Marvel–breedsmic 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–breedsmic equivalence continuum is a continuum of 7-limit temperament families which equate a number of breedsmas (2401/2400) with a marvel comma (225/224). This continuum is theoretically interesting in that these are all 7-limit temperament families supported by septimal miracle temperament.

All temperaments in the continuum satisfy (2401/2400)ⁿ ~ 225/224. Varying n results in different temperament families listed in the table below. It converges to breedsmic 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 sensi (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 10.69267, and temperaments having n near this value tend to be the most accurate ones.

Temperament families in the continuum

n	Temperament family	Comma
		Ratio	Monzo
−3	31 & 41 & 244	1977326743/1966080000	[−20 −1 −4 11⟩
−2	Quince	823543/819200	[−15 0 −2 7⟩
−1	Gamelismic	1029/1024	[−10 1 0 3⟩
0	Marvel	225/224	[−5 2 2 −1⟩
1	Mirkwai	16875/16807	[0 3 4 −5⟩
2	31 & 41 & 179	40500000/40353607	[5 4 6 −9⟩
3	31 & 41 & 278cd		[10 5 8 −13⟩
4	31 & 41 & 336cd		[15 6 10 −17⟩
…	…	…	…
∞	Breedsmic	2401/2400	[−5 −1 −2 4⟩

We may invert the continuum by setting m such that m − 1/n = 1. This may be called the marvel–gamelismic equivalence continuum, which is essentially the same thing. The just value of m is approximately 1.09352, and temperaments having n near this value tend to be the most accurate ones.

Temperaments with integer m

m	Temperament family	Comma
		Ratio	Monzo
−3	Ampersand	34171875/33554432	[−25 7 6⟩
−2	12d & 31 & 41	1063125/1048576	[−20 5 4 1⟩
−1	Mirwomo	33075/32768	[−15 3 2 2⟩
0	Gamelismic	1029/1024	[−10 1 0 3⟩
1	Breedsmic	2401/2400	[−5 −1 −2 4⟩
2	Mirkwai	16875/16807	[0 3 4 −5⟩
3	31 & 41 & 133d	3796875/3764768	[−5 5 6 −6⟩
4	31 & 41 & 114d	854296875/843308032	[−10 7 8 −7⟩
…	…	…	…
∞	Marvel	225/224	[−5 2 2 −1⟩