Ecludian sequencers

In electronic music, you may come across Ecludian Rhythms.
They mostly are associated with patterns and beats.

They were only discovered in 2005 by the Canadian mathematian Godfried Toussaint.
His Book, "The Geometry of Musical Rhythm" discusses this in detail.
And this paper from McGill Uni, Montreal is a good read too.

In practical terms, ecludian sequencers are thus usually trigger sequencers (not melodic).
Patterns are derived from algorithms which come down to us from Euclid's Elements.

Eulcid was a Greek Mathematician who lived in Alexandria in 300BC.
In Ancient Greek, the word Elements is Στοιχεῖα Stoicheia.
This is a name of a very cool trigger sequencer (Eurorack)

The algorithms compute the greatest common divider of 2 numbers.
Ie, the highest number by which both can be divided.
GCD -  Greatest Common Dividers.
For example, the gcd of 8 and 12 is 4

They are interesting as they illustrate a close relationship between Maths & Music.
In a earlier post I discussed the mathematical idea of Lowest Common Multiples (LCM) and how one can also use them in music.

Euclidean Rhythms have the property that their onset patterns and numbers are distributed as evenly as possible.

This idea of distributing numbers evenly is found in many world music patterns from Cuba to Africa.

Ecludian rhythms are calculated from two numbers a larger value & a smaller value.
Larger = length of pattern
Smaller = the number of beats... these need to be distributed evenly across the pattern.
(A third consideration when making a pattern is the offset & rotation of the beats)

Examples: (where x=beat and 0=rest)
2,8 =  x000x000
2,5 = x0x00 (Tchaikovsky’s Symphony No. 6 & Brubeck’s Take Five)
3,4 = x0xx (Trinidad)
5, 16 =  x000x00x00x00x00
5, 8 = x0xx0xx0 (common Cuban cinquillo rhythm)
Here is a variation of the 5, 8 = 0x0xx0xx (Spanish Tango?)

5,16 = 00x00x00x00x00x0 (Brazillian Bosanova)
7,12 = x0xx0x0xx0x0 (West African)
4,9 = x0x0x0x00 (Turkish)
9,16 = x0xx0x0x0xx0x0x0 (central Africa)

If you overlay rhythms of different lengths you can create polyrhythms
By rotating and offsetting the beats and rests you can create new patterns.

Though the Standard use for a Euclidean sequencer is to trigger beats and melodies you can use them to
trigger an envelope. If you then run the envelope through a quantiser you can create melodic sequences.
 The Στοιχεῖα Stoicheia. module also has a companion tonic module which can create melodies from gates.

List of my Fav Ecludian Sequencers
+ Buchla 252e

 The Euclidian Library.
There is a built in Euclidian pattern library... to help generate rhythmic patterns.
 To generate pulse patterns using the Euclidian algorithm, select a ring using the “beats/cycl”
encoder, and press the “RHYTHM SELECT” button so that the Euclidian LED lights up.
Any pulse data in the selected ring will be erased and replaced by the Euclidian solution

rendered in red pulses (however, other pulses can be added).
Turn the “pattern #” encoder to generate each of the Euclidian pulse patterns for the number of
beats in the selected ring. The number of pulses in the current solution will be shown on the
segmented display to the left of the encoder.
To exit Euclidian mode, press the “RHYTHM SELECT” button, or turn the “beats/cycl” encoder
to select a different ring.



+ Pittsburgh Modular Game System
+ Στοιχεῖα Stoicheia.
+ Klasmata - single channel of stocheia but with cv control.
+ ALM Pamelas Workout
+ Pulsar  - Qu-Bit Electronix
+ MI - Grids & Yarns
+ Euclidean Circles v2 from Vladimir Pantelic Musikelektronik (6 tracks)
+ Addac heuristic
+ Snazzy Ardcore
+ 2HP Euclid

Links
+ LCMs - combining two Korg SQ-10s
+ LCM Vs GCD
+ Steve Reich - Clapping Music (Scrolling)