Syntonic–kleismic equivalence continuum
The syntonic-enneadecal equivalence continuum is a continuum of 5-limit temperaments which equate a number of syntonic commas (81/80) with the enneadeca ([-14 -19 19⟩).
All temperaments in the continuum satisfy (81/80)n ~ [-14 -19 19⟩. Varying n results in different temperaments listed in the table below. It converges to meantone as n approaches infinity. If we allow non-integer and infinite n, the continuum describes the set of all 5-limit temperaments supported by 19edo (due to it being the unique equal temperament that tempers both commas and thus tempers all combinations of them). The just value of n is approximately 0.1309..., and temperaments having n near this value tend to be the most accurate ones.
| n | Temperament | Comma | |
|---|---|---|---|
| Ratio | Monzo | ||
| 0 | Enneadecal | [-14 -19 19⟩ | |
| 1 | Countermeantone | [-10 -23 20⟩ | |
| 2 | Sensi | 78732/78125 | [2 9 -7⟩ |
| 3 | 19 & 169c | [2 31 -22⟩ | |
| 4 | 19 & 162c | [-2 35 -23⟩ | |
| 5 | Unicorn | 1594323/1562500 | [-2 13 -8⟩ |
| … | … | … | … |
| ∞ | Meantone | 81/80 | [-4 4 -1⟩ |
Examples of temperaments with fractional values of n:
- Parakleismic (n = 0.5)
- 19 & 506 (n = 1/3 = 0.3)
19 & 506
Commas: [38 61 -58⟩
POTE generator: 505.1394 cents
Map: [<1 26 28|, <0 -58 -61|]