The Euclidean algorithm computes greatest common divisor of two numbers. Godfried Toussaint discovered its musical applications and published them in "The Euclidean Algorithm Generates Traditional Musical Rhythms". Applied to music, the algorithm takes
k pulses (notes) and distributes them as equidistant as possible in
It turns out that equidistant distribution is a key to creating rhythms that are inherently musical. An extraordinary number of traditional rhythms can be generated through this simple process, see the Example Rhythms section.
This is a Max Package. The best way to install it is to simply
git clone this repo it into your Max Packages folder.
bjorklund.litcoffee: This follows the description of the algorithm verbatim from "Structural properties of Euclidean rhythms", but it also gets slightly different results than the examples listed in Toussaint's other paper.
toussaint.litcoffee: This version is slightly more elegant, and it's corrected to get identical results to the examples listed in "The Euclidean Algorithm Generates Traditional Musical Rhythms".
The sequencer uses the
toussaint version by default.
"The Euclidean Algorithm Generates Traditional Musical Rhythms" contains a number of example inputs (
n) that result in traditional rhythms:
E(2,3) = [x . x]is a common Afro-Cuban drum pattern. For example, it is the conga rhythm of the 6/8-time Swing Tumbao. It is also common in Latin American music, as for example in the Cueca.
E(2,5) = [x . x . .]is a thirteenth century Persian rhythm called Khafif-e-ramal. It is also the metric pattern of the second movement of Tchaikovsky’s Symphony No. 6. When it is started on the second onset (
[x . . x .]) it is the metric pattern of Dave Brubeck’s Take Five as well as Mars from The Planets by Gustav Holst.
E(3,4) = [x . x x]is the archetypal pattern of the Cumbia from Colombia , as well as a Calypso rhythm from Trinidad. It is also a thirteenth century Persian rhythm called Khalif-e-saghil, as well as the trochoid choreic rhythmic pattern of ancient Greece.
E(3,5) = [x . x . x], when started on the second onset, is another thirteenth century Persian rhythm by the name of Khafif-e-ramal, as well as a Rumanian folk-dance rhythm.
E(3,7) = [x . x . x . .]is a Ruchenitza rhythm used in a Bulgarian folk-dance. It is also the metric pattern of Pink Floyd’s Money.
E(3,8) = [x . . x . . x .]is the Cuban tresillo pattern discussed in the preceding.
E(4,7) = [x . x . x . x]is another Ruchenitza Bulgarian folk-dance rhythm. >
E(4,9) = [x.x.x.x..]is the Aksak rhythm of Turkey. It is also the metric pattern used by Dave Brubeck in his piece Rondo a la Turk.
E(4,11) = [x . . x . . x . . x .]is the metric pattern used by Frank Zappa in his piece titled Outside Now.
E(5,6) = [x . x x x x]yields the York-Samai pattern, a popular Arab rhythm, when started on the second onset.
E(5,7) = [x . x x . x x]is the Nawakhat pattern, another popular Arab rhythm.
E(5,8)=[x . x x . x x .]is the Cuban cinquillo pattern discussed in the preceding. When it is started on the second onset it is also the Spanish Tango and a thirteenth century Persian rhythm, the Al-saghil- al-sani.
E(5,9) = [x . x . x . x . x]is a popular Arab rhythm called Agsag-Samai. When started on the second onset, it is a drum pattern used by the Venda in South Africa, as well as a Rumanian folk-dance rhythm.
E(5,11) = [x . x . x . x . x . .]is the metric pattern used by Moussorgsky in Pictures at an Exhibition.
E(5,12) = [x . . x . x . . x . x .]is the Venda clapping pattern of a South African children’s song.
E(5,16) = [x..x..x..x..x....]is the Bossa-Nova rhythm necklace of Brazil. The actual Bossa-Nova rhythm usually starts on the third onset as follows:
[x . . x . . x . . . x . . x . .]. However, there are other starting places as well, as for example
[x . . x . . x . . x . . . x . .].
E(7,8) = [x . x x x x x x]is a typical rhythm played on the Bendir (frame drum), and used in the accompaniment of songs of the Tuareg people of Libya.
E(7,12) = [x . x x . x . x x . x .]is a common West African bell pattern. For example, it is used in the Mpre rhythm of the Ashanti people of Ghana.
E(7,16) = [x . . x . x . x . . x . x . x .]is a Samba rhythm necklace from Brazil. The actual Samba rhythm is
[x . x . . x . x . x . . x . x .]obtained by starting
E(7,16)on the last onset. When
E(7,16)is started on the fifth onset it is a clapping pattern from Ghana.
E(9,16) = [x.xx.x.x.xx.x.x.]is 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. When it is started on the penultimate onset it is the bell pattern of the Ngbaka-Maibo rhythms of the Central African Republic.
E(11,24) = [x . . x . x . x . x . x . . x . x . x . x . x .]is a rhythm necklace of the Aka Pygmies of Central Africa . It is usually started on the seventh onset.
E(13,24) = [x . x x . x . x . x . x . x x . x . x . x . x .]is another rhythm necklace of the Aka Pygmies of the upper Sangha. It is usually started on the fourth onset.