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Discrete Fourier Transform of Distributions

  • Emmanuel AmiotEmail author
Chapter
Part of the Computational Music Science book series (CMS)

Summary

This chapter gives the basic definitions and tools for the DFT of subsets of a cyclic group, which can model for instance pitch-class sets or periodic rhythms. I introduce the ambient space of distributions, where pc-sets (or periodic rhythms) are the elements whose values are only 0’s and 1’s, and several important operations, most notably convolution which leads to ‘multiplication d’accords’ (transpositional combination), algebraic combinations of chords/scales, tiling, intervallic functions and many musical concepts. Everything is defined and the chapter is hopefully self-contained, except perhaps Section 1.2.3 which uses some notions of linear algebra: eigenvalues of matrices and diagonalisation. Indeed it is hoped that the material in this chapter will be used for pedagogical purposes, as a motivation for studying complex numbers and exponentials, modular arithmetic, algebraic structures and so forth. The important Theorem 1.11 proves that DFT is the only transform that simplifies the convolution product into the ordinary, termwise product.

Keywords

Discrete Fourier Transform Inverse Fourier Transform Convolution Product Circulant Matrix Music Theory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Laboratoire de Mathématiques et PhysiqueUniversité de Perpignan Via DomitiaPerpignanFrance

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