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 19-comma ([-30 19⟩).

All temperaments in the continuum satisfy (81/80)^k ~ [-30 19⟩. Varying k results in different temperaments listed in the table below. It converges to meantone as k approaches infinity. If we allow non-integer and infinite k, 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 k is approximately 6.376..., and temperaments having k near this value tend to be the most accurate ones.

This continuum used to be expressed as the relationship between 81/80 and the enneadeca ([-14 -19 19⟩). That is, (81/80)ⁿ ~ [-14 -19 19⟩. In this case, n = 3k - 19.

Temperaments in the continuum
k	Temperament	Comma
k	Temperament	Ratio	Monzo
0	19edo	1162261467/1073741824	[-30 19⟩
1	Lalayo	71744535/67108864	[-26 15 1⟩
2	Lala-Yoyo	4428675/4194304	[-22 11 2⟩
3	Stump	273375/262144	[-18 7 3⟩
4	Negri	16875/16384	[-14 3 4⟩
5	Magic	3125/3072	[-10 -1 5⟩
6	Hanson	15625/15552	[-6 -5 6⟩
7	Sensi	78732/78125	[2 9 -7⟩
8	Unicorn	1594323/1562500	[-2 13 -8⟩
9	19 & 51c	129140163/125000000	[-6 17 -9⟩
…	…	…	…
∞	Meantone	81/80	[-4 4 -1⟩