The Philosophy of Rhythm
Aesthetics, Music, Poetics
Edited by PETER CHEYNE, ANDY HAMILTON, AND MAX PADDISON
Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide. Oxford is a registered trade mark of Oxford University Press in the UK and certain other countries.
Published in the United States of America by Oxford University Press 198 Madison Avenue, New York, NY 10016, United States of America.
© Oxford University Press 2019
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, by license, or under terms agreed with the appropriate reproduction rights organization. Inquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above.
You must not circulate this work in any other form and you must impose this same condition on any acquirer.
Library of Congress Cataloging-in-Publication Data
Names: Cheyne, Peter. | Hamilton, Andy, 1957– | Paddison, Max.
Title: The philosophy of rhythm : aesthetics, music, poetics / edited by Peter Cheyne, Andy Hamilton, and Max Paddison.
Description: New York, NY : Oxford University Press, [2019] | Includes bibliographical references.
Identifiers: LCCN 2019004326 | ISBN 9780199347773 (cloth) | ISBN 9780199347780 (pbk.) | ISBN 9780199347896 (oxford scholarship online)
Subjects: LCSH: Musical meter and rhythm. | Music—Philosophy and aesthetics. | Musical perception. | Music—Psychological aspects.
Classification: LCC ML3850 .P55 2019 | DDC 781.2/24117—dc23 LC record available at https://lccn.loc.gov/2019004326
1 3 5 7 9 8 6 4 2
Paperback printed by Marquis, Canada
Hardback printed by Bridgeport National Bindery, Inc., United States of America
“This remarkable collection of essays brings together philosophical and empirical approaches to the significance of rhythm across the arts. The approach is refreshingly interdisciplinary. Anyone concerned with the place of rhythm and metric structure in the arts, and—more generally—within the wider domain of human practices will find this an extraordinarily helpful volume.”
—Robert Kraut, Professor of Philosophy, The Ohio State University
“Fascinating and mysterious, rhythm is at the heart of music, dance, poetry, sociology, and neuroscience. This inspired volume engages, enlightens, and is the first to explore rhythm across a broad range of philosophical, aesthetic, and perceptual domains. This book is required reading for anyone concerned with time and rhythm in contemporary life.”
—Peter Nelson, University of Edinburgh
“This wonderful collection considers questions about rhythm from a wide variety of angles, perspectives, and disciplines—among them analytic and continental philosophy, musicology, art history, poetics, and neuroscience. Like the dialogue that opens the book, The Philosophy of Rhythm supports no particular line of thought or argument but enormously deepens our understanding of a topic so palpable and yet so mysterious.”
—Christoph Cox, Hampshire College
Preface
This project began in the mists of time, as a collaboration between Andy Hamilton and Will Montgomery. Will had to pull out and Max Paddison took his place—but Will remained as a contributor and his essay on rhythm in poetry is invaluable. Max has worked on musical time since his contributions to the 2004 special edition of Musicae Scientiae on spatialization and temporality in music, while Andy’s first publication on rhythm was for Proceedings of the Aristotelian Society in 2011. Max’s expertise in Continental philosophical traditions has been a necessary corrective to Andy’s more analytic background, and they organized a workshop in Durham in 2013, at which many contributions were presented. Besides his contributed essay, Peter Cheyne has been involved in an editorial role from an early stage. He reorganized the material, making it thematic rather than discipline-centered, and closely edited each chapter.
Acknowledgments are gratefully given to Laura Dearlove for diligently checking the style of several chapters; Anthony Parton for advice on artwork permissions; Suzanne Ryan, Jamie Kim, and Dorian Mueller at OUP for their work in helping to bring the volume to press; the anonymous reader for careful criticisms; Brian Marley for invaluable assistance in helping compile the index; and Durham University and the British Society of Aesthetics for their support for the workshop. Later-stage work was supported by JSPS Kakenhi grant number 19K00143. Finally, a sincere apology is due to the patient contributors. This volume has taken much longer in preparation than was originally anticipated.
Illustrations
8.1. Sulzer’s Schlagfolge 127
8.2. Koch’s Schlagreihe 128
8.3. Koch’s Schlagreihe with resting points (Ruhepuncten) 129
8.4. Model of the drift from rhythm to punctuation 129
8.5. Koch’s schema of a sonata-form exposition 133
8.6. Mozart, String Quartet in C major, K. 465, “Dissonance,” bars 23–6 135
8.7. Mozart, bars 67–97 136
8.8. Introduction, bars 1–4 137
9.1. Michael Jackson, “Billie Jean,” cabasa, drums, bass guitar, and synthesizer, timing c.00:20–00:24 144
9.2. The double backbeat 146
11.1 Kanizsa triangle, organized array (left panel); disorganized array (right panel) 172
11.2. Beat interpolation in the opening theme of Brahms’ Symphony No. 4 in E-minor, Op. 98 177
11.3. A “metrically malleable” melody notated in (a) 4/4, and (b) 9/8, showing alternate listening construals 178
11.4. The “Standard Pattern” (or “Bell Pattern”) timeline found in many styles and genres of African music. Upper system: construed as a three- (or six-) beat pattern. Bottom system: construed as a four-beat pattern 178
15.1. Keats, “Hymn to Pan,” first stanza (from Endymion, Book 1, lines 232–46), annotated. 236
15.2. The two sides of “projection” 242
15.3. One version, among many, of the nursery rhyme “Humpty Dumpty” as an illustration of the determinacy of tetrameter (and also as an example of the pattern long-long-short-short-long) 244
15.4. The first line said as strict iambic pentameter (five isochronous beats each of the form “weak-strong”) in obvious violation of the line’s complexity 245
15.5. The first line said as five more or less isochronous beats, but now allowing for complexities of “weak” and “strong” 246
15.6. The first line said as five “flexibly isochronous” beats whose flexibility or variability is determined (or “controlled”) by higher levels of complexity 246
15.7. Beat-to-beat projections in which each new event is focally aware simply of its immediate predecessor 249
15.8. A representation of projective complexity showing the resistance of a five-beat line to the determinacies of four, or rather two, beats (Attridge’s “doubling”). NB: this diagram is not meant to represent the complexity of most “pentameter” lines where a reduction to two is not an issue. (Indeed, in some “pentameter” nine- or ten-syllable lines there are four beats, but these situations are hardly “square.”)
18.1. Husserl’s structure of time-consciousness
19.1. Sonia Delaunay, Rythme coleur n° 1076 (1939). Centre national des arts plastiques. © Pracusa S.A./Cnap/Photograph: Yves Chenot
19.2. Dactyl illustration (Creative Commons)
19.3. Raphael, The Miraculous Draught of Fishes (1515–16), bodycolour over charcoal underdrawing, mounted on canvas, 320 x 390cm, Victoria and Albert Museum. © The Royal Collection, HM The Queen/Victoria and Albert Museum, London
19.4. Nicholas Dorigny, The Miraculous Draught of Fishes (1719), etching and engraving on paper, 51 x 65cm, Victoria and Albert Museum. © Victoria and Albert Museum, London
19.5. Joseph-Marie Vien, St. Denis Preaching in Gaul (1767), oil on canvas, 660 x 393cm, Église Saint-Roche, Paris. © The Art Archive
19.6. Ten percent of all saccades of forty viewers (twenty art experts and twenty non-experts) beholding Vien’s painting for two minutes each. Adapted version © Laboratory for Cognitive Research in Art History, University of Vienna
250
297
310
313
314
316
322
323
19.7. Visualization of frequent saccadic transitions between fixation clusters for Vien’s St. Denis Preaching in Gaul (average of forty viewers, twenty art experts and twenty non-experts, whilst viewing for two minutes each). Adapted version © Laboratory for Cognitive Research in Art History, University of Vienna 324
19.8. Visualization of frequent saccadic transitions between fixation clusters for Doyen’s The Miracle of St. Anthony’s Fire (average of forty viewers, twenty art experts and twenty non-experts, whilst viewing for two minutes each).
Adapted version © Laboratory for Cognitive Research in Art History, University of Vienna 325
21.1. Opening lines of John Milton’s Paradise Lost displayed as prose in David Charles Bell and Alexander Melville Bell’s Standard Elocutionist (London, 1878), 426 352
21.2. Graphic record of various rhythmical sounds, from Edward Wheeler Scripture, Elements of Experimental Phonetics (New York, 1902), 509 356
bpm beats-per-minute
Abbreviations
EDM electronic dance music
fMRI functional magnetic resonance imaging
HKB Haken–Kelso–Bunz [equation]
OED Oxford English Dictionary
Notes on Contributors
Aili Bresnahan is Associate Professor of Philosophy at the University of Dayton (USA). She specializes in aesthetics, particularly in applied philosophy of dance, improvisation, interpretation, and the philosophy of mind and motor cognition as it relates to the performing arts. She is also the founder and moderator of Dance Philosophers, an interdisciplinary research and networking Google group. More information can be found on her professional website: https://www.artistsmatter.com. Contact:abresnahan1@ udayton.edu.
Peter Cheyne is Associate Professor at Shimane University, and Visiting Fellow in Philosophy at Durham University. He is leading two international projects, one on the Aesthetics of Perfection and Imperfection, the other on the Seventeenth- to Nineteenth-Century Philosophy of the Life Sciences. Published in journals including Intellectual History Review and the Journal of Philosophy of Life, and editor and co-author of Coleridge and Contemplation (OUP, 2017), he recently completed a monograph on Coleridge’s Contemplative Philosophy (OUP, forthcoming 2020).
Martin Clayton is Professor in Ethnomusicology at Durham University. His publications include Time in Indian Music (OUP, 2000) and Experience and Meaning in Music Performance (OUP, 2013, co-edited with Laura Leante and Byron Dueck). He is currently pursuing research on entrainment in musical performance within Durham’s Music and Science Lab (https://musicscience.net).
Víctor Durà-Vilà is Lecturer at the University of Leeds. In aesthetics, he works on Humean aesthetics, aesthetic experience, ethics and aesthetics, aesthetic cognitivism, as well as on interdisciplinary projects in music and dance. Other research interests include applied ethics (parental obligations; autonomy and paternalism) and philosophy of physics. His work has been published in journals such as Analysis, Journal of Value Inquiry, Journal of Aesthetics and Art Criticism, and British Journal of Aesthetics
Jason Gaiger is Associate Professor of Contemporary Art History and Theory at the Ruskin School of Art and a Fellow of St. Edmund Hall at the University of Oxford. His principal research interests are in aesthetics and art theory from the mid-seventeenth century through to the present day; he also works on theories of depiction and visual meaning, and on twentieth-century and contemporary art practice and theory.
Ted Gracyk teaches philosophy at Minnesota State University Moorhead, and is co-editor of the Journal of Aesthetics and Art Criticism. He is the author of several philosophical books on music, including Rhythm and Noise: An Aesthetics of Rock Music (Duke University Press, 1996), Listening to Popular Music (University of Michigan Press, 2007), and On Music (Routledge, 2013). With Andrew Kania, he co-edited The
Routledge Companion to Philosophy and Music (Routledge, 2011). Most recently, he co-authored Jazz and the Philosophy of Art (Routledge, 2018).
Garry L. Hagberg is James H. Ottaway Professor of Philosophy and Aesthetics at Bard College. Author of numerous papers at the intersection of aesthetics and the philosophy of language, his books include Art as Language: Wittgenstein, Meaning, and Aesthetic Theory (Cornell University Press, 1995), and Describing Ourselves: Wittgenstein and Autobiographical Consciousness (OUP, 2008). He is editor of Art and Ethical Criticism (Wiley Blackwell, 2008); Fictional Characters, Real Problems (OUP, 2016); and Wittgenstein on Aesthetic Understanding (Palgrave Macmillan, 2017). Editor of the journal Philosophy and Literature, Hagberg is currently writing Living in Words: Literature, Autobiographical Language, and the Composition of Selfhood.
Jason David Hall is Associate Professor of English at the University of Exeter. He has written the books Seamus Heaney: Poet, Critic, Translator (Palgrave Macmillan, 2007), Seamus Heaney’s Rhythmic Contract (Palgrave Macmillan, 2009), and NineteenthCentury Verse and Technology: Machines of Meter (Palgrave Macmillan, 2017), and is contributing author and editor of Meter Matters: Verse Cultures of the Long Nineteenth Century (Ohio University Press, 2011) and Decadent Poetics: Literature and Form at the British Fin de Siecle (Palgrave Macmillan, 2013, with co-editor Alex Murray).
Andy Hamilton teaches Philosophy at Durham University, UK. He specializes in aesthetics, philosophy of mind, political philosophy, and history of nineteenth- and twentiethcentury philosophy, especially Wittgenstein. His books are Aesthetics and Music (Continuum, 2007), The Self in Question: Memory, the Body and Self-Consciousness (Palgrave Macmillan, 2013), and Routledge Philosophy Guidebook to Wittgenstein and On Certainty (Routledge, 2014). He also teaches aesthetics and history of jazz at Durham, and published Lee Konitz: Conversations on the Improviser’s Art (University of Michigan Press, 2007).
Christopher Hasty is Walter W. Naumburg Professor of Music at Harvard University where he teaches music theory. His research interests center on questions of time and rhythm understood from perspectives of process and event formation. Recent publications include essays in Thought and Play in Musical Rhythm: Multidimensional Perspectives on African, Asian, and EuroAmerican Musics (co-edited with Richard Wolf and Steven Blum, OUP, 2019) and an essay on “Time” in The Oxford Handbook of Western Music and Philosophy (ed. McAuley, Nielsen, and Levinson, OUP, 2020).
John Holliday has a PhD in philosophy from the University of Maryland and is currently Lecturer in the Philosophy Department at Stanford University, where he supports Stanford’s initiative in Philosophy and Literature. His research centers on issues of literary value and has appeared in the British Journal of Aesthetics and the Journal of Aesthetics and Art Criticism.
Salomé Jacob holds a PhD on philosophy from the University of Durham. Her research lies at the intersection between philosophy of perception, aesthetics, and phenomenology. She focuses on the nature of musical movement.
Jenny Judge is PhD candidate in philosophy at NYU. She also holds a PhD in musicology from the University of Cambridge, as well as degrees from University College Cork
and the Cork School of Music. Her research explores the resonances between musical experience and the philosophy of mind. Her doctoral dissertation defends and elaborates the thesis that music represents attitudes. Judge is also an active musician.
Justin London is Andrew W. Mellon Professor of Music, Cognitive Science, and the Humanities at Carleton College (USA). He received his PhD from the University of Pennsylvania where he worked with Leonard Meyer. His research interests include rhythm and timing in non-Western music, beat perception, sensorimotor synchronization and joint action, and musical aesthetics. He has served as President of the Society for Music Theory (2007–9) and President of the Society for Music Perception and Cognition (2016–18).
David Macarthur is Associate Professor at the University of Sydney. He has published articles on liberal naturalism, pragmatism, metaphysical quietism, skepticism, common sense, Wittgenstein’s philosophy of language, perception, and philosophy of art—especially concerning architecture, photography, and film. He has co-edited three collections of papers with Mario De Caro: Naturalism in Question (Harvard University Press, 2004); Naturalism and Normativity (Columbia University Press, 2010); and Philosophy in an Age of Science (Harvard University Press, 2012); and recently edited Hilary and Ruth-Anna Putnam, Pragmatism as a Way of Life (Harvard University Press, 2017).
Will Montgomery teaches contemporary poetry at Royal Holloway, University of London. He is the author of The Poetry of Susan Howe (Palgrave, 2010); co-edited (with Robert Hampson) Frank O’Hara Now (Liverpool University Press, 2010); and co-edited (with Stephen Benson) Writing the Field Recording (Edinburgh University Press, 2018); and has published numerous articles on contemporary and twentieth-century poetry. His monograph on short form in American poetry is forthcoming. He has a long-standing involvement in experimental music and field recording and has released several CDs.
Matthew Nudds is Professor of Philosophy and Chair of the Faculty of Social Sciences at the University of Warwick. His work is principally in the philosophy of perception and he has a particular interest in the non-visual senses and auditory perception.
Max Paddison is Emeritus Professor of Music Aesthetics at the University of Durham. He works in critical theory, philosophy, contemporary music, and popular music. His publications include Adorno’s Aesthetics of Music (CUP, 1993), Adorno, Modernism and Mass Culture (Kahn & Averill, 1996), and Contemporary Music: Theoretical and Philosophical Perspectives (co-edited with Irène Deliège, Ashgate, 2010). He has recently contributed essays to The Palgrave Handbook of Critical Theory (Palgrave Macmillan, 2017), The Routledge Companion to the Frankfurt School (Routledge, 2018), and The Oxford Handbook of Western Music and Philosophy (forthcoming 2019).
Deniz Peters is Professor for Artistic Research in Music at the University of Music and Performing Arts Graz, Austria, where he also directs the Doctoral School for Artistic Research. His research concerns philosophical questions, such as the concept of musical expression, listening modes, ensemble empathy, and the epistemic potential of artistic research through music. His explorative pianistic practice is part of his research method.
Peter Simons is Professor Emeritus of Philosophy at Trinity College Dublin. He is the author of the monograph Parts (OUP, 2000) and some 300 essays on pure and applied ontology, philosophy of language, logic and mathematics, the history of early analytic philosophy and of Central European philosophy (mainly Austrian and Polish) in the nineteenth and twentieth centuries. He is a member of the British, European, Irish, and Polish Academies.
Michael Spitzer is Professor of Music at the University of Liverpool and Editorial Chair of the Society for Music Analysis. He inaugurated the International Conferences on Music and Emotion (Durham, 2009), and co-organized the International Conference on the Analysis of Popular Music (Liverpool, 2013). His publications explore interactions between music theory, philosophy, and psychology, and include Metaphor and Musical Thought (Chicago University Press, 2004); Music as Philosophy (Indiana University Press, 2006); A History of Emotion in Western Music (OUP, forthcoming 2020); and The Musical Human (Bloomsbury, forthcoming 2021).
Roger Squires works in areas opened up by the mid-twentieth-century revolution in philosophy of mind brought about by Wittgenstein and Ryle. Publications include: “Depicting,” Philosophy, 44 (1969); “Memory Unchained,” Philosophical Review, 77.2 (1969); “On One’s Mind,” Philosophical Quarterly, 20 (1970); “Silent Soliloquy,” Understanding Wittgenstein, Royal Institute of Philosophy Lectures, 7 (1973); “The Problem of Dreams,” Philosophy, 48 (1973); “Mental Arithmetic,” Ratio, 1 (1994).
Alison Stone is Professor of European Philosophy at Lancaster University. She is the author of Petrified Intelligence: Nature in Hegel’s Philosophy (University of Notre Dame Press, 2004), Luce Irigaray and the Philosophy of Sexual Difference (CUP, 2006), An Introduction to Feminist Philosophy (Polity, 2007), Feminism, Psychoanalysis and Maternal Subjectivity (Routledge, 2011), and The Value of Popular Music (Palgrave Macmillan, 2016). She edited the Edinburgh Critical History of Nineteenth-Century Philosophy (Edinburgh University Press, 2011) and co-edited the Routledge Companion to Feminist Philosophy (Routledge, 2017).
Michael Tenzer is Professor of Music at the University of British Columbia. His books include Gamelan Gong Kebyar: The Art of Twentieth Century Balinese Music (Chicago University Press, 2000) and the edited volumes Analytical Studies in World Music (OUP, 2006) and Analytical and Cross Cultural Studies in World Music (OUP, 2011, with coeditor John Roeder). His compositions are available on New World and Cantaloupe Records. Recent articles include the cross-cultural study of world “Polyphony” in the Oxford Handbook of Critical Concepts in Music Theory (OUP, 2018).
Matthew Tugby is Associate Professor at Durham University. He has published on a range of topics in contemporary metaphysics and co-edited Metaphysics and Science (OUP, 2013).
Rebecca Wallbank is PhD candidate in the Philosophy Department at Uppsala University, specializing in aesthetics and the philosophy of art. In addition to research on rhythm and the philosophy of literature she also has strong interest in the philosophy of trust and its relation to aesthetic testimony. She is Editorial Assistant to the British Journal of Aesthetics
Udo Will is Professor of cognitive ethnomusicology at The Ohio State University. He has studied music, sociology, and neuroscience, holds a PhD in both musicology and neurobiology, and his research focuses on cognitive aspects of music performances in oral cultures. He leads projects on physiological entrainment to music, on cultural effects on cognitive processing of prosodic components in music and language in Asian and African tone language cultures, and on cross-cultural studies of rhythm perception, movement and the concept of time.
Rachael Wiseman is Lecturer in Philosophy at University of Liverpool. She works on Wittgenstein, early analytic philosophy, and philosophy of mind, action, and ethics, and wrote the Routledge Guidebook to Anscombe’s Intention (Routledge, 2016). Her articles have been published in the Journal of Philosophy, American Catholic Philosophical Quarterly, and Philosophical Topics
Introduction
Philosophy of Rhythm
Peter Cheyne, Andy Hamilton, and Max Paddison
This volume brings together philosophical and empirical approaches to offer critical perspectives on the philosophy of rhythm. The editors have not imposed theoretical or interpretational prescriptions, except that contributors should examine concrete manifestations of rhythm in the various arts and in human activity. Our aim is to locate fruitful questions and stimulate lively discussion of them. Contributors offer definitions and theories of rhythm in music and prosody that are often opposed, referring to meter, pulse, stress, and accent as constituent elements of rhythm, or at least as key concepts in understanding it; lines of dispute are examined from different perspectives throughout the book. As well as examining the case of music, essays explore possibilities or hypotheses of rhythm in non-musical and nonprosodic (non-poetic) arts.
As the essays are generally contemporary in scope, Section 1 outlines some key points in the history of rhythm in philosophy, not in the pretence of providing a comprehensive survey in such a short space, but to offer some historical precedents for the problems addressed. Section 2 discusses the extent of recent attention to rhythm and the puzzling neglect of the field, especially in philosophy. Section 3 gives an outline of the chapters, describing the conceptual space of the book.
1. Historical Considerations
Recent neglect notwithstanding, philosophical traditions have long acknowledged the importance of rhythm across the arts and in everyday life. However defined, it is readily agreed that rhythm is fundamental to those arts that directly involve duration and temporality: dance, music, drama, and recited poetry. These arts were closely associated in classical Greece. They all include rhythm, the animating, flowing factor it is the purpose of this book to explore, along with the associated phenomena of movement, measure, pattern, and repetition.
Before Parmenides and Plato, Heraclitus ascribed to rhythm a universal significance in holding that “everything flows [panta rhei].” This stream of thought
Peter Cheyne, Andy Hamilton, and Max Paddison, Dialogue on Rhythm In: The Philosophy of Rhythm. Edited by: Peter Cheyne, Andy Hamilton, and Max Paddison. Oxford University Press (2019). © Oxford University Press. DOI: 10.1093/oso/9780199347773.001.0001
continues in contemporary process philosophy, influencing thinkers, including some in this book, who employ the concept of rhythm as flow. Less cosmic, more socio-cultural, Plato’s consideration of rhythm, in contrast, focuses primarily on music and its effects on culture and mood. In the Republic, he has Socrates discuss the various rhythms and regulations of meter as modeling different virtues (courageous, self-controlled, active, graceful) and vices (lamenting, drunken, idle, graceless), reprising the theme in his later dialogue, the Laws. It is also in the Laws in a discussion of the ability to control and order one’s bodily movements and speech— that Plato has “the Athenian” give the definition by which: “Order within movement is called ‘rhythm.’ ”1
A core motivation for this collection of essays is to explore rhythm across the arts. Connections between the different arts are addressed in Aristotle’s Poetics via three related concepts: mimēsis, metaphora, and poiēsis. Aristotle saw mimesis as a dynamic, performative impulse to “mimic” actions, processes, emotions, and gestures through different media and art forms. Mimesis, he said, “is natural to us,” and in the opening pages of the Poetics he specifically refers to rhythm as a medium for mimesis:
the medium of imitation [mimēsis] is rhythm, language, and melody, but these may be employed either separately or in combination. For example, music for pipe and lyre . . . uses melody and rhythm only, while dance uses rhythm by itself and without melody (since dancers too imitate character, emotion and action by means of rhythm expressed in movement).2
Clearly, Aristotle describes our capacity for an embodied mimesis that enables us to move rhythmically in space and move together in time with others. His discussion links music, poetry, and dance and anticipates the theory of entrainment discussed in several contributions to this volume.
Another theme in this book is the contemporary debate between proponents of the dynamic thesis, who hold that music literally moves, and those on the other side, who conform to the thesis that movement in music is metaphorical. We return to this debate in Section 2, but should note here the three categories of rhythm distinguished by Aristides Quintilianus in his Peri musikês:
The term ‘rhythm’ is used in three ways. It is applied to bodies that do not move, as when we speak of a statue having ‘good rhythm’; to anything that moves, as when we speak of someone walking with ‘good rhythm’; and it has a specific application to sound . . . . [viz.] a systēma of durations [chronoi] put together in some kind of order.3
1 Plato, Republic, Bk 3, 397a–401a; Laws, Bk 7, 798d–802e; Laws, Bk 2, 665a.
2 Aristotle, Poetics, 3–4.
3 Aristides, On Music, Bk 1, Ch. 13.
For Aristides, then, rhythm applies to proportionality in static objects, physical movement, and music. Distinguishing rhythm in things that move from rhythm in music, it seems he stands on the movement-as-metaphor rather than the dynamic side of the debate.
Philosophical theories of rhythm in the modern era include Rousseau in the eighteenth century, with his entry on rhythm in Diderot and d’Alembert’s Encyclopedia (1751), and his later Dictionary of Music (1768).4 In the Encylopedia he states that:
Rhythm can be defined generally as the proportion that the parts of a measure, parts of a movement, or even parts of a whole have with each other: in music it is the difference in movement which results from speed or slowness, from respective length or brevity of the notes.5
In 1802 (though published posthumously, another half-century later), Schelling proposed that art first “breaks through into the world of representation” via the expression in music of “the primal rhythm of nature.”6 It is, he claimed, “through rhythm” that humans “impose variety or diversity onto everything,” thereby finding pleasure in “an entire unity within a particular multiplicity,” often transforming “an essentially meaningless succession into a meaningful one.”7 Schelling thus argued that with rhythm music transforms the atomic or disparate into the organic, its basic units forming larger groups which in turn cohere in a variegated whole. In its articulative capacity to transform experience, rhythm is, in Schelling’s view, the dominant of the three powers in music—rhythm, melody, and modulation. Because articulating or “informing . . . unity into multiplicity” is for him the essence of music, and since rhythm effects “this informing within music itself,” he concludes that “Rhythm is the music within music.” Nietzsche’s early lecture “Rhythmic Researches” (1870–2) distinguished what he saw as Greek mathematical rhythms from the fluid, living rhythms of the body, anticipating his influential Apollonian–Dionysian distinction.8 Influenced by Schopenhauer, who distinguished music as “entirely apart” from all other arts in reaching further than mimetic representation and into a “serious and deeper significance . . . referring to the innermost essence of the world and of our selves,”9 Nietzsche finds the primality of will in rhythm and dance. Thus this philosopher, for whom “Without music life would be an error,”10 affirms the Dionysian necessity of rhythm: one must dance to enter fully, bodily, into the life of the world. Thus too he declared: “I would only believe in a god who knew how to dance.”11
4 Rousseau, Dictionnaire
5 Rousseau, “Rhythme.”
6 Schelling, Philosophy of Art, 17.
7 Schelling, Philosophy of Art, 110–11.
8 Elaine P. Miller, “Harnessing Dionysus.”
9 Schopenhauer, Will and Presentation, 306.
10 Nietzsche, Twilight of the Idols, 10 (“Epigrams and Arrows” §33).
11 Nietzsche, Thus Spake Zarathustra, 29.
Drawing from such nineteenth-century sources, Bergson’s distinctly modernist writings on time, duration, and continuity had a remarkable influence on music and philosophy in France from the 1890s up to the 1930s. The phenomenologist Gaston Bachelard’s The Dialectic Of Duration (1936; revised 1950), for example, arose from a critique of Bergson’s concepts of duration and continuity. Against Bergson’s notion of continuity, and indeed against the tradition that since the ancient Greek philosophers has regarded musical rhythm and melody as “flow,” Bachelard argues that “music’s action is discontinuous; it is our emotional resonance that gives it continuity.”12 Bachelard regards continuity and duration in music as an elaborate metaphor “reconstructed in reverse” by the experiencing subject.
2. Recent Times: Attention and Neglect
Given its importance in ancient and modern philosophy, the neglect of rhythm as an area of inquiry in contemporary philosophical aesthetics is puzzling. This lack of interest is not only from aesthetics, however. Poetics is also marked by a neglect of rhythm; there is a corresponding lack of interest from prosody, the area of linguistics concerned with patterns of stress and intonation. In the case of musicology, the neglect has been relatively less evident but nevertheless noticeable, given that, in contrast to popular music, rock music, and jazz, the dominant focus in the theory and analysis of Western art music has tended to be on the parameter of pitch in relation to harmony, as opposed to rhythm as such.
Grosvenor Cooper and Leonard B. Meyer’s groundbreaking work on rhythmic structure commented on the “moribund state” of its topic.13 Subsequently, Christopher Hasty, in an ambitious work, analyzed the experience of music as an irreducibly temporal phenomenon, as opposed to the spatialized representation assumed by many theorists and by ordinary thinking.14 Philosophically influenced by William James, phenomenologists such as Edmund Husserl, and process thinkers such as Henri Bergson and A. N. Whitehead, Hasty argued that music should be regarded as a process of becoming rather than a record of what has become, rejecting the image of meter as an artifact of a system of representation—that is, of notation.
There has also been a neglect of the relationship of rhythm and larger-scale form and structure. Aspects of this relationship occur in the work of Heinrich Koch in the eighteenth century, as Michael Spitzer has observed.15 In non-Western music theory and practice, however, notably that of North Indian classical music, rhythm and its relation to extended improvisation has an ancient and long-standing,
12 Bachelard, The Dialectic of Duration, 124.
13 Cooper and Meyer, Rhythmic Structure of Music
14 Hasty, Meter as Rhythm
15 Spitzer, Metaphor and Musical Thought, 69, further discussed at 243–59.
fundamental significance, as has been emphasized by Martin Clayton’s work on the relation of meter, duration, and structure in this tradition.16
In English-language philosophy, John Dewey, Leonard B. Meyer, Roger Scruton, Andy Hamilton, and Andrew Kania are among the small number of philosophers to address rhythm to any great extent. This blind spot is particularly unfortunate because rhythm is a phenomenon that is immediately evident to everyone, and is a topic on which philosophical progress can be made without expert technical knowledge. The need for greater attention to rhythm provides a major motivation for this volume, which sets out to rectify the oversight. We have sought to do this not only through commissioning contributions from philosophical aesthetics, but also— albeit with a philosophical perspective—from other disciplines like neuroscience, psychology, musicology, ethnomusicology, poetics, literary studies, dance, and art history. In taking an interdisciplinary approach, this volume facilitates lines of inquiry that investigate whether rhythm (and related concepts including meter and duration) should be restricted to music, dance, and poetry, or, by contrast, should be extended to non-poetic literature and theatre, as well as painting and the visual arts, and also architecture.
The attempt to apply the concept of rhythm across the arts raises problematic philosophical issues, and the term “metaphor” is often employed rather loosely. It is, in any case, hard to define. Might there remain in all the arts something—perhaps even some dimension, such as movement, the immediately spatial, or the immediately temporal—that can only be discussed in metaphorical terms? This suggestion raises a number of interesting further questions in relation to rhythm and has led to much recent debate. Music is a time-based art and has duration, but can we say that music really moves, and if so, what does talk of “movement in musical space” mean? The debate arises among those who hold movement in music to be metaphorical— such writers include Roger Scruton, who draws from Victor Zuckerkandl. Scruton concludes that the sense of movement is, though vastly important, only metaphorical in terms of the physical space in which bodies move, that is, the sounds of music “are ordered in space only apparently, and not in fact.”17 Zuckerkandl’s position involves the further sophistication that while, as he concludes, music transcends physical and geometrical space, it does not transcend spatiality completely, for it testifies to a space that remains in the absence of physical objects and geometry.18
What can be said today of rhythm in arts besides music? In the case of poetry— and indeed literature in general—one can say that duration is involved, in that it takes time to read it, but, as with music, what might be meant by movement in poetic or literary space? The visual arts of painting, sculpture, and architecture, which literally occupy space, also endure through time, and take time to view and walk around or within—but are they generically different in terms of temporality from
16 Clayton, Time in Indian Music
17 Scruton, Aesthetics of Music, 14.
18 Zuckerkandl, Sound and Symbol, 292.
artworks that take time to unfold in their entirety, such as performed music and recited poetry? These questions are taken up in a number of essays in this volume.
Simply to say that we can discuss an “absent dimension” only metaphorically is also to underestimate the importance of metaphor in its relation to mimesis and poiesis (creative, artistic production), whether in our experience of the arts and of nature (which has traditionally been the domain of aesthetics), in our attempts to understand, explain, and interpret the arts (which is the domain of hermeneutics), or in the making of art (which was traditionally the domain of poetics). As Aristotle says: “A metaphor is the application of a [word] which properly applies to something else.”19 He refers to metaphor as a “transference” from one sphere to which it belongs to another where it is not normally encountered.
Thus this collection aims to provide both an overview of an often neglected but vital aspect of aesthetic experience, and an examination of formal affinities between historically interconnected fields of music, dance, poetry and literature, and also the visual and spatial arts, addressing key concepts such as embodiment, movement, entrainment, and performance. We have attempted to avoid an overemphasis on music, and have sought also to stress structural parallels between different art forms and their aesthetics. An essential aim has been intelligibility across disciplines. While the volume draws on a wide range of disciplines, contributors were encouraged to present their ideas non-technically as far as possible, and to engage in cross-disciplinary dialogue, in part through the insights of philosophical aesthetics.
3. Outline of Chapters
Enhancing its interdisciplinary ambition, this book is organized not territorially, into academic disciplines, but thematically, into aspects and questions concerning rhythm. With this arrangement, the editors not only encourage connections between the disciplines and a closer exchange of perspectives, but also see a conceptual map of the philosophy of rhythm taking shape. The five thematic parts that make up the volume arose naturally, as the project progressed, revealing a spread of concerns among current scholars regarding rhythm, suggesting also the shape of the conceptual space itself.
Part One, “Movement and Stasis,” addresses conceptual questions that include: Does rhythm necessarily involve movement, or is this a matter of metaphor only? Is rhythm as a literal phenomenon restricted to human activities and actions, or does it extend to natural and mechanical phenomena such as ocean waves and the sound of a train on a track? How is rhythm experienced through the senses—is it recognized or projected?
19 Aristotle, Poetics, 34.
The opening chapter, collated and edited by Andy Hamilton, is a dramatized dialogue in the long philosophical tradition of that form. The debate poses the dynamic conception—that rhythm involves movement—against the view that nothing relevant in the music moves literally, that is, spatially. Hamilton’s dynamic conception characterizes rhythm as “[a primitive] order within human bodily movement or movement-in-sound,” and opposes static accounts in terms of order-in-time and Scruton’s metaphorical conception. Most dialogue participants support a dynamic conception of some kind, but Macarthur denies that rhythm “moves in a literal but non-spatial sense.” Squires and Wiseman develop Hamilton’s account, arguing that the movement criterion should be expressed as a capacity and not a disposition.
Matthew Nudds’ “Rhythm and Movement” continues this theme, arguing that we can experience literal movement in rhythm. The argument depends on the claim that our experience of musical grouping involves experiencing sounds as produced by extra-musical events that include movement, and that musical grouping is central to our experience of rhythm in music, hence our experience of rhythm involves the experience of movement. The view defended rejects the suggestion that movement can only be heard in music in a metaphorical sense.
In “The Ontology of Rhythm,” Peter Simons defends, in contrast, a static conception of rhythm. Investigating its complex ontology, he sets out the types of entity on which rhythm is founded and their relationships with rhythm itself. No single characterization will work, Simons argues; rather, a series of types branches off from simple paradigms. Rhythm in music is characterized in its simplest form by a repetitive temporal pattern, which forms the basis for variations generating the whole range of musical rhythms. In music, but not rhythm in general, this range is limited (though not constituted) by anthropological constraints concerning pitch, tempo, volume, and complexity.
Jenny Judge’s chapter on “ ‘Feeling the Beat’ ” argues that the experience of musical meter is multimodal: it involves the binding to a common sensory individual of auditory and proprioceptive content. One hears the beat, and feels it, too. She further claims that a consideration of this multimodal content undermines the seeming necessity of the appeal to “metaphorical perception” as a way of accounting for the experience of movement in the case of musical meter.
Next, in “Dance Rhythm,” Aili Bresnahan proposes a theory of dance rhythm as distinct from rhythm in dance. Distinguishing natural from intentional rhythm, she defends this account by exploring musical and non-musical connections between rhythm and dance. She argues that dance rhythm can arise in conjunction with music; follow music; set the musical rhythm; or be completely independent of music, though natural or internal bodily rhythms can underpin both. Finally, she asserts the existence of dance that might be naturally rhythmic, but not in a way essential to dance qua dance.
Part Two, “Emotion and Expression,” considers the relation of rhythm to human feeling and covers topics including: the deep significance of rhythm deriving from its being “a universal scheme of existence”; the use of rhythm in empathetic
communication and composition; rhythm at the base of cognitive and linguistic meaning; the creativity involved in bodily responding to musical rhythms such as those found in popular music; and entrainment and the recognition of expression in music.
In “Theories of Rhythm,” Garry Hagberg poses the question: Why does rhythm speak to us so deeply? Patterns of percussive sound that move us are meaningful, yet we find it hard to say what associations or connotations create that meaning. What is required is something more elemental and universal than personal or idiosyncratic associations. Hagberg argues that John Dewey’s Art as Experience has important insights on this question. Focusing on examples from jazz improvisation, Hagberg suggests that both player and listener are very like Dewey’s broader conception of the live organism interacting within its environment.
Deniz Peters’ “Rhythm, Preceding its Abstraction” takes a non-reductive approach to the understanding of musical rhythm based on reflections on his musical practice, arguing that, preceding its abstraction, rhythm centrally resides in “doings” and “happenings” in our bodies and interactions between each other. Further, it resides in our somatic and cognitive awareness of these “doings” and “happenings” by way of experience and attention. The line of thought Peters develops stems from a number of related observations concerning how “lived rhythm,” unlike “represented rhythm,” comes into being via interpersonal- and self-attention.
In “Mozart’s ‘Dissonance’ and the Dialectic of Language and Thought in Classical Theories of Rhythm,” Michael Spitzer discusses Mozart and eighteenth-century theories of rhythm. Challenging the abstract conception of musical rhythm, Spitzer argues for its intrinsic expressiveness, supporting his claim with reference to a rich eighteenth-century tradition. He argues that eighteenth-century rhythmic theory was cognitive, in tune with the cognitive qualities of classical music. Another expressive aspect of classical rhythm was its linguistic character. It imitated the nature of primitive grammar as imagined by contemporary linguists. Spitzer concludes by showing how these ideas can enhance our understanding—and hearing—of a piece by Mozart.
Next, in “Rhythm and Popular Music,” Alison Stone explores how rhythm functions and affects us in popular music. She considers explicit rhythm as a constant layer of percussion that has no precise pitch. Relative to this layer, the rhythmic qualities the other layers of sound are heightened, emphasizing beats that fit in or pull against those emphasized by the percussion. Referring to Michael Jackson’s “Billie Jean,” Stone discusses how this pronounced rhythmic character of popular music appeals to our bodies to move in time with the emphases sounded in different layers.
In “Rhythms, Resemblance, and Musical Expressiveness,” Ted Gracyk argues for the plausibility of entrainment accounts of musical expression in holding that the ability to hear expressiveness in rhythmic sounds is logically prior to hearing some musical patterns as expressive gestures. Although Gracyk does not endorse arousalism as a general account of musical expression, understanding the role of
rhythm in expressiveness supports a combination of the “resemblance” and “contagion” accounts of musical expressiveness, blending what are often treated as mutually opposed accounts.
Part Three, “Entrainment and the Social Dimension,” expands on the concept of entrainment raised in Gracyk’s chapter and discusses rhythm in psychology, neuroscience, and biology. They aim less to make a particular scientific contribution, than to assess the nature and viability of scientific approaches to rhythm. Justin London begins Part Three with “Metric Entrainment and the Problem(s) of Perception,” discussing the limits and mechanisms of our perceptual faculties for auditory rhythm. His consideration reveals that the perceptual process is not a linear chain of information from the external world, but an active interplay between mind and world. Yet while considering our senses as cross-modal perceptual systems solves some problems of perception, it creates other, perhaps deeper ones, he argues. In music, our rhythmic percepts are often non-veridical, as we add accents, beats, and grouping structure to otherwise undifferentiated stimuli.
Martin Clayton’s “Entrainment and the Social Origin of Musical Rhythm” discusses the social nature and origin of musical rhythm. The argument draws on Halbwachs’ idea of rhythm as a social rather than a natural phenomenon, and Schütz’s critique of Halbwachs in his famous essay “Making Music Together.” Clayton argues that rhythm in fact emerges spontaneously both in individuals and (crucially) in interactions between them, and that it is therefore both natural, in the sense of physiological, and social in origin.
Michael Tenzer’s “How Many Kinds of Rhythm Are There?” offers an ethnomusicological perspective on the indefinite varieties of rhythm, examining the contrasts between musical and linguistic rhythm, anthropological categories, perception, and technology. Partitioning the universe of rhythm typologically, Tenzer views the potential of rhythm along various continua: via comparison with language; in the development of human culture; in the life of an individual’s experience, perception, and cognitive prowess; and in the non-human natural world.
Udo will considers the physiological, psychological, and social origins of rhythm in “Temporal Processing and the Experience of Rhythm: A Neuro-Psychological Approach.” Reviewing data from Australian Aboriginal music, he argues for dynamic neural models that challenge abstract conceptions of rhythm. Will holds that instrumental rhythms and vocal rhythms in speech and music derive from different ways of interacting with our environment and are controlled by different temporal mechanisms. Thus, he argues, instrumental music should be considered in parallel to vocal music, not as derived from it.
Part Four, “Time and Experience: Subjective and Objective Rhythm,” considers subjective and objective conceptions of time; phenomenological, process philosophy, and empiricist perspectives; rhythmic duration; and—reprising the main theme of Part One for this experiential point of view—whether movement is a necessary criterion for rhythm. In “Complexity and Passage: Experimenting with Poetic Rhythm,” Christopher Hasty begins by treating rhythm as the shaping of
events and their succession, rather than as a pre-existent order of isochronous division. He argues for rhythm as flow, as the fluid, active, and characterful creation of things or events, rather than of a homogeneous substance (“time”). He relates this concept to poetry by reading the opening of Keats’s “Hymn to Pan,” analyzing the continuing “life” of the vocal impulse along the lines and through the word-sounds taken as “mouth events”—a reading after the manner of M. H. Abrams (2012).
Peter Cheyne defends an unprioritized ontology regarding the subjectivity and objectivity of rhythm in “Encoded and Embodied Rhythm: An Unprioritized Ontology,” and thus argues against writers such as Christopher Hasty and Nicholas Cook, who prioritize the subjectivity of rhythm as flow. Cheyne argues that because rhythm is perceived through the senses as patterned temporality evoking emotional response, it has both objective and subjective qualities according to Lockean criteria. He further argues that the intricacy of actual rhythm neither excludes its description in objective form, nor its subsequent performance by other skilled performers who are present and listening attentively.
In “Time, Rhythm, Subjectivity: The Aesthetics of Duration,” Max Paddison argues that rhythm must be considered in relation to time and subjectivity, understood within a larger concept of “rhythmicized duration” as form. Drawing on Bachelard’s phenomenology of duration, he argues that aesthetic concepts of temporality, movement, and rhythm in music and the performing arts are subject to change, development, and displacement, and have functioned normatively and metaphorically in different historical periods. He concludes that our experience of rhythm as structured duration is both subjectively experimental and historically contingent.
Salomé Jacob examines the implications of Husserl’s model of temporal consciousness on the experience of musical rhythm in “Husserl’s Model of TimeConsciousness, and the Phenomenology of Rhythm.” Husserl’s framework, when applied to rhythm, suggests that listeners retain the just-past sounds and anticipate the sounds-to-come in the light of what has been heard. Besides, Husserl’s model helps to frame a rich embodied phenomenology of rhythm. One’s experience encompasses the perception of musical rhythm but also a bodily awareness of one’s own movements, where both aspects share the same temporal structure.
In “Pictorial Experience and the Perception of Rhythm,” Jason Gaiger considers whether a painting can have a rhythm. Rhythmic structure unfolds in time, but if rhythm is essentially durational, he asks, how can a static configuration of marks and lines be rhythmic? Gaiger argues that although viewing a picture takes place in time, and thus is successive, it cannot be temporally structured in a sufficiently determinate manner to sustain the attentional focus required for the communication of even simple rhythmic patterns. Graphic art is non-sequential and this has important consequences for picture perception.
Víctor Durà-Vilà then engages Gaiger’s essay in “Soundless Rhythm,” to develop a notion of rhythm that is independent of sound and can include all senses. Durà-Vilà argues against the theoretical proposal that music is required
to conceptualize rhythm. Moreover, he contends that rhythm in painting can be experienced in a non-metaphorical way. Finally, he examines some potential implications of his thesis for incipient art practices involving senses other than sight and hearing.
Part Five, “Reading Rhythm,” addresses the role of rhythm in reading and thus focuses on poetry and prose. The stressed–unstressed model of metrical analysis, and its variants, now seems obsolete as a means of describing the patterns of emphasis in poetry. However, although in the twentieth century verse loosened its relationship to meter, these essays show how rhythm remains an essential, though less easily described, feature of literary language.
Jason Hall presents a genealogy of metrical abstraction in nineteenth- and twentieth-century literature in “Rhythm, Meter, and the Poetics of Abstraction,” surveying approaches to metrical abstraction that have shaped the modern metrical imagination, taking abstract meter as their starting point. Hall examines the early twentieth-century return to theories that tried to avoid the complications introduced by emphasizing voiced particularities of rhythm; these theories resisted earlier syntheses of meter with music. For the New Critics, at least, the reader is in “a better position” to offer a rhythmically “meaningful” reading if he or she “recognizes the meter.”
In “The Not-So-Silent Reading . . . ,” Rebecca Wallbank asks: What does it mean to say that we appreciate rhythm in literature? By raising this question, she aims, first, to illuminate the modes of attention to rhythm in literature, and second, to call for a re-evaluation of certain common assumptions concerning literary aesthetic experience and appreciation. She analyzes the impact of different forms of attention within aesthetic experiences, and through this aims to expose and illuminate the overlooked roles of rhythmic auditory-imagining within our experiences of literary works.
Will Montgomery shows in “Leaving it Out: Rhythm and Short Form in the Modernist Poetic Tradition,” how in the modernist era rhythm was no longer a stable background pattern, but became part of the overall acoustic texture of the poem—with short-form poetry the most powerful vehicle for rhythmic innovation. Montgomery focuses on the Poundian line of influence, with particular emphasis on the writing of the American poet Robert Creeley. Montgomery argues that brevity and ellipsis are integral to a modernism best approached through the modernist dictum Dichten = condensare (to poetize is to condense).
Finally, in “Hearing it Right: Rhythm and Reading,” John Holliday addresses the neglect of sound and rhythm within prose literature, arguing that poetry is not more rhythmic than prose. He argues that works of prose have rhythm, to which the pauses, inflections, stresses, and pronunciation of its language all contribute. As such, prose literature, like poetry, should be considered musical. While poetry is lineated and prose is not, Holliday argues that this distinction does not result in poetry being more rhythmic. He concludes that rhythm in prose literature generally deserves attention for the different roles it plays.
Works
Cited
Abrams, M. H., The Fourth Dimension of a Poem, and Other Essays (New York, 2012).
Aristides Quintilianus, On Music (c.350?), tr. R. P. Winnington-Ingram, in Greek Musical Writings II: Harmonic and Acoustic Theory, ed. Andrew Barker (Cambridge, 1989).
Aristotle, Poetics (c.335 bc), tr. Malcolm Heath (London, 1996).
Bachelard, Gaston, The Dialectic of Duration, tr. Mary McAllester Jones (Manchester, 2000).
Clayton, Martin, Time in Indian Music: Rhythm, Metre and Form in North Indian Rag Performance (Oxford, 2000).
Cooper, Grosvenor and Leonard B. Meyer, The Rhythmic Structure of Music (Chicago, 1960). Hasty, Christopher, Meter as Rhythm (Oxford, 1997).
Nietzsche, Friedrich, Twilight of the Idols (1889), tr. Richard Polt (Indianapolis, 1997).
Nietzsche, Friedrich, Thus Spake Zarathustra: A Book for All and None (1883–92), tr. and ed. Adrian del Caro, co-ed. Robert B. Pippin (Cambridge, 2006).
Miller, Elaine P., “Harnessing Dionysus: Nietzsche on Rhythm, Time, and Restraint,” Journal of Nietzsche Studies, 17 (1999), 1–32.
Plato, The Republic (c.380 bc), tr. G. M. A. Grube, rev. C. D. C. Reeve, in Complete Works, ed. John M. Cooper (Indianapolis, 1997).
Plato, The Laws (c.360 bc), tr. Trevor J. Saunders, in Complete Works, ed. John M. Cooper (Indianapolis, 1997).
Rousseau, Jean-Jacques, “Rhythm” (1765), The Encyclopedia of Diderot & d’Alembert Collaborative Translation Project, tr. Valerie Porcello (Ann Arbor, 2005): http://hdl.handle.net/2027/spo. did2222.0000.491.
Rousseau, Jean-Jacques, Dictionnaire de la musique (Paris, 1768).
Schelling, F. J. W., Philosophy of Art (1802–3; published 1859), tr. and ed. Douglas W. Stott (Minneapolis, 1989).
Schopenhauer, The World as Will and Presentation, tr. and ed. Richard E. Aquila (London, 2008).
Scruton, Roger, The Aesthetics of Music (Oxford, 1997).
Spitzer, Michael, Metaphor and Musical Thought (Chicago, 2004).
Zuckerkandl, Victor, Sound and Symbol: Music and the External World, tr. Willard R. Trask (Princeton, 1956).
