Status: Some notes for a longer article.
Usually we think of the basic atom of music as being the note. But when listening to music, we always infer a tonic for the current passage. This means that every note we hear is, at least implicitly, an interval. Therefore the basic atom of music is really the interval.
In 12-TET music, there are twelve possible intervals, but these intervals can be coloured in many different way. Therefore it's a good idea to think of the twelve intervals as just being basic intervals, rather than contextual intervals. (For example, the b5 has a different quality when used in the Lydian mode compared to when it's used as an addition to the Phrygian mode.)
Theory of ratios. The perfect fifth is 3/2, and the perfect third is 5/4. After that, things get a bit more complicated. For example, 6/5 is a minor third, and 7/4 is a minor seventh, though a sweet sounding one. To explain this, the concept of limits was introduced. But this doesn't explain many things, such as why 12-TET works at all.
Lots of ways to organise, or categorise the chromatic intervals. Best is probably:
Of course the 3rd can also be moved beyond major/minor and into a sus2 or a sus4 which feels like it need resolution. Similarly, in diminished chords, the seventh can be double flatted to form a diminished seventh, which is enharmonically a major sixth.
Allan Holdsworth says he thinks of chords as all being part of a scale family. But he also modulates the families, changing scales, and in fact he has a sort of "crowded family" that helps him to do this. I tend to find modal harmonic playing in this way not so much my cup of tea, and I notice that classical music that I enjoy seems to have accidentals in the parts that I enjoy the most. It's using them for good effect that's the hard part. Holdsworth himself makes great music, but he does change scales and do all sorts of very advanced stuff.
A guitarist who worked with Steve Vai said he considers the basic notes of an arpeggio from a chord to be the first level of notes; the tones from the scale to be the next level; and then the whole chromatic range to be the third level. George Benson notes that we shouldn't forget the chromatic level, because every single note has a kind of story to tell.