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A Structural Theory of Rhythm Notation Based on Tree Representations and Term Rewriting Read chapter Read first chapter

2015 | OriginalPaper | Chapter

Applications of DFT to the Theory of Twentieth-Century Harmony

Author : Jason Yust

Published in: Mathematics and Computation in Music

Publisher: Springer International Publishing

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Abstract

Music theorists have only recently, following groundbreaking work by Quinn, recognized the potential for the DFT on pcsets, initially proposed by Lewin, to serve as the foundation of a theory of harmony for the twentieth century. This paper investigates pcset “arithmetic” – subset structure, transpositional combination, and interval content – through the lens of the DFT. It discusses relationships between interval classes and DFT magnitudes, considers special properties of dyads, pcset products, and generated collections, and suggest methods of using the DFT in analysis, including interpreting DFT magnitudes, using phase spaces to understand subset structure, and interpreting the DFT of Lewin’s interval function. Webern’s op. 5/4 and Bartok’s String Quartet 4, iv, are discussed.

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previous chapter Symbolic Music Similarity Using Neuronal Periodicity and Dynamic Programming
next chapter Utilizing Computer Programming to Analyze Post-Tonal Music: Contour Analysis of Four Works for Solo Flute
Footnotes
1
I am indebted to Emmanuel Amiot for pointing this out and helping me improve upon a previous less elegant proof.
 
Literature
1.
2.
Amiot, E.: Discrete Fourier transform and Bach’s good temperament. Mus. Theor. Online 15 (2009)
3.
Amiot, E.: The Torii of phases. In: Yust, J., Wild, J., Burgoyne, J.A. (eds.) MCM 2013. LNCS, vol. 7937, pp. 1–18. Springer, Heidelberg (2013) CrossRef
4.
Amiot, E.: Viewing diverse musical features in Fourier space: a survey. Paper presented to the International Congress on Music and Mathematics, Puerto Vallarta, 28 November 2014
5.
6.
Antokoletz, E.: Transformations of a special non-diatonic mode in twentieth-century music: Bartók, Stravinsky, Scriabin and Albrecht. Mus. Anal. 12, 25–45 (1993)CrossRef
7.
Boulez, P.: Boulez on Music Today. Translated by Bradshaw, S., Bennett, R.R., Harvard University Press, Cambridge (1971)
8.
Burkhart, C.: The symmetrical source of Webern’s Opus 5, No. 4. In: Salzer, F. (ed.) Music Forum V, pp. 317–334. Columbia University Press, New York (1980)
9.
Callender, C.: Continuous harmonic spaces. J. Mus. Theor. 51, 277–332 (2007)CrossRef
10.
Cohn, R.: Transpositional combination and inversional symmetry in Bartók. Mus. Theor. Spectr. 10, 19–42 (1988)
11.
Forte, A.: A theory of set-complexes for music. J. Mus. Theor. 8, 136–183 (1964)CrossRef
12.
Forte, A.: The Structure of Atonal Music. Yale University Press, New Haven (1973)
13.
Koblyakov, L.: Pierre Boulez: A World of Harmony. Harwood, New Haven (1990)
14.
Lewin, D.: Re: intervallic relations between two collections of notes. J. Mus. Theor. 3, 298–301 (1959)CrossRef
15.
Lewin, D.: Special cases of the interval function between pitch-class sets X and Y. J. Mus. Theor. 45, 1–29 (2001)CrossRef
16.
Lewin, D.: Generalized Musical Intervals and Transformations, 2nd edn. Yale University Press, New Haven (2007) CrossRef
17.
Perle, G.: Serial Composition and Atonality: An Introduction to the Music of Schoenberg, Berg, and Webern, 6th edn. UC Press, Berkeley (1991)
18.
Quinn, I.: General equal-tempered harmony (in two parts). Perspectives of New Mus. 44, pp. 114–159 and 45, pp. 4–63 (2006)
19.
Yust, J.: Schubert’s harmonic language and Fourier phase space. J. Mus. Theor. 59, pp. 121–181 (2015)
Metadata
Title
Applications of DFT to the Theory of Twentieth-Century Harmony
Author
Jason Yust
Copyright Year
2015
Book
Mathematics and Computation in Music

Print ISBN: 978-3-319-20602-8

Electronic ISBN: 978-3-319-20603-5

Copyright Year: 2015

DOI
https://doi.org/10.1007/978-3-319-20603-5_22

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