The Pentahelix: A Four Dimensional Realization of the Spiral Array
The Proceedings of the AMS Special Session on Mathematical Techniques in Musical Analysis Sep Sep
We propose to extend Chew's spiral array (2000), a geometric model for tonality, to four dimensions using simplical complexes so as to represent tetrachords. The spiral array represents pitch classes, triads, and keys as discretized helices embedded in three-dimensional space. Any discretized spiral, such as the pitch class helix in the spiral array, can be mapped into a tetrahelix. This map is not always one to one; the pitch class spiral, where adjacent pitches are seven half steps apart, maps to one tetrahelix, but a spiral in which adjacent pitch classes are only one half step apart maps to two tetrahelices. The building block of a tetrahelix is a tetrahedron, a 3-simplex. One full turn of the pitch class spiral corresponds to one full