Polyrhythms 3:2, 4:3, 5:4, 7:4: calculation and how to play them

Definition

A polyrhythm superimposes two different divisions of the same duration: "3 against 2" means three equal notes in the same time in which two are played. This entry is practical: how to calculate the positions, how to play them with your hands or program them, how to use them in a piece. (The general theory is in the entry "Polyrhythm, hemiola and polymeter.")

The least common multiple method

Divide the shared duration into LCM(a, b) pulses; voice A plays every LCM/a pulses, voice B every LCM/b.

(X = both together, A = voice A only, B = voice B only, "." = rest.)

In milliseconds

At 120 BPM one bar of 4/4 lasts 2000 ms.

• 4 notes: every 500 ms (quarter notes).
• 3 notes over the bar (3:4 over the bar, half-note triplet): every 666.7 ms.
• 5 notes: every 400 ms. 7 notes: every 285.7 ms.

Useful for setting free-time delays or for checking a programmed polyrhythm.

How to play it with your hands (or with pads)

• 3:2: count "1-2-3-4-5-6" slowly; right hand on 1, 3, 5, left on 1 and 4. The combined pattern is long-short-short-long...: better to say the grid out loud and speed up only when it is automatic.
• 4:3: count 12, right hand on 1-4-7-10, left on 1-5-9. The combined pattern has a recognizable "galloping" profile. It is the same ratio as the famous passage in Chopin's Fantaisie-Impromptu Op. 66 (sixteenths against triplets).
• 5:4 and 7:4: they can't be counted well on a grid at real speed; you learn them by hearing the odd voice as a melody of its own over a fixed pulse. Practice: metronome on the quarter notes, sing 5 equal syllables ("ta-ka-di-mi-ta" of Indian konnakol) per bar until the last always arrives before the new 1.
• General strategy: one voice at a time, then the two together at 50% of the tempo, then increase by 5% at a time.

How to program it in FL Studio

• 3:2 and 4:3 are simple: use "1/3 beat" snap for eighth-note triplets, "1/6 beat" for sixteenth-note triplets; the half-note triplet (3 over the bar) is written with notes of 4/3 beat (snap 1/3 beat, length 4 units).
• 5 and 7 are not on the standard grid: the project resolution (PPQ) must be raised to 960 in the project settings; a bar of 3840 ticks divided by 5 gives exactly 768 ticks. For 7 (548.57 ticks) round or work with "None" snap while watching the position.
• Creative alternative: create a pattern of a different length (e.g. a 5-step pattern) and extend it in the Playlist: it is a polyrhythm by polymeter (see below).

Polyrhythm vs polymeter

In a polyrhythm the two voices restart together at each cycle (3 notes and 2 notes in the same duration). In polymeter, two cycles of different length with the same unit (an arpeggio of 3 sixteenths over a bar of 16) realign only after LCM pulses (48 sixteenths = 3 bars).

Musical example in A minor

At 110 BPM: kick and bass on the quarter notes with Am – F; a marimba arpeggiating A–C–E in half-note triplets (3 notes per bar = 3:4 over the bar); hi-hat in sixteenths. The result: an ostinato that "floats" above the grid. In the last bar before the chorus have the whole band play the same triplet to create a hemiola fill.

Use in real tracks

• Afro-Cuban and Afrobeat: 3:2 is the basis of 6/8 against 4 (bell and congas against the step in 2). Fela Kuti builds entire grooves on interlockings of this kind.
• Trap: hi-hat rolls in sixteenth-note triplets over a sixteenth-note grid = momentary 3:2.
• Chopin, Op. 66 (4:3) and many pages of Scriabin and Brahms (3:2 between the hands).

Exercise

1. Program 4 bars of 4:3 with two different percussion sounds (rim and cowbell) and record yourself playing it on the pads.
2. Add a voice in 5 on a filtered tom and listen to see after how many bars the pattern seems to "repeat."

Related entries

"Polyrhythm, hemiola and polymeter," "Triplets and irregular groupings," "African timelines: the bell pattern in 12/8," "Metric modulation." Texts: Toussaint, "The Geometry of Musical Rhythm"; Gary Chester, "The New Breed" for limb independence.