What is the lowest common multiple in a polyrhythm?

It is the smallest number of equal subdivisions that both pulses divide evenly. For 3 against 4 the LCM is 12, so a bar split into 12 ticks lets you place 3 evenly spaced notes and 4 evenly spaced notes on the same grid.

How do I program a polyrhythm from this?

Set your sequencer to the LCM number of steps per bar. Place the first pulse every LCM divided by the first number of steps, and the second pulse every LCM divided by the second number. Both lines restart together at step one.

Why do the two pulses only meet at the start?

When the two numbers share no common factor, their notes never overlap except at the very beginning of the cycle. That sparse coincidence is what gives a true polyrhythm its rolling, unsettled feel.

What if the two numbers share a common factor?

Then the LCM is smaller than the simple product. For 4 against 6 the product is 24 but the LCM is 12, because they share the factor 2. The pulses coincide more often, giving a less complex cross-rhythm.

Does this work for more than two pulses?

This tool handles two pulses, which covers almost all musical polyrhythms. For three or more layers you take the LCM of all the numbers in turn, applying the same gcd-based method pair by pair.

Polyrhythm & LCM Calculator

A polyrhythm layers two pulses that divide the same span of time into different numbers of equal parts, such as three beats against four. They never quite line up, which is what makes them sound complex and rolling. To sequence one cleanly you need the single grid both pulses share, and that grid is found with the lowest common multiple.

How it works

Take a polyrhythm of a against b. The two pulses align perfectly only when the bar is divided into a number of ticks that both a and b divide evenly. The smallest such number is the lowest common multiple, computed from the greatest common divisor:

gcd(a, b) = repeated remainder (Euclid's algorithm)
lcm(a, b) = (a × b) / gcd(a, b)

On a grid of lcm ticks:

the first pulse lands every lcm / a ticks,
the second pulse lands every lcm / b ticks,
and both coincide only at tick 0, the downbeat, because the numbers share no factor beyond their gcd.

Worked example

For 3 against 4: gcd(3, 4) = 1, so lcm = (3 × 4) / 1 = 12. Divide the bar into 12 ticks. The three-pulse line hits ticks 0, 4 and 8. The four-pulse line hits ticks 0, 3, 6 and 9. They meet only at tick 0, then drift apart for the rest of the bar — the classic three-against-four sound.

Contrast 4 against 6: gcd(4, 6) = 2, so lcm = 24 / 2 = 12. The shared factor of 2 means the pulses coincide more often and the cross-rhythm is gentler.

Tips and notes

The number of coincidence points inside one cycle equals the gcd of the two numbers (counting the downbeat). Coprime pairs such as 5-against-7 produce the busiest grids and the most striking effect. For three-layer polyrhythms, fold the LCM in pair by pair. All of this runs locally in your browser.