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Asymmetric Rhythmsand Tiling Canons

Rachel W. Hall and Paul KlingsbergSaint Joseph’s University

IRCAM, July 2005

[email protected] • [email protected]://www.sju.edu/!rhall

1

Rhythm patterns

We define rhythm patterns as sequences of attacks,accents, or changes in tone color. A note is theinterval between successive attacks, also called noteonsets.

We assume there is some invariant beat that cannotbe divided, so that every note onset occurs at thebeginning of a beat.

Notation

The following are equivalent.

standard.œ .œ œ or

jœ ‰ ‰ jœ ‰ ‰ jœ ‰additive 3 + 3 + 2

drum tablature x..x..x.

binary 10010010

2

Asymmetric rhythm patterns

Asymmetric rhythm patterns cannot be broken intotwo parts of equal duration, where each part startswith a note onset. Arom (1991, p. 245-6) classifiesmany Central African rhythms as asymmetric.

Regular asymmetric rhythms

Irregular asymmetric rhythms

3

Rhythm cycles

We say that two periodic rhythms are equivalent ifone is a shift of the other. For example,

!! !!"# $ $ "# $ $ "# $=

!! !!$ "# $ $ "# $ $ "#where repeat signs indicate “infinite repeats.”

We call the equivalence class consisting of all cyclicshifts of a rhythm pattern a rhythm cycle.

4

Binary necklaces

A binary necklace is an equivalence class of finitebinary sequences, where two sequences are equivalentif one is a cyclic shift of the other:

0010011 " 1001100

Binary necklaces are also represented as necklaces ofblack and white beads equivalent under rotation.Turning over a necklace (inversion) is not allowed.

Every rhythm cycle is represented by a unique binarynecklace.

10010010 #$ #$ !! !!"# $ $ "# $ $ "# $

Many mathematical results on binary necklaces haveinteresting applications to rhythm cycles.

5

Asymmetric rhythm cycles

Asymmetric rhythm cycles are equivalence classes ofasymmetric patterns—they cannot be delayed so thatnote onsets coincide with both of the strong beats inthe measure. Many periodic rhythms from Africa areasymmetric cycles (Arom, 1991; Chemillier, 2002).

The rhythm cycle |: x..x..x. :| is asymmetric.

However, the cycle |: x.x...x. :| is not, as

|: x.x...x.:| = |: x...x.x. :|

Rhythmic oddity

Asymmetric rhythm cycles composed only of notes oflength 2 and 3 are of particular significance inArom’s study. The question of counting cycles withthis property has been answered by Chemillier andTruchet (2003).

6

Rhythms as functions

A rhythm pattern is represented by a function

f : Z $ {0, 1},

where

f (x) =

!1 if there is a note onset on pulse x0 otherwise

A periodic function with period p can be identifiedwith a function

f : Zp $ {0, 1}.

A rhythm cycle is an equivalence class of functions onZp modulo rotation. That is, f1 is equivalent to f2 iff1(x) = f2(x % k) holds for some k and for all x.

The asymmetry condition

A rhythm pattern of period p = 2n is asymmetric if

f (x) = 1 =& f (x + n) = 0

for all x ' Z2n.

7

Counting asymmetric rhythm patterns

Note that asymmetric patterns only occur in duplemeter.

Let Sn2 = {asymmetric patterns of length 2n}.

Then the size of Sn2 is 3n (written |Sn

2 | = 3n).

Partition the elements of Z2n into n pairs{{0, n}, . . . , {n % 1, 2n % 1}}; choose either zero orone element of each pair to be mapped to 1.

Let Sn2 (r) ( Sn

2 be the subset of asymmetricpatterns with r note onsets (0 ) r ) n).

Then |Sn2 (r)| =

"n

r

#2r.

Choose r of the pairs above; choose one element ofeach pair to be mapped to 1.

Let Pn2 ( Sn

2 and Pn2 (r) ( Sn

2 (r) represent primitiveasymmetric patterns—those whose primitive periodis 2n. We also have formulas that count primitiverhythm patterns.

8

Burnside’s Lemma

The cyclic group Z2n acts on patterns of length 2nby cyclic shift.

Precisely, an element m ' Z2n acts on a function fin Sn

2 by shifting it through m positions.

For example, if f corresponds to 1001100 andm = 1, m · f corresponds to 0100110.

We’ll count equivalence classes in each set ofpatterns modulo cyclic shifts. These classes areorbits induced by a group action. Therefore, we canapply Burnside’s Lemma.

Burnside’s Lemma Let a finite group G act on afinite set S; for each ! ' G, define fix(!) to be thenumber of elements s in S such that ! · s = s. Thenthe number of orbits that G induces on S is given by

1

|G|$

!'G

fix(!).

9

The number of asymmetric rhythm cycles

Theorem 1 The number of asymmetric rhythmcycles of period 2n is

1

2n

%

&&'$

d|n

"(2d) +$

d|nd odd

"(d)3n/d

(

))* ,

where "(d) is Euler’s totient function (i. e., "(d) isthe number of integers 1 ) x ) d that are relativelyprime to d).

Proof. For each divisor d of 2n, find the elements !of order d and determine fix(!), which depends onlyon d. Note that for each d, the number of ! of orderd is "(d).

10

Case 1. d divides 2n and d is even.

The only asymmetric element preserved by a rotationof even order is the empty rhythm f " 0, sofix(!) = 1.

Case 2. d divides 2n and d is odd.

An element preserved by a rotation of odd order isasymmetric if f (x) = 1 & f (x + n/d) = 0. Theelements 0, . . . 2n/d of Z2n are partitioned into n/dpairs {{0, n/d}, . . . , {n/d % 1, 2n/d % 1}}; eitherzero or one element of each pair is sent to 1.

!

11

The number of r-note asymmetric rhythmcycles

Theorem 2 If 1 ) r ) n, then the number ofasymmetric rhythm cycles with r note onsets is givenby

1

2n

$

d| gcd(n,r)d odd

"(d)

"n/d

r/d

#2r/d.

12

Proof. For each divisor d of 2n, we will find theelements ! of order d and determine fix(!).

Case 1. d divides 2n and d is even.

Since r * 1, fix(!) = 0.

Case 2. d divides 2n and d is odd.

(a) d divides r

The elements 0, . . . , 2n/d of Z2n are partitioned inton/d pairs {{0, n/d}, . . . , {n/d % 1, 2n/d % 1}}.Choose r/d of these pairs and send one of theelements to 1.

(b) d does not divide r

Since r * 1, fix(!) = 0.

!

We also have formulas that count primitive rhythmcycles.

13

Asymmetry in other meters

Asymmetry is defined by Arom with respect to duplemeter. But a similar situation can occur in anymeter. . .

Suppose we divide a measure of M beats into #equal parts and place a strong beat at the beginningof each part, creating “#-tuple meter.”

Rhythm cycles are asymmetric in #-tuple meter if,even when shifted, they cannot be broken into #parts of equal duration, where more than one partstarts with a note onset.

14

Definition of #-asymmetry

We say that a periodic rhythm of period #n is#-asymmetric if when position x contains a noteonset, then all other positions y, where y " x(mod n), do not contain note onsets.

For example, the 12-periodic rhythm cycle

|: x.....x..x.x :|

is 3-asymmetric (n = 4). All of its shifts contain nomore that one onset among the three principal beatsin triple meter.

Note that our previous definition of asymmetrycorresponds to #-asymmetry when # = 2.

15

Counting #-asymmetric rhythm cycles

Theorem 3 The number of #-asymmetric rhythmcycles of length M = #n is

1

M

%

&&&'$

d|Mgcd(d,#)>1

"(d) +$

d|ngcd(d,#)=1

"(d)(# + 1)n/d

(

)))*.

The number of r-note #-asymmetric rhythmcycles

Theorem 4 For 1 ) r ) n, the number of#-asymmetric rhythm cycles of length M = #n withr onsets is given by

1

M

$

d| gcd(n,r)gcd(d,#)=1

"(d)

"n/dr/d

##r/d.

16

Primitive #-asymmetric rhythm cycles

In what follows µ signifies the classical Mobiusfunction: µ(1) = 1 and

µ(d) =

+,,,,,,-

,,,,,,.

0 if x is divisible by the squareof any prime

1 if x is the product of an evennumber of distinct primes

%1 if x is the product of an oddnumber of distinct primes

Theorem 5 The number of primitive #-asymmetricrhythm cycles of length M = #n is

1

M

$

d|ngcd(d,#)=1

µ(d)[(# + 1)n/d % 1].

Theorem 6 If 1 ) r ) n, then the number ofprimitive #-asymmetric rhythm cycles of lengthM = #n with r onsets is given by

1

M

$

d| gcd(n,r)gcd(d,#)=1

µ(d)

"n/dr/d

##r/d.

17

Example

We now list the rhythm cycles of length 12 which are3-asymmetric and have four note onsets. (# = 3 andr = n = 4). There are eight:

1. |: xxxx........ :|

2. |: xxx....x.... :|

3. |: xx....xx.... :|

4. |: xx.x......x. :|

5. |: xx.x..x..... :|

6. |: x.....x..x.x :|

7. |: x.x..x.x.... :|

8. |: x..x..x..x.. :|

Patterns 3 and 8 are not primitive.

Patterns 5 and 6 are inversions of each other; allother patterns are symmetric with respect toinversion.

18

Applications to rhythmic canons

Messaien (1992) coined the term rhythmic canon,which is produced when each voice plays a rhythmpattern (the inner rhythm), and the voices are o!setby amounts determined by a second pattern (theouter rhythm).

The terms inner rhythm and outer rhythm comefrom (Andreatta et al., 2002).

Example

inner rhythm = x x . . x . x . xouter rhythm = e . e . . e

The canon:

Voice 1: x x . . x . x . xVoice 2: x x . . x . x . xVoice 3: x x . . x . x . x

Entries: e . e . . e

19

Complementary canons

Messiaen described the sound of a rhythmic canon asa sort of “organized chaos” (Messiaen, 1992, p. 46).In Harawi, “Adieu,” he uses the inner rhythm

x..x....x.......x....x..x...x..x......x..x...x.x.x..x....x

together with the outer rhythm e.e.e to generatethe canon

V.1: x..x....x.......x....x..x...x..x......x..x...x.x.x..x....xV.2: x..x....x.......x....x..x...x..x......x..x...x.x.x..x....xV.3: x..x....x.......x....x..x...x..x......x..x...x.x.x..x....x

Ent: e.e.e

A rhythmic canon is called complementary if, oneach beat, no more than one voice has a note onset.

Messiaen’s canon is almost complementary.

20

Canons of periodic rhythms

In this paper we study canons of periodic rhythms.This is consistent with (Vuza, 1991–1993; Andreattaet al., 2002). For example,

inner rhythm = |: x.....x....x :|outer rhythm = |: e...e...e... :|.

defines a 12-beat rhythmic canon with threeequally-spaced voices:

x . . . . . x . . . . x |: x . . . . . x . . . . x :|

x . . . . . x . |: . . . x x . . . . . x . :|

x . . . |: . . x . . . . x x . . . :|

e . . . e . . . e . . . |: e . . . e . . . e . . . :|

The inner rhythm is a 3-asymmetric rhythm cycle oflength 12.

A rhythmic canon of # voices, each spacedby n notes from the previous, is

complementary+&

its inner rhythm is #-asymmetric.

21

Rhythmic tiling canons

A rhythmic tiling canon is a canon of periodicrhythms that has exactly one note onset perbeat (Vuza, 1991–1993; Andreatta et al., 2002).

In order to satisfy the tiling condition when # = 3and n = 4, for each x ' Z12, the positions x, x + 4,and x + 8 in the inner rhythm must contain exactlyone onset. Thus, the inner rhythm must be a 4-note3-asymmetric cycle.

A rhythm of period #n = M forms a tilingcanon of # equally-spaced voices

+&it is #-asymmetric and has n note onsets.

22

Problems

Tiling canons with unequally-spaced voices.For example,

inner rhythm = |: x.x..... :|outer rhythm = |: ee..ee.. :|.

How can we classify and count these canons?

Maximal category. Tiling canons of maximalcategory are tiling canons where both the innerrhythm and the outer rhythm are primitive. Noneexist for periods less than 72 (Vuza, 1991–1993).

Inversion. The problem of finding tiling canonsusing one rhythm and its inversion (see BeethovenOp. 59, no. 2—the patterns are xx..x. and..xx.x.) is equivalent to one studied byMeyerowitz (2001). He showed that any rhythm withthree onsets must tile in this way, but the generalquestion remains open. We have analogues ofTheorems 4 and 6 for the cases in which the fulldihedral group acts on the set of #-asymmetricpatterns.

24

Tilings of the integers. Rhythmic tiling canonsare, in fact, one-dimensional tilings of the integersusing a single tile. All such tilings areperiodic (Newman, 1977). Our results on tilingcanons give the number of tilings of Z byequally-spaced placements of a single tile. There aremany open questions on one-dimensional tilings.

25

References

Moreno Andreatta, Carlos Agon, and Emmanuel Amiot. Tiling problemsin music composition: theory and implementation. In Proceedings ofthe International Computer Music Conference, pages 156–163,Goteborg, 2002.

Simha Arom. African polyphony and polyrhythm: musical structure andmethodology. Cambridge University Press, Cambridge, 1991.

Marc Chemillier. Ethnomusicology, ethnomathematics. The logicunderlying orally transmitted artistic practices. In Mathematics andmusic (Lisbon/Vienna/Paris, 1999), pages 161–183. Springer, Berlin,2002.

Marc Chemillier and Charlotte Truchet. Computation of words satisfyingthe rhythmic oddity property (after Simha Arom’s works). InformationProcessing Letters, 86:255–261, 2003.

Olivier Messiaen. Traite de rythme, de couleur, et d’ornithologie. Editionsmusicales Alphonse Leduc, Paris, 1992.

Aaron Meyerowitz. Tiling the line with triples. In Discrete models:combinatorics, computation, and geometry (Paris, 2001), DiscreteMath. Theor. Comput. Sci. Proc., AA, pages 257–274 (electronic).Maison Inform. Math. Discret. (MIMD), Paris, 2001.

Donald J. Newman. Tesselation of integers. J. Number Theory, 9(1):107–111, 1977.

Dan Tudor Vuza. Supplementary sets and regular complementaryunending canons I–IV. Perspectives of New Music, 29(2)–31(1),1991–1993.

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