Rastmic–ptolemismic equivalence continuum: Difference between revisions
Created page with "The '''rastmic–ptolemismic equivalence continuum''' is a continuum of temperaments which equate a number of rastmas (243/242) with the ptolemisma (100/99). All temperaments in the continuum satisfy {{nowrap|(243/242)<sup>''n''</sup> ~ 100/99}}. 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 allo..." Tags: Mobile edit Mobile web edit |
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Examples of temperaments with fractional values of ''n'': | |||
* [[Tetracot|Tetracot expansion]] ({{nowrap|''n'' {{=}} {{frac|1|2}} {{=}} 0.5}}) | |||
* 7 & 34 & 164 ({{nowrap|''n'' {{=}} {{frac|3|2}} {{=}} 1.5}}) | |||
* 7 & 34 & 453 ({{nowrap|''n'' {{=}} {{frac|7|3}} {{=}} 2.333…}}) | |||
* 7 & 34 & 301 ({{nowrap|''n'' {{=}} {{frac|5|2}} {{=}} 2.5}}) | |||
[[Category:Tetracot]] | [[Category:Tetracot]] | ||
[[Category:Equivalence continua]] | [[Category:Equivalence continua]] | ||
Latest revision as of 00:36, 31 January 2026
The rastmic–ptolemismic equivalence continuum is a continuum of temperaments which equate a number of rastmas (243/242) with the ptolemisma (100/99).
All temperaments in the continuum satisfy (243/242)n ~ 100/99. 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 tetracot 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 approximately 2.4372, and temperaments near this tend to be the most accurate ones.
| n | Temperament | Comma | |
|---|---|---|---|
| Ratio | 2.3.5.11 monzo | ||
| −2 | 7 & 34 & 40 | 164025/161051 | [0 8 2 -5⟩ |
| −1 | Tetracot-diesic (7 & 9 & 34) | 1350/1331 | [1 3 2 -3⟩ |
| 0 | Ptolemismic (7 & 12 & 15) | 100/99 | [2 -2 2 -1⟩ |
| 1 | Tetracot-kleismic (7 & 34 & 46) | 2200/2187 | [3 -7 2 1⟩ |
| 2 | Anthill (7 & 34 & 118) | 532400/531441 | [4 -12 2 3⟩ |
| 3 | 7 & 34 & 183 | 129140163/128840800 | [-5 17 -2 -5⟩ |
| … | … | … | … |
| ∞ | Neutral expansion | 243/242 | [-1 5 0 -2⟩ |
Examples of temperaments with fractional values of n:
- Tetracot expansion (n = 1⁄2 = 0.5)
- 7 & 34 & 164 (n = 3⁄2 = 1.5)
- 7 & 34 & 453 (n = 7⁄3 = 2.333…)
- 7 & 34 & 301 (n = 5⁄2 = 2.5)