In formulating the notion concerning the unity of musical space I relied on the assertion that had already been made by previous theoreticians, namely: chords are the vertical product of the overtones, but the scale is the horizontal product. I carried this thought to its conclusion and consequently arrived at the concept whereby the vertical and the horizontal, harmonic and melodic, the simultaneous and the successive were in reality comprised within one unified space.
The Cambridge History of Western Music Theory (Nineteenth-century harmonic theory)
Tonality for Schoenberg is not merely a certain collection of pitches of a scale, but more importantly, a kind of centricity. All pitches of a key-collection are related to a single tonal center, each in a specific way. The function of a single tone is signified by the degree of the scale it represents. The function of a chord depends upon its root, which is, in turn, the scalar degree upon which the chord is constructed. Tonality, then, is a set of functions of scalar degrees.
Schoenberg on Tonal Function