Archytas–chromatic equivalence continuum
Jump to navigation
Jump to search
| This page presents a novelty topic. It may contain ideas which are less likely to find practical applications in xenharmonic music, or numbers that are impractically large, exceedingly complex, or chosen arbitrarily. Novelty topics are often developed by a single person or a small group. As such, this page may also contain idiosyncratic terms, notation, or conceptual frameworks. |
The Archytas–chromatic equivalence continuum, or septimal–chromatic equivalence continuum, is a continuum of 7-limit temperaments which equate a number of Archytas commas (64/63) with the Pythagorean apotome (2187/2048).
All temperaments in the continuum satisfy (64/63)n ~ 2187/2048. Varying n results in different temperaments listed in the table below. It converges to archy as n approaches infinity. The just value of n is 4.169771, and temperaments near this tend to be the most accurate ones.
| n | Temperament | Comma | |
|---|---|---|---|
| Ratio | Monzo | ||
| 0 | Whitewood | 2187/2048 | [-11 7⟩ |
| 1 | Flattone | 137781/131072 | [-17 9 0 1⟩ |
| 2 | 8680203/8388608 | [-23 11 0 2⟩ | |
| 3 | 546852789/536870912 | [-29 13 0 3⟩ | |
| 4 | Schismatic (Sesquiquartififths) |
34451725707/34359738368 | [-35 15 0 4⟩ |
| 41⁄6 | [very long] | [-216 92 0 25⟩ | |
| … | … | … | |
| ∞ | Archy | 64/63 | [6 -2 -1⟩ |