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.
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.
Obviously, the rewrites don't look as elegant as the original tuplets, but they give you the tools to accurately place each