Abstract
This paper deals with the structure of a musical piece. The score is modeled as an Abstract Syntactic Tree (AST) to account for the hierarchy of its elements. Formal definitions of harmony, texture and instrumentation are proposed and constitute the main components of the model. Concatenation and parallelization operators are then proposed to combine these components and organize them in a tree structure. This approach is illustrated on some examples.
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Notes
- 1.
The standard way to do that is to associate each pitch with its corresponding MIDI number, i.e., C3 \(\rightarrow \) 60, C\(\sharp \)3 \(\rightarrow \) 61, D3 \(\rightarrow \) 62, C4 \(\rightarrow \) 72, etc.
- 2.
The question that naturally arises is how these chords are chosen, but this will be the subject of the next sections.
- 3.
We may model ties by the sum of two rational numbers and tuplets by using different numbers in the denominator.
- 4.
We may even allow for negative numbers for the case of anacrusis.
- 5.
While the term section usually involves instruments of the same family, in this case a section means the instruments that play the same pattern in a fragment.
- 6.
The dependence on I is omitted to simplify notations.
- 7.
We use the O for instrumentation since I is already used and it refers to the term orchestration, usually taken as synonym.
- 8.
The term tensor contraction comes from the similarity of this operation with tensor contraction, where the contraction operator is the union instead of the sum.
- 9.
\(A_i\) corresponds to the bar \(\lfloor \tfrac{i}{2} \rfloor + 1\) and to the first half if i is even and to the second half if i is odd.
- 10.
This is only a simplification, the model can handle several instances of an instrument like Vln. I and Vln. II.
- 11.
We need to use the disjoint union for the case where there are common members between \(N_1\) and \(N_2\).
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Romero-García, G., Agón, C., Bloch, I. (2024). A Model of Scores as Abstract Syntactic Trees. In: Noll, T., Montiel, M., Gómez, F., Hamido, O.C., Besada, J.L., Martins, J.O. (eds) Mathematics and Computation in Music. MCM 2024. Lecture Notes in Computer Science, vol 14639. Springer, Cham. https://doi.org/10.1007/978-3-031-60638-0_21
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