1997 Abridged Notes
Keywords: scale, music education, music theory, information theory, graph theory, set theory, visualization.
Edgar Matias, University of Toronto
James McGowan, Eastman School of Music1997
IntroThis is an extension of an extention of a paper written in 1993. We recommend that you read the original paper first.
Note also that while that original paper was quite basic and easy to understand, this extension is very technical and cryptic. Unless you have some knowledge or training in pitch class set theory, you're not likely to understand much of it.
UpdateWe have generated component breakdown tables for M0 to M6. They are summarised in a single table available online.
A table showing the pcsets with similar component breakdowns is also available.
The next logical step is to remove the whole lateral / diagonal distinction, and form pcsets by interval classes alone, without restriction. With 6 interval classes, that would result in 64 component combinations (tables).
ReferencesAgmon, Eytan (1989). A Mathematical Model of the Diatonic System. Journal of Music Theory, 29:249-270
Clough, John and Gerald Myerson (1985). Variety and Multiplicity in Diatonic Systems. Journal of Music Theory, 29: 249-270.
Copyright 1997 Edgar Matias & James McGowan