All-(Generalized-)Interval(-System) Chords

We survey the all-interval chords of small order and the interval systems in which they are situated. We begin with an examination of traditional all-interval chords in chromatic pitch-class spaces, and extend the notion of their structure to their counterparts in David Lewin’s Generalized Interval Systems. Mathematically, we observe that these chords belong to three categories of difference sets from the field of combinatorics: (v, k, 1) planar difference sets, (v, k, 2) non-planar difference sets, and (v, k, 1, t) almost difference sets. Further, we explore sets of all-interval chords in group-theoretical terms, where such sets are obtained as orbits under the action of the normalizer of the interval group. This inquiry leads to a catalog of the 11,438 all-interval chords of order k, where 2 6 k 6 8. We conclude with remarks about future work and open questions.