Syntonic–diatonic equivalence continuum

The syntonic-diatonic equivalence continuum is a continuum of temperaments which equate a number of syntonic commas (81/80) with the limma (256/243).

All temperaments in the continuum satisfy (81/80)ⁿ ~ 256/243. 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 5edo (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 4.1952…, and temperaments near this tend to be the most accurate ones.

256/243 has the advantage of being the characteristic 3-limit comma tempered out in 5edo. For each case, we notice that n equals the order of harmonic 5 in the corresponding comma, and equals the number of generators to obtain a harmonic 3 in the MOS scale. However, if we let k = n + 1 (meaning n = k - 1) so that k = 0 means n = -1, k = 1 means n = 0, etc. then the continuum corresponds to (81/80)^k = 16/15, which might be a preferred way of conceptualising it because:

16/15 is the diatonic semitone, notable in the 5-limit as the difference between 4/3 and 5/4, so this shifted continuum could also logically be termed the "syntonic-diatonic equivalence continuum". This means that at k = 0, 4/3 and 5/4 are mapped to the same interval while 81/80 becomes independent of 16/15 (meaning 81/80 may or may not be tempered) because the relation becomes (81/80)^0 = 1/1 = 16/15.
k = 1 and upwards (up to a point) represent temperaments with (the potential for) reasonably good accuracy as equating at least one 81/80 with 16/15 seems like a good lower bound for a temperament intended to model JI. A good upper bound might be rodan (k = 4), with the only exception being meantone (n = k = (unsigned) infinity). (Temperaments corresponding to k = 0, -1, -2 are comparatively low-accuracy to the point of developing various intriguing structures and consequences.)
16/15 is the simplest ratio to be tempered in the continuum.

Temperaments in the continuum
k = n + 1	n = k − 1	Temperament	Comma
k = n + 1	n = k − 1	Temperament	Ratio	Monzo
-3	-4	Laquadgu	177147/160000	[-8 11 -4⟩
-2	-3	Laconic	2187/2000	[-4 7 -3⟩
-1	-2	Bug	27/25	[0 3 -2⟩
0	-1	Father	16/15	[4 -1 -1⟩
1	0	Blackwood	256/243	[8 -5⟩
2	1	Superpyth	20480/19683	[12 -9 1⟩
3	2	Immunity	1638400/1594323	[16 -13 2⟩
4	3	Rodan	131072000/129140163	[20 -17 3⟩
5	4	Vulture	10485760000/10460353203	[24 -21 4⟩
6	5	Pental		[-28 25 -5⟩
7	6	Hemiseven		[-32 29 -6⟩
…	…	…	…
∞	∞	Meantone	81/80	[-4 4 -1⟩