Syntonic–chromatic equivalence continuum: Difference between revisions
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The '''syntonic-chromatic equivalence continuum''' is a continuum of temperaments which equate a number of [[81/80|syntonic commas (81/80)]] with the [[2187/2048|apotome (2187/2048)]]. | The '''syntonic-chromatic equivalence continuum''' is a continuum of temperaments which equate a number of [[81/80|syntonic commas (81/80)]] with the [[2187/2048|apotome (2187/2048)]]. | ||
All temperaments in the continuum satisfy (81/80)<sup>''n''</sup> ~ 2187/2048. Varying ''n'' results in different temperaments | All temperaments in the continuum satisfy (81/80)<sup>''n''</sup> ~ 2187/2048. Varying ''n'' results in different temperaments listed in the table below. It converges to [[meantone]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all 5-[[limit]] temperaments supported by [[7edo]]. The just value of ''n'' is 5.2861… | ||
Note that if you let k=n-2 (meaning n=k+2) so that k=0 means n=2, k=-1 means n=1, etc. then the continuum corresponds to (81/80)^k = 25/24, which might be a preferred way of conceptualising it because: | |||
* 25/24 is the chromatic semitone, highly notable in the 5-limit as the difference between 5/4 and 6/5, so this shifted continuum could ''also'' be termed the "syntonic-chromatic equivalence continuum". | |||
* k=0 being n=2 could be more intuitive as when (81/80)^0 = 25/24 is used, 81/80 no longer becomes dependent on 25/24, and 25/24 is tempered (thus ''equating'' 5/4 and 6/5) and so it represents the border at which 5/4 and 6/5 swap places in the circle of fifths. | |||
* k=1 and upwards (up to a point) represent temperaments of reasonably good accuracy as equating at least one 81/80 with 25/24 seems like a good lower bound for a temperament intended to model JI. A good upper bound might be Gravity (k=4) or at most Absurdity (k=5), with the only exception being Meantone (n = k = (unsigned) infinity) which is in a sense a simplicity that is the reverse of Dicot's as in Meantone, 81/80 is tempered and 25/24 is no longer dependent on 81/80. | |||
* 25/24 is the simplest ratio to be tempered in the continuum. | |||
{| class="wikitable center-1 center-2" | {| class="wikitable center-1 center-2" | ||
|+ Temperaments in the continuum | |+ Temperaments in the continuum | ||
|- | |- | ||
! rowspan="2" | ''n'' | ! rowspan="2" | ''k'' = ''n'' - 2 | ||
! rowspan="2" | ''n'' = ''k'' + 2 | |||
! rowspan="2" | Temperament | ! rowspan="2" | Temperament | ||
! colspan="2" | Comma | ! colspan="2" | Comma | ||
| Line 13: | Line 20: | ||
! Monzo | ! Monzo | ||
|- | |- | ||
| -2 | |||
| 0 | | 0 | ||
| [[Whitewood]] | | [[Whitewood]] | ||
| Line 18: | Line 26: | ||
| {{monzo| -11 7 }} | | {{monzo| -11 7 }} | ||
|- | |- | ||
| -1 | |||
| 1 | | 1 | ||
| [[Mavila]] | | [[Mavila]] | ||
| Line 23: | Line 32: | ||
| {{monzo| -7 3 1 }} | | {{monzo| -7 3 1 }} | ||
|- | |- | ||
| 0 | |||
| 2 | | 2 | ||
| [[Dicot]] | | [[Dicot]] | ||
| Line 28: | Line 38: | ||
| {{monzo| -3 -1 2 }} | | {{monzo| -3 -1 2 }} | ||
|- | |- | ||
| 1 | |||
| 3 | | 3 | ||
| [[Porcupine]] | | [[Porcupine]] | ||
| Line 33: | Line 44: | ||
| {{monzo| 1 -5 3 }} | | {{monzo| 1 -5 3 }} | ||
|- | |- | ||
| 2 | |||
| 4 | | 4 | ||
| [[Tetracot]] | | [[Tetracot]] | ||
| Line 38: | Line 50: | ||
| {{monzo| 5 -9 4 }} | | {{monzo| 5 -9 4 }} | ||
|- | |- | ||
| 3 | |||
| 5 | | 5 | ||
| [[Amity]] | | [[Amity]] | ||
| Line 43: | Line 56: | ||
| {{monzo| 9 -13 5 }} | | {{monzo| 9 -13 5 }} | ||
|- | |- | ||
| 4 | |||
| 6 | | 6 | ||
| [[Gravity]] | | [[Gravity]] | ||
| Line 48: | Line 62: | ||
| {{monzo| -13 17 -6 }} | | {{monzo| -13 17 -6 }} | ||
|- | |- | ||
| 5 | |||
| 7 | | 7 | ||
| [[Absurdity]] | | [[Absurdity]] | ||
| Line 58: | Line 73: | ||
| … | | … | ||
|- | |- | ||
| Inf | |||
| Inf | | Inf | ||
| [[Meantone]] | | [[Meantone]] | ||