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.
Alternatively, we can express the continuum in terms of (81/80)k ~ [-30 19⟩; in this case, k = (n + 19) / 3.
| n | k | Temperament | Comma | |
|---|---|---|---|---|
| Ratio | Monzo | |||
| -19 | 0 | 19edo | 1162261467/1073741824 | [-30 19⟩ |
| -16 | 1 | Lalayo | 71744535/67108864 | [-26 15 1⟩ |
| -13 | 2 | Lala-Yoyo | 4428675/4194304 | [-22 11 2⟩ |
| -12 | 7/3 | 19 & 8ccc | [-62 29 7⟩ | |
| -11 | 8/3 | 19 & 1cc | [-58 25 8⟩ | |
| -10 | 3 | Latriyo | 273375/262144 | [-18 7 3⟩ |
| -9 | 10/3 | 19 & 6c | [-50 17 10⟩ | |
| -8 | 11/3 | 19 & 11c | [-46 13 11⟩ | |
| -7 | 4 | Negri | 16875/16384 | [-14 3 4⟩ |
| -6 | 13/3 | 19 & 23b | 296630859375/274877906944 | [-38 5 13⟩ |
| -5 | 14/3 | 19 & 35b | 18310546875/17179869184 | [-34 1 14⟩ |
| -4 | 5 | Magic | 3125/3072 | [-10 -1 5⟩ |
| -3 | 16/3 | 19 & 78 | 152587890625/146767085568 | [-26 -7 16⟩ |
| -2 | 17/3 | 19 & 109 | 762939453125/743008370688 | [-22 -11 17⟩ |
| -1 | 6 | Hanson | 15625/15552 | [-6 -5 6⟩ |
| 0 | 19/3 | Enneadecal | [-14 -19 19⟩ | |
| 1 | 20/3 | Countermeantone | [-10 -23 20⟩ | |
| 2 | 7 | Sensi | 78732/78125 | [2 9 -7⟩ |
| 3 | 22/3 | 19 & 169c | [2 31 -22⟩ | |
| 4 | 23/3 | 19 & 162c | [-2 35 -23⟩ | |
| 5 | 8 | 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|]