Music Theory 2018-04-23

By Max Woerner Chase

In this post:

Okay, let's talk about ratios of small integers. For my initial attempt to recapitulate the history of musical intervals, I'm going to use intervals rooted at 440 Hz, below 880 Hz, and with and without 880 Hz above. The intervals will include factors of 2 up to the third power, 3 up to the second power, 5, 7, and 11. I'll be using pure sine waves because I'm lazy.

List of intervals:

I look over my terribly-articulated subjective reactions, and I wonder what kind of scale I could build using 11:7. It looks like that's a somewhat stretched 3:2. Binary logarithm shows it to be quite close to 2^(13/20), so now I'm curious about 20edo.

Suppose I try to build a diatonic equivalent in 20edo, containing degrees 12 and 13. Actually, maybe that's the wrong approach. I liked the major chord in first inversion, and if you look at what would be inverted to get degree 13, you get degree 7.

The frequencies for 0-7-12-20 on 440 in 20edo are: 440, 561, 667, 880. Let's try that in root position and first inversion. I'm liking the result. The xenharmonic wiki calls this cord "downmajor". I clearly don't grasp all the theory going into this, because it surprises me that degree 8 is both a major third and a perfect fourth. Looks like if I want to mess with this, I want the Blackwood Major decatonic scale, which hits a few other intervals I liked. Looks like the chords in this are: either minor and upminor, and major and downmajor. There are two interlocking circles of fifths.

I'm going to pivot a little now from considering music theory, to trying to do algorithmic composition, or at least composition on the computer, now that I've decided that Blackwood Major decatonic sounds like it could be interesting.

I don't know if I'll accomplish anything with this in this entry; there are some problems with the uploader that I need to get worked out by Monday night. But let's see about making composition stuff myself, since it seems really tricky to get the features that I want, together with... actually working.

Let's have the idea of a "scale". I'll represent scales using a sequence of interval sizes, combined with the period (which defaults to 2). I'm going to focus on equal temperament for now. The wiki describes various other types of tunings, some of which I don't understand yet.

(In between these paragraphs, I fixed the uploader.)

Okay, time to make another project. Call it... tiny_music. Pulled a bunch of bits out of Homunculus, and the result is a decent starting point for an arbitrary Python project. I should make a cookie-cutter for this layout sometime, because I don't like the cookie-cutters I've seen so far.

In fact, I'm going to focus on gettin things right for a bit, before I start writing code. Things kind of blew up in our face at work because we weren't pinning requirements properly, and I'd rather not face that again. So, I'll wrap up here for now, and try to get back to 20ed2 decatonic goodness next week, once I've completely-foreseeable-circumstances-proofed my code.

Next week, very basic synthesis, common scales, blackwood scales. I should probably also dip my toes into different synthesis techniques, because using pure sine waves like I did above is extremely limiting.