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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