Formal comma

A formal comma (also called a mapping comma^{[idiosyncratic term]}) for a prime number p > 3 is a comma in some musical notation that maps every 2.3.p-subgroup interval to a nearby conventional 3-limit interval. For example, 81/80 maps every 5-limit interval to the 3-limit.

A formal comma can be identified by the prime p and the 3-limit interval that p/1 (octave-reduced) maps to. The 3-limit interval is named conventionally. Thus both "prime 5 = major third" and "5/4 = M3" unambiguously indicate 81/80.

A mapping comma's monzo or prime-count vector always has a p-count of ±1. The 2-count and 3-count are almost always non-zero, and all other counts are always zero. For example, the usual mapping comma for prime 19 is 513/512 ([-9 3 0 0 0 0 0 1⟩).

Usage in JI notations

A JI notation will typically have for each prime p a pair of inflections that raise/lower a note by p's formal comma. Thus 5/4 from C is notated as an inflected E, 7/4 as an inflected B♭, etc. Ratios like 35/32 and 49/48 are inflected twice, and the commas accumulate, so complex ratios may be quite distant from the uninflected 3-limit note.

Each JI notation assumes certain formal commas. The notations largely agree but do diverge for certain primes, because the exact method for choosing the best formal commas is disputed. Ideally, both the 3-count and the size in cents is minimized, and there are other considerations as well. The choice for neutral-sounding primes like 11 (P4 vs. A4) and 13 (m6 vs. M6) is particularly tricky. Certain choices map the ratio 13/11 = 289 ¢ to either M2 or M3. Some notation systems allow one to replace the formal comma for such a prime with its chromatic and/or enharmonic counterparts.

Todo: add ratios

Formal commas used by various JI notations

	5	7	11	13	17	19
Color notation	M3	m7	P4/A4	m6/M6	m2	m3
Functional Just System	M3	m7	P4	m6	m2	m3
Helmholtz–Ellis notation	M3	m7	P4	M6	A1	m3
Prime-factor Sagittal	M3	m7	P4	M6	m2	m3

See also

• Solfege string, a concise way to indicate a series of formal commas.

External links

• The Sagittal Forum | Prime-factor Sagittal JI notation (one symbol per prime)