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 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 12.0078623975..., and temperaments having n near this value tend to be the most accurate ones – indeed, the fact that this number is so close to 12 reflects how small Kirnberger's atom (the difference between 12 schismas and the Pythagorean comma) is.
| n | Temperament | Comma | |
|---|---|---|---|
| Ratio | Monzo | ||
| 0 | Enneadecal | [-14 -19 19⟩ | |
| … | … | … | … |
| ∞ | Meantone | 81/80 | [-4 4 -1⟩ |