“There is geometry in the humming of the strings, there is music in the spacing of the spheres.” — Pythagoras
Counting, rhythm, scales, intervals, patterns, symbols, harmonies, time signatures, overtones, tone, pitch. The notations of composers and sounds made by musicians are connected to mathematics. The next time you hear or play classical, rock, folk, religious, ceremonial, jazz, opera, pop, or contemporary types of music, think of what mathematics and music have in common and how mathematics is used to create the music you enjoy.
With apologies to the musical Grease, mathematics and music go together like rama lama lama ke ding a de dinga a dong. You need to look no further than Plusto see how the links between the two have fascinated researchers for centuries — see all Plus articles tagged with the mathematics and music tag.
Their article Geometrical Music Theory, published in the April 18 edition of Science, outlines their theory that musical operations, such as transpositions, can be expressed as symmetries of n-dimensional space.
They categorise sequences such as chords, rhythms and scales into mathematical “families”. The families can be represented by points on the complex plane, and different types of categorisation produce different geometrical spaces. The authors argue that through this method researchers will be able to analyse music more deeply and understand how music has changed over time. They also say that the theory will allow the comparison of many kinds of Western music, although only some non-Western styles. This is because the theory is based on concepts such as the “chord”, which are present in nearly all Western styles, but not all non-Western styles.
The basis of geometrical music theory is that it provides a unified mathematical framework for musical events that are described differently depending on the scenario, but are fundamentally the same. For example, a “C” followed by the “E” and “G” above it may be described as a “C major chord,” “an ascending C major arpeggio,” “a major chord” and the list goes on. The authors describe five different methods of categorising such collections of notes. These are called “OPTIC symmetries,” with each letter of OPTIC representing a different categorisation method that may, for example, look at what octaves the notes are in, their order of play, or how many times each note is played. Each categorisation focuses on one aspect of the music, ignoring the others. The five symmetries can then be combined together to produce different musical concepts, some familiar, some new. Three-note chords end up on a triangular doughnut while other chord types live on the surface of a cone.
Tymoczko believes that their theory can be used to investigate the differences between musical styles. “Our methods are not so great at distinguishing Aerosmith from The Rolling Stones,” he said. “But they might allow you to visualise some of the differences between John Lennon and Paul McCartney. And they certainly help you understand more deeply how classical music relates to rock or is different from atonal music.”
The authors even hope that through their work, new musical instruments may be developed. “You could create new kinds of musical instruments or new kinds of toys,” said Tymoczko. He also envisaged new visual shows that could accompany the music. “Imagine going to a classical music concert where the music was being translated visually.”
So next time you go see a visually spectacular Madonna or Kylie show, look at the big screens — you might learn some maths!
The new shape of music: Music has its own geometry, researchers find
The figure shows how geometrical music theory represents four-note chord-types — the collections of notes form a tetrahedron, with the colors indicating the spacing between the individual notes in a sequence. In the blue spheres, the notes are clustered, in the warmer colors, they are farther apart. The red ball at the top of the pyramid is the diminished seventh chord, a popular 19th-century chord. Near it are all the most familiar chords of Western music. Credit: Dmitri Tymoczko, Princeton University
The connection between music and mathematics has fascinated scholars for centuries. More than 200 years ago Pythagoras reportedly discovered that pleasing musical intervals could be described using simple ratios.
And the so-called musica universalis or “music of the spheres” emerged in the Middle Ages as the philosophical idea that the proportions in the movements of the celestial bodies — the sun, moon and planets — could be viewed as a form of music, inaudible but perfectly harmonious.
Now, three music professors – Clifton Callender at Florida State University, Ian Quinn at Yale University and Dmitri Tymoczko at Princeton University — have devised a new way of analyzing and categorizing music that takes advantage of the deep, complex mathematics they see enmeshed in its very fabric.
Writing in the April 18 issue of Science, the trio has outlined a method called “geometrical music theory” that translates the language of musical theory into that of contemporary geometry. They take sequences of notes, like chords, rhythms and scales, and categorize them so they can be grouped into “families.” They have found a way to assign mathematical structure to these families, so they can then be represented by points in complex geometrical spaces, much the way “x” and “y” coordinates, in the simpler system of high school algebra, correspond to points on a two-dimensional plane.
Different types of categorization produce different geometrical spaces, and reflect the different ways in which musicians over the centuries have understood music. This achievement, they expect, will allow researchers to analyze and understand music in much deeper and more satisfying ways.
The work represents a significant departure from other attempts to quantify music, according to Rachel Wells Hall of the Department of Mathematics and Computer Science at St. Joseph’s University in Philadelphia. In an accompanying essay, she writes that their effort, “stands out both for the breadth of its musical implications and the depth of its mathematical content.”
The method, according to its authors, allows them to analyze and compare many kinds of Western (and perhaps some non-Western) music. (The method focuses on Western-style music because concepts like “chord” are not universal in all styles.) It also incorporates many past schemes by music theorists to render music into mathematical form.
“The music of the spheres isn’t really a metaphor — some musical spaces really are spheres,” said Tymoczko, an assistant professor of music at Princeton. “The whole point of making these geometric spaces is that, at the end of the day, it helps you understand music better. Having a powerful set of tools for conceptualizing music allows you to do all sorts of things you hadn’t done before.”
Like what?
“You could create new kinds of musical instruments or new kinds of toys,” he said. “You could create new kinds of visualization tools — imagine going to a classical music concert where the music was being translated visually. We could change the way we educate musicians. There are lots of practical consequences that could follow from these ideas.”
“But to me,” Tymoczko added, “the most satisfying aspect of this research is that we can now see that there is a logical structure linking many, many different musical concepts. To some extent, we can represent the history of music as a long process of exploring different symmetries and different geometries.”
Understanding music, the authors write, is a process of discarding information. For instance, suppose a musician plays middle “C” on a piano, followed by the note “E” above that and the note “G” above that. Musicians have many different terms to describe this sequence of events, such as “an ascending C major arpeggio,” “a C major chord,” or “a major chord.” The authors provide a unified mathematical framework for relating these different descriptions of the same musical event.
The trio describes five different ways of categorizing collections of notes that are similar, but not identical. They refer to these musical resemblances as the “OPTIC symmetries,” with each letter of the word “OPTIC” representing a different way of ignoring musical information — for instance, what octave the notes are in, their order, or how many times each note is repeated. The authors show that five symmetries can be combined with each other to produce a cornucopia of different musical concepts, some of which are familiar and some of which are novel.
In this way, the musicians are able to reduce musical works to their mathematical essence.
Once notes are translated into numbers and then translated again into the language of geometry the result is a rich menagerie of geometrical spaces, each inhabited by a different species of geometrical object. After all the mathematics is done, three-note chords end up on a triangular donut while chord types perch on the surface of a cone.
The broad effort follows upon earlier work by Tymoczko in which he developed geometric models for selected musical objects.
The method could help answer whether there are new scales and chords that exist but have yet to be discovered.
“Have Western composers already discovered the essential and most important musical objects?” Tymoczko asked. “If so, then Western music is more than just an arbitrary set of conventions. It may be that the basic objects of Western music are fantastically special, in which case it would be quite difficult to find alternatives to broadly traditional methods of musical organization.”
The tools for analysis also offer the exciting possibility of investigating the differences between musical styles.
“Our methods are not so great at distinguishing Aerosmith from the Rolling Stones,” Tymoczko said. “But they might allow you to visualize some of the differences between John Lennon and Paul McCartney. And they certainly help you understand more deeply how classical music relates to rock or is different from atonal music.”
The deep connection between music and mathematics was recognized at least as early as the time of Pythagoras. Now, Ian Quinn, an assistant professor in Yale’s music department and its cognitive science program, and his colleagues have devised a new mathematical means of understanding music. This “geometrical music theory” can translate the language of music theory into that of contemporary geometry and create visual representations of music’s underlying mathematical structure.
In the April 18 issue of Science, they describe five ways (“symmetries”) of categorizing groups of notes that are similar but not identical: the same note in different octaves, or the same group of notes in a different order. Then they show how these symmetries can be combined to map musical works in coordinate space where, for instance, two-note chords take the shape of a Mobius strip, three-note chord types take the shape of a three-dimensional cone, and four-note chord types somewhat resemble a pyramid.
“We can put any music into the model,” Quinn says, “and visualize the structure behind similarities and differences among musical styles—why Chopin, for instance, sounds different from Mozart.” Or Lennon from McCartney.
The translation of music theoretical terms into precise geometrical language provides a framework for investigating contemporary music-theoretical topics, Quinn says. It can also be useful in analysis, composition, pedagogy, and even the design of new kinds of instruments. Adds Quinn, “My students have used the models to write in the styles of various composers. Somewhat to my surprise these complex topics are fairly easily taught.”
Geometry shapes sound of music, FSU professor says
Through the ages, the sound of music in myriad incarnations has captivated human beings and made them sing along, and as scholars have suspected for centuries, the mysterious force that shapes the melodies that catch the ear and lead the voice is none other than math.
The space of three-note chord types is a cone. Numbers refer to pitch classes, with 0 = C, 1 = C#, etc. Points represent equivalence classes of transpositionally related chords. Thus, (C, D, E) and (D, E, F#) are both instances of 024. (Image made with Dmitri Tymoczko’s “ChordGeometries” program available here.)
It’s geometry, to be more precise, and now, a trio of 21st-century music professors from Florida State University, Yale University and Princeton University have analyzed and categorized in brand-new ways the mathematics intrinsic to musical harmony. Their cutting-edge collaboration has produced a powerful tool they call “geometrical music theory,” which translates the language of music theory into that of contemporary geometry.
The research is described in the April 18 issue of the journal Science, where the publication of work by music theorists and composers is rare if not unprecedented, said Clifton Callender, an assistant professor of composition in FSU’s College of Music. Callender is co-author of the paper “Generalized Voice-Leading Spaces” with Ian Quinn of Yale and Dmitri Tymoczko of Princeton.
“Our research offers a variety of tools for understanding and exploring music by drawing upon contemporary mathematics in natural and musically relevant ways,” Callender said. “It also provides a way to compare chords, and represents all possible combinations of pitches, including those found in non-Western music and avant-garde works that don’t conform to the traditional scales of Western music.”
As a result, composers could explore all sorts of uncharted musical possibilities; musicians may well be trained differently; new types of toys and musical instruments might be created; and music could be manifested visually (and geometry manifested aurally) in previously unimagined ways.
Geometrical music theory represents a culminating moment in the longstanding marriage of music and math. That marriage began when Pythagoras described pleasing musical intervals with simple mathematical ratios more than 2,600 years ago and further evolved during the Middle Ages when deep thinkers used those same ratios to model the “music of the spheres”—what many at that time believed to be the literally harmonious movements of the sun, moon and planets.
Understanding and interpreting music, say the authors of the study, is a process of discarding information—which in turn is the key to discovering its underlying mathematical structure.
A chord, for example, may be variously described as “the opening chord of Bach’s G minor Sonata for Unaccompanied Violin,” “G minor triad,” “minor triad” or simply “triad.”
“Each of these terms can refer to the same musical object at different levels of abstraction,” Callender said.
“We also experience a sense of distance when moving from one chord to another,” he said. “Changing one note just a little feels like a small motion between similar chords, while changing many notes by large amounts feels like a large motion between dissimilar chords.
“So, building on my own research and that of my Princeton colleague, our research modeled these spatial intuitions about chords at various levels of abstraction geometrically, using what mathematicians call ‘quotient spaces.’ Most of those spaces are warped and twisted such that they contain multiple ‘straight’ paths connecting any pair of points,” Callender said.
“Imagine being near the peak of a mountain and needing to get to the immediately opposite location,” he said. “You could proceed clockwise around the peak, counter-clockwise, or directly over the peak. These same three paths represent unique types of motions between major and minor triads in the space of three-note chord types, which is a cone. In fact, these motions and chords have been ubiquitous in Western music since medieval times to the present day.”
At each level of abstraction, musical objects are grouped into families of chords or melodies. Mathematical structure is assigned to the “families” so that they can be represented as points within complex geometrical spaces in much the same way that “x” and “y” coordinates correspond to points on a two-dimensional plane in simple high school algebra. The different families produce an exotic maze of diverse geometrical spaces such as twisted triangular donuts and pinched cones—and even some spaces that mathematicians haven’t dreamed up names for yet.
“My fellow researchers and I have found it thrilling to discover unexplored areas of mathematics in the course of solving musical problems,” Callender said.
“Professor Callender and his colleagues at Yale and Princeton are working at the forefront in this rarified area of music theory,” said Don Gibson, dean of the FSU College of Music. “Their research—and its publication in Science—represents a signal achievement in the discipline.”
A few years ago, Princeton University music theorist and composer Dmitri Tymoczko was sitting in the living room of his home playing with a piece of paper. Printed on the sheet were rows and columns of dots representing all the two-note chords that can be played on a piano — AA, AB b , AB, and so on for the rows; AA, B b A, BA, and so on for the columns. It was a simple drawing, something a child could make, yet Tymoczko felt that the piece of paper was trying to show him something that no one ever had seen before.
Suddenly Tymoczko (pronounced tim-OSS-ko) realized that if he cut two triangles from the piece of paper, turned one of the triangles upside down, and reconnected the two triangles where the chords overlapped, the two-note chords on one edge of the resulting strip of paper would be the reversed versions of those on the opposite edge. If he then twisted the paper and attached the two edges, the chords would line up. “That’s when I got a tingly feeling in my fingers,” he says.
Tymoczko had discovered the fundamental geometric shape of two-note chords. They occupy the space of a Möbius strip, a two-dimensional surface embedded in a three-dimensional space. Music is not just something that can be heard, he realized. It has a shape.
He soon saw that he could transform more complex chords the same way. Three-note chords occupy a twisted three-dimensional space, and four-note chords live in a corresponding but impossible-to-visualize four-dimensional space. In fact, it worked for any number of notes — each chord inhabit ed a multidimensional space that twisted back on itself in unusual ways — a non-Euclidean space that does not adhere to the classical rules of geometry. A physicist friend told him that these odd multidimensional spaces were called orbifolds — a name chosen by the graduate students of Princeton mathematician William Thurston, who first described them in the 1970s. In the 1980s, physicists found a few applications for orbifolds in arcane areas of string theory. Now Tymoczko had discovered that music exists in a universe of orbifolds.
Tymoczko’s insight, made possible through a research collaboration with Clifton Callender from Florida State University and Ian Quinn from Yale University, has created “quite a buzz in Anglo-American music-theory circles,” says Scott Burnham, the Scheide Professor of Music History at Princeton. His work has “physicalized” music. It provides a way to convert melodies and harmonies into movements in higher dimensional spaces. It has given composers new tools to write music, has revealed new ways to teach music students, and has revealed surprising musical connections between composers as distant as Palestrina — the Italian Renaissance composer — and Paul McCartney.
In a book to be published in March by Oxford University Press, A Geometry of Music: Harmony and Counterpoint in the Extended Common Practice, Tymoczko uses the connection between music and geometry to analyze the music of the last millennium and position modern composers in a new landscape. He rejects the idea that music can be divided into distinct genres. As Tymoczko sees it, medieval polyphony, the high classical music of Beethoven and Mozart, the chromatic romanticism of Wagner and Debussy, the jazz improvisations of Bill Evans, and the Beatles’ Sgt. Pepper’s Lonely Hearts Club Band all are built on the same handful of principles. Tymoczko writes in the preface: “It would make me happy to think that these ideas will be helpful to some young musician, brimming with excitement over the world of musical possibilities, eager to understand how classical music, jazz, and rock all fit together — and raring to make some new contribution to musical culture.”
The link between geometry and music has deep roots. Sometime between 530 and 500 B.C., in the town of Kroton on the rocky southern coast of Italy, Pythagoras and his followers made one of the most consequential discoveries in the history of science. If the string of a harp is shortened by half, it creates a tone one octave above that of the unshortened string. If the original string is shortened by two-thirds, the resulting tone is separated from the octave tone by a euphonious interval we know today as a fifth. Further experimentation showed that dividing the string into four parts produces intervals now known as fourths, with fur ther divisions of the string producing the familiar 12-note chromatic scale that the Greeks bequeathed to history.
Pythagoras and his followers thought big. The rational division of the musical scale was not just beautiful or pleasing — it was a sign that the universe was constructed on a rational basis and could be understood. “It was the first consistent realization that there is a mathematical rationality in the universe and that the human mind can make sense of that rationality,” says Kitty Ferguson, the author of The Music of Pythagoras: How an Ancient Brotherhood Cracked the Code of the Universe and Lit the Path from Antiquity to Outer Space.
Two-and-a-half millennia later, fifths and fourths are still the basis not just of three-chord rock-and-roll, but of much basic music theory. Students learn how to recognize intervals and relate those intervals to different kinds of scales, like major and minor scales. They practice transposing, inverting, and modulating melodies and chords. They absorb, perhaps without fully realizing it, the mathematician Gottfried Leibniz’s injunction that music is the “unknowing exercise of our mathematical faculties.”
Tymoczko falls squarely into the mathematical tradition in music. His father, Thomas, was a well-known philosopher of mathematics at Smith College who was fascinated by the use of computers in mathematics. His sister, Julianna, is a mathematician specializing in algebraic geometry at the University of Iowa.
But Tymoczko, growing up in the 1980s in Northhampton, Mass., spent more time listening to the Talking Heads, John Coltrane, and Brian Eno than solving equations. He swapped his piano lessons for guitar lessons and began playing in bands. He entered Harvard intending to study music, but the abstract and atonal music his professors preferred left him cold, and he switched from composition to philosophy. After studying philosophy at Oxford on a Rhodes scholarship, he kicked around Harvard for a few years as a teaching assistant, composing on the side and dabbling in journalism. Finally he decided to become serious about music again and enrolled in music graduate school at the University of California, Berkeley.
Tymoczko got interested in the mathematics of music for what he calls a “selfish reason” — he wanted his music to sound better. His music is distinctly modern but also “tonal” — a term that he defines in his new book as music that adheres to several basic principles. In compositions such as “Four Dreams” and “The Agony of Modern Music” (termed “brilliant” by music critic Alex Ross), he combines the free-flowing vibe of jazz, the strangeness of 20th-century music, and the raw exuberance of the rock ‘n’ roll he listened to growing up. Combining such different kinds of music is a delicate balancing act, and to do it well, Tymoczko decided that he needed to understand music at a more fundamental level.
Early on, Tymoczko recognized that tonal music has two major distinguishing features. First, melodies tend to move short distances from note to note — a characteristic known in music theory as “efficient voice leading.” Think of the tunes you can hum off the top of your head. They probably have notes that are fairly close to each other, rather than strings of notes that jump wildly up and down. Melodies with close-together notes are easier to sing and to play on most instruments, and they’re easier to listen to.
A second important feature of tonal music is that it uses harmonies that sound good rather than bad. For reasons that aren’t entirely clear, humans (including infants) tend to prefer certain kinds of chords to other kinds of chords. Echoing the findings of Pythagoras, we tend to like chords that divide the octave almost, but not completely, evenly. Triads, seventh chords, ninths, elevenths — the archetypal chords of everything from early polyphony to jazz — divide the octave into ever-smaller but approximately equal-sized segments, much as the vertical lines on a ruler divide feet and inches into equal-sized intervals. In contrast, notes that are next to each other sound harsh and dissonant when played together, like a child banging his fists on a piano.
These two properties may seem to be unrelated. But the “amazing and mysterious” thing about music, Tymoczko says, is that each requires the other. Three singers can go from a pleasing C-major chord to the complementary and more plaintive A-minor chord by moving just one note: changing from CEG to CEA. Someone playing “Hey Jude” on the piano can move his or her fingers very little while moving from one sonorous chord to another. “Miraculously, the chords that sound good together and the ones that produce efficient voice leading are the same,” Tymoczko says.
Tymoczko and other music theorists knew that these obser vations must have a mathematical representation, and previ ous theorists had captured some of these properties using geo metric ideas. He developed an intense e-mail relationship with Quinn, Callender, and several other theorists who were working on the same general problem: How can music be captured mathematically in the most powerful possible way? “People had been coming up with ad hoc solutions,” says Yale’s Quinn. “We wanted to do something general.”
Initially, Tymoczko took an algebraic approach, representing chord progressions by equations. But a critical contribution came from Callender. In a paper published in Music Theory Online, he explained how you could arrange two-note chords on a two-dimensional surface, so that changes from one chord to another could be represented by movements of a point on the surface. However, this surface had some very odd properties. For example, if a point moving on the surface hit an edge it would essentially bounce off, like a billiard ball careening off a bumper. In mathematical terms, the surface had a discontinuity or “singularity.”
Callender, Quinn, and Tymoczko all knew that the unusual properties of the space meant that it somehow harbored additional, non-Euclidean dimensions. But how were those dimensions configured? This is where Tymoczko’s checkered academic history came in. From his time at Harvard and Berkeley, he had lots of mathematician and physicist friends. One suggested making a physical representation of the space and playing with it. That’s the piece of paper he printed out, stared at in his living room, and used to discover the Möbius strip representation of two-note chords. Tymoczko’s scientist friends also told him that when they make a big discovery they submit the work to Science. A 2006 paper by Tymoczko and a 2008 paper by all three collaborators became the first on music theory in Science’s 129-year history.
Tymoczko’s discovery of the orbifold structure of musical space immediately had an important consequence: It explained his earlier observations about efficient voice leading and euphonious chords. When orbifolds are used to represent musical sounds, the chords that most evenly divide the octave reside in the central regions of the space. For example, in the Möbius strip representation of all two-note chords, fourths and fifths occupy the central area of the strip, while dissonant chords of closely spaced notes huddle near the edge of the strip. Composers can move from one euphonious chord to another while moving short distances in the central region of a musical space. Movements of short distances correspond to notes that are close together, producing singable melodies.
The implications for composers are momentous, says Tymoczko. “Imagine that you’re a blind man in a city who knows how to get from your home and to a coffee shop. Suddenly your vision is restored. You realize that not only are the post office and coffee shop very close to each other, but that you can get there many other ways, and many of them are better than the way you knew.”
The geometric representation of music also provides a powerful way to analyze past compositions. In his new book, Tymoczko uses geometric concepts to “retell the history of Western music.” For example, on the Möbius strip representation of two-note chords, medieval composers of two-voice polyphony tended to remain near the center of the space, with occasional leaps to its edge whenever the two singers were to voice the same note. By the high classical period of Bach, Mozart, and Beethoven, composers had become experts at crafting harmonious pieces that fully exploited these central regions of musical orbifolds. Later composers then began to branch out, exploring new regions of musical space. By the 20th century, minimalist composers like Steve Reich were relying on burbling arpeggios of closely spaced notes right on the edges of orbifolds, while jazz composers like Miles Davis and Bill Evans were further exploring the chromatic spaces between the centers and edges of orbifolds pioneered by Wagner and Debussy.
This way of visualizing music provides new insights into how composers wrote some of the world’s most beautiful music. In his living room, Tymoczko plays Chopin’s “Prelude in E” through the speakers of his laptop as the computer mon itor displays a three-dimensional projection of a four-dimensional orbifold. With each new chord, a ball moves through a latticework of points on the screen. Repeatedly the prelude returns to a particular point in the lattice — representing a diminished seventh chord — from which it branches first to a chord on the immediate right and then to a chord on the immediate left. But at other points in the prelude, the ball moves freely along the lattice, a kind of improvisation more commonly associated with modern music. “Composers in the 19th century had an intuitive understanding of the bizarre geometry of musical chord space,” Tymoczko says. “In fact, they had a better feel for non-Euclidean, higher-dimensional spaces than did their mathematical contemporaries.”
Tymoczko’s approach also makes it possible to compare very different kinds of music. At various times in his new book he compares Schumann’s “Chopin” movement in Carna val to Nirvana’s “Heart-Shaped Box,” Shostakovich’s “G-Minor Piano Quintet” to the Black Sabbath song “Sabbath, Bloody Sabbath,” and Philip Glass’ opera Einstein on the Beach to the opening of the TV show Battlestar Galactica. “Music that superficially seems quite different, like Renaissance music and jazz, make remarkably similar use of musical space,” he says.
In fact, Tymoczko argues, the music of the past millennium in the West, and much of non-Western music as well, constitutes an “extended common practice” characterized by the continued broadening of usable musical space. That history has, of course, been shaped by the creativity and idiosyncrasies of individual composers. But it has followed certain broad paths because of how musical orbifolds are configured. It’s like a mountaineer ascending a rock face, he says. In principle, the climber is free to move in any direction. But the structure of the rock provides certain natural routes, offering handholds and footholds along the way. “The trick for the historian is to make room for historical contingency while also capturing the way in which music history sometimes follows the path of least resistance, like a climber ascending a cliff by way of a particularly inviting chute.”
Tymoczko’s work falls into the category of basic research — it doesn’t have a guaranteed payoff. Yet already it’s finding lots of applications. Among the first enthusiasts were composers who use computers to produce musical ideas — and in some cases, complete compositions. Composer Michael Gogins, for example, uses orbifolds to add harmonic and melodic structure to notes generated randomly by the computer. “It gives you more power,” he says.
Some music theorists have begun using orbifolds to teach composition and theory to undergraduates, a trend Tymoczko hopes his new book will encourage. The value of the idea, says Quinn at Yale, is that a geometric approach demonstrates both the possibilities and constraints of music. “Once composers said, ‘Let’s get out of the box,’ that’s when they began wrapping themselves around the singularities and other weird parts of the space.”
Other applications of orbifolds are still on drawing boards. Tymoczko envisions orbifold-inspired children’s toys that could teach chord structure through play. A dancer could move through a space wired to generate the chords associated with each location. Composers could write music by moving through orbifolds rather than writing notes on staves.
Has the discovery of music’s geometry achieved Tymoczko’s original purpose — to make his music sound better? It helps, he says, but it can’t replace inspiration. Composition has both a mechanical phase and a mysterious phase. The use of geometric ideas can suggest melodies and harmonies — it frees a composer “from repeating the formulas of the past.” But when he’s writing music, says Tymoczko, who is working on an album of jazz, funk, and classical fusion pieces, the mechanical part must be followed by the mysterious part, when a composer “learns what a piece is really about.”
The saxophone genius Charlie Parker put it a bit differently: Learn all the theory you can, he said, and then forget it when you play.
Steve Olson, based in Seattle, has written about genetics, race, evolution, climate change, talent, and punk-rock music, among other things. His most recent book, Anarchy Evolution: Faith, Science, and Bad Religion in a World Without God, co-written by Greg Graffin, was published in September by itbooks.
How is the Beatles’ “Help!” similar to Stravinsky’s “Dance of the Adolescents?” How does Radiohead’s “Just” relate to the improvisations of Bill Evans? And how do Chopin’s works exploit the non-Euclidean geometry of musical chords?
In this groundbreaking work, author Dmitri Tymoczko describes a new framework for thinking about music that emphasizes the commonalities among styles from medieval polyphony to contemporary rock. Tymoczko identifies five basic musical features that jointly contribute to the sense of tonality, and shows how these features recur throughout the history of Western music. In the process he sheds new light on an age-old question: what makes music sound good?
A Geometry of Music provides an accessible introduction to Tymoczko’s revolutionary geometrical approach to music theory. The book shows how to construct simple diagrams representing relationships among familiar chords and scales, giving readers the tools to translate between the musical and visual realms and revealing surprising degrees of structure in otherwise hard-to-understand pieces.
Tymoczko uses this theoretical foundation to retell the history of Western music from the eleventh century to the present day. Arguing that traditional histories focus too narrowly on the “common practice” period from 1680-1850, he proposes instead that Western music comprises an extended common practice stretching from the late middle ages to the present. He discusses a host of familiar pieces by a wide range of composers, from Bach to the Beatles, Mozart to Miles Davis, and many in between.
A Geometry of Music is accessible to a range of readers, from undergraduate music majors to scientists and mathematicians with an interest in music. Defining its terms along the way, it presupposes no special mathematical background and only a basic familiarity with Western music theory. The book also contains exercises designed to reinforce and extend readers’ understanding, along with a series of appendices that explore the technical details of this exciting new theory.
Table of Contents
PREFACE
PART I. Theory
CHAPTER 1. Five Components of Tonality 1.1 The five features. 1.2. Perception and the five features. 1.3 Four Claims. A. Harmony and counterpoint constrain each other. B. Scale, macroharmony, and centricity are independent. C. Modulation involves voice leading. D. Music can be understood geometrically. 1.4 Music, magic, and language. 1.5 Outline of the book, and a suggestion for impatient readers.
CHAPTER 2. Harmony and Voice Leading 2.1 Linear pitch space. 2.2 Circular pitch-class space. 2.3 Transposition and inversion as distance-preserving functions. 2.4 Musical objects. 2.5 Voice leadings and chord progressions. 2.6 Comparing voice leadings. 2.7Voice-leading size. 2.8 Near identity. 2.9 Harmony and counterpoint revisited. 2.10 Acoustic consonance and near-evenness
CHAPTER 3. The Geometry of Chords 3.1 Ordered pitch space. 3.2 The Parable of the Ant. 3.3 Two-note chord space. 3.4 Chord progressions and voice leadings in two-note chord space. 3.5 Geometry in analysis. 3.6 Harmonic consistency and efficient voice leading. 3.7 Pure parallel and pure contrary motion. 3.8 Three-dimensional chord space. 3.9 Higher-dimensional chord spaces. 3.10 Voice leading lattices. 3.11 Triads are from Mars, seventh chords are from Venus. 3.12 Two musical geometries. 3.13 Study guide.
CHAPTER 4. Scales 4.1 A scale is a ruler. 4.2 Scale degrees, scalartransposition, scalar inversion. 4.3 Evenness and scalar transposition. 4.4 Constructing common scales. 4.5 Modulation and voice leading. 4.6 Voice leading between common scales . 4.7 Two examples. 4.8 Scalar and interscalar transposition. 4.9 Interscalar transposition and voice leading. 4.10 Combining interscalar and chromatic transpositions.
CHAPTER 5. Macroharmony and Centricity 5.1 Macroharmony. 5.2 Small-gap macroharmony. 5.3 Pitch-class circulation. 5.4 Modulating the rate of pitch-class circulation. 5.5 Macroharmonic consistency. 5.6 Centricity. 5.7 Where does centricity come from? 5.8 Beyond “tonal” and “atonal.”
PART II. History and Analysis
CHAPTER 6. The Extended Common Practice 6.1Disclaimers. 6.2 Two-voice medieval counterpoint. 6.3 Triads and the Renaissance. 6.4 Functional harmony. 6.5 Schumann’s Chopin. 6.6 Chromaticism. 6.7 Twentieth-century scalar music. 6.8 The extended common practice.
CHAPTER 7. Functional Harmony 7.1 The thirds-based grammar of elementary tonal harmony. 7.2 Voice leading in functional harmony. 7.3 Sequences. 7.4 Modulation and key distance. 7.5 The two lattices. 7.6 A challenge from Schenker. CHAPTER 8. Chromaticism 8.1 Decorative chromaticism. 8.2 Generalized augmented sixths. 8.3 Brahms and Schoenberg. 8.4 Schubert and the major-third system. 8.5 Chopin’s tesseract. 8.6 The Tristan Prelude. 8.7 Alternative approaches. 8.8Conclusion
CHAPTER 9. Scales in Twentieth-Century Music 9.1 Three scalar techniques. 9.2 Chord-first composition. A. Grieg’s “Drömmesyn,” (Vision), Op. 62 no. 5 (1895). B. Debussy’s “Fetes” (1899). C. Michael Nyman’s “The Mood That Passes Through You” (1993). 9.3 Scale-first composition. A. Debussy’s “Des pas sur la neige” (1910). B. Janácek’s “On an Overgrown Path,” Series II, no. 1 (1908). C. Shostakovich’s Fs minor Prelude and Fugue, Op. 87 (1950). D. Reich’s “New York Counterpoint” (1985). E. Reich’s “The Desert Music,” movement 1 (1984). F. The Who’s “Can’t Explain” (1965) and Bob Seger’s “Turn the Page” (1973). 9.4 The Subset Technique. A. Grieg’s “Klokkeklang,” (Bell Ringing), Op. 54 no. 6 (1891). B. “PetitAirs,” from Stravinsky’s Histoire du Soldat (1918). C. Reich’s “City Life” (1995). D. Stravinsky’s “Dance of the Adolescents” (1913). E. The Miles Davis Group’s “Freedom Jazz Dance” (1966). 9.5 Conclusion.
CHAPTER 10. Jazz. 10.1 Basic jazz voicings. 10.2 From thirds to fourths. 10.3 Tritone substitution. 10.4 Altered chords and scales. 10.5 Bass and upper-voice tritone substitutions. 10.6 Polytonality, sidestepping, and “playing out.” 10.7 Bill Evans’s “Oleo.” 10.8 Jazz as modernist synthesis.
CONCLUSION
APPENDIX A. Measuring voice-leading size APPENDIX B.Chord geometry: a technical look. APPENDIX C.Discrete voice leading lattices. APPENDIX D. The interscalar interval matrix. APPENDIX E.Scale, macroharmony, and Lerdahl’s “basic space.” APPENDIX F. Some study questions, problems, and activities.
Prelude: transposition along a collection 2. Rock logic 1. A melodic principle 2. A harmonic principle 3. A first chord-loop family 4. Two more families 5. Shepard-tone passacaglias 6. Minor triads and other trichords 7. A fourth family 8. Other modalities 9. Function and retrofunction 10. Continuity or reinvention?
Prelude: the Tinctoris transform 3. Line and configuration 1. The imperfect system 2. Voice exchanges 3. Other intervals 4. The circle of diatonic triads 5. Voice exchanges and multiple chord types 6. Four-voice triadic counterpoint 7. Counterpoint within the chord 8. Seventh chords 9. Harmony and counterpoint
Prelude: sequence and function 4. Repetition 1. Repetition reimagined 2. Repeating contrapuntal patterns 3. The geometry of two-voice sequences 4. Three voices and the circle of triads 5. Three voices arranged 2+1 6. Four voices 7. Contrary-motion sequences 8. Melodic sequences and near sequences 9. Near sequences 10. Sequences as reductional targets
Prelude: three varieties of analytical reduction 5. Nonharmonic tones 1. The first practice and the SNAP system 2. Schoenberg’s critique 3. Monteverdi’s “Ohimè” 4. The standardized second practice 5. A loophole 6. After nonharmonicity
Prelude: functional and scale-degree analysis 6. The origins of functional harmony 1. The logical structure of protofunctionality 2. Similarities and differences 3. Origin and meaning 4. Harmony and polyphony 5. The Pope Marcellus Kyrie 6. A broader perspective 7. “I Cannot Follow”
Prelude: could the Martians understand our music? 7. Functional progressions 1. A theory of harmonic cycles 2. A more principled view 3. Rameau and Bach 4. Functional melody, functional harmony 5. Fauxbourdon and linear idioms 6. Sequences 7. Bach the dualist
Prelude: chromatic or diatonic? 8. Modulation 1. Two models of key distance 2. Enharmonicism and loops in scale space 3. Minor keys 4. Modulatory schemas 5. Up and down the ladder 6. Modal homogenization and scalar voice leading 7. Generalized set theory
Prelude: hearing and hearing-as 9. Melodic strategies 1. Strategy and reduction 2. Two models of the phrase 3. Chopin and the Prime Directive 4. An expanded vocabulary of melodic templates 5. Simple harmonic hierarchy 6. The four-part phrase 7. Grouping, melody, harmony 8. Beyond the phrase: hierarchy at the level of the piece
Prelude: why Beethoven? 10. Beethoven theorist 1. Meet the Ludwig 2. From schema to flow 3. The Tempest 4. The Fifth Symphony 5. The “Pastorale” sonata, op. 28 6. Schubert’s Quartettsatz 7. The prelude to Lohengrin
From musical chords to twin primes / Jack Douthett, David Clampitt & Norman Carey
Hypercubes and the generalized Cohn cycle / Jack Douthett, Peter Steinbach & Rick Hermann
Associahedra, combinatorial block designs and related structures / Franck Jedrezejewski
Rhythmic and melodic l-canons / Jeremy Kastine
The Fibonacci sequence es metric suspension in Luigi Nono’s Il canto sospeso / Jon Kochavi
Note samba : navigating notes and their meanings within modes and exo-modes / Thomas Noll
Difference sets and all-directed-interval chords / Robert W. Peck
Harmonious opposition / Richard Plotkin
Orbifold path models for voice leading : dealing with doubling / James R. Hughes
Remarks on the geometry of chords / Thomas A. Ivey
Theoretical physics and category theory as tools for analysis of musical performance and composition / Maria Mannone
Intuitive musical homotopy / Aditya Sivakumar & Dmitri Tymoczko
Geometric generalizations of the Tonnetz and their relation to Fourier phases space / Jason Yust
Deterministic geometries : a technique for the systematic generation of musical elements in composition / Brent A. Milam
Flamenco music and its computational study / Francisco Gómez
Examining fixed and relative similarity metrics through jazz melodies / David J. Baker & Daniel Shanahan
In search of arcs of prototypicality / Daniel Shanahan.
Publisher’s summary Questions about variation, similarity, enumeration, and classification of musical structures have long intrigued both musicians and mathematicians. Mathematical models can be found from theoretical analysis to actual composition or sound production. Increasingly in the last few decades, musical scholarship has incorporated modern mathematical content. One example is the application of methods from Algebraic Combinatorics, or Topology and Graph Theory, to the classification of different musical objects. However, these applications of mathematics in the understanding of music have also led to interesting open problems in mathematics itself.The reach and depth of the contributions on mathematical music theory presented in this volume is significant. Each contribution is in a section within these subjects: (i) Algebraic and Combinatorial Approaches; (ii) Geometric, Topological, and Graph-Theoretical Approaches; and (iii) Distance and Similarity Measures in Music. remove. (source: Nielsen Book Data)
Questions about variation, similarity, enumeration, and classification of musical structures have long intrigued both musicians and mathematicians. Mathematical models can be found from theoretical analysis to actual composition or sound production. Increasingly in the last few decades, musical scholarship has incorporated modern mathematical content. One example is the application of methods from Algebraic Combinatorics, or Topology and Graph Theory, to the classification of different musical objects. However, these applications of mathematics in the understanding of music have also led to interesting open problems in mathematics itself.
The reach and depth of the contributions on mathematical music theory presented in this volume is significant. Each contribution is in a section within these subjects: (i) Algebraic and Combinatorial Approaches; (ii) Geometric, Topological, and Graph-Theoretical Approaches; and (iii) Distance and Similarity Measures in Music.
remove Contents:
Section I:
From Musical Chords to Twin Primes (Jack Douthett, David Clampitt and Norman Carey)
Hypercubes and the Generalized Cohn Cycle (Jack Douthett, Peter Steinbach and Richard Hermann)
Associahedra, Combinatorial Block Designs and Related Structures (Franck Jedrzejewski)
Rhythmic and Melodic L-canons (Jeremy Kastine)
The Fibonacci Sequence as Metric Suspension in Luigi Nono’s II Canto Sospeso (Jon Kochavi)
One Note Samba: Navigating Notes and Their Meanings Within Modes and Exo-modes (Thomas Noll)
Difference Sets and All-Directed-Interval Chords (Robert W Peck)
Harmonious Opposition (Richard Plotkin)
Section II:
Orbifold Path Models for Voice Leading: Dealing with Doubling (James R Hughes)
Reflections on the Geometry of Chords (Thomas A Ivey)
Theoretical Physics and Category Theory as Tools for Analysis of Musical Performance and Composition (Maria Mannone)
Intuitive Musical Homotopy (Aditya Sivakumar and Dmitri Tymoczko)
Geometric Generalizations of the Tonnetz and Their Relation to Fourier Phases Spaces (Jason Yust)
Deterministic Geometries: A Technique for the Systematic Generation of Musical Elements in Composition (Brent A Milam)
Section III:
Flamenco Music and Its Computational Study (Francisco Gómez)
Examining Fixed and Relative Similarity Metrics Through Jazz Melodies (David J Baker and Daniel Shanahan)
In Search of Arcs of Prototypicality (Daniel Shanahan)
Readership: Students and researchers in Mathematical Music Theory.Mathematics and Music;Algebra;Geometry;Topology;Graph Theory;Combinatorics;Distance and Similarity Measures;Discrete Fourier Transform0 Key Features:
It includes the most prominent authors in the field
It gathers a gamut of the most recent work in the field, which is something very difficult to find in one volume
It will appeal to mathematicians, music theorists, and computer scientists. Within mathematics, it offers a variety of areas and techniques related to musical phenomena that cannot be found together in other volumes
1 Department of Mathematics and Computer Science, Saint Joseph’s University, Philadelphia, PA 19131, rhall@sju.edu
Music theorists have frequently invoked geometry in modeling musical objects such as chords, rhythms, and scales; however, no unified geometric perspective has hitherto emerged. On page XXX of this issue, Callender, Quinn, and Tymoczko (1) demonstrate that many musical terms can be understood as expressing symmetries of n -dimensional space. Identifying— “gluing together”—points related by these symmetries produces a family of non-Euclidean quotient spaces that subsume a large number of geometric models proposed in the literature. The use of mathematics to describe, analyze, and create music goes back millennia. Questions have previously emerged in music theory that are appealing, nontrivial, and, in several cases, connected to other scientific fields (2,3,4). Math inspired composers such as Schoenberg, Messaien, and Xenakis; musical investigations have even motivated mathematical discoveries (5). However, the “geometrical music theory” proposed by Callender, Quinn, and Tymoczko stands out both for the breadth of its musical implications and the depth of its mathematical content. It suggests a wealth of new techniques for studying music theoretical topics, including chord similarity and melodic contour, and provides attractive visualizations of harmonic relationships. Although the authors’ use of contemporary geometry departs from the discrete mathematics normally used to describe music, they make a convincing argument for the validity of their models. Musicians think in degrees of abstraction: While “middle C” is a particular pitch (frequency measured on a logarithmic scale), the letter name (or pitch class) “C” refers to any pitch that is a whole number of octaves away from it. Any collection of C, E, and G notes is a “C major chord.” A “major chord” is any transposition of a C major chord; a “consonant triad” is any three-note chord containing a major third, a minor third, and a perfect fifth. Music theorists invoke the mathematical concept of equivalence class —a set of objects that are “the same” if we ignore certain information—to describe these musical structures. Callender, Quinn, and Tymoczko show that many musical terms describe equivalence classes under combinations of five basic “OPTIC” relations: Octave shifts, permutation (reordering), transposition (the relation between pitches sharing the same succession of intervals, regardless of their starting note), inversion (turning a sequence “upside down”), and cardinality equivalence (ignoring repetitions). The authors go beyond traditional music theory by showing that there are two distinct ways in which these symmetries can apply to progressions of musical objects: “uniformly,” where the same symmetry applies to each object in a progression, and “individually,” where distinct symmetry operations apply to the harmonies in a progression. This distinction allows them to formalize the relationships among a large number of musical terms, as in table S1 of their paper. Following Tymoczko (6), they develop geometrical models of these relationships. Pitches correspond to real numbers; each point in the Euclidean n -dimensional “configuration” space represents a sequence of n pitches. Points near each other differ by microtones (a C major chord
The Geometry of Music
A composer has taken equations from string theory to explain why Bach and bebop aren’t so different
When you first hear them, a Gregorian chant, a Debussy prelude and a John Coltrane improvisation might seem to have almost nothing in common–except that they all include chord progressions and something you could plausibly call a melody. But music theorists have long known that there’s something else that ties these disparate musical forms together. The composers of these and virtually every other style of Western music over the past millennium tend to draw from a tiny fraction of the set of all possible chords. And their chord progressions tend to be efficient, changing as few notes, by as little as possible, from one chord to the next.
Exactly how one style relates to another, however, has remained a mystery–except over one brief stretch of musical history. That, says Princeton University composer Dmitri Tymoczko, “is why, no matter where you go to school, you learn almost exclusively about classical music from about 1700 to 1900. It’s kind of ridiculous.”
But Tymoczko may have changed all that. Borrowing some of the mathematics that string theorists invented to plumb the secrets of the physical universe, he has found a way to represent the universe of all possible musical chords in graphic form. “He’s not the first to try,” says Yale music theorist Richard Cohn. “But he’s the first to come up with a compelling answer.”
Tymoczko’s answer, which led last summer to the first paper on music theory ever published in the journal Science, is that the cosmos of chords consists of weird, multidimensional spaces, known as orbifolds, that turn back on themselves with a twist, like the Möbius strips math teachers love to trot out to prove to students that a two-dimensional figure can have only one side. Indeed, the simplest chords, which consist of just two notes, live on an actual Möbius strip. Three-note chords reside in spaces that look like prisms–except that opposing faces connect to each other. And more complex chords inhabit spaces that are as hard to visualize as the multidimensional universes of string theory.
But if you go to Tymoczko’s website music.princeton.edu/~dmitri) you can see exactly what he’s getting at by looking at movies he has created to represent tunes by Chopin and, of all things, Deep Purple. In both cases, as the music progresses, one chord after another lights up in patterns that occupy a surprisingly small stretch of musical real estate. According to Tymoczko, most pieces of chord-based music tend to do the same, although they may live in a different part of the orbifold space. Indeed, any conceivable chord lies somewhere in that space, although most of them would sound screechingly harsh to human ears.
The discovery is useful for at least a couple of reasons, says Tymoczko. “One is that composers have been exploring the geometrical structure of these maps since the beginning of Western music without really knowing what they were doing.” It’s as though you figured out your way around a city like Boston, for example, without realizing that some of your routes intersect. “If someone then showed you a map,” he says, “you might say, ‘Wow, I didn’t realize the Safeway was close to the disco.’ We can now go back and look at hundreds of years of this intuitive musical pathmaking and realize that there are some very simple principles that describe the process.”
Computational geometric aspects of rhythm, melody, and voice-leading
Godfried Toussaint 1 School of Computer Science and Center for Interdisciplinary Research in Music Media and Technology, McGill University, Montréal, Québec, Canada
Many problems concerning the theory and technology of rhythm, melody, and voice-leading are fundamentally geometric in nature. It is therefore not surprising that the field of computational geometry can contribute greatly to these problems. The interaction between computational geometry and music yields new insights into the theories of rhythm, melody, and voice-leading, as well as new problems for research in several areas, ranging from mathematics and computer science to music theory, music perception, and musicology. Recent results on the geometric and computational aspects of rhythm, melody, and voice-leading are reviewed, connections to established areas of computer science, mathematics, statistics, computational biology, and crystallography are pointed out, and new open problems are proposed.
Exploring Musical Spaces: A Synthesis of Mathematical Approaches
By Julian Hook
Oxford, 2022
Exploring Musical Spaces is a comprehensive synthesis of mathematical techniques in music theory, written with the aim of making these techniques accessible to music scholars without extensive prior training in mathematics. The book adopts a visual orientation, introducing from the outset a number of simple geometric models―the first examples of the musical spaces of the book’s title―depicting relationships among musical entities of various kinds such as notes, chords, scales, or rhythmic values. These spaces take many forms and become a unifying thread in initiating readers into several areas of active recent scholarship, including transformation theory, neo-Riemannian theory, geometric music theory, diatonic theory, and scale theory.
Concepts and techniques from mathematical set theory, graph theory, group theory, geometry, and topology are introduced as needed to address musical questions. Musical examples ranging from Bach to the late twentieth century keep the underlying musical motivations close at hand. The book includes hundreds of figures to aid in visualizing the structure of the spaces, as well as exercises offering readers hands-on practice with a diverse assortment of concepts and techniques.
A guided tour of the mathematical principles inherent in music.
Taking a “music first” approach, Gareth E. Roberts’s From Music to Mathematics will inspire students to learn important, interesting, and at times advanced mathematics. Ranging from a discussion of the geometric sequences and series found in the rhythmic structure of music to the phase-shifting techniques of composer Steve Reich, the musical concepts and examples in the book motivate a deeper study of mathematics.
Comprehensive and clearly written, From Music to Mathematics is designed to appeal to readers without specialized knowledge of mathematics or music. Students are taught the relevant concepts from music theory (notation, scales, intervals, the circle of fifths, tonality, etc.), with the pertinent mathematics developed alongside the related musical topic. The mathematics advances in level of difficulty from calculating with fractions, to manipulating trigonometric formulas, to constructing group multiplication tables and proving a number is irrational.
Topics discussed in the book include
• Rhythm • Introductory music theory • The science of sound • Tuning and temperament • Symmetry in music • The Bartók controversy • Change ringing • Twelve-tone music • Mathematical modern music • The Hemachandra–Fibonacci numbers and the golden ratio • Magic squares • Phase shifting
Featuring numerous musical excerpts, including several from jazz and popular music, each topic is presented in a clear and in-depth fashion. Sample problems are included as part of the exposition, with carefully written solutions provided to assist the reader. The book also contains more than 200 exercises designed to help develop students’ analytical skills and reinforce the material in the text. From the first chapter through the last, readers eager to learn more about the connections between mathematics and music will find a comprehensive textbook designed to satisfy their natural curiosity.
physics and philosophy, but the most important is music, for in the patterns of the arts are the keys to all learning.
Plato, c. 428-347 BC
The first concrete argument for a fundamental link between mathematics and music was perhaps made by the early philosopher and mathematician Pythagoras (569-475 BC), often referred to as the “father of numbers.” He can also be considered the “father of harmony,” given that his discovery of the overtone series and analyses of the acoustics and ratios involved in music have served as the foundation of harmony in western-hemisphere music composition ever since. The Pythagorean, Quadrivium and Platonic classifications of mathematics were based on hierarchical dimensions, starting with arithmetic, then geometry, astronomy and finally music.
Reportedly, Pythagoras experimented with the tones produced when plucking strings of different lengths. He found that some specific ratios of string lengths created pleasing combinations (“harmonies”) and others did not. Based on his careful observations, Pythagoras identified the physics of intervals, or distances between notes, that form the primary harmonic system which is still used today (Parker, 2009, pp. 3-5).
Music is based on proportional relationships. The mathematical structure of harmonic sound begins with a single naturally occurring tone, which contains within it a series of additional frequencies above its fundamental frequency (“overtones”), of which we are normally unaware on a conscious level. Within this harmonic or overtone series, there is a mathematical relationship between the frequencies – they are specific integer multiples of each other. For example, if the slowest frequency (the “fundamental”) were 100 Hz, then the overtones would be 2 x 100 (200 Hz), 3 x 100 (300 Hz) and so forth. (The overtone series is often referred to as harmonics.)
Pythagoras observed several ratios of sound wave frequencies and the corresponding intervals between them, including 4:3 (known to musicians as the interval of a perfect fourth, or two pitches that are five semitones apart from each other) and 3:2 (a perfect fifth, seven semitones apart). Note that pitch is the frequency or rate of vibration of a physical source such as a plucked string.
The most prominent interval that Pythagoras observed highlights the universality of his findings. The ratio of 2:1 is known as the octave (8 tones apart within a musical scale).
When the frequency of one tone is twice the rate of another, the first tone is said to be an octave higher than the second tone, yet interestingly the tones are often perceived as being almost identical. For example, a woman’s voice may fluctuate around 220 Hertz while a man’s voice is around 110 Hertz, approximately half the frequency of the woman’s. However, if they sing together, it may sound as though they are singing the same melody together in unison, even though they are actually an octave apart. This 2:1 ratio is so elemental to what humans consider to be music, that the octave is the basis of all musical systems that have been documented – despite the diversity of musical cultures around the world. Moreover, this physical phenomenon is so fundamental that even non-human species such as monkeys and cats recognize it (Levitin, 2008, p. 31).
The inherent properties of physics and mathematics within music, perceived so long ago by Pythagoras, may help to explain why many physicists and mathematicians are also musicians. This point is illustrated by a quotation from Einstein: “The theory of relativity occurred to me by intuition, and music was the driving force behind that intuition….My new discovery was the result of musical perception” (Suzuki, 1969, 90).
*Portions of this article are adapted from Edel Sander’s chapter in Musik i forskola och tidiga skolar (2015), a Swedish textbook for music educators.
10 – Music theory and mathematics
from PART II – SPECULATIVE TRADITIONS
Published online by Cambridge University Press: 28 March 2008
In Chapter 6 of The Manual of Harmonics (early second century CE), Nicomachus of Gerasa narrates the legendary story of Pythagoras passing by the blacksmith’s shop, during which in an epiphany of sonorous revelation, he discovered the correlation of sounding intervals and their numerical ratios. According to Nicomachus, Pythagoras perceived from the striking of the hammers on the anvils the consonant intervals of the octave, fifth, and fourth, and the dissonant interval of the whole tone separating the fifth and fourth. Experimenting in the smithy with various factors that might have influenced the interval differences he heard (force of the hammer blows, shape of the hammer, material being cast), he concluded that it was the relative weight of the hammers that engendered the differences in the sounding intervals, and he attempted to verify his conclusion by comparing the sounds of plucked strings of equal tension and lengths, proportionally weighted according to the ratios of the intervals.
Physical and logical incongruities or misrepresentations in Nicomachus’s narrative aside, the parable became a fixture of neo-Pythagorean discourse because of its metaphoric resonance: it encapsulated the essence of Pythagorean understanding of number as material or corporeal, and it venerated Pythagoras as the discoverer of the mathematical ratios underlying the science of harmonics. The parable also established a frame of reference in music-theoretical thought in the association between music and number, or more accurately, music theory and mathematical models, since it is not through number alone but through the more fundamental notions of universality and truth embedded in Pythagorean and Platonic mathematics and philosophy that one can best begin to apprehend the broad range of interrelationships between music theory and mathematics.
A new way of visualizing Western music theory could inspire innovations in everything from musical instrument design to music composition techniques—even to new kinds of toys.
Western Music Moves in Three and Even Four (!) Dimensional Spaces: How the Pioneering Research of Princeton Theorist Dmitri Tymoczko Helps Us Visualize Music in Radical, New Ways
Supplementary Sets and regular complementary unending canons,
Dan Tudor Vuza,
Perspectives of New Music, Numbers 29(2), 30(1), 30(2), and 31(1). Journal of Mathematics and Music, Special Issue on Tiling Problems in Music, Vol. 3, No. 2
Computational Geometric Aspects of Rhythm, Melody, and Voice-Leading
Godfried Toussaint∗
School of Computer Science
and Center for Interdiisciplinary Research in Music Media and Technology McGill University Montr ́eal, Qu ́ebec, Canada
To appear in: Computational Geometry: Theory and Applications, 2009. doi:10.1016/j.comgeo.2007.01.003
Viewpoints which Matter 