Marvel–syntonic equivalence continuum

The marvel-syntonic equivalence continuum is a continuum of 7-limit rank-3 temperament families which equate a number of marvel commas (225/224) with a syntonic comma (81/80). This continuum is theoretically interesting in that these are all 7-limit rank-3 temperament families supported by septimal meantone temperament.

All temperaments in the continuum satisfy (81/80)ⁿ ~ 64/63. Varying n results in different temperament families listed in the table below. It converges to marvel as n approaches infinity. If we allow non-integer and infinite n, the continuum describes the set of all 7-limit temperament families supported by septimal meantone (due to it being the unique rank-2 temperament that tempers both commas and thus tempers all combinations of them). The just value of n is approximately 2.788851128489581..., and temperaments having n near this value will be more accurate.

Temperament families in the continuum

n	Temperament family	Comma
		Ratio	Monzo
0	Didymus	81/80	[-4 4 -1 0⟩
1	Starling	126/125	[1 2 -3 1⟩
2	Hemimean	3136/3125	[6 0 -5 2⟩
2.75	1205 & 612 & 1236	[39 -6 -26 11⟩
3	Meter	703125/702464	[-11 2 7 -3⟩
∞	Marvel	225/224	[-5 2 2 -1⟩