Compton family: Difference between revisions

(4 intermediate revisions by 2 users not shown)

Line 1:

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 ~~will be that~~ of 12edo, ~~two~~ [[~~cent~~]]~~s flat~~, ~~the tuning of the larger primes is not so constrained, and the point of~~ these temperaments is to ~~improve on it~~.

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 ==

Line 7:

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 ~~known~~ as ''aristoxenean'' in [[Tonalsoft Encyclopedia]].

This temperament is documented as ''aristoxenean'' in [[Tonalsoft Encyclopedia]].

[[Subgroup]]: 2.3.5

Line 30:

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]].

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 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 ~~known~~ as ''waage'' in [[Graham Breed]]'s [https://x31eq.com/temper/ temperament finder].

Septimal compton is catalogued as ''waage'' in [[Graham Breed]]'s [https://x31eq.com/temper/ 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.

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

Line 130:

== 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]].

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

Line 158:

Optimal tunings:

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

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

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

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

Line 173:

Optimal tunings:

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

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

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

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

Line 188:

Optimal tunings:

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

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

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

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

Badness (Sintel): 1.14

Line 203:

Optimal tunings:

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

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

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

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

Line 415:

[[Category:Temperament families]]

~~[[Category:Pages with mostly numerical content]]~~

[[Category:Compton family| ]]

[[Category:Compton| ]]

[[Category:Rank 2]]

@@ Line 1: / Line 1: @@
 {{Technical data page}}
-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 will be that of 12edo, two [[cent]]s flat, the tuning of the larger primes is not so constrained, and the point of these temperaments is to improve on it.
+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 ==
@@ Line 7: / Line 7: @@
 -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 known as ''aristoxenean'' in [[Tonalsoft Encyclopedia]].
+This temperament is documented as ''aristoxenean'' in [[Tonalsoft Encyclopedia]].
 [[Subgroup]]: 2.3.5
@@ Line 30: / Line 30: @@
 {{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]].
+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 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 known as ''waage'' in [[Graham Breed]]'s [https://x31eq.com/temper/ temperament finder].
+Septimal compton is catalogued as ''waage'' in [[Graham Breed]]'s [https://x31eq.com/temper/ 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.
+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
@@ Line 130: / Line 130: @@
 == 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]].
+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
@@ Line 158: / Line 158: @@
 Optimal tunings:
-* CTE: ~16/15 = 99.8542{{c}}, ~7/4 = 975.8519{{c}} (~64/63 = 22.6896{{c}})
+* WE: ~16/15 = 99.8542{{c}}, ~7/4 = 975.8519{{c}} (~64/63 = 22.6896{{c}})
-* POTE: ~16/15 = 100.0000{{c}}, ~7/4 = 976.4125{{c}} (~64/63 = 23.5875{{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 }}
@@ Line 173: / Line 173: @@
 Optimal tunings:
-* CTE: ~16/15 = 99.8791{{c}}, ~7/4 = 970.9614{{c}} (~64/63 = 27.8300{{c}})
+* WE: ~16/15 = 99.8791{{c}}, ~7/4 = 970.9614{{c}} (~64/63 = 27.8300{{c}})
-* POTE: ~16/15 = 100.0000{{c}}, ~7/4 = 972.2549{{c}} (~64/63 = 27.7451{{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 }}
@@ Line 188: / Line 188: @@
 Optimal tunings:
-* CTE: ~16/15 = 99.8519{{c}}, ~7/4 = 965.7912{{c}} (~64/63 = 32.7275{{c}})
+* WE: ~16/15 = 99.8519{{c}}, ~7/4 = 965.7912{{c}} (~64/63 = 32.7275{{c}})
-* POTE: ~16/15 = 100.0000{{c}}, ~7/4 = 965.8666{{c}} (~64/63 = 34.1334{{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, 72ce }}
+{{Optimal ET sequence|legend=0| 12, 24, 36 }}
 Badness (Sintel): 1.14
@@ Line 203: / Line 203: @@
 Optimal tunings:
-* CTE: ~16/15 = 99.8308{{c}}, ~7/4 = 961.1391{{c}} (~40/39 = 37.1694{{c}})
+* WE: ~16/15 = 99.8308{{c}}, ~7/4 = 961.1391{{c}} (~40/39 = 37.1694{{c}})
-* POTE: ~16/15 = 100.0000{{c}}, ~7/4 = 961.1435{{c}} (~40/39 = 38.8565{{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 }}
@@ Line 415: / Line 415: @@
 [[Category:Temperament families]]
-[[Category:Pages with mostly numerical content]]
 [[Category:Compton family| ]] <!-- main article -->
 [[Category:Compton| ]] <!-- key article -->
 [[Category:Rank 2]]