# tempering out the syntonic comma

[Joe Monzo]

(This is a detailed analysis of a discussion in the HEWM notation definition, and also provides a detailed illustration of the promo concept.)

Pythagorean tuning has two types of semitone: the "diatonic semitone" of about 90 cents (also called "limma"), and the "chromatic semitone" of about 114 cents (also called "apotome").

5-limit Just-Intonation, on the other hand, because it is 2-dimensional, provides a myriad of different sizes for both of the semitones, depending on which exponents of prime-factor 5 are involved in the composition of the ratios.

On the lattice diagrams below, I use three different thickness of lines to represent the vectors of the different intervals. The thinnest line is the vector for the "chromatic semitone", the medium-thickness line is for the "diatonic semitone", and the thickest line is for the "whole-tone". In all cases it is plain to see how adding the two different semitones, in either order, always results in the 9:8 "whole-tone" [2 0].

First, here is the structure in pythagorean tuning, which is the basis of our notation:

Here are two different pairs of 5-limit "chromatic" and "diatonic" semitones:

There are many more examples, but these three are the most relevant, as they are the closest to the center of the lattice, and thus the most likely to be used or implied in typical diatonic musical pieces.

In any temperament in which the syntonic comma vanishes -- such as meantone, well-temperament, or any of these equal-temperaments: 12-edo, 19-edo, 24-edo, 26-edo, 31-edo, 36-edo, 38-edo, 43-edo, 45-edo, 50-edo, and 55-edo -- the tempering-out of this interval has the effect of broadening the meaning of # (sharp) and b (flat) so that it may imply any of the 5-limit combinations, including the three illustrated above.

So these intervals are all functionally identical in these types of tunings, and so, depending on the harmonic context, # and b can "mean" an intonational inflection by any of these 5-limit intervals. This is one important aspect of the promo, of which the syntonic-comma is probably the primary example.

This is an important consideration for anyone who studies the standard Euro-centric repertoire, because it applies for the vast majority of that music, which never distinguishes this interval in notation, so that A is always simply A, B is always simply B, etc.

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