Formal comma: Difference between revisions

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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 [[~~Monzo|~~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>.

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 ===

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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.

{| class="wikitable"

|+Mapping commas used by various JI notations

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 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 [[Monzo|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>.
+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 ===
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 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.
+{{Todo|inline=1|text=add ratios}}
 {| class="wikitable"
 |+Mapping commas used by various JI notations