Melodic Morphing Algorithm in Formalism

概要

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Up to now, experiments run with many subjects are almost the only way for demonstrating the validity of a musical algorithm. In contrast, we employ formalism to verify that a melodic morphing algorithm correctly works as specified. The way we present in the paper is formal, theoretical, and viable. We provide a full-fledged feature structure for representing a time-span tree of ``A Generative Theory of Tonal Music'' (GTTM). The key idea here is that the GTTM reduction is identified with the subsumption or the ``is-a'' relation, which is the most fundamental relation in knowledge representation. Since we obtain the domain in which a partial order is defined, we introduce the join and meet operations based on unification. In such an algebraic framework, as preliminaries of the proof of a melodic morphing algorithm, we introduce the following notions: reduction path, melodic complexity, similarity, and interpolation. These notions are defined using the subsumption relation, join and meet, in a formal way. Thenwe take the melodic morphing algorithm proposed by Hamanaka et al. (2008) because it is constructed using the join and meet operations. Finally we prove the theorem that the melodies generated by the algorithm are the interpolations of two given melodies.
2011-06-18