Abstract
Euler’s speculum musicum is a finite selection of tones from the two dimensional tone lattice known as the Tonnetz. The idea of representing larger or smaller collections of tones as finite subsets of the Tonnetz reappears in the scholarly discourse in various contexts. However, formal rules for such selections that would satisfactorily reflect musical reality are not known: those proposed in the past are either too restrictive (not allowing all musically relevant tone systems to enter the model) or too loose (not preventing musically irrelevant tone systems from entering the model). The paper offers a formal framework that yields selections satisfactorily reflecting the musical reality. The framework draws methods from the Minkowski geometry of numbers. It is shown that only selection bodies of very specific shapes called (skewed) selection polygons lead to relevant selections. Manifold music-theoretical examples include chromatic, superchromatic, and subchromatic tone systems.
This paper was supported by EURIAS Junior Fellowship at NIAS awarded to the author.
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Žabka, M. (2013). The Minkowski Geometry of Numbers Applied to the Theory of Tone Systems. In: Yust, J., Wild, J., Burgoyne, J.A. (eds) Mathematics and Computation in Music. MCM 2013. Lecture Notes in Computer Science(), vol 7937. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39357-0_18
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