Hamiltonian Cycle: Saxophone
alto-saxophone accompanied by wood-block
Hamiltonian Cycle: Saxophone, written in 1979 for the composer-saxophonist David Mott, is one of a projected series of works whose compositional design is based on the properties of one of the five Platonic solids. Platonic solids are those three-dimensional polyhedra that possess equal-sized and equal-shaped faces. (The cube and pyramid are examples.) The work at hand is determined by the dodecahedron, a twenty-edged, twelve-faced (each a pentagon) form.
In order to generate a musical form, I labeled the dodecahedron's faces with the basic kinds of musical materials to be used in the composition. Then I traced a Hamiltonian cycle (named after the 19th century Irish mathematician, William Hamilton) over the vertices (corners) and edges of the solid. As each vertex and edge occurs once and only once in a Hamiltonian cycle, the cycle determines a unique and efficient ordering of all the musical materials. As I moved from vertex to vertex along the solid's edges, the materials assigned to faces adjoining each vertex determined the character of a musical passage associated with that vertex.
Thus, the piece proceeds by passing through a series of musical "states" each set-off by a "crack" from the woodblock (here a Japanese hyoshigi, used in Kabuki performances) which corresponds to the arrival at a new vertex.
The details of pitch organization within each section is governed by the mutual relationships among a family of related pitch sequences. The resulting compositional structures are articulated by the use of harmonics, sub-tones, micro-tones, timbre-trills and the like. As in other works I have written for saxophone(s), the lines, rhythms, and articulations are meant to suggest the flavor of jazz improvisation.
In this piece I regard the entire Hamiltonian process to be only partially cyclic; it serves to transform the opening F natural into the final A-flat. This can be considered as a fading in from then out to the atemporal space of anahatanad, or "unstruck sound" of Indian music philosophy.
David Mott has recorded Hamilton Cycle: Saxophone on Music Gallery Editions, MGE-35.