Microtonality and the Tuning Systems of Erv Wilson

This book explores the emerging area of microtonality through an exam- ination of the tuning theories of Erv Wilson. It is the first publication to offer a broad discussion of this influential theorist whose innovations have far-reaching ramifications for microtonal tuning systems. This study ad- dresses the breadth and complexity of Wilson’s work by focusing on his mi- crotonal keyboard designs as a means to investigate his tuning concepts and their practical applications. Narushima examines materials ranging from historical and experimental tunings to instrument design, as well as musical applications of mathematical theories and multidimensional geometry. The book provides an analysis of some of Wilson’s most significant theoretical ideas, including the Scale Tree, Moments of Symmetry, Constant Structures, and Combination-Product Sets. These theories offer ways to conceptualize musical scales as patterns with structural integrity and whose shapes can be altered to produce infinitely varying forms. The book shows how these structural properties can be used to map scales onto a microtonal keyboard by providing step-by-step guidelines and clearly illustrated examples. Most importantly, it brings together theoretical and practical methods of tuning to enable composers, performers, and instrument designers to explore pre- viously uncharted areas of microtonality, making a significant contribution to the fields of music theory, composition, and music technology.

Terumi Narushima is a senior lecturer in music at the University of Wo­ llongong, Australia. She is a composer, performer, and sound designer specialising in microtonal tuning systems. Her projects include acoustic and electronic music, works for film and theatre, and a research collaboration to develop microtonal flutes using 3D printing. Routledge Studies in Music Theory

Music and Twentieth-Century Tonality Harmonic Progression Based on Modality and the Interval Cycles Paolo Susanni, Elliott Antokoletz

Reconceiving Structure in Contemporary Music New Tools in Music Theory and Analysis Judy Lochhead

Microtonality and the Tuning Systems of Erv Wilson Terumi Narushima Microtonality and the Tuning Systems of Erv Wilson

Terumi Narushima First published 2018 by Routledge 2 Park Square, Milton Park, Abingdon, Oxon OX14 4RN and by Routledge 711 Third Avenue, New York, NY 10017 Routledge is an imprint of the Taylor & Francis Group, an informa business © 2018 Terumi Narushima The right of Terumi Narushima to be identified as author of this work has been asserted by her in accordance with sections 77 and 78 of the Copyright, Designs and Patents Act 1988. All rights reserved. No part of this book may be reprinted or reproduced or utilised in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers. Trademark notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging-in-Publication Data Names: Narushima, Terumi author. Title: Microtonality and the tuning systems of Erv Wilson/Terumi Narushima. Description: bingdon, Oxon; New York, NY: Routledge, 2018. | Series: Routledge studies in music theory | Includes bibliographical references and index. Identifiers: LCCN 2017024419 | ISBN 9781138857568 (hardback) | ISBN 9781315718583 (ebook) Subjects: LCSH: Musical intervals and scales. | Wilson, Erv, 1928–2016. | Microtones. Classification: LCC ML3809 .N16 2018 | DDC 781.2/69—dc23 LC record available at https://lccn.loc.gov/2017024419

ISBN: 978-1-138-85756-8 (hbk) ISBN: 978-1-315-71858-3 (ebk) Typeset in Times New Roman by codeMantra Bach musicological font developed by © Yo Tomita For Kraig Grady, and in memory of Erv Wilson who sadly passed away before this book went to print. This page intentionally left blank Contents

List of figures ix List of tables xi Foreword xii Acknowledgements xv List of abbreviations xvi

1 Microtonality and the enigma of Erv Wilson 1

2 Microtonal keyboards 13

3 Mapping linear scales on Wilson’s generalized keyboard 29

4 Moments of Symmetry and the Scale Tree 59

5 Mapping Constant Structures and navigating new pathways 109

6 Cross-sets, Diamonds, and Combination-Product Sets 140

7 Conclusion 187

Appendix 195 Glossary 203 Index 209 This page intentionally left blank Figures

3.1 Linear series of fifths; Scale Tree; Gral Keyboard Guide; mapping the root, octave, and generator on Wilson’s generalized keyboard 32 3.2 Mapping notes of a 7/12 scale on the 7/12 keyboard 40 3.3 Layout of notes for a 7/12 scale on the 4/7 keyboard (showing modes) 48 3.4 Layout of notes for a 7/12 scale on the 3/5 keyboard (showing pentatonic scales), 2/3 keyboard, 1/2 keyboard, and 1/1 keyboard 54 4.1 Linear series of fourths as a spiral, chart, and Horogram; the Tanabe Cycle; 5-note Secondary MOS derived from a 7-note Pythagorean MOS 63 4.2 Parallelogram from the Tanabe Cycle; keyboard layouts of pentatonic scales with interval structure L-s-L-s-s from the bottom block of the Parallelogram 74 4.3 Scales of the Parallelogram from the Tanabe Cycle mapped on a 2/5 keyboard; Left-Right pathways and ratios of consecutive Fibonacci numbers on the Scale Tree 77 4.4 Straight Line Patterns of the Scale Tree; Golden Horogram 2; Scale Tree segment showing the zigzag pattern for the Noble Mediant between 1/6 and 1/7 88 4.5 Interval patterns for MOS from the Scale Tree (Adult/baby Rabbit Sequence); Noble MOS of Golden Horogram 2; chart and Horogram showing MOS for generator 8/7 93 4.6 1/x routine and zigzag pattern for determining MOS for generator 8/7; 5/26 MOS scale for generator 8/7 mapped onto a 1/6 keyboard 96 5.1 Mapping Kraig Grady’s 12-tone Centaur scale on the 4/7 keyboard 111 5.2 17- and 19-tone extensions of the Centaur scale; extending the 2)5 1-3-7-9-11 Dekany to a 12-tone Constant Structure 125 5.3 Keyboard mappings of the Bohlen-Pierce scale, Harry Partch’s 43-tone scale, and Erv Wilson’s 36-tone D’alessandro tuning 133 x Figures 6.1 Various cross-sets and reciprocal cross-sets; 5-limit Lambdoma; 5-limit Tonality Diamond; 1-3-5 Triadic Diamond; 1-3-5-7 Tetradic Diamond 141 6.2. 2)4 1-3-5-7 Hexany and its four pairs of harmonic and subharmonic triads; the Hexany as a self-mirroring structure; symmetry of intervals within the Hexany 152 6.3 1-3-5-7 Hexany Diamond; reciprocal cross-set of the 1-3-5-7 Hexany; Stellate Hexany; Pascal’s Triangle showing Combination-Product Sets (CPS) 157 6.4 Lattices of the 1-3-5-7-9-11 Eikosany based on an irregular but symmetrical hexagon, centred pentagon, and pentagonal asterisk; tetrads and Hexanies of the Eikosany 163 6.5 Dekanies of the Eikosany; intersection of the 1-3-5-7-9-11 Diamond and Eikosany; 0)6 through 6)6 1-3-5-7-9-11 CPS; Hexany compared with Euler-Fokker genus 171 6.6 Keyboard layout, CPS subsets, and various lattices for the D’alessandro tuning 178 A.1 Harmonic and subharmonic series in staff notation 196 Tables

3.1 12-tone Pythagorean scale generated from a chain of fifths above and below 1/1 in the positive and negative directions, with 1/1 designated as C 31 3.2 12-tone Pythagorean linear scale 34 3.3 Notes in the 7/12 scale showing linear position, letter name and scale degree 42 4.1 A linear chain generated from the interval 8/7 60 4.2 Small and large intervals of the 12-tone Pythagorean scale 69 4.3 Linear chain of 8/7s extended to 26 places, with 10 in the positive direction and 15 in the negative direction 102 4.4 5/26 MOS scale showing cent values 103 5.1 Kraig Grady’s 12-tone Centaur scale with 1/1 designated as C 110 5.2 All intervals found between pitches of Kraig Grady’s 12-tone Centaur scale 113 5.3 List of all intervals in Kraig Grady’s 12-tone Centaur scale arranged according to size 115 5.4 Scale degrees of harmonics found in the Centaur scale 120 5.5 9-tone Bohlen-Pierce (Lambda) scale in JI 132 5.6 13-tone Bohlen-Pierce scale in JI 135 6.1 Farey series of order 5 (F5) with superparticular intervals between each ratio 144

1  2  6.2   1, 7, 93 ×   , 5, 11 partitioned cross-set, 3  3  showing tones of the 1-7-9 harmonic triad and 3-5-11 subharmonic triad in the 1-3-5-7-9-11 Eikosany 165 6.3 Pitches of the full D’alessandro tuning, including suggestions for note names and frequency calculations 182 A.1 1-3-5-7 Hexany Diamond in 19ET, 22ET and 31ET 200 Foreword

Ervin M. Wilson was arguably the most creative music theorist in the world of alternative scales and tunings. He was also one of the least accessible because he chose to work alone and to share his prodigious discoveries and inventions personally, rather than teaching courses, writing formal papers, or publishing textbooks on his remarkably fruitful musical theories. This long-awaited book remedies this situation, for Dr Narushima is both an ac- ademic musicologist and an accomplished composer who fully understands and utilizes Wilson’s theories in her writings and compositions. At her sug- gestion, I will recount how I met Ervin Wilson and had the good fortune to study and collaborate with him. I first met Erv, as he preferred to be called, in the early 1960s when I was a graduate student in Biology at the University of California, San Diego. At that time, UCSD was located on the beach at the Scripps Institution of Oceanography, and the atmosphere was so intense that graduate students were given keys to the university library stacks. One evening I got tired of reading about molecular biology and genetics, so I decided to peruse the holdings of the Journal of the Acoustical Society of America. Soon I discov- ered that a certain Tillman Schafer had constructed a 19-tone electrome- chanical musical instrument at Mills College in the late 1940s, and that he had worked for the U.S. Navy in Point Loma, a suburb of San Diego. Hop- ing that he still lived in the San Diego area, I called the facility and asked to speak to him. As it turned out, Schafer was still in San Diego and invited me to his home where I saw the instrument, but more importantly, was given the names of other people interested in microtonal music. One of these was Ivor Darreg, who then lived in Los Angeles; so I drove to LA and met Ivor, who in turn referred me to Ervin Wilson. Just prior to this meeting, I had attended a seminar at the Salk Institute by David Rothenberg, a composer and theorist from New York City, who is best known for his theories on the perception of musical tones in scalar contexts. Remarkably, David also knew Wilson and was visiting him the same time I was. So, through this network of associa- tions, I met Erv Wilson and became one of the fortunate few who have had the opportunity to learn firsthand of his work. Foreword xiii As a graduate student, I had access to a large mainframe computer, so in 1968 I asked Erv if there were any computations I could do for him. We decided that a table of all the equal temperaments from 5 to 120 tones per octave would be useful, especially if I also computed the errors in a set of small-number ratios for each system. I did this, and at Erv’s suggestion, sent a copy to Professor Fokker in The Netherlands. Fokker was the leading pro- ponent of 31-tone equal temperament in Europe, and the author of a number of articles as well as a composer in that system. Having finished this study, Erv and I also compiled a very large table of just intervals and distributed multiple copies to other workers. Other projects included equal divisions of 3/1 and other integers, as these can approximate divisions of the octave with improved representations of certain harmonically important intervals. In recent years, other composers such as Heinz Bohlen, John Pierce, Kees van Prooijen, Enrique Moreno, and others have also become interested in divisions of 3/1. At this point, I had to stop and finish my dissertation, which was on the genetics of the tryptophan pathway in Neurospora crassa, if I were ever going to graduate from UCSD. (One of my graduate advisors had started referring to me as “graduate student emeritus”.) After getting my doctorate, I moved around the country, and became involved in research and teaching in the fields of microbial genetics, industrial microbiology, biochemistry, and biotechnology, eventually returning to UCSD. Through all of these peregri- nations, I continued to correspond with Erv and pored over his sometimes puzzling letters, intriguing keyboard diagrams, and cryptic worksheets. An invitation from Larry Polansky to spend a summer writing at Mills College followed by a part-time research position again at U.C. Berkeley allowed me to complete my book, Divisions of the Tetrachord, a task that was originally suggested by Lou Harrison. Much of this book was directly concerned with Erv’s theories and musical discoveries. So, it gives me great pleasure to see that Erv’s theories are finally made available to musicians and musicologists who don’t have access to his origi- nal papers and diagrams as I and a few other lucky students have had. Central to Erv’s work and among his first discoveries are his keyboard de- signs. To play microtonal music with the same skill and expression as is done in 12-tone equal temperament, one needs instruments that are designed for alternative tunings. Dr Narushima has done a magnificent job of explaining Erv’s keyboard designs and how they may be applied to many different tun- ing systems. This is one of the most valuable aspects of this book, not only for understanding Wilson’s contributions, but as a guide for designers and builders of instruments to play music outside the standard 12-tone equal temperament. Wilson was also one of the most intuitive mathematicians one is likely ever to meet outside of academia, though he had little or no formal training in the subject. In addition to being able to visualize and geometrically plot musical scales as objects in higher dimensional space, he rediscovered or xiv Foreword reinvented a method equivalent to the approximation of irrational numbers by continued fractions (The Scale Tree), and characteristically applied it as an organized system for discovering new musical scales of the type he terms Moments of Symmetry (MOS). Dr Narushima’s explanation of this mathematical process is crystal clear without losing sight of its musical sig- nificance, particularly as it applies to keyboard design. Other key concepts of Wilson’s musical theories are Constant Structures, which are scales in which every occurrence of a given interval is always di- vided by the same number of smaller intervals. These may be considered as a generalization of MOS and have comparable structural stability. They exist in both equal temperaments and ratiometric tunings (extended Just Intonation) as do another class of scales, the Combination-Product Sets. CPS are found by multiplying a set of n harmonic generators, m at a time. The prototype is the Hexany, a six-note set generated from four integers rep- resenting harmonic functions such as 1, 3, 5, and 7, two at a time. The result- ing set of pitches is partitionable into four pairs of (generalized) triads and their inversions. Others are the Dekany (two out of five or three out of five), and the Eikosany, four out of six. These in turn may be divided into smaller CPS: Dekanies into Hexanies, and Eikosanies into both. These structures are especially fascinating because they are harmonic without being centric as any note or none can function as the tonic. They are also defined in equal temperaments, but are generated by addition rather than multiplication. Needless to say, there is much more in this comprehensive exposition of Wilson’s contributions to music theory. Dr Narushima, as well as Kraig Grady and Warren Burt, among others, have composed innovative and aes- thetically significant music based on materials invented and discovered by Ervin M. Wilson, thus proving that his work is not empty speculation and audibly imperceptible theoretical invention.

John H. Chalmers, PhD. Author of Divisions of the Tetrachord Founding Editor of Xenharmonikôn Rancho Santa Fe, CA, USA Acknowledgements

I gratefully acknowledge permission from the incomparable Erv Wilson to reproduce copyrighted material from his vast collection of papers. Thank you, Erv, for your enthusiastic support for the publication of this book. I hope it meets your approval. The bulk of the research presented in this volume is based on my PhD thesis which I completed at the University of Wollongong in 2013. I wish to thank my supervisors, Associate Professor Greg Schiemer who piqued my interest in microtonal music in the very beginning and has continued to sup- port my work, and Professor Catherine Cole for guiding me through the writ- ing process and for her wonderful mentorship. I would also like to express my deep gratitude and respect to my examiners, Professor Bill Alves and Dr John Chalmers, for their generous and insightful comments which gave me the impetus to pursue the writing of this book. Thanks also to Heather Jamieson and the anonymous reviewers who gave valuable feedback on my book proposal, and to my editors at Routledge, Emma Gallon and Annie Vaughan, for their attention to detail. I gratefully acknowledge the Faculty of Law, Humanities, and the Arts at the University of ­Wollongong for grant- ing me study leave to complete this manuscript. My sincere appreciation also goes to members of the tuning community, especially Stephen Taylor, Marcus Hobbs, Warren Burt, Michael Dixon, David Doty, David Finnamore, Neil Haverstick, Mykhaylo Khramov, and Dante Rosati, who have at various times responded to my queries or en- gaged in tuning-related discussions. And most importantly, my thanks to Kraig Grady for permission to in- clude his diagrams and lattices of his Centaur tuning and the Bohlen–Pierce scale (Chapter 5), his tireless effort in making Erv Wilson’s work accessible to the public through the Wilson Archives, for unselfishly sharing his knowl- edge and answering my questions on tuning with patience and thoughtful- ness, for assembling the many diagrams for this book, for constant feedback and encouragement, and for being an inspiring and supportive partner. Abbreviations

COMP complement CPS Combination-Product Set ET equal temperament GEN generator JI just intonation MOS Moments of Symmetry 1 Microtonality and the enigma of Erv Wilson

Microtonality is a rapidly growing field among contemporary musicians who are searching for new melodic and harmonic resources to expand their musical vocabulary. The label “microtonal” suggests music with very small intervals, but it can also refer to music that uses any intervals not found in the standard Western system of 12-tone equal temperament. The latter ­interpretation of microtonality encompasses not only experimental tuning systems but scales found in different musical cultures around the world, as well as historic intonation systems, from ancient Greek scales to temperaments that predate the gradual adoption of 12 equal divisions of the octave in the West. An even broader, more pluralistic approach is to view microtonality as a musical continuum that embraces “all intervals and tun- ing systems”, of which 12-tone equal temperament is “only one of the myriad of possibilities” (Schulter 2012). The chief instigator for a renewed interest in alternative tuning systems in the twentieth century was the iconoclastic composer, theorist and instrument maker Harry Partch (1901–1974). During his lifetime, Partch built a large ensemble of instruments tuned to a scale he developed with 43 tones per octave and these were used to perform his musical and the- atrical works. He also wrote Genesis of a Music, a seminal book which was to inspire subsequent generations of musicians to explore microtonal scales. While Partch was a proponent of just intonation – a tuning based on intervals found in the natural harmonic series – other musicians such as Ivor Darreg have experimented with scales that divide the octave into different numbers of equally spaced intervals other than 12. These include scales that divide the semitone into smaller intervals such as quartertones, sixth tones and eighth tones (e.g., Julián Carrillo, Alois Hába and Ivan ­Wyschnegradsky), as well as scales that closely approximate just intervals such as 19-tone equal temperament (e.g., Joseph Yasser, Joel Mandelbaum) and 31-tone equal temperament (e.g., Adriaan Fokker), to name but a few. Still other musicians have proposed scales that are non-octave-based, such as the Bohlen–Pierce scale (Bohlen 1978; Mathews, Roberts & Pierce 1984) or Wendy Carlos’ Alpha, Beta and Gamma scales (Carlos 1987, pp. 42–43). Such examples of microtonal tuning systems provide musicians with a vast 2 Microtonality and the enigma of Erv Wilson range of pitch resources with which to play. They offer a palette of vibrant colours that would otherwise not be available from the “black and white” notes of conventional tuning. Although a majority of contemporary musi- cians still work with 12-tone equal temperament as their standard tuning, there is a growing awareness of alternate scales as more musicians are se- duced by the endless possibilities offered by microtonal tuning systems. A parallel concern for microtonalists has been the development of new instruments that are capable of performing music in different scales. This has led to the invention of a variety of keyboards – including R.H.M. ­Bosanquet’s generalized keyboard, Adriaan Fokker’s 31-tone organ, George Secor’s generalized keyboard for the Motorola Scalatron and Erv Wilson’s MicroZone – as well as refretted guitars, tuned percussion, electronic and other original instruments. It has also offered new explorations with wind instruments and non-Western instruments. Significant advances have been made in the development of computer music programs that can play micro- tonal pitches, of which Manuel Op de Coul’s Scala tuning software (2016) deserves special mention, alongside Marcus Hobbs’ Wilsonic app (Satellite 2016) which is specifically designed for users to explore the tuning systems of Erv Wilson. These developments are discussed in greater detail in Chapter 2, with a particular focus on microtonal keyboard instruments. Indeed, a study of Wilson’s keyboard designs and their potential applications in the explo- ration of new territories in microtonal tuning forms the basis of this book. I first became interested in microtonal music in the late 1990s as a com- position student at the Sydney Conservatorium of Music. At the time I was writing music with quartertones and other pitch inflections in an attempt to draw on harmonic resources beyond 12 tones per octave, so a study of more systematic approaches to tuning seemed a logical step for me to take. Through a series of introductory lectures given by composer and teacher Greg Schiemer, I was exposed to the pioneering work of Harry Partch as well as other contemporary musicians working in non-standard tuning systems, such as Jacques Dudon, Mamoru Fujieda, Ellen Fullman, Kraig Grady, David Hykes, Larry Polansky, Terry Riley, Carter Scholz and William Sethares. For my final Masters project at the Conservatorium I explored the relationship between tuning and timbre by creating a sound installation for synthesized bells whose overtones related to the scales in which they were played. The tuning system I used was a set of scales called Tritriadics which were developed by contemporary American theorist John Chalmers. My work was influenced by the new tuning ideas to which I was exposed and this creative engagement made me want to explore microtonal composition further. Of the musical examples I heard, I was particularly attracted to the work of Kraig Grady (1952–), a composer, performer and instrument builder be- longing to the Californian group of experimental musicians who continue Partch’s legacy today. Grady’s music is directly informed by the ideas of Los Angeles-based tuning theorist Ervin Wilson (1928–2016), an influential but Microtonality and the enigma of Erv Wilson 3 enigmatic figure in microtonal music, who has been described as “one of the most prolific and innovative inventors of new musical materials extant” (Chalmers 1993, p. 3). Wilson, a skilled draftsperson, assisted Harry Partch with the second edition of Genesis of a Music by producing new illustrations and diagrams for his book (Partch 1974, p. 267, note 9). The breadth of Wilson’s contribution to tuning is conveyed in the follow- ing description:

Since the 1950s, Wilson has categorized and catalogued equal tem- peraments and developed generalized Bosanquet-type keyboards and notations for them. He has also discovered and characterized many different new classes of just intonation scales, and his discovery of the Combination-Product Set method of generating just tunings has been called “a giant step forward” for just intonation theory. Since that time, Wilson has explored aliquot scales, tunings based on Pascal’s triangle, and many different numerical series. His work continues today with un- abated imagination and originality. (McLaren 1997)

Wilson’s innovations in tuning have far-reaching ramifications, not only from a theoretical or analytical perspective, but by providing practical tools for the performance and creation of new music. His scales offer novel ap- proaches to melody and harmony that can be applied to a variety of musi- cal contexts and styles. Testament to this is the diverse range of musicians who have studied with Wilson or have been influenced by his ideas. These include – but are not restricted to – composers, performers and instrument makers working in experimental music, jazz, rock, free improvisation, film composition, electronic music and world music. People who studied with him at different times include Gary David, Kraig Grady, Scott Hackleman, Jose L. Garcia, Marcus Hobbs, Craig Huxley, Chuck Jonkey, Todd Manley, Rod Poole, Glen Prior, Michael Stearns, Stephen James Taylor and ­Daniel Wolf. Others who have been influenced by Wilson’s ideas in­ clude Lydia ­Ayers, Warren Burt, David Finnamore, Neil Haverstick, Dave Keenan, Pete McRae, Andrew Milne, Paul Rapoport, Greg Schiemer, Margo Schulter, Ron Sword and Robert Walker. In addition to Partch, Wilson also ­collaborated with various people, including tuning theorist and astrobiol- ogist John Chalmers, North Indian musician Amiya Dasgupta, instrument designer Larry Hanson, physicist Walter O’Connell, percussionist Emil Richards and musician George Secor. One of Wilson’s leading protégés is the aforementioned Kraig Grady who studied with Wilson for 30 years. According to Grady, Wilson was a remarkable teacher and mentor who tailored his material to benefit each individual according to their specific needs and creative interests, and as a result many of his students learned different things from him. Wilson’s the- ories were often based on a re-evaluation of existing systems of tuning, both 4 Microtonality and the enigma of Erv Wilson historical practices as well as examples from the various musical cultures of the world (Grady 2012). What impressed Grady about Wilson’s teaching was his ability to extrapolate the inherent structural properties of the systems that hitherto may have been overlooked, and to transform their patterns into general principles that could then be applied and extended to gener- ate new scales. This was a very powerful assertion. Whereas some theorists tried to prescribe all the rules within a specific closed system, Wilson’s work was especially appealing for its dynamism and open-endedness: his theories were “not just some mathematical games…. [They] were actually filled with immense beauty” (Grady quoted in Taylor 2011) and had the potential to inspire countless interpretations and creative applications. I met Wilson for the first time at MicroFest 2001, a conference and festival of music in alternate tunings held at the Claremont Colleges, California. MicroFest, which was organized by composer Bill Alves, was a formative event in my development as a musician, as it gave me the chance to meet with and hear the work of the chief exponents of contemporary microtonality. At the conference, Wilson gave a perplexing presentation on multidimensional tone lattices. To represent these structures, he had built a series of physical models from molecular model kits, the type normally used for chemistry lessons. Intriguingly, he proceeded to show each model, holding them up in his hands and rotating them one by one, but the presentation offered few verbal clues or explanations. I, like many others in the audience, was left bewildered. Despite my early enthusiasm, I quickly realized that a great deal of research would be required on my part if I wished to gain a more sophisticated understanding of Wilson’s work. Following the conference, a face-to-face meeting was arranged with ­Wilson and Grady, as well as Stephen James Taylor, a film and television composer who had also studied with Erv Wilson. Upon discovering my Jap- anese cultural heritage, Wilson recounted his experience of hearing sam- ples of Japanese court music for the first time through a meeting with the Japanese musicologist, Dr Hisao Tanabe. This took place soon after World War II when Wilson was a young man in the US Air Force in Japan. He particularly remembered hearing the interval 8/7 played on the flute, and he was deeply affected by the experience. Although I was baffled by Wilson’s earlier presentation at the conference, I was struck by his personal charisma during this initial encounter and felt compelled to pursue his ideas further. Exactly what I would discover I was not sure, but I was certain that I would learn something new and that this would somehow influence the direction of my own work. Shortly after I returned to Australia, I received from Wilson a single sheet of paper with a diagram labelled “The Tanabe Cycle” on one side, and a chart of intervals titled “Parallelogram from the Tanabe Cycle” on the re- verse side. At first the information I was able to decipher seemed trivial but the diagram also seemed to suggest a puzzle whose deeper significance was hidden beneath the surface. This sheet of paper became an invitation for me Microtonality and the enigma of Erv Wilson 5 to delve further. Since that time I have actively sought to improve my under- standing of Wilson’s work by studying his charts and diagrams, reading any written material I could find about his theories and communicating with other musicians who had engaged with his ideas. However, it has taken me several years to appreciate the full implications of Wilson’s paper, and my findings are discussed later in Chapter 4.

A brief biography of Erv Wilson Unfortunately, not a great deal has been written about Wilson’s life, but the following short biography aims to give an overview of what information is available. Ervin McDonald Wilson was born on 11 June 1928 in Colonia Pacheco, a remote mountain village northwest of Chihuahua, Mexico. He was the sec- ond son of a large Mormon family who had migrated from Utah to Mex­ ico by wagon in 1926. Colonia Pacheco was a small community ­originally ­established by Mormon exiles fleeing from the United States in the l880s. His childhood was spent helping with the family ranch, tending goats and other animals as well as cultivating crops, while attending grade school. Life during this period was probably arduous as it was the time of the Great Depression. Wilson cites as his early musical influences Mormon hymns and popular Mexican songs. He also learned to read music and play the reed organ from his mother. Reed organs were “common on the frontier because they were light, easy to transport and tended to remain in tune” (Wilson, G 2009, p. 3). Apparently Wilson was eager to compose music from a young age, but was confounded when some of the notes he could hear clearly in his head, such as the Mexican vocal tunes, could not be reproduced on the organ. He claims that this germinal experience launched his lifelong inquiry into mu- sical scales (Wilson 1989). At age 15, Wilson’s family moved to Oregon in the USA, where he attended high school for a year before being sent to live with his aunt in Salt Lake City, Utah. As a teenager he developed an interest in Indian music, and started studying the subject on his own by borrowing and reading books from the library. Eventually Wilson joined the US Air Force and served with the US occupation force in Japan at ages 19 and 20. In Japan, “a chance meeting with a total stranger introduced him to musical harmonics, which changed the course of his life and work” (Wilson 2001). Legend also has it that test results from around this time revealed his extraordinary powers of visualiza- tion which were applied to the analysis of aerial photographs. These abilities were later put to good musical use for visualizing complex tuning structures. After serving in the US Air Force, Wilson briefly studied music theory at Brigham Young University, but being unable to find an explanation for “his concept of ‘missing’ keys in the musical scale” (Wilson, G 2009, p. 4), he was lured away to California which by comparison was a hive of musical activity 6 Microtonality and the enigma of Erv Wilson at the time. There he continued his musical education through self-study and interaction with other microtonalists. He filled countless notebooks with charts, diagrams and written explanations as a record of his ideas and activities over several decades. These reveal his meticulously systematic and thorough approach to problem solving, and also an incredibly fertile mind that could perform breathtaking leaps of imagination. He also kept copies of letters he sent to various people with whom he corresponded about music and tuning theory. One of his chief correspondents was John Chalmers, his long-time friend and colleague, as well as founder and editor of the journal Xenharmonikôn (1974–1979, 1991–1998), and author of Divisions of the Tetra- chord (1993). In the 1960s, Wilson and his father formed a small culinary and medici- nal herb business that distributed herbs from the Sierra Madre of Northern Mexico to various outlets in the US (Burns 1986, p. 2). Through this work he “became intensely interested in the cultivation and propagation of various medicinal herbs” (Wilson, G 2009, p. 4), and this led to his ongoing preoccu- pation with collecting and breeding plants, an endeavour that has persisted alongside his musical pursuits. Especially significant was his work with high-lysine corn which is a hybrid developed to improve the protein in corn over ordinary varieties. These efforts were motivated by his desire to help people living in countries like Mexico where corn is a staple of their diet. Influenced by his work in breeding plants, Wilson “began to think of the musical scale as a living process” (Wilson 2001). He saw each kind of scale as a plant species that had the potential for growth and transformation, but was also subject to the forces of evolution where some species might survive while others may not. As Grady explains, “Erv is the great seed scatterer… he’s scattered all these different seeds and he’s not really sure which one of these plants will take hold and continue to grow. I think that’s the nature of his work” (Grady quoted in Taylor 2011). For most of his adult life Los Angeles was Wilson’s home, thus placing him firmly with the American West Coast movement of microtonality that in- cludes Partch, Lou Harrison (1917–2003), La Monte Young (1935–) and Terry Riley (1935–), many of whom he knew personally. He became a magnet for musicians from a diverse range of backgrounds and styles who came to visit him, or study and be mentored by him for varying periods of time. Wilson says that the goal of his research into tuning was to make scales “musically accessible to the composer and the listener…. I sculpt in the architecture of the scale. Other people come along and animate it” (Wilson 2001). Between the years 1974 and 1989, Wilson published a dozen or so articles in Xenharmonikôn, an informal journal established by fellow theorist John Chalmers for the purpose of communicating ideas relating to tuning “among active workers in the field of experimental music” (Chalmers 1974, p. 2). These articles covered many areas of Wilson’s early research, including tetrachordal modulations, Combination-Product Sets (CPS), notation sys- tems, microtonal keyboards and other instrument designs. Today, most of Microtonality and the enigma of Erv Wilson 7 Wilson’s documents on tuning theory are readily available online through the Wilson Archives,1 a vast collection of articles, letters, charts and metic- ulously hand-drawn diagrams. The Archives are managed by Kraig Grady who has remained one of Wilson’s most dedicated advocates. Grady is re- sponsible for digitizing, collating and publishing Wilson’s files through his own anaphoria.com website as an on-going research commitment. Initially Wilson was reluctant to have his work disseminated through such a channel because he preferred the oral tradition of teaching: he would give his papers to individuals as an illustration to accompany a verbal discussion. Accord- ing to Grady, problems began when Wilson found that copies of his work were being distributed freely without his consent, and he was concerned that his ideas were being misinterpreted and misappropriated by others with- out proper acknowledgement. After much persuading, Wilson eventually agreed to have his papers published online and as a result, people such as myself, who were not able to have individual lessons with Wilson in Los Angeles, can now benefit from his work. In 2010, due to unforeseen changes in his living circumstances, it became necessary for Wilson’s enormous collection of papers and personal posses- sions to be consolidated. Several of his supporters, including Grady, Taylor and myself, were enlisted to assist. My task in the archival process was to make digital scans of nearly 300 of Wilson’s documents on tuning, includ- ing papers that had not yet been published. This undertaking gave me a unique appreciation of the immense breadth and depth of Wilson’s oeuvre, the rigour of his endeavours and the intricate beauty of his many charts and diagrams which number in the thousands, if not tens of thousands. With such an abundance of Wilson’s material now available to the public, one might expect his ideas to have reached a wider musical audience than they have so far: but why has this not been the case? First, the often discour- aging obscurity of his work has meant that many musicians continue to find his material difficult, and choose to ignore rather than engage with his ideas. As Brian McLaren says, “Reading a Wilson article is like being hauled up from the bottom of the Cayman Trench in a bathyscaphe – rapidly. The effort to decompress Wilson’s gnostic piths and gists provokes acute ver- tigo” (1998, p. 80). Even as a committed supporter of Wilson’s ideas, I too found that his theoretical papers were often highly esoteric and mostly written for a small circle of microtonalists who were already familiar with his musical language. As a result, much of his work remains impenetrable to a reader who has not been initiated into his ideas. In his PhD thesis, Algorithms, Microtonality, Performance: Eleven Musical Compositions, the composer Warren Burt playfully describes Wilson as a “non-academic re- searcher”, whose “work does not indulge in either the clear explications or aesthetic justifications of academic writing”. Burt also notes, however, that “the bulk of microtonal research in the 20th century was, indeed, carried out by non-academic researchers”, and only “in recent years has an inter- est in microtonality become academically respectable…” (2007, pp. 37–38). 8 Microtonality and the enigma of Erv Wilson Second, any difficulty in understanding Wilson’s work is also augmented by the breadth of his research which ranges from historical and world music scales, experimental tunings, instrument designs, as well as musical applica- tions of mathematical theories and multidimensional geometry. The scope of his output resists easy categorization. Third, the interconnection of all his ideas into what Grady describes as “a unified field theory for microtonal- ity” (quoted in Taylor 2011) makes it difficult to find a simple access point through which to enter Wilson’s musical cosmos. As I began to comprehend the extent of Wilson’s prolific output, it became apparent that some sort of intermediary guide was needed to help any inter- ested musicians navigate their way through Wilson’s daunting, mysterious and thrilling musical universe. I have therefore decided to focus this book on one aspect of Wilson’s work, namely his microtonal keyboard designs, as a means to explore some of his tuning theories and their practical applica- tions. What can be gained from studying his keyboard designs? How do they relate to his tuning theories? What is their significance in the context of his overall research? These questions bring forward an age-old quest that has preoccupied musicians for centuries: the search for an instrument capable of playing music in different tunings. A major concern for microtonal musicians is the difficulty in finding a single interface that can accommodate more than one tuning system. ­Wilson’s solution to this problem was his generalized microtonal keyboard, an ­instrument that underpins many aspects of his work in tuning and whose development has been an ongoing project through much of his life. Wi­ lson’s microtonal keyboard consists of an array of hexagonal keys that are ­arranged in a skewed honeycomb pattern. It allows musicians to perform in a number of different tuning systems on the one keyboard without having to learn how to play a new instrument for each new scale. It is also a practical device that allows composers to hear and experiment with different tunings, and an analytical tool that offers insights into the deeper structural proper- ties of a scale. Furthermore, Wilson’s keyboard mapping system might also be applied to new designs for musical instruments. Given that Wilson’s keyboard has the potential to be such a valuable tun- ing resource, why does it remain largely unrecognized? How robust and ex- tensible is his keyboard system, and what does it offer that other systems do not? How can this unusual interface help musicians to apprehend ­Wilson’s body of work? These are the issues that provide the foundation for this book as it sets out on a research journey to understand more fully Wilson’s notions of tuning, particularly in relation to the untapped potential of his keyboard designs, and their influence on contemporary music-making and beyond. As part of the investigation, this book examines Wilson’s idiosyncratic ter- minology and notation as well as a number of his documents that have not been discussed previously. In so doing, it aims to offer a broad discussion of this enigmatic theorist’s ideas as a way to encourage other musicians to further explore his profound and highly original body of work. Microtonality and the enigma of Erv Wilson 9 Chapter 2, “Microtonal keyboards”, introduces Wilson’s keyboard de- signs as part of a broader discussion on the importance of musical interfaces for the development and realization of new tuning systems. How did Wilson devise an instrument such as his generalized microtonal keyboard and by what creative processes did his ideas take shape? Who influenced his design and what did he reject of their work to come up with what was uniquely his own? The chapter begins with a brief overview of the early evolution of keyboard instruments in relation to historical developments in tuning systems. It concentrates especially on instruments that display the property of transpositional invariance, or the ability to transpose a chord or sequence of notes to any key while maintaining the same geometric fingering pattern. This discussion leads to the invention of R.H.M. Bosanquet’s generalized keyboard, and ends with Wilson’s innovations in the design of generalized keyboards for which he holds two patents. The main focus of the chapter is his MicroZone keyboard, an instrument manufactured by Starr Labs, fea- turing an array of skewed hexagonal keys. The details of this design are examined with regard to physical considerations, such as hand span and space optimization, and more importantly in terms of its adaptability in accommodating a variety of different types of scales. Chapter 3, “Mapping linear scales on Wilson’s generalized keyboard”, demonstrates how to map a mono-dimensional scale onto Wilson’s micro- tonal keyboard, a process that requires considerable practice and explana- tion. It describes how to build a linear scale from a chain of a generating interval (such as a perfect fifth) whose pitches are reduced to within a period (typically an octave). Steps involved in mapping a simple and familiar ex- ample, namely a 12-note Pythagorean scale, are explained in detail using some of Wilson’s notational conventions. The same scale is then mapped onto several different keyboard layouts to show the versatility of Wilson’s system, and the pros and cons of different layouts are discussed in relation to practical musical concerns. These principles form the core of Wilson’s keyboard mapping system. In Chapter 4, “Moments of Symmetry and the Scale Tree”, the structural principle of linear scales is applied as a method for generating new scales. Whereas the previous chapter focuses on a conservative example generated from a chain of fifths (3/2), this chapter extends the concept to build scales from alternative intervals such as a septimal whole tone (8/7). This process of generating new scales raises the question of how far the linear chain should be extended. How does one know when to stop? The answer lies at the heart of one of Wilson’s most sustaining concepts, his theory of Moments of Sym- metry (MOS). This chapter discusses the importance of MOS as a guiding principle for building robust scales with structural integrity, and these ideas are linked to other key concepts in Wilson’s theoretical framework such as the Scale Tree. These theories work together to support Wilson’s keyboard mapping system, and conversely the keyboards also reinforce these tuning principles in a reciprocal relationship. An often overlooked category of 10 Microtonality and the enigma of Erv Wilson scales, called the Secondary Moments of Symmetry, are also illustrated with examples mapped onto the Wilson keyboard. Chapter 5, “Mapping Constant Structures and navigating new pathways”, presents Wilson’s theory of Constant Structures in order to address the prob- lem of how to map nonlinear scales onto the generalized keyboard layout. It argues that Constant Structures are a useful way of looking at scales that go beyond a single limit. Unlike MOS scales, Constant Structures are not based on a single generating interval but instead typically use different har- monics to imitate a single linear chain. In this way, they mimic the arche- typal form of MOS scales in that they display a certain consistency of shape between intervals. This structural property is exploited to map Constant Structures onto the generalized keyboard layout using an ingenious system devised by Wilson. This technique of mapping is more complex than pre- vious examples and it involves the use of harmonic templates and tuning lattices. A ­variety of examples, including Partch’s 43-tone scale, Wilson’s D’alessandro ­tuning, the Bohlen–Pierce scale and extensions of Grady’s Centaur scale, are mapped onto the keyboard to show the versatility and extensibility of Wilson’s system. Lastly, Chapter 6, “Cross-sets, Diamonds and Combination-Product Sets”, provides an in-depth analysis of Wilson’s harmonically based tuning structures. It focuses mainly on CPS, Wilson’s best-known contributions to tuning, which include the Hexany, Dekany, Eikosany and Hebdomekon- tany, among others. The discussion compares CPS with Diamonds which are theoretical counterparts that can be built from the same set of harmonic ­factors but develop into different complementary forms. Whereas Di­ amonds are centred around a tonic, CPS are centreless structures that suggest a new approach to creating non-tonal harmonies using relatively consonant ­sonorities. Wilson illustrates CPS using multidimensional ge­ ometric ­lattices to represent the many harmonic relationships that are found within these structures. The chapter also explains how to interpret these lattices as well as other types of charts and diagrams, including cross-sets and Lambdomas. Finally, it examines Wilson’s solutions for mapping CPS onto the general- ized keyboard through the example of his D’alessandro keyboard mapping. It should be noted that this book is not a complete treatise covering all of Wilson’s work, for such an undertaking would require many volumes. It does, however, aim to provide a thorough introduction to his most fun- damental tuning concepts in order to whet the appetite and enable musi- cians to explore his work for themselves. I have chosen to concentrate on just intonation (JI) scales to illustrate Wilson’s concepts because much of the work described in the book was originally conceived in JI. (For readers who are unfamiliar with the basic principles of JI, an overview is presented in the Appendix to the book.) This is not to say that Wilson excluded equal temperaments (ETs). On the contrary, much of his early work investigated properties of countless ETs, particularly 22-, 31- and 41-tone ETs (Wilson 1957–2006), and also included non-octave-based scales (Wilson 1962–2001). Microtonality and the enigma of Erv Wilson 11 He collaborated with John Chalmers in the 1960s to produce exhaustive ­computer-generated tables of ETs from 5 to 120 tones per ­octave ­(Chalmers & Wilson 1982). Wilson’s personal preference was 41ET. For ­instance, he often used a 41ET instrument to try out various musical ideas on the fly, but he also “promoted 31ET as a practical alternative because of the progress being made in the Fokker 31 tone school” (Grady 2016) in the hope of developing an international exchange of microtonality. It seems fair to say that most of Wilson’s ideas discussed in this book are best presented in JI; however, once the framework or process for constructing a scale is ­understood, it can often be reconfigured as an ET by substituting each JI ratio with the nearest interval in the ET scale. An example of this is also included in the ­Appendix. Where relevant, the reader will be directed to Wilson’s related papers on ETs. To supplement the figures that appear in this book, many additional di- agrams have been made available online as Web Figures. These can be ac- cessed through the Wilson Archives on a dedicated web page for the book at http://anaphoria.com/wilsonbook.html. For any references to Web Figures found throughout the text, please refer to this URL. Finally, a Glossary of terms is also provided at the end of the book to assist the reader. One of the motivations for writing this book is my belief that Wilson’s extensive work in tuning deserves a much wider musical audience beyond a field of aficionados of microtonality. His revolutionary ideas have the potential to advance our understanding of not only the scales that shape our melodies and harmonies, but how to invent new scales in order to cre- ate a kind of music whose sounds have not yet been heard or imagined. By taking advantage of the valuable resources Wilson has provided, I hope that future generations of musicians will have the opportunity to forge ahead into musical domains that have not yet been discovered, and that much greater dialogue can be shared between composers, performers and instrument designers. Perhaps this will eventually lead to a disintegration of the boundaries that separate microtonal and non-microtonal music, and that alternative tuning systems will be embraced as part of a continuous spectrum of sonic possibilities.

Note 1 Refer to http://anaphoria.com/wilson.html.

References Bohlen, H 1978, ‘13 Tonstufen in der Duodezime’, Acustica, vol. 39, no. 2, pp. 76–86. Burns, B 1986, ‘Seed Savers in Their Own Right: Erv Wilson’, The Seedhead News- letter, no. 16, p. 2. Burt, W 2007, ‘Algorithms, Microtonality, Performance: Eleven Musical Composi- tions’, PhD thesis, University of Wollongong. 12 Microtonality and the enigma of Erv Wilson Carlos, W 1987, ‘Tuning: At the Crossroads’, Computer Music Journal, vol. 11, no. 1, pp. 29–43. Chalmers, J 1974, ‘Editorial and Prospectus’, Xenharmonikôn, vol. 1, 2 pages. ——— 1993, Divisions of the Tetrachord, Frog Peak Music, Lebanon, NH. Chalmers, J & Wilson, E 1981, ‘Combination Product Sets and Other Harmonic and Melodic Structures’, Proceedings of the 7th International Computer Music Confer- ence, North Texas State University, Denton, TX, pp. 348–362. Grady, K 2012, About the Wilson Archives, viewed 5 Dec 2012, http://anaphoria. com/wilsonabout.html. ——— 2016, Comments on Equal Temperament Papers, viewed 26 Jun 2016, http:// anaphoria.com/etcomment.html. Mathews, MV, Roberts, LA & Pierce, JR 1984, ‘Four New Scales Based on Nonsuccessive-Integer-Ratio Chords’, Journal of the Acoustical Society of Amer- ica, vol. 75, p. S10(A). McLaren, B 1997, Microtonality in the United States, viewed 4 May 2007, http://daschour.club.fr/micromegas/mclaren.html. ——— 1998, ‘A Brief History of Microtonality in the Twentieth Century’, Xenhar- monikôn, vol. 17, pp. 57–110. Op de Coul, M 2016, Scala Home Page, viewed 28 Jun 2016, www.huygens-fokker. org/scala/. Partch, H 1974, Genesis of a Music: An Account of a Creative Work, Its Roots and Its Fulfillments, 2nd edn, Da Capo Press, New York. Satellite, M 2016, Wilsonic, version 1.7.3, mobile app, viewed 28 Jun 2016, https:// itunes.apple.com/us/app/wilsonic/id848852071?mt=8. Schulter, M 2012, Huh? Microtonal Music? A Guide for the Perplexed, UnTwelve, viewed 11 Aug 2012, http://untwelve.org/what. Taylor, SJ 2011, Kraig Grady Interview, online video, The Sonic Sky, viewed 28 Sep 2012, www.thesonicsky.com/video/kraig-grady-interview/. Wilson, E 1957–2006, Papers on Various Equal Divisions, The Wilson Archives, viewed 26 Jun 2016, http://anaphoria.com/wilsonbasic.html#et. ——— 1962–2001, Various Papers on Non-Octave Scales, The Wilson Archives, viewed 28 Jun 2016, http://anaphoria.com/nonoctave.pdf. ——— 1989, A Biography of Erv Wilson, The Wilson Archives, viewed 24 Jun 2016, http://anaphoria.com/wilsonbio.html. ——— 2001, Biography, MicroFest 2001 Conference and Festival of Music in Alter- nate Tunings: Biographies of Conference Participants, viewed 24 Jun 2016, http:// pages.hmc.edu/alves/microfestbios.html#wilson. Wilson, G 2009, Ervin’s Life – Younger Years, The Wilson Archives, viewed 30 Mar 2016, http://anaphoria.com/ErvinsLife.pdf. Microtonality and the enigma of Erv Wilson Bohlen, H. 1978, ‘13 Tonstufen in der Duodezime’, Acustica, vol. 39, no. 2, pp. 76–86. Burns, B. 1986, ‘Seed Savers in Their Own Right: Erv Wilson’, The Seedhead Newsletter, no. 16, p. 2. Burt, W. 2007, ‘Algorithms, Microtonality, Performance: Eleven Musical Compositions’, PhD thesis, University of Wollongong. 12 Carlos, W. 1987, ‘Tuning: At the Crossroads’, Computer Music Journal, vol. 11, no. 1, pp. 29–43. Chalmers, J. 1974, ‘Editorial and Prospectus’, Xenharmonikôn, vol. 1, 2 pages. Chalmers, J. 1993, Divisions of the Tetrachord, Frog Peak Music, Lebanon, NH. Chalmers, J. & Wilson, E. 1981, ‘Combination Product Sets and Other Harmonic and Melodic Structures’, Proceedings of the 7th International Computer Music Conference, North Texas State University, Denton, TX, pp. 348–362. Grady, K. 2012, About the Wilson Archives, viewed 5 Dec 2012, http://anaphoria.com/wilsonabout.html. Grady, K. 2016, Comments on Equal Temperament Papers, viewed 26 Jun 2016, http://anaphoria.com/etcomment.html. Mathews, M.V. , Roberts, L.A. & Pierce, JR 1984, ‘Four New Scales Based on Nonsuccessive- Integer-Ratio Chords’, Journal of the Acoustical Society of America, vol. 75, p. S10(A). McLaren, B. 1997, Microtonality in the United States, viewed 4 May 2007, http://daschour.club.fr/micromegas/mclaren.html. McLaren, B. 1998, ‘A Brief History of Microtonality in the Twentieth Century’, Xenharmonikôn, vol. 17, pp. 57–110. Op de Coul, M. 2016, Scala Home Page, viewed 28 Jun 2016, www.huygens-fokker.org/scala/. Partch, H. 1974, Genesis of a Music: An Account of a Creative Work, Its Roots and Its Fulfillments, 2nd edn, Da Capo Press, New York. Satellite, M. 2016, Wilsonic, version 1.7.3, mobile app, viewed 28 Jun 2016, https://itunes.apple.com/us/app/wilsonic/id848852071?mt=8. Schulter, M. 2012, Huh? Microtonal Music? A Guide for the Perplexed, UnTwelve, viewed 11 Aug 2012, http://untwelve.org/what. Taylor, SJ 2011, Kraig Grady Interview, online video, The Sonic Sky, viewed 28 Sep 2012, www.thesonicsky.com/video/kraig-grady-interview/. Wilson, E. 1957–2006, Papers on Various Equal Divisions, The Wilson Archives, viewed 26 Jun 2016, http://anaphoria.com/wilsonbasic.html#et. Wilson, E. 1962–2001, Various Papers on Non-Octave Scales, The Wilson Archives, viewed 28 Jun 2016, http://anaphoria.com/nonoctave.pdf. Wilson, E. 1989, A Biography of Erv Wilson, The Wilson Archives, viewed 24 Jun 2016, http://anaphoria.com/wilsonbio.html. Wilson, E. 2001, Biography, MicroFest 2001 Conference and Festival of Music in Alternate Tunings: Biographies of Conference Participants, viewed 24 Jun 2016, http://pages.hmc.edu/alves/microfestbios.html#wilson. Wilson, G. 2009, Ervin’s Life – Younger Years, The Wilson Archives, viewed 30 Mar 2016, http://anaphoria.com/ErvinsLife.pdf.

Microtonal keyboards Allen, JS 1997, The General Keyboard in the Age of MIDI, viewed 3 Sep 2012, www.bikexprt.com/music/bosanqet.htm. Altoft, S. & Bousted, D. 2012, Microtonal Trumpet, viewed 30 Aug 2012, www.microtonaltrumpet.com/. Alves, B. 2012, Bill Alves, viewed 17 Sep 2012, www.billalves.com/home.html. Bosanquet, RHM 1874–1875, ‘Temperament; or, the Division of the Octave [Part I]’, Proceedings of the Musical Association, 1st Sess., pp. 4–17. Bosanquet, R.H.M. 1876, An Elementary Treatise on Musical Intervals and Temperament, Macmillan & Co., London. Burt, W. 2012, My History with Music Technology Part 2: San Diego, viewed 30 Aug 2012, www.warrenburt.com/my-history-with-music-tech2/. Carlos, W. 2012, Wendy Carlos Homepage, viewed 30 Aug 2012, www.wendycarlos.com. Çoğulu, T. 2016, Adjustable Microtonal Guitar, viewed 28 Jun 2016, www.microtonalguitar.org/. Crab S. 2014, ‘The Motorola Scalatron. Herman Pedtke & George Secor. USA, 1974’, 120 Years of Electronic Music, viewed 29 Jun 2016, http://120years.net/the-motorola-scalatron-herman- pedtke-george-secor-usa-1974/. Dixon, M.H. 2011, ‘Tuning the Horn: A Composer-Performer’s Perspective on Using Extended Just Intonation’, DCA thesis, University of Wollongong. Doty, D. 1994, The Just Intonation Primer: An Introduction to the Theory and Practice of Just Intonation, 2nd edn, Just Intonation Network, San Francisco, CA. Forster, C. 2010, Musical Mathematics: On the Art and Science of Acoustic Instruments, Chronicle Books, San Francisco, CA. Fujieda, M. 2010, Mamoru Fujieda Official Site, viewed 17 Sep 2012, www.fujiedamamoru.com/. Grady, K. 2012, Musical Instruments and Tunings, viewed 30 Aug 2012, http://anaphoria.com/musinst.html. Hackleman, JS 2001, ‘The Hackleman-Wilson 19-Tone Clavichord’, abstract of conference paper, MicroFest 2001, viewed 23 Jul 2017, http://pages.hmc.edu/alves/microfestabstracts.html#hackleman. 26 Hanson, L.A. 1989, ‘Development of a 53-Tone Keyboard Layout’, Xenharmonikôn, vol. 12, pp. 68–85. Harmoniumnet 2011, viewed 2 Sep 2012, www.harmoniumnet.nl/museum-saltaire-ENG.html. Harrison, L. 1992, Lou Harrison Gamelan Music, CD, Musicmasters Classics 01612-67091-2. Haverstick, N. 2012, Microstick, viewed 6 Dec 2012, www.broadlandsmedia.com/microstick/. Helmholtz, H. 1954, On the Sensations of Tone as a Physiological Basis for the Theory of Music, 4th German edn, 1877; trans., rev., corrected by A.J. Ellis , Dover Publications, Inc., New York. Hopkin, B. 1994, ‘Tubulonia’, Experimental Musical Instruments, vol. 10, no. 2, pp. 26–27. Hunt, A. 2016, Tonal Plexus, H-Pi Instruments, viewed 29 Jun 2016, http://hpi.zentral.zone/tonalplexus. Huygens-Fokker Foundation 2010, Fokker Organ, viewed 2 Sep 2012, www.huygens- fokker.org/instruments/fokkerorgan.html. Hykes, D. 2015, David Hykes & The Harmonic Presence Foundation, viewed 28 Jun 2016, www.harmonicworld.com. Keislar, D. 1987, ‘History and Principles of Microtonal Keyboards’, Computer Music Journal, vol. 11, no. 1, pp. 18–28. Published with corrections as ‘History and Principles of Microtonal Keyboard Design’, 1988, viewed 5 Dec 2012, https://ccrma.stanford.edu/files/papers/stanm45.pdf. Kepler Quartet 2016, Kepler Quartet, viewed 28 Jun 2016, www.keplerquartet.com. ‘Keyboard’ n.d., in Oxford Dictionary of Music, 2nd ed. rev., Oxford Music Online, viewed 31 Aug 2012, www.oxfordmusiconline.com/subscriber/article/opr/t237/e5570. KeyboardMag 2011, ‘Starr Labs Microzone U-648’, Keyboard, viewed 17 Sep 2012, www.keyboardmag.com/miscellaneous/1265/starr-labs-microzone-u-648/30901. Khramov, M. 2010, ‘Pam and Phil Fluke’, Project CommaTor, viewed 7 Jul 2010, https://sites.google.com/site/commator/flukes. Kurzweil 2012, Product: K2000 | Kurzweil, viewed 17 Sep 2012, http://kurzweil.com/product/k2000/. Lindley, M. 2009, ‘Temperaments’, in Grove Music Online. Oxford Music Online, viewed 1 Sep 2012, www.oxfordmusiconline.com/subscriber/article/grove/music/27643. Luedtke, H. 17 Nov 1936, Design of Keyboards or Fields of Sounding Bodies, US Patent 2061364. Mandelbaum, J. 1974, ‘Review: Toward the Expansion of Our Concepts of Intonation’, Perspectives of New Music, vol. 13, no. 1, pp. 216–226. Meeùs , N n.d., ‘Keyboard’, in Grove Music Online. Oxford Music Online, viewed 31 Aug 2012, www.oxfordmusiconline.com/subscriber/article/grove/music/14944. Op de Coul, M. 2016, Scala Home Page, viewed 28 Jun 2016, www.huygens-fokker.org/scala/. Partch, H. 1974, Genesis of a Music: An Account of a Creative Work, Its Roots and Its Fulfillments, 2nd edn, Da Capo Press, New York. Rapoport, P. 1983, ‘Towards the Infinite Expansion of Tonal Resources’, Tempo, New Series, no. 144, pp. 7–11. 27 Reinhard, J. 2011, American Festival of Microtonal Music, viewed 30 Aug 2012, www.afmm.org/. Satellite, M. 2016, Wilsonic, version 1.7.3, mobile app, viewed 28 Jun 2016, https://itunes.apple.com/us/app/wilsonic/id848852071?mt=8. Schiemer, G. 2017, Satellite Gamelan, viewed 23 Jul 2017, http://satellitegamelan.net/. Schneider, J. 2013, John Schneider, MicroFest, viewed 28 Jun 2016, http://microfest.org/about- john-schneider/. Science Museum 2012, Keyboard of Bosanquet’s Enharmonic Harmonium c 1876, Image, Science & Society Picture Library, viewed 30 Nov 2012, www.scienceandsociety.co.uk/results.asp?image=10213657. Scott, X.J. 2011, Glossary of Music Tuning Definitions, viewed 29 Oct 2012, www.nonoctave.com/tuning/glossary.html. Secor, G. 1975, ‘The Generalized Keyboard Scalatron’, Xenharmonikôn, vol. 3, 3 pages. Secor, G. 2010, Re: Some Alternative Halberstadt Layouts, Alternate Tunings Mailing List, 3 Dec, viewed 5 Jul 2012, http://launch.groups.yahoo.com/group/tuning/message/94964. Sherriff, A. 2016, Our Staff: Adrian Sherriff, Melbourne Polytechnic, viewed 28 Jun 2016, www.melbournepolytechnic.edu.au/our-staff/adrian-sherriff. Starr Labs 2015, MicroZone U-648, viewed 28 Jun 2016, www.starrlabs.com/product/microzoneu648/. StarrLabsZone 2011, Starr Labs Wilson-MicroZone U648 Layout & Tuning Demo, viewed 3 Sep 2012, www.synthtopia.com/content/2011/05/05/starr-labs-wilson-microzone-u648-layout-tuning- demo/. Synthtopia 2015, Microtonal Music, Synthtopia, viewed 28 Jun 2016, www.synthtopia.com/content/tag/microtonal-music/. Taylor, S.J. 2007, Coprime Colors, viewed 28 Sep 2012, http://stephenjamestaylor.com/sjt/microtonality.html. Taylor, S.J. 2011, Coming Soon » Erv Wilson iPad/iPhone Application by Marcus Satellite, The Sonic Sky, viewed 30 Nov 2012, www.thesonicsky.com/coming-soon/erv-wilson-ipadiphone- application-by-marcus-satellite/. Wilkinson, S.R. 1988, Tuning In: Microtonality in Electronic Music: A Basic Guide to Alternate Scales, Temperaments, and Microtuning Using Synthesizers, H. Leonard Books, Milwaukee, WI. Wilson, E.M. 12 Dec 1961, Musical Instrument, US Patent 3012460. Wilson, E.M. 1965–2002, ‘Various Mallet Designs Pt 1: Non-Bosanquet Designs’, in K. Grady (ed.), The Wilson Archives, viewed 31 Aug 2012, http://anaphoria.com/MalletDesigns.pdf. Wilson, E.M. 19 Sep 1967, Musical Instrument Keyboard, US Patent 3342094. Wilson, E.M. c. 1968, The Act of Scale Formation (in a letter to Gary David), viewed 24 Oct 2012, http://anaphoria.com/wilsonabout.html. Wilson, E.M. 1970, Letter to Bob Moog, The Wilson Archives, Feb, viewed 31 Aug 2012, http://anaphoria.com/LetterBobMoog.pdf. Wilson, E.M. 1974a, ‘The Bosanquetian 7-Rank Keyboard after Poole and Brown’, Xenharmonikôn, vol. 1, 9 pages. Wilson, E.M. 1974b, ‘Bosanquet – A Bridge – A Doorway to Dialog’, Xenharmonikôn, vol. 2, 7 pages. Wilson, E.M. 1974c, Letter to Richard Harasek, 6 Dec [unpublished], in KbdTxt.pdf, pp. 6–10, The Wilson Archives. 28 Wilson, E.M. 1975, ‘On the Development of Intonational Systems by Extended Linear Mapping’, Xenharmonikôn, vol. 3, 15 pages. Wilson, E.M. 1984, Guitar Frettings, The Wilson Archives, viewed 1 Sep 2012, http://anaphoria.com/guitarfrettings.pdf. Wilson, E.M. 1987, Multi-Keyboard Gridiron, The Wilson Archives, annotated by Wilson 9 Aug 1989, viewed 21 Mar 2010, http://anaphoria.com/keygrid.pdf. Wilson, E.M. 1995, Some Notes on a Scroll-Scan Matrix, 25 Oct [unpublished], in KbdText.pdf, p. 21, The Wilson Archives. Wilson, E.M. 2001a, Triangulation of Uath Octave [unpublished], in EllipticHexagon.pdf, p. 12, The Wilson Archives. Wilson, E.M. 2001b, Matrix: With Pingalas Meru Prastāra and Uath Gral Keyboard, The Wilson Archives, p. 1, viewed 6 Oct 2012, http://anaphoria.com/22tonescales.pdf. Xenharmonic Wiki 2016, Microtonal Keyboards, viewed 28 Jun 2016, http://xenharmonic.wikispaces.com/Microtonal+Keyboards. Young, L.M. 1987, The Well-Tuned Piano: 81 x 25 (6:17:50-11:18:59 PM NYC), CD, Gramavision 18-8701-2. Ztarland 2009, Wilson MicroZone U-648 Array Keyboard from Starr Labs, Starr Labs, 17 Mar, viewed 17 Sep 2012, www.youtube.com/watch?v=H7bsYzhmdv0&feature=related.

Mapping linear scales on Wilson’s generalized keyboard Bosanquet, R.H.M. 1876, An Elementary Treatise on Musical Intervals and Temperament, Macmillan & Co., London. Doty, D. 1994, The Just Intonation Primer: An Introduction to the Theory and Practice of Just Intonation, 2nd edn, Just Intonation Network, San Francisco, CA. Helmholtz, H. 1954, On the Sensations of Tone as a Physiological Basis for the Theory of Music, 4th German edn, 1877; trans., rev., corrected by AJ Ellis, Dover Publications Inc., New York. Hirsh, P. 2002, Interval Consistency, Intuitive Instruments for Improvisers, viewed 15 Nov 2010, http://improvise.free.fr/def/interval.html. Kolinski, M. 1961, ‘The Origin of the Indian 22-Tone System’, Studies in Ethnomusicology, vol. 1, pp. 3–18. Peirce, C.S. 1933, Collected Papers of Charles Sanders Peirce, vol. 4, Charles Hartshorne & Paul Weiss (eds), Harvard University Press, Cambridge, MA. Schulter, M. 1998, Pythagorean Tuning and Medieval Polyphony, viewed 23 Jul 2017, http://www.medieval.org/emfaq/harmony/pyth.html. Sethares, W. 1998, Tuning, Timbre, Spectrum, Scale, Springer, London. Starr Labs 2010, Wilson Generalized Keyboard, viewed 15 Dec 2010, http://cgi.ebay.com/Starr- Labs-MicroZone-U-648-Generalized-Keyboard-/330506274900#shId. Wilson, E. 1975, Letter to Chalmers Pertaining to Moments of Symmetry / Tanabe Cycle, 26 Apr, The Wilson Archives, viewed 3 Oct 2010, http://anaphoria.com/mos.pdf. Wilson, E. 1987, Multi-Keyboard Gridiron, The Wilson Archives, annotated by Wilson 9 Aug 1989, viewed 21 Mar 2010, http://anaphoria.com/keygrid.pdf. Wilson, E. 1989, ‘D’alessandro, Like a Hurricane’, Xenharmonikôn, vol. 12, pp. 1–38. Wilson, E. 1994a, Scale Tree (Peirce Sequence), The Wilson Archives, viewed 28 Jun 2007, http://anaphoria.com/sctree.pdf. Wilson, E. 1994b, The Gral Keyboard Guide, The Wilson Archives, viewed 21 Mar 2010, http://anaphoria.com/gralkeyboard.pdf. Wilson, E. 1999a, A Spectrum of Keyboards from the Gral Keyboard Guide, The Wilson Archives, viewed 21 Mar 2010, http://anaphoria.com/gralspectrum.pdf. Wilson, E. 1999b, Letter to Fred Kohler, 3 Aug [unpublished], in KbdText.pdf, p. 8, The Wilson Archives. Wilson, E. 2000a, Diophantine Triplets of Temperament Derived Intervals, The Wilson Archives, viewed 6 Jan 2016, http://anaphoria.com/DiophantineTripletsTEMPER.pdf. Wilson, E. 2000b, Pecan-Tree Patterns, in a Nut-Shell, The Wilson Archives, viewed 22 Nov 2012, http://anaphoria.com/peach.pdf. Wilson, E. 2001, Triangulation of Uath Octave [unpublished], in EllipticHexagon.pdf, p. 12, The Wilson Archives. Wilson, E. n.d.1, Ref: Nicolas Faber’s Halberstadt Organ, etc. [unpublished], in More XYs.pdf, pp. 4–6, The Wilson Archives. Wilson, E. n.d.2, MIDI Treats the Standard (Halberstadt) Keyboard as a Slightly Convoluted Monaxial Series… [unpublished], in KbdText.pdf, p. 19, The Wilson Archives. Xenharmonic Wiki 2011, Secor 5_23TX, viewed 7 Dec 2012, http://xenharmonic.wikispaces.com/Secor5_23TX.

Moments of Symmetry and the Scale Tree Austin, D. 2012, Trees, Teeth, and Time: The Mathematics of Clock Making, American Mathematical Society, viewed 18 Jun 2012, www.ams.org/samplings/feature-column/fcarc- stern-brocot. Bogomolny, A. 2012a, Stern-Brocot Tree and Continued Fractions, from Interactive Mathematics Miscellany and Puzzles, viewed 11 Jul 2012, www.cut-the- knot.org/blue/ContinuedFractions.shtml. Bogomolny, A. 2012b, Stern-Brocot Tree Introduction, from Interactive Mathematics Miscellany and Puzzles, viewed 18 Jun 2012, www.cut-the-knot.org/blue/Stern.shtml. Borowski, E.J. & Borwein, J.M. 1989, Dictionary of Mathematics, Collins, Glasgow. Burk, P. Polansky, L. Repetto, D. Roberts , M & Rockmore, D. 2011, Music and Computers: A Theoretical and Historical Approach, viewed 11 Jul 2012, http://music.columbia.edu/cmc/musicandcomputers/chapter1/01_03.php. Burt, W. 2007, ‘Algorithms, Microtonality, Performance: Eleven Musical Compositions’, PhD thesis, University of Wollongong. Carey, N. & Clampitt, D. 1989, ‘Aspects of Well-Formed Scales’, Music Theory Spectrum, vol. 11, no. 2, pp. 187–206. Carey, N. & Clampitt, D. 1996, ‘Regions: A Theory of Tonal Spaces in Early Medieval Treatises’, Journal of Music Theory, vol. 40, no. 1, pp. 113–147. Chalmers, J. 1975, ‘Cyclic Scales’, Xenharmonikôn, vol. 4, 9 pages. Chalmers, J. 1993, Divisions of the Tetrachord, Frog Peak Music, Lebanon NH. Chalmers, J. 1996, Subject: Kornerup, online post, 2 May, viewed 12 Nov 2012, www.microtonal.freeservers.com/mclaren/post102.html. Clough, J. & Douthett, J. 1991, ‘Maximally Even Sets’, Journal of Music Theory, vol. 35, no. 1–2, pp. 93–173. Clough, J. & Myerson, G. 1985, ‘Variety and Multiplicity in Diatonic Systems’, Journal of Music Theory, vol. 29, no. 2, pp. 249–270. Erlich, P. 2006, ‘A Middle Path between Just Intonation and the Equal Temperaments’, Xenharmonikôn, vol. 18, pp. 159–199. Finnamore, D.J. 2003, Golden Horagrams, viewed 19 Oct 2012, www.elvenminstrel.com/music/tuning/horagrams/horagram_intro.htm. 107 Grady, K. 1995, ‘A Rhythmic Application of the Horagrams’, Xenharmonikôn, vol. 16, pp. 94–98. Grady, K. 2012a, An Introduction to the Moments of Symmetry, viewed 11 Jul 2012, http://anaphoria.com/wilsonintroMOS.html. Grady, K. 2012b, More on Horogram Rhythms, viewed 12 Nov 2012, http://anaphoria.com/horo2.pdf. Grady, K. 2012c, One Person’s Introduction and Digressions to the Works of Erv Wilson, viewed 11 Jul 2012, http://anaphoria.com/wilsonintro.html. Hayes, B. 2000, ‘On the Teeth of Wheels’, American Scientist, vol. 88, no. 4, pp. 296–300. Helmholtz, H. 1954, On the Sensations of Tone as a Physiological Basis for the Theory of Music, 4th German edn, 1877; trans., rev., corrected by A.J. Ellis , Dover Publications Inc., New York. Knott, R. 2010, The Fibonacci Numbers and Golden Section in Nature – 1, viewed 7 Mar 2012, www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat.html. Knott, R. 2011a, Fractions in the Farey Series and the Stern-Brocot Tree, viewed 10 Jul 2012, www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fractions/fareySB.html. Knott, R. 2011b, The Fibonacci Rabbit Sequence – the Golden String, viewed 7 Mar 2012, www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibrab.html. Kornerup, T. 1934, Acoustic Methods of Work in Relation to Systematic Comparative Musicology: Including Some Acoustic Tables, trans. M Baruel, Author, Copenhagen. Mandelbaum, J. 1961, ‘Multiple Division of the Octave and the Tonal Resources of the 19-Tone Temperament’, PhD thesis, University of Indiana. Milne, A.J. , Carlé, M. , Sethares, W.A. , Noll, T. , & Holland, S. 2011, ‘Scratching the Scale Labyrinth’, in C. Agon , E. Amiot , M. Andreatta , G. Assayag , J. Bresson , & J. Mandereau (eds), Mathematics and Computation in Music: Third International Conference, MCM 2011, Paris, France, June 2011, Lecture Notes in Computer Science, vol. 6726, Springer-Verlag, Berlin Heidelberg, pp. 180–195. ‘Moment of Symmetry, MOS ’, 2005, in Tonalsoft Encyclopedia of Microtonal Music Theory, J. Monzo (ed.), viewed 18 Oct 2012, www.tonalsoft.com/enc/m/mos.aspx. O’Connell, W. 1993, ‘The Tonality of the Golden Section’, Xenharmonikôn, vol. 15, pp. 3–18. Op de Coul, M. 2016, Scala Home Page, viewed 28 Jun 2016, www.huygens-fokker.org/scala/. Rothenberg, D. 1978, ‘A Model for Pattern Perception with Musical Applications’, Mathematical Systems Theory, 11, pp. 199–234. Satellite, M. 2016, Wilsonic, version 1.7.3, mobile app, viewed 28 Jun 2016, https://itunes.apple.com/us/app/wilsonic/id848852071?mt=8. Schroeder, M.R. 2009, Number Theory in Science and Communication: With Applications in Cryptography, Physics, Digital Information, Computing, and Self-Similarity, Springer, Berlin, Heidelberg. Schulter, M. & Keenan, D. 2007, The Noble Mediant: Complex Ratios and Metastable Musical Intervals, viewed 20 Nov 2012, www.dkeenan.com/Music/NobleMediant.txt. Taylor, S.J. 2011, John Chalmers Interview, online video, The Sonic Sky, viewed 22 Oct 2012, www.thesonicsky.com/video/john-chalmers-interview/. Taylor, S.J. 2015, How to Make Music on the Wilsonic iPad App – Moments of Symmetry, online video, The Sonic Sky, viewed 5 Jul 2016, http://thesonicsky.com/wilsonic-ipad-app/how-to- make-music-on-the-wilsonic-ipad-app-moments-of-symmetry/. Temes, L. 1970, Golden Tones?, viewed 30 Nov 2012, http://anaphoria.com/temes.pdf. van Ravenstein, T. 1988, ‘The Three Gap Theorem (Steinhaus Conjecture)’, Journal of the Australian Mathematical Society (Series A), vol. 45, pp. 360–370. 108 Walker, R. 2008, Making Moments of Symmetry and Hyper-MOS Scales, viewed 7 Oct 2012, http://robertinventor.com/software/tunesmithy/help/hypermos.htm. Weisstein, E.W. 1999, CRC Concise Encyclopedia of Mathematics, CRC Press, Boca Raton, FL. Wildberger, N. 2012, MF97: The Stern-Brocot Tree, Matrices and Wedges, online video, 29 May 2012, viewed 11 Jul 2012, www.youtube.com/watch?v=qPeD87HJ0UA&feature=relmfu. Wilson, E. 1963–1969, Various Unpublished Letters to John Chalmers (including references to MOS on 21 Aug 1965 and 19 Apr 1969) [unpublished], in LettersChalmersMisc.pdf, 8 pages, The Wilson Archives. Wilson, E. 1975, Letter to Chalmers Pertaining to Moments of Symmetry/Tanabe Cycle, 26 Apr, The Wilson Archives, viewed 3 Oct 2010, http://anaphoria.com/mos.pdf. Wilson, E. 1989 & 1998, Moments of Symmetry of Equal Divisions of the Octave, The Wilson Archives, viewed 1 Jul 2016, http://anaphoria.com/MOSedo.pdf. Wilson, E. 1991, Tree of 2-Interval Chain-Patterns (After Fibonacci’s Adult/Baby Rabbit Sequence, The Wilson Archives, viewed 21 Feb 2012, http://anaphoria.com/RabbitSequence.pdf. Wilson, E. 1994, Scale Tree (Peirce Sequence), The Wilson Archives, viewed 28 Jun 2007, http://anaphoria.com/sctree.pdf. Wilson, E. 1995, Letter to John Clough, 9 Feb [unpublished], SecondMOS(afterLtoJC).pdf, 11 pages, The Wilson Archives. Wilson, E. 1996a, MOS of 8/7, The Wilson Archives, viewed 31 Aug 2012, http://anaphoria.com/chain7.pdf. Wilson, E. 1996b, Straight Line Patterns of the Scale Tree from 0/1 to 1/2, The Wilson Archives, viewed 28 Jun 2007, http://anaphoria.com/line.pdf. Wilson, E. 1997, The Golden Horograms of the Scale Tree, The Wilson Archives, viewed 16 Jan 2012, http://anaphoria.com/hrgm.pdf. Wilson, E. 1998, The Tanabe Cycle and Parallelogram from the Tanabe Cycle, The Wilson Archives, pp. 11–12, viewed 22 Aug 2008, http://anaphoria.com/mos.pdf. Wilson, E. 2000, Scale Tree and Triangle, The Wilson Archives, viewed 12 Nov 2012, http://anaphoria.com/scaletreetriangle.pdf. Wilson, E. 2001, 26-Tone Chains of Sevens, The Wilson Archives, viewed 19 Nov 2012, http://anaphoria.com/chain7.pdf. Wilson, E. 2005, The 3-Gap Theorem (Steinhaus Conjecture) Revisited, The Wilson Archives, viewed 16 Dec 2011, http://anaphoria.com/steinhaus.pdf. Wilson, E. n.d.1, How to Grow a Scale Tree [unpublished], in ScaleTreeText.pdf, pp. 1–5, The Wilson Archives. Wilson, E. n.d.2, I First Began the Scale Tree Experiment in 1949–1950… [unpublished], in ScaleTreeText.pdf, pp. 6–10, The Wilson Archives. Wilson, E. n.d.3, The 3 Important Things I Learned Early from Joseph Yasser [unpublished], in ScaleTreeText.pdf, pp. 50–51, The Wilson Archives. Wright, D. 2009, Mathematics and Music, Mathematical World, vol. 28, American Mathematical Society, Rhode Island. Xenharmonic Wiki 2012a, MOS Scales, viewed 7 Oct 2012, http://xenharmonic.wikispaces.com/MOSScales. Xenharmonic Wiki 2012b, Mathematics of MOS, viewed 18 Oct 2012, http://xenharmonic.wikispaces.com/Mathematics+of+MOS. Yasser, J. 1932, A Theory of Evolving Tonality, American Library of Musicology, New York.

Mapping Constant Structures and navigating new pathways Bohlen, H. 1978, ‘13 Tonstufen in der Duodezime’, Acustica, vol. 39, no. 2, pp. 76–86. Bohlen–Pierce Site 2010, Huygens-Fokker Foundation, viewed 13 Mar 2013, www.huygens- fokker.org/bpsite/. Ensemble Offspring 2013, Between the Keys, CD, Ensemble Offspring, viewed 6 Jul 2016, http://ensembleoffspring.com/releases/between-the-keys-cd-2/. Grady, K. 1986, ‘Combination-Product Set Patterns’, Xenharmonikôn, vol. 9, 4 pages. Grady, K. 1999, ‘Re: CS’, Alternate Tunings Mailing List, 4 Oct, viewed 23 Nov 2012, http://launch.groups.yahoo.com/group/tuning/message/5244. Grady, K. 2011, Centaur: A 7-Cap Tuning, viewed 23 Nov 2012, http://anaphoria.com/centaur.html. Grady, K. 2012, Musical Instruments and Tunings, viewed 30 Aug 2012, http://anaphoria.com/musinst.html. Grady, K. & Narushima, T. 2010, A Select Group of Dekanies and Constant Structures in which They are Found, viewed 26 Jun 2016, www.anaphoria.com/dekanyconstantstructures.pdf. Harrison, L. 1993, Music Primer (Japanese edition), trans. T. Kakinuma & M. Fujieda , Jesuku Ongaku Bunka Shinkokai, Tokyo. Mandelbaum, J. 1961, ‘Multiple Division of the Octave and the Tonal Resources of the 19-Tone Temperament’, PhD thesis, University of Indiana. Mathews, M.V. , Roberts, L.A. & Pierce, J.R. 1984, ‘Four New Scales Based on Nonsuccessive- Integer-Ratio Chords’, Journal of the Acoustical Society of America, vol. 75, p. S10(A). Op de Coul, M. 2016, Scala Home Page, viewed 28 Jun 2016, www.huygens-fokker.org/scala/. Partch, H. 1974, Genesis of a Music: An Account of a Creative Work, Its Roots and Its Fulfillments, 2nd edn, Da Capo Press, New York. Prooijen, K.V. 1978, ‘A Theory of Equal-Tempered Scales’, Interface, vol. 7, no. 1, pp. 45–56. 139 Walker, E. 2011, Bohlen–Pierce Scale: Modes and Chords, viewed 23 Nov 2012, www.ziaspace.com/elaine/BP/Modes_and_Chords.html. Wilson, E. 1962–2001, Various Papers on Non-Octave Scales, The Wilson Archives, viewed 28 Jun 2016, http://anaphoria.com/nonoctave.pdf. Wilson, E. 1963–1969, Various Unpublished Letters to John Chalmers (including references to MOS on 21 Aug 1965 and 19 Apr 1969) [unpublished], in LettersChalmersMisc.pdf, 8 pages, The Wilson Archives. Wilson, E. 1964, Letter to John from Erv Wilson, 19 Oct [unpublished], in PartchMappedTo41.pdf, p. 1, The Wilson Archives. Wilson, E. 1975a, ‘On the Development of Intonational Systems by Extended Linear Mapping’, Xenharmonikôn, vol. 3, 15 pages. Wilson, E. 1975b, Letter to Chalmers Pertaining to Moments of Symmetry / Tanabe Cycle, 26 Apr, The Wilson Archives, viewed 3 Oct 2010, http://anaphoria.com/mos.pdf. Wilson, E. 1987, Multi-Keyboard Gridiron, The Wilson Archives, annotated by Wilson 9 Aug 1989, viewed 21 Mar 2010, http://anaphoria.com/keygrid.pdf. Wilson, E. 1989a, ‘D’alessandro, Like a Hurricane’, Xenharmonikôn, vol. 12, pp. 1–38. Wilson, E. 1989b, Hebdomekontany Notes, The Wilson Archives, viewed 24 Nov 2012, http://anaphoria.com/Hebdomekontany.pdf. Wilson, E. 1999, Untitled (Polar Version of the Partch Diamond) [unpublished], 25 Aug, in Partch.pdf, p. 10, The Wilson Archives. Wilson, E. , Smith, S. & Grady, K. 1989, ‘Notes on a New Marimba, its Tuning, and its Music’, Xenharmonikôn, vol. 11, pp. 46–60.

Cross-sets, Diamonds, and Combination-Product Sets Canright, D. 1995, ‘Superparticular Pentatonics’, 1/1: Journal of the Just Intonation Network, vol. 9, no. 1, pp. 10–13. Chalmers, J. 1994, ‘An Anticipation of the Partch Diamond in Augusto Novaro’s Sistema Natural Base del Natural-Aproximado (1927)’, 1/1: Journal of the Just Intonation Network, vol. 8, no. 2, pp. 9–10. Chalmers, J. & Wilson, E. 1981, ‘Combination Product Sets and Other Harmonic and Melodic Structures’, Proceedings of the 7th International Computer Music Conference, North Texas State University, Denton, TX, pp. 348–362. Duncan, D. 1993, ‘Why Superparticular?’, 1/1: Journal of the Just Intonation Network, vol. 8, no. 1, pp. 1, 4–10. Forster, C. 2010, Musical Mathematics: On the Art and Science of Acoustic Instruments, Chronicle Books, San Francisco, CA. Gilmore, B. 1998, Harry Partch: A Biography, Yale University Press, New Haven & London. Grady, K. 1986, ‘Combination-Product Set Patterns’, Xenharmonikôn, vol. 9, 4 pages. Grady, K. 1991, ‘Erv Wilson’s Hexany’, 1/1: Journal of the Just Intonation Network, vol. 7, no. 1, pp. 8–11. Hero, B. 1999, ‘The Lambdoma Matrix and Harmonic Intervals’, IEEE Engineering in Medicine and Biology, vol. 8, no. 2, pp. 61–73. Novaro, A. 1927, Teoría de la Musica: Sistema Natural Base del Natural-Aproximado, Author, Mexico, D.F. Partch, H. 1974, Genesis of a Music: An Account of a Creative Work, Its Roots and Its Fulfillments, 2nd edn, Da Capo Press, New York. Rapoport, P. 1994, ‘Just Shape, Nothing Central’, Musicworks, vol. 60, pp. 42–49. Satellite, M. 2016, Wilsonic, version 1.7.3, mobile app, viewed 28 Jun 2016, https://itunes.apple.com/us/app/wilsonic/id848852071?mt=8. Schiemer, G. 2017, Satellite Gamelan, viewed 23 Jul 2017, http://satellitegamelan.net/. Schoenberg, A. 1975. Style and Idea: Selected Writings of Arnold Schoenberg, L. Stein (ed.), Faber, London. 186 Taylor, S.J. 2015, Microtonal Music (aka Transcendent Tonality), viewed 28 Sep 2016, http://stephenjamestaylor.com/sjt/microtonality.html. Wilson, E. 1964–1970, Early Pages on 22ET, The Wilson Archives, viewed 28 Jun 2016, http://anaphoria.com/22ET.pdf. Wilson, E. 1967–1997, Hexany Stellates and Expansions, The Wilson Archives, viewed 8 Jan 2016, http://anaphoria.com/HexanyStellatesExpansions.pdf. Wilson, E. 1967–1999, The Hexany, The Wilson Archives, viewed 8 Feb 2015, http://anaphoria.com/hexany.pdf. Wilson, E. 1967–2001, Constructing the Euler Genera, The Wilson Archives, viewed 13 Jan 2016, http://anaphoria.com/Eulergenera.pdf. Wilson, E. c. 1968, The Act of Scale Formation (in a letter to Gary David), viewed 24 Oct 2012, http://anaphoria.com/wilsonabout.html. Wilson, E. 1970, Some Diamond Lattices (and Blanks), The Wilson Archives, viewed 31 Mar 2013, http://anaphoria.com/diamond.pdf. Wilson, E. 1970, Letter to Adriaan Fokker, Oct, The Wilson Archives, viewed 23 Mar 2011, http://anaphoria.com/FOKKER1.pdf. Wilson, E. 1999, Larger CPS Structures Mainly Illustrated in Blanks, The Wilson Archives, viewed 11 Apr 2013, http://anaphoria.com/LargerCPS.pdf. Wilson, E. 1971, Letter to John Chalmers Part 1, 4 Apr, The Wilson Archives, viewed 6 Jun 2016, http://anaphoria.com/CPStoC-pt1.pdf. Wilson, E. 1986, Letter to David Doty, 9 Mar [unpublished], CPS_LetterToDoty.pdf, 3 pages, The Wilson Archives. Wilson, E. 1989a, ‘D’alessandro, Like a Hurricane’, Xenharmonikôn, vol. 12, pp. 1–38. Wilson, E. 1989b, Hebdomekontany Notes, The Wilson Archives, viewed 24 Nov 2012, http://anaphoria.com/Hebdomekontany.pdf. Wilson, E. 1996, So-Called Farey Series, Extended 0/1 to 1/0 (Full Set of Gear Ratios), and Lambdoma, The Wilson Archives, viewed 6 Feb 2006, http://anaphoria.com/lamb.pdf. Wilson, E. n.d., Combination-Product Sets [unpublished], in PreEiko.pdf, pp. 34–36, The Wilson Archives.

Conclusion Grady, K 1999, ‘Re: CS’, Alternate Tunings Mailing List, 4 Oct, viewed 23 Nov 2012, http://launch.groups.yahoo.com/group/tuning/message/5244. Grady, K 2012, One Person’s Introduction and Digressions to the Works of Erv Wilson, viewed 11 Jul 2012, http://anaphoria.com/wilsonintro.html. Satellite, M 2016, Wilsonic, version 1.7.3, mobile app, viewed 28 Jun 2016, https://itunes.apple.com/us/app/wilsonic/id848852071?mt=8. Taylor, SJ 2012, ‘Surfing the Sonic Sky’ Awarded 2012 Worldfest Houston Special Jury Award, The Sonic Sky, viewed 30 Nov 2012, www.thesonicsky.com/uncategorized/surfing-the-sonic- sky-awarded-2012-worldfest-houston-special-jury-award/. Wilson, E c. 1968, The Act of Scale Formation (in a letter to Gary David), viewed 24 Oct 2012, http://anaphoria.com/wilsonabout.html. Wilson, E c. 1975, ‘On the Development of Intonational Systems by Extended Linear Mapping’, Xenharmonikôn, vol. 3, 15 pages.

References Doty, D 1994, The Just Intonation Primer: An Introduction to the Theory and Practice of Just Intonation, 2nd edn, Just Intonation Network, San Francisco, CA. Gann, K 2005, Just Intonation Explained, viewed 26 Aug 2016, www.kylegann.com/tuning.html. Schlesinger, K 1939, The Greek Aulos: A Study of Its Mechanism and of Its Relation to the Modal System of Ancient Greek Music, Followed by a Survey of the Greek Harmoniai in Survival or Rebirth in Folk-Music, Methuen, London. Wilson, E 1962–1973, Diaphonic Cycles, The Wilson Archives, viewed 14 Aug 2016, http://anaphoria.com/diaphonicset.pdf. Bogomolny, A 2012, Farey Series and Euclid’s Algorithm, from Interactive Mathematics Miscellany and Puzzles, viewed 6 Feb 2006, www.cut-the-knot.org/blue/Farey.shtml. Chalmers, J 1993, Divisions of the Tetrachord, Frog Peak Music, Lebanon NH. 208 Grady, K 1986, ‘Combination-Product Set Patterns’, Xenharmonikôn, vol. 9, 4 pages. Grady, K 1999, ‘Re: CS’, Alternate Tunings Mailing List, 4 Oct, viewed 23 Nov 2012, http://launch.groups.yahoo.com/group/tuning/message/5244. Hero, B 1999, ‘The lambdoma matrix and harmonic intervals’, IEEE Engineering in Medicine and Biology, vol. 8, no. 2, pp. 61-73. Knott, R 2010, The Fibonacci Numbers and Golden Section in Nature – 1, viewed 7 Mar 2012, www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat.html. Nelson, K n.d., Music-Science Glossary, viewed 6 Feb 2006, www.music- science.net/Glossary.html. Novaro, A 1927, Teoría de la Musica: Sistema Natural Base del Natural-Aproximado, Author, Mexico, D.F. Schroeder, MR 2009, Number Theory in Science and Communication: With Applications in Cryptography, Physics, Digital Information, Computing, and Self-Similarity, Springer, Berlin, Heidelberg. Wilson, E 1969–1970, Some Diamond Lattices (and Blanks), The Wilson Archives, viewed 31 Mar 2013, http://anaphoria.com/diamond.pdf. Wilson, E 1989, ‘D’alessandro, Like a Hurricane’, Xenharmonikôn, vol. 12, pp. 1–38. Wilson, E 1994, The Gral Keyboard Guide, The Wilson Archives, viewed 21 Mar 2010, http://anaphoria.com/gralkeyboard.pdf. Wilson, E 1996, So-Called Farey Series, Extended 0/1 to 1/0 (Full Set of Gear Ratios), and Lambdoma, The Wilson Archives, viewed 6 Feb 2006, http://anaphoria.com/lamb.pdf. Wilson, E 1998, The Tanabe Cycle and Parallelogram from the Tanabe Cycle, The Wilson Archives, pp. 11–12, viewed 22 Aug 2008, http://anaphoria.com/mos.pdf. Wilson, E 2000, Diophantine Triplets of Temperament Derived Intervals, The Wilson Archives, viewed 6 Jan 2016, http://anaphoria.com/DiophantineTripletsTEMPER.pdf. Xenharmonic Wiki 2012, MOS Scales, viewed 7 Oct 2012, http://xenharmonic.wikispaces.com/MOSScales.