Table Of ContentComputational Music Science
Series editor Eva Li
Music
Numinous and
Mathematics
Form and Analysis
Through Fourier Space
Edited by Eva Li
"Music: Numinous and Mathematics. Form and Analysis Through Fourier Space"
Edited by Eva Li
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Contents
Part I. Music and the Numinous
Introduction
1 The Transcendental and Rational Discourse Music
2 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 PATTERN:
2 The Shaping Factors PROPORTION: The
3 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 Saliency
4 Continuous Spaces, Continuous FT
5 Phases of Fourier Coefficients
6 Conclusion
7 Annexes and Tables
8
Music and the Numinous
by Richard Elfyn Jones
Contents
Preface 5
Introduction 11
Chapter 1: The Transcendental and Rational Discourse 21
Chapter 2: Music as Sublime Organism 43
Chapter 3: Process Philosophy 63
Chapter 4: Music and Process 73
Chapter 5: A Whiteheadian Aesthetic and a Musical Paradigm 85
Chapter 6: Music, the Other Arts and Process 95
Conclusion 107
Selected Bibliography 117
Index 119
Preface
The lines of thought that joined my interest in music and the arts
with my enquiries into metaphysics began to emerge some twelve
years ago. However, since childhood I have been aware that, as a
keyboard player, making music was for me essentially a form of
philosophising (carried out, I hope, spontaneously and instinctively
rather than in any over-ambitious manner). Early memories of
recorded performances by Schnabel, Solomon, Arrau and Brendel
served to instil a restless curiosity about what comprises the real
essence of the musical art. This book therefore aims to explore music,
and to a lesser degree the other arts, in a somewhat unusual way by
relating it directly to its ontological roots.
For many years I have been a proponent of Process philosophy.
Process philosophy is derived from and inspired by the writings, grand
in scope and bold in application, of Alfred North Whitehead (1861-
1947). In works like Religion in the Making, Adventures of Ideas,
Science and the Modern World and especially Process and Reality
he formulated a metaphysics applicable to all aspects of life. His
system is intrinsically interdisciplinary and interrelational. All
individual beings, from God to the most insignificant thing, are
explained through Whitehead’s set of metaphysical concepts. The
philosophy’s influence has been wide-ranging, especially in the USA.
From the middle of last century many American theologians embraced
Process as a tool for developing a new Christian natural theology
whereby ideas of God as an absolute predestinator, or as a “being-
itself”, was rejected in favour of a more world-affirming theism. The
concept of the numinous is obviously central to theology as well as to
other contemporary expressions of Christian faith. But this book has
little to say about theological perspectives. The numinous is defined
by me as pertaining to the numen or the supernatural. Although this
6 Preface
essentially Kantian idea of the numinous as a category for
understanding religion was developed forcefully and somewhat
mystically by the theologian Rudolf Otto, this book avoids
theological arguments. On the contrary it adheres to the basic tenets of
Whitehead’s metaphysics as expounded in Process and Reality. In
choosing not to examine any Biblical and confessional gloss on
Process that appeared after Whitehead’s time I accept the general
character of Whitehead’s teaching and apply that to a specific example
of reality which we value, namely music. Naturally, such is the
generality of Whiteheadian Process that other aspects of reality could
just as easily have been chosen. It is inevitable therefore that our
discussion should be as broad based as possible.
The reason Whitehead is not discussed until Chapter 3 and
afterwards is because the Introduction and the first Chapter provide a
historical preview recalling the pervasiveness of ideas of process in
Western thought since the time of the Greeks. In providing these two
introductory chapters we are also reminded that it was Whitehead
himself who proclaimed that famous judgement on Western
philosophy, namely, that it is all merely footnotes to Plato.
The Introduction presents an oblique approach to the subject
through links between art and mystical and esoteric theories. Joscelyn
Godwin’s Harmonies of Heaven and Earth is a rich compendium of
ancient beliefs about music’s magical powers and the connections
between its mathematical substructures and the architecture of the
universe. I am indebted to this source for many intriguing facts that
are discussed in this chapter. The Introduction provides a critical pot-
pourri of what Godwin terms “speculative music” preparatory to the
more important approaches which will be discussed later. In Chapter 1
“The Transcendental and Rational Discourse”, there is an inevitably
succinct review of Process’s indebtedness to Plato, Plotinus, St.
Augustine, St. Thomas Aquinas and some later writers before
focussing on Schopenhauer’s important hypotheses on music’s
capacity to lead to a transcendental reality.
It was Mendelssohn who, in a famous letter, pointed out that, far
from being an abstract art music is the most particular. Our
examination of the materials of Western music must therefore be
technical, but without analysing pieces of music in a conventional
manner. Chapter 2, “Music as Sublime Organism”, provides a detailed
examination of theories by the Austrian musicologist Victor
