Formal comma

A formal comma is a comma used as an accidental in a just intonation notation system. Usually, there is a single formal comma for each prime, which is in the 2.3.p subgroup for prime p and contains one factor of p, and thus a formal comma serves as an assignment of a prime harmonic to a particular Pythagorean interval. The functional structure of a just intonation notation system essentially comes down to defining a set of formal commas. The 2.3 subgroup here may be replaced with 2.sqrt(3) as in Neutral FJS.

A formal comma may be denoted either by its ratio, as with any just interval (e.g. 81/80) or in the 2.3.p or 2.sqrt(3).p case by assigning a reduced prime harmonic to a Pythagorean interval (e.g. 5/4 = M3).

Prime	Functional Just System		HEJI notation	Color notation**	Sagittal notation		Ben Johnston notation*
	Neutral	Pythagorean			Prime factor	Full** (prime intervals***)
5	81/80					81/80, 32805/32768	81/80*
7	64/63						36/35
11	sqrt(243/242)	33/32		33/32, 729/704	33/32	33/32, 729/704	33/32
13	sqrt(512/507)	1053/1024	27/26	27/26, 1053/1024	27/26	27/26, 1053/1024	65/64
17	4131/4096		2187/2176	4131/4096		2187/2176, 4131/4096	51/50
19	513/512						96/95
23	736/729			16767/16384	736/729		46/45

*Ben Johnston notation uses the 5-limit Zarlino scale for nominals, which means that 81/80 has a different role in the system than other formal commas. As a result, Ben Johnston formal commas are generally 2.3.5.p instead of 2.3.p.

** Color notation and sagittal notation have multiple formal commas for some primes.

*** Sagittal notation has formal commas for some composite intervals, allowing intervals such as 14/11 to be notated as the diatonic intervals they are near melodically. More info available on the Sagittal notation page.