Music Through Fourier Space

1. Front Matter

   Pages I-XV

   PDF

2. Discrete Fourier Transform of Distributions

   • Emmanuel Amiot

   Pages 1-26

3. Homometry and the Phase Retrieval Problem

   • Emmanuel Amiot

   Pages 27-49

4. Nil Fourier Coefficients and Tilings

   • Emmanuel Amiot

   Pages 51-89

5. Saliency

   • Emmanuel Amiot

   Pages 91-133

6. Continuous Spaces, Continuous FT

   • Emmanuel Amiot

   Pages 135-155

7. Phases of Fourier Coefficients

   • Emmanuel Amiot

   Pages 157-178

8. Conclusion

   • Emmanuel Amiot

   Pages 179-181

9. Annexes and Tables

   • Emmanuel Amiot

   Pages 183-197

10. Back Matter

    Pages 199-206

    PDF

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, nil Fourier coefficients and tilings, saliency, extrapolation to the continuous Fourier transform and continuous spaces, and the meaning of the phases of Fourier coefficients.

This is the first textbook dedicated to this subject, and with supporting examples and exercises this is suitable for researchers and advanced undergraduate and graduate students of music, computer science and engineering. The author has made online supplementary material available, and the book is also suitable for practitioners who want to learn about techniques for understanding musical notions and who want to gain musical insights into mathematical problems.

• Discrete Fourier Transform (DFT)
• Music Theory
• Cyclic Groups
• Tiling
• Homometry
• Saliency

“This book is a short exposition of important material in the field of mathematics and music. It describes a number of topics to which Amiot has contributed significant research. ... This book should prove to be a real delight for aficionados of mathematics and music. It contains a wealth of important material and is certain to drive further investigations in this fascinating interdisciplinary field.” (James Stephan Walker, Mathematical Reviews, June, 2018)

“Amiot’s nice book gives the state of the art in the usage of DFT of abstract musical structures such as rhythms, scales, chords, pitch class distributions, and so on. … This is possibly the first textbook on the topic. … The target readership includes graduates and advanced undergraduates of computational music science and engineering. Researchers in music should also find it of interest.” (Soubhik Chakraborty, Computing Reviews, May, 2017)

• Laboratoire de Mathématiques et Physique, Université de Perpignan Via Domitia, Perpignan, France

  Emmanuel Amiot

Emmanuel Amiot teaches mathematics at the Lycée François Arago in Perpignan, he is a researcher in the Laboratoire de Mathématiques et Physique (LAMPS) of Université de Perpignan Via Domitia, and he is a regular collaborator with researchers at the Institut de Recherche et Coordination Acoustique/Musique (IRCAM), Paris. He is a pioneer of the techniques described in this textbook, with considerable research and teaching experience in the related areas, geometry, topology, and applied mathematics.

Policies and ethics