Abstract
Mathematical Morphology provides powerful tools for image processing, analysis and understanding. In this paper, we apply these tools to analyze scores, that are image-like representations of Music. To do that, we consider chroma rolls, a representation of scores similar to piano rolls that use chromas instead of pitches. Endowing this representation with a lattice structure, one can define Mathematical Morphology operators, and setting a group structure to the Time-Frequency plane allows us to use the notion of structuring element. We show throughout some examples that this relates with the notion of pitch-class set and chord progressions, and we analyze two Chopin’s Nocturnes with this technique.
This research is supported by European Research Council ERC-ADG-883313 REACH, and Agence Nationale de la Recherche ANR-19-CE33-0010 MERCI.
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Notes
- 1.
- 2.
\(\psi :L_1 \rightarrow L_2\) is said to be increasing if \(\forall X, Y \in L_1\), \(X \le _1 Y \Rightarrow \psi (X) \le _2 \psi (Y)\).
- 3.
\(\forall X \in L\), \(\psi (X) \le X\).
- 4.
\(\psi ^2 = \psi \).
- 5.
\(\forall X \in L\), \(X \le \psi (X)\).
- 6.
The stabilizer of a subset \(A \subseteq \mathbb {Z}_{12}\) is defined by \(\text {Stab}(A) = \{ n \in \mathbb {Z}_{12} : T_{n}A = A\}\) and is a subgroup of \(\mathbb {Z}_{12}\).
- 7.
This number comes from the number of subsets of \(\mathbb {Z}_{12}\) that is equal to \(2^{12}\).
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Romero-García, G., Bloch, I., Agón, C. (2022). Mathematical Morphology Operators for Harmonic Analysis. 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_21
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