Maximal Area Sets and Harmony

A musical scale can be viewed as a subset of notes or pitches taken from a chromatic universe. For the purposes of this paper we consider a chromatic universe of twelve equally spaced pitches. Given integers (N,K) with N > K we use particular integer partitions of N into K parts to construct distinguished sets, or scales. We show that a natural geometric realization of these sets have maximal area, so we call them maximal area sets. We then discuss properties of maximal area sets for the integer pairs (12,5) (12,6) (12,7) and (12,8) with the obvious relevance to scales in our normal chromatic collection of 12 pitches. Complementary maximal area sets are those sets where the chosen K notes realize maximal area, and the complementary N −K notes also realize maximal area. The complementary maximal area sets closely match a significant collection of scales identified in a book on jazz theory by Mark Levine [9].