Tap your foot to almost any pop song and your foot lands in the right place. Now listen to a bell player in a West African ensemble. The bell strikes a short, irregular pattern, seven strokes spread across twelve beats of pulse, and if you try to tap the bell you are lost. The bell is not the pulse. It sits on top of it, like a signature, and the drummers, singers and dancers all place themselves against it.
That is the lesson of this course in one picture. In Western notation, rhythm is organized into bars with a strong beat at the start. Much of the rest of the world organizes it into cycles, repeated patterns whose shape is the identity of the music. Both are systems, with rules a trained musician hears. This piece teaches both, as equals, and then asks what groove is, and where it lives. It builds on Rhythm You Can Feel (09/06), which has a drum machine with the clave and a swing slider; here we go further into why those patterns work, and into the Indian and Arab systems.
Start with the pulse: the steady, even ticks you could tap to. Pulse alone is not yet metre. Metre appears when you group pulses, so that some are felt as stronger than others. The grouping unit is almost always two or three. A march is 2+2. A waltz is 3. 6/8 is 3+3, two beats of three. And because you can mix twos and threes, you can make odd lengths: 5 is 2+3 or 3+2; 7 is 2+2+3 or its rotations; 9 can be 3+3+3 or 2+2+2+3.
The Ottoman and Balkan traditions have a name for the lopsided groupings: aksak, from a Turkish word meaning roughly "limping." Constantin Brăiloiu popularized the term for ethnomusicology in a 1951 article. In Turkish theory the word strictly names the nine-pulse pattern 2+2+2+3, which appears in many Bulgarian dances (such as the Daychovo horo) and in Dave Brubeck's jazz piece "Blue Rondo à la Turk." To someone raised on 4/4 these patterns can feel as if they stumble. To the dancers they are as steady as walking, because the unequal groups are what they step to.
Try the builder below. Pick the number of pulses, then a way of grouping them. Only the first pulse of each group is accented, and the sound is the same everywhere else. Eight pulses grouped 2+2+2+2 and grouped 3+3+2 are the same number of notes in the same time; what changes is where you feel the beat.
Tall boxes are the accented pulse at the start of each group. The outline moves with the sound.
A bar is a measure of time with a strong start, and the music flows through it. A cycle (or timeline) is a pattern that goes round, and has no strong "one" in the Western sense: it can be entered, or felt, from several places. The West African standard bell pattern is a famous one. In twelve pulses it has seven strokes, with gaps of 2-2-1-2-2-2-1. Scholars such as Gerhard Kubik describe such key patterns as the structural core of a piece; David Locke, who studied Ewe drumming in Ghana, points out that the bell's span establishes the metre; and Kofi Agawu has cautioned against describing African rhythm as merely "additive" (adding up twos and threes), arguing that the concept is problematic.
The pattern travelled. In Cuba, where West and Central African traditions met Spanish music, it became the family of clave patterns. The word means "key," as in keystone. Son clave, the most common in Cuban popular music, has five strokes in sixteen pulses: three, then two. Rumba clave moves one stroke a sixteenth later. The 3-2 and 2-3 distinction is the same cycle entered at the other side. The idea of moving a song between the two sides came out of 1940s New York, where Mario Bauzá, music director of Machito and his Afro-Cubans, developed it and Tito Puente learned it from him; most salsa today is in 2-3, though Cuban musicians often don't use that terminology. The clave in the Brazilian bossa nova is a close cousin of son clave with its last stroke delayed one sixteenth.
Agbekor, a dance and drum style of the Ewe and Fon peoples of West Africa, once a war dance and now performed in places such as Dzogadze near Akatsi in Ghana's Volta Region, shows how a timeline works inside a community: a lead drummer calls, dancers and the other drums answer, and the pattern underneath holds everything in place.
The gold dots are the key pattern; the small blue dots are the main beats, played as a low drum. The line sweeps once per cycle. Bell: pulses 1, 3, 5, 6, 8, 10, 12. Clave patterns in sixteen pulses follow the standard grids; all numbers computed in node.
Slide the start of the bell around the ring. Nothing about the pattern changes, only where you enter it, and each starting point is a different rhythm you might meet in a different dance. This is the main idea of a cycle: it has a shape that you can start anywhere.
Many of the world's favorite rhythms share a mathematical property: their strokes are spread as evenly as possible around the cycle. The gaps between strokes are all the same, or differ by at most one. Godfried Toussaint, a computer scientist and musician, noticed in 2004 that Euclid's ancient algorithm for the greatest common divisor, run in the right way, generates such patterns, and published the finding in a 2005 paper, "The Euclidean Algorithm Generates Traditional Musical Rhythms." Write E(k, n) for k strokes spread over n pulses. E(3, 8) is the Cuban tresillo, 3-3-2. E(5, 8) is the cinquillo. E(7, 12) is a rotation of the standard bell pattern.
Here is a pleasing coincidence that has been argued over. The bell's gaps, 2-2-1-2-2-2-1, are the step pattern of the major scale, whole, whole, half, whole, whole, whole, half (see Music Fundamentals, 35/63), if one pulse is a semitone. Jeff Pressing pointed out in 1983 that pitch patterns and rhythm patterns from West Africa, the Balkans and Western tonality share structures. Whether that reflects anything deep about how brains work is debated: Justin London (2002) argued the analogy between pitch and time breaks down; a 2019 study by Wen and Krumhansl found that the bell and the scale follow different perceptual hierarchies even if they share some principles. It is a neat fact, not a proven law.
Generated with Bjorklund's algorithm in the page. The patterns shown start on a stroke; rotate them mentally. Only the matches listed in Toussaint's paper and checked in node are named.
A polyrhythm sets two or more different divisions of the same stretch of time against each other. The classic pair is 3 against 2: three evenly spaced hits in the time of two. They only meet at the start. The smallest grid that contains both is the least common multiple of 3 and 2, six pulses; one voice hits on pulses 0, 2, 4 and the other on 0, 3. Cross-rhythms like 3:2 and 4:3 are basic to many West and Central African traditions and to the Afro-American music that grew from them; in the Western classical tradition 3:2 appears as the hemiola, and four against three drives the right and left hands of Chopin's Fantaisie-Impromptu (composed 1834).
The hard part is not counting; it is feeling both at once. Teachers often start by speaking one layer and tapping the other, or by hearing the combined rhythm as a single pattern, which is what the "both" line below prints.
Gold (outer) is the higher drum, blue (inner) the lower. The line sweeps once per cycle.
Indian classical music counts time in a tala, a cycle of beats grouped into sections. In Hindustani (north Indian) music each section is a vibhag. The first beat of the cycle is sam, "the most important and heavily emphasised," the point where phrases resolve; some sections are marked with a hand clap (tali), and an empty beat (khali) with a wave. Teentaal has 16 beats in 4+4+4+4 with sam on 1, claps on 5 and 13, and a wave on 9. A tabla player plays a skeleton called the theka in spoken syllables, bols: Dha Dhin Dhin Dha, Dha Dhin Dhin Dha, Dha Tin Tin Ta, Ta Dhin Dhin Dha. Jhaptal is 10 beats (2+3+2+3), and Rupak 7 beats (3+2+2). The tempo, laya, is free: a performance moves through slow (vilambit), medium and fast (drut) with the same tala.
The key difference from a bar is arrival. A musician improvises for many cycles and aims every phrase at sam; the audience hears the landing. A Western downbeat is a point the music passes. Sam is a point the music is going to. Carnatic (south Indian) music has its own system of talas built from short units. Adi tala has 8 beats: a laghu of 4 (a clap, then finger counts) and two drutams of 2 (a clap and a wave).
The Arab world counts rhythm in iqa'at (singular iqa'; also wazn, plural awzan): cycles remembered through the syllables dum (a low stroke from the centre of the drum) and tak or tek (a high one from the edge). Reportedly there are over a hundred; they run from two beats to 48 or more, and one can last 176 units. Three common four-beat ones: maqsum (dum tak, tak dum tak), baladi (dum dum, tak dum tak) and saidi (dum tak, dum dum tak). Players slide between them within a piece.
Javanese gamelan marks time in a different way, called colotomic structure: instruments such as the big gong, the kenong and the kempul divide the cycle into nested sections, like a clock with hour, minute and second hands. In a standard lancaran form, a 16-beat cycle ends on the gong, the kenong sounds on beats 4, 8, 12 and 16, and the ketuk keeps the faster count between them. The idea is the same as sam: the music is built to arrive at the gong. See Music Traditions of the World (35/69).
Ring labels: X = sam, numbers = claps, 0 = wave (Hindustani); C = clap, 1 2 3 = finger counts, W = wave (Carnatic); D = dum, T = tak (Arab). The sounds are simple stand-ins for drum strokes, not recordings; the drum mode and hand-clap mode apply to Hindustani talas only, and Adi tala is always shown with hand beats. Groupings and thekas follow standard references.
Swing is the unequal division of the beat. Two eighth notes in a beat are played long-short instead of even. A swing ratio of 2:1 is the long-short of a triplet (the off-beat note lands two-thirds of the way through); 1:1 is straight. Jazz swing is not a fixed ratio. In jazz it typically lies between 1:1 and 3:1 and varies a lot, and Friberg and Sundström's 2002 study of jazz drummers found swing ratios wider at slow tempos and narrower at fast ones. One reason is arithmetic: at 240 beats a minute a 2:1 ratio leaves the short note about 83 ms long.
Groove is the feeling of being drawn to move, and it is harder to pin down. Two findings are worth knowing. First, in a 2014 study by Maria Witek and colleagues, 66 participants rated 50 drum breaks; those with medium amounts of syncopation produced the most desire to move and the most pleasure, an inverted U (too regular is dull; too off-beat loses the pulse). Second, whether tiny timing deviations ("microtiming") are what make a groove is disputed: musicians often describe playing "behind" or "ahead of" the beat, yet in one 2013 experiment on a simple drum pattern the strongest groove response was to the version without deviations. Groove may need more than timing.
A loop of hi-hat eighths, a low drum on 1 and 3, and a snare on 2 and 4. Move the sliders while it plays. Computed from the tempo: beat = 60,000 / tempo milliseconds.
Western notation was built for music with a bar and a downbeat, so it writes timelines by forcing them into bars: a clave is "3-2" because it fits a 4/4 bar, and the bell is written in 12/8. That works well enough for communication, and it quietly gives the pattern a "one" it may not have in the tradition. Agawu's warning about "additive" accounts is partly this: describing an African cycle as 3+3+2 is a Western way of counting it, and the players may think in terms of the whole pattern, the dance step or the drum language. The same caution applies to the "clave" and "downbeat" in Latin music, where Cuban musicians often don't use the 3-2 and 2-3 terms.
The question of who records and describes other people's music is live in ethnomusicology itself. Alan Lomax (1915–2002) recorded folk music across the US and Europe and founded the Association for Cultural Equity in 1983, which works to share and return his archive; the Library of Congress and the Association released it digitally in 2012. Credit and authorship in such fieldwork have been questioned: the English singer Shirley Collins said her part in his 1959 Southern Journey recordings was barely acknowledged in his later writing. Teachers today increasingly teach other musical systems on their own terms, from practitioners, rather than as footnotes to European theory.
Rhythm is a steady pulse cut into groups of twos and threes, and traditions differ in what they hang on it: a bar with a downbeat, a timeline that can be entered anywhere, a tala that points at sam, an iqa' of dum and tak. Swing and groove are the same system felt rather than counted, and nobody yet fully agrees on what makes a groove.
Try this: this week, tap a steady pulse on a table with one hand. With the other, tap three evenly spaced taps in the time of every two of the first. Take it slowly. It is hard for a minute and then it clicks, and you have just played 3 against 2.
Sources. Bell pattern (strokes 1, 3, 5, 6, 8, 10, 12 of 12; Kubik, Locke, Agawu): encyclopedia summary citing Kubik, Theory of African Music; Agawu, Representing African Music (2003); Locke, Drum Gahu. Clave grids, son-to-Havana, Bauzá and Puente: encyclopedia summary; bossa nova clave as "son clave with the last stroke delayed one subdivision". Euclidean rhythms: Toussaint (2004 discovery; BRIDGES, Banff, 2005); the patterns and their matches were generated and checked in node. Pressing (1983, Studies in Music 17); London (2002); Wen and Krumhansl (2019). Aksak: Brăiloiu (1951), encyclopedia summary. Agbekor: encyclopedia summary. Talas, sam, khali, laya: encyclopedia summaries of tala, teental, jhaptal and rupak; rupak notation varies between sources. Iqa'at: encyclopedia summary and Maqam World; transcriptions of maqsum, baladi and saidi vary between sources. Colotomic structure and lancaran: encyclopedia summary. Friberg and Sundström (2002, Music Perception); Witek et al. (2014, PLOS ONE); Frühauf, Kopiez and Platz (2013, Musicae Scientiae). Lomax dates and the 2012 digital release: encyclopedia summary. Chopin's Fantaisie-Impromptu (composed 1834) and the hemiola are standard music history. All interactive numbers were computed in node.