Syntonic–rastmic equivalence continuum: Difference between revisions
Created page with "The '''syntonic-rastmic equivalence continuum''' is a continuum of temperaments which equate a number of syntonic commas (81/80) with the [[243/242|rastma (243/242)]..." |
Okay, I had to fix the initial sentence... Given that this continuum is about equating a certain number of rastmas with the syntonic comma, I'm wondering if we need to rename both this page and syntonic-rastmic subchroma notation to better fit with this reality. |
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The '''syntonic-rastmic equivalence continuum''' is a continuum of temperaments which equate a number of [[ | The '''syntonic-rastmic equivalence continuum''' is a continuum of temperaments which equate a number of [[243/242|rastmas (243/242)]] with the [[81/80|syntonic comma (81/80)]]. | ||
All temperaments in the continuum satisfy 81/80 ~ (243/242)<sup>''n''</sup>. Varying ''n'' results in different temperaments listed in the table below. It converges to the 2.3.5.11 subgroup temperament of {243/242} as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all 2.3.5.11 subgroup temperaments supported by [[mohaha]] due to it being the unique temperament that tempers out both commas and thus tempers out all combinations of them. The just value of ''n'' is 3.0125…, and temperaments near this tend to be the most accurate ones. | All temperaments in the continuum satisfy 81/80 ~ (243/242)<sup>''n''</sup>. Varying ''n'' results in different temperaments listed in the table below. It converges to the 2.3.5.11 subgroup temperament of {243/242} as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all 2.3.5.11 subgroup temperaments supported by [[mohaha]] due to it being the unique temperament that tempers out both commas and thus tempers out all combinations of them. The just value of ''n'' is 3.0125…, and temperaments near this tend to be the most accurate ones. | ||
Revision as of 02:07, 11 December 2021
The syntonic-rastmic equivalence continuum is a continuum of temperaments which equate a number of rastmas (243/242) with the syntonic comma (81/80).
All temperaments in the continuum satisfy 81/80 ~ (243/242)n. Varying n results in different temperaments listed in the table below. It converges to the 2.3.5.11 subgroup temperament of {243/242} as n approaches infinity. If we allow non-integer and infinite n, the continuum describes the set of all 2.3.5.11 subgroup temperaments supported by mohaha due to it being the unique temperament that tempers out both commas and thus tempers out all combinations of them. The just value of n is 3.0125…, and temperaments near this tend to be the most accurate ones.
The continuum is very significant for the syntonic-rastmic subchroma notation, as each member of it entails a distinct way of notation.
| n = k + 2 | Temperament | Comma | |
|---|---|---|---|
| Ratio | Monzo | ||
| 0 | Meantone expansion | 81/80 | [-4 4 -1⟩ |
| 1 | Protomere | 121/120 | [-3 -1 -1 2⟩ |
| 2 | Deuteromere | 14641/14580 | [-2 -6 -1 4⟩ |
| 3 | Tritomere | 1771561/1771470 | [-1 -11 -1 6⟩ |
| … | … | … | |
| ∞ | Neutral expansion | 243/242 | [-1 5 0 -2⟩ |
This series of temperament names derives from Greek ordinal prefixes + -mere, which derives from ancient Greek meros, meaning "part".