"Music: Numinous and Mathematics. Form and Analysis Through Fourier Space" ed. by Eva Li

This volume brings new ideas to the conundrum by taking up certain philosophers not usually cited in connection with music and the classical Greek notion of process.
The book explains the state of the art in the use of the discrete Fourier transform (DFT) of musical structures such as rhythms or scales.

The book opens up an original approach to the transcendent and, to many, the sacred quality heard in music, drawing both upon authorities concerned with the numinous (that feeling of awe and attraction behind religious experience) and upon his own lifelong engagement with music as scholar, teacher and composer.

From Ancient Greek times, music has been seen as a mathematical art, and the relationship between mathematics and music has fascinated generations. This collection of wide ranging, comprehensive and fully-illustrated papers, authorized by leading scholars, presents the link between these two subjects in a lucid manner that is suitable for students of both subjects, as well as the general reader with an interest in music.
Physical, theoretical, physiological, acoustic, compositional and analytical relationships between mathematics and music are unfolded and explored with focus on tuning and temperament, the mathematics of sound, bell-ringing and modern compositional techniques.

Understanding the way music unfolds to the listener is a major key for unlocking the secrets of the composer’s art. Musical Form and Analysis, highly regarded and widely used for two decades, provides a balanced theoretical and philosophical approach that helps upper-level undergraduate music majors understand the structures and constructions of major musical forms. Spring and Hutcheson present all of the standard topics expected in such a text, but their approach offers a unique conceptual thrust that takes readers beyond mere analytical terminology and facts. Evocative rather than encyclopedic, the text is organized around three elements at work at all levels of music: time, pattern, and proportion. Well-chosen examples and direct, well-crafted assignments reinforce techniques. A 140-page anthology of music for in-depth analysis provides a wide range of carefully selected works.

The book 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.

Contents
Part I. Music and the Numinous
Introduction
1 The Transcendental and Rational Discourse
2 Music as Sublime Organism
3 Process Philosophy
4 Music and Process
5 A Whiteheadian Aesthetic and a Musical Paradigm
6 Music, the Other Arts and Process

Part II. From Pythagoras to Fractals
Music and mathematics: an overview
1 Music and mathematics through history
2 The mathematics of musical sound
3 Mathematical structure in music
4 The composer speaks

Part III. Form and Analysis: Time, Pattern, Proportion
PRELUDE: Why Analysis?
1 TIME: The Motivating Forces
2 PATTERN: The Shaping Factors
3 PROPORTION: The Distinguishing Features
POSTLUDE: METAFORM: Beyond Formal Analysis
ANTHOLOGY: MUSIC FOR ANALYSIS

Part IV. Discrete Fourier Transform in Music Theory
1 Discrete Fourier Transform of Distributions
2 Homometry and the Phase Retrieval Problem
3 Nil Fourier Coefficients and Tilings
4 Saliency
5 Continuous Spaces, Continuous FT
6 Phases of Fourier Coefficients
7 Conclusion
8 Annexes and Tables