Compton family: Difference between revisions
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The '''compton family''', otherwise known as the '''aristoxenean family''', tempers out the [[Pythagorean comma]] | 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 == | == 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. 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 | {{Main| 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. 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 | [[Subgroup]]: 2.3.5 | ||
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== Septimal compton == | == Septimal compton == | ||
Septimal compton is also known as ''waage''. In terms of the normal list, compton adds 413343/409600 | {{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. | 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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== Catler == | == 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 | 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 | [[Subgroup]]: 2.3.5.7 | ||
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== Hours == | == 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 | 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 | [[Subgroup]]: 2.3.5.7 | ||
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== Decades == | == Decades == | ||
The decades temperament has a period of 1/36 octave and tempers out the [[gamelisma]] (1029/1024) and the stearnsma (118098/117649). The name | 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 named for the reason that the period is 1/36 octave and there are roughly 36 decades (''ten days'') per year (12 months × 3 decades per a month). | ||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||