low-dimensional musical pitch and chord spacesthe pitch-class distance between x = 11 and y = 2 is...
TRANSCRIPT
Low-dimensional musical pitch and chord spaces
Jordan Lenchitz
Indiana University
December 7th, 2015
Jordan Lenchitz (Indiana University) Low-dim’l pitch and chord spaces December 7th, 2015 1 / 28
Outline
1 Foundational definitions
2 Modeling pitch spaces
3 Some metrics
4 The geometry of 2- and 3-chord spaces
Jordan Lenchitz (Indiana University) Low-dim’l pitch and chord spaces December 7th, 2015 2 / 28
Definitions
Frequency
Pitch
Chroma
Octave
Pitch class
n-chord
Jordan Lenchitz (Indiana University) Low-dim’l pitch and chord spaces December 7th, 2015 3 / 28
1-dimensional pitch space: a pitch-class model [Z12]
Jordan Lenchitz (Indiana University) Low-dim’l pitch and chord spaces December 7th, 2015 4 / 28
Aural motivation
Functional perception of pitch
Understanding tonal / atonal music
Jordan Lenchitz (Indiana University) Low-dim’l pitch and chord spaces December 7th, 2015 5 / 28
Measuring distance between pitches: d1
Definition
Let x , y be pitch classes. Then the function d1 : Z12 × Z12 → {0, . . . , 6}given by d1(x , y) = min(x − y mod 12, y − x mod 12) gives theirpitch-class distance.
Proposition
d1 is a metric on Z12.
Example
The pitch-class distance between x = 11 and y = 2 ismin(2− 11 mod 12, 11− 2 mod 12) = min(−9 mod 12 , 9 mod 12 ) =min(3, 9) = 3.
Jordan Lenchitz (Indiana University) Low-dim’l pitch and chord spaces December 7th, 2015 6 / 28
Measuring distance between pitches: d1
Definition
Let x , y be pitch classes. Then the function d1 : Z12 × Z12 → {0, . . . , 6}given by d1(x , y) = min(x − y mod 12, y − x mod 12) gives theirpitch-class distance.
Proposition
d1 is a metric on Z12.
Example
The pitch-class distance between x = 11 and y = 2 ismin(2− 11 mod 12, 11− 2 mod 12) = min(−9 mod 12 , 9 mod 12 ) =min(3, 9) = 3.
Jordan Lenchitz (Indiana University) Low-dim’l pitch and chord spaces December 7th, 2015 6 / 28
Measuring distance between pitches: d1
Definition
Let x , y be pitch classes. Then the function d1 : Z12 × Z12 → {0, . . . , 6}given by d1(x , y) = min(x − y mod 12, y − x mod 12) gives theirpitch-class distance.
Proposition
d1 is a metric on Z12.
Example
The pitch-class distance between x = 11 and y = 2 ismin(2− 11 mod 12, 11− 2 mod 12) = min(−9 mod 12 , 9 mod 12 ) =min(3, 9) = 3.
Jordan Lenchitz (Indiana University) Low-dim’l pitch and chord spaces December 7th, 2015 6 / 28
n-chord spaces
Definition
The n-chord space is the nth Cartesian product of 1-dimensional pitchspace mod permutations of its ordinates, ie (Z12)n/
∑n.
Example
Equivalency classes in 3-chord space
Jordan Lenchitz (Indiana University) Low-dim’l pitch and chord spaces December 7th, 2015 7 / 28
n-chord spaces
Definition
The n-chord space is the nth Cartesian product of 1-dimensional pitchspace mod permutations of its ordinates, ie (Z12)n/
∑n.
Example
Equivalency classes in 3-chord space
Jordan Lenchitz (Indiana University) Low-dim’l pitch and chord spaces December 7th, 2015 7 / 28
Measuring distance between chords: dn
Definition
Let x = (x1, . . . , xn), y = (y1, . . . , yn) be n-chords. Then the functiondn : (Z12)n/
∑n× (Z12)n/
∑n→ {0, . . . , 6n} given by
dn(x , y) = minσ∈Sn
∑i
d1(xi , yσ(i)) gives their chord-class distance.
Proposition
dn is a metric on (Z12)n/∑
nfor 2 ≤ n ≤ 4.
Conjecture
dn is a metric on (Z12)n/∑
nfor n ≥ 5.
Jordan Lenchitz (Indiana University) Low-dim’l pitch and chord spaces December 7th, 2015 8 / 28
Measuring distance between chords: dn
Definition
Let x = (x1, . . . , xn), y = (y1, . . . , yn) be n-chords. Then the functiondn : (Z12)n/
∑n× (Z12)n/
∑n→ {0, . . . , 6n} given by
dn(x , y) = minσ∈Sn
∑i
d1(xi , yσ(i)) gives their chord-class distance.
Proposition
dn is a metric on (Z12)n/∑
nfor 2 ≤ n ≤ 4.
Conjecture
dn is a metric on (Z12)n/∑
nfor n ≥ 5.
Jordan Lenchitz (Indiana University) Low-dim’l pitch and chord spaces December 7th, 2015 8 / 28
Measuring distance between chords: dn
Definition
Let x = (x1, . . . , xn), y = (y1, . . . , yn) be n-chords. Then the functiondn : (Z12)n/
∑n× (Z12)n/
∑n→ {0, . . . , 6n} given by
dn(x , y) = minσ∈Sn
∑i
d1(xi , yσ(i)) gives their chord-class distance.
Proposition
dn is a metric on (Z12)n/∑
nfor 2 ≤ n ≤ 4.
Conjecture
dn is a metric on (Z12)n/∑
nfor n ≥ 5.
Jordan Lenchitz (Indiana University) Low-dim’l pitch and chord spaces December 7th, 2015 8 / 28
Measuring distance between chords: dn
Example
Let x = (0, 3) and y = (5, 7).Then d2(x , y) = min(d1(0, 5) + d1(3, 7), d1(0, 7) + d1(3, 5)) =min(5 + 4, 5 + 2) = min(9, 7) = 7.
Jordan Lenchitz (Indiana University) Low-dim’l pitch and chord spaces December 7th, 2015 9 / 28
Benefits and limitations of dn
Benefits
Measuring dissonanceUnderstanding atonal musicWorks for arbitrarily large n-chords
Limitations
CoarsenessMeaningless distances between degenerate n-chords.
Jordan Lenchitz (Indiana University) Low-dim’l pitch and chord spaces December 7th, 2015 10 / 28
Benefits and limitations of dn
Benefits
Measuring dissonanceUnderstanding atonal musicWorks for arbitrarily large n-chords
Limitations
CoarsenessMeaningless distances between degenerate n-chords.
Jordan Lenchitz (Indiana University) Low-dim’l pitch and chord spaces December 7th, 2015 10 / 28
Benefits and limitations of dn
Benefits
Measuring dissonanceUnderstanding atonal musicWorks for arbitrarily large n-chords
Limitations
CoarsenessMeaningless distances between degenerate n-chords.
Jordan Lenchitz (Indiana University) Low-dim’l pitch and chord spaces December 7th, 2015 10 / 28
Building a tonal dictionary
Definition
For any specific permutation representation of an n-chord (x1, . . . , xn) withn ≥ 2, its interval vector is the ordered (n − 1)-tuple〈x2 − x1, . . . , xn − xn−1〉.
Example
The interval vector of (0, 3, 8) is 〈3, 5〉, while the interval vector of (8, 0, 3)is 〈4, 3〉. In general, n-chords in the same equivalence class do not usuallyall have the same interval vector.
Jordan Lenchitz (Indiana University) Low-dim’l pitch and chord spaces December 7th, 2015 11 / 28
Building a tonal dictionary
Definition
For any specific permutation representation of an n-chord (x1, . . . , xn) withn ≥ 2, its interval vector is the ordered (n − 1)-tuple〈x2 − x1, . . . , xn − xn−1〉.
Example
The interval vector of (0, 3, 8) is 〈3, 5〉, while the interval vector of (8, 0, 3)is 〈4, 3〉. In general, n-chords in the same equivalence class do not usuallyall have the same interval vector.
Jordan Lenchitz (Indiana University) Low-dim’l pitch and chord spaces December 7th, 2015 11 / 28
Building a tonal dictionary
Definition
For any specific permutation representation of an n-chord (x1, . . . , xn) withn ≥ 2, its interval vector is the ordered (n − 1)-tuple〈x2 − x1, . . . , xn − xn−1〉.
Example
The interval vector of (0, 3, 8) is 〈3, 5〉, while the interval vector of (8, 0, 3)is 〈4, 3〉. In general, n-chords in the same equivalence class do not usuallyall have the same interval vector.
Jordan Lenchitz (Indiana University) Low-dim’l pitch and chord spaces December 7th, 2015 11 / 28
Building a tonal dictionary (cont.)
Example
Dictionary of 3-chords [triads]: a subset of (Z12)3/∑
3
Major: 〈4, 3〉
Minor: 〈3, 4〉
Diminished: 〈3, 3〉
Augmented: 〈4, 4〉
Jordan Lenchitz (Indiana University) Low-dim’l pitch and chord spaces December 7th, 2015 12 / 28
Building a tonal dictionary (cont.)
Example
Dictionary of 4-chords [seventh chords]: a subset of (Z12)4/∑
4
〈4, 3, 3〉
〈4, 3, 4〉
〈3, 4, 3〉
〈3, 4, 4〉
〈3, 3, 4〉
〈3, 3, 3〉
〈4, 4, 3〉
Jordan Lenchitz (Indiana University) Low-dim’l pitch and chord spaces December 7th, 2015 13 / 28
Toward a tonal metric: the root of a 3- or 4-chord
Definition
The root x of a 3- or 4-chord x in the dictionary is any pitch class in thechord for which there exists a permutation σ such that σ(x1) = x and(x , . . . , xσ(n)) has interval vector consisting solely of 3s and/or 4s.
Example
The root of x = (0, 3, 8) is 8 because (8, 0, 3) has interval vector 〈4, 3〉.
Example
The root of y = (3, 7, 11) is 3, 7, or 11 since y has interval vector 〈4, 4〉 soany permutation of y ’s ordinates has interval vector 〈4, 4〉.
Jordan Lenchitz (Indiana University) Low-dim’l pitch and chord spaces December 7th, 2015 14 / 28
Toward a tonal metric: the root of a 3- or 4-chord
Definition
The root x of a 3- or 4-chord x in the dictionary is any pitch class in thechord for which there exists a permutation σ such that σ(x1) = x and(x , . . . , xσ(n)) has interval vector consisting solely of 3s and/or 4s.
Example
The root of x = (0, 3, 8) is 8 because (8, 0, 3) has interval vector 〈4, 3〉.
Example
The root of y = (3, 7, 11) is 3, 7, or 11 since y has interval vector 〈4, 4〉 soany permutation of y ’s ordinates has interval vector 〈4, 4〉.
Jordan Lenchitz (Indiana University) Low-dim’l pitch and chord spaces December 7th, 2015 14 / 28
Toward a tonal metric: the root of a 3- or 4-chord
Definition
The root x of a 3- or 4-chord x in the dictionary is any pitch class in thechord for which there exists a permutation σ such that σ(x1) = x and(x , . . . , xσ(n)) has interval vector consisting solely of 3s and/or 4s.
Example
The root of x = (0, 3, 8) is 8 because (8, 0, 3) has interval vector 〈4, 3〉.
Example
The root of y = (3, 7, 11) is 3, 7, or 11 since y has interval vector 〈4, 4〉 soany permutation of y ’s ordinates has interval vector 〈4, 4〉.
Jordan Lenchitz (Indiana University) Low-dim’l pitch and chord spaces December 7th, 2015 14 / 28
Toward a tonal metric: the circle of fifths
Jordan Lenchitz (Indiana University) Low-dim’l pitch and chord spaces December 7th, 2015 15 / 28
Circle-of-fifths distance [for 3- or 4-chords in the dictionary]
Definition
The circle-of-fifths distance dC between two 3- or 4-chords x and y in thedictionary with roots x ≡ 7j and y ≡ 7k is given by dC (x , y) = d1(j , k). Ifat least one of the chords does not have a unique root, then dC isunderstood to be the minimum over all possible roots of the chord(s).
Example
The Star Wars Theme: x = (7, 11, 2), y = (4, 8, 11), z = (10, 2, 5) sox = 7, y = 4 and z = 10, all with interval vector 〈4, 3〉.Now x ≡ 7(1) mod 12, y ≡ 7(4) mod 12 and z ≡ 7(10) mod 12 sodC (x , y) = d1(1, 4) = min(−3, 3) = 3 anddC (y , z) = d1(4, 10) = min(−6, 6) = 6.
Jordan Lenchitz (Indiana University) Low-dim’l pitch and chord spaces December 7th, 2015 16 / 28
Circle-of-fifths distance [for 3- or 4-chords in the dictionary]
Definition
The circle-of-fifths distance dC between two 3- or 4-chords x and y in thedictionary with roots x ≡ 7j and y ≡ 7k is given by dC (x , y) = d1(j , k). Ifat least one of the chords does not have a unique root, then dC isunderstood to be the minimum over all possible roots of the chord(s).
Example
The Star Wars Theme: x = (7, 11, 2), y = (4, 8, 11), z = (10, 2, 5) sox = 7, y = 4 and z = 10, all with interval vector 〈4, 3〉.Now x ≡ 7(1) mod 12, y ≡ 7(4) mod 12 and z ≡ 7(10) mod 12 sodC (x , y) = d1(1, 4) = min(−3, 3) = 3 anddC (y , z) = d1(4, 10) = min(−6, 6) = 6.
Jordan Lenchitz (Indiana University) Low-dim’l pitch and chord spaces December 7th, 2015 16 / 28
Embedding 2-chord space in a Mobius Strip
Geometric intuition
(S1 × S1)/∑
2
Jordan Lenchitz (Indiana University) Low-dim’l pitch and chord spaces December 7th, 2015 17 / 28
Embedding 2-chord space in a Mobius Strip
Geometric intuition
(S1 × S1)/∑
2
Jordan Lenchitz (Indiana University) Low-dim’l pitch and chord spaces December 7th, 2015 17 / 28
Embedding 2-chord space in a Mobius Strip
Jordan Lenchitz (Indiana University) Low-dim’l pitch and chord spaces December 7th, 2015 18 / 28
Embedding 2-chord space in a Mobius Strip
Jordan Lenchitz (Indiana University) Low-dim’l pitch and chord spaces December 7th, 2015 19 / 28
Embedding 2-chord space in a Mobius Strip
Jordan Lenchitz (Indiana University) Low-dim’l pitch and chord spaces December 7th, 2015 20 / 28
Embedding 2-chord space in a Mobius Strip
Jordan Lenchitz (Indiana University) Low-dim’l pitch and chord spaces December 7th, 2015 21 / 28
Embedding 2-chord space in a Mobius Strip
Jordan Lenchitz (Indiana University) Low-dim’l pitch and chord spaces December 7th, 2015 22 / 28
Embedding 2-chord space in a Mobius Strip
Jordan Lenchitz (Indiana University) Low-dim’l pitch and chord spaces December 7th, 2015 23 / 28
Embedding 2-chord space in a Mobius Strip
Jordan Lenchitz (Indiana University) Low-dim’l pitch and chord spaces December 7th, 2015 24 / 28
Embedding 2-chord space in a Mobius Strip
Jordan Lenchitz (Indiana University) Low-dim’l pitch and chord spaces December 7th, 2015 25 / 28
What’s the boundary?
Jordan Lenchitz (Indiana University) Low-dim’l pitch and chord spaces December 7th, 2015 26 / 28
What’s the boundary?
Jordan Lenchitz (Indiana University) Low-dim’l pitch and chord spaces December 7th, 2015 26 / 28
Embedding 3-chord space in a twisted triangular torus
Jordan Lenchitz (Indiana University) Low-dim’l pitch and chord spaces December 7th, 2015 27 / 28
Further applications
Voice-leading
Modulation in 19th century harmony
Comparing Western and Eastern music
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