the distance geometry of deep rhythms and scales by e. demaine, f. gomez-martin, h. meijer, d....

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


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Rhythms and Scales


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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

Diatonic scale or Bembé

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D

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FG

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its geometric representation is similar to that of a rhythm

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Erdős 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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Erdős Distance & Rhythms

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

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2 7 4 1 6 5

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2 7 4 1 6 5

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2 7 4 1 6 5

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

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2 7 4 1 6 5

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77

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

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2 7 4 1 6 5

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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The term deep was first introduced by Winograd in 1966 in an unpublished class term paper.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 Erdős 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

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FG

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n = 12k = 7

6 1 4 3 2 5

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Examples of Deep Rhythms

Cuban Tresillo

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Cuban TresilloHelena Paparizou

Eurovision 2005“My Number One”

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Cuban Tresillo Cuban Cinquillo

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Bossa–Nova

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Characterization

Erdős-deep rhythms consist of :

1. Dk,n,m = {i.m mod n | i = 0, …, k}

2. F = {0, 1, 2, 4}6

- m and n are relatively prime

- k ≤ n/2 + 1

n = 6k = 4

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Characterization

Erdős-deep rhythms consist of :

1. Dk,n,m = {i.m mod n | i = 0, …, k}

2. F = {0, 1, 2, 4}6

- m and n are relatively prime

- k ≤ n/2 + 1

n = 6k = 4

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Characterization: Example D7,16,5

n = 16k = 7 ≤ 9m = 5

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n = 16k = 7 ≤ 9m = 5

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n = 16k = 7 ≤ 9m = 5

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n = 16k = 7 ≤ 9m = 5

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n = 16k = 7 ≤ 9m = 5

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n = 16k = 7 ≤ 9m = 5

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n = 16k = 7 ≤ 9m = 5

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n = 16k = 7 ≤ 9m = 5

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Main Theorem

A rhythm is Erdős-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 Erdős-deep rhythm R is a sequence s1, s2, …, sk of onsets in R such that R – {s1, s2, …, si} is a

Erdős-deep rhythm for i = 0, …, k.0

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Deep Shellings



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Deep Shellings



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Deep Shellings



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Deep Shellings



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Corollary: Every Erdős-deep rhythm has a shelling

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Thank you

the distance geometry of deep rhythms and scales by e. demaine, f. gomez-martin, h. meijer, d....

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