The Shape of Music

pagina 1 van 2 Seed: The Shape of Music j SEEDMAGAZINE.COM RENOLUTID & : A Seven young innovatars working at the intersection of science, design, and architecture 7 [trs 24 2: Dow - Seo tvs world through ino eyes of . the Human Element SEEDAAGAZINE. CON PHYSICS & MATH The Shape of Music How do harmony and melody combine to make music? by DMITRI TYMOCZKO « Posted Katy 9, 2008 04 03 PM Roughly 2,500 years ago, Pythagoras observed that objects, such as the anvils he purportedly CR studied, produced harmonious sounds while ing at frequencies N in simple whole ratios. More complex ratios gave rise to more dissonant sounds, which indicated that human beings were unconsciously sensitive to relationships ink werld could be described th ical math in nature. By showing that the ically, Pythagoras not only provided an important inspiration for physics, but he also discovered a particular affinity between mathematics and music~one that Gottfried Leibniz was to invoke centuries later when he described music as the “unknowing exercise of our mathematical faculties.” For a thousand years, Western musicians have end: d to satisfy two fundamental constraints in their compositions. The first is that melodies should, in general, move by short distances. When played on apiano, melodies typically move to nearby keys rather than take large jumps across the keyboard. The second is that music should use chords { of si y ded notes) that are audibly similar. Rather than leap willy-nilly between completely unrelated sonorities, musicians typically restrict themselves to small portions of the musical particularly Western: Many non-Westem styles either reject chords altogether, using only one note at a time or build entire pieces around a single unchanging harmony. Zed. YMoczko,D. TT IA So universe, for instance by using only major and minor chords. While the melodic constraint is nearly universal, the harmonic constraint is more Together these ints ensure a two-di ional coh in Western music analogous to that of a woven cloth. Music is a collection of simultaneously occurring melodies, parallel horizontal threads that are held together tightly by short-distance motion. But Western music also has a vertical, or harmonic, coherence. If we consider only the notes sounding at any one instant, we find that they form familiar chords related to those that sound at other instants of time. These basic requirements impose nontrivial constraints on composers--not just any sequence of chords we imagine can g a collection of short-di melodies. We might therefore ask, how do we combine harmony and melody to make music? In other words, what makes music sound good? To answer these questions, we need > just as Pythagoras supposed. But as 1 and other music theorists have recently shown, we need a kind of mathematics that Pythagoras could not have imagined: the geometry and topology of what math icians call “quotient spaces” or “orbifolds.” These exotic spaces contain singularities-- “unusual” points that are analogous to the black holes of Einstein's general relativity—that can be described using only very recent mathematics. Western music can ultimately be represented as a series of points and tine segments on abstract shapes in higher di ions. If we can und d their structure, then the deep principles underlying Western music will finally be revealed. To tum music into math, we begin by numbering the keys on the piano from low to high. Musicians typically number the 88 piano keys so that the lowest is 2! and the highest is 108, with middle C at 60. Mathematically, these numbers are the legarithms of the slowest frequency at which the piano string is vibrating. In principle we can assign numbers even to notes that are not found on the keyboard, with 60.5 referring to the note halfway between middle C and the next-highest key. These numbers refer to pitches. Next we mode! the phenomenon of “octave equivalence”: the fact that notes 12 keys apart sound similar. (As Maria teaches in The Sound of Music, “ti” brings us back to "do.") To do this mathematically, we divide our piano key numbers by 12 and keep only the remainder. In this way each of the 88 piano keys is assigned a number less than 32: the °C" keys 48, 60, and 72 are represented by 0, while the "C-sharp" (or "D-flat*), keys 49, 61, and 73 are all represented by 3, and so on. Musicians say that these numbers refer to pitch classes, representing the intrinsic “character” or “color” ofthe note. G ically, pitch cl clock--though "12" on this clock refers to "0." Musically, the order ofa group of notes is less imp all live on a circle divided into 12 equal parts, exactly like the face of an ordinary than its The ordered to E-G-C, or 4-7-12; musicians consider both to be “C major chords.” A chord is therefore C-E-G, or 12-4-7, on the clock, is audibly related defined as an dered collection ofpitch cl. http://www.seedmagazine.com/news/2008/07/the_shape_of_music.php?page=all&p=y

Seed: The Shape of Music ponding gi pagina 2 van 2 d set of points on a circle like hours on a clock face. ically to an Chords that are related by rotation on the clock face all sound similar. For example, take the C major chord (12, 4, 7), and move each of the notes clockwise two places. This is the D major chord (2, 6, 9 on the clock), which sounds very much like the C major. In fact, a chord is a major chord if and only if it can be obtained by rotating the C major (so 3, 7, 10 would be another one, the E-flat major chord). The reason these chords all sound alike is that the human ear is more sensitive to the distances between notes than their absolute position on the clockface. Rotating each of the hands of a clock together doesn't change the distance between them and doesn't alter the chord's quality. We can use this clock analogy to understand the two constraints of Wester music mentioned earlier. To satisfy the harmonic constraint, composers nced to use chords that are related by rotation, or at least approximately so. This successive chord stay pretty much constant. To satisfy the melodi that the di p b the notes in cach the notes of successive chords by short distances. For example, one could connect the C major chord (12, 4, 7) to the F major chord (5, 9, 12) by keeping the 12 fixed, moving the 4 one place clockwise to 5, and moving the 7 two clockwise places to 9. This represents a much mere efficient alternative to moving each note five places clockwise. Western music is built out of a seq of such mapping: fe ing a two-dimensional sonic tapestry. The final stage in the process of translating music into math is to pass into the clock's configuration space: Rather than representing chords using multiple points on a one-dimensional circle, we construct an equivalent, higher-dimensional space in which every chord is a different point. The term “configuration space” refers to the fact that points in the higher-di ional space rep "configurations" (or arrangements) of the points on the lower-dimensional circle. These spaces are considerably more interesting than the plain-vanilla spaces of ordinary Euclidean geometry. Here, a complexity arises because the notes in a chord are dered, wh the di of ag: ical point are typically ordered. Recall from high school geometry that a Cartesian plane is used to model ordered pairs of real numbers (x, y). To create the space of unordered points on a circle, we can just “fold” the familiar Cartesian spaces (representing ordered points on a line) in various ways. In two dimensions (when there are two notes in each chord), we first wrap around each axis, x and y, so that they become circles rather than lines. The resulting space is a doughnut, or in mathematical parlance, a torus. Second, we glue together all the points in the doughnut whose coordinates are related by reordering--so in two dimensions, (x, y) and (y, x) become the same point. In three dimensions (for three notes in each chord), the process is much trickier; we must glue together all six permutations of (x, y. 2), and so on. When, the dust settles, two-note chords live on a Mobius strip, three-note chords live on a solid, on higher-di ional analogues, twisted triangular doughnut, and larger notes live whose shapes become difficult to describe nonmathematically. The boundary of each space, or shape, is geometrically unusual ("singular”)--line segments appear to “bounce off” the boundary, rather like billiard balls reflecting off the edge of a pool table. The structure of these spaces, representing all possible chords, shows us exactly how the two elemental properties of Western music can be combined. Structurally similar chords live on circles that wind through the spaces multiple times (these circles can be understood as lines that retum back upon themselves like the Earth's equator). Melodic connections between chords--such as “hold 12 constant, move 4 one unit clockwise to become 5, and move 7 two units ise to b 9°--are rep did line segments in the space that may retum back on themselves, or bounce off the space's boundaries. Our original musical ion about g harmony and meledythusb age ical q about finding circles that are “close to themselves”—~that is, circles containing two points that can be connected by short line segments. The most direct way to combine melody and harmony is to use chords that divide the 12 positions on our clockface of notes nearly (but not precisely) evenly, such as the C major chord (12, 4, 7), which divides the clockface into three roughly equal parts. These harmonies occupy the center of our musical spaces, and are thus able to take effective advantage ofits non-Euclidean twists, Remarkably, in the 12-tone system of notes, these are precisely the chords that Pythagoras identified almost 2,500 years ago: the chords that sound intrinsically harmonious. Far from arbitrary ot haphazard, scales and chords come close to being the unique solutions to the problem of creating two-dimensional musical coherence. Contrary to the hopes of generations of avant-garde composers, it follows that the goal of developing robust alternatives to tonality may be extremely difficult, if not impossible, to achieve. The shapes of the space of chords we have described also reveal deep connections between a wide range of musical genres. It tums out that superficially different styles—Renaissance music, classical and Romantic music, jazz, rock, and other popular forms~all make remarkably similar use of the geometry of chord space. Traditional techniques for manipulating musical scales tum out to be closely analogous to those used to connect individual chords. And some composers have displayed a profound understanding of the higher-dimensional geometry of musical chords. In fact, one can argue that Romantic composers such as Chopin had an intuitive feel for non-Euclidean higher-dimensional spaces that exceeded the explicit und ding of their mathematical contemp The ideas I have been describing were first published in an article | wrote in Science in 2006. More recently, Clifton Callender, lan Quinn, and | have shown that the connection between music theory and geometry is in fact much deeper and more comprehensive than even my earlier work indicated: There are in fact large families of g ical spaces ponding to a wide range of musical terms, some of which are considerably more exotic than those described here. (For instance, three-note chord types-such as "major chord" or "minor chord*~live on a cone containing two different flavors of singularity.) Seen in the light of this new geometrical perspective, a wide number of traditional music-theorctical questions become tractable. In some sense musicians have been doing geometry all along without ever realizing it. The mathematician Rachel Hall and I are also exploring some i: ing bi b music theory and ics, Similar g ical spaces appear in both disciplines, and questions about how to measure distances between musical chords are very similar to questions about how to measure the distance between economic states. This may seem implausible until one reflects that the geometrica} operations we have been discussing are very general. Ultimately, the geometry of music is a branch of the geometry of unordered coll and dered coll are basic enough to have applicationsin a wide range of fields. Pythagoras was correct more than two and a half millennia ago: Music provides aes, of ihe sdsatest examples of ranch deep tplation between mathematics and human experience. The Shape ot Music, written by Omitr Tymoczio, posted on July 8, 2008 04 03 PM, io in the category Physica & Math. View bog reactions http://www.seedmagazine.com/news/2008/07/the_shape_of_music.php?page=all&p=y _ 11-7-2008