How to Calculate and Master Polyrhythms

Welcome to my article on polyrhythms. Right now, you might be thinking that figuring out the the polyrhythm staring back at you from your sheet music is an imposible task. Or maybe you're frustrated at being unable to recreate the rhythms that you hear. Or maybe, just maybe, you're like I was 20 years ago; fresh home from a lesson and trying to figure out how to follow my teacher's guidance ("just fit 5 on top of 4") while also trying to slow the music down enough to overcome the physical and technical challenges of the music. 

I remember finding my teacher's advice absolutely useless. I wasn't one of these "started so young" musicians. I didn't start playing guitar until I was 16 (I did play cello and piano for a year each in elementary school, but that went nowhere).  I didn't have some amazing sense of rhythm built from years of experience. Heck, for grades K-12, I went to a private religous school that held a grand total of 2 dances. 

Every time I sat down to practice that rhythm, it was like banging my head against a wall. Nothing seemed to work. I didn't think I was ever going to get that rhythm right with my teacher's advice. I came to accept that I was a mediocre musician who couldn't just "fit 5 on 4" (while playing classical guitar. So playing 4 with the thumb while playing 5 with the fingers). And I figured out a way to teach myself that rhythm (and any polyrhythm) with 100% accuracy. 

My teacher didn't think much of my method, but I nailed it.

Years later, I was reading through backissues of Soundboard Magazine and found out that my method wasn't unique or special. And it most defintely wasn't for mediocre musicians. It was there that "my" method was that of no other than Daniel Tannenbaum, virtuoso classical guitarist and the most well-known performer of "new classical" works for guitar (where polyrhythms about).

Here I'm going to explain the same fool-proof method that master musicians use to play perfect polyrhythms. No matter your starting point, this article will help you understand and master your music. By the end of this article, you’ll understand the simple LCM method (hint: LCM is an acronym for a common math concept that you learned in grade school. It's the key to easy polyrhythms). I’ll even include an LCM calculator and number line generator so you can practice your polyrhythms right now. 

What is a polyrhythm?

A polyrhythm is when

• Two rhythms are played simultaneously 

AND

• Those two rhythms can not be divided into one another (without a remainder).

To understand this more clearly, let’s compare some primary rhythms to several different polyrhythms. 

NoT POlyrhythms

Basic Rhythms

2:4

In this example, we have eighth notes and sixteenth notes. The eighth notes divide the beat into 2, while the sixteenth notes divide it into 4. In other words, there are two eighth notes for every beat and four sixteenth notes for every beat. 

Every other eighth note will line up precisely with a sixteenth note when playing this. This is because 4 is divided by 2 (4/2=2, so every other note will line up). In fact, you can see the sixteenth and eighth notes line up perfectly in the written music. 

3:6

In this example, in compound time (3 eighth notes per beat), we have 3 eighth and 6 sixteenth notes per beat. Using simple division to see if this is a polyrhythm, we see it is not: 3/6 = 3. Looking at the music, we can see that the notes line up every third beat. 

Polyrhythms

3:2

In this example, both rhythms are written as eighth notes. However, one has a “3” written under it, indicating that it is a triplet (3 notes per beat). The other set of eighth notes is only 2 notes per beat. 

No matter how you divide these (3/2 or 2/3), you do not get a nice, neat, even division. These note groups will only line up at the beginning of the beat. Once they get going, they don’t line up again (unless the rhythm is repeated, in which case they'll line up at the beginning again). 

4:3

In this example, we have 3 quarter notes fitting into the same amount of time as 4 quarter notes. This gives us a 4:3 (or 3:4) polyrhythm. Obviously, they won't line up (except at the beginning). In fact, you can visually see this in the written notation:

3 over 4: How to talk about polyrhythms

When reading about polyrhythms, you often see a colon between two numbers. But how do you actually talk about it?

Polyrhythms are described as being “over” the fundamental rhythm, called the pulse. The polyrhythm 3:4 is called “3 over 4.” 

The pulse is the basic rhythm in a piece. Adding a counterpulse is what makes it into a polyrhythm. When you combine these two pulses, it's called a composite rhythm. 

How do you know which is the pulse and which is the counterpulse? Like much of music, it depends on the context. 

Here, we have two examples. The first is in 4|4. If you know how to read time signatures, you know that in 4|4, the eighth note divides the beat into 2. Adding a triplet makes this a polyrhythm; this is 3 over 2. 

Next, we see the same polyrhythm (a combination of 2 and 3) in 6|8. Since the beats in 6|8 are usually divided into 3, the division of 2 is unusual. Therefore, this rhythm is “2 over 3.” 

While these two rhythms have slightly different names, depending on musical context, they are fundamentally the same thing; 2 beats and 3 beats happening in the same amount of time. Therefore, they’re performed (almost) exactly the same way. 

These naming conventions are only helpful when discussing situations where a meter is clearly defined and/or the polyrhythms appear infrequently. In situations where the polyrhythm is more consistent (cross-rhythmic music), one person may feel it as 3 over 2. In contrast, another may feel it as 2 over 3. Both are legitimate interpretations. 

UNDERSTANDING ANY Polyrhythm

Listen First

As a dedicated Suzuki Guitar Method teacher, I’ve seen how my students progress when they acquire music as a language; by listening before "speaking". The easiest way to develop a new skill is to listen first. The more the better. 

This is just as true for polyrhythms (although they’re much less engaging than listening to a new song you want to learn). So, I’ve provided several common polyrhythms here for you to listen to, along with a video showing the beats in action. 

Using a Polymetric Metronome

The above examples were created using the Tempo Pro metronome on iOS. This metronome is my go-to metronome due to the simplicity of setting up subdivisions and polyrhythms. 

Using a polyrhythmic metronome can be helpful, especially when you already have a basic sense of the rhythm and/or you’re only playing one part of the rhythm (while another musician plays the other part). 

However, a polymetric metronome won't be helpful if you don’t understand the rhythm. Instead, you’ll want to use the LCM Method to develop a clear understanding of the polyrhythm.

Using the LCM Method

While listening to other performances and using a polymetric metronome can help develop your polyrhythmic abilities, the LCM method is the only way to break polyrhythms down and work on them accurately at a slow tempo. Only once you’ve done this can you speed them up and confidently know you’re performing them accurately. 

Here's how to do it:

1. Find the LCM

The LCM is the Least Common Multiple. For those who need a basic math review, the Least Common Multiple is the smallest number that two different numbers (in this case, your polyrhythm numbers) can be multiplied into. 

For example, the common 2:3 polyrhythm has an LCM of 6. By dividing the beat into 6, we can feel a steady subdivision that allows us to simultaneously count in 2 and 3. 

How can you figure out the LCM? There are a few ways. You can watch this Kahn Academy video and brush up on your math. Or you can google “What is the Least Common Multiple of X and Y?”.

Or, better yet, you can use my Polyrhythm Calculator, which will generate the proper LCM number line for you to use for your practice sessions.

2. Make a number line

Write a number line for the LCM. Our least common multiple is 6, so we'd write: 

1 2 3 4 5 6

Easy Peasy Lemon Squeezy. Next…

3. Figure out Spacing

Now that we have the Least Common Multiple (6) of our 2:3 polyrhythm, we need to figure out spacing. To do this, we divide the pulse and counterpulse into the Least Common Multiple:

• LCM/subdivision = Spacing.
• 6/2 (eighth notes) = 3. A pulse will happen every third beat. 
• 6/3 (triplets) = 2. A pulse will happen every other beat.

4. Write it on the Number Line

The next step is determining what numbers to circle on our number line. Using the spacing numbers from the previous step, we know that:

For the eighth notes, a beat happens on every third number. For the triplet, a beat happens on every second number. 

I'm going to use a TUBS (Time Unit Box System) diagram for this here, just to help you understand with a visual aid. You don't need to do this, and I'm going to go over this all again below without the TUBS system to make sure its 100% clear how to do it without. (I first learned about TUBS in the fantastic book A Rhythmic Vocabulary, a book which went a long way to improving my rhythmic abilities.)

While the TUBS system is a fantastic way of visualizing and understanding rhythms, I prefer a simple number line when practicing. That way, my eyes don't need to move up and down as much to see what I'm doing (a problem I also talk about in my Ultimate Guide to Note Reading article). Here you can see how that works:

And, of course, it's also much faster to write out a number line and circle and/or underline the appropriate numbers. 

Making an LCM numberline without a TUBS box

Alright, now here's the sumarized version, without TUBS. Let's use the same polyrhtythm from before. 3:2.

The Least Common Multiple is 6. So we'll write a numberline from 1-6:

1 2 3 4 5 6

Let's start with the triplets. 6 / 3 triplets = 2. So you'll start on 1 and circle every second number on the the line (I bolded them here). That will give you your triplet division of the beat. 

1 2 3 4 5 6

Next, let's do the eighth notes. 6/2 = 3. Again, you'll start on 1 and then circle (I underlined) every 3rd note .

1 2 3 4 5 6

1 2 3 4 5 6

1 2 3 4 5 6

1 2 3 4 5 6

Performing THE 3:2 Polyrhythm using A Numberline

The easiest way to learn any rhythm is away from the instrument. You'll do this by clapping your hands. It's best to start by clapping the composite rhythm (reminder: that's everything all together). To do this, count "1 2 3 4 5 6" and clap your hands on each circled (or, on this digital page, bolded and underlined) number.

1 2 3 4 5 6

Now it's time to level-up. For this,  your left hand will slap your left leg, while your right hand will slap your right leg. Again, you'll count "1 2 3 4 5 6", but this time, your left hand will slap on the bolded numbers, while your right hand will slap on the underlined numbers.

Once this is comfortable, reverse it; left hand on underlines, right hand on bolds. 

This step is especially important for polyphonic players (piano and classical guitar), drums, and any other instrument where separate parts of your body (hands, or fingers/thumb for classical guitar) are playing either the pulse and counterpulse.

Next, use a metronome to speed it up! Start with one click for every number. Slowly speed it up, just like you would do with any other problematic musical passage. With enough practice, you will get it fast enough that it all blends together into one beautiful composite rhythm. If this is enough to get you playing as fast as you need to, great! Stop here, or maybe just go down and try out some of the other common polyrhythms I've shown below. 

However, if you're still struggling to get things up to speed, jump down to the section on Neely Number Charts below. It's really just a different way of creating a number line, but it can help a lot with solidifying the rhythm. 

Other Common Polyrhythm Number lines

How to count a 3:4 or 4:3 Polyrhythm

The least common multiple of 3 and 4 is 12. The spacings are: 

• 3: 12/3=4
• 4: 12/4=3

The number line is: 

1 2 3 4 5 6 7 8 9 10 11 12

How to count a 4:5 or 5:4 Polyrhthm

The least common multiple of 4 and 5 is 20. The spacings are: 

• 4: 20/4=5
• 5: 20/5=4

The number line is: 

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Note: All the common examples on this page have a least common multiple that is simply our two polyrhythmic pulses multiplied together= x * y. This makes it nice and simple.

However, consider the polyrhythm of 6:4. 6*4 = 24, but the LCM is actually 12! You have to look out for this kind of problem. But even moreso, you have to look out for ways to simplify your practice. Why do I say that? I'd never practice a 6:4 polyrhythm, because it's precisely the same as a 3:2 rhythm played twice in a row. 

Another Pro Tip

Learn to feel how each part of the composite rhythm divides into the other. This will allow you to develop even more speed for two reasons. First, you’ll feel a bigger beat, allowing you to move faster. Second, you won’t need to count as high; simplifying the count can help you play faster. 

To do this, create a Neely Chart (I’ve named this after Adam Neely because I learned this method from his video on polyrhythms, which is linked below. Naming this will help us differentiate it from the LCM Number Line method, used above). 

Neely Number Chart Examples

Note: I use the Tabuteau Rhythm Counting Method for the reasons described in the above tip.

Let’s use 3:4 as an example. To make a Neely Number Chart, do the following

1. Take the counterpulse of your polyrhythm (4 in our example) and write a number line that long. 

4 1 2 3

2. Repeat step 1 for the number of pulses in your polyrhthm. (3 in our 3:4 example)

3. Underline the first number in each line. This represents the pulse. As you can see, this happens 3 times in this example of a 3:4 polyrhthm. 

4. Take the pulse in your polyrhythm (3). Starting with the first number in your number line, circle (here, bolded) every third number. As you can see, this happens four times in our pattern; this gives us the counterpulse of the polyrhythm (4 in the 3:4 example). 

To practice this, count so that the metronome lines up with the first number of each line. To help you differentiate between the two versions of the polyrhythm (3:4 and 4:3), accent the first number of each line (which lines up with the metronome). 

Need a visual? You can see Adam Neely himself do this here:

You’ll also want to practice the reverse rhythm to deepen your understanding of this rhythm. For our example of 3:4, this means practicing 4:3. Here is the Neey Chart for 4:3

Comparing Neely Charts with LCM Number Lines

To aid your understanding, I wrote both Neely Charts (3:4 and 4:3) in a single row so you can compare them to the LCM Number Line. As you can see, all the accented beats line up, no matter how you count them (however, whether a note is underlined or bolded will depend on whether you’re counting in 3:4 or 4:3, even though the two sound exactly the same in isolation).

Once you are comfortable counting using Neely Charts, you’re ready to use a polyrhythmic metronome (I prefer Tempo Advance for its ease of use) to help you speed up the rhythm. Set up your polyrhythm and start with a tempo slow enough that you can count along (note that the metronome will NOT play every single number; it will only play the numbers you clap on, that is, the underlined and circled/bolded numbers). 

Start ramping up the tempo once you’re comfortable at a slow speed. You’ll likely need to get up to a speed that’s too fast to count every single number; you’ll need to be able to “feel” the polyrhythm. While this sounds daunting, the polyrhythmic metronome can help you do this gradually.

1. Mastery: Feeling the Big Beat 
2. Polyrhythms are (usually) complex subdivisions; to play them accurately, you’ll want to be able to play them with a metronome only happening on the “big beat.” Whether you’re counting using an LCM Numberline or a Neely Chart, the metronome only happens on the first number in the chart or line. Here, I’ve shown this by making that number larger. 

Before you begin this practice stage, you should have a solid understanding of what the polyrhythmic feels like, even if it is very slow. At this point, mnemonics can help you “lock in” the rhythm at a faster speed. Here are mnemonics for common polyrhythms. 

• 3:2, 2:3: Hot Cuppa Tea
• 3:4, 4:3: Give Advice to Mother (or, more typically, Pass the Goddamn butter)
• 4:5, 5:4: I'm Looking for a Home To Buy

You can also use a polyrhythmic metronome (I prefer Tempo Advanced for its ease of use) to help you speed up the rhythm. However, remember that to truly “lock in” the rhythm, you must be able to perform it using a basic metronome. The polyrhythmic metronome is helpful only to help you speed up a polyrhythm you can already count and get it to a speed where you can begin to feel it.

Conclusion

This method of learning polyrhythms is near and dear to my heart, as it was a method I “discovered” on my own (in the days before you could find such niche information on the web or YouTube) when my college professor gave me absolutely useless advice when I needed to play a 5:4 rhythm ("Just Play 4 and fit 5 in between" ?!?).

I knew that was absolutely useless, so I finally figured out the LCM method by brute force. I shared my discovery with my professor; however, he was dismissive. Whatever, it worked for me. 

Years later, I got extreme validation. I read an article describing the same method in Soundboard Magazine, Volume 34, no. 2 (2008). The article was written by renowned new-music guitarist David Tanenbaum. Of course, today, this solution is easy to find on the web; it’s basically common knowledge! Hopefully, everyone will learn this method when it’s time to play their first polyrhythms.

Please let me know if this was helpful to you and what other rhythm questions you may have in the comments below!