## Counting Connected Tetrads in the Decatonic Dynamic Major Scale

This is a Decatonic Dynamic Major scale analysis. My information on this scale comes from Paul Erlich’s paper here, Figure 7, and I’m using the non-tempered, just version here. I’m assuming that C is 1/1 and this falls on vertex 4 of the undirected graph.

The notation is in Mixed Sagittal; the playback is from Dorico in 224EDO, which is reasonably close enough to just that the harmonic colors are clear.

These chords are listed in order of otonal simplicity, and categorized by how many edges a chord has based on treating each tetrad as a 4-vertex subgraph of the graph interpretation of this scale. Thus, K4 is a complete graph and arguably the most “consonant”; and K4-e, C4, paw, K1,3, and P4 refer to the other possible 4-vertex connected graphs.

I used some graph theory tools in SageMath to help count all these tetrads.

Thoughts and observations:

In a 10-note scale, there are 10 choose 4, or 210 possible tetrads.

Given this particular interpretation of the scale, there are 114 ‘connected’ tetrads. These tetrads break down into: 4 K4 (complete/consonant; 6 edges), 20 K4-e (diamond; 5 edges), 2 C4 (square; 4 edges), 46 paw (4 edges) , 6 claw (3 edges), and 36 P4 (path; 3 edges).

While I initially thought that this categorization of subgraphs would clearly break down tetrads by levels of dissonance, this seems only to be a very general rule and not a useful distinction. However dissonant a chord is, the more ‘edges’ the chord has means that they are more easily tuned by ear. This is interesting in that some dissonant chords may be as easy to tune as some consonant chords, which on the surface seems counter-intuitive.

While these chords are not categorized easily by dissonance, to my ears the more edges each chord has, the clearer the harmonic color of the chord. Thus, I find the K4, and K4-e chords particularly ‘sharp’ whether they are consonant or not, with a very general lessening of this effect as the chords have less direct ‘consonant’ intervals in them. This implies that it may be a good idea to generally favor these chords in a composition. I’m interested to see if others have a similar perception.

This scale does not contain notes to form a typical ii-V-I progression. Thus, this specific progression would require chords taken from outside the pitch space, or modulation.

Along the idea of having narrow intervals resolving to a consonant chord as a basis for having ‘strong’ harmonic motion, the narrowest interval is 20:21, between A and B-flat, and between F-sharp and G. The next-smallest interval 15:16, between E and F, and B and C. The next-smallest then is 14:15, between C and C-sharp, E-flat and E, F and F-sharp, and B-flat and B.

I7-IV7 is decently strong, having E-F and Bflat-A resolving in opposite directions, plus E-Eflat resolving downwards. Thus, this progression has one of each type of these ‘narrow’ intervals resolving.

While strong, I7-IV7 is not uniquely strong in terms of voice leading. There are several other tetrads in this list that resolve 3 notes in contrary motion to a consonant chord. This means that the options for flexible voice leading and harmonic color are plentiful, and if (in addition to small-interval motions) a general motion from dissonant harmony to consonant harmony is desired, there are several better options than I7-IV7.

The ‘square’ chords (m. 28-29 in the video playback) each have an interesting strong, yet ambiguous, resolution to one of two perfect fifths.

The chords that resolve all 4 notes to a consonant chord are not in this list, which means that (least in just intonation) they are unconnected chords and will be difficult-to-tune dissonant.

This is not the only way to interpret consonances in this scale. For example, if 5:7/7:10 (the edges between vertices 0-5, 1-6, 2-7, and 4-9) is removed, the resulting set of connected tetrads totals 76, with 0 K4, 11 K4-e, 0 C4, 28 paw, 4 claw, and 33 P4 tetrads.