This article was originally posted 2015/02/25 on my personal website and has been culled from the ever-awesome wayback machine and a recent cache of images found on a recovered harddrive. Enjoy!
Ever get stuck and find yourself tapping away at a beat grid to find something new and exciting? Just looking for something to get you musically unstuck? Today I’m going to give you a rundown of creating new beats using the Euclidean Algorithm. Don’t stop reading, the math behind this post is going to be kept to a minimum. In fact, we’ll forget it almost as fast as it’s explained. First, I’ll cover the basics of the Euclidean Algorithm. Next I’ll get right to showing you a quick way to create a beat using the process defined in the algorithm (with no math!). From there we’ll branch of into various ways to view and manipulate the results for alternative beats using what is called polygon notation. Finally, I’ll show you some pretty interesting things that can happen when you take these methods and combine them across kit pieces in your favorite DAW/Drum editor.
Euclid and the Euclidean Algorithm
Euclid was a Greek mathematician who lived between the 4th and 3rd century BC and is generally considered the primary creator of geometry. Among his many contributions is the Euclidean Algorithm which is used to calculate the greatest common divisor of two numbers. Essentially you take two numbers and replace the larger of the two by their difference until both are equal. The final result is the greatest common divisor. For instance given 6 and 16:
- 16-6 = 10
- 10-6 = 4
- 6-4 = 2
- 4-2 = 2
The greatest common divisor of 16 and 7 is 2. Now what does this math have to do with creating a beat? Not much. It’s the steps in this process I just showed that we are interested in more than the mathematics.
Your First Beat
Let’s take a simpler approach to the problem above using the numbers 3 and 8. Using the above algorithm, the breakdown is as follows:
- 8-3 = 5
- 5-3 = 2
The easiest way to approach this in a visual/musical way is to grab 3 dimes and 5 pennies to total 8 coins and lay them out as shown below.
Now this represents that we want to have 3 hits (the dimes) in an 8th note pattern (the complete measure). We always start with the number of beats followed by the difference to fill the measure. Implementing the same steps in the algorithm, we drag the coins down to subtract.
Eight minus three leaves five in the top row
That five minus the three below equals two
There are no more coins to subtract. We pull apart the resulting columns.
Now rotate each column counter-clockwise into a single row.
How about now?
You’ve just created your first Euclidean Rhythm. Play with the above process using various numbers of dimes and study the outcome.
If you read a previous post about my random music creation system you will recall the discussion about dice used to create beats. In that post a die containing 2s and 3s (the foundations of musical rhythm) was rolled X times until a total measure was filled. This is a similar method however it favors an even distribution of beats over the measure finalizing with a remainder. Generally your first beats are usually going to be evenly spaced and the last beat(s) shorter at the tail end of the measure unless the total beats are evenly divided by the desired beat count (i.e. 16/4=4). The method of even distribution provided by the Euclidean algorithm lends itself well to a balanced beat however using the algorithm always produces the same result when you provide the algorithm the same numbers. Let’s now look at this beat in a different way to see how we can switch up things a little.
The image below is the same drumbeat shown in polygon notation along with our coin-based representation.
In polygon notation, the number of beats are evenly distributed around a circle while the rhythmic pulses are points plotted on the circle. The points can be connected to form a shape. Polygon notation is a rather easy method for visualizing drum beats and we’ll use it along with the coin example to show how to alter the beat through rotation. Sight reading polygon notation is simply counting the beats with a pat of the hand for each red dot.
To perform the rotation, imagine that the circle is similar to a clock face and the triangle rotates at the center much like an hour hand. We want to rotate the triangle clockwise so the beat on point 7 moves forward to point 1. This rotation in turn pushes beat 1 to point 3 and beat 2 to point 6. It is the equivalent of moving the last two coins to the front of our coin chain.
Now we have a totally new pulse/beat for the measure. It carries the same balanced rhythm, we’ve just offset the pulses through rotation. As we have 3 hits in the 8 beat measure, we can perform the rotation once more for a final variation. So your number of hits represents the number of rotations, or unique beats, that are possible. This is not the case for fully balanced rhythms like a solid quarter note hit for each beat in a 4/4 measure because each rotation simply returns you to the same rhythm.
Mixing It Up
So now that we have the fundamentals down for creating a basic beat, let’s explore some other concepts by repeating the algorithm, combining rotations across kit pieces, and also combining different algorithm results into the same polygon notation structure. As mentioned earlier, the Euclidean Algorithm is well suited (along with rotations) when you want a balanced beat. If you want to mix things up a little, there are two options at your disposal.
Scrambling the Source
The first option is to start by scrambling the initial “coins” in various ways. For instance shifting coins 2 and three
Results in the following beat with two strong quarter notes, a rest then a final quarter.
You can also get a little more uncomfortable by removing the first beat like this:
Which results in a starting rest but otherwise strongly 4/4 aligned beat with an 1/8th accent
Re-Folding The Results
The other method at your disposal is simply performing the same Euclidean Algorithm on the final result, a process I call re-folding. So you’ll take the initial seed of 3 and 5:
Perform the algorithm to create the beat:
Then perform the algorithm once again on the result which provides us with this beat:
Combining Results for Drum Beats
Finally, you can use all of these processes in various ways to create full drum beats. In the following examples the red lines are the kick, the blue is the snare, and the green is the high hat. These are all Euclidean derived beats mixed together to create interactions across the pieces of the kit. This example is the same rhythm with one rotation per instrument.
This example combines a E(3,8) (shorthand notation for a Euclidean 3, 8 rhythm), E(5,8) and E(4,8) into one rhythm:
You can also, of course, expand into E(7,16) and other frameworks. See? Told you there was no math. Now I realize these final examples aren’t very musical but once you start to experiment with the rotations you’ll start to uncover beats that aren’t as “lock step” and have a unique sense of fluidity.
I hope this has given you some great ideas on creating new beats to experiment with in your own work.