Transcript File - Euclidean Sequencing

Using A Euclidean Sequence
For Rhythmic Improvisation
Rob Conrad
Music Theory IV
Mesa College
Introduction
In the 3rd century BC, in his book The Elements, the Greek
mathematician Euclid described an algorithm for determining the
greatest common denominator (GCD) of two integers.
1. Take two positive integers. Assign the Highest to M
(measure)
and the lowest to N (notes).
2. Divide M by N and assign the remainder to R
3. If R is <> 0 then repeat step 2.
4. If R=0 then the current value of N is the GCD.
EUCLID(m,n)
1. if n= 0
2. then return m 3.
else return EUCLID(n,m mod n)
Wait....what?
Nobody said this was going to involve math...
What does that have to do with music?
Music is math!
As it turns out, this algorithm can create patterns similar to
the way people have divided beats into rhythms all across
the globe, from ancient to modern times.
In 2005, Godfried Toussaint analyzed the relationship
between rhythms and the Euclidean algorithm in his paper
"The Euclidean Algorithm Generates Traditional Musical
Rhythms".
In the paper, Touissant describes how the GCD of two
numbers can be used to rhythmically calculate the number of
beats and rests, generating a myriad of musically useful
rhythms. A primary characteristic of Euclidean patterns is that
the beats in the resulting rhythms are as equidistant as
possible.
From Sambas and Bossa Nova, to Bulgarian folk dances and
Burundi beats, to Gamelan orchestras and Phillip Glass
minimalism, these rhythms can easily be calculated and
translated into a binary representation for the digital realm of
composition and performance.
The formula is a simple method to distribute a number of
notes as evenly as possible over a period of time, where time
is divided in equal parts. A very basic example: Say you have
one measure of sixteen sixteenth notes and there are four
notes to be played, then this is how those four notes would be
equally distributed by the algorithm E[16,4] :
x...x...x...x...
In binary code, that can be represented in 16 bits as :
1000100010001000
In standard notation, it looks like this :
More rhythmic complexity happens when the numbers don’t
divide so easily. Look at five beats on sixteen steps - the pattern
becomes more irregular:
x---x--x--x--x-These irregular patterns can create interesting syncopations,
especially when several different patterns are chained or layered.
If the patterns are of different M lengths - If pattern 1M is 16 and
pattern 2M is 10, you can generate complex polyrhythms playing
against each other.
Additional complexity can be introduced by shifting a pattern to
the right or left by a variable number of steps. A shift of two steps
right alters the previous pattern to this:
--x---x--x--x--x
Using a sixteen step pattern, there are 65,553 possible beat
variations!
Examples of Euclidean Rhythms
Calypso rhythm from Trinidad - E(3,4) [x . x x]
Metric pattern of the second movement of Tchaikovsky’s Symphony No.
6,
and of Dave Brubeck’s Take Five - E(2,5) [x . x . .]
Cuban cinquillo pattern. Started on the second onset it is also the
Spanish Tango, and a 13 century Persian rhythm, the Al-saghil- al-sani E(5,8) [x . x x . x x .]
Widespread pattern used frequently and with different onsets in Central
and West African music. In Cuba it is the bell pattern of the Sarabanda
rhythm associated with the Palo Monte cult - E(7,12) [x.xx.x.xx.x.]
A rhythm necklace used in the Central African Republic. When it is
started on the fourth onset it is a rhythm played in West and Central
Africa, as well as a cow-bell pattern in the Brazilian samba - E(9,16)
Courtesy Rebel Technologies and Gottfried Toussaint
[x.xx.x.x.xx.x.x.]
So , wouldn’t it be nice if we had the ability to generate,
modify, interact and perform with these rhythmic patterns
live, in real time?
We Can!
Let’s take a look at a Euclidian Sequencer
Rebel Technologies Stoicheia
PatternShift
Bar Length
Note Density
Text
Mute/Chain/ Reset
Reset
Gate/Trigger
Clock In
Pattern A
Pattern B