Composing microtonal

You see, music composition, such as any other area, is an area of researchers: before common-practice composers researched melodies a lot (their system was prone for doing so, making way for the argument that Dean brough up above with regards to music learning); during tonality's reign we had a deep research in harmony and structure (a good deal of form names that come from music we inherited from this period); after the emancipation of dissonance, even though we had an array of different systems, the research is somewhat more prominent in timbre, color and texture - and also on melody, harmony and structure. This research is what gives a musician's work its purpose. One can appreciate anyway he/she'd like a certain atonal/microtonal piece of music, but any endeavors should bring into account that such a piece's intent or its richness is NOT in the "appeal" of its harmonic system (unless this IS THE CASE), since it's reason to be might have been motivated by another aspect of music, another topic of research. In other words, if you listen to Arkana from Varèse, or Metastaseis from Xenakis, you probably won't find one single diatonic melodic line (a line that would fit in the "pleasant" archetype of the OP), but both pieces are masterpieces of 20th century music, pieces that can be appreciated just like a Beethoven symphony. What changes is the purpose.

It seems to me that when we approach this topic to pop music, we will find examples, and good ones, of bands that bend tonality rules or don't apply them whatsoever, choosing atonal or noise atmospheres instead. However, this didn't become a tendency of pop music, given that it's completely regulated by cultural industry. I'm sure if the commercial music of the 70s (prog included) had not been overwhelmed by punk rock, we would have some pretty crazy stuff out there still being released.

One to which I've not got an answer other than "learn by example" as it seems to me that a lot of Indian and Arabic music is traditionally passed down - it's what you culturally play, rather than following hard and fast rules. Or so it seems to me. 

When I started this topic, I was first looking for a way to compose music which uses microtonality but is pleasant at first hearing (or maybe a few) for "normal" western ears, but I'm still not sure if that's a paradox in itself. The Wyschnegradsky etudes come close, but it's actually the atonality which disturbs the beauty. Though I found some of the chord progressions in the video I posted truly beautiful at first hearing.

Whether you are using whole tones, half-tones or quarter-tones, or diatonic or chromatic scales, only some combinations of notes will be harmonious (unlike the pentatonic scales where all note combinations are harmonious). Using diatonic scales we already know which combinations work (tritones for example) and which ones to avoid and the same is true in chromatic scales and microtonal scales using quarter tones. In Dave's example of Arabic Maqam (a quarter-tone micrononal system) only certain combinations of notes can be used to create chords (and harmony) since Maqam is a melodic system rather than a harmonic system. Suffice to say the chords that "work" in that system basically follow the same triatone rules of Western Music for exactly the same (mathematical) reason that the ancient Greeks figured out 2½ thousand years ago. If the chord works then a melody composed using those note-intervals will be harmonious.

Where it all tends to get complicated is when the microtonal scales are not based upon binary divisions of an octave, but this is rare as all the world's music systems are based upon octaves and "the twelfth root of two" (or "the twenty-forth root of two" for quarter tone systems).

Cage also took a lot of ideas from Satie and Webern too.

I love Ligeti's later works far more than his 60s actually and I don't like Penderecki's later works 

Xenakis is my favorite composer (so surprise), his best work though I think starts from 1974. Many of his works actually do utilize microtonal elements but it wasn't a focus of his. The early works are amazing too but once his music fully matured and grew, he became something incredible and unique. 

There is quite a bit of "beautiful music" from the spectral composers and I would also recommend Wyschnegradsky with his quarter-tone music:

Ligeti's orchestral and ensemble works are often very beautiful, also Scelsi may be of interest?

Partch is one of my favorite composers, so is Ives. 

What about the whole (so called) spectral movement? Grisey, Murail, Dufourt, Harvey, Haas and Lindberg? 

All of their music utilities microtonality and the whole spectralism thing is based around the harmonic series, another thing for the OP to look into  

Here it's just chords that sound beautiful, but using this as basis for a song could sound very fresh I thought and I wonder how she actually makes these microtonal chord progressions work, that's my interest.