Abstract
Compositions in the Hindustani music of North India are regulated by parent modes called thaats. This article presents the thirty-two theoretically possible thaats as vertices of a five-dimensional hypercube. Minimal paths between vertices enumerate the notes differing between corresponding modes. Modern Hindustani music uses ten thaats selected on aesthetic grounds by the musicologist V. N. Bhatkhande a century ago. Nine of these ten occupy a great cycle of the hypercube, but the tenth mode breaks the pattern. This article proposes two more mathematically consistent Hindustani musical systems, both based on minimal pitch change. The first is based on cycles of ten modes that complete great cycles of the musical hypercube, and the second is based on a Hamiltonian cycle of the hypercube tracing through all thirty-two thaats. The article then examines the geometry of the melakartasystem of seventy-two modes, originally codified for South Indian Carnatic music by Venkatamakhin in the seventeenth century, and demonstrates a method for generating a large family of Hamiltonian cycles encompassing the seventy-two melakartas.
