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

: mapping generators: ~36/35, ~5

[[Optimal tuning]]s:  

* [[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}})

: error map: {{val| 0.000 -1.955 -2.242 -2.898 }}

{{Optimal ET sequence|legend=1| 24, 48, 72, 312bd, 384bcdd, 456bcdd, 528bcdd, 600bccdd }}

Subgroup: 2.3.5.7.11

{{Mapping|legend=0| 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

 

 

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}})

 

 

 

== Gamelstearn ==

 

 

[[Subgroup]]: 2.3.5.7

 

 

 

 

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

 

 

 

 

 

 

 

 

 

 

 

 

: [[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

[[Category:Compton family| ]] <!-- main article -->

[[Category:Compton| ]] <!-- key article -->

[[Category:Temperament families]]

[[Category:Catalogs of rank-2 temperaments]]