Syntonic–rastmic equivalence continuum: Difference between revisions
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The ''' | The '''syntonic–rastmic equivalence continuum''' is a [[equivalence continuum|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. | All temperaments in the continuum satisfy {{nowrap|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. | ||
The continuum is very significant for the [[ | The continuum is very significant for the [[syntonic–rastmic subchroma notation]], as each member of it entails a distinct way of notation. | ||
{| class="wikitable center-1" | {| class="wikitable center-1" | ||
|+ Temperaments in the continuum | |+ style="font-size: 105%;" | Temperaments in the continuum | ||
|- | |- | ||
! rowspan="2" | ''n'' = ''k'' + 2 | ! rowspan="2" | {{nowrap|''n'' {{=}} ''k'' + 2}} | ||
! rowspan="2" | Temperament | ! rowspan="2" | Temperament | ||
! colspan="2" | Comma | ! colspan="2" | Comma | ||
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[[Category:Mohaha]] | [[Category:Mohaha]] | ||
[[Category:Equivalence continua]] | [[Category:Equivalence continua]] | ||
Latest revision as of 02:10, 4 January 2025
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".