Gareth Montanarello

How to Count Polyrhythms: Part 1

The last post I put out mentioned the bar of 11/8 that Bloch so conveniently slapped over a bar of 4/4 in Schelomo. I half-jokingly included this rewrite as the accurate solution to playing said measure, but now I actually want to talk about how I came up with the rewrite and how I approach learning polyrhythms. T

A while back I read Arthur Weisberg's demystifying book "Performing Twentieth-Century Music: A Handbook for Conductors and Instrumentalists." I recommend picking up a copy partially because it answers a ton of questions and partially so I don't feel bad about stealin... er... adapting his genius for my blog posts.

Hemiolas and the Basic Steps

The hemiola (3 beats in the space of 2) is perhaps the most common polyrhythm, and odds are you have some intuitive awareness as to how to play it. In 6/8 (a duple meter) it might

look like this:

And if I asked how to subdivide and group notes for the 3 beats and the 2 beats, you would probably know that eighth notes are the common note value for both sets of beats and come up with something like this:

Finding the common note value, or as Weisburg calls it, the least common denominator, is essential for figuring out any polyrhythm/cross-rhythm/tuplet, so let's practice with some more examples.

Here are the steps to keep in mind (adapted from pg. 22 if you have a copy of the book):

1. Determine the rhythmic ratio (3:2, 4:3, 5:4, 7:12, etc.)

2. Determine the common note value (least common denominator) and how many of that note value fit in the duration of the polyrhythm. You can find this value by simply multiplying both sides of the ratio: (3x2=6; 4x3=12; etc.)

3. Write out a "rhythmic grid" with the corresponding number of note values and group them according to the *measure's* meter. (3:2 means 6 notes in the space of two beats giving you two eight note triplets assuming the measure is in 2/4)

4. Tie groups of notes together until you have equal groupings. The number of groupings should be the same as whatever the first number of the ratio is.

5. Rewrite and simply the rhythm by getting rid of slurred notes where appropriate.



  • 2:3 Here is a 2:3 polyrhythm in two different meters: 3/4 and 3/2. I laid out each step for each meter, though for 3/2 you can see the rewrite is in its simplest form before the last step.

A 2:3 cross-rhythm makes up much of the Battle Scene in Ein Heldenleben, and Strauss uses two different notations to show the same rhythmic values.

These duplets use half notes, not quarters. Same difference.
  • 4:3 and 3:4 Here are two more common polyrhythms. We'll do 4 notes in the space of 3 and the inverse, 3 notes in the space of 4.

If the final rewrite is symmetrical then it's correct.

Obviously, the rewrites don't look as elegant as the original tuplets, but they give you the tools to accurately place each note and ensure each note is the same length.

Central Park in the Dark is filled with 3:4 and 5:4 (rewrite below) polyrhythms. Too bad he only gave the bass whole notes...

  • 5:4 and 4:5 I have seen 5:4 more often than 4:5, but here are the solutions to both. These get more complicated since we start delving into subdividing with quintuplets.

  • 7:3 Here's another tricky one: 7 quarters in the space of 3 half notes.

I can't think of any examples of this rhythm off the top of my head, but I'll update the post if I think of one.

  • 11:4 Here's the aforementioned 11/8 over 4/4. I don't think Bloch expected perfect rhythmic accuracy, so I didn't bother trying to play the rewrite. You can read a bit more about it in the original post.

The rewrite just for fun:



That should lay the basic groundwork for approaching polyrhythms. In part 2 I'll talk about how I practice this stuff and share a shortcut that bypasses most of the rewriting without sacrificing accurate results. Please leave a comment if you have any questions and please share this post if it was helpful!


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