Syntonic–chromatic equivalence continuum: Difference between revisions
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m apotome is more unambiguous. chromatic semitone could mean 25/24 or an interval around that size. |
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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| | 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 such as [[whitewood]], [[mavila]], [[dicot]], [[porcupine]], [[tetracot]], [[amity]], [[gravity]], and [[absurdity]]. It converges to [[meantone]] as ''n'' approaches infinity. The just value of ''n'' is 5.2861… | All temperaments in the continuum satisfy (81/80)<sup>''n''</sup> ~ 2187/2048. Varying ''n'' results in different temperaments such as [[whitewood]], [[mavila]], [[dicot]], [[porcupine]], [[tetracot]], [[amity]], [[gravity]], and [[absurdity]]. It converges to [[meantone]] as ''n'' approaches infinity. The just value of ''n'' is 5.2861… | ||
Revision as of 03:11, 17 January 2021
The syntonic-chromatic equivalence continuum is a continuum of temperaments which equate a number of syntonic commas (81/80) with the apotome (2187/2048).
All temperaments in the continuum satisfy (81/80)n ~ 2187/2048. Varying n results in different temperaments such as whitewood, mavila, dicot, porcupine, tetracot, amity, gravity, and absurdity. It converges to meantone as n approaches infinity. The just value of n is 5.2861…
| n | Temperament | Comma | |
|---|---|---|---|
| Ratio | Monzo | ||
| 0 | Whitewood | 2187/2048 | [-11 7⟩ |
| 1 | Mavila | 135/128 | [-7 3 1⟩ |
| 2 | Dicot | 25/24 | [-3 -1 2⟩ |
| 3 | Porcupine | 250/243 | [1 -5 3⟩ |
| 4 | Tetracot | 20000/19683 | [5 -9 4⟩ |
| 5 | Amity | 1600000/1594323 | [9 -13 5⟩ |
| 6 | Gravity | 129140163/128000000 | [-13 17 -6⟩ |
| 7 | Absurdity | 10460353203/10240000000 | [-17 21 -7⟩ |
| … | … | … | … |
| Inf | Meantone | 81/80 | [-4 4 -1⟩ |
Also fractional values of n: enipucrop (n = 1.5), seville (n = 2.3), sixix (n = 2.5), sevond (n = 3.5), brahmagupta (n = 5.25).