Compton family: Difference between revisions

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{{Technical data page}}~~<br><br>~~

The '''compton family''', otherwise known as the '''aristoxenean family''', tempers out the [[Pythagorean comma]], 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 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 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.

== Compton ==

5-limit compton is also known as ''aristoxenean''. It 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 ~~- that is, it is equivalent to the root12(2).5 subgroup with 2^(7/12) mapped to 3/2~~. 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 &~~amp;~~ 72 temperament, and [[72edo]], [[84edo]] or [[240edo]] make for good tunings.

5-limit compton is also known as ''aristoxenean''. It 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, it is the {{nowrap| 12 & 72 }} temperament, and [[72edo]], [[84edo]] or [[240edo]] make for good tunings.

[[Subgroup]]: 2.3.5

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== Septimal compton ==

Septimal compton is also known as ''waage''. In terms of the normal list, 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]].

Septimal compton is also known as ''waage''. In terms of the normal list, 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 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.

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~~{{Multival|legend=1| 0 12 24 19 38 22 }}~~

[[Optimal tuning]]s:

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

~~Wedgie: {{Multival| 0 12 24 36 19 38 57 22 42 18 }}~~

Optimal tunings:

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

~~Wedgie: {{Multival| 0 12 24 36 48 19 38 57 76 22 42 67 18 46 33 }}~~

Optimal tunings:

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

~~Wedgie: {{Multival| 0 12 24 36 -24 19 38 57 -38 22 42 -100 18 -156 -216 }}~~

Optimal tunings:

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== 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 &~~amp;~~ 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

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: mapping generators: ~16/15, ~7

~~{{Multival|legend=1| 0 0 12 0 19 28 }}~~

[[Optimal tuning]]s:

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== 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.

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 named for the reason that the period is 1/24 octave and there are 24 hours per day.

[[Subgroup]]: 2.3.5.7

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: mapping generators: ~36/35, ~5

~~{{Multival|legend=1| 0 24 -24 38 -38 -123 }}~~

[[Optimal tuning]]s:

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* CTE: ~36/35 = 50.000, ~5/4 = 384.226 (~121/120 = 15.774)

* POTE: ~36/35 = 50.000, ~5/4 = 384.054 (~121/120 = 15.946)

~~{{Multival|legend=1| 0 24 -24 0 38 -38 0 -123 -83 83 }}~~

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

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* CTE: ~36/35 = 50.000, ~5/4 = 385.420 (~121/120 = 14.580)

* POTE: ~36/35 = 50.000, ~5/4 = 384.652 (~121/120 = 15.348)

~~{{Multival|legend=1| 0 24 -24 0 24 38 -38 0 38 -123 -83 -33 83 156 83 }}~~

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Badness (Smith): 0.026931

== ~~Decades~~ ==

== Gamelstearn ==

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)~~.

The gamelstearn temperament has a period of 1/36 octave and tempers out the [[gamelisma]] (1029/1024) and the [[stearnsma]] (118098/117649).

It used to be named "decades".

[[Subgroup]]: 2.3.5.7

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: mapping generators: ~49/48, ~5

~~{{Multival|legend=1| 0 36 0 57 0 -101 }}~~

[[Optimal tuning]]s:

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: mapping generators: ~100/99, ~13

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

[[Optimal tuning]]s:

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[[Category:Temperament families]]

[[Category:Pages with mostly numerical content]]

[[Category:Compton family| ]]

[[Category:Compton| ]]

[[Category:Rank 2]]

@@ Line 1: / Line 1: @@
-{{Technical data page}}<br><br>
+{{Technical data page}}
-The '''compton family''', otherwise known as the '''aristoxenean family''', tempers out the [[Pythagorean comma]], 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 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 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.
 == Compton ==
--limit compton is also known as ''aristoxenean''. It 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 - that is, it is equivalent to the root12(2).5 subgroup with 2^(7/12) mapped to 3/2. 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 &amp; 72 temperament, and [[72edo]], [[84edo]] or [[240edo]] make for good tunings.
+{{Main| Compton }}
+-limit compton is also known as ''aristoxenean''. It 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, it is the {{nowrap| 12 & 72 }} temperament, and [[72edo]], [[84edo]] or [[240edo]] make for good tunings.
 [[Subgroup]]: 2.3.5
@@ Line 24: / Line 26: @@
 == Septimal compton ==
-Septimal compton is also known as ''waage''. In terms of the normal list, 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]].
+{{Main| Compton }}
+Septimal compton is also known as ''waage''. In terms of the normal list, 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 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.
@@ Line 35: / Line 39: @@
 {{Mapping|legend=1| 12 19 0 -22 | 0 0 1 2 }}
-{{Multival|legend=1| 0 12 24 19 38 22 }}
 [[Optimal tuning]]s:
@@ Line 54: / Line 56: @@
 Mapping: {{mapping| 12 19 0 -22 -42 | 0 0 1 2 3 }}
-Wedgie: {{Multival| 0 12 24 36 19 38 57 22 42 18 }}
 Optimal tunings:
@@ Line 71: / Line 71: @@
 Mapping: {{mapping| 12 19 0 -22 -42 -67 | 0 0 1 2 3 4 }}
-Wedgie: {{Multival| 0 12 24 36 48 19 38 57 76 22 42 67 18 46 33 }}
 Optimal tunings:
@@ Line 103: / Line 101: @@
 Mapping: {{mapping| 12 19 0 -22 -42 100 | 0 0 1 2 3 -2 }}
-Wedgie: {{Multival| 0 12 24 36 -24 19 38 57 -38 22 42 -100 18 -156 -216 }}
 Optimal tunings:
@@ Line 130: / Line 126: @@
 == 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 &amp; 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 139: / Line 135: @@
 : mapping generators: ~16/15, ~7
-{{Multival|legend=1| 0 0 12 0 19 28 }}
 [[Optimal tuning]]s:
@@ Line 307: / Line 301: @@
 == 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.
+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 named for the reason that the period is 1/24 octave and there are 24 hours per day.
 [[Subgroup]]: 2.3.5.7
@@ Line 316: / Line 310: @@
 : mapping generators: ~36/35, ~5
-{{Multival|legend=1| 0 24 -24 38 -38 -123 }}
 [[Optimal tuning]]s:
@@ Line 339: / Line 331: @@
 * CTE: ~36/35 = 50.000, ~5/4 = 384.226 (~121/120 = 15.774)
 * POTE: ~36/35 = 50.000, ~5/4 = 384.054 (~121/120 = 15.946)
-{{Multival|legend=1| 0 24 -24 0 38 -38 0 -123 -83 83 }}
 {{Optimal ET sequence|legend=1| 24, 48, 72, 312bd, 384bcdd, 456bcdde, 528bcdde }}
@@ Line 356: / Line 346: @@
 * CTE: ~36/35 = 50.000, ~5/4 = 385.420 (~121/120 = 14.580)
 * POTE: ~36/35 = 50.000, ~5/4 = 384.652 (~121/120 = 15.348)
-{{Multival|legend=1| 0 24 -24 0 24 38 -38 0 38 -123 -83 -33 83 156 83 }}
 {{Optimal ET sequence|legend=1| 24, 48f, 72, 168df, 240dff }}
@@ Line 363: / Line 351: @@
 Badness (Smith): 0.026931
-== Decades ==
+== Gamelstearn ==
-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).
+The gamelstearn temperament has a period of 1/36 octave and tempers out the [[gamelisma]] (1029/1024) and the [[stearnsma]] (118098/117649).
+It used to be named "decades".
 [[Subgroup]]: 2.3.5.7
@@ Line 373: / Line 363: @@
 : mapping generators: ~49/48, ~5
-{{Multival|legend=1| 0 36 0 57 0 -101 }}
 [[Optimal tuning]]s:
@@ Line 409: / Line 397: @@
 : mapping generators: ~100/99, ~13
-{{Multival|legend=1| 0 0 0 0 72 0 0 0 114 0 0 167 0 202 249 }}
 [[Optimal tuning]]s:
@@ Line 423: / Line 409: @@
 [[Category:Temperament families]]
+[[Category:Pages with mostly numerical content]]
 [[Category:Compton family| ]] <!-- main article -->
 [[Category:Compton| ]] <!-- key article -->
 [[Category:Rank 2]]