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Incarnations of the Blaring Bluesblinger

By David Becker

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INCARNATIONS of the

BLARING BLUESBLINGER

A Multimedia Mathmantra on Manifestations of Mutuality in Music,

Molecules, and Morphogenesis

By David A. Becker, Ph.D.

Copyright © 2011 by David A. Becker. All rights reserved. No part of this publication may be reprinted, reproduced, transmitted, or utilized in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfi lming, and recording, or in any information retrieval system without the written permission of University Readers, Inc.

First published in the United States of America in 2011 by University Readers, Inc.

Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identifi cation and explanation without intent to infringe.

15 14 13 12 11 1 2 3 4 5

Printed in the United States of America

ISBN: 978-1-609279-55-4

CONTENTS

Acknowledgments vii

CHAPTER IMusic in the Third Dimension 1

CHAPTER IIA Dozen Quinary Denizens of a Cube 7

CHAPTER IIIProjecting the Tonal Voices 27

CHAPTER IVOne Shape Does Not Fit All 31

CHAPTER VA Rock’s Bilateral Symmetry Rocks: Tritones Align While a Vital Bluesblinger’s Groups of Five Make Jumping Jive 37

CHAPTER VIFurther Compositions and Analyses of These andOther Pertinent Songs 55

Seven Bridges 57Notes In Space 77Cephalogenesis 91Harmathonia 103

Clock Voices 115Midnight Diamonds 127The Faces of a Face 141Partners in Perpetunity 148Connectivity 150Music Box 169Twelve Tone Road 181Net Walks 203Form of the Funktion 221Code Of Blues 227Music for “All Blues” Rendition 230Paint These Headsets Blue 231Score for Bluesblingerinium Chromate 243

CHAPTER VIIAncient Beauty Still Tuning Heads: The Cadherin Cadenza 253

CHAPTER VIIIPassages, Reorganizations, and a Rendition of the Head to Pat Metheny’s “(It’s Just) Talk” On a (Talking) Head 265

CHAPTER IXNo Similar Tricks With Six 283

CHAPTER XExpansions and Contractions 285

CHAPTER XITaking Another Reading 297

CHAPTER XIICode Hearing: The Strikingly Structured Twin Dodecahedral Progeny of Music’s Timeless Circle 303

HARMONIC CEPHALOMETRY?

This project, like countless scientifi c endeavors, took the investigator along a path that emerged in completely unexpected and intriguing destinations. Having started out many years ago as an eff ort by a music lover to shed light on the structure of the harmonies, melodies, and chord

progressions of modern tonal music in mathematical terms, the work here presented makes forays not only to the fi elds of music theory and composition, but also to the fi elds of anatomy and bio-chemistry. At times, the encountered concepts are framed in a personal context as the interpretation of prominent fi ndings assists in the coming to grips with challenging life circumstances. Th roughout, it is geometry that plays the central role.

ACKNOWLEDGMENTS

The ideas, compositions, and conclusions in this book would never have come to fruition with-out the large network of caring people who have supported me, in every way necessary to foster a path towards growth, over the many years of inquiry invested in this project.

To my loving family, both those whose geometry and essence remain recognizable in a form I have relished on a daily basis, and those that every day I relish in contemplating their connectivity in higher forms, I could never thank you enough.

To my immeasurably beloved daughter Melanie, you will always be the most beautiful embodiment of creativity that parents could ever wish for. You, your mother, and I, along with all the many others who love you, are eternally linked in this wondrously orchestrated universe.

To sweetest Susan, my new wife as of the end of the month of this writing, your patiently nourish-ing love has acted as a bridge that allows passage to higher ground and to higher vistas. I love you.

To delightful Tristan, you have in me a person who is committed to fulfi lling the role of a father in every way. Th ank you for being enthusiastic about the merging of our two families into one. I love you.

To my beloved, geometrically-evolved Elise, your enlightening energy shines incessantly and has supplied me with the strength to face dissonance in the face—and to win that staring contest. Th is work as a whole is especially dedicated to you and Melanie, as well as to the two future Marcus members of our immediate family.

To my father, Stanley, who selfl essly postponed his higher education until after seeing to it that his children would have theirs, thank you for instilling in me the value of hard work and for promoting in me the kind of perseverance that is necessary to properly conduct research. I love you and will always be grateful for all of your eff orts on my behalf.

To my mother, Helen, thank you for always knowing the diff erence between when I needed some-one to listen to me and when I needed to listen to myself. I love you.

To my brother Steve, thank you for providing me as solid a big brother role model that any younger brother could ask for. I love you.

viii Incarnations of the Blaring Bluesblinger

To my sister, Jeannie, thank you for always staying in touch through thick and thin. I love you.Much love is also sent to my wonderful nieces and nephews, aunts and uncles, cousins, and the

entire Gilbert, Marcus, Th ierer, and Applebaum families. (A big woof goes to Brandy, Cookie, and Madison.)

Many musical aspects of this project were brilliantly organized by one of the top musicians in con-temporary jazz, the incomparable Randy Bernsen. His faith in my hopes for this work and his extreme diligence in making them happen were always beyond my expectations. In addition to his gifted ar-rangements of the compositions, the recruitment of outstanding instrumentalists, the extensive editing and sound engineering, and, of course, his amazing guitar playing on the tracks, he, over the course of our long friendship, elevated this author time and time again to places never thought reachable. Not only have I become a better musician because of knowing him, but, more importantly, a better person.

I would also like to express special thanks to the following educators who, in addition to Randy, profoundly expanded my horizons in music. Huge waves of gratitude are sent to the geometrically-evolved Herb Pomeroy, with whom I had hoped to share parts of this work, but instead will dedicate to his boundlessly upbeat nature and kind spirit.

My interest in music would not be what it is today without the exceptional tutelage of Jamesville DeWitt High School’s Ron Nuzzo, and Jamesville DeWitt Middle School’s Vic Russo. I believe that it is precisely the strong eff orts of grade school music instructors such as these that do much to equip young students with the cognitive tools necessary for success in many fi elds.

No better measure of the quality of the Jamesville DeWitt School District’s music program can be found than in the stellar career of saxophonist Walt Weiskopf. Walt’s willingness to participate in this project in between his teaching obligations at the Eastman School, and his demanding touring/studio schedule with Steely Dan will always be cherished by this fellow Nuzzo-ite. Hearty thanks go to Tom Helf for his invaluable help in the actualization of Walt’s appearance within this work.

Th e highest praise is also sent to pianist extraordinaire Michael Gerber, yet another of this project’s musicians who has played with a plethora of my childhood musical idols, and whose fabulous contribu-tions will be, by me, continuously appreciated. I am honored to have recently become one of Michael’s students and even more so by having become a close friend of his (and of his dear wife Atlanta).

In addition to the aforementioned music teachers, I wish to personally thank the many other musi-cians (and in some cases, also music educators) that have played key roles in this project’s realization. To fi rst do so, it is at this point a good time to delineate the instrumentalists appearing on each of the thirteen tracks of the accompanying CD entitled “Jammy Codecrackers”.

SEVEN BRIDGES TRACK #1Guitar – Randy Bernsen; Tenor Sax – Walt Weiskopf; Piano – Bob Grozier; Bass – Javier Carrion;

Drums – Mark Griffi th; Composer – Dave Becker; Arranger – Randy Bernsen

CEPHALOGENESIS TRACK #2Guitar – Randy Bernsen; Tenor Sax – Walt Weiskopf; Keyboards – Ray Lyon; Trumpet – Jason

Carder; Bass – Javier Carrion; Drums – Mark Griffi th; Composer – Dave Becker; Arranger – Randy Bernsen and Ray Lyon

ACKNOWLEDGMENTS ix

HARMATHONIA TRACK #3Tenor Sax – Walt Weiskopf; Trumpet – Jason Carder; Piano – Mike Levine; Bass – Javier Carrion;


CONNECTIVITY TRACK #4Tenor Sax – Andy Middleton; Trumpet – Jason Carder; Synthesized Percussion – Randy Bernsen;

Keyboard – Dave Becker; Composer – Dave Becker; Arranger – Randy Bernsen

NOTES IN SPACE TRACK #5Guitar – Randy Bernsen; Tenor Sax – Walt Weiskopf; Bass – Javier Carrion; Drums – Mark Griffi th;

Piano – Dave Becker; Composer – Dave Becker; Arranger – Randy Bernsen

CODE OF BLUES TRACK #6Guitar – Randy Bernsen; Piano – Michael Gerber; Tenor Sax – Gary Campbell; Bass – Javier

Carrion; Drums – Mark Griffi th; Composer – Dave Becker; Arranger – Randy Bernsen

CLOCK VOICES TRACK #7Tenor Sax – Ed Maina; Synthesized Percussion – Randy Bernsen; Keyboard – Dave Becker;

Composer – Dave Becker; Arranger – Randy Bernsen

PARTNERS IN PERPETUNITY TRACK #8Piano – Michael Gerber; Composer – Dave Becker; Arranger – Michael Gerber

THE FACES OF A FACE TRACK #9Tenor Sax – Walt Weiskopf; Trumpet – Jason Carder; Piano – Jim Gasior; Bass – Jamie Ousley;


MUSIC BOX TRACK #10Orchestration – Randy Bernsen; Piano – Dave Becker; Composer – Dave Becker; Arranger – Randy

Bernsen

NET WALKS TRACK #11Tenor Sax – Andy Middleton; Harmonica – Randy Singer; Piano – Jim Gasior; Bass – Jamie

Ousley; Drums – Mark Griffi th; Composer – Dave Becker; Arranger – Randy Bernsen

x Incarnations of the Blaring Bluesblinger

TWELVE TONE ROAD TRACK #12Guitar – Randy Bernsen; Keyboards – Ray Lyon; Soprano Sax – Andy Middleton; Bass – Javier

Carrion; Drums – Mark Griffi th; Composer – Dave Becker; Arranger – Randy Bernsen

MIDNIGHT DIAMONDS TRACK # 13Piano – Dave Becker; Composer – Dave Becker

Utmost thanks go to all the marvelously talented musicians taking part in these recordings. Th anks also to Javier Carrion (mixing and mastering), Mike Couzzi (mastering), and Ivan Zervigon (engineering).

Melanie and I have been extremely fortunate to benefi t from a lasting friendship with Prof. Alan Davison and Lynne Davison, who have always been there to inspiringly bring out my best and to get me back on my feet when things have knocked me down.

Th e following individuals also deserve many thanks for contributing to my wellness and devel-opment either before the start of this project or during its various stages (I apologize profusely in advance for what I fear will be the inevitable inadvertent exclusion of the names of pertinent helpers): Tom Helf, Mark Schmuckler, Larry Rosenblum, Scott Schaff er, Gary Stockman, Rick Rosoff , Maria DeAngelis, Andy Middleton, John Rangel, Howard Zucker, Mike Gengos, Bob Segura, David Atlas, Jim Kronauge, Jinnie Kim, Jamshied Sharifi , Gilbert Stork, Tomas Hudlicky, Josie Hudlicky, Ed Hampton, Mike Natchus, Curtis Jones, Rusty Lewis, Angelo Diana, Ward and Elaine Kremer, Pete Trias, Pete Iannacone, Archie Pena, Nicole Yarling, Tim Ravenna, Jonathan Joseph, Frank Richard, Brad Keller, Kenny Ruf, George Garcia, Jackson Bunn, Lorraine Faina, Sam Chiodo, David Nizri, Uzi Nizri, Clay Kremer, John Yarling, Bobby Th omas Jr., Martha Ramey Bernsen, Roger Bernsen, Ducth Bernsen, Lee Bernsen, Brian Fisher, Tom Brown, and Daniela Rosati.

Helpful discussions with the following academicians are gratefully acknowledged: Prof. Raphael Atlas, Prof. Mark Schmuckler, Prof. Lawrence Rosenblum, Prof. Walt Weiskopf, Prof. Gary Campbell, Prof. Andy Middleton, Chancellor Th omas F. George, Prof. Douglas Norton, Prof. Charles Cooper, Prof. Kari Vilonen, Prof. David Chatfi eld, and Prof. Charles Berlin.

Warmest thanks to Len Pace and Florida’s public radio NPR-affi liate WLRN for giving this project’s music and science its world premiere on November 4, 2009. Michael Gerber’s PR eff orts in this regard as well as his and Randy Bernsen’s participation in the broadcast will never be forgotten by this author.

Graphic Arts by Tim Ravenna and Ravenna Design were integral to the success of this endeavor.Th e moral support of Drew and Tammy Cummings throughout the preparation of this work was

invaluable as was advice from Randy DeWitt.Much positivity goes out to Ingrid, Julius, and Felix Pastorius.Th e stalwart friendship of Lisa and Mayer Gattegno, Jeff Malken, Lila Malken, Rich and Julie

Ehrlich, Andy Cohen, Evy and geometrically-evolved Marv Malikow, Guy and Naomi Seligman, Kenny and Erica Beberman, the geometrically-evolved Randy Trotman, Roberto Santos, Michael and Jill Previti, Jason Frybergh, and Dale Stevens is noted in gratitude.

Th anks to Pat Metheny, Jack DeJohnette, and Dave Holland for their incredible days of devoted teaching in December/January of 1981/1982.

My deepest thanks to Dean Lesley Northup and Dean Juan Carlos Espinosa of the Honors College at Florida International University for their interest and support of my upcoming 2010-2011 course

ACKNOWLEDGMENTS xi

on this work entitled “Heady Harmonocosms”. Th anks in advance to the intellectually adventurous FIU Honors College students and all others who may choose to study this material.

Last but certainly not least, copious thanks are directed to Bassim Hamadeh, Carolyn Carter, Brent Hannify, Monica Hui, and their coworkers at Cognella Academic Publishing for trusting in me and for enthusiastically promoting the fi nalization of this project in the form of this book and accompanying CD.

MUSIC IN THE THIRD DIMENSION

“Nature geometrizeth and observeth order in all things”— Sir Th omas Browne

Much of the work in this project concerns the use of tangible, fundamental, and often familiar polyhedral three-dimensional objects as tools to assist in the composition and analysis of tonal music in a variety of genres that exhibit Western harmony. Why the emphasis on

three-dimensional objects? Every house ever built, every molecule ever synthesized, is a careful as-semblage of parts in 3-D space, and so, why should this not also be true for the parts needed to construct tonal music? As an organic chemist, I can vouch for the position that the 3-D structure of matter matters a great deal in the proper functioning of biological processes on the molecular level. Our bodies (and especially our brains) are sophisticated 3-D networks of specialized cell types. Could it be that many forms of music, including songs with complex jazz harmonies, can successfully be shown to possess simple shapes with three-dimensional architecture?

Since childhood, every time I heard the perfect cadence IV-V7-I, something intuitively led me to believe that there must be a basic mathematical explanation to account for the fact that so many people are fond of this classic chord progression. As an undergraduate chemistry student at the University of Rochester from 1979-1983, I had the pleasure of being allowed to take music theory at the Eastman School of Music and the displeasure of fi nishing the course feeling far less than enlightened with re-spect to understanding the mathematical basis of chord progressions in contemporary Western tonal music. Oddly enough, on November 16, 2007, while Dmitri Tymoczko at the annual meeting of the Society for Music Th eory was holding a special introductory session on Geometrical Music Th eory in Baltimore, Maryland, I was working carefully with Javier Carrion during a six hour period as Javier put

CHAPTER I

2 Incarnations of the Blaring Bluesblinger

the fi nishing touches on the mixes for eleven of the tracks of the compositional portion of this project. I composed these pieces between March 2005 and May 2006, and was not familiar with Tymoczko’s fascinating work until the latter half of 2006. Interestingly, it was a small blurb in a fall 2006 issue of the Technology Review entitled “Seeing Music” that alerted me to Tymoczko’s research at Princeton. As an alumnus and former visiting professor at MIT, I regularly receive the Technology Review. I noted with great interest that Tymoczko had published the fi rst work ever on music theory in the long history of the journal Science in July of 2006 and his work relating orbifolds from string theory to music was featured in Time magazine in the winter of 2007. Curiously, the author of that article in Time, science writer Michael Lemonick, had scribed a piece approximately four months earlier in Time magazine entitled “Th e Unraveling of String Th eory” in which Columbia University mathematician Peter Woit, a vocal critic of string theory, is quoted with respect to his misgivings for the approach. As noble and elegant as Tymoczko’s goal is in attempting to create a “map of all chords”, I feel that for most people, its abstract higher dimensional topologies make the identifi cation of important relationships between such harmonically complex entities as pentachords a far less tractable and accessible problem than the three-dimensional methodologies developed in this project. (A recent 2009 paper by Tymoczko entitled “Th ree Conceptions of Musical Distance“ seems to suggest that he is aware that there are currently a number of diff erent approaches aimed at clarifying the concept of distance between chords and that further work in the area is necessary.)

In March of 2005, after roughly ten years of research, I managed to fi nd what to me was an eye-opening music/geometry relationship that, to my knowledge (and to the knowledge of several trust-worthy academicians), had not been previously described and remains to the 2008 day of this writing undisclosed elsewhere. I contacted Mr. Jacob Lasky, Editorial Administrator at Scientifi c American, and attempted to publish these fi ndings in July of 2005. At that time, I had written a musical composition based on this particular tonal geometry entitled “Midnight Diamonds”. Mr. Lasky told me to submit a synopsis of the work to the editor of Scientifi c American. I agreed and also off ered to submit a recording of the corresponding musical piece, but was instructed not to do so. Accordingly, I sent the synopsis—and heard nothing. Perhaps it was wishful thinking to pursue publication of the work in Scientifi c American at that early juncture since that particular magazine is not known for covering work that has not been previously peer-reviewed. However, I explained to the editor in a cover letter accompanying my submission, that prior to submitting the work to Scientifi c American, I had (in mid-June of 2005) presented the fi ndings and musical composition to Professor Raphael Atlas of the Music Th eory Department at Smith College in Northampton, Massachusetts—coincidentally, the same school where Dmitri Tymoczko’s late father, Th omas Tymoczko, had been a mathematically active professor of philosophy. Raphael has recently told me, in what turns out to be an even more striking coincidence, that Dmitri Tymoczko, while growing up in Northampton, interacted enthusiastically with him at Smith College on matters related to their mutual interests in music theory.

I was in Boston to attend a retirement party for my close friend Professor Alan Davison of the MIT Chemistry Department on June 17, 2005, and I had earlier arranged to meet Raphael at his home in Northampton during the same trip. Not surprisingly, there was some serious brainpower at the party, and part of me wanted to discuss the geometry/music fi ndings with the talented group of scientists gathered there. In addition to Alan, an eminent scientist and pioneer in developing highly useful organo-technetium radiopharmaceuticals for heart imaging, many of his MIT col-leagues throughout his career as well as colleagues from all over the world were in attendance, including chemistry Nobel laureates Barry Sharpless and Richard Schrock. Sharpless had been one

MUSIC IN THE THIRD DIMENSION 3

of my teachers at MIT, and although I hadn’t seen him in seventeen years, I was pleased to fi nd that he had not forgotten me. After thoroughly enjoying the lavish feting of Alan (and refraining from inappropriately blabbing about math and music), I headed off to see Raphael in Northampton the next day.

Raphael earned his doctorate in music theory at Yale under the supervision of renowned music theorist David Lewin who subsequently moved to Harvard before his passing in 2003. After listen-ing to my preamble and examining the tonally encoded rhombic dodecahedron that I gave to him, Raphael informed me that he had not seen the fi nding elsewhere. I felt fortunate that a music theorist with Raphael’s credentials was willing to inspect the model. It was also fortunate that I had fi rst met Raphael in the mid 1980’s in Boston at a fi ne dinner prepared by Raphael’s mom. I was a doctoral student in chemistry at MIT at that time and was invited to the dinner by Raphael’s brother David, a friend I had made at the University of Rochester. David had a degree from UR in optics and, in Rochester, had worked for Kodak. He grew up in Boston and later earned a degree in business from the Sloan School at MIT.

In the spring of 2005, just before my visit with Raphael in Northampton, I had phoned two of my grade school buddies, Larry Rosenblum and Mark Schmuckler, both of whom, like myself, have become academicians. As kids, Larry, Mark, Tom Helf, Rick Rosoff and I were all friends and classmates active in music within the Jamesville DeWitt school district in the suburbs east of Syracuse, New York, where I was born and raised. We, under the direction of Ron Nuzzo, played together in our high school jazz band along with Walt Weiskopf, now an accomplished jazz musician, a professor at the Eastman School, and a saxophonist with Steely Dan.

My phone call with Larry had me reminding him of a conversation the two of us had during breakfast one morning in the beautiful mountain town of Julian, California in the summer of 1997. It was then, while I was away from my home in Florida to negotiate off ers from two west coast biotech companies to license some intellectual property I had developed at FIU, that the idea of thinking about music on discrete dodecahedral surfaces fi rst occurred to me. Not knowing whether any 3-D work along these lines had been previously described (although, in eff ect, I had already been investigat-ing this topic in light of the fact that the two-dimensional rendering of the network that I had been exploring since 1995 can be, as we will soon examine, shown to relate to certain three-dimensional twelve-faced objects), I wondered about the prospects of placing one of the twelve tones on each face of a pentagonal dodecahedron in such a way that groups of neighboring tones might give rise to logical chord progressions as a path were followed from face to face. I had fi rst come across the pentagonal dodecahedron during my undergraduate days in Rochester when Professor Leo Paquette of the Ohio State University Chemistry Department delivered a lecture on the synthesis of the hydrocarbon known as dodecahedrane. As one of the so-called Platonic Solids (the other four are the tetrahedron, cube, octahedron, and icosahedron), the pentagonal dodecahedron has received scientifi c attention for thousands of years. It is also worth mentioning that the number twelve shows up conspicuously in all fi ve Platonic Solids (the tetrahedron has twelve rotational symmetries; the cube has twelve edges; the octahedron has twelve edges; the pentagonal dodecahedron has twelve faces; and the icosahedron has twelve vertices).

I proceeded on the phone to tell Larry that an important musical code did indeed exist on the faces of a dodecahedron. However, it was not the Platonic dodecahedron, it was Kepler’s rhombic dodecahedron! In 1619, the luminary scientist Johannes Kepler fi rst published a description of the rhombic dodecahedron in his treatise “Harmonice Mundi” (“Th e Harmony of the World”). Th at


aptly titled opus touches upon a diverse array of topics from polyhedra, to astronomy, to, yes, music. Kepler believed that planetary motions and relative positions had musical/geometrical relationships (sometimes referred to as “Th e Music of the Spheres”). While Kepler’s published musical ideas don’t yield logical connections between such complex harmonic entities as pentachords, his rhombic do-decahedron certainly does. Note that a rhombic dodecahedron of a second kind (a so-called golden rhombic dodecahedron with a ratio of rhombic diagonals equal to the golden ratio instead of the square root of two) was described many years after Kepler’s 1619 publication.

Figure 1

MUSIC IN THE THIRD DIMENSION 5

I encourage the interested reader to construct an origami rhombic dodecahedron. In order to do this, one can cut out the requisite shape from a piece of paper that contains the template shown in Figure 1. Once the scissoring is done, fold the paper away from you along every line that is shared by a rhombus on either side. After the folds have been made, use some adhesive tape to hold in place the three-dimensional structure that is obtained when the paper is curled up on itself somewhat like a ball. In Figure 1, each of the twelve diamond-shaped faces has been assigned one of the twelve tones.

Keep in mind that there are eleven other harmonically equivalent arrangements to that in Figure 1 that could be represented by transposition. Th e tonal arrangement on the origami rhom-bic dodecahedron resulting from Figure 1 (hereafter referred to as the Figure 1-derived rhombic dodecahedron) is one of nearly one million possible distinct tonal arrangements (my calculations lead me to believe that the exact number is 831,600 distinct tonal arrangements). To calculate this, imagine the probability of sequentially placing each one of the twelve tones on the proper face of a rhombic dodecahedron. Th e probability of placing the fi rst tone is 12/12 since all twelve faces of the empty rhombic dodecahedron are equivalent. Th e probability of placing the second tone correctly is 4/11 since four of the remaining eleven empty rhombi share an edge with the rhombus that bears the fi rst placed tone (for the purposes of this analysis, we’ll assume the equivalency of mirror images, although in biochemistry, the absolute confi guration is often extremely important). Th e probability of placing the third tone correctly is 1/10, the fourth is 1/9, the fi fth is 1/8, the sixth is 1/7, the seventh is 1/6, the eighth is 1/5, the ninth is 1/4, the tenth is 1/3, the eleventh is 1/2, and the twelfth is 1/1. Multiplying the probabilities of each of these events together we get four divided by eleven factorial (4/11!) or 1/9,979,200. Th us, the probability of arriving at the exact tonal confi guration of the Figure 1-derived rhombic dodecahedron (or its mirror image) is about one in ten million. Accounting for the other eleven similar arrangements obtained by transposition we get 12/9,979,200 or 1/831,600.

So far, we have not said anything about why the tonal arrangement of the Figure 1-derived rhombic dodecahedron is fundamentally relevant in the context of music theory—let alone the methodology employed to arrive at this particular juxtaposition of the twelve tones. Before we get to those issues, it is important to recognize that the twelve points in space corresponding to the center of each of the twelve rhombic faces of a rhombic dodecahedron also describe the location of the twelve vertices or corners of a properly sized cuboctahedron. A cuboctahedron or truncated cube can be obtained from a cube by cleaving off all eight corners in a specifi c manner. Th e cuboctahedron possesses fourteen faces (six square and eight triangular) and constitutes what is known as the dual of the rhombic dodecahedron (a dual polyhedron is that polyhedron obtained by connecting all face centers of a given polyhedron). Although less widely known, these same twelve points in space also describe the location of the center of each of the twelve edges of a properly sized cube. Moreover, these same twelve points defi ne the edge centers of a properly sized octahedron. It is beyond question that a cube is one of the most familiar and fundamental three-dimensional objects. A cube is one of the fi ve Platonic solids, and the orthogonality of any two adjacent faces of a cube is of obvious architectural signifi cance (the vast majority of houses and buildings in the world possess walls that are perpendicular to the fl oor). Cubes are seen in nature as well, such as in crystals of sodium chloride (table salt) and the mineral pyrite (a compound of iron and sulfur with molecular formula FeS2 that is also known as Fool’s Gold because of its lustrous gold appearance). Th e rhombic dodecahedron can be seen in natural garnet crystals and, concerning the cuboctahedron, researchers at the Scripps Institute in La Jolla have recently discovered the fi rst known protein with a subunit possessing cuboctahedral topology. Any organic chemist will tell you that the

tetrahedron is integral to many carbon-containing molecules. Suffi ce it to say that nature makes use of these polyhedral structures and we will consider these and more such structures in subsequent sections of the book.

A DOZEN QUINARY DENIZENS OF A CUBE

Five is rarely the fi rst number that people think of when contemplating a cube, but it is there in a quintessential way. To my knowledge, prior to the discovery in early 2005 of the tonal relationship within Figure 1 (with its twelve tones occupying the twelve face centers of a

rhombic dodecahedron as well as the twelve edge centers of a cube), music theorists had not described any fundamental tonal geometry in which all twelve tones appear upon a single cube.

In 1998, Douthett and Steinbach published a relationship entitled “Cube Dance” that makes use of four interconnected stretched cubes. Each of the cubes possesses two augmented triads, three major triads, and three minor triads at a given cube’s eight vertices as shown in Figure 2 (edges connect the triads). As a jazz afi cionado, while I like and have nothing against triads and tetrads (that at times seem perfectly called for—even among more complex sonorities), I have long been fascinated with more intricate chords that frequently contain fi ve distinct tones (and with chord progressions that utilize these entities). To be sure, there are certain six-tone chords that I fi nd equally fascinating, but fewer of these are obligatory in tonal music in comparison to the pentatonic counterparts. (To confi rm this, simply ask a competent solo jazz guitarist whether the employment of chords in which all six strings of the guitar are sounding six diff erent representatives from among the twelve tones is more frequent than the use of pentatonic chords.)

Seven and eight-tone sonorities are even less common (with the possible exception of chords comprised of the seven tones of the major scale and its related modes or chords built with all eight tones of a diminished scale). A pentatonic chord is often suffi cient to imply the harmonic intent of a chord comprised of more than fi ve tones. We will later have more to say with respect to the assignment of a scale or mode containing more than fi ve tones to a given pentatonic chord within a progression.

Incidentally, the actual number of hexatonic chord qualities in a given key (irrespective of voicing and with the same bass note as root) is 462, whereas for pentatonic chord qualities, the number, 330, is lower. Th ese numbers can be found in the twelfth row of what is known as Pascal’s Triangle (named

CHAPTER II


after the famous mathematician Blaise Pascal), a versatile mathematical tool displayed in Figure 3. Note that an interior number in any row is the sum of the two numbers above right and above left of that interior number. Pascal’s Triangle fi nds use in a range of applications involving combinatorics. For example, most sophomore organic chemistry students at colleges are taught that the numbers within an appropriate row of Pascal’s Triangle describe the relative peak areas within a given proton signal of a compound’s nuclear magnetic resonance spectrum. To make use of Pascal’s Triangle in music, imagine that you possess a piano with only twelve specifi c keys. Th e twelve keys consist of the seven white keys that comprise the fi rst seven notes of an ascending C major scale commencing on middle C and the fi ve black keys within the octave defi ned by middle C and the C above middle C. How many sounds that include the tone of C could this piano make? Th e answer, 211 (or 2048), can be obtained by calculating the sum of the numbers in the twelfth row of Pascal’s Triangle. Th us, 1 + 11 + 55 + 165 + 330 + 462 + 462 + 330 + 165 + 55 + 11 + 1 = 2048.

Of course, many of these 2048 sonorities would be extremely dissonant and violate tertian harmony by containing three or more consecutive semitones. Nevertheless, of the 2048 sonorities, there are (after eliminating the single one-tone sonority and the eleven two-tone sonorities) 2036 distinct chord qualities possible in any given key (as we see here in the key of C with the tone C as bass note and root) when one uses the defi nition of a chord as three or more of the twelve tones played simultaneously (2036= 211-1-11).

Figure 2

A DOZEN QUINARY DENIZENS OF A CUBE 9

Looking closer at the numbers in the twelfth row of Pascal’s Triangle, we fi nd that there is only one way on this twelve-key piano to play the tone C all by itself (a one-tone sonority), there are eleven ways to play the tone C with one of the other eleven keys (a two-tone sonority), there are fi fty fi ve ways to play the tone C with two of the other eleven keys to constitute a three-tone sonority (m

nC=n!/(n-m)!m! with n=11 and m=2 becomes 11!/9!2!=55), and so forth. By the way, the twelfth row of Pascal’s Triangle also supplies the numbers of possible polytonic scales in a given key. For example, just as there are 330 pentatonic chord qualities in any given key, there are 330 distinct pentatonic ascending (or descending) scales within a single octave in any given key (in the key of C, these scales contain the tone C and four of the eleven remaining tones from the original group of twelve tones). Likewise, there are 462 such hexatonic scales (and 462 such heptatonic scales). Interestingly, within Nicolas Slonimsky’s classic book “A Th esaurus of Musical Scales and Patterns” (which happens to be one of my favorite books given to me as a gift years ago by my close friend, the renowned musician and producer of this project, Randy Bernsen), one fi nds 49 of the 330 possible pentatonic scales and 54 of the 462 possible heptatonic scales. Arnold Schoenberg is quoted on the jacket of this Slonimsky book as follows: “I have looked through your whole book and was very interested to fi nd that you have in all probability organized every possible succession of tones. Th is is an admirable feat of mental gymnastics. But as a

Figure 3


composer, I must believe in inspiration rather than mechanics.” I believe Pascal (who, like Schoenberg, knew a thing or two about putting things in rows) would have taken Schoenberg to task with regard to this issue. I also believe that inspiration and mechanics are not mutually exclusive. It is my sincere hope that the work presented herein will cause others to concur.

Now, without further ado, let’s get back to the signifi cance of the tonal geometry of the Figure 1-derived rhombic dodecahedron and how this relates to quinary aspects of a cube. Examination of the origami rhombic dodecahedron that is constructed from Figure 1 reveals twelve distinct pentatonic sets. Each of these twelve sets is obtained by combining a selected tone with the four tones on faces that share an edge with the face of the selected tone. Performance of this exercise with respect to all twelve tones generates all twelve sets.

To illustrate, if the tone B is selected, we notice that the four other tones that are on faces that share an edge with the rhombic face that bears the tone B are D#, E, F#, and G#. Th ese are the fi ve tones that are visible if one places the Figure 1-derived rhombic dodecahedron on a table such that the face bearing the tone D is fl ush with the surface of the table and the rhombic dodecahedron is viewed by looking straight down upon it from above. It is interesting to note that the tone B is the center rhombus of this set of fi ve rhombuses and that the tone B is also the center tone (and fi fth) of the resulting E major ninth pentad (1, 3, 5, 7, 9 = E, G#, B, D#, F#). Likewise, two of the remaining eleven pentatonic sets constitute the chords A major ninth and B major ninth with the fi fth of A major ninth (E) and the fi fth of B major ninth (F#) occurring at the center rhombus of their respective fi ve-rhombus X-shaped (quincuncial) aggregates.

Th e presence of these three particular major ninth chords (E, A, and B) is, as any competent music theorist would tell you, noteworthy because of their 1, 4, 5 relationship (here seen in the key of E while a similar 1, 4, 5 relationship is present in any one of the eleven equivalent transpositions). In basic music theory, the major triad designated with the Roman numeral I corresponds to the fi rst note of the major scale in a key corresponding to the piece or passage and is referred to as the tonic. Th e IV chord is known as subdominant and the V chord as dominant. Th e I, IV, V triumvirate is a cornerstone of Western harmony. Indeed, many well-known songs in a wide variety of genres are constructed solely of these three bedrock components.

Th ere remain nine encoded pentatonic sets that we have not yet discussed and these also display rational (though not in all cases easily predictable) relationships. Th us, three of these nine sets can be viewed as examples of a minor ninth chord (another inarguably fundamental sonority) and are, specifi cally, C minor ninth (centered about G), F minor ninth (centered about C), and G minor ninth (centered about D). Once again, the 1, 4, 5 relationship is observed, but in this instance, in minor ninths in the key of C. Yet another 1, 4, 5 relationship is evident in three of the remaining six pentatonic sets. Th ese three can each be viewed as a pentatonic chord consisting of a major triad that also possesses a fl at ninth and a fl at thirteenth. Interestingly, one can consider these dissonant chords as hybrids between major and minor triads that are separated by a semitone. For example, inspection of the face of the Figure 1-derived rhombic dodecahedron that bears the tone D# reveals that the four tones on the faces that share an edge with the D# rhombus are B, C, F#, and G. Th ese fi ve notes together (B, C, D#, F#, G) may be viewed as a B major triad (B, D#, F#) with the fl at ninth (C) and fl at thirteenth (G). It may also be viewed as a C minor triad with a major seventh (B) and a sharp eleventh (F#). I like to think of this fascinating yin-yang pentatonic sonority as a major triad sounded with a minor triad in which both triads share the same third. Later, we will discuss bona fi de major and minor triads that emerge from this tonal geometry with fundamental relationships as well as hybrid major/minor tetrads that are present. Th e occurrence of several other important tetrads will also be


noted. In addition to the pentatonic set that, as we have just seen, can be considered to represent a B major triad with a fl at ninth and fl at thirteenth, the same chord with roots of E (from the rhombus bearing G# as the center of the fi ve-rhombus aggregate) and F# (from the rhombus bearing A# as the center of the fi ve-rhombus aggregate) is present to complete this 1, 4, 5 relationship in the key of B.

Th e three remaining pentatonic sets, unlike the three triply occurring chord qualities within the three 1, 4, 5 relationships of the aforementioned nine sets, constitute three distinct chord qualities devoid of an obvious 1, 4, 5 relationship to each other. From the rhombus bearing the tone A, a

Figure 4


pentatonic set that can be classifi ed as a D minor ninth with a major seventh is seen (D, F, A, C#, E). Th e pentatonic set that is generated from the rhombus bearing the tone C# can be classifi ed as that of an F# dominant seventh chord with a sharp ninth (F#7+9). Th e presence of this pentatonic sonority is extremely signifi cant in the context of a perfect cadence (IV to V7 to I) and constitutes another (and perhaps the most historically signifi cant) 1, 4, 5 relationship that here occurs in the key of B when grouped with the previously mentioned B major ninth and E major ninth sonorities. In other words, an elaborated perfect cadence (IV major ninth to V7+9 to I major ninth) in the key of B (E major ninth to F#7+9 to B major ninth) is present on the rhombic dodecahedron derived from Figure 1 by traveling in what one sees is a continuous path from the rhombus that bears the tone B (E major ninth) to that which bears C# (F#7+9) to that which bears F# (B major ninth).

In going directly from the B rhombus to the C# rhombus, the path traverses one of the fourteen vertices of the rhombic dodecahedron, while the path traverses one of the twenty-four edges of the rhombic dodecahedron in going directly from the rhombus that bears C# to that which bears F#. Incidentally, the perfect cadence I put into the piece “Midnight Diamonds” was later deleted by Randy. He wryly said something to the eff ect of “that chord progression has been played to death for the last three hundred years and the piece does not need it”. I completely agreed.

Th e last of the twelve pentatonic sets from the Figure 1-derived rhombic dodecahedron is gleaned by observation of the rhombus assigned to the tone F (an infamous tritone interval from the B tonality of the encoded perfect cadence). Th is set consists of the tones C, D, F, G#, and A. My expectation is that very few people would regard any chord comprised of these fi ve tones as consonant (regardless of the voicing). However, in what seems to constitute a phenomenon in stark contrast to Tymoczko’s assertion that neighboring chords of a consonant chord should also be consonant, this dissonant pentatonic set directly neighbors a highly consonant A major ninth on the Figure 1-derived rhombic dodecahedron as a direct path exists from the F rhombus to the E rhombus via traversing the vertex shared by these two rhombic faces.

Similarly, the aforementioned dissonant pentatonic set C, E, F, G#, B (from the rhombus that bears G# as the center of the respective fi ve-rhombus aggregate) is an even more direct neighbor to the consonant A major ninth chord (and contains three tones in common with an A major ninth chord whereas the C, D, F, G#, A set only has two such common tones) as a direct path exists from the G# face to the E face by traversing the edge shared by these two faces (the sharing of an edge between two faces encoding two neighboring sets guarantees that the two sets possess at least two common tones). Despite their dissonance, judiciously chosen voicings of the tense C, E, F, G#, B and C, D, F, G#, A sonorities do release (resolve) nicely into the consonant A major ninth chord and these progressions occur several times in the piece “Midnight Diamonds”.

Th e C, D, F, G#, A sonority has been classifi ed as a G# thirteenth with a fl at fi fth and fl at ninth that lacks a seventh and eleventh while, within the piece, it is played over a bass note of A (the fl at ninth). Unquestionably, thirteenth chords with fl at fi fths and fl at ninths are among the important tension producing sonorities of Western music, and this jazz fan enjoys their complex nature and their subsequent release into more consonant neighboring chords. Th e fi ve tones of this chord (C, D, F, G#, and A) are all part of the octatonic diminished scale containing the tones (C, D, D#, F, F#, G#, A, and B). Th e occurrence of these particularly dissonant pentatonic sets among the other indisputably coherent pentatonic sets on the Figure 1-derived rhombic dodecahedron suggests that this dissonance is of fundamental signifi cance. It is my opinion that nature off ers some very compelling arguments to rationalize the existence of dissonance in the midst of consonance. Figure 4 summarizes the twelve pentatonic sets that are obtained from the rhombic dodecahedron corresponding to Figure 1.


It can be seen in Figure 5a that we can carefully assign (according to the rhombic dodecahedron obtained from Figure 1) one of the twelve tones to each of the twelve edge centers of a cube to arrive at the depicted tonal geometry. One could think of the edges of a cube as musical strings. Or, if you like to extrapolate, the vibrating network of strings in the form of cube edges can also be imagined as a vibrating network of membranes in the form of rhombic dodecahedral faces. It is not so far-fetched to envision that if the tension on each cube edge or string were to correspond to the musical tone to which it is assigned (with all twelve tones within one octave), then the structural integrity of the cube would crucially depend on the existence of tensile balance.

Indeed, the lauded architect and geometer, R. Buckminster Fuller (whose work inspired the con-struction of the “Spaceship Earth” geodesic dome at Epcot Center in the Disney World theme park in Orlando), was enamored with the balanced nature of the cuboctahedron and elaborated ideas along these lines in his so-called tensegrity considerations of cuboctahedral vector equilibrium. We have already seen that the twelve edge centers of a cube can also describe the twelve vertices of a cuboctahe-dron, and thus, it is reasonable to look for signs of balance that may carry over from the cuboctahedron to the cube.

Figure 5a


Fuller’s work is applicable to many fi elds and it is worth noting that he is held in especially high esteem by chemists and biologists. For example, in chemistry, fullerenes that bear his name are a rela-tively new class of molecules comprised of carbon with spheroid and ellipsoidal structures unlike more familiar allotropes of carbon such as graphite and diamond. Buckminsterfullerene is composed of sixty carbon atoms with each carbon at each of the sixty vertices of a truncated icosahedron. A truncated icosahedron arises by slicing off each of the twelve vertices of an icosahedron in a particular manner. Th e shape of a truncated icosahedron is that of a soccer ball with twenty hexagonal faces and twelve pentagonal faces. In biology, viruses are known to possess protein shells termed capsids (consisting of subunits referred to as capsomeres). Th e overall topology of these capsids is frequently icosahedral and capsids may have layers not unlike the layers of an onion. Fuller’s studies, to the surprise of virologists, led him to accurately predict many geometrical structures within the family of capsid coats.

In light of the preceding discussion, it is interesting to note that we can fi nd some inherent elements of balance in the cube depicted in Figure 5a. Th us, by assigning each tone a numerical value (such that C=1, C#=2, D=3, D#=4, E=5, F=6, F#=7, G=8, G#=9, A=10, A3=11, and B=12), we fi nd that the sum of the four edges constituting three of the six square faces of this cube is identical. For the sake of clarity, let us focus on one of the six square faces of the Figure 5a cube; namely, that with tones of C#, A, D, and A# (the top face as shown). Notice that the sum for this face is 26 (C#=2, A=10, D=3, A#=11, and accordingly, 2+10+3+11=26). Interestingly, the sum for the bottom face (B, G#, C, D#)

Figure 5b


is 26 and the sum for the back face (C#, E, B, F#) is also 26. Th e sum of the left face edges is 30 and the sum of the right face edges also is 30. Th e front face edges add up to a sum of 18. When these six values for the sums of each of the six square faces of the cube are added together, we must, regardless of the arrangement of the twelve tones, get 156 because 2(1+2+3+4+5+6+7+8+9+10+11+12)=156. Th e factor of 2 in the preceding expression arises because precisely two cube faces share any edge. Th us, 26+26+26+30+30+18 does indeed add up to 156. Note that 156 divided by 6 is 26. In other words, according to this methodology, the average cube face tension (with cube face tension defi ned as the sum of the four tone values that comprise the four edges of any given face) of a cube assigned one of the twelve tones per edge, is 26.

Of the six square faces of the cube shown in Figure 5a, three such faces (50%) possess sums of edges that add up to precisely 26. Th is (along with the symmetrical occurrence of those faces that harbor equivalent sums) is indicative of the presence of a substantial degree of tensile balance. Th e twelve pentatonic sets that we previously discussed as twelve X-shaped fi ve-rhombus aggregates on the rhombic dodecahedron can be viewed as fi ve-edge aggregates on the cube that resemble sawhorses (Figure 5b). To see such a sawhorse on the cube, select a tone and identify the edge to which that tone is assigned. Th at particular edge touches exactly four other edges. Th e pentatonic set consists of the selected tone and the four tones of the four edges that touch the selected edge.

Acknowledgment of fundamental non-pentatonic harmonic aspects of the Figure 1-derived rhom-bic dodecahedron is also important. Let us focus our attention on the fourteen vertices of the rhombic dodecahedron. Eight of these fourteen vertices are in contact with groups of three edges (and thus, each of these eight is shared by three particular faces) while six of the fourteen are in contact with groups of four edges (and thus, each of these six is shared by four particular faces). Th e aforementioned eight vertices possess a spatial relationship corresponding to the eight vertices of a cube (while the other six vertices defi ne the vertices of an octahedron). Each one of these eight vertices defi nes a distinct triad comprised of the three tones that are assigned to the faces that share the selected vertex.

It is blatantly evident that the tonal arrangement in the Figure 1-derived rhombic dodecahedron not only accounts for a fundamental collection of pentatonic sets (vide supra), but also accounts for a historically cherished collection of tritonic sets. Accordingly, we see that one of the eight three-faced (or three-edged) vertices of the Figure 1-derived rhombic dodecahedron shares the faces assigned the tones of an E major triad (E, G#, B). Likewise, three of the remaining seven such vertices correspond to the following major triads: an F# major triad (F#, A#, C#), an A major triad (A, C#, E), and a B major triad (B, D#, F#). Th e other four such vertices defi ne four minor triads; namely, a C minor triad (C, D#, G), a D minor triad (D, F, A), an F minor triad (F, G#, C), and a G minor triad (G, A#, D).

Th erefore, we now have four neighboring major triads defi ning two 1, 4, 5 relationships (in the key of E and B) that, relative to E, include the tonic (E major triad as the I chord), the supertonic (F# major triad as the II or V of V chord), the subdominant (A major triad as the IV chord), and the dominant (B major triad as the V chord) while we also have four neighboring minor triads defi ning two 1, 4, 5 relationships (in the key of C and G) that, relative to C, include the tonic (C minor triad as the i chord), the supertonic (D minor triad as the ii or v of v chord), the subdominant (F minor triad as the iv chord), and the dominant (G minor triad as the v chord). It is fascinating that these four major triads can be viewed as occurring on the top four vertices of a cube and the four minor triads at the bottom four vertices. Th e proximity of the D minor triad to the A major triad also makes sense in the context of a iv to I progression in the key of A. Also interesting is that the major triads are closer to the point of maximal tensile balance (the center of the central face of the three Figure 5a cube faces


with edges that sum up to 26) than are the minor triads. Th ese sets, with their 1, 4, 5 and 1, 2, 4, 5 relationships are summarized in Figure 6.

With regard to encoded tetratonic sets, there are six of these to be found at the six octahedrally oriented vertices of the rhombic dodecahedron. For the Figure 1-derived rhombic dodecahedron, at one of these vertices we fi nd the important set C#, E, F#, B (containing the tonic, supertonic, subdominant, and dominant tones in the key of B) that can be classifi ed as an F#7sus4 chord. If one orients that vertex such that it is at the top, then the bottom vertex encodes the tetratonic set C, D, F, G that can be classifi ed as a G7sus4 chord. While maintaining the top and bottom relationship of these two vertices, one can orient the rhombic dodecahedron such that the front vertex encodes the dissonant tetratonic set D#, F#, G, A#. Th is set can be interpreted as another yin-yang hybrid sonority containing all elements of both a D# major triad and a D# minor triad. Th e back vertex encodes the dissonant set E, F, G#, A that can be interpreted as an F major seventh chord with a sharp ninth that is devoid of a fi fth. Th e other two octahedrally disposed vertices encode another yin-yang major/minor hybrid set comprised of the tones C, D#, G#, B (a blend of a G# major and G# minor triad), and another set comprised of the tones C#, D, A, A# (an A# major seventh chord with a sharp ninth that is devoid of a fi fth).

I feel obliged to interject at this point a pertinent magazine article that arrived in the mail dur-ing the course of writing this manuscript. Th e article is entitled “Th e Psychoacoustics of Harmony Perception” by Norman D. Cook and Takefuni Hayashi and appears in the July/August 2008 issue of American Scientist. Th e authors present fi ndings on the consonance or dissonance of particular triads (but do not touch upon chord progressions) and they extend their analysis to tetrads while mentioning barbershop quartets. Two quotes that I consider especially relevant to the present discussion are the following: “accomplished [barbershop] quartets can lend the impression that there are actually fi ve

Figure 6


voices” and “On rare occasions-such as in barbershop quartet singing-the upper partials may reinforce each other to such an extent that they are almost as strong as the fundamentals, and this creates the much-coveted illusion of a ‘fi fth voice’.” I am also a proponent of fi ve-part harmony. Daniel Levitin, in his popular book, “Th is Is Your Brain on Music”, includes a discussion of the fi fth-voice phenomenon known as the quintina.

Chemists and artists that draw develop skills that enable them to represent three-dimensional objects on a fl at (two-dimensional) surface. Whether it is a Newman projection of a particular conformation of an organic molecule such as ethane or the uncanny accuracy of Canaletto’s seemingly photographic paintings of Venice, geometry dictates the proper positions in the plane to capture aspects of the three-dimensional structure. We can represent the twelve face centers of a rhombic dodecahedron in two dimensions by utilizing the presented network (Figure 7) known to mathematicians as a cuboctahedral graph. In addition to the face centers of a rhombic dodecahedron, the twelve points in a cuboctahedral graph also represent the twelve vertices of a cuboctahedron (and the twelve edge centers of both a cube and an octahedron). Th is type of polyhedral graph is sometimes referred to as a Schlegel diagram. One can see the fourteen faces of a cuboctahedron within a cuboctahedral graph in the eight triangles and six quadrilaterals (the large quadrilateral in Figure 7 that establishes the perimeter is one of the six quadrilaterals).

A cuboctahedral graph can also be described (by a branch of mathematics known as graph theory) as an example of a quartic planar graph on twelve vertices. Each vertex of a regular quartic

Figure 7

graph is joined to four other vertices. A planar graph is a graph that can be drawn on a fl at surface such that it contains no crossing edges. I think it is fair to say that the term “cuboctahedral graph” has not hit the mainstream in the music theory literature although I suspect that it soon will become more familiar. As a matter of fact, the very fi rst diagram in Jeff rey Johnson’s 1997 book “Graph Th eoretical Models of Abstract Musical Transformation” is a cuboctahedral graph although it looks somewhat diff erent in that it is constructed via the intersection of four circles as shown in Figure 8. While this diagram occurs several times in the 1997 book, Johnson never refers to it as a cuboctahedral graph.

Johnson, a professor of music at the University of Bridgeport in Connecticut and a graduate of the Eastman School, applies the cuboctahedral graph to construct a brief three-bar atonal quartet by assigning one of four instruments (or voices) to each of the four circles in the diagram in Figure 8 and having them each play the six tones on their respective circles starting at prescribed vertices and proceeding in either clockwise or counterclockwise directions (the rhythms for each voice are at the discretion of the composer). At no point in the book does Johnson use a cuboctahedral graph for generating music with fi ve-part harmony and the code for the tonal geometry of the Figure 1-derived rhombic dodecahedron in the form of a cuboctahedral graph (Figure 9) does not appear (in any key) among the various arrangements of the twelve tones on the cuboctahedral graphs in the book nor does it appear in any other form.

After coming across the tonal geometry of the Figure 1-derived rhombic dodecahedron in early 2005, I set out to determine if this tonal geometry had already been reported and it was then that I first found Johnson’s book. Some years prior to that (circa 1996), the only other twentieth century work on music and graph theory I had found was an article by Prof.

Figure 8


James Bennighof of the Department of Music Theory at Baylor University entitled “Set Class Aggregate Structuring, Graph Theory and some Compositional Strategies” (Journal of Music Theory, 1987, vol 31, pages 51-98). Footnote 2 of the manuscript expresses gratitude to David Lewin (Raphael Atlas’ mentor) for his comments on the work. Around the same time that this manuscript appeared, Lewin and Lewin’s student, Henry Klumpenhouwer, were making strides in musical networks. Cuboctahedral graphs are absent from the 1987 Bennighof article and there is no discussion of any familiar convex polyhedra since the graphs are all drawn with crossing edges. As in Johnson’s work, Bennighof describes no relationships between the depicted graphs and tonal music.

Long before the work of Bennighof and Johnson, another music theorist had made signifi cant head-way in terms of music and geometry. Th at music theorist, the great Swiss mathematician, Leonhard Euler, is also considered to be the founder of graph theory. Within the span of a few years from 1736 to 1739, Euler made landmark contributions to the foundations of graph theory and to graph theoretical approaches to harmony.

In 1736, Euler determined that it was not possible to visit all locations on either side of the seven bridges in the Prussian town of Konigsberg (now in Russia) via a path that returned to the point of departure without crossing one of the bridges twice. Euler’s proof has come to be known as “Th e Seven Bridges of Konigsberg” and has spawned many interesting mathematical “traveling salesman” type problems. Clearly having broken new mathematical ground in 1736, Euler set his formidable neuronal

Figure 9


pathways on music. In 1739, he published a treatise on music theory entitled “Tentamen Novae Th eoriae Musicae” (“Essay on a New Th eory of Music”) that contains the tone network shown in Figure 10. Th is network can be shown with a repeating two-dimensional geometry now referred to as the “tonnetz” and can be easily interpreted to provide useful information on the harmonic connectivity of major and minor triads. Th e same previously noted 1, 2, 4, 5 relationships between major and minor triads encoded upon the Figure 1-derived rhombic dodecahedron can be seen to occur within neighboring triads of the tonnetz. Th e hybrid major/minor tetrads previously noted on the Figure 1-derived rhombic dodecahedron are also easily discerned in the tonnetz with quadrilateral perimeters.

Figure 10


However, as we will later see, the tonnetz does not account for the geometric aggregate regularity of all twelve pentatonic sets of the Figure 1-derived rhombic dodecahedron.

Examination of the current music theory literature reveals a striking dearth of investigations con-cerning the generation of pentachord maps. Yet, in my experience, it is precisely with pentachords that the most compelling relationships emerge. Th us far, while Euler’s two-dimensional tonnetz provides guidance with triads and while such music theorists as Douthett, Steinbach, and Tymoczko have dealt with maps of triads and tetrads, pentads are noticeably absent. Tymoczko, in his 2006 article in Science, has briefl y indicated his contention that the most consonant pentad is a dominant ninth chord and that the next most consonant pentad is a major ninth chord. I would argue that the major ninth chord is more consonant than a dominant ninth chord in view of the fact that all fi ve of the tones of a major ninth chord are within the major scale corresponding to the root.

Tymoczko’s orbifold ideas put forth that the most consonant examples of chords are chords with tones that divide the octave nearly evenly. He further posits that consonant chords that connect well to each other lie toward the center of the orbifold, with dissonant chords at the orbifold’s periphery. Th is would seem to beg the question: “What is considered to be the boundary between an orbifold’s center and its periphery?” With pentachords, the invoked orbifold is in an intangible fi ve-dimensional space. Regardless of the validity of this assertion, Tymoczko has as of this writing published little (if any) work on the connectivity of pentachords.

While, as Tymoczko and coworkers claim, the tonal geometry of earlier models such as the tonnetz and “Cube Dance” may very well be contained in his orbifold map of triads, I have serious reservations about whether they can convincingly demonstrate that the tonal geometry of the Figure 1-derived rhombic dodecahedron is contained toward the “center” of a fi ve-dimensional orbifold since, as we have observed, some of the pentatonic sets are undeniably dissonant (although, of course, not as dis-sonant as sets with tone clusters containing three or more consecutive semitones) yet directly neighbor and resolve well to extremely consonant pentachords on the three-dimensional object. We will later consider logical musical passages that obey tonal geometries in which dissonant sonorities with tones that divide the octave highly unevenly exist as direct neighbors to highly consonant sonorities with tones that divide the octave as nearly evenly as possible.

Let us return to the dissection of a perfect cadence while taking into consideration the tonal ge-ometry of the Figure 1-derived rhombic dodecahedron. Interestingly, in the key of B, the face centers giving rise to the IV major ninth pentachord (the subdominant E major ninth from the B face) and V7+9 pentachord (the dominant F#7+9 from the C# face) are equidistant from that of the I major ninth chord (the tonic B major ninth from the F# face). Th e presence of the sharp ninth in the F#7+9 chord is clearly not part of a traditional perfect cadence (although it can work well as an embellishing passing tone), but the typical exclusion of this tone is perhaps refl ected in the fact that the tone of A is clearly the farthest tone of the fi ve tones of this chord from the vertex that shares the three faces bearing the three tones that constitute the tonic B major triad. If anything, one would expect the natural ninth to be harmonically more appropriate for the dominant seventh chord of an elaborated perfect cadence, and we will later address an alternative reading of the code on the Figure 1-derived rhombic dodecahedron to account for this expectation.

A musician’s intuition would also lead one to expect that a dominant seventh chord (V7) is closer to the tonic triad (I) than the subdominant triad (IV). I would argue that the tonal geometry of the Figure 1-derived rhombic dodecahedron supports this expectation although I believe this notion can be better demonstrated with an often-used, already mentioned, variant of the V7 chord—a V7sus4


chord. Th us, ordered movement around a specifi c set of three of the four vertices of the rhombic face assigned to the tone B (those three vertices closest to the previously discussed point of maximal tensile balance on the corresponding cube) gives us the following highly viable I to IV to V7sus4 to I chord progression: B to E to F#7sus4 to B (a slightly altered and commonly employed alternative to a perfect cadence). In this case, the distance from what we can call the tonic vertex (that which shares the three faces bearing the tones of a B major triad) to the vertex sharing the three faces bearing the tones of an E major triad (the subdominant vertex) is in fact greater than the distance from the tonic vertex to the vertex sharing the four faces bearing the tones of an F#7sus4 tetrad (the dominant vertex). We see that one can fi nd, on a single three-dimensional object, logical code for chord progressions that consist of chords comprised of three, four, and fi ve tones.

Earlier, I had described a 2005 telephone conversation with my grade school friend Larry Rosenblum (a professor of psychology at the University of California, Riverside). Subsequent to that discussion, Larry suggested that I call another previously mentioned Syracuse friend, Mark Schmuckler (a professor of psychology at the University of Toronto). Mark has long been interested in the psychology of music and obtained his doctorate at Cornell University under the supervision of Carol Krumhansl whose research interests include psychological and mathematical aspects of harmony perception. Krumhansl earned her doctorate at Stanford University with Roger Shepard and Shepard’s 1982 paper on the modifi cation of the Eulerian/Riemannian tonnetz into a double helix is cited in Tymoczko’s 2006 Science paper.

I called Mark in the spring of 2005 and he was extremely helpful in confi rming that he had not previously seen the fi ndings I described to him. His comments prompted me to think harder about the circle of fi fths and also alerted me to the fact that while the tonal geometry of the Figure 1-derived rhombic dodecahedron is indeed distinct from that of the tonnetz, some similarities between the two geometries can be found. For example, Mark pointed out that one can inscribe the three major ninth and the three minor ninth pentads from the Figure 1-derived rhombic dodecahedron inside

Figure 11


of congruent trapezoids on the tonnetz as shown in Figure 11. Moreover, a congruent trapezoid also inscribes both the F#7+9 pentad and the G#13 fl at fi fth, fl at ninth chord (devoid of the seventh and the eleventh) comprised of the tones C, D, F, G#, and A. However, in what clearly constitutes a de-parture from the tonnetz geometry, the D minor ninth major seventh pentad (C#, D, E, F, A) and the three pentads that constitute major triads accompanied by the respective fl at ninth and fl at thirteenth cannot be inscribed in a congruent trapezoid. Th ese pentads can be inscribed from the tonnetz lattice in rectangles as seen in Figure 12. Remember that all twelve of the aforementioned pentatonic sets from the Figure 1-derived rhombic dodecahedron possess the same geometry in terms of their identical fi ve-rhombus X-shaped aggregates. Th e fact that two diff erent quadrilaterals are necessary to inscribe all twelve of these pentatonic sets within the tonnetz whereas all twelve such sets possess the same geometry on the Figure 1-derived rhombic dodecahedron is demonstrative of the nonequivalence of these two tonal arrays.

Another obvious diff erence between the two is that the positions of the face centers of the four tones of the F#7sus4 tetrad (B, C#, E, F#—the tonic, supertonic, sub-dominant, and dominant tones in B major) on the Figure 1-derived rhombic dodecahedron constitute a square whereas these four tones in the tonnetz have a collinear confi guration (as shown in Figure 13). We will later examine yet another important diff erence between these two tonal geometries with a special emphasis on symmetry considerations.

It is now time to address the issue of describing the methodology used to arrive at the tonal ge-ometry of the Figure 1-derived rhombic dodecahedron. In March of 2005, I constructed an origami rhombic dodecahedron and proceeded to install two pentachords upon it. Th e two pentachords that I selected were C minor ninth (C, D, D#, G, A#) and G minor ninth (D, F, G, A, A#). Th ese two were chosen because I happen to be especially fond of this particular two-chord vamp. Th e two selected pentachords possess three common tones; namely, D, G, and A#.

Th e fact that three common tones must be shared by two pentatonic sets on a rhombic dodecahe-dron wherein such sets consist of tones within the two X-shaped fi ve-rhombus aggregates encoded by

Figure 12


two rhombic faces with a common edge (when every face receives one tone and no tone can be used more than once) and the fact that no other confi guration produces two pentatonic sets with three com-mon tones means that two of these three common tones must necessarily occur on adjacent rhombic faces that share an edge and that each of these two tones will constitute the center rhombus within the overall X-like shape of each of their respective fi ve-rhombus aggregates. Th e remaining common tone must occur on a rhombic face that shares one edge with the face containing one of the two central rhombic common tones and another edge with the face containing the other central rhombic common tone.

In view of the fact that both G and D respectively constitute the fi fth and central tone of the chords C minor ninth and G minor ninth, it seemed logical to assign these two common tones to the center rhombus of both of their corresponding fi ve-rhombus aggregates. With the preceding stipulation con-cerning the selection of G and D as the central common tones, we will see (in the next chapter) that there are four possible confi gurations of the seven tones necessary to spell out the C minor ninth to G minor ninth chord progression on a rhombic dodecahedron remembering that, in this methodology, all twelve tones are eventually to be used one per face. Without limiting the two central common tones to G and D, there are twelve possible confi gurations. I arbitrarily selected one of the four knowing full well that some tones would most likely need to be shuffl ed later if unacceptable dissonances involving any pentatonic set with three or more consecutive semitones in any possible voicing arose (tone clusters).

Upon selecting the initial confi guration for the C minor ninth to G minor ninth chord progression, the remaining fi ve tones (C#, E, F#, G#, and B) needed to be somehow assigned to the remaining fi ve vacant rhombic faces of the rhombic dodecahedron. Th ere are a total of 120 (5!) diff erent ways to accomplish this without regard for the avoidance of the aforementioned unacceptably dissonant tone clusters. Th us, in attempting to place the remaining fi ve tones, a tinkerer’s favorite procedure

Figure 13


involving trial and error was pursued. To my utter astonishment, after a lengthy series of such trials and errors, the culmination of years of inquiry took shape in the form of the tonal geometry of the Figure 1-derived rhombic dodecahedron that had been at long last tenaciously dislodged from the determined clutch of the powerfully obfuscating polyhedral keeper of the secret…I did not sleep a wink that night while contemplating the inherent beauty of the object that at times I thought would never materialize. Little did I know how much more exploring there was to be done…

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