Abstract
Music theorists have proposed two very different geometric models of musical objects, one based on voice leading and the other based on the Fourier transform. On the surface these models are completely different, but they converge in special cases, including many geometries that are of particular analytical interest.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
Throughout this paper we assume that pitch classes are labeled with a continuous map from the space of notes’ fundamental frequencies to \(\mathbb {R}\), with the integer values of traditional music theory arising as a discretization of this mapping. This permits labelings like f(C\() = 0\), f(C\(\sharp ) = 1\), f(D\()=2\), but not f(C\()=0\), f(G\()=1\), f(D\()=2\).
- 2.
Such smoothly changing distributions might occur in algorithmic composition or in statistical analysis. Yust [20] touts this approach as a “cardinality-flexible,” “common-tone-based” conception of musical distance, because it can relate chords of different sizes based on shared pitch-class content. It is possible that a something similar might be achieved by non-bijective cross-cardinality voice leading.
- 3.
Note that we define these crossfade paths for real-valued pitch classes, assuming each chord only has a finite number of non-zero weighted pitch classes.
- 4.
This is a reflection of the fact that a loop enclosing the circular dimension of voice-leading space sends each note in a chord up or down by one chordal step.
- 5.
It remains an open question whether there exist forms of enharmonicism, perhaps arising from extended just intonation, that cannot be captured by scalar context.
- 6.
Amiot [1], pp. 145–9, subsequently demonstrated this analytically.
References
Amiot, E.: Music Through Fourier Space. Discrete Fourier Transform in Music Theory. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-45581-5
Callender, C.: Continuous transformations. Mus. Theory Online 10(3) (2004)
Callender, C.: Continuous harmonic spaces. J. Mus. Theory 51(2), 277–332 (2007)
Callender, C., Quinn, I., Tymoczko, D.: Generalized voice-leading spaces. Science 320(5874), 346–348 (2008)
Hook, J. Signature transformations. In: Hyde, M., Smith, C. (eds.) Mathematics and Music: Chords, Collections, and Transformations, pp. 137–60. University of Rochester Press (2008)
Hughes, James R.: Using fundamental groups and groupoids of chord spaces to model voice leading. In: Collins, Tom, Meredith, David, Volk, Anja (eds.) MCM 2015. LNCS (LNAI), vol. 9110, pp. 267–278. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-20603-5_28
Hughes, J.R.: Orbifold path models for voice leading: dealing with doubling. In: Monteil, M., Peck, R.W. (eds.) Mathematical Music Theory: Algebraic, Geometric, Combinatorial, Topological and Applied Approaches to Understanding Musical Phenomena, pp. 185–94. World Scientific (2018)
Milne, A.J., Bulger, D., Herff, S.A.: Exploring the space of perfectly balanced rhythms and scales. J. Math. and Mus. 11(3), 101–133 (2017)
Quinn, I.: A unified theory of chord quality in equal temperaments. University of Rochester, PhD dissertation (2004)
Quinn, I.: General equal-tempered harmony: parts two and three. Persp. New Mus. 45(1), 4–63 (2006)
Sivakumar, A., Tymoczko, D.: Intuitive musical homotopy. In: Monteil, M., Peck, R.W. (eds.) Mathematical Music Theory: Algebraic, Geometric, Combinatorial, Topological and Applied Approaches to Understanding Musical Phenomena, pp. 233–51. World Scientific (2018)
Tymoczko, D.: Voice leadings as generalized key signatures. Mus. Theory Online 11(4) (2005)
Tymoczko, D.: The geometry of musical chords. Science 313, 72–74 (2006)
Tymoczko, D.: Scale theory, serial theory, and voice leading. Mus. Anal. 27(1), 1–49 (2008)
Tymoczko, D.: Set class similarity, voice leading, and the Fourier transform. J. Mus. Theory 52(2), 251–272 (2008)
Tymoczko, D.: A Geometry of Music: Harmony and Counterpoint in the Extended Common Practice. Oxford University Press, Oxford (2011)
Tymoczko, D.: Tonality: An owner’s manual. Unpub. MS
Yust, J.: A space for inflections: following up on JMTs special issue on mathematical theories of voice leading. J. Math. Mus. 7(3), 175–193 (2013)
Yust, Jason: Applications of DFT to the theory of twentieth-century harmony. In: Collins, Tom, Meredith, David, Volk, Anja (eds.) MCM 2015. LNCS (LNAI), vol. 9110, pp. 207–218. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-20603-5_22
Yust, J.: Schubert’s harmonic language and Fourier phase space. J. Mus. Theory 59(1), 121–181 (2015)
Yust, J.: Special collections: renewing set theory. J. Mus. Theory 60(2), 213–262 (2016)
Yust, J.: Organized Time: Rhythm, Tonality, and Form. Oxford University Press, Oxford (2018)
Yust, J.: Ganymed’s heavenly descent. Mus. Anal. 38 (Forthcoming)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Tymoczko, D., Yust, J. (2019). Fourier Phase and Pitch-Class Sum. In: Montiel, M., Gomez-Martin, F., Agustín-Aquino, O.A. (eds) Mathematics and Computation in Music. MCM 2019. Lecture Notes in Computer Science(), vol 11502. Springer, Cham. https://doi.org/10.1007/978-3-030-21392-3_4
Download citation
DOI: https://doi.org/10.1007/978-3-030-21392-3_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-21391-6
Online ISBN: 978-3-030-21392-3
eBook Packages: Computer ScienceComputer Science (R0)