(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:
Spoiler:
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.