Euclidean rhythm

In Toussaint's paper the task of distributing $k$ beats within $n$ time steps is considered. It is given that , so there are fewer beats than steps. The question arises of how to distribute these beats such that they are as equidistant as possible. This is easy when $n$ is divisible by $k$—in this case we distribute the beats such that they are $n/k$ steps away from their neighbour. As an example, below is a euclidean rhythm for $n\; =\; 16$ and $k\; =\; 4$. These beats are 4 steps away from each other because $n/k\; =\; 16/4\; =\; 4$.

[ x . . . x . . . x . . . x . . . ]

Here "x" represents a beat and "." represents a silence.

The problem becomes more complicated when $k$ does not divide $n$. In this case the formula $n/k$ doesn't produce an integer, so some beats must be slightly closer to one neighbour than the other. Because of this the beats are no longer perfectly equidistant. As an example, take the case when $n\; =\; 13$ and $k\; =\; 5$. A naive algorithm may place the beats like this:

[ x . x . x . . x . . x . . ]

Although the beats are technically distributed with ideal spacing between the beats—they are either two steps apart or three—we still have a problem where the beats are "clumped" at the start and spaced out at the end. A more concrete definition of "equidistant" might ask that these spacings ("x ." and "x . .") are also distributed evenly.

Toussaint's observation is that Euclid's algorithm can be used to systematically find a solution for any $k$ and $n$ that minimizes "clumping". Taking the previous example where $n\; =\; 13$ and $k\; =\; 5$ we perform Euclid's algorithm:

:$\backslash begin\{align\}$

& & n = 13,\ k = 5 \\

n &= q_0 k + r_0 \implies&q_0 = 2,\ r_0 = 3 \\

k &= q_1 r_0 + r_1 \implies&q_1 = 1,\ r_1 = 2 \\

r_0 &= q_2 r_1 + r_2 \implies&q_2 = 1,\ r_2 = 1 \\

r_1 &= q_3 r_2 + r_3 \implies&q_3 = 2,\ r_3 = 0 \\

\end{align}

Toussaint's algorithm first constructs the following rhythm.

[ x x x x x . . . . . . . . ]

Then, using the sequence $t\; =\; k,\; r\_0,\; r\_1,\; r\_2,\; ...$ we iteratively take $t\_n$ columns off the right of the sequence and place them at the bottom. Starting with $t\_0\; =\; k\; =\; 5$, we get

[ x x x x x . . .

. . . . . ]

Next is $t\_1\; =\; r\_0\; =\; 3$:

[ x x x x x

. . . . .

. . . ]

Next is $t\_2\; =\; r\_1\; =\; 2$:

[ x x x

. . .

. . .

x x

. . ]

The process stops here because , i.e. there is only one column to move. The final beat pattern is read out from top to bottom, left to right:

[ x . . x . x . . x . x . . ]

In the 17th century Conrad Henfling writing to Leibniz about music theory and the tuning of musical instruments makes use of the Euclidean algorithm in his reasoning.[http://plus.maths.org/content/os/issue40/features/wardhaugh/index Musical pitch and Euclid's algorithm]

Viggo Brunhttps://anaphoria.com/brun-euclideanalgo.pdf Euclidean Algorithms and Musical Theory investigated the use of Euclidean Algorithm in terms of constructing scales up to 4 different size intervals. Erv Wilson explored both usinghttps://anaphoria.com/viggo3.pdf A sequence of Constant Structures ratios andhttps://anaphoria.com/viggo2.pdf Viggo's Brun's algorithm applied scale steps of which Kraig Grady applied tohttps://anaphoria.com/ViggoRhythm.pdf Applying Viggo Brun's Algorithm to Rhythm rhythms within long meters.

• Algorithmic composition
• Polyrhythm

{{Reflist}}

• G. T. Toussaint, [http://cgm.cs.mcgill.ca/~godfried/publications/banff.pdf The Euclidean algorithm generates traditional musical rhythms], Proceedings of BRIDGES: Mathematical Connections in Art, Music, and Science, Banff, Alberta, Canada, July 31 to August 3, 2005, pp. 47–56.
• {{cite web |url = http://ruinwesen.com/blog?id=216 |title = Generating African rhythms using the euclidean algorithm |author = Phil Baljeu and Manuel Odendahl (Ruin & Wesen) |archiveurl = https://web.archive.org/web/20131114124454/http://ruinwesen.com/blog?id=216 |archivedate = 2013-11-14}}
• {{cite web |url = http://plus.maths.org/content/os/issue40/features/wardhaugh/index |title = Music and Euclid's algorithm |author = Benjamin Wardhaugh |date = 1 September 2006}}
• Links to videos about and a Flash app for experimenting with [http://www.hisschemoller.com/2011/euclidean-rhythms/ Euclidean rhythms]
• [https://dbkaplun.github.io/euclidean-rhythm/ Euclidean rhythm demo] — interactive browser-based tool for experimenting with Euclidean rhythms
• A tutorial on [https://web.archive.org/web/20130211035641/http://cgm.cs.mcgill.ca/~mcleish/644/Projects/DerekRivait.1/ The Euclidean Algorithm Generates Traditional Musical Rhythms] by Derek Rivait
• [https://www.soundhelix.com/ SoundHelix] is a free software for algorithmic random music composition that supports Euclidean rhythms
• [http://www.iniitu.net/Euclidian_Erdös_Deep_Aksak_rhythms.html Euclidian rhythms list] - a list of all Euclidian rhythms E(i,2 to 32), indicating if they are Winograd-deep, Erdős-deep, Authentic Aksak, Quasi-Aksak or Pseudo-Aksak
• XiiixxiQ : Roundels is a unique, and free, Euclidean sequencer that employs summed on the subject of Euclidian algorithm it is often stated that its functions to find the most equal divisions of a cycle, yet i cannot find that in any of the historical papers on he subject. it seems to be nothing more how a certain number of intervals divide a cycle. It is unclear where this modern assumption come from. rhythms to drive a non-linear step sequencer [https://www.kvraudio.com/product/roundels-by-xiiixxiq ]
• https://anaphoria.com/journal.html#rhythm Papers of Kraig Grady on Brun's Euclidian Algorithm and related MOS patterns applied to rhythm.

{{Rhythm and meter}}

rhythm

Category:Music theory

Category:Musical analysis