The Rhythm of The Primes #some2

[Music] have you ever stared at the prime numbers and thought i wonder what these would sound like as a giant insane musical polyrhythm well if so you're in luck because we're about to explore the rhythms of the primes one of the simplest ways of finding prime numbers is called the sieve of eratosthenes the basic ideas that you start out with a list of positive integers greater than one and then repeat the following process first circle the next available number in this case two then cross out all of its multiples in this case the even numbers then all we do is repeat those steps over and over again we circle the next available number which is three then we cross out the multiples of three next we circle five and cross out its multiples then 7 then 11 and so on and so on in each case the circled number is prime because if it had a divisor other than one or itself it would have been crossed out already it's a simple elegant algorithm that results in all the prime numbers being circled and all of the composite numbers being crossed out not only that but if you use a different color for each prime the colors with which any given composite number is crossed out reveals its prime factorization for example 15 is 3 times 5. 34 is 17 times 2 and 36 is made up of only twos and threes okay but what does any of this have to do with the giant insane polyrhythm well as a musician i look at these numbers and i see a series of beats within this series of beats the multiples of each prime form a series of regular accents for example here are the multiples of two and here are the multiples of three and the multiples of five if we layer these rhythms on top of each other and speed it up a little we get a pretty interesting polyrhythm by the way did you hear the gaps in this pattern well the first gap appeared on beat seven since it's a prime number and therefore not a multiple of two three or five in fact every prime number after five creates a gap in the two by three by five polyrhythm it's not until beat 49 that we finally get a gap on a composite number since 49 is 7 times 7 and we're only playing cycles of 2 3 and 5. so what happens if we start filling in these cracks taking our cue from eratosthenes and adding a new rhythmic cycle every time we reach a gap in the previous cycle well here's a polyrhythm of 2 3 5 and 7. sounds pretty exciting but let's try again with a cycle of 11 added in and maybe the cycle of 13. [Music] heck let's go wild and add in the cycles of 17 19 23 and 29 anyway you can probably see where this is going what if we just kept on adding new time rhythmic cycles going off towards infinity what would that sound like [Music] one of the problems with infinity is that you do rather tend to run out of percussion sounds so let's try a different approach and use a single instrument like the piano and have the pitch change with each new prime it's pretty typical in music for the lower parts to move slower and the higher parts to move faster so let's try mapping the small prime numbers to high pitches and have each new prime play at a lower and lower pace so far so good but as we go lower and lower we don't want to run out of notes and fall off the bottom of the piano keyboard we need a mapping that doesn't get too low too fast and i've got just the one an inverted harmonic series the basic idea of an inverted harmonic series is that unlike a harmonic series where you multiply a given fundamental frequency by bigger and bigger integers here you divide your starting frequency by bigger and bigger integers as you can hear the descending intervals get narrower and narrower which is exactly what we want so here are the pitches we'll use for the different prime numbers if they sound a bit unusual perhaps even out of tune that's because they fall in between the cracks of the piano keys just like the notes of the regular harmonic series do anyway i hope you like them so are you ready for the giant insane infinite polyrhythm well here we go one of the fun things about listening to the prime numbers like this is that you can hear some well-known properties of the primes in the result for example it's well known that the primes become less and less dense the higher up you go in our sonification each new prime enters as a new lowest note and you can hear the density of these notes slowly drop off over time another thing you can hear is the twin primes pairs of primes like 11 and 13 that fall on successive odd numbers just listen for two accented low notes entering one right after the other like these two or these two it's theorized that there are infinitely many twin primes but this is still unproven one other thing that the musicians among you may notice is that the rhythm sounds a little bit like it's in 3 4 time the reason for this is that aside from the number 3 all prime numbers have a remainder of either 1 or 5 when divided by 6. they can't have a remainder of 0 2 or 4 because that would mean they're even and they can't have a remainder of 3 because any number that's 3 more than a multiple of 6 is divisible by 3. since each prime number is a low accented note this means that we end up with an accent pattern that reinforces a simple triple meter like this [Music] of course not all the numbers falling on beats 1 or 3 will be prime but the point is that none of the primes can fall anywhere other than beat 1 or 3 of a perceived 3 4 time signature what other properties of prime numbers can you hear in the rhythm i hope it's clear at this point that though the rhythm itself is built into the structure of the primes the exact sound of the music depends a lot on how we represent each prime number sonically for example we could use a regular ascending harmonic series or we could use different scales [Music] or different tone colors it's also important to consider tempo and the non-uniformity of human rhythmic perception at different speeds for example we are most sensitive to regular pulses around 100 beats per minute try to clap along pretty easy right but what if i slow it down to 15 beats per minute try clapping along now it's a lot harder because our brains and bodies just aren't built to move at this speed the effect of this differing sensitivity is that it's fairly hard to perceive the cycles of the larger prime numbers since they move so slowly but what if we sped things up a little or maybe a lot [Music] can you hear the cycles of the larger primes come into focus now anyway there are plenty of other ways to map prime rhythmic cycles to music each mapping not only offers a different aesthetic quality but also highlights and makes audible different properties of the prime numbers what other ways can you think of to turn the prime numbers into music and what properties of the prime numbers will your mapping reveal the possibilities like the prime numbers themselves are truly endless [Music] you

Channel: Marc Evanstein / music․py

Views: 833,694

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Id: 8x374slJGuo

Length: 10min 11sec (611 seconds)

Published: Mon Aug 15 2022

Reddit Comments

[u/setecordas]

From the music.py YouTube channel, a video on polyrhythms with primes and microtonal mappings. It's a great musical idea generator, and a good follow on to 3B1B's Music and Measure Theory, one of my favorite videos on microtonality.

[u/DopplerDrone]

Awesome!

[u/dcollett]

Thanks! This was excellent.