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Some Mathematical and Computational Relations Between Timbre and Color

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Mathematics and Computation in Music (MCM 2022)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 13267))

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Abstract

In physics, timbre is a complex phenomenon, like color. Musical timbres are given by the superposition of sinusoidal signals, corresponding to longitudinal acoustic waves. Colors are produced by the superposition of transverse electromagnetic waves in the domain of visible light. Regarding human perception, specific timbre variations provoke effects similar to color variations, for example, a rising tension or a relaxation effect. We aim to create a computational framework to modulate timbres and colors. To this end, we consider categorical groupoids, where colors (timbres) are objects and color variations (timbre variations) are morphisms, and functors between them, which are induced by continuous maps. We also sketch some gestural variations of this scheme. Thus, we try to soften the differences and focus on the similarity of structures.

J.S. Arias-Valero—Independent researcher.

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Notes

  1. 1.

    Loudness in music performance can affect timbre characteristics [6].

  2. 2.

    A first experiment, where participants were asked to associate colors, color bands, and timbres, confirmed a non-negligible perceptive correlation [26].

  3. 3.

    We mix colors in printing and painting with the subtractive model, a sort of dual of the additive one.

  4. 4.

    We do not have a proof of this continuity.

  5. 5.

    The wave is periodic if the quotient between carrier and modulator frequencies is a rational number.

  6. 6.

    This relation is not a proper morphism but it can be achieved by pasting two suitable 2-simplices in \(\text {Sing}(X)\), which is a higher gesture according to [2]. It is also a hypergesture in the sense of [31].

  7. 7.

    This defines a continuous map \([0,20]\times \mathbb {R}\longrightarrow \mathbb {R}\), so the exponential transpose \([0,20]\longrightarrow \mathbb {R}^\mathbb {R}\), which is a path, is continuous.

  8. 8.

    The audio file, in which we identify the increasing modulation index with time (in seconds), can be accessed from the link https://soundcloud.com/maria-mannone/fm-path/s-cFJ7kNrqJjs?.

  9. 9.

    In essence, it is continuous because each component (composition with f for arrows and f for vertices) is.

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Author information

Authors and Affiliations

  1. Department of Engineering, University of Palermo, Palermo, Italy

    Maria Mannone

  2. Dipartimento di Scienze Ambientali, Informatica e Statistica (DAIS) and European Centre for Living Technology (ECLT), Ca’ Foscari University of Venice, Venezia, Italy

    Maria Mannone

  3. Bogotá, Colombia

    Juan Sebastián Arias-Valero

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  1. Maria Mannone
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Corresponding author

Correspondence to Maria Mannone .

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Editors and Affiliations

  1. Georgia State University, Atlanta, GA, USA

    Mariana Montiel

  2. Technological University of the Mixteca, Huajuapan de León, Mexico

    Octavio A. Agustín-Aquino

  3. Universidad Politécnica de Madrid, Madrid, Spain

    Francisco Gómez

  4. Life University, Marietta, GA, USA

    Jeremy Kastine

  5. National Autonomous University of Mexico, Mexico City, Distrito Federal, Mexico

    Emilio Lluis-Puebla

  6. Georgia State University, Atlanta, GA, USA

    Brent Milam

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Mannone, M., Arias-Valero, J.S. (2022). Some Mathematical and Computational Relations Between Timbre and Color. In: Montiel, M., Agustín-Aquino, O.A., Gómez, F., Kastine, J., Lluis-Puebla, E., Milam, B. (eds) Mathematics and Computation in Music. MCM 2022. Lecture Notes in Computer Science(), vol 13267. Springer, Cham. https://doi.org/10.1007/978-3-031-07015-0_11

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