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Published in:
2013 | OriginalPaper | Chapter
The Minkowski Geometry of Numbers Applied to the Theory of Tone Systems
Author : Marek Žabka
Published in: Mathematics and Computation in Music
Publisher: Springer Berlin Heidelberg
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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.