User:VectorGraphics/Diatonic and chromatic steps

Diatonic semitones, chromatic semitones, enharmonic dieses, and subchromatic commas are four types of steps that can be found when using diatonic intervals, defined in terms of the diatonic and chromatic scales. However, there are some important considerations:

• Here, "diatonic scale" refers not explicitly to the diatonic scale but to the 7 interval classes it defines, such that a diminished third, minor third, major third, and augmented third are all in the same diatonic interval class ("third", or 2\7).
• Similarly, "chromatic scale" does not refer explicitly to the chromatic scale, but to the 12 interval classes it defines, such that a diminished fourth, a double-augmented second, and a major third are all in the same chromatic interval class (4\12). So it does not matter if m-chromatic or p-chromatic is used; both of them work the same in this context.

Here is a list of the types of steps, and their correspondence to the diatonic and chromatic scale:

• diatonic "semitones" (alterations by 1 diatonic step and 1 chromatic step, i.e. E to F)
• chromatic "semitones" (alterations by 0 diatonic steps and 1 chromatic step, i.e. E to E#)
• enharmonic "dieses" (alterations by 1 diatonic step and 0 chromatic steps, i.e. E to Fb)
• subchromatic "commas" (alterations by 0 diatonic steps and 0 chromatic steps, i.e. E to ^E)

A more in-depth explanation follows.

Enharmonic and chromatic steps

For starters, there is a formal meaning of "the diatonic scale" and "the chromatic scale" being used here. The chromatic scale is 12edo, and the diatonic scale is 7edo, such that any interval can be mapped onto these two systems and thus identified in a rank-2 structure

Somewhat counterintuitively, the generators for this structure are the chromatic semitone (augmented 1sn) and the enharmonic diesis (diminished 2nd), rather than the diatonic semitone, but it makes sense when you look at how they're defined. The chromatic semitone is one chromatic step and zero diatonic steps, and the enharmonic diesis is one diatonic step and zero chromatic steps. So, any interval in this structure can be generated by stacking these two intervals, and the "enharmonic diesis" and "diatonic semitone" are simply badly named.

It can be seen that these work as generators for meantone temperament, as any meantone interval can be created by stacking enharmonic dieses (d2) and chromatic semitones (A1). For example:

Interval	Class	Chromatic semitones	Enharmonic dieses
~3/2	Perfect fifth	7	4
~5/4	Major third	4	2
~5/3	Major sixth	9	5
~9/5	Minor seventh	10	6
~125/72	Augmented sixth	10	5

In fact, it can be seen that the number of chromatic semitones required to reach an interval is exactly the same as its mapping in 12edo, and the number of enharmonic dieses required to reach an interval is exactly the same as its mapping in 7edo, which represents the generic interval class of the interval (though, minus one because mappings start from 0 and interval regions start from 1), such as "third" (2 steps), "fourth" (3 steps), and "sixth" (5 steps).

This is because the enharmonic diesis (as the diminished second) is mapped to 1 step of 7edo and 0 steps of 12edo, and the chromatic semitone is mapped to 1 step of 12edo and 0 steps of 7edo, which is a simple feature of both tunings' diatonic scales.

As such, we can create a meantone mapping matrix representing this set of generators, which could perhaps be considered more intuitive than the standard mapping, as it relates directly to notes' position in the scale:

[math]\displaystyle{ \begin{bmatrix} 7 & 11 & 16 \\ 12 & 19 & 28 \end{bmatrix} }[/math]

This simply stacks the vals for 7edo and 12edo, which are the edos we provided previously as representing the "diatonic" and "chromatic" scales.

A "diatonic semitone" is simply the combination of an enharmonic and chromatic step, such that the perfect fifth of 7 chromas and 4 dieses can instead be characterized as 4 diatonic semitones and 3 additional chromas, which corresponds to the combination of the 7edo and 5edo mappings (3 steps of 5edo, which each increase by one chroma without increasing by any diatonic semitones, and 4 steps of 7edo, which increase by diatonic semitones without increasing by any chromas).

Subchromatic steps

To extend a rank-2 structure defined this way to a rank-3 structure, along with enharmonic and chromatic steps, one can add an additional set of categories, defined by a subchromatic "comma" interval. For example, if we want to generate 5-limit JI in a diatonic framework, rather than using a meantone mapping, we can choose three edos, for example 7, 12, and 15. To highlight the distinction, here are some mappings with 15edo introduced:

Interval	Class	Chromatic class	Diatonic class	15edo class
~81/64	Major third	4	2	6
~5/4	Downmajor third	4	2	5
~3/2	Perfect fifth	7	4	9
~40/27	Down fifth	7	4	8

As you can see, 15edo makes a distinction between intervals tempered by a syntonic comma, where the other two edos don't. As such, the space of intervals represented is three-dimensional, and is in fact the 5-limit, though notice that our classes no longer correspond as neatly to standard intervals.

Other temperaments

Let's take a look at the temperament defined by 5edo and 12edo, a.k.a. 2.3.7 superpyth. 5edo provides a "pentatonic" interval category, while 12edo defines our chromatic category as before.

Interval	Class	Chromatic class	Pentatonic class
~3/2	Perfect fifth	7	3
~9/7	Major third	4	2
~12/7	Major sixth	9	4
~7/4	Minor seventh	10	4

And we can also take a look at temperaments of a much-higher prime limit, such as 2.3.5.7.11 porcupine, or 7 & 15:

Interval	Onyx class	15edo class
~3/2	4	9
~5/4	2	5
~7/4	6	12
~11/8	3	7