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2021 | OriginalPaper | Chapter

Modal Logic for Tonal Music

Author : Satoshi Tojo

Published in: Perception, Representations, Image, Sound, Music

Publisher: Springer International Publishing

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Abstract

It is generally accepted that the origin of music and language is one and the same. Thus far, many syntactic theories of music have been proposed, however, all these efforts seem mainly to concern the generative syntax. Although such syntax enables us to construct hierarchical tree, the mere tree representation is not sufficient in representing mutual references in music. In this research, we propose the annotation of tree with modal logic, by which the reference from each pitch event to harmonic regions are clarified. In addition, while the conventional generative syntax constructs the tree in the top-down way, the modal interpretation gives the incremental construction according to the progression of music. Therefore, we can naturally interpret our theory as the expectation–realization model that is more familiar to our human recognition of music.

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previous chapter End-to-End Classification of Ballroom Dancing Music Using Machine Learning

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Koelsch [15] distinguished the three levels of the reference to the outer worlds; (i) the simple imitation of sounds by instruments, (ii) the implication of human emotions, and (iii) the artificial connection to our social behavior.

We need to take care of the ambiguity of what semantics means. In Montagovian theory, the syntax of natural language is those written by categorial grammar and the semantics is written by logical formuale, while in mathematical logic the formal language (logic) has its own syntax and its semantics is given by set theory or by algebra.

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Note that notion of head resides also in Combinatory Categorial Grammar (CCG), as \(X \rightarrow X/Z ~Z\) implies that X/Z is the principle constituent of X.

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As for ‘\(\cap \)’ sets are finite; if we admit infinite sets, e.g.. in Euclidean space, \(\cap _{n \in \mathbb {N},\ge 1}[0,1/n)=[0]\) is not an open set.

\(\mathcal {N}(w)\) is a filter when for any \(U \in \mathcal {N}(w)\) there exists \(V (\subset U) \in \mathcal {N}(w)\) and for all \(w'\in V, U\in \mathcal {N}(w')\).

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\(x \le y\) and \(y\le z\) implies \(x\le z\).

It is shown that every context-free rule is transformed into this binary branching, known as Chomsky Normal Form (CNF).

\(\forall x[\phi \supset \psi ]\) versus \(\exists x[\phi \wedge \psi ]\). We obtain one from the other by negating the whole formula.

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Title: Modal Logic for Tonal Music
Author: Satoshi Tojo
Publisher: Springer International Publishing
Book: Perception, Representations, Image, Sound, Music
Print ISBN: 978-3-030-70209-0

Electronic ISBN: 978-3-030-70210-6

Copyright Year: 2021
DOI: https://doi.org/10.1007/978-3-030-70210-6_8

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