Abstract
This paper considers three conceptions of musical distance (or inverse “similarity”) that produce three different musico-geometrical spaces: the first, based on voice leading, yields a collection of continuous quotient spaces or orbifolds; the second, based on acoustics, gives rise to the Tonnetz and related “tuning lattices”; while the third, based on the total interval content of a group of notes, generates a six-dimensional “quality space” first described by Ian Quinn. I will show that although these three measures are in principle quite distinct, they are in practice surprisingly interrelated. This produces the challenge of determining which model is appropriate to a given music-theoretical circumstance. Since the different models can yield comparable results, unwary theorists could potentially find themselves using one type of structure (such as a tuning lattice) to investigate properties more perspicuously represented by another (for instance, voice-leading relationships).
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References
Callender, C.: Continuous Transformations. Music Theory Online 10(3) (2004)
Callender, C.: Continuous Harmonic Spaces. Journal of Music Theory 51(2) (in press) (2007)
Callender, C., Quinn, I., Tymoczko, D.: Generalized Voice-Leading Spaces. Science 320, 346–348 (2008)
Cohn, R.: Properties and Generability of Transpositionally Invariant Sets. Journal of Music Theory 35, 1–32 (1991)
Cohn, R.: Maximally Smooth Cycles, Hexatonic Systems, and the Analysis of Late-Romantic Triadic Progressions. Music Analysis 15(1), 9–40 (1996)
Cohn, R.: Neo-Riemannian Operations, Parsimonious Trichords, and their ‘Tonnetz’ Representations. Journal of Music Theory 41(1), 1–66 (1997)
Cohn, R.: Introduction to Neo-Riemannian Theory: A Survey and a Historical Perspective. Journal of Music Theory 42(2), 167–180 (1998)
Cohn, R.: As Wonderful as Star Clusters: Instruments for Gazing at Tonality in Schubert. Nineteenth-Century Music 22(3), 213–232 (1999)
Douthett, J., Steinbach, P.: Parsimonious Graphs: a Study in Parsimony, Contextual Transformations, and Modes of Limited Transposition. Journal of Music Theory 42(2), 241–263 (1998)
Hall, R., Tymoczko, D.: Poverty and polyphony: a connection between music and economics. In: Sarhanghi, R. (ed.) Bridges: Mathematical Connections in Art, Music, and Science, Donostia, Spain (2007)
Hoffman, J.: On Pitch-class set cartography (unpublished) (2007)
Lewin, D.: Re: Intervallic Relations between Two Collections of Notes. Journal of Music Theory 3, 298–301 (1959)
Lewin, D.: Special Cases of the Interval Function between Pitch-Class Sets X and Y. Journal of Music Theory 45, 1–29 (2001)
Quinn, I.: General Equal Tempered Harmony (Introduction and Part I). Perspectives of New Music 44(2), 114–158 (2006)
Quinn, I.: General Equal-Tempered Harmony (Parts II and III). Perspectives of New Music 45(1), 4–63 (2007)
Robinson, T.: The End of Similarity? Semitonal Offset as Similarity Measure. In: The annual meeting of the Music Theory Society of New York State, Saratoga Springs, NY (2006)
Straus, J.: Uniformity, Balance, and Smoothness in Atonal Voice Leading. Music Theory Spectrum 25(2), 305–352 (2003)
Straus, J.: Voice leading in set-class space. Journal of Music Theory 49(1), 45–108 (2007)
Tymoczko, D.: Scale Networks in Debussy. Journal of Music Theory 48(2), 215–292 (2004)
Tymoczko, D.: The Geometry of Musical Chords. Science 313, 72–74 (2006)
Tymoczko, D.: Scale Theory, Serial Theory, and Voice Leading. Music Analysis 27(1), 1–49 (2008a)
Tymoczko, D.: Voice leading and the Fourier Transform. Journal of Music Theory 52(2) (in press) (2008b)
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Tymoczko, D. (2009). Three Conceptions of Musical Distance. In: Chew, E., Childs, A., Chuan, CH. (eds) Mathematics and Computation in Music. MCM 2009. Communications in Computer and Information Science, vol 38. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02394-1_24
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DOI: https://doi.org/10.1007/978-3-642-02394-1_24
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