Marvel–syntonic equivalence continuum

This page presents a topic of primarily mathematical interest. While it is derived from sound mathematical principles, its applications in terms of utility for actual music may be limited, highly contrived, or as yet unknown.

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 (225/224)ⁿ ~ 81/80. 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 2.788851, 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	[very long]	[39 -6 -26 11⟩
3	Metric	703125/702464	[-11 2 7 -3⟩
∞	Marvel	225/224	[-5 2 2 -1⟩