Semaphore–chromatic 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 semaphore–chromatic equivalence continuum is a continuum of 7-limit rank-3 temperament families which equate a number of semaphore commas (49/48) with a classic chromatic semitone (25/24). This continuum is theoretically interesting in that these are all 7-limit rank-3 temperament families supported by decimal temperament.

All temperaments in the continuum satisfy (49/48)ⁿ ~ 25/24. Varying n results in different temperament families listed in the table below. It converges to semaphore 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 decimal (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 1.9797965913603088..., and temperaments having n near this value will be more accurate. As this value is so close to 2, temperaments tempering out the breedsma (2401/2400) are unusually accurate. It is even closer to 196/99, but the equivalent comma, while tiny even for an unnoticeable comma at 0.004907 ¢, is unreasonably complex, with a monzo of [-487 -97 -198 392⟩.

Temperaments in the continuum

n	Temperament	Comma
		Ratio	Monzo
−1	4 & 10 & 12d	1225/1152	[-7 -2 2 2⟩
0	Dicot expansion	25/24	[-3 -1 2 0⟩
1	Jubilismic	50/49	[1 0 2 -2⟩
2	Breed	2401/2400	[-5 -1 -2 4⟩
3	46 & 50 & 60	117649/115200	[-9 -2 -2 6⟩
…	…	…	…
∞	Semaphoresmic	49/48	[-4 -1 0 2⟩

Temperaments with non-integer n

n	Temperament	Comma
		Ratio	Monzo
196/99	4 & 10 & 3299cd	664 digits	[-487 -97 -198 392⟩