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RELATIVE SATURATION OF INTERVAL AND SET CLASSES: A NEW MODEL FOR UNDERSTANDING PCSET COMPLEMENTATION AND RESEMBLANCE

RELATIVE SATURATION OF INTERVAL AND SET CLASSES: A NEW MODEL FOR UNDERSTANDING PCSET... mation than does my AS(/X/, /Y/).7 EMB(/X/, /Y/) returns the number of distinct forms of /X/ that are embedded in some Y of /Y/. If, for example, /X/ = sc(3-1) [012] and Y = {1,2,3,6,7,8,a}, EMB(/X/, /Y/) = 2 because both {1,2,3} and {6,7,8} are forms of [012] and are included in {1,2,3,6,7,8,a}. Given that {1,2,3} (or any form of [012]) is an inversionally symmetrical set, it can map onto itself through two distinct operations: the identity operation (T0{1,2,3} = {1,2,3}); and under transposition and inversion (T4I{1,2,3} = {1,2,3}). These two transformations yield the same pcset and will be considered a single distinct form of 3-1. Abstract inclusion vectors, including n-class vectors,8 such as the #2 subset-class vector (which is equivalent in appearance to the ICV) can be derived by performing EMB(/X/, /Y/) for each distinct /X/ where #X = n.9 When #X = 2, EMB(/X/, /Y/) returns an argument of the #2 subset-class vector of /Y/ (ICV(Y)). Formally, each argument (i) in the 2CV can be defined as follows: 2CV(X)i = EMB(i, /X/).10 Later in this article, I will introduce a new inclusion function, SATEMB(/X/, /Y/), which returns two arguments that reflect a comparison between EMB(/X/, /Y/) http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Music Theory Duke University Press

RELATIVE SATURATION OF INTERVAL AND SET CLASSES: A NEW MODEL FOR UNDERSTANDING PCSET COMPLEMENTATION AND RESEMBLANCE

Journal of Music Theory , Volume 45 (2) – Jan 1, 2001

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Publisher
Duke University Press
Copyright
Copyright 2001 by Yale University
ISSN
0022-2909
eISSN
1941-7497
DOI
10.1215/00222909-45-2-263
Publisher site
See Article on Publisher Site

Abstract

mation than does my AS(/X/, /Y/).7 EMB(/X/, /Y/) returns the number of distinct forms of /X/ that are embedded in some Y of /Y/. If, for example, /X/ = sc(3-1) [012] and Y = {1,2,3,6,7,8,a}, EMB(/X/, /Y/) = 2 because both {1,2,3} and {6,7,8} are forms of [012] and are included in {1,2,3,6,7,8,a}. Given that {1,2,3} (or any form of [012]) is an inversionally symmetrical set, it can map onto itself through two distinct operations: the identity operation (T0{1,2,3} = {1,2,3}); and under transposition and inversion (T4I{1,2,3} = {1,2,3}). These two transformations yield the same pcset and will be considered a single distinct form of 3-1. Abstract inclusion vectors, including n-class vectors,8 such as the #2 subset-class vector (which is equivalent in appearance to the ICV) can be derived by performing EMB(/X/, /Y/) for each distinct /X/ where #X = n.9 When #X = 2, EMB(/X/, /Y/) returns an argument of the #2 subset-class vector of /Y/ (ICV(Y)). Formally, each argument (i) in the 2CV can be defined as follows: 2CV(X)i = EMB(i, /X/).10 Later in this article, I will introduce a new inclusion function, SATEMB(/X/, /Y/), which returns two arguments that reflect a comparison between EMB(/X/, /Y/)

Journal

Journal of Music TheoryDuke University Press

Published: Jan 1, 2001

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