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A Geometry of Music
Dmitri Tymoczko
Princeton University
[email protected]
http://music.princeton.edu/~dmitri/SIAM.ppt
http://music.princeton.edu/~dmitr
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.
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
(C4,write:
E4, G4)(C4, F5, A5) or
(60, 64, 67)(60, 65, 69)
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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 Rn.
• Example 1: two-note chords.
– Start with the Cartesian plane, R2.
– 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
Second note
• Mapping a score to
R2.
First note
• Rotating the axes.
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Geometry (2)
• Example 2: three-note chord types (OPT)
– Start with the Cartesian plane, R3.
– 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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(C4, E4, G4)(C4, F5, A5)
(E3, G4, C5)(F3, A4, C5)
• 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.
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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 Lp
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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 polynomialtime algorithm for determining the distance between two chordtypes?
– It’s easy for L1 and L∞, but hard for L2.
– 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 nonEuclidean 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 build new
instruments.
• We can provide
simultaneous, realtime visualization of
complex musical
structures (at a
concert, say).
• New educational
paradigms.
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Thank you!
D. Tymoczko, “The Geometry of Musical
Chords.” Science 313 (2006): 72-74.
C. Callender, I. Quinn, and D. Tymoczko,
“Generalized Voice-Leading Spaces.”
Science 320 (2008): 346-348.
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