Properties and Generative Methods of Scales and Sets

Many theorists in the second half of the twentieth century examined familiar scales and sets in order to ascertain the unique properties that may help explain why composers favor these collections. The methods that can be used to generate these sets also yield interesting information about the sets themselves. The properties andgenerative methods studied by scholars can be organized within two branches of set theory: chromatic and diatonic. Although the approaches scholars have taken in organizing properties can be classified as diatonic or chromatic, the sets themselves do not necessarily map completely into one or the other, which is to say that a single set may be generated by more than one method. In chromatic set theory, sets are derived from asingle-tiered universe of twelve equally spaced pitches within the octave. Properties from this group include inversional symmetry, transpositional combination, and transpositional invariance. Generative methods for sets featuring these properties include combination and complete-interval-cycle generation. In the second branch of set theory, the diatonic, sets are viewed within a two-tiered approach where some subset of the 12-pcsystem is placed against the background of the chromatic universe. This subset, generically called a diatonic set, may be the usual diatonic scale (013568T), but may alternatively be any other subset of the chromatic scale. Properties derived from diatonic set theory include: partitioning, cardinality equals variety, structure implies multiplicity, Myhill's property, semi-reducedness, reducedness, consecutivity property, maximal evenness, deepness, and well-formedness. Generative methods for sets featuring these properties include algorithms and partial-interval-cycle generation. When all of these properties are compiled and defined, they may be organized into a table that shows which sets feature which properties. Such a table should prove useful to composers and analysts seeking to learn more about intervallic relationships, structures, and properties within and among particular sets.

Name thesis:degree_name

Master of Arts

Level thesis:degree_level

masters

Discipline thesis:degree_discipline

Music

Grantor dc:publisher

The Ohio State University

Year dc:date

2005

Author dc:creator

• Miller, John Gabriel

Contributors dc:contributor

• Dobos, Lora

• Music

Statement dc:rights

• unrestricted
• This thesis or dissertation is protected by copyright: all rights reserved. It may not be copied or redistributed beyond the terms of applicable copyright laws.

Language dc:language

English

Repository record dc:identifier

http://rave.ohiolink.edu/etdc/view?acc_num=osu1364292301

OAI identifier oai:identifier

oai:etd.ohiolink.edu:osu1364292301