Music Theory 2018-04-16

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By Max Woerner Chase

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Okay, this is a little rough... I got a lot of post written, before I realized I wasn't sure where I was going with all this. I guess what I want to figure out, in terms of music theory, is how to use symmetries in a musical structure, to simplify composition on a computer. Like, when I learned music theory, one thing we went over was repeated phrasal structure. But every composition software I've used has me entering in every note; I can't tell it "use a common section here, and I'll fill in the last measure or so, so they have different cadences".

Music is normally understood to consist of some sounds, each of which has a recognizable pitch, spaced out according to a rhythm. I mean, it's a spectrum. But anyway.

A lot of music theory derives from the relationships between tones. Which relationships are important? Well, it's possible to get bogged down in too much detail, which a previous version of this post did. How tricky can it all be? Well...

The details of how hearing works are very elaborate. In the nineteenth century, Ohm and Helmholtz believed that human the human ear separated sounds into component frequencies of sine waves. This is part of the story. The ear, in addition to separating out frequencies, can selectively alter its sensitivity to frequencies. Furthermore, the brain processes related frequencies together, which can create the perception of tones that a simple spectral breakdown doesn't contain. And our ability to distinguish small differences in frequency is greater for overlaid tones, in contrast to consecutive tones.

So, psychoacoustics is complicated. All the same, we can identify musical notes with a fundamental frequency, and consider the relationships between notes in terms of the relationships between the sine waves.

There are several philosophies of choosing relationships between frequencies:

Equal temperament has gotten more popular over time. Ultimately, consonance of a ratio comes from how close it is to a small integer ratio.

The choice of instrumentation can constrain the choice of tuning, and because the ear is more sensitive to dissonance in overlaid tones, the musical texture of a piece can influence which choice of tuning is acceptable. A piece of music can have different textures at different times. Because it's possible for the tuning to change in different parts of a piece (unaccompanied vocals vs vocals plus piano), a scale should ideally fit into multiple systems of tuning.

Anyone who's learned music has had to deal with musical scales. These are usually presented as a collection of notes to play in sequence, but the reason why those frequencies are chosen, and not some other, lies in what happens when some of those notes are picked out, and played together.

Integer ratios between frequencies sort of fade into the lowest frequency, because the sounds made by a musical instrument are a linear combination of integer multiples of the note's fundamental frequency. Therefore, instead of sounding like "another pitch", adding in a 2:1 ratio alters the character of a note, but not its perceived pitch. With this in mind, 2:1 is the simplest ratio at which to "loop" a tuning system, and 3:2 is the simplest ratio to produce distinct but harmonious sounds. This is why most chords are based on a ratio that contains exactly one factor of three, opposite some power of two.

In any case, when building a scale "above" some pitch, you want to fill out the pitches to twice that pitch. You want to include the 3:2 ratio.

Simple ratios below 3:2 include:

One interesting thing about some of these ratios is that 5:4 times 6:5 equals 3:2; they fit together into the simpler ratio. Combining them with the 3:2 ratio gives the smaller ratio the "support" of the simpler one. The results of this are known as the "major" and "minor" chords.

A fixed scale extends the 3:2 ratio, or an approximation, around the scale, except for one note, which has its note 3:2 above "outside" the scale. Within a given (approximate) 3:2 ratio, the note above the bottom is either (approximately) 6:5 up, or 5:4 up.

The most common scale in western music, the major scale, has 3:2 intervals (called fifths, because they are four notes above; thus, a fifth is two thirds [1]) above six of the notes, but not the seventh note in the scale.

One thing that I feel like should be more obvious to me how to do, is electronic composition outside of 12-TET. I'd like to see something with an easier interface to, like, 31-TET, or using arbitrary rational numbers untethered from octaves.

I'll be honest, I look over this and I'm not really sure where I'm going with all of this. I kind of want to take the different styles of composition, and the different ways of designing a scale, and have some formalism for converting that into a specific set of principles for composition using a set of related scales.

Next week, I think I'll probably noodle around with rational numbers and powers of primes.

[1]You can mark that square off on your "STEM guy is smug about music theory" bingo card now.