I realized that neither harmony nor counterpoint showed up explicitly in that list of "fundamental concepts" I was considering yesterday. The closest I got was that counterpoint arises from the superposition of melodies, from which one may then conclude that harmony arises from the simultaneity of intervals that arises when the melodies are superposed. This may be a useful departure from how most of us were taught to think about harmony. While I do not want to throw the baby of the tonic-dominant relationship out, the bathwater of assigning "chord labels" to any set of notes that happens to be sounding at the same time may deserved to be sent down the drain. After all, a lot of the "dissonances" of twentieth-century modernism may simply be products of sounding a bunch of triads at the same time (regardless of whether or not there is any "harmonic relationship" between pairs of triads or whether some kind of "polytonality" is in play).
It is all very well and good to attach label to all the steps of the scale, to represent those labels by Roman numerals, and then to elaborate on that representation with Arabic numerals through some technique such as figured bass. The question is whether or not we are simply replacing the simultaneity of notes that we see on the score page with an alternative set of abstract symbols and then assuming that "harmony" involves reasoning about the symbols rather than about what those symbols designate. Suppose that, as Heinrich Schenker did, we go back to the basics of the harmonic series, not for the sake of trying to define an Ursatz but as an axiomatic assertion that the ear does best when trying to detect perfect fifths (the interval between the third and second harmonic) and major thirds (the interval between the fifth and fourth harmonic).
If we were to begin with that axiom, than, rather than trying to label chords, we would try to "parse" any simultaneity as a composition of interval relationships using nothing more than the first five harmonics as a point of departure. Clearly, this will not take us very far. We do not want to derive all of our intervals through long migrations down the circle of fifths. However, Schenker may have been on to something in his harmony textbook when he suggested the idea of combining major and minor scales. Thus, for example one would "parse" the minor third from C to E-flat as the result of taking the major third from C to E and then shifting the underlying scale from C major to C minor. (Note, as an aside, that I am trying very hard not to make this sound like a "generative" system. I have nothing against generative grammars; but they often "generate" complex objects that can be analyzed with pencil and paper but would elude the demands of "real-time thinking" when listening to spoken language.) I would suggest that this approach to parsing a large number of simultaneities would tell us more about what the ear responds do than any amount of traditional labeling.
I would now like to go a bit further out on a limb and suggest that such parsing already takes place among those who actual perform music, rather than just analyze the symbols on score pages. One you depart from keyboard instruments, just about every instrument has some capacity for subtle adjustments to pitch through intonation. Thus, we have had studies about whether or not, for example, string players orient themselves with respect to the fifth in the natural harmonic series rather than the seven-semitone span of an equal-tempered chromatic scale. Intonation amounts to adjusting the pitch you are sounding to the pitch that another instrument is sounding, which means you have to decide what interval your are trying to create and with respect to what other pitch.
This would be a more "action-based" approach to harmony, concerned more with what you do during the immediacy of performance than with how the marks on the score page can be reduced to some abstract structure. It would also be consistent with a story about Thelonious Monk that shows up in Robin D. G. Kelley's biography. In that book we read that Monk really disliked beginners who would improvise over the chord changes, doing little more than run up and down arpeggio patterns. He insisted that the melody should always be the basis for improvisation. If the player respected that principle, then the harmony would take care of itself. To be more crude about it, if the listener can follow what the improviser is doing to the melody, then (s)he will not give a rat's ass whether or not it gets stuck on the ambiguity of a tritone!
All of this is still far from being even half-baked. However, I figure it is about time that I take the name of this blog seriously again. Research is always a matter of rehearsing ideas before subjecting their claims to the rigors of argumentation. The "rehearsal studio" is where those ideas first emerge; and this document is a "laboratory notebook" of "unkempt thoughts." More serious thinking can kick in once is seems as if those thoughts have some "legs."