[FabHK]

I haven't found an introduction to music theory that makes sense to me.

I vaguely understand that complications arise because we want nice harmonics, ie frequencies whose ratio is a "nice" rational number, such as 2/3 or 3/5 or so.

But our chosen notes should be invariant under doubling of frequencies ("shifting by an octave"), because that's basically the same note.

The problem then is that roots of 2 are irrational, that is, one cannot find (p/q)^2 = 2, or (p/q)^n = 2, or even (p/q)^n = 2^m. Therefore, one cannot find a "nice" interval that, applied several times, wraps around to an octave (or multiple octaves).

However, in a neat coincidence, (3/2)^12 = 129.7463378906... which is close to 2^7 = 128. So, based on that ("Pythagorean comma"), something something something, and we end up with 12 half notes that are basically of frequency f_i = f_0 * 2^(i/12), which are all horribly irrational, but apparently sound "nice" enough, largely (because they are close enough to some "nice" fractions), but only if we pick out some specific 7 of them.

And then the question becomes, which 7 of the 12 do we pick, approximately uniformly distributed. (Why not 6? Every second? I don't know.)

And then, you can transpose them somehow (ie multiply frequencies by 2^(j/12) for some j, but then you change the names for some reason, and everything gets complicated and tonic and Mixolidian double-sharp.

Also, instead of frequencies of the form f_i = f_0 * 2^(i/12) (which, clearly, have the advantage that any multiplication by a power of 2^(1/12) is just a shifting of the index i), you could also use non-equal tuning, with the powers of the 12th root of 2 replaced by some "nice" fraction, which means that any shifting then subtly changes the character of everything, I assume.

This is complicated, admittedly, but for me the nomenclature obscures, rather than elucidates, the issue.

ETA: I sympathise with what irrational wrote: "Is this what it is like when I talk to people who don’t know anything about programming about my work? Pure gibberish?"

[zarmin]

If you're trying to learn music theory by thinking about the mathematical relationship between frequencies, you are severely overcomplicating things for yourself. I have a tendency to do the same thing.

What question are you trying to answer with what you just wrote?

[FabHK]

I guess you're right. I was trying to understand why it's so complicated, and how one could express it in a simpler manner (that refers to the underlying reality, namely frequencies).