Peculiar ~ Prime ~ Possibilities

I’ve started to become very interested in using prime numbers in the context of music. Obviously Mr. Eno did a lot of theorising about this already but I have had a lot of ‘duhhh’ moments this past month while putting prime numbers to the test in different areas of my productions.

Probably the least surprising is offsetting your notes by prime numbers to keep away those pesky repetitions but I realised that using prime numbers for your extremely short delays can also change the tonality a surprising amount. I went and did some searching around and found this very interesting post on the old MusicPlayer forums circa 2004 from the legendary EQ obsessive George Massenburg saying —

I use prime numbers because they’re indivisible by 2. Imagine if you set a delay of 8ms and hit it with a transient-kind-of event. you’d hear something like a 125Hz-based tone. So if you’ve got a bunch of delays and they’re set, say, 8, 4, 2, 1, you’re going to have a processing context that favors 125Hz and it’s overtones (250Hz & etc). I use this as a worst case, and most of us don’t use delays that short (although I can remember important times that I have ). But even at longer delays, I get a vague sense of a “tone” even without regen. By using prime numbers you 1> are more likely to “dove-tail” several different harmonic series, and 2> are more likely to have these vague “tones” land away from musically significant harmonics. Maybe I get a sense that it sounds more “neutral”.

Has anyone else messed around with using prime numbers in other aspects of production?


My friend Steve MacLean from Berklee released a record called Prime based on prime numbers.

I downloaded it from Apple Music the other day and am looking forward to listening to it soon…

1 Like

Immediately thought about this Steevio piece, specifically the breakdown video, where all the sequence components (maybe intervals, too?) are derived from prime numbers:


Yes! I like using melodic figures/loops of prime lengths to avoid repetition – or, to be more precise, it suffices for the loop lengths to be coprime (or at least to have a small GCD). This means that you don’t get a repeating pattern until you reach the LCM of your individual loop lengths. For example, three loops of length 5, 8, and 9 apiece, playing simultaneously, won’t hit a repeated pattern until beat 360. (You can take it further, Eno-style, by having non-integer lengths.)



I have used this polymetric coprime trick in the past, but it never works as well as I imagine, probably because I push it too far. 7 Vs 8 Vs 9 Vs 11 Vs 13 sound great on paper: LCM of 72072, or 5544 iterations of the longest pattern; at 120 BPM that’s no repeats for 5 hours. The problem I run into is that the result, though not technically repeating, ends up feeling repetitious. Even with much longer scale modulations it feels like it’s not moving enough; in a way I feel like it’s intellectually interesting but not necessarily musically enjoyable. What do you do to make sure your music of this type has intent and meaning, and is more than just an academic or intellectual exercise?


That’s a really good observation… I struggle a lot with this too. I’ve tried two things to get this sort of thing to be musically satisfying:

  1. Make the loop lengths relatively short (so that it doesn’t take too long for them to repeat together) and add a strongly rhythmic/repeating element to ground the whole thing, eg a 4/4 loop. With this, you give up having a long non-repeating pattern, but it’s easier to make it musically interesting.
  2. Make the loop lengths longer, make them interesting by themselves (ie, any individual loop in isolation is musically interesting too), and don’t add any other elements. This is easier said than done since this technique is essentially “be Brian Eno.” The gold standard is Music for Airports, where the individual loops are very compelling on their own as well as together. This is really hard! To be honest I haven’t had success pulling it off in a satisfying manner, but it’s a lot of fun to try. The one thing I’ve found that helps is to bring some loops in and out over time, which creates motion.

So this is going to veer slightly off the prime topic, but I would argue that these are not the aspects of “non-repetitious” rhythms that are musically interesting. Breaking down a polymetric relationship with data points like these focus your attention on the numbers and not how the numbers affect a listener. Listening to a large number of short polymeteric phrases like the example you listed will feel repetitious because the cycles are so small, and while the listener will be able to feel different sets of 2-3 cycles against each other, they are still listening to the same short phrases over and over again. In my experience the key for polymeters to be really interesting is, & here’s the dirty word of the topic: subdivisions. Blasphemy, I know, but prime numbers are absolutely still applicable.

Take, for example, an instrument playing in 11. This could be subdivided a million ways, but say you break it up into a 6->5 feel:
| x _ _ x _ _ | x_ x _ _ |

Now take an instrument playing in 17. There’s a myriad of ways this can be broken up as well, but if you include your 11 pattern from above somewhere in that stretch, you’ll get a shifting relationship of the same pattern phasing against itself. & rather than subdivide the remaining 6 beats in two equal groups of 3, arrange them in an uneven way that will allow more space between the phasing.

| x _ _ _ | x _ _ x _ _ | x _ x _ _ | _ x |

The rest on the first beat of that last group of two can also psychologically blur the fact that you’ve repeated the 11 pattern within the 17 pattern. Now add a single beat phrase of 5, but offset to not make the start point so explicit:

| _ _ x_ _ |

This might only take the three instruments 935 beats to come back full circle, but now you have an interplay based on a common phrase and a straight prime cycle to keep interest beyond the pure “non-repetition”. This interplay, in my opinion, is what’s truly interesting about polymeters within or outside the realm of prime number.


I arrived at an love for primes after my research into Partch in the early 1990s and my love for LaMonte Young’s installation, Dream House. The notion of the twin primes substantiated my cosmic ideas built upon the research Of Hans Kayser “Akroasis”, Barbara Hero and also filtered down into my love of hindustani classical music initially via texts by Alain Danielou. The Prime being a “new identity” [my own particular research 2001 included scales and harmonic lattices based on the 17/16, Shankar’s “SA” and my chosen Fundamental.

recently, another prime lover/mathematician on twitter shared this with me

1 Like