{"id":4734,"date":"2021-04-17T09:24:21","date_gmt":"2021-04-17T09:24:21","guid":{"rendered":"http:\/\/desres20.netornot.at\/?p=4734"},"modified":"2021-04-17T09:24:21","modified_gmt":"2021-04-17T09:24:21","slug":"euclidean-rhythms","status":"publish","type":"post","link":"http:\/\/desres20.netornot.at\/?p=4734","title":{"rendered":"Euclidean Rhythms"},"content":{"rendered":"\n

Music and mathematics are closely linked and an example of their common path are the so-called Euclidean rhythms.<\/p>\n\n\n\n

As the name suggests, its roots go back to the Greek mathematician Euclid (around 300 BC). In his text “Elements” a revolutionary algorithm is described to efficiently find the greatest common divisor (GCF) of 2 integers.<\/p>\n\n\n\n

Here an example on how it works [2]<\/p>\n\n\n\n

[<\/p>\n\n\n\n

The Euclidean Algorithm for finding GCD(A,B) is as follows:<\/p>\n\n\n\n

If A = 0 then GCD(A,B)=B, since the GCD(0,B)=B, and we can stop.<\/p>\n\n\n\n

If B = 0 then GCD(A,B)=A, since the GCD(A,0)=A, and we can stop.<\/p>\n\n\n\n

Write A in quotient remainder form (A = B\u22c5Q + R)<\/p>\n\n\n\n

Find GCD(B,R) using the Euclidean Algorithm since GCD(A,B) = GCD(B,R)<\/p>\n\n\n\n

Example:<\/strong><\/p>\n\n\n\n

Find the GCD of 270 and 192<\/p>\n\n\n\n

A=270, B=192<\/p>\n\n\n\n

A \u22600<\/p>\n\n\n\n

B \u22600<\/p>\n\n\n\n

Use long division to find that 270\/192 = 1 with a remainder of 78. We can write this as: 270 = 192 * 1 +78<\/p>\n\n\n\n

Find GCD(192,78), since GCD(270,192)=GCD(192,78)<\/p>\n\n\n\n

A=192, B=78<\/p>\n\n\n\n

A \u22600<\/p>\n\n\n\n

B \u22600<\/p>\n\n\n\n

Use long division to find that 192\/78 = 2 with a remainder of 36. We can write this as:<\/p>\n\n\n\n

192 = 78 * 2 + 36<\/p>\n\n\n\n

Find GCD(78,36), since GCD(192,78)=GCD(78,36)<\/p>\n\n\n\n

A=78, B=36<\/p>\n\n\n\n

A \u22600<\/p>\n\n\n\n

B \u22600<\/p>\n\n\n\n

Use long division to find that 78\/36 = 2 with a remainder of 6. We can write this as:<\/p>\n\n\n\n

78 = 36 * 2 + 6<\/p>\n\n\n\n

Find GCD(36,6), since GCD(78,36)=GCD(36,6)<\/p>\n\n\n\n

A=36, B=6<\/p>\n\n\n\n

A \u22600<\/p>\n\n\n\n

B \u22600<\/p>\n\n\n\n

Use long division to find that 36\/6 = 6 with a remainder of 0. We can write this as:<\/p>\n\n\n\n

36 = 6 * 6 + 0<\/p>\n\n\n\n

Find GCD(6,0), since GCD(36,6)=GCD(6,0)<\/p>\n\n\n\n

A=6, B=0<\/p>\n\n\n\n

A \u22600<\/p>\n\n\n\n

B =0, GCD(6,0)=6<\/p>\n\n\n\n

So we have shown:<\/strong><\/p>\n\n\n\n

GCD(270,192) = GCD(192,78) = GCD(78,36) = GCD(36,6) = GCD(6,0) = 6<\/p>\n\n\n\n

GCD(270,192) = 6<\/strong><\/p>\n\n\n\n

] [2]<\/p>\n\n\n\n

Its application in music and their formal discovery is attributed to the computer scientist Godried Toussint in 2004.<\/p>\n\n\n\n

He found that this algorithm could be translated into rhythms.<\/p>\n\n\n\n

But how? The GCF could be used to distribute a determined number of onsets across a determined number of step.<\/p>\n\n\n\n

These rhythms are often represented with a circular visualization.<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

So, as we can see, the Euclidean rhythm needs a number of onset (filling) uniformly distributed over a number of steps (per measure).<\/p>\n\n\n\n

Here’s an example of what it might sound like:<\/p>\n\n\n\n

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