It'd be nice to be able to use MIDI for more diverse music than just that based on western scales. Are there any MIDI extensions that support quater tones, or other sub-semi-tone pitch divisions? I know things can be done with pitch bend automation, but that's really a pain..
You can go microtonal using MIDI! You don't need an extension. The question is: does your instrument/device (hardware or software) allows it?
MIDI can handle microtonality from the control surface to the program interpreting it.
One example of MIDI allowing microtonality in the interpreting side is Native Instrument's Absynth. You can set the instrument to 19, 24, 24, 31, 48, or 96 semi-tones per octave. In fact, you can edit the tuning as you wish, and it can be played with any MIDI control surface. You can set each note in the MIDI piano (or whatever MIDI device you are using) to any frequency.
MIDI can also handle microtonality from the control surface, all you need is a control surface that implements it.
There are different solutions for microtonal control. If you can't tune your MIDI device, you can use a microtonal MIDI keyboard converter, like this one: http://www.h-pi.com/TBX1intro.html
There are MIDI control surfaces specifically designed for microtonality: http://www.h-pi.com/TPX28buy.html
Or if your device allows it, personalize the tuning.
If MIDI isn't getting the work done, you can look into OSC, but I think MIDI has you covered on this one.
To suppliment the answers that explain in detail how MIDI works, I would like to point out that practically all dedicated hardware keyboard synthesizers and sampler instruments, particularly digital pianos, have a built-in capability to let the user select from among several different preset tunings and temperaments rather than 12-tone equal temperament. Technically this has nothing to do with MIDI, but it provides one with a keyboard that can achieve many different tunings and temperaments. Most such keyboards provide historical Western temperaments of 12 (unequally-spaced) pitches to the octave, for Western common-practice tonality.
However, there are also keyboard instruments from several manufacturers designated "Oriental" which provide scales and tunings usable in ethnic music from places like Turkey, the Arabic nations, India and China. These instruments also provide built-in preset sounds designed for use in music of these ethnicities; for example, the oud and the nay.
These instruments have an extra one-octave miniature keyboard positioned above the main one on the left which can be used by the left hand to modify the pitches played on the main keyboard to produce the quarter-tones required by Arabic music.
From the Korg web site:
This Pa600QT (Quarter Tone) version contains all the functionality of the standard Pa600, and adds a mini-keyboard on the left side, which can be used to customize quarter tone scales and quickly access scale presets.
Midi is a data file/transfer format. The software on both ends generate and interpret the midi data. The software should at first need to support such perform and just translate to/from midi using the method that you describe. That is probably a pain to debug when you create it but i think that it should not be that hard to create a function that calculates there values.
I do know that there are keyboards where you can select another mode that allows to easily play such notes and these have midi out too. Those probably create midi messages with that pitch bend method.