(I'm not sure Microtonal is the right term for what I'm talking about. Wikipedia suggests "xenharmonics" and I would have just named it "non-standard tunings".)
Here's the wikipedia page if you want to dip your toes in. Basically, the common western 12-tone scale is not the only one that produces consonant harmonies. In fact, the equal tempered 12-tone scale - which we're trained to accept - is pretty bad at it!
My feeble attempt at explaining why tuning systems are more complex than most people think:
An octave above any particular note is calculated by multiplying the base frequency by two. So if the base note is 100hz, the octaves above that are 200hz, 400hz, 800hz, 1600hz, etc. Now you can see that there are overtones between the octaves: 300hz, 500hz, 600hz, 700hz, 900hz, etc. That means that when you play a note, you are playing overtones that are neither the base note nor octaves of the base note. The overtones are new notes that all sound "consonant" when played with the base note.
OK, let's define 100hz as a G note, since it's pretty close. Then 200hz and 400hz are both G's as well. The second overtone, 300hz, is close to a D(perfect fifth), and the fourth overtone, 500 hz, is close to a B (major third). I don't think it's a coincidence that those first two non-G overtones form the very pleasing major G chord... And unsurprisingly, the next few overtones also correspond with other notes in the 12-tone scale. 700hz is an F (minor seventh), 900hz is an A (major second), etc, etc, etc.
Except here's the problem. If you define a semitone by dividing an octave into 12 equal intervals - like we commonly do in western music; it's called equal-tempering - you can't get any of the non-octave overtone frequencies by going up and down by semitones. If your base frequency is 100hz, the closest you can get to 300hz by using semitones is 299.7hz. Which, granted, is pretty darn close. But the closest you can get to 500hz is 504.0hz, which is not really close at all!
The end result of all this is that when you play G and B together, you're playing 100hz, 504hz, and all of the overtones, including 500hz. Therefore it sounds like you are playing 500hz and 504.0hz together, which sounds out of tune even though it's precisely in tune for how we've defined the tuning system. If you are a good performer you will tend to bend the pitches slightly to compensate, but only if you are using an instrument that can pitch-bend, like your voice or a wind instrument. But on a piano, for example, bending the pitch is not going to happen.
So why do we use the equal tempered 12 tone scale? It's actually pretty arbitrary IMO. It approximates the overtone frequencies better than other numbers of equally spaced tones, up until the 53 tone scale which is nearly spot on. But it's hard to tell the difference between the semitones in the 53 tone scale, not to mention inconvenient.
Instead of defining a semitone by dividing the octave equally, we could define every note by using the frequencies in the harmonic overtone series (called "just tuning"). But then when we move to a new base note (e.g. playing a song in the key of A instead of the key of G), every single note needs to change frequencies slightly in order be in tune.
On top of all that crap, there are harmonic overtones that we don't even attempt to hit with the 12-tone scale. Relatively consonant harmonies that you may have never even heard before.
My favorite nonstandard tuning is the Bohlen-Pierce scale. It sounds totally alien at first, but after I get reoriented it clicks into place and can become hypnotic. And the theory behind it fascinates me. Plus, it's super convenient how the equal tempered version is very close to the just version.
Unfortunately, it appears as if I've run out of Bohlen Pierce music to listen to! Youtube and google seem to be tapped out; I'm not sure if that's because google has tailored my searches or because there's very little music out there. I keep running into Elaine Walker ("Stick Men" is pretty good), and Richard Boulanger's "Solemn Song for Evening" was recommended in the book I'm currently reading. There are a few excellent clarinet solos on youtube. There are a couple versions of Canon on youtube (which IMO are neither creative nor pleasant to listen to). There's a dubstep Bohlen-Pierce song on youtube (Sevish - Mako Haze) that is amazing. And my favorite has been some of the stuff off of Charles Carpenter's record "Splat". You can listen to a few songs here.
Has anyone dabbled in composing with microtonal systems? I'm trying, but the only method I've come up with is to have Audacity (awesome free program) generate sine waves which I import into my main recording software (Cubase). It would be nice to use my midi controller but I don't know how to set the keys to play audio files. Pretty sure it can do it. Maybe I'll sit down with the manual this week and figure it out.