Music Through Fourier Space Discrete Fourier Transform in Music Theo
This book explains the state of the art in the use of the discrete Fourier transform (DFT) of musical structures such as rhythms or scales. In particular the author explains the DFT of pitch-class distributions, homometry and the phase retrieval problem,
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Emmanuel Amiot
Music Through Fourier Space Discrete Fourier Transform in Music Theory
Computational Music Science Series editors Guerino Mazzola Moreno Andreatta
More information about this series at http://www.springer.com/series/8349
Emmanuel Amiot
Music Through Fourier Space Discrete Fourier Transform in Music Theory
123
Emmanuel Amiot Laboratoire de Mathématiques et Physique Université de Perpignan Via Domitia Perpignan France
ISSN 1868-0305 Computational Music Science ISBN 978-3-319-45580-8 DOI 10.1007/978-3-319-45581-5
ISSN 1868-0313
(electronic)
ISBN 978-3-319-45581-5
(eBook)
Library of Congress Control Number: 2016954630 © Springer International Publishing Switzerland 2016 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
Introduction
This book is not about harmonics, analysis or synthesis of sound. It deals with harmonic analysis but in the abstract realm of musical structures: scales, chords, rhythms, etc. It was but recently discovered that this kind of analysis can be performed on such abstract objects, and furthermore the results carry impressively meaningful significance in terms of already well-known musical concepts. Indeed in the last decade, the Discrete Fourier Transform (DFT for short) of musical structures has come to the fore in several domains and appears to be one of the most promising tools available to researchers in music theory. The DFT of a set (say a pitch-class set) is a list of complex numbers, called Fourier coefficients. They can be seen alternatively as pairs of real numbers, or vectors in a plane; each coefficient provides decisive information about some musical dimensions of the pitch-class set in question. For instance, the DFT of CEGB is (4, 0, 0, 0, 4e4iπ/3 , 0, 0, 0, 4e2iπ/3 , 0, 0, 0) where all the 0’s
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