Formal comma
A mapping comma (also called a formal comma) for a prime number P > 3 is a comma that maps every 2.3.P interval to a nearby conventional 3-limit interval. For example, 81/80 maps every 5-limit interval to the 3-limit.
A mapping 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 3rd" and "5/4 = M3" unambiguously indicate 81/80.
A mapping comma's prime-count vector or monzo 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 mapping comma. Thus 5/4 from C is notated as an inflected E, 7/4 as an inflected B flat, 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 mapping commas. The notations largely agree but do diverge for certain primes, because the exact method for choosing the best mapping 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! Color notation allows one to replace the mapping comma for such a prime with its apotome complement.
| 5 | 7 | 11 | 13 | 17 | 19 | |
|---|---|---|---|---|---|---|
| Color notation | M3 | m7 | P4/A4 | m6/M6 | m2 | m3 |
| FJS notation | M3 | m7 | P4 | m6 | m2 | m3 |
| HEJI notation | M3 | m7 | P4 | M6 | A1 | m3 |
| Prime-factor Sagittal | M3 | m7 | P4 | M6 | m2 | m3 |
See also
https://forum.sagittal.org/viewtopic.php?t=99
Solfege string, a concise way to indicate a series of mapping commas.