Euclidean rhythm

In Toussaint's paper[3] the task of distributing ${\displaystyle k}$ beats within ${\displaystyle n}$ time steps is considered. It is given that ${\displaystyle k<n}$, 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 ${\displaystyle n}$ is divisible by ${\displaystyle k}$—in this case we distribute the beats such that they are ${\displaystyle n/k}$ steps away from their neighbour. As an example, below is a euclidean rhythm for ${\displaystyle n=16}$ and ${\displaystyle k=4}$. These beats are 4 steps away from each other because ${\displaystyle n/k=16/4=4}$.

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

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

The problem becomes more complicated when ${\displaystyle k}$ does not divide ${\displaystyle n}$. In this case the formula ${\displaystyle n/k}$ 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 ${\displaystyle n=13}$ and ${\displaystyle k=5}$. 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 ${\displaystyle k}$ and ${\displaystyle n}$ that minimizes "clumping." Taking the previous example where ${\displaystyle n=13}$ and ${\displaystyle k=5}$ we perform Euclid's algorithm:

${\displaystyle {\begin{aligned}&&n=13,\ k=5\\n&=q_{0}k+r_{0}\implies &q_{0}=2,\ r_{0}=3\\k&=q_{1}r_{0}+r_{1}\implies &q_{1}=1,\ r_{1}=2\\r_{0}&=q_{2}r_{1}+r_{2}\implies &q_{2}=1,\ r_{2}=1\\r_{1}&=q_{3}r_{2}+r_{3}\implies &q_{3}=2,\ r_{3}=0\\\end{aligned}}}$

Toussaint's algorithm first constructs the following rhythm.

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

Then, using the sequence ${\displaystyle t=k,r_{0},r_{1},r_{2},...}$ we iteratively take ${\displaystyle t_{n}}$ columns off the right of the sequence and place them at the bottom. Starting with ${\displaystyle t_{0}=k=5}$, we get:

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

Next is ${\displaystyle t_{1}=r_{0}=3}$.

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

Next is ${\displaystyle t_{2}=r_{1}=2}$.

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

The process stops here because ${\displaystyle t_{3}<2}$, 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.[4] Viggo Brun[5] investigated the use of Euclidean Algorithm in terms of constructing scales up to 4 different size intervals. Erv Wilson explored both using[6] ratios and[7]scale steps of which Kraig Grady applied to[8]rhythms within long meters.

• Algorithmic composition
• Polyrhythm

• G. T. Toussaint, 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.
• Phil Baljeu and Manuel Odendahl (Ruin & Wesen). "Generating African rhythms using the euclidean algorithm". Archived from the original on 2013-11-14.
• Benjamin Wardhaugh (1 September 2006). "Music and Euclid's algorithm".
• Links to videos about and a Flash app for experimenting with Euclidean rhythms
• Euclidean rhythm demo — interactive browser-based tool for experimenting with Euclidean rhythms
• A tutorial on The Euclidean Algorithm Generates Traditional Musical Rhythms by Derek Rivait
• SoundHelix is a free software for algorithmic random music composition that supports Euclidean rhythms
• 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://anaphoria.com/journal.html#rhythm Papers of Kraig Grady on Brun's Euclidian Algorithm and related MOS patterns applied to rhythm.