Over dinner I learned all sorts of things about music from madmadammim and phoebus. One of them is Augmented Sixth Chords, which show up all over the place in Angel Eyes, and in some blues I play, but I never knew what they were called until now. I mostly use the "German" style, although there are apparently "Italian", "French", and controversially "Swiss" variants.
Addtionally mark explained (at least one reason why) the ionian scale is actually special, which makes me sleep better at night, because western music seems so damned contingent, you know? But I feel like now I can reconstruct at least the major scale from first principles with only a few arbitrary postulates required.
So, here's my current understanding of How To Invent (a very small portion of) Western Music.
Look at a vibrating string, and take the sequence of harmonics up to 5 or 6 or so. If the fundamental is C, this is C C G C E G. For whatever reason, stop short of the 7th harmonic. Take factors of 2 in freqency (i.e. what we know as octaves) as tonally identical, and in a moment we will consider factors of 1/3 (equivalently 2/3, what we call fifths) as "related". Notice already the major triad C E G necessarily follows from two arbitrary choices, (a) octaval identification and (b) arbitrarily not thinking about 7th or higher harmonics.
Now let's invent scales. Start with some note, say C, and move away from it up and down by fifths. Somewhat arbitrarily we're going to stop at seven distinct pitches, but this is a point I need to think about further. Things start getting "crowded" in a certain sense if you have eight or more notes in a scale, but I'm trying to get a better handle on quantitative ways of saying this. The argument given at dinner is that you have two half-step intervals in a row, but the notion of "half-step" seems to only really make sense later in the development, and I'd be happier if there were a more primitive notion of "crowded" that I could appeal to.
Here the question is, if it's going to be 7 total, how many of those should be up and how many should be down? This winds up being a choice about what mode (Ionian, Dorian, Phrygian, Lydian, Mixolydian, Aeolian, Locrian) we're talking about. Ionian scale in fifths is
F C G D A E B
i.e. one down from the tonic and five up. I thought this was really strange that the most basic, default scale is so asymmetric! Dorian "makes more sense", being balanced at three up and three down. However, Ionian uniquely has a peculiar property: (this is the thing that phoebus pointed out) both the dominant and subdominant major triads fit in it!
Thus if I take the perspective that the thing I care about is where the three major triads available to me fall on the circle of fifths, then Ionian is the one scale that symmetrically puts the tonic in the middle of these three. Also Aeolian is the one that uniquely centers the three available minor triads on the tonic. Crazy!