Sensitive interval property for scales as words in the free group F2

Abstract

The sensitive interval property is a special feature of musical scales that generalize the diatonic Ionian (major) and Aeolian (minor) modes: specifically, ascending authentic Ionian and descending plagal Aeolian. This discussion is situated in a music-theoretic interpretation of algebraic combinatorics on words over two-letter alphabets. The present paper provides an introduction to this approach, but relies on results from a number of recent papers in this area. While previous studies have restricted attention to the free monoid of words on two letters, the present one extends consideration to F2, the free group with two generators. This permits treatment of ascending and descending modal varieties of musical scales, together with rising or falling circle-of-fifths presentations (or their generalizations), within a unified mathematical framework. The special property investigated herein positions the diatonic major third (and its generalizations) as of structural significance within the theory. © 2011 Springer-Verlag.