First, there was the tone. This tone creates overtones, the images of which create the triadic chord, called the Chord of Nature. Most contemporary theorists ignore the chord of nature part of Schenker's theory, for reasons that I listed here. Instead, we accept that Schenker's tonality is based upon the triad, and upon major or minor modes. These postulates are accepted because the music literature of 1600-1880(ish) exhibit this behavior and the analyses based upon Schenker's theory work with this starting postulate. Recent theories suggest that jazz tonality is based upon the seventh chord rather than the triad.
This triadic chord is unfolded through time by stepwise descending motion in the upper voice, from either the third or fifth* of the chord to the root tonic, which Schenker calls the fundamental tone**. Schenker equates this melodic motion, the fundamental line(Urlinie) with our own life-impulses, striving towards a goal. The melodic motion is accompanied by an arpeggiation in the lower voice up a fifth and back down again, as a foundational counterpoint. Why the fifth? From the overtone series, but also from contrapuntal practice: in three-voice strict counterpoint, if the closing scale degrees 2 and 7 are in upper voices, only scale degree 5 is allowable to complete the triad, and was the only leaping bass line found at the close (same as a cadence) in modal counterpoint.
Schenker has a few terms for this counterpoint: background, fundamental structure (Ursatz), and diatony. This fundamental structure is further expanded through middleground and foreground levels by transformations, prolongations, and elaborations, until the actual musical composition is realized (the surface level). The fundamental structure provides the unity, the primary identity of the tonal piece, by marking the goal and the direct path to that goal. Schenker defines "tonality" as the sum of the fundamental structure with all the elaborations and prolongations.
I'll stop here for the first part, quoting from p. 5 of Free Composition:
As the image of our life-motion, music can approach a state of objectivity, never, of course, to the extent that it need abandon its own specific nature as an art. Thus, it may almost evoke pictures or seem to be endowed with speech; it may pursue its course by means of associations, references, and connectives; it may use repetitions of the same tonal succession to express different meanings; it may simulate expectation, preparation, surprise, disappointment, patience, impatience, and humor. Because these comparisons are of a biological nature, and are generated organically, music is never comparable to mathematics or to architecture, but only to language, a kind of tonal language.
Next time I will describe dissonance as it affects the fundamental structure, and begin to explain the transformations that elaborate this fundamental structure.
* Schenker does allow for the traversal of an octave, from root down to root, but later analyses show that he doesn't think it is very common.
**Or at least that is how Ernst Oster translates it from the german.