What is a Polyrhythm?

If we break down the word, it's composed of "poly-" (which means more than one) and "rhythm" (a pattern in music). It doesn't take much to derive a simple definition: more than one rhythm occurring in music.

This definition is sufficient for the English dictionary, but it doesn’t completely capture the intention and use of polyrhythms. A musician may still ask: What does a polyrhythm accomplish? How are these contrasting rhythms constructed? What constitutes a contrasting rhythm? What purpose does this fulfill for the composition? I do not consider myself qualified to alter or create the definition for this word, but I do believe I have some insight to help fellow musicians better understand it.

A polyrhythm occurs when two or more different rhythms are played simultaneously, interweaving to create a complex, layered musical texture. Each rhythm maintains its own regular spacing—its own time interval between beats—while preserving the integrity of the overarching time unit. The rhythms do not necessarily align with each other in the same way as they would in simpler rhythmic patterns.

This long winded definition covers all bases necessary to understanding polyrhythms. Let’s break it down.

Two or more different rhythms must occur simultaneously, yet they do not need to align as rhythms might in simpler rhythmic textures. The rhythmic grids of the underlying rhythms do not need to be filled in, but they do need to be implied. More on this later.
Words like “interweave” and “overarching time unit” imply that the rhythms do indeed start and end together. Whether this alignment happens over beats, measures, or entire compositions does not matter, which is why the term "overarching" is used. It is important to note that these starting and ending alignments may be intentionally omitted through purposeful composition.
Each rhythm that constructs the polyrhythm must be equally spaced within itself. If one of the rhythms divides the overarching time unit into three beats, then these three beats need to be equally spaced. Similarly, if the other rhythm divides the time unit into five beats, these five beats must also be equally spaced. When these two rhythms interweave, our ears should be able to pick out two distinct rhythms, each with equal spacing.
Maintaining the integrity of the overarching time unit means that the listener should be able to tap their foot (or headbang, for those who prefer it) to the foundational time unit while experiencing the complexity of these interwoven rhythms. The keyword is "should," as it may require a musically mature ear to feel the foundational time unit while hearing a polyrhythm. This can be used to the advantage of the composer who wants the listener to lose the sense of time. However, a listener should still be able to comprehend the texture if accompanied by a metronome (or any other steady beat-providing device/part).

Now that the intention and use of polyrhythms have been made clear, it is essential to go over how to construct polyrhythms so that they adhere to the 'rules'.

Constructing Polyrhythms

Before constructing polyrhythms, we must understand that they are most commonly measured in ratios. These ratios are written out as 2:3, 3:2, 4:5, 6:4, etc. There are many different ways to describe these ratios, with the most common being "2 over 3" or "2 against 3." Both phrases mean the same thing but may carry different implications. The order of the numbers is crucial, as it determines how we construct the polyrhythm.

The process is tedious yet fruitful in its results, as you can create a polyrhythm of any desired length (with "length" referring to the amount of vertical time the polyrhythm will occupy).

The Process

Decide what polyrhythm it is that you want to construct. For the following examples, we will construct a 2:3 (verbalized as 2 over 3 or 2 against 3).

Since the 2 is being played against the 3, we first need to establish the 3. This will serve as the foundation of the polyrhythm. By using a measure of 3/4, we already have this foundation in place.

Now we must create a rhythm in which the measures are simultaneously divided into two equal beats. To do this correctly, we must subdivide each foundational beat into two. Since we are dealing with quarter notes, we can simply divide each note into two eighth notes. The second number in the ratio is always divided by the first number. We will see how this works with more complicated polyrhythms later.

It is shown that each measure is divided by three beats, and each subsequent beat is divided into two equal notes (resulting in eighth notes for this example). To finally create the polyrhythm, we must accent every third subdivision. This results in two groupings of three eighth notes. These two groups of equally spaced/valued notes divide the measures into two, thus creating the polyrhythm. The overarching time is divided into three, but our ears could be fooled into thinking that the measures are divided into two.

Now that we have the grid, we can create any rhythmic variations of the polyrhythm. The simplest would be to reduce both rhythms to their basic units. The division of three will be reduced to quarter notes, and the division of two, since subdivided into groupings of three, will be reduced to dotted quarter notes.

(Note that both rhythms realign at the beginning of each measure, visually representing the start of a new polyrhythmic cycle)

Breakdown of Ratios

All of the information needed for the process presented above can be retrieved from the ratio alone. Here is a list of how that is so, which also serves as an abbreviated form of the process above. 2:3 will still be used for this example.

The second number is the foundation of the polyrhythm, in this case 3. That means we must have a foundation of three. The value of the foundational notes do not matter, so long as they divide the macro-unit of time into three equal beats.
The first number decides what meter is going to counter the foundation. It also decides how the foundational beats will be divided. In this case it is two, meaning the counter rhythm will divide the macro-unit into two equal beats, and the foundational rhythm will be divided into 2 subdivisions (whole notes will subdivide into half notes; sixteenth notes into thirty-second notes, etc.)
The second number indicates how many subdivisions are within each grouping determined by the first number—or the spacing of accents after the initial accent. In this case, the second number tells us that an accent is applied to every third subdivision.

One More Example

This will be an expedited example of creating a 5:4 polyrhythm, which is a little more complex.

Second number states the unit of time must be divided into four. Quarter notes will be used.

The first number decides how these four beats will be divided. The quarter notes will be divided into quintuplets.

The second number determines how many notes are required for the groupings of the first number—or it determines the interval between accents. In this case, every fourth sixteenth note of the quintuplets, following the first accent, will receive an accent.

With the grid that has been created, a simplest form can be derived. The division of four will be simplified to quarter notes, and the division of five simplified as best as possible.

(Sometimes the simplified form of the polyrhythm is not the best suitable for being read)

Having Fun with Polyrhythms

Thus far, we’ve reviewed what polyrhythms are, how to measure them, and how to construct them. The excitement behind polyrhythms continues as the complexity increases. Simple polyrhythms are interesting to listen to and perform and can be a powerful tool for a composer. To increase the complexity, a composer can imply a polyrhythm without actually providing the grid. This can involve omitting notes from the subdivided grid as well as omitting notes from the simplified grid. Similarly, we can take these grids and change note values, resulting in less notes, or in the case of making the values smaller, adding more notes.

Omitting Notes from the Grid

For this example we will use a 4:3 polyrhythm. Below are the grids:

Subdivided:

Simplified:

Now with the omitted notes:

Subdivided:

By removing a few notes and inserting rests, the polyrhythm becomes more intricate and rhythmically engaging. While the accents still emphasize the strong beats of the 4, the omission of notes from the grid introduces a varied texture. This example represents a common polyrhythmic groove in rock and metal music. In some instances, this groove is utilized in multi-layered polyrhythms, where the bass, guitar, and drums follow different rhythmic patterns yet intertwine to suggest the overarching time unit.

Simplified:

This example creates a texture that has more space, making it harder to hear the implied polyrhythm. A composer would be able to use this to their advantage in portraying perhaps a more ominous or uncertain feeling.

Adding Notes to the Grid:

Subdivided:

When the top line is beginning to look like a snare etude, you’re on the right track. For this example, the subdivided grid added notes by shortening the values of those present, and removed notes due to the lengthening of note values. It is evident that as we further deviate from the grid, the polyrhythm is harder to hear, yet the integrity of the overarching time unit remains.

Simplified:

Again, notes are omitted due to lengthening of values, and notes are added due to shortening values.

Note: Both lines of the polyrhythm can undergo these processes. I refrained from altering them for the sake of transparency in the examples. The more you deviate from the established grid, the more challenging it becomes to understand the polyrhythm. This simple fact can either enhance your composition or detract from it.

Quick Note on Layered Polyrhythms

The main purpose of these examples was to create polyrhythms using two rhythms. It's important to note that polyrhythms can be stacked. This can be achieved by either dividing the macro-time unit further or by subdividing already existing rhythms. Dividing the macro-time unit might involve triplets against sixteenth notes against quintuplets, for example, written as 3:4:5. Subdividing already existing rhythms becomes more intricate and involves creating polyrhythms against the already present rhythms. For instance, if you have a 4:5, you can create a 3:4 using the existing division of 4.

Polyrhythms have immense potential as a compositional device, capable of enhancing a piece by adding depth and altering textures. However, when used disproportionately, they can diminish the overall impact of the piece. Nonetheless, as a percussionist myself, these are some of the most satisfying musical textures to listen to, perform, and create.