Rhythm Playground

Western notation writes rhythm as a line that ends and starts again. A repeating rhythm is really a cycle, and drawing it as points around a circle — a rhythmic necklace — makes properties visible that a line hides: rotations become turns of the same shape, two rhythms related by a shift become the same necklace, and evenness becomes a matter of how close the onsets come to the vertices of a regular polygon.

Beats, Tempo, and Swing

Beats / cycle is the meter the wheel is measured against: how many beats one full turn lasts. It is not a property of any ring — it is the frame all of them are read in, and it does four things at once. It sets the cycle’s duration at the current tempo (six beats take half again as long as four, so the wheel turns more slowly); it draws the spokes; it decides which cells in the box notation count as beats; and it fixes the strong beats that the syncopation figure measures distance from. The same twelve-step ring is a bar of triplets against four beats and a bar of four groups of three against three beats — the hemiola, available by moving one slider, with the spokes and the analysis both following.

Swing answers one question: where inside the beat does the offbeat land? Eighth notes written evenly are rarely played evenly. Straight playing puts the second of each pair exactly halfway through the beat, at 50 %; swung playing pushes it later, so the pair is heard as long-then-short rather than even-even. The percentage is that landing point. The scale is the one Roger Linn devised for the drum machines that made swing a knob, and which the Akai MPC and Logic still use: 50 % straight, 66.7 % the triplet shuffle behind most jazz and blues, and 75 % — the MPC’s upper limit, and this page’s — a hard three-to-one long-short. It is a convention, not a law: Ableton Live, FL Studio, and Cubase count the same effect from zero instead, where 0 % is straight. At 75 % the pair is dotted-eighth-plus-sixteenth when the swung pair is eighths; on a finer grid the same ratio lands on smaller values.

The on-beat onsets never move; only the offbeats slide, and everything between 50 % and 75 % is available continuously. The percentages under the first pair are the two halves of the beat: at 50 % they are equal, at 66.7 % the first is twice the second, at 75 % three times.

Written as notes, the same three settings are three familiar ways of dividing one beat. Only the middle one involves triplets, and the “dotted” one is a dotted eighth — the note lasting half a beat, lengthened by half again to three-quarters of the beat — not a dotted half.

Reading the rows: straight is two ordinary eighth notes, each half a beat. Triplet divides the beat into three equal parts instead of two and gives the first note two of them and the second note one — notated as a quarter note and an eighth note bracketed with a small 3, which is what the “swing” instruction at the head of a jazz chart asks for. Dotted divides the beat into four and gives the first note three of them: a dotted eighth followed by a sixteenth, the long-short limp of a shuffle blues. The slider passes continuously through every value between, including the shallower feels around 54–62 % that most swung playing actually sits in.

Swing is visible in the instrument, not merely audible: raising it slides the offbeat dots around the wheel and shifts the offbeat cells in the box notation, so the displacement can be seen as well as heard. It applies only to rings whose steps divide the beat evenly — a twelve-step ring against four beats is already a triplet grid, so swinging it would swing a shuffle twice, and those rings are left alone.

Polyrhythm and Polymeter

The two words are often used interchangeably, and the difference is exactly the choice this instrument puts on a menu. Polyrhythm is one span of time cut into different numbers of equal parts: three against four, where both parts start together, finish together, and disagree about everything in between. Polymeter is a shared pulse grouped into cycles of different lengths: a seven-step pattern and a sixteen-step pattern counting the very same unit, coming apart because they run out at different moments and only realigning after their least common multiple. Polyrhythm is a disagreement about division; polymeter is a disagreement about length.

The terms are not used this cleanly everywhere. Broader definitions treat “polyrhythm” as any simultaneous rhythms not heard as deriving from one another, wide enough to swallow polymeter; scholarship on African music generally prefers cross-rhythm for the three-against-two case and keeps polyrhythm for denser textures; and there is a standing question, raised by work on rhythmic perception, of whether listeners hear two meters at all rather than extracting one composite and hearing the rest as ground — something this instrument makes easy to test on oneself.

Each ring chooses which it takes part in. Under fill the cycle, the ring’s steps divide one cycle evenly, so a ring of 3 and a ring of 4 complete together and the classic three-against-four results — polyrhythm. Under one unit per step, every step takes the same grid unit regardless of ring size, so a ring of 7 finishes early and starts again out of alignment — polymeter; the pattern of coincidences takes several cycles to repeat, and the analysis panel reports how many under Composite repeat. The grooves filed under Polyrhythm & polymeter load one of each to compare, and the cyclic view is where the difference is easiest to see: a polyrhythm’s rings meet at twelve o’clock every bar, while a polymeter’s drift past one another. Phase drift pushes one ring fractionally faster than the rest, the mechanism behind Steve Reich’s tape pieces: identical patterns slide out of phase, pass through every intermediate alignment, and eventually lock again.

Euclidean Rhythms

Distributing k onsets as evenly as possible among n steps has one natural answer, computed by the same algorithm Euclid gave for the greatest common divisor. The result, E(k,n), turns out to be a remarkable number of the world’s traditional rhythms: E(3,8) is the tresillo, E(5,8) the cinquillo, E(2,5) a Persian and Balkan commonplace. The analysis panel reports whether the current ring is a rotation of its Euclidean pattern — and the interesting cases are the famous ones that are not, the son and rumba claves among them, which sit one step away from maximal evenness.

Performing Clapping Music

The cell is not arbitrary. Of all 4096 twelve-beat patterns, 4020 are aperiodic — a pattern that repeats inside itself would sound the same under more than one shift. Counting only one representative per rotation leaves 335; keeping those with four to eight claps leaves 287; and forbidding two rests in a row leaves exactly eleven. Reich’s cell is one of them, and one of only two whose runs of claps between rests are all different lengths. All eleven are offered below the transport, derived in the page rather than transcribed; the starred one is Reich’s. Each loads as two clapping parts, panned apart, the second shifting one step every twelve repeats — the interval the score specifies.

Clap along. Reich’s instruction is to play the piece with a partner, and the hardest part is holding one line while the other moves against it. Clap along silences the selected ring and leaves the rest sounding, so that part can be practised alone: press C, or click or tap the wheel, on each clap. Every clap is timed against the onset it should have landed on — measured on the audio clock, since a frame-based clock drifts by more than the window being judged — and reported as early, late, or dead on, with a running count and streak.

Steve Reich’s Clapping Music (1972) is a single twelve-quaver pattern — eight claps in twelve, the gaps running 1-1-2-1-2-2-1-2 — clapped by two performers. They begin in unison; one then shifts by a single quaver every twelve bars, and after twelve shifts the pair arrives back where it started. The Clapping Music groove loads both rings with the pattern and arms the second ring’s auto-shift, which advances that ring by one step every four cycles, so the piece performs itself. Four cycles is an accelerated demonstration speed; the score calls for a shift every twelve bars, which auto-shift set to 12 reproduces. Either way, after twelve shifts the two rings are back in unison. To take the shifts by hand instead, set auto-shift to off, select the second ring (click or tap its row under Rings, or press 2), and click or tap ↻ Rotate or press → once per shift; the score’s own interval is one shift every twelve bars, which auto-shift will also do if set to 12. Each shift produces a different composite, and the ear reorganizes what it hears at every stage even though nothing about either part has changed. Setting a small phase drift on one ring instead performs the continuous version of the same idea, the tape-loop phasing of It’s Gonna Rain and Piano Phase, where the shift happens gradually rather than in steps.

Two rings playing the same voice would otherwise fuse into a single composite, so the two clappers are placed left and right — far enough apart that either part can be followed on its own, close enough that the resultant pattern between them still coheres, which is the effect the piece is written for. Every ring has the same Pan control; the grooves use it lightly, putting hats and shakers off to one side and leaving kick and snare centered. Turning Spatialization on hands placement to the listener model instead, and the pan control steps aside rather than applying twice.

Standing Inside the Ring

With step angle → azimuth chosen, the listener is placed at the center of the wheel and every onset is rendered from the bearing that step occupies on the circle — so the wheel is not a diagram of the sound field, it is the sound field, heard from inside. A pattern travelling clockwise round the ring is heard travelling clockwise round the head; two rings whose onsets fall at different angles separate in direction as well as in time. The alternative, per-ring azimuth, keeps each ring whole at one fixed bearing, which is closer to a drum kit laid out in front of a player.

Head-track completes the illusion. Without it the sound field is glued to the head: turning to look at a ring takes the ring along with it, and the space feels like headphones rather than a room. With the camera tracking, the rings hold still in the world while the head turns inside them, and looking towards a ring brings it round to the front — the same head-tracking used in the chapter 9 narrowcasting sandbox. The video never leaves the browser and is released the instant tracking stops or the tab is hidden; While it runs, a small mirrored camera preview sits in the corner of the wheel — a live camera should always be visible as such — and the listener appears at the hub as a head with a nose and a facing wedge, turning as the head turns while the rings stay put. The angle is read out in degrees beneath it. Recenter declares the current head position to be straight ahead. Headphones are essential for any of this — over speakers the room fixes the directions itself.

Reading the Analysis

• Intervals — the gaps between successive onsets, the 3-3-4-2-4 reading of the son clave. Rotating a necklace rotates this string; two rhythms with the same cyclic interval string are the same necklace.
• Interval vector — how many onset pairs are separated by each possible distance. A rhythm whose multiplicities are all different is called deep.
• Evenness — the perimeter of the polygon through the onsets, divided by the perimeter of the regular polygon with the same number of vertices. A value of 1 means the onsets are as evenly spread as the grid allows.
• Off-beatness — how many onsets fall on positions that no regular subdivision of the cycle can reach. Those positions are unavailable to any simple meter, which is part of why they sound the way they do.
• Rhythmic oddity — the property, named by Simha Arom for the Aka rhythms of Central Africa, that no two onsets split the cycle into two equal halves.
• Nearest timeline — the closest match among the classic timelines over every rotation, with the number of steps that separate the two.

Hearing the Circle

With step angle → azimuth selected, the picture stops being a metaphor: each onset is rendered through the listener model at the compass bearing it occupies on the wheel, so a pattern sweeps around the head as it plays and two rings at different angles separate spatially as well as rhythmically. The alternative places each ring as a whole at a fixed bearing, which is closer to how a drum kit is laid out in front of a listener.

Working the Instrument

• The timelines each occupy a single ring, so any of them can be put on any ring, and clicking or tapping a rhythm’s name rather than one of its cells loads it into every ring at once, in unison — the starting position for a shifting piece. The grooves, by contrast, are whole kits and do not all use the same number of rings: most use three (kick, snare, and a timekeeper), Bo Diddley uses three of a different make-up, and Clapping Music uses two, because the piece is written for two performers. Rings can be added or removed at any time regardless of what was loaded.
• The grooves row loads a whole kit at once — rock, shuffle, disco, funk, boom bap, reggae one-drop, a clave-based rock beat, swing ride, bossa nova, and Reich’s Clapping Music. A named timeline occupies one ring; a groove is a relationship between rings, so these fill the wheel.
• The named-timeline grid holds eighteen classic rhythms — tresillo, cinquillo, the son and rumba claves, bossa nova, shiko, soukous, gahu, samba, fume-fume, bembé, Balkan aksak and ruchenitza, and the plain backbeat patterns. One row per rhythm, one column per ring: click or tap a cell to load that rhythm into that ring, and click or tap the lit cell again to clear it. A lit cell also reports what a ring is holding, so the grid doubles as a readout — edit a ring by hand and its cell goes dark the moment the pattern stops matching.
• Click or tap a step to toggle it, on the wheel or in the box notation below. Drag up or down on a step for velocity; shift-click cycles ghost → normal → accent.
• Press Space to start and stop, 1–8 to select a ring, ← / → to rotate it, M and S to mute and solo, T to tap. The shortcuts are printed on the controls they drive, and work wherever the focus happens to be on the page.
• Clone copies the selected ring’s pattern onto a fresh ring one step further round and panned opposite, which is the two-line texture of a shifting piece in a single click or tap.
• Timelines vs. grooves. A groove is a whole kit — each ring gets its own part. A timeline is a single line: son clave, tresillo, the bembé bell. In the Timelines table a cell loads one timeline into one ring; clicking the name arranges that line across the whole kit at once — the line itself on the bell voices, the kick on the strong beats it lands on, a backbeat snare, and the pulse on the hat. (Clicking the name does not clone the line onto every ring: that would put the whole kit in unison, which is the opposite of what a timeline is for.) Clicking the lit name again clears the kit.
• Two ways to sort a dance. The social-dance folders group by rhythm — waltz time, march time, ballroom swing, and the Latin ballroom family — because rhythm is what the wheel can show. Ballroom competition sorts the very same dances a second way, by carriage and costume, into Modern Standard and Latin American; where a dance belongs to one of those, its tooltip says so. Keeping the two apart matters because they disagree: the tango is a Modern Standard dance built on a habanera, and the jive is a Latin American one that is rhythmically a shuffle — a near relative of the rock & roll groove filed under Backbeat. Samba stays with its Brazilian relatives under Caribbean & Latin for the same reason, since the batucada on the wheel is not the ballroom dance that borrowed its name.
• Footfall. A dance step is a rhythm: slow-quick-quick is 2 + 1 + 1 beats, which the wheel can hold like any other part. Loading a dance groove draws its step timing twice over: as a dashed amber ring inside the wheel, marked footfall at the hub, and as a Steps row beneath the box notation. Either way each footfall carries the dancer’s own count — S and Q for the slows and quicks, or the numbers and the & and a of a syncopated one. Both are annotations and make no sound — the wheel’s ring is dashed while that is all it is. + Footfall ring, among the Rings controls and shown only for a dance, promotes the pattern to a ring of its own so the feet can be heard against the drums; the dashed ring then goes solid and the hub reads footfall — sounding. Space for that ring is held open even while it is off, so switching it on moves nothing else on the wheel. It stays on, too: every dance groove loaded afterwards brings its own footfall with it, which is what makes a quickstep and a foxtrot worth loading one after the other. It is worth hearing why the cha-cha-chá has five steps in four beats, or that the quickstep and the foxtrot share one footfall and differ only in tempo. Where a basic figure runs longer than one bar — the tango’s does, and the jive’s — the row shows its first bar and the note says so.
• Ghost and accent. A ghost is a stroke played much softer than the others — audible as texture rather than as a beat; an accent is one played harder than the others. Drummers use both to shape a line that would otherwise be mechanical. Shift-clicking a step cycles it through ghost, normal, and accent — on the wheel or in the box notation, which take the same gestures. Both displays say it the same way, in the ring’s own color: a ghost is drawn hollow, an accent carries an outline just outside the shape, and a normal stroke is plain. Only the shape differs — discs on the wheel, rectangles in the box notation. That matches the parentheses and > of drum notation.
• Shift-click a ring to add it to the selection, or shift-click a selected one to drop it. Every slider, menu, and transform then applies to the whole selection at once — retune three rings together, Euclidean-fill two at a time — while the readouts describe the primary ring, marked with a bar down its left edge. Shift with 1–8 does the same from the keyboard.
• Mute and solo follow the site’s narrowcasting convention: solo on any ring silences the rest, and shift-click extends the solo set.
• Store four kits in the banks and type a chain such as AABC to step through them, one per cycle.
• Copy link puts the entire kit — every ring, every step, every velocity, and every knob — into the address as a ?w= query parameter, so a groove can be posted or mailed and arrives set up exactly as it was left. Shift-click the same button for a readable link instead, in which every ring is a named parameter — ?bpm=96&beats=4&kick=1000100010001000&snare=0000100000001000 — which a recipient can read, edit by hand, or generate from a script. Modifiers attach with dots: kick=1000100010001000.rot2.pan-0.4. Opening such a link rebuilds the wheel before anything is played. MIDI and WAV export write four bars.

Sources & Inspiration

• The Rhythm Circle — Paul Lascabettes, Corentin Guichaoua, & Moreno Andreatta (2025), An Interactive Open-Source Web Environment Based on the Circular Representation, Int. Computer Music Conf., Boston. The concentric-ring instrument at rhythm-circle.com, from the Mathémusique project, is the model this page extends; its preset timelines are the starting point of the library here.
• Godfried T. Toussaint, The Geometry of Musical Rhythm (CRC Press, 2013), and “The Euclidean Algorithm Generates Traditional Musical Rhythms” (Bridges, 2005) — the necklace representation, the Euclidean-rhythm result, and the evenness, off-beatness, and oddity measures reported here.
• E. Bjorklund, “The Theory of Rep-Rate Pattern Generation in the SNS Timing System” (2003) — the algorithm, written for a particle accelerator, that generates the Euclidean patterns.
• Simha Arom, African Polyphony and Polyrhythm (Cambridge, 1991) — the rhythmic-oddity property and the Aka repertoire it describes.
• Drum notation — the score pane follows the Percussive Arts Society conventions: each voice on its customary line or space, an accent written > above the note, and a ghost stroke set in parentheses. For a plain-language walk through the whole convention — which drum each line and space stands for, why the metals take a × notehead and the drums a •, and what the accent and the bracketed ghost note mean — see Beat‑Note’s introduction to drum notation.

Taking It Further

• PolyNome Pro (Joe Crabtree / PolyNome Ltd) — where this page is for seeing how rhythms are built, PolyNome is for drilling them: a sample-accurate practice metronome with two independent sequencers, accents and ornaments, automated tempo ramps, and a log of everything practised. The natural next step once a pattern here is one worth being able to play.
• Musica! — Math meets Music (Jürgen Richter-Gebert, free for iPad) — three of its activities are close relatives of this page. A Sequencer builds rhythms on spinning discs: each disc carries evenly-spaced bars whose darkness is their strength, and an instrument sits as a point on the rim, sounding each time a bar sweeps past it — the same circle, with the pattern turning under a fixed pickup rather than a hand turning over a fixed pattern. n over m Rhythms is the polyrhythm case on its own. Steve Reich’s Clapping Music is the shifting piece, likewise made interactive.
• Clapping Music (Touchpress) — Reich’s piece as a play-along trainer, which asks the hands to do what the Clapping Music groove here does automatically.