Semaphore–chromatic equivalence continuum

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 cents, is unreasonably complex, with a monzo of [-487 -97 -198 392⟩.

Temperaments in the continuum

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

Temperaments with non-integer n

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