Download Music Through Fourier Space: Discrete Fourier Transform in by Emmanuel Amiot PDF

By Emmanuel Amiot

This e-book explains the state-of-the-art within the use of the discrete Fourier remodel (DFT) of musical constructions similar to rhythms or scales. particularly the writer explains the DFT of pitch-class distributions, homometry and the section retrieval challenge, nil Fourier coefficients and tilings, saliency, extrapolation to the continual Fourier rework and non-stop areas, and the which means of the stages of Fourier coefficients.

this is often the 1st textbook devoted to this topic, and with aiding examples and workouts this is often compatible for researchers and complex undergraduate and graduate scholars of song, computing device technological know-how and engineering. the writer has made on-line supplementary fabric to be had, and the booklet can be compatible for practitioners who are looking to find out about innovations for knowing musical notions and who are looking to achieve musical insights into mathematical problems.

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Extra resources for Music Through Fourier Space: Discrete Fourier Transform in Music Theory

Example text

In this chapter, we will focus on the magnitude of the Fourier transform | f | instead of d 2 ( f ). 1. For the sake of simplicity, we stick to Zn in the present section and refer the more curious readers to the bibliography. 1. A distribution u ∈ CZn ≈ Cn is a spectral unit iff its Fourier transform is unimodular: ∀t ∈ Zn |u(t)| = 1. We will denote the set of spectral units on Zn as Un (or Un (K) if we restrict ourselves to coefficients in a subfield K ⊂ C). 2. Two distributions f , g in CZn ≈ Cn are homometric iff there exists a spectral unit u such that f = u ∗ g.

0), neutral for the convolution product, is the vector f itself. 30. Let us define the matrix of a distribution s as the circulating matrix S whose first column is the set of values of s, and the scale matrix of a pc-set as the matrix B of the characteristic function b of B. e. the distribution of the opposite/inverse pc-set). Then C = A T ×B and hence whenever A is invertible, B = C × A T 15 −1 . Meaning that any operation in one structure is echoed by a similar operation in the other structure.

On the other hand, elucidating the relationships between the elements of an homometry class is extremely useful for composers: for instance, the aforementioned class is composed of one orbit under the affine group, two orbits under T/I ({0, 1, 4, 6} and {0, 1, 3, 7}) and four under T (adding {0, 2, 5, 6} and {0, 4, 6, 7}, see Fig. 13). More about the computations of these groups can be found in [41], hinting at some compositional applications by Tom Johnson. A rich example uses paths between the 108 homometric sets with size 5 in Z12 , computed by Franck Jedrzejewski and drawn by Johnson in Fig.

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