The Distance Geometry of Deep Rhythms and Scales

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The Distance Geometry of Deep Rhythms and Scales

• by E. Demaine, F. Gomez-Martin, H. Meijer, D.
  Rappaport, P. Taslakian, G. Toussaint, T.
  Winograd, D. Wood

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Rhythms and Scales

• A rhythm is a repeating pattern of beats that is
  a subset of equally spaced pulses.

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Rhythms and Scales

• A rhythm is a repeating pattern of beats that is
  a subset of equally spaced pulses.

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Rhythms and Scales

• A rhythm is a repeating pattern of beats that is
  a subset of equally spaced pulses.

clave Son
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Rhythms and Scales

• A scale is a collection of musical notes sorted
  by pitch.

Diatonic scale
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Rhythms and Scales

• Pitch intervals in a scale are not necessarily
  the same
• Similar to a rhythm, a scale is cyclic

? its geometric representation is
similar to that of a rhythm
Diatonic scale or Bembé
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Erdos Distance Problem (1989)

• Find n points in the plane s.t. for every i
  1,, n-1, there exists a distance determined by
  these points that occurs exactly i times.
• Solved for 2 n 8

(0, 2)
(1, 0)
(1, 0)
(0, 1)
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Erdos Distance Rhythms

• A rhythm that has the property asked by Erdos is
  called a Erdos- deep rhythm

0
15
14
Multiplicity
4
5
10
2
7
4
1
6
5
9
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Erdos Distance Rhythms

• A rhythm that has the property asked by Erdos is
  called a Erdos- deep rhythm

0
15
14
4
Multiplicity
4
5
10
2
7
4
1
6
5
9
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Erdos Distance Rhythms

• A rhythm that has the property asked by Erdos is
  called a Erdos- deep rhythm

0
15
14
4
6
Multiplicity
4
5
10
2
7
4
1
6
5
9
11
Erdos Distance Rhythms

• A rhythm that has the property asked by Erdos is
  called an Erdos- deep rhythm

0
15
14
Multiplicity
4
7
7
5
10
2
7
4
1
6
5
9
12
Erdos Distance Rhythms

• A rhythm that has the property asked by Erdos is
  called an Erdos- deep rhythm

0
15
14
4
4
Multiplicity
4
5
4
10
2
7
4
1
6
5
9
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Winograd Deep Scales

• The term deep was first introduced by Winograd in
  1966 in an unpublished class term paper.
• He studied a restricted version of the Erdos
  property in musical scales
• He characterized the deep scales with n intervals
  and k pitches, with k ?n/2? or k
  ?n/2? 1

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The Diatonic Scale is Deep
C
B
D
A
Multiplicity
E
G
F
6
1
4
3
2
5
n 12 k 7
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Examples of Deep Rhythms
Cuban Tresillo
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Examples of Deep Rhythms
Helena Paparizou Eurovision 2005 My Number One
Cuban Tresillo
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Examples of Deep Rhythms
Cuban Cinquillo
Cuban Tresillo
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Examples of Deep Rhythms
BossaNova
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Characterization

• Erdos-deep rhythms consist of
• Dk,n,m i.m mod n i 0, , k
• F 0, 1, 2, 46
• - m and n are relatively prime
• - k ?n/2? 1

n 6 k 4
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Characterization

• Erdos-deep rhythms consist of
• Dk,n,m i.m mod n i 0, , k
• F 0, 1, 2, 46
• - m and n are relatively prime
• - k ?n/2? 1

n 6 k 4
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Characterization Example D7,16,5
n 16 k 7 9 m 5
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Characterization Example D7,16,5
0
n 16 k 7 9 m 5
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Characterization Example D7,16,5
0
n 16 k 7 9 m 5
5
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Characterization Example D7,16,5
0
n 16 k 7 9 m 5
5
10
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Characterization Example D7,16,5
0
15
n 16 k 7 9 m 5
5
10
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Characterization Example D7,16,5
0
15
n 16 k 7 9 m 5
4
5
10
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Characterization Example D7,16,5
0
15
n 16 k 7 9 m 5
4
5
10
9
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Characterization Example D7,16,5
0
15
14
n 16 k 7 9 m 5
4
5
10
9
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Main Theorem

• A rhythm is Erdos-deep if and only if it is a
  rotation or scaling of F or the rhythm Dk,n,m for
  some k, n, m with
• k ?n/2? 1,
• 1 m ?n/2? and
• m and n are relatively prime.

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Deep Shellings

• A shelling of a Erdos-deep rhythm R is a sequence
  s1, s2, , sk of onsets in R such that R s1,
  s2, , si is a Erdos-deep rhythm for i 0, ,
  k.

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Deep Shellings

• A shelling of a Erdos-deep rhythm R is a sequence
  s1, s2, , sk of onsets in R such that R s1,
  s2, , si is a Erdos-deep rhythm for i 0, ,
  k.

0
15
14
4
5
10
9
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Deep Shellings

• A shelling of a Erdos-deep rhythm R is a sequence
  s1, s2, , sk of onsets in R such that R s1,
  s2, , si is a Erdos-deep rhythm for i 0, ,
  k.

0
15
14
4
5
10
9
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Deep Shellings

• A shelling of a Erdos-deep rhythm R is a sequence
  s1, s2, , sk of onsets in R such that R s1,
  s2, , si is a Erdos-deep rhythm for i 0, ,
  k.

0
15
14
4
5
10
9
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Deep Shellings

• A shelling of a Erdos-deep rhythm R is a sequence
  s1, s2, , sk of onsets in R such that R s1,
  s2, , si is a Erdos-deep rhythm for i 0, ,
  k.

0
15
14

• Corollary Every Erdos-deep rhythm has a shelling

4
5
10
9
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Open Problem

• Given the frequency of each distance, reconstruct
  the onsets of the deep rhythm (i.e. find m and
  n).This is a special case of the Beltway
  problem

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Thank you