t T he essays in Music Theory and he editors are Jack Douthett, “These essays, by leading American music theorists, continue the Douthett Music Theory Mathematics: Chords, Collections, a mathematician and music development of some of the most important research of the last twenty Hyde and Transformations define the theorist who collaborated years into mathematical models of basic musical structures. These Smith state of mathematically oriented music extensively with Clough; Martha M. Hyde, and Mathematics theory at the beginning of the twenty- models are elegant in the abstract, but they are also shown to have a distinguished scholar of twentieth- first century. This book is dedicated to many practical applications in explaining a wide range of art music. century music; and Charles J. Smith, a John Clough (1928–2003), the long-time recognized expert in tonal theory. The Several of the contributions are bound to be classics in this literature.” holder of the Slee Chair of Music Theory contributors are all prominent scholars, M Chords, Collections, and Transformations at the State University of New York at teaching at institutions such as Harvard, —John Roeder, professor of music theory, u Buffalo, whose groundbreaking work Yale, Indiana University, and the University of British Columbia s has inspired all the contributors. The University at Buffalo (SUNY). Six of them i volume includes essays in diatonic set (Clampitt, Clough, Cohn, Douthett, Hook, c edited by theory, transformation theory, and neo- and Smith) have received the Society for “Music Theory and Mathematics is a fitting memorial to John Clough, T Riemannian theory—the newest and most one of music theory’s great pioneers. Clough was among the first Music Theory’s prestigious Publication h Jack Douthett | Martha M. Hyde | Charles J. Smith exciting fields in music theory today. Award, and one (Hyde) has received the scholars to introduce non-trivial mathematics into what has emerged The essays constitute a close-knit e ASCAP Deems Taylor Award. The collection as diatonic set theory or scale theory. This volume consists of essays body of work—a family in the sense o includes the last paper written by Clough by important theorists on a variety of topics ranging from scale and of tracing their descent from a few key r before his death, as well as the last paper Riemannian theory to analysis of works by Bartók and Schoenberg. y breakthroughs by John Clough, David written by David Lewin, an important Building on Clough’s research, Music Theory and Mathematics poses Lewin, and Richard Cohn in the 1980s and music theorist also recently deceased. a 1990s. More important, they are integrated new questions and approaches to what are perhaps the most exciting n by the ongoing dialogue they conduct with directions in music theory today.” d one another. For example, both Clough Contributors: David Clampitt, John and Julian Hook exploit mathematical Clough, Richard Cohn, Jack Douthett, —Robert Morris, professor of composition, M group theory, the former to investigate a Nora Engebretsen, Julian Hook, Martha Eastman School of Music (University of Rochester) a particular relationship among Riemannian M. Hyde, Timothy A. Johnson, Jonathan t transformations, the latter to reconsider Kochavi, David Lewin, Charles J. Smith, h important elements of tonal theory. Some and Stephen Soderberg. e parts of Jack Douthett’s modeling of scales m and chords with dynamical systems are related to Hook’s essay, others to Clough’s. a In addition, Hook’s essay has important t connections with Timothy Johnson’s i c pedagogical approach to diatonic set s theory. The geometric devices used by Johnson can be related to the dynamical University of Rochester Press approach taken by Douthett. And so forth, through the entire volume. An 668 Mt. Hope Avenue, Rochester, NY 14620-2731 introduction by the editors places the Jacket Image and Design: Geri McCormick P.O. Box 9, Woodbridge, Suffolk IP12 3DF, UK essays in historical context, and supplies www.urpress.com essential bibliographical background. Music Theory and Mathematics Eastman Studies in Music Ralph P. Locke, Senior Editor Eastman School of Music Additional Titles in Music Theory and in Music since 1900 Analyzing Wagner’s Operas: Alfred Lorenz Pentatonicism from the Eighteenth and German Nationalist Ideology Century to Debussy Stephen McClatchie Jeremy Day-O’Connell Aspects of Unity in J. S. Bach’s The Pleasure of Modernist Music: Listening, Partitas and Suites: An Analytical Study Meaning, Intention, Ideology David W. Beach Edited by Arved Ashby CageTalk: Dialogues with and about Representing Non-Western Music in John Cage Nineteenth-Century Britain Edited by Peter Dickinson Bennett Zon Concert Music, Rock, and Jazz since 1945: Ruth Crawford Seeger’s Worlds: Essays and Analytical Studies Innovation and Tradition in Edited by Elizabeth West Marvin Twentieth-Century American Music and Richard Hermann Edited by Ray Allen and Ellie M. Hisama Elliott Carter: Collected Essays and Lectures, 1937–1995 Schumann’s Piano Cycles and the Novels of Edited by Jonathan W. Bernard Jean Paul Erika Reiman Explaining Tonality: Schenkerian Theory and Beyond The Sea on Fire: Jean Barraqué Matthew Brown Paul Griffiths The Music of Luigi Dallapiccola The Substance of Things Heard: Writings Raymond Fearn about Music Paul Griffiths Opera and Ideology in Prague: Polemics and Practice at the National Theories of Fugue from the Age of Josquin to Theater, 1900–1938 the Age of Bach Brian S. Locke Paul Mark Walker A complete list of titles in the Eastman Studies in Music Series, in order of publication, may be found at the end of this book. Music Theory and Mathematics Chords, Collections, and Transformations EDITED BY JACK DOUTHETT, MARTHA M. HYDE, AND CHARLES J. SMITH UNIVERSITY OF ROCHESTER PRESS Copyright © 2008by the Editors and Contributors All rights reserved.Except as permitted under current legislation, no part of this work may be photocopied, stored in a retrieval system, published, performed in public, adapted, broadcast, transmitted, recorded, or reproduced in any form or by any means, without the prior permission of the copyright owner. First published 2008 University of Rochester Press 668Mt. Hope Avenue, Rochester, NY 14620, USA www.urpress.com and Boydell & Brewer Limited PO Box 9, Woodbridge, Suffolk IP123DF, UK www.boydellandbrewer.com ISBN-13: 978–1–58046–266–2 ISBN-10: 1–58046–266–9 ISSN: 1071–9989 Library of Congress Cataloging-in-Publication Data Music theory and mathematics : chords, collections, and transformations / edited by Jack Douthett, Martha M. Hyde, and Charles J. Smith. p. cm. — (Eastman studies in music, ISSN 1071–9989; v. 50) Memorial volume for John Clough. Includes bibliographical references (p. ) and index. ISBN-13: 978-1-58046-266-2(hardcover : alk. paper) ISBN-10: 1-58046-266-9(hardcover : alk. paper) 1. Music theory— Mathematics. I. Douthett, Jack M. (Jack Moser) II. Hyde, Martha M. III. Smith, Charles J., 1950– IV. Clough, John (John L.) MT6.M96204 2008 781.2—dc22 2007033160 A catalogue record for this title is available from the British Library. This publication is printed on acid-free paper. Printed in the United States of America. Contents Preface vii Charles J. Smith Introduction 1 Norman Carey, Jack Douthett, and Martha M. Hyde 1 “Cardinality Equals Variety for Chords” in Well-Formed Scales, with a Note on the Twin Primes Conjecture 9 David Clampitt 2 Flip-Flop Circles and Their Groups 23 John Clough 3 Pitch-Time Analogies and Transformations in Bartók’s Sonata for Two Pianos and Percussion 49 Richard Cohn 4 Filtered Point-Symmetry and Dynamical Voice-Leading 72 Jack Douthett 5 The “Over-Determined” Triad as a Source of Discord: Nascent Groups and the Emergent Chromatic Tonality in Nineteenth-Century German Harmonic Theory 107 Nora Engebretsen 6 Signature Transformations 137 Julian Hook 7 Some Pedagogical Implications of Diatonic and Neo-Riemannian Theory 161 Timothy A. Johnson 8 A Parsimony Metric for Diatonic Sequences 174 Jonathan Kochavi vi ❧ contents 9 Transformational Considerations in Schoenberg’s Opus 23, Number 3 197 David Lewin 10 Transformational Etudes: Basic Principles and Applications of Interval String Theory 222 Stephen Soderberg Works Cited 245 List of Contributors 253 Index 257 Preface In the fall of 2001, John Clough announced his intention to retire from his fac- ulty position at the University at Buffalo. As a celebration of John’s central role in the field of music theory, a collection of essays by scholars who had worked and studied with him seemed timely and appropriate. Martha Hyde and I (John’s music theory colleagues at UB) first discussed this idea at the Society for Music Theory annual meeting that fall in Philadelphia, and then began to put together a roster of possible contributors and to issue invitations. Sadly, John’s severe health problems appeared at the beginning of the next semester, in January 2002. What had been planned as an orderly transformation of the UB theory program became a mad scramble to cover John’s teaching, advising, and departmental duties. After his January surgery, he seemed well enough; nonetheless, a slow, inexorable process of diminishment had begun. The dissertation defenses of Nora Engebretsen and Jonathan Kochavi, both in the spring of 2002, were the last he participated in, and virtually his last public appearances in the department. By the summer, the decline had become impos- sible to ignore. John’s children had to manage the difficult task of looking after him in Buffalo, which they did with courage and determination. His UB col- leagues continued to visit, but every time we saw him, there seemed to be less and less of him. Finally, John was moved to be nearer his family, and died in the late summer of 2003—less than two years after making those retirement plans. What was originally conceived as a collection honoring one of the great lights in the field of music theory, on the occasion of his retirement, was first recon- ceived as a get-well offering, and now is appearing as a post-mortem tribute. In this context, it seemed more appropriate to Martha Hyde and me that we pres- ent a collection of papers in the areas of John’s specialization, rather than a more general offering from friends and colleagues. Since neither of us has much expertise in mathematical music theory, we decided to withdraw our own planned contributions, and invited Jack Douthett, a professional mathematician and one of John’s best friends, as well as his frequent co-author, to share the edi- torial duties. Jack has done most of the work getting John’s own piece, his last music theoretical paper, ready for publication. He did the same for David Lewin’s paper, since, by tragic coincidence this great scholar has also left us, shortly after submitting his piece for this collection. viii ❧ preface Martha compiled the master bibliography for the entire collection, negotiated with publishers, and co-wrote the introduction along with Jack and Norman Carey. I made most of the initial contacts with contributors, and formatted the manuscripts according to the publisher’s specifications. It has also fallen to me to preface the collection—perhaps because I had known John Clough since 1972, the longest of any of the co-editors. I am left with an almost impossible task—to try to communicate some sense of a remarkable person to those of you not privileged to have known him, and (much harder!) to do justice to a central figure in the lives of almost everyone who did. It’s relatively easy to honor John professionally—he transformed music theory, though not single-handedly (because none of us works alone). His contributions to the the- ory of diatonicism in all its manifestations have revolutionized the way we all think about the basic elements of music. Expressions like “maximally even” and “cardi- nality equals variety,” popularized by John, are now indispensable to theorists grap- pling with tonality and post-tonality. The three Buffalo Symposia on neo-Riemannian transformations, with their many distinguished invited partici- pants, were John’s, both in conception and execution. They reflected his commit- ment to and excitement about this emerging field. David Lewin and Richard Cohn may have been the actual parents of transformation theory, but John Clough was its midwife. One of the central themes of John’s long music-theoretical career, from its leading-tone beginnings (the first article in the first issue of Journal of Music Theory!) to its neo-Riemannian cadence, was his refusal to espouse any theoretical orthodoxies. All his life, he remained a skeptic about clubs of all sorts—especially those whose main purpose seemed to be that of excluding the uninitiated from their ranks. He never could stand the idea of doing something just because everyone else did it; his advice was always to think things out on their own terms. I well remember expressing dismay at finding myself in some theoretical quandary (a frequent occurrence!), at which John’s response would be “Perhaps you’re asking the wrong question,” or “Perhaps you haven’t thought through your assumptions carefully enough.” Also notable was his commitment to precision and rigor—not least in his life- long fascination with the interaction between music and mathematics. John turned to math not to avoidtalking about the details of music; rather, he saw how certain important things about music could be said precisely and effectively only with the help of mathematics. In the process, he created theoretical structures of great elegance, fascination, and beauty—objects that are deserving of aes- thetic appreciation and study in their own right. My primary duty here, however, is to honor John Clough personally, because it’s the person we miss more than the theorist. I had known John since my days in the graduate program at the University of Michigan, where he was teacher, supervisor, and mentor for so many of us. John became the Slee Chair of Music Theory at the State University of New York at Buffalo (as it was then called), and I was later priv- preface ❧ ix ileged to join him on that faculty, where I was a Clough colleague for over fifteen years. John pulled his professional and collegial weight; he was, in fact, one of the most respected senior faculty figures in Buffalo’s history. The academic landscape is littered with his theoretical progeny—Buffalo and Michigan students he advised, others he helped, and colleagues he encouraged and inspired. As a music theorist, John’s work was primarily motivated by his ready love of music—music of all flavors, from every place and group. He had a sophisticated but firm taste in jazz, and played the old established standards on the piano with resolute and solid enthusiasm. Another important backdrop for his life was recreational mathematics and puzzles of all kinds (not to mention his love for blackjack, a game in which he could adjust the odds by his considerable skill at card-counting)—passions that would help him to congeal many of the most important friendships of his life.1 Finally, though, John Clough’s real importance to those who knew him well was his all-encompassing integrity. I don’t think I’ve ever known anyone with as fine and deep a natural sense of fair play. If I had to pick a literary character to capture John’s essence, my choice would be Atticus Finch, in Harper Lee’s To Kill a Mockingbird— especially in Gregory Peck’s screen realization of that character.2John never played favorites, and never advocated his own interests above those of the department or whatever community he was working within. He was unfailingly generous with his time and his encouragement, but had no tolerance for empty rhetoric. If he had something critical or unpleasant to say to you, he simply said it to your face, without rancor. In short, who he was in public was exactly the same as who he was in private; there was no role-playing, no hypocrisy, no attempt to escape from the standards he had set for himself. John never sought to impress anyone; I don’t think he ever felt the need to do so. I can’t think of a single serious scholar in the world of music the- ory who has a bad word to say about him. His UB students knew this better than any of us, and loved him for it; to them, he was simply “Papa Clough.” I knew John as my colleague for many years, and as my teacher for many years before that. I’m sure that I’ll never know another like him: father figure, the most civilized of colleagues, dear friend. This, then, is the dedication of this book: to Papa Clough. Charles J. Smith Coordinator of Music Theory and Chair of the Music Department The University at Buffalo Notes 1. One of many world-puzzles I showed John over the years particularly delighted him as a diatonic set theorist. As a tribute to his sense of play and for those who care about such things, I have embedded that puzzle within this paragraph....