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A Geometry of Music Dmitri Tymoczko Princeton University

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What makes music sound good? Melodies move by short distances (horizontal) –Auditory streaming Harmonies sound similar (vertical) –chords, whatever they are, are structurally similar Chords sound intrinsically good (“acoustic consonance”) “Limited macroharmony” –Music is limited to 5-8 pitch-classes over moderate stretches of time. Centricity (one note sounds “stable” or restful) Universality vs. cultural conditioning; possible contributions of biology

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Elementary Music Theory I Notes can be represented numerically by the logarithms of their fundamental frequencies. (If done right, this corresponds to numbering piano keys.) We experience frequency space as being periodic, with notes an octave apart sounding similar

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Elementary Music Theory II A musical staff is a two-dimensional graph

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Elementary Music Theory III A musical object is an ordered sequence of notes. The ordering can represent ordering in time, or by instrument. = (64, 67, 70, 74)

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Elementary Music Theory IV A voice leading is a transition from one musical object to another. These are the atomic constituents of musical scores. (C4, E4, G4) (C4, F5, A5) or (60, 64, 67) (60, 65, 69) Here, the music moves from a C major chord to an F major chord, articulating three individual melodies: G moves to A, E moves to F, while C stays fixied. We can write:

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Symmetry and Music Musicians are in the business of combining similar objects. What does it mean to say two objects are similar (or “functionally equivalent”)? Or in other words, what sorts of symmetries do musicians habitually recognize?

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The OPTIC symmetries We’ve defined a musical object as an ordered series of pitches –e.g. (C4, E4, G4) Musicians recognize five “OPTIC” symmetries, which allow you to transform an object without changing its essential identity. –Octave: (C4, E4, G4) ~ (C5, E4, G2) –Permutation: (C4, E4, G4) ~ (E4, G4, C4) –Transposition (or translation): (C4, E4, G4) ~ (D4, F#4, A4) –Inversion (or reflection): (C4, E4, G4) ~ (G4, Eb4, C4) –Cardinality change: (C4, E4, G4) ~ (C4, E4, E4, G4)

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Common Musical Terms expressed in terms of the OPTIC symmetries Chord (e.g. “C major”) = OPC equivalence class Chord type (e.g. “major chord”) = OPTC equivalence class Many others: pitch, pitch-class, chord progression, voice leading, tone row, set, set class, etc.

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Geometry Each combination of OPTIC symmetries produces its own geometrical space. These spaces are quotients of R n. Example 1: two-note chords. –Start with the Cartesian plane, R 2. –Identify all points (x, y), (y, x), (x+12, y) NB: (x, y) ~ (y, x) ~ (y+12, x) ~ (x, y + 12) –The result is an orbifold: a Möbius strip whose singular “boundary” acts like a mirror.

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Example 1 Mapping a score to R 2. First note Second note Rotating the axes.

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Geometry (2) Example 2: three-note chord types (OPT) –Start with the Cartesian plane, R 3. –Identify (x, y, z) with: (x+12, y, z) (y, x, z), (y, z, x) (x + c, y + c, z + c) –The result is a cone whose “boundary” acts as a mirror, and whose tip is singular. –Mathematically, this is the leaf space of a foliation of the bounded interior of a twisted triangular 2-torus.

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Interpreting the Geometry Points represent equivalence classes: –Chords, chord-types, and so on Ordered pairs of points sequences of objects with no implied mappings between their elements – e.g. C major G major –Musicians call these “chord progressions.” Images of line segments represent particular mappings between chords’ elements –For instance, in two note space, there is a line segment (C, E) (D, F) that represents the event in which C moves up by two semitones to D, and E moves up by one semitone to F. –Musicians call these transitions “voice leadings.”

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Here are two different voice leadings. In both cases, C stays fixed, E moves up by semitone to F, and G moves up by two semitones to A. In three-note OP space these are represented by the same line segment. (C4, E4, G4) (C4, F5, A5) (E3, G4, C5) (F3, A4, C5)

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Some interesting math Our spaces are quotients of product spaces: we take several “copies” of R, representing the space of possible pitches, and then apply equivalence relations to the result. A musical scale provides a metric on the underlying one-dimensional space. (It tells us how to go up one.) This means that for a given path in the space, we can compute a set of distances, representing the total distances moved by each individual musical voice. However, this does not give a metric on the product space, since there are many ways to calculate a single distance from a given a set of distances (e.g. the L p norms).

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Some interesting math In other words, we’re somewhere between topology and geometry: we can assign a set of “distances” to every path in the space, but not a single distance. Which metric did Mozart use???

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Some interesting math Rather than choosing a metric arbitrarily, we can try to find reasonable constraints that any acceptable distance metric must obey. One such constraint is that voice crossings never make a path shorter. A restricted form of the triangle inequality.

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Some interesting math The principle that crossings not make a path shorter is (non-obviously) equivalent to the submajorization partial order, which originated in economics. An optimal redistribution of wealth never reorders individuals in terms of wealth. It’s been known for a century, and appears throughout mathematics. It provides us with the ability to compare some distances in our space: a partial-order geometry.

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An open problem Define a chord type as an equivalence class under OPTC: a set of points on the circle modulo transposition (major chord, minor chord, diatonic scale, etc.). Define the distance between chord types as the shortest path between them in OPTC space; these paths can involve duplications. –E.g. the shortest path between (0, 4, 6) and (6, 10, 0) is (0, 0, 4, 6) (10, 0, 6, 6) Q: Given the Euclidean metric, can you construct a polynomial- time algorithm for determining the distance between two chord- types? –It’s easy for L 1 and L ∞, but hard for L 2. –This is a very practical problem, from a musical point of view; it amounts to quantifying a natural notion of similarity.

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OK, so what? For 600+ years, Western music has a two-dimensional coherence. –Melodies move by short distances –Chords (simultaneously sounding notes) are heard to be structurally similar. Q: how is this possible?

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Chords and Scales Furthermore, for 300+ years, Western music has been hierarchically self-similar, combining these two kinds of coherence in two different ways. When a classical composer moves from the key of D major to the key of A major, the note G moves up by semitone to G#, linking two structurally similar scales (D and A major) by a short melodic motion (G G#).

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Western music is hierarchically self-similar, using the same procedures (short melodic motions linking structurally similar harmonies) at two time scales (that of the chord and that of the scale).

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If we can understand how this is possible, we can perhaps demarcate the space of coherent musics — that is the range of possible styles exhibiting melodic and harmonic consistency!

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Geometry to the rescue! Once you understand the geometry of the OPTIC spaces, it is obvious how to combine melodic and harmonic consistency. It’s a matter of exploiting the non- Euclidean features of chord space!

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We can use visual pattern recognition to uncover interesting Musical relationships!

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Chopin moves systematically through this four-dimensional space!

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Using geometry, we can generalize many contemporary music-theoretical concepts. For instance, music theorists have been talking about lines and planes in the OPTIC spaces, without realizing it.

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We can use geometrical distance to construct a notion of similarity that is more flexible than are traditional conceptions

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More practically … We can provide simultaneous, real- time visualization of complex musical structures (at a concert, say). New educational paradigms. We can build new instruments.

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Thank you! D. Tymoczko, “The Geometry of Musical Chords.” Science 313 (2006): C. Callender, I. Quinn, and D. Tymoczko, “Generalized Voice-Leading Spaces.” Science 320 (2008): BOOK NOW AVAILABLE FOR PREORDER AT AMAZON.COM!!!!

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