The Geometry of Music
By Julie J. Rehmeyer
The connection between mathematics and music is often touted in awed, mysterious tones, but it is grounded in hard-headed science. For example, mathematical principles underlie the organization of Western music into 12-note scales. And even a beginning piano student encounters geometry in the "circle of fifths" when learning the fundamentals of music theory.
But according to Dmitri Tymoczko, a composer and music theorist at Princeton University, these well-known connections reveal only a few threads of the hefty rope that binds music and math. To grasp the true structure of music, he says, we need to understand the geometry of hyperdimensional objects. Doing so has given him new ways of understanding pieces of music that have long baffled theorists and even led him to new insights into the history of music.
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This shows a three-dimensional representation of a four-dimensional space that captures many essential features of musical structure. |
Tymoczko compares the structure of music to the shape of a rock face that a rock-climber is scrambling up. "If you know the conditions of the rock face, you can predict the motions of the climber," he says. "The structure of the space makes certain choices overwhelmingly natural or convenient. There's something similar that goes on with music. When you think about things abstractly, you can come to understand that the directions that music went aren't completely arbitrary. Composers are exploring the possibilities that musical space presents them with."
Tymoczko built on familiar geometrical analogs for music. For example, musical pitch is often imagined as lying on a line with low notes to the left and high notes to the right. Furthermore, as pitches go higher and higher, the notes repeat in different octaves, such that a low C, a middle C, and a high C all sound very similar. Often, the exact octave of a particular note doesn't matter very much in music. Instead, musicians commonly visualize a "pitch class circle," which comes from the original line by gluing together each point of the line that represents the same note in different octaves. So low C, middle C, and high C, for example, would all be glued together.
Applying the same kind of reasoning to complete pieces of music, Tymoczko created a geometric space in which he could analyze a piece of music with two notes being played simultaneously. He started with a piece of paper and made the horizontal direction represent the pitch of one note and the vertical direction represent the pitch of the other. A piece of music with two voices would correspond to dots moving around in this space.
Then he modified the space to embed musical structure within it. First, Tymoczko used the same method musicians used to create the pitch circle. He glued the left edge of the page to the right edge, turning the horizontal lines into circles and creating a cylinder from the whole page. Then he glued the bottom end of the cylinder to the top, turning the vertical lines into circles as well and creating a donut shape from the entire page.
Next, he noted that the order of the notes in a chord doesn't much matter. That means that the point on his page that has C in the horizontal direction and E in the vertical direction is really the same as the point that has E in the horizontal direction and C in the vertical direction. So he took his space and glued all those points together. It takes a bit of effort to visualize it, but for two simultaneous notes, this turns the donut shape into a Möbius strip.
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Two-note chords correspond to points on a Möbius strip. |
Tymoczko used the same method to create geometrical spaces to model pieces with any number of simultaneous notes. A piece with three notes, for example, would correspond to points in three-dimensional space. When he wrapped the space around to form circles and identified chords with the same pitches in a different order, he created a twisted prismatic donut. More notes require more than three dimensions, which gets hard to picture but not so hard to describe mathematically.
Having constructed his spaces, he began translating musical principles into their geometric equivalents. He noted that if he plotted a major chord in his geometrical space and reflected it, as if across a mirror down the middle of the space, it turned the chord into a minor one. Rotating a chord to a different spot in the space corresponded to transposing the chord into a different key. Composers need to choose sequences of chords that the ear can make sense of harmonically, and Tymoczko noted that this tends to be accomplished by transitioning between chords using combinations of these geometric rotations and reflections, or approximations to them.
Composers also need to write music in such a way that our minds can link the sounds into simultaneous, overlapping melodies. This is this easiest when each individual melody moves only in fairly small steps.
The trick for composers is to accomplish both those goals, harmonic and melodic consistency, at the same time. "We've just translated that into a math problem," Tymoczko says. "The solution is to use sequences of points close together that are related by rotation, or nearly so."
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In the video above, Chopin's E-minor prelude can be visualized as multiple points moving around on a circle. Note that the notes tend to move short distances. (Click here or on the image above to play the video in a new window.) |
Music theorists have long found Chopin's E minor prelude puzzling. Although the chord progressions sound smooth to the ear, they don't quite follow the traditional rules of harmony. When Tymoczko looked at the piece and watched the composition's motion through his geometrical space, he saw that Chopin was moving in a systematic way among the different layers of the four-dimensional cubes. "It's almost as if he's an improviser with a set of rules and set of constraints," Tymoczko says.
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Another way of visualizing Chopin's composition is through a three-dimensional projection of a four-dimensional space, as in the video above. The chords primarily cluster in the center of the space, usually moving through small distances. (Click here or on the image above to play the video in a new window.) |
What's particularly amazing, Tymoczko says, is that the mathematics needed to describe these spaces wasn't even developed in Chopin's time. Nevertheless, he says, "it is unquestionable that he had some cognitive representation of the space. So there was this period of history where the only way Chopin could express this abstract knowledge was through music. His knowledge of four-dimensional geometry was most efficiently expressed through piano pieces."
References:
Tymoczko, D. 2006. The geometry of musical chords. Science 313(July 7):72-74. Available at www.sciencemag.org/cgi/content/full/313/5783/72.
Tymoczko has much more information, including his original compositions, software and additional videos, at www.music.princeton.edu/~dmitri/.
Comments
Of course, someone should have told the author that Chopin (and so many others!)improvised a lot of his music, especially the shorter piano works. A pianist can almost feel Chopin stretching out through his or her hands (and heart)to reveal his chromatic progressions and enduring eloquence. Reading this article was a bit like reading an explanation of what makes a beautiful flower "tick" or a sculpture "breathe"! It's a crazy world where scientists seek explanations (and legitimacy) for the higher and worthier mysteries of human endeavour. But I suppose it's inevitable.
Posted by: Faith | March 8, 2008 04:31 AM
Julie J. Rehmeyer
I've commented on your stuff before:
I generally like your topics.
I've been composing algorithmic stuff since about 1993
( using basic on an Amiga).
I've lately been working on graphs of hyperdimensional figures
as chord geometries to make Jazz chord sequences
using graph substitutions.
I'm not a ph.d. or music theorist but you music sounds very similar to
mine.
Only played by real instruments instead of midi...
Actually I seem to have composed more in a short time
than he has as I use Mathematica to do my calculations of notes.
I think he probably inspired this work of mine indirectly
as I remember having run across you work.
in a Google search.
I had been doing hyperdimensional sequences and substitutions
looking to Cartan and group theory ends
and topological graphs.
I was more into the idea of getting fractal / chord algorithms
and Jazz than classical or Norgard type stuff.
Really ugly some of the stuff comes out.
I prefer my own compositions to the algorithmic stuff.
Sequences online:
http://www.research.att.com/~njas/sequences/?q=bagula+chords&language=english&go=Search
music online:
http://www.archive.org/search.php?query=roger%20bagula
http://www.archive.org/details/NoIdea_117
Here is a pentaonic algorithmic:
http://ia341039.us.archive.org/3/items/PiecesOfEight/FractalFugue.mp3
I just did some Norgard sequences:
http://www.research.att.com/~njas/sequences/?q=bagula+norgard&sort=0&fmt=0&language=english&go=Search
I don't think any body with an instrument has ever played any of my music.
Pentatonic sequence avoid the problems of synchronizing that you get
with chord algorithms
and sometimes have a nice oriental sound.
The Greek and I think the ancient Egyptians used pentatonic scales
so that it is a reversion to a more primitive music.
Paul Bourke has two of my compositions up: ( from about 2004 or so )
http://local.wasp.uwa.edu.au/~pbourke/podcast/cantor_cold_blues.mp3
http://local.wasp.uwa.edu.au/~pbourke/podcast/the_hurricane_ticktock.mp3
Roger Bagula
Posted by: Roger Bagula | March 12, 2008 04:25 PM
i've been an SN reader off-and-on since the 60's. i'm realy happy to see these articles by Julie Rehmeyer As a math guy, i've long been annoyed by ivars petersen's style of writing, in which he conveyed only a gee-whiz attitude about math, without actually presenting the math content very well. rehmeyer does a nice job of presenting the math with some simplifying clarity, while not bleeding the mathematical life out of the story. i'm looking forward to reading her stuff regularly.
Posted by: Don Davis | March 17, 2008 10:51 PM
Faith,
understanding something does not make it less beautiful, it makes it more beautiful.
Richard Feynman put it much better than I
(from http://www.fieldstudy.com/Classes/Announcements/feynmanquote.htm)
(As quoted from the "Best Mind Since Einstein" NOVA Video)
"I have a friend who’s an artist and he’s some times taken a view which I don’t agree with very well. He’ll hold up a flower and say, 'look how beautiful it is,'and I’ll agree, I think. And he says, 'you see, I as an artist can see how beautiful this is, but you as a scientist, oh, take this all apart and it becomes a dull thing.' And I think he’s kind of nutty.
First of all, the beauty that he sees is available to other people and to me, too, I believe, although I might not be quite as refined aesthetically as he is. But I can appreciate the beauty of a flower.
At the same time, I see much more about the flower that he sees. I could imagine the cells in there, the complicated actions inside which also have a beauty. I mean, it’s not just beauty at this dimension of one centimeter: there is also beauty at a smaller dimension, the inner structure…also the processes.
The fact that the colors in the flower are evolved in order to attract insects to pollinate it is interesting – it means that insects can see the color.
It adds a question – does this aesthetic sense also exist in the lower forms that are…why is it aesthetic, all kinds of interesting questions which a science knowledge only adds to the excitement and mystery and the awe of a flower.
It only adds. I don’t understand how it subtracts."
Posted by: Burton MacKenZie | March 22, 2008 05:53 AM
This sort of vacuous pseudoscience reappears periodically in music journals and has a long history. For details, read Les theories scientifiques de la musique, Paris: Librairie philosophique J. Vrin, 1996, by Sorbonne professor of music Laurent Fichet.
Of course, neither Julie Rehmeyer nor Dmitri Tymoczko have heard of Prof. Fichet, nor have they read his summary of the follies and fallacies which pass for "scientific music theory" over the past 200 years.
"The final section of the text is a twenty-page conclusion, which Fichet commences by noting the yawning chasm between the hopes raised by the theories reviewed and the actual progress they have made to music theory, and that this shortcoming is particularly apparent for those theories from the preceding century. In fact, Fichet concludes that the only theorist from that period to have made any real contribution to music theory was Helmholtz, with his theory of dissonance. He notes that, while notions of what is considered "scientific" have changed over the last two centuries, this nevertheless does not excuse the number of mathematical and logical errors and inconsistencies contained in these purportedly "scientific" theories of music, and that several theorists (notably Ansermet) attempted to give a scientific flavor to their writings in an attempt merely to give them some credence of rigor and certitude." [Perrott, David, review of Les theories scientifiques de la musique," Online Journal of Music Theory, Volume 5, Number 4 September, 1999]
Of course Fichet remains only the most recent academic to debunk pseudoscience in music; the list of papers goes all the way back to "Pseudo-Science in Music," John Backus, Journal of Music Theory, Vol. 4, No. 2, 221-232, Nov., 1960, and before that, to L. S. Lloyd's "Pseudo-science in Music `Theory'" from Proceedings of the Royal Academy Of Music, 1943. (Can Rehmeyer or Tymoczko cite the first 5 references from the bibliography of the Lloyd paper? Can they cite the second paragraph of the Backus journal article? Of course not, they've never read these peer-reviewered journal article, nor have they even heard of them. Like so many practitioners of the hard sciences, they erroneously assume that their expertise in science or mathematics carries over into music, which of course it does not. Scholars who venture outside their specialized field often wind up embarrassing themselves, as is the case here.)
Naturally, nothing I've said is original or novel: many hard-headed skeptical musicians and scholars of music history have made exactly the same points I'm making, which is why this comment must be censored and will surely not be allowed to see the light of day. I'm merely reiterating Laurent Fichet's well-worn points and citing the same logical errors identified by Fichet, Backus, Lloyd, and, most recently, Darrell in
"Some Questions the Music Mathematicians Forgot To Ask"
It's worth taking a moment to specifically debunk the fraudulent claims made in Rehmeyer's article, since they perpetuate a well-worn series of deceptions, misrepresentations and false pseudoscientific claims made about the allegedly "mathematical" nature of Western music.
[1] Rehmeyer claims "[music] is grounded in hard-headed science. For example, mathematical principles underlie the organization of Western music into 12-note scales."
The 12 note equal tempered scale can be described mathematically, but it is not based on mathematics. This is the same fallacy as the claim that because a row of tract houses are labelled with numbers, the people who live in those tract houses live there because of mathematics. That's obviously foolish and false. The map is not the territory, a fact which Rehmeyer et al seem to have forgotten. In reality, a study of music history shows us that Greek music theory derives from vacuous idle speculations about gods and souls. The Pythagoreans, who originally asserted that music was based on mathematics, spewed out a variety of wacky beliefs, including the claim that some numbers are male and other numbers are female, that all music was based on tetraktys (a numerological set consisting of various arrangements of the first 4 integers) and that their cult leader Pythagoras could hear a "music of the spheres" which no one else could detect. The Pythagorean cult never provided any hard evidence for any of their bizarre claims, and no one today takes these claims seriously.
So let's ask Rehmeyer and Tymoczko: what evidence would they require to disprove the claim that "mathematical principles underlie the organization of Western music into 12-note scales"?
Please cite the specific double-blind experiment whose results would suffice to disprove that claim.
Of course they won't answer, since they're spewing vacuous pseudoscience, so let me interrupt the vast echoing silence formed by their lack of reply to point out that in actual fact the division of the Western octave into 12 logarithmically equal parts is a quite recent innovation. From circa 1490 to 1700, Western music used meantone tuning, which is quite different matheamtically from the 12 logarithmically equal parts vaunte dby Rehmeyer et al. From circa 600 A.D. to 1490, Western music used 12 notes of Pythagorean tuning, which is once again entirely mathematically different from the 12 logarithmically equal pitches we use today. And before that, from circa 500 B.C. to around 600 A.D., Western music used the three Greek genera formed from conjunct or disjunct tetrachords, a system which has nothing in common, mathematically speaking, with the 12 logarithmically pitches described by Rehmeyer et al.
So it seems that Rehmeyer and Tymoczko are desperately ignorant of the fundamentals of Western musical history. This is not a good place from which to launch grandiose assertions that music "is grounded in hard-headed science." Hint to music theorists: before you make sweeping claims about Western music, make sure you know something about Western music history.
[2] Tymoczko claims: "The structure of the space makes certain choices overwhelmingly natural or convenient. There's something similar that goes on with music. When you think about things abstractly, you can come to understand that the directions that music went aren't completely arbitrary. Composers are exploring the possibilities that musical space presents them with."
Please provide us with a set of double-blind objective experiments which would suffice to disprove that claim.
What's that?
You can't?
Hm. Well, in that case, your claims appear to be unfalsifiable, don't they?
And what do we know about unfalsifiable theories? They're one of the prime hallmarks of pseudoscience.
Astrologers and experts in gematria and other crackpot paractitioners of empty numerology consistently fail to provide us with a way of disconfirming their hypotheses... Just as Tymoczko fails to provide us with a way to disconfirm his vacuous claims about alleged connections between mathematics and music.
[3] Rehmeyer gushes: "Next, he noted that the order of the notes in a chord doesn't much matter."
Sorry, but this is just grossly ignorant of basic musicianship. Any musician learns in the Introduction To Music 101 course that a second inversion triad sounds "wrong" and peculiar in the finalis of a cadence. A I-IV-V-I progression which ends with the final I triad with the fifth in the bass is one of the basic prohibitions in common practice period music. Tymoczko's claim that a cadential 6-4 chord sounds or functions the same as a root triad simply betrays Tymoczko's gross ignorance of basic music theory.
I quote from a standard text which ought to be well-known to any so-called music "theory," especially one who teaches at Princeton:
These are the least stable of the inverted chords. Because of historical and traditional reasons, the interval of a fourth above the bass has been treated as a mild dissonance in much of music of the common practice period. Second inversion chords contain this interval and because of this, require special treatment.
From his statement, it's clear that Tymoczko is not only pervasively uninformed about the scholarly literature concenring music and mathematics, he's not only bereft of knowledge of Western music history, but he's also systematically ignorant of the basics of Western music practice during the common practice period (ca. 1500 - 1900).
In summary, what we have got here is a guy who churns out some numerology and whips up pretty pictures to support it. Alas, there's nothing special about that. Astrologers have done the same thing for millenia, and so have numerolgoists, palmists, ufologists, and all manner of psychic surgeons and crackpot "sonic healing" gurus.
In fact, Tymoczko's diagrams and numerology prove distressingly similar to the "Sympathetic Vibratory Physics" of the 19th century pereptual motion machinist and musical crackpot John W. Keely.
I see no meaningful difference between Keely's "sacred geometry" pseudoscience and Tymoczko's numerological twaddle. In both cases, the "theorists" provide us with set of experiments which would suffice to disconfirm their hypotheses; in both cases, the subject matter of their idle speculations involves diagrams so vacuous and so nebulous that it is impossible to discern what concrete specific hypothesis they are actually putting forth. In both cases, these "theorists" systematically avoid making specific predictions which allow anyone to test their claims using the scientific method in the real world. (Viz., the conditions under which listeners will hear musical dissonance).
Lastly, both these "theorists" studiously ignore non-Western music. The Javanese gamlean, for example, uses a comleteloy different tuning from the Western 12-equal system, as does the xylophone of the Kwaiker Indians, the Harp of the people of Ghana, the mbiras fo Central Africa, and so on. By systematically ignoring the music of 80% of the world's population, both these so-called msuic "theorists" leave us with the impression that Western music is only practice worthy of the name "music," a white supremist ideology so long discredited and so laughably obsolete that it cannot be taken seriously.
Once upon a time, crica 1890, German music theorists made these sorts of claims: today, as Jaap Junst and Mantle Hood have long pointed out, we know better. But apparently Tymoczko and Rehmeyer still find themselves stuck in the year 1890.
Why is it that serious mathematicians and reputable physicists, who ought to know better, always lose sight of the most basic elements of the scientific method as soon as they foot in the realm of music theory?
If a claim is to be scientific, it must be [1] meaningful; [2] we must be able to disconfirm it by experiment;; and [3] the claim must be quantifiable.]
Where are the quantifiable and objectively testable claims made by Rehmeyer's and Tymoczko? Where is the evidence tha vague idle speculations like "Composers also need to write music in such a way that our minds can link the sounds into simultaneous, overlapping melodies. This is this easiest when each individual melody moves only in fairly small steps" are even meaningful? This is laughably foolish on its face. The first movement of Beethoven's Moonlight Sonata uses triad arpeggions -- is Tymoczko claiming that piece isn't good music because the individual melody doesn't move "only in fairly small step"?
This entire article is deeply embarrassing...proof once again that the human brain is a marvellous organ which begins working the instant we're born, and only stops when a scientist begins talking about music theory.
Posted by: mclaren | March 22, 2008 11:20 PM
Julie: Thank you for this introductory overview of Tymoczko's fascinating work. While musical inspiration cannot be reduced to math algorithms, the structure of music is amenable to math modeling, as shown in Tymoczko's work.
The difference between life (as in musical inspiration) and form (as in musical structure) should be clearly outlined in any debate about the validity of Tymoczko's approach. True, theoretical explorations into the structure of musical compositions is like dissecting cadavers to learn anatomy, but medical sciences have made great progress with the information gained from autopsies. The scientific study of anatomy, while not the same as physiology, is a valid scientific research approach which sheds light on physiology. Likewise for the exploration of math models in music.
The charge by devotees of the "scientificist" doctrine claiming that double-blind studies are the only and final word on the truth of anything is misleading. Since Euclid, never has mathematics used such empirical approach to prove its theorems and corollaries. While asserting the true merits of the scientific method, these modern "debunking" Inquisitors fail to see the limits of such "blind" approach. As shown by Khun, the scientific method is an evolving standard of consensual beliefs. If reality were to be judged only by its measure, magnetic fields did not exist before our physical senses (and their scientific extensions) were able to measure them.
Music and mathematics do share a common core of principles. Whoever denies the geometrical beauty of a perfect fifth must suffer some sort of congenital blindness. The convenient logarithmic approximations of our twelve-tone equal temperament scale should not cloud the fact that physical ratios of standing wave patterns (harmonics) underlies musical pitches and intervals. We should welcome Tymoczko's explorations of this psychological and geometrical space. Thank you. -JB
Posted by: JB | March 30, 2008 09:33 AM
mclaren has misunderstood the purpose of a mathematical model. Mathematical models don't get proved. They are only useful or not useful. They are like a "map" of a conceptual domain. Does this map provide useful information or not? That is the only test.
Just because lots of conjectured maps of the southern hemisphere in centuries gone by were false doesn't mean that the idea of having a map is foolish.
Another way to think of mathematics is as a language. Do the words and verbs in some language, that is, do the concepts used to describe the phenomenom at hand, give any insight? If not, then that particular set of concepts is not useful. The fact that one set of concepts is not useful does not show that all sets of concepts are pointless!!
Further, double blind experiments are used in experiments involving humans, usually in drug trials, to avoid the subjects being tested figuring out if their pills are the placebo or not by the unconscious actions of the people carrying out the trial. You do not use double blind experiments in physics or chemistry. maclaren's insistence on double blind experiments show a a serious lack of understanding of the purpose of mathematics in science. The emotive language used by maclaren shows only a fear of mathematics; there are no serious objections to using mathematics raised, only to not very good theories in ages past.
Mathematics, as an entity, is conceptually beautiful: it is clean, pure thought. No-one is attempting to describe emotional responses in listeners by mathematics, that would be silly. But to understand a little of the structure of a composition using mathematics only adds to the magic. It can never detract, because there is always another question to be answered. The questions only get more profound, never less.
Posted by: eliz | April 22, 2008 11:59 AM