Warning, long post. Main stuff highlighted.

Let’s face it, western music theory obfuscates the simple geometric reasons that explain why sounds are harmonious. Pitches whose frequencies are pairwise in simple ratios (such as 5:4 or 3:2) sound good together. If you’ve seen the 3B1B video on music theory he explains how 2^{1/12} helps design musical instruments. Basically this limits the pitches you can play. A true musician might prefer a fretless because he knows what sounds good, so he’s not limited at all. I’m interested in limiting the available pitches even more, ie giving even less than 12 notes (but eventually permitting more), to really understand what’s going on.

My idea is simple. I choose a set of fractions in [1,2] to represent pitches relative to a tonic, call the set P for pitches. If I want I can choose a separate set H of fractions to be the harmonious intervals. Usually I just let P=H. Then, I form all subsets of S such that the subset is pairwise harmonious, call this subset of powerset(P) to be C. **Two pitches are harmonious if their ratio is in H**. Then, I order C in a partial order relation of inclusion, and just take all the maximal elements and look at those. Each element of C I call a “scale”, or a “triad” or “dyad” or “tetrad” or whatever polyad. An element of C could be an A major triad for example.

The next part of this is dissonance. Dissonance is used to escape from one scale and enter another. Music gets interesting when you use dissonance in a minimal sort of way, sort of like adding salt to a stew. You want some salt but not too much. Technically I use dissonance like this: **once I have my maximal polyads, I take a pair of them, and call one A the primary and the other B the secondary. I then get B’ by choosing a pitch p in A and letting B’ = {px : x in B}. I then take A union B’ and see how dissonant it is**, i.e., what the worst fraction that was introduced was. If it’s a fraction like 15/8 then that’s ok. If it’s a fraction like 25/16 then that’s bad. I want the worst introduced fraction to be as simple as possible, and that’s what I mean when I say this process is like salting a stew. Musically what this means is that the available pitches are A union B’. We could even abandon A and just stick with B’, and even treat B’ as the new primary.

The next part of this is the pathway or trail of pitch sets that get created. Really it’s just a graph of vertices and edges. We had pitch set A and we then created B’, so that’s 2 vertices. We can repeat the process and sort of map out the least dissonant pitch set changes. In musical language, this is something like going from an Amajor chord to a C#minor chord.

So in practice I did this, and I found choosing P = {1/1, 6/5, 5/4, 4/3, 3/2, 5/3, 8/5, 2/1} = H was a good choice. I then looked for all dissonant ways out, and strangely enough a path I traced out was what we call the circle of 5ths. Except I went through the circle of 5ths in higher resolution. I found in getting from an Emajor chord to a Bmajor chord, I could go through G#minor. So the progression was E G#m B D#m F# and so on. Though I still ran into the same problem Pythagorus had, because (3/2)^{12} is ~=129.7, not 2^{something} so technically it is not a full circle.

**The maximal elements under the partial order on C are:**

{1/1, 6/5, 3/2, 2/1}

{1/1, 6/5, 8/5, 2/1}

{1/1, 5/4, 3/2, 2/1}

{1/1, 5/4, 5/3, 2/1}

{1/1, 4/3, 5/3, 2/1}

{1/1, 4/3, 8/5, 2/1}

**So for example**, the available notes might be A = {1/1, 5/4, 3/2, 2/1}. To break out of that, I might choose {1/1, 4/3, 8/5, 2/1}, and have these be relative to the 5/4. So, multiplying them all by 5/4 gives B’={5/4, 5/3, 2/1, 5/2}, which are the notes relative to the original tonic. Basically this is a chord change from Cmaj to Aminor.

Then, we can choose {1/1, 6/5, 3/2, 2/1}, and have those all be relative to the 5/2 in B’. So we multiply all of them by 5/2 to get B”={5/2, 6/2, 15/4, 5}. Which in standard terminology is an Eminor chord, or the mediant.

Then we can choose {1/1, 5/4, 3/2, 2/1} and have those all be relative to the 6/2 in B”. So, we multiply all of them by 6/2 to get B”’={6/2, 15/4, 9/2, 6}. In standard terminology this is a G chord.

Then finally we can choose {1/1, 4/3, 5/3, 2/1} and have all these be relative to the 6/2 in B”’. So we multiply all of them by 6/2 to get {6/2, 4, 5, 6}, which is again a Cmaj chord. And then we have truly come full circle, exactly (because of octave equivalence. Everything can just be divided by 4 to get the starting polyad {3/4, 1, 5/4, 3/2}).

So we found one cycle on the graph: triads of C, then Am, then Em, then G, then C.

I’m already a decent musician so I know what notes sound good and how to make transitions. But for example, I played the rules of the game I described above, and came up with the chord progression E G#m B D#m, which is something I would not normally have come up with. That’s why I’m interested in playing this game with the fractions.

I think this is a simple enough game. Clearly what I’ve created is a giant graph of vertices and edges, with a root (tonic/key) vertex, and there exist (as I showed in the example) cycles, and there exist paths that just go off and never return to the root vertex. Referencing my above example there’s a vertex for A, then there’s vertex B’ adjacent to it, then there’s vertex B” adjacent to B’, and then there’s B”’ adjacent to B”’, and B”’ happens to be adjacent to A. When a musician considers doing something unexpected, he chooses an edge, and the available notes to him are the union of the adjacent vertices. For example the first edge between A and B’ has notes {1/1, 5/4, 3/2, 2/1, 5/4, 5/3, 2/1, 5/2}. Then the musician transitions to B’ and only uses {5/4, 5/3, 2/1, 5/2}.

The problem here of course is the degree of the vertices in this graph. The degree is huge, it takes forever to do the calculations, and even longer to interpret it as piano keys. So the problem which I’ve created is to analyze that graph somehow, and see what chord changes it gives. And of course for each choice of P and each choice of H, a new graph is created. I’m pretty confident that what I gave as P is the happy medium of having enough pitches to create interesting music, without it being bland.

I’m just posting this in case it interests anyone.