Poverty and Polyphony a connection between economics and music

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Poverty and Polyphonya connection between
economics and music
BRIDGES, Donostia, Spain, July 24, 2007
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Time and money

• In which year did incomes change the most?

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Question

• In which year did incomes change the most?

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Question

• In which year did incomes change the most?
• What assumptions are we making?
• Should we make these assumptions?
• How robust are our conclusions?

no inflation
inflation
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Poverty and polyphony
Economic changes
How should we measure simultaneous changes?
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Harmony and counterpoint

• The rules of harmony dictate which notes can be
  sounded together at one time.
• The rules of counterpoint dictate how the notes
  in a sequence of chords should be distributed to
  voices to form simultaneous, independent
  melodies.

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Voice leadings

• Composers usually want voices to move by short
  distances in pitch
• instrumentalists can play the music more easily
• hearers can parse the music into simultaneous,
  independent melodies
• Voice leadings assign notes to voices in a way
  that satisfies the demands of both harmony and
  counterpoint.
• What does it mean for one voice leading to be
  more efficient than another?

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Preliminaries

• Pitch is frequency measured on a logarithmic
  scale. There are twelve units of pitch to an
  octave.
• The equivalence class of a pitch modulo octave
  shift is its pitch class.
• Integer pitch classes are notes in twelve-tone
  equal temperament (Z/12Z).

A chord is an unordered multiset of points on the
pitch class circle. Example C major C, E,
G 0, 4, 7
R/12Z
pitch class circle
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Voice leadings

• A voice leading is a bijection from the notes of
  one chord to the notes of another.
• Its displacement multiset is the multiset of its
  arc lengths.
• Example the voice leading
• C?C, E?F, G?A has
• displacement multiset 0, 1, 2.
• Efficient voice leadings are
• collections of short paths linking two
• sets of points on the circle.

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Voice leading inChopins E minor prelude
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The comparison problem

• We may have a clear intuition that one voice
  leading is smaller than another.
• The displacement multiset 2, 0, 0 is
  definitely smaller than 8, 4, 3.
• However, our intuitions do not allow us to choose
  a particular measure for these collections of
  paths.
• Analogy measuring income volatility. We can
  compare individual income changes, and we have
  some intuitions about overall income volatility,
  but we can't single out a particular measure.

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The distribution constraint

• What general principles must any method of
  comparing voice leadings obey?
• DT (2006) proposed the distribution constraint.
• consistent with the voice leading behavior we
  observe in actual music
• partial order on the space of displacement
  multisets

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Distribution constraint

• Decreasing the distance traveled by any voice
  should not make a voice leading larger.
• Eliminating voice crossings should not make a
  voice leading larger.
• Efficient voice leadings preserve the order of
  the voices.
• These same principles apply in economics.

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Questions

• Is there an efficient algorithm for comparing
  voice leadings (or economic changes) according to
  the distribution constraint?
• How should we measure voice leadings?

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Submajorization and the distribution constraint

• Proposition. The partial order on displacement
  multisets determined by the distribution
  constraint is equivalent to the submajorization
  partial order.
• Computation. Easy!
• Start with two equally-sized multisets of
  nonnegative numbers.
• Compare the largest elements of each set, the sum
  of the largest two elements, the sum of the
  largest three elements, etc.
• Submajorization means that all comparisons agree.

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Submajorization

• Submajorization is a weakened form of
  majorization, originally proposed by Lorenz in
  1905 as a way of comparing the inequality level
  of two societies.
• We say that a real-valued function on multisets
  that respects the distribution constraint is an
  acceptable measure.
• One multiset submajorizes another if and only if
  every acceptable measure agrees that it is as
  least as large.

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Examples of acceptable measures

• Sum of changes 4
• Largest change 3
• Square root of the sum of the squared changes
• Many, many more!

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And now for something completely different
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The geometry of chords

• We have represented chords by multisets of points
  on the pitch class circle and voice leadings by
  collections of paths on the pitch circle.
• Chords with n notes can also be represented by
  points in an n-dimensional space. The shape of
  this space is determined by musical symmetries.
• What is the shape of chord space?
• What is the role of voice leadings?

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Representation in Rn

• An ordered n-tuple of pitches corresponds to a
  point in n-dimensional space (Rn ). A musical
  score determines a path.

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Symmetries of 2D pitch space
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More symmetries
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Tilings and orbifolds

• Any path through a tiling can be represented on
  one tile with edge identificationsan orbifold.

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Two-note chord space(Möbius strip)
The orbifold is T2/S2 the 2-torus T2 (from
octave identification) modulo the symmetric group
S2 (were ignoring the order of the voices).
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Chopin revisited
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Three-note chord space

• T3/S3

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Distance????

• What is the meaning of distance in chord space?
• How should we measure it?
• A line segment in Rn determines a voice leading.
  
• Well say that the distance between any two
  chords is determined by the shortest line between
  them in chord space.
• This leads to some disagreement

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The taxicab and the crow

• In real life, we sometimes disagree on how to
  measure and compare distances.

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Acceptable measures

• Every acceptable voice-leading measure gives us a
  different geometry. (different circles, lines,
  etc.)
• Sum of changes Taxicab distance
• Square root of the sum of the squared changes
  Euclidean distance (as the crow flies)
• Each measure gives a different meaning of
  closeness. When do they agree? When do they
  disagree?

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The geometry of submajorization
Submajorization tells us when all acceptable
measures will agree, and when some will disagree.
determined by a family of polytopes
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Comparing chord types

• We can do the same thing in other quotient
  spaces
• Musicians sometimes think about types of chords
  like major chords, minor chords, or dominant
  seventh chords.
• Chord types are multisets of points on the pitch
  class circle, modulo rotation (musical
  transposition).
• We consider some chord types to be fairly similar
  and others to be very different.
• Example Major chords seem similar to augmented
  triads, and not so similar to clusters.

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Geometry of chord types

• Chord-type space is a flattening of chord
  space.
• (It is obtained by projection from chord space
  along the line of transposition.)
• Points in this space represent chord types. Line
  segments represent voice leadings modulo the
  individual transposition of either chord.

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Chord-type space for trichords
modding out by transposition, permutation, and
octave equivalence
modding out by transposition and permutation
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Chord-type space for trichords
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Distance between chord types

• Voice-leading size gives us a notion of distance
  between chords. Can we also use voice-leading
  size to explain our intuitions about distance
  between chord types?
• Analogy measuring income volatility in a way
  that is insensitive to global inflation.

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Closeness of chord types

• If we have chosen a measure of voice leading
  size, we can use the quotient to measure
  similarity.
• The distance between the chord types of X and Y
  is the size of the minimal voice leading from X
  to any transposition of Y.
• (This is like finding the distance between two
  lines.)

Minimizes the largest change.
Minimizes the sum of the changes.
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T-closeness

• But we want to avoid choosing a particular
  measure.
• Suppose X, Y, and Z are chord types. We say that
  X is T-closer to Y than Z is to Y if every
  acceptable measure agrees that the minimal
  distance from X to Y is smaller than the minimal
  distance from Z to Y.
• We have an algorithm (related to submajorization).

Y
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Evenness ordering on trichords
We can draw contours in chord-type space for
trichords along which closeness to the augmented
triad increases in every acceptable measure.
(half of orbifold shown)
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Other neat facts

• All acceptable measures agree on how to minimize
  an inversionally symmetric voice leading.
• Therefore, they agree about the minimal voice
  leadings between two perfect fifths, two major or
  two minor triads, and two dominant or two
  half-diminished sevenths.
• These are among the most common voice leadings we
  find in Western tonal music.

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Application income volatility with inflation

• Given any acceptable measure, we define the
  relative volatility to be the minimal size of a
  mapping from one income distribution to a
  translation of the other. (If we use log-dollar
  space, this is a scaling.)
• We state an algorithm that determines whether all
  acceptable measures agree that the relative
  volatility is smaller in year 1 than in year 2.

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Application income inequality

• Given any acceptable measure of income changes,
  we define the inequality index of a society to
  be the minimal overall change (using this
  measure) to an even division of incomes.
• For example, suppose we add absolute income
  changes. Then
• 20K, 50K, 60K, 100K has inequality index
  90.
• 50K, 50K, 60K, 70K has inequality index 30.
• We say that society X has more inequality than
  society Y if all acceptable measures agree that
  the inequality index of X is greater than that of
  Y.

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Conclusion

• We can use submajorization to compare distances
  between chords and chord types without
  arbitrarily choosing a particular measure.
• On the one hand, our conclusions are robust
  because all acceptable measures agree with them.
• On the other hand, acceptable measures may
  disagree!
• for further research
• Are there perceptual facts, or facts about how
  composers write their music, that lead us to
  choose a particular voice-leading metric? Are
  any of these facts inconsistent with the
  distribution constraint?

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For further reading

• Rachel W. Hall and Dmitri Tymoczko, Poverty
  and polyphony. Preprint, 2007. Available at
  www.sju.edu/rhall.
• Dmitri Tymoczko, The geometry of musical
  chords. Science 313 (2006) 72-74.
• Clifton Callender, Ian Quinn, and Dmitri
  Tymoczko, Geometrical music theory. Preprint,
  2007. Available at music.princeton.edu/dmitri.
• Download ChordGeometries.

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