mathematical model

Deleuze uses the mathematical model of three dimensional space to describe the manner in which the musical concepts of harmony and melody relate to each other. He states, "Returning to the opposition [between the smooth and the striated], the striated is that which intertwines fixed and variable elements, produces an order and succession of distinct forms, and organizes horizontal melodic lines and vertical harmonic planes." By equating harmony to a greater level of geometric complexity, he is drawing the distinction between the two elements such that he concludes harmony is the determination of melody plus some other elusive multiplicity. He is somehow trying to suggest that there is a greater level of conceptual depth within the construction of harmony than there is in melody. I chose to highlight this use of a mathematical model because I believe his analogy fails.

Harmony and melody are bound and ruled by the same lines and differ only in the mechanics in which they portray the same underlying complexity. Harmony portrays the (traidic) chord structure of a piece through the use of unison, melody through the use of sequence and arpeggiation. In melody interstitial tones are inserted not only to create stepwise motion along the scale, which may lead one to incorrectly conclude that melody and harmony are fundamentally different, but also as way to create delays or rhythmic way stations that allow the composer to hold off from using a tone which would establish the next chord of the harmonic progression before the chord infrastructure dictates. On average throughout a piece, these interstitial tones will appear more often on the off beat of the rhythmic meter, where as the three tones that make up a chord (or base triad of a seventh chord in jazz) will more often occupy the primary positions of the rhythmic structure. These primary tones make up the very same notes of the backing harmonies played by whatever music ensemble backs up the lead melody.

I would more view the concepts of tonal and rhythmic orders within a piece of music to be the multiplicities that act in different, often contradictory, ways which are the parameters whose intensities a composer will modulate to create the fundamental nature of their creation.