Why microtonal music?
The question 'why microtonal music?' really begins when we investigate 'why 12 standard pitches to the octave?'. When this question is answered, we understand a whole lot more, and we also understand that there are many other viable choices for music-making, and it even becomes clear that we are indeed surrounded by microtonality were we least expect it.
As it turns out, the smallest musical step size, the semitone (on the piano, 'C' to 'C-sharp' is an example, has an infinite universe between its two defining tones. Anyone familiar with certain ethnic world musics, including the obvious example of traditional Middle Eastern music, already has experience of this, whether they know it or not. This universe in the middle of the semitone has enormous potential for expression of new moods and colors, which can be exciting for composers, performers, and audiences alike.
One important concept in the idea of tuning is called Just Intonation, and it is exemplified by the idea that any two pitches that are in a rather simple numerical relationship in their vibration speeds have a noticeable 'stability' to their sound. Two pitches which are the same are in a 1:1 ratio. The well known octave is the ratio of 2:1.The perfect 5th in the ratio of 3:2. Another might be the sound, often described as 'sweet', of the pure major 3rd, in a ratio of 5:4. The standard 12-note per octave tuning is used because it provides a relatively concise set of pitches that approximate the important interval of the 'perfect 5th' pretty well, while having less sonorous and sweet sounding major and minor thirds. Some historical tunings, a notable example being meantone, aim for the purity of the thirds at the slight expense of the purity of the perfect 5th.
These historical meantone systems have counterparts in divisions of the octave into equally spaced step sizes, called 'equal divisions of the octave' or 'edos' for short. One such example that has become popular is 19-edo, because it has new resources that have a foothold in history as well, and we only need 7 more pitches per octave.
Another really exciting approach is to ignore the ratios of the harmonic series, even the octave (2:1), and explore the strange and wonderfully exotic sounds of the inharmonic universe, which is infinitely larger than the infinitely large harmonic universe. One ethnic precedent for this sonic world would be the sounds of the Indonesian gamelan, and some other non-Western ethnic musics.
Many independent composers on the web from around the world have used microtones, and some to great effect. If you are curious to hear some of the various pieces composed using this technique, check out the xenharmonic wiki's listening list.