Mto.99.5.2.Soderberg
Volume 5, Number 2, March 1999 Copyright © 1999 Society for Music Theory Stephen Soderberg REFERENCE: MTO Vol. 4, No. 3 KEYWORDS: John Clough, Jack Douthett, David Lewin, maximally even set, microtonal music, hyperdiatonic system, Riemann system ABSTRACT: This article is a continuation of White Note Fantasy, Part I, published in MTO Vol. 4, No. 3 (May 1998). Part II combines some of the work done by others on three fundamental aspects of diatonic systems: underlying scale structure, harmonic structure, and basic voice leading. This synthesis allows the recognition of select hyperdiatonic systems which, while lacking some of the simple and direct character of the usual (“historical”) diatonic system, possess a richness and complexity which have yet to be fully exploited. II. A Few Hypertonal Variations 5. Necklaces 6. The usual diatonic as a model for an abstract diatonic system 7. An abstract diatonic system 8. Riemann non-diatonic systems 9. Extension of David Lewin’s Riemann systems References 5. Necklaces [5.1] Consider a symmetric interval string of cardinality n = c+d which contains c interval i’s and d interval j’s. If the i’s and j’s in such a string are additionally spaced as evenly apart from one another as possible, we will call this string a “necklace.” If s is a string in Cn and p is an element of Cn, then the structure ps is a “necklace set.” Thus <iijiij>, <ijijij>, and <ijjijjjijjj> are necklaces, but <iijjii>, <jiijiij>, and <iijj>, while symmetric, are not necklaces. i and j may also be equal, so strings of the 1 of 18 form <iii...> are valid necklaces.
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