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Poverty and Polyphonya connection between

economics and music

BRIDGES, Donostia, Spain, July 24, 2007

Time and money

- In which year did incomes change the most?

Question

- In which year did incomes change the most?

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

Poverty and polyphony

Economic changes

How should we measure simultaneous changes?

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.

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?

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

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.

Voice leading inChopins E minor prelude

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.

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

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.

Questions

- Is there an efficient algorithm for comparing

voice leadings (or economic changes) according to

the distribution constraint? - How should we measure voice leadings?

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.

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.

Examples of acceptable measures

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

And now for something completely different

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?

Representation in Rn

- An ordered n-tuple of pitches corresponds to a

point in n-dimensional space (Rn ). A musical

score determines a path.

Symmetries of 2D pitch space

More symmetries

Tilings and orbifolds

- Any path through a tiling can be represented on

one tile with edge identificationsan orbifold.

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).

Chopin revisited

Three-note chord space

- T3/S3

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

The taxicab and the crow

- In real life, we sometimes disagree on how to

measure and compare distances.

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?

The geometry of submajorization

Submajorization tells us when all acceptable

measures will agree, and when some will disagree.

determined by a family of polytopes

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.

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.

Chord-type space for trichords

modding out by transposition, permutation, and

octave equivalence

modding out by transposition and permutation

Chord-type space for trichords

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.

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.

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

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)

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.

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.

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.

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?

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.

Contact

Thank you!

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