A Dictionary of Exotic Rhythms With a brief note on their combinatorial properties

Mike Keith





Introduction

Many musical compositions are characterized by a simple rhythmic pattern, consisting of some number n of beats per measure, which repeats over and over. Common values are n=4 in jazz and rock music and n=3 in certain types of music such as the waltz. The value n=2 is also ubiquitous, but since each pair of adjacent 2's adds up to 4, this is perceptually similar to n=4. This basic rhythmic pattern is called the meter or time signature of the music.

Meters with n>4 are quite uncommon (except for those in which n = 2n or n = 3n, because those are very similar to n=2 (or 4) and n=3) but certainly do appear from time to time, and not just in "modern" music. The goal of this page is to collect examples of exotic meters (from well-known pieces of music, where possible) and present a few results about the combinatorics of meter.



Meters and the Fibonacci Series

One facet of meter which is important, both perceptually and mathematically, is that meters with n>4 tend to be interpreted by a listener as combinations of 2's and 3's. For example, n=5 can be heard as either 3+2 or 2+3. In some meters a case can be made for a "1", and in others it can be argued that the basic unit is larger than 2 or 3 (and not a power of 2 or 3), but for simplicity we focus on just those (which we believe to be most) that can be heard as combinations of 2's and 3's.

The basic combinatorial question is: for a given n, how many distinct meters are there? This is the same as asking how many ways there are to represent n as an ordered sum of 2's and 3's; let's call this function r(n).

A simple formula for r(n) follows immediately by considering the problem recursively. The ordered partitions of n into 2's and 3's can be divided into two groups: those that start with a 2 and those that start with a 3. How many are there of the first kind? Well, the sum of the rest of the partition, after the initial 2, is n-2, and there are by definition r(n-2) ways of writing this as a sum of 2's and 3's. So, this means there are r(n-2) partitions of the first kind. Similarly, there are r(n-3) of the second kind. Thus,

r(n) = r(n-2) + r(n-3).

Note the similarity between this formula and the defining relation for the Fibonacci numbers:

F(n) = F(n-1) + F(n-2).

We might call r(n) the delayed Fibonacci sequence. Here are the first few values of r(n), from r(2) to r(22):

1 1 1 2 2 3 4 5 7 9 12 16 21 28 37 49 65 86 114 151 200 ...



And now...The List

The table below attempts to collect examples of as many of these different rhythms as possible, preferably from well-known (popular or classical) pieces of music - though more obscure pieces are quite acceptable in order to help snag some of the more unusual meters. The table is exhaustive up through n=12 (i.e., it lists all possible divisions into 2's and 3's, with the hope of finding at least one example for each one) but non-exhaustive thereafter. Where it seemed warranted I have listed more than one piece for the same meter, but again the goal in that case isn't to list them all but rather to give some of the most well-known examples.

Please email me if you have additions to suggest, especially for meters with n < 12 for which I do not currently have any examples. Send as many of {title, artist, and album name} as you know.

As you can see, a lot of these different exotic meters can be found out there. Three jazz musicians and composers appear frequently in the table as a result of their extensive experimentation with exotic meters: trumpter Don Ellis, pianist Dave Brubeck, and guitarist John McLaughlin.

Thanks to Matt Endahl, Richard Sabey and Barry Brake for significant contributions to the table.

A Dictionary of Meters

Illustrated with Musical Examples

n Partition Example 2 2 ubiquitous 3 3 ubiquitous 4 2 2 ubiquitous 5 3 2





2 3 "Take Five", Paul Desmond/Dave Brubeck, Time Out

"Indian Lady", Don Ellis, Electric Bath

"Conference of the Birds", David Holland, same album

"Symphony No. 6, 2nd Movement", Tchaikovsky 6 3 3

2 2 2 common

common 7 2 2 3

2 3 2

3 2 2 "Money", Pink Floyd

"Turkish Bath", Don Ellis/Ron Myers, Electric Bath

"The Remembering", Yes, Tales from Topographic Oceans 8 2 2 2 2

3 3 2

3 2 3

2 3 3 ubiquitous

Traditional bluegrass banjo rhythm; also the "samba"

Bass riff, Benny Goodman's Sing Sing Sing

"From Bohemian Woods and Fields", Smetena (opening) 9 3 3 3

3 2 2 2

2 3 2 2

2 2 3 2

2 2 2 3 common

"Strawberry Soup", Don Ellis, Tears of Joy

"Vesuvius" (main theme), Frank Tichelli, school band piece

"Final Analysis", Don Ellis, Live at Fillmore

"Blue Rondo a la Turk", Dave Brubeck, Time Out 10 2 2 2 2 2

3 3 2 2

2 2 3 3

3 2 3 2

3 2 2 3

2 3 3 2

2 3 2 3 "Scratt and Fluggs", Don Ellis, Autumn

"Indian Lady", Don Ellis, Electric Bath





"String Quartet #5, 3rd movemt, trio", Bartok [note 1]



"Study, op. 35, no. 12", Alkan 11 3 3 3 2

3 3 2 3

3 2 3 3



2 3 3 3

2 2 2 2 3

2 2 2 3 2

2 2 3 2 2

2 3 2 2 2

3 2 2 2 2 "Outside Now", Frank Zappa, Broadway the Hard Way



"Eleven Four", Paul Desmond/Dave Brubeck, Countdown

"Promenade" from Pictures at an Exhibition, Moussorgsky



"Follow Your Heart", John McLaughlin, Extrapolation

"I Say a Little Prayer For You", Bacharach/David, chorus 12 2 2 2 2 2 2

3 3 3 3

3 3 2 2 2

3 2 3 2 2

3 2 2 3 2

3 2 2 2 3

2 3 3 2 2

2 3 2 3 2

2 3 2 2 3

2 2 3 3 2

2 2 3 2 3

2 2 2 3 3 common

common

"America" from West Side Story, Leonard Bernstein











Afro-cuban bembe rhythm [note 2]



"Rock Odyssey", Don Ellis/Hank Levy, Live at Fillmore 13 2 3 3 2 3

3 3 2 2 3

3 3 2 3 2 "Vingt Regards sur l'Enfant-Jesus: movemt 20", Messiaen

"Chain Reaction", Hank Levy, Don Ellis' Connection

"Great Divide", Don Ellis, Live at Fillmore 14 3 3 2 2 2 2 "Three to Get Ready", Dave Brubeck quartet, Time Out 15 3 3 3 3 3

2 2 2 2 2 2 3

2 2 3 2 2 2 2 2 "Dream", J. McLaughlin, Between Nothingness and Eternity

"No Motion Picture", Eberhard Weber, Colours of Chloe

"Tubular Bells", Mike Oldfield 17 2 2 3 2 2 3 3

2 3 2 3 2 2 3 "Changes", Yes, 90125

"New Horizons", Don Ellis, Electric Bath 19 3 3 2 2 3 2 2 2 "33 222 1 222", Don Ellis, Live at Monterey 20 3 3 3 3 3 3 2 "Dance of Maya" (solos), John McLaughlin, Birds of Fire 25 2 2 2 2 3 2 2 2 2 3 3 "Tanz" from Carl Orff, Carmina Burana 32 2 3 2 2 3 2 2 2 3 2 2 2 2 3 "Variation for Trumpet" (pt. 4), Don Ellis, Autumn 33 2 2 2 2 3 2 2 2 2 3 2 3 2 2 "Bulgarian Bulge", Don Ellis, Don Ellis Goes Underground

Notes:

[1] Even within the trio, not all measures are 3 2 2 3 (e.g., some are 2 3 2 3). But since most are, and since 3 2 2 3 is so hard to snag, we put this piece here.