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

By Emmanuel Amiot

This e-book explains the state-of-the-art within the use of the discrete Fourier rework (DFT) of musical constructions resembling rhythms or scales. specifically 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 that means of the levels of Fourier coefficients.

this can be the 1st textbook devoted to this topic, and with helping examples and workouts this is often compatible for researchers and complicated undergraduate and graduate scholars of track, machine technology and engineering. the writer has made on-line supplementary fabric to be had, and the e-book can be appropriate for practitioners who are looking to find out about suggestions for figuring out musical notions and who are looking to achieve musical insights into mathematical problems.

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Additional info for Music Through Fourier Space: Discrete Fourier Transform in Music Theory

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In this case, it is unique: 1 1 1 1 1 1 1 1 1 1 u = ( , , 0, , − , − , , , 0, , − , ). 19. The Fourier coefficients of u are unit-sized as expected: iπ iπ 5iπ 5iπ iπ iπ u = 1, e 6 , e− 3 , i, 1, e 6 , −1, e− 6 , 1, −i, e 3 , e− 6 . 3 In Z12 , {0, 1, 3, 7} = 5 × {0, 1, 4, 6} − 5. 5. If two pc-sets A, B are homometric then there exists a spectral unit u with rational coefficients such that 1A = u ∗ 1B . 6 (Rosenblatt). Two distributions f , g in QZn ≈ Qn are homometric iff there exists a spectral unit u with values in Q such that f = u ∗ g.

In this light one may remember that moving from a major to a minor triad was a ‘local’ transformation, depending on both triads. Actually it IS the affine group itself, permutating the interval vector without changing it since these are all-interval sets. 42 2 Homometry and the Phase Retrieval Problem Fig. 4. Paths between homometric 5-sets drawn by T. 1 = Constant For instance, the intervallic contents of Tristan’s chord {3, 5, 8, 11} and its complement are (4, 0, 1, 2, 1, 2, 0, 2, 1, 2, 1, 0) and (8, 4, 5, 6, 5, 6, 4, 6, 5, 6, 5, 4); the difference between those two vectors is constant and equal to 4, the cardinality difference.

This additional information compensates for the missing one. 29. The melodic minor scale B = {0, 2, 3, 5, 7, 9, 11} is one of Lewin’s special cases: FB (2)(= FB (10)) = 0. Assume IFunc(A, B) = (2, 2, 2, 1, 2, 3, 0, 3, 1, 2, 2, 1) where A remains to be found. e. the elements of IFunc(A, B), one gets 21, meaning that A has three elements to B’s seven. e. all except FA (2), FA (10), dividing the coefficients of the DFT of IFunc(A, B) by the conjugates of those of B. 366i, 1, [and conjugates]). The secret weapon at this juncture is Parseval-Plancherel’s formula: ∑ |FA (k)|2 = n #A in the case of a pc-set.

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