Awesome video! Can you help with a similar music project? Planets like Saturn have rings of asteroids. They remind me of a vinyl record or CD. If we had a high-res image of Saturn, could it set to music?
Quite a few years ago I made an array of primes and then using bitwise math triggered midi notes in the same key as bits 3 through 12 or so turned on in the integers as the array was cycled through, the bits acting like the pegs in a music box. It played an interesting song. I also painted a picture with the bits of primes in a strip. Low bits at the top. If the bit were on I drew a small line on the screen down to about 24 bits or so. The ribbon of primes was ghastly looking, bizarre, never repeating. With the same algorithm if you painted all integers the strip looks like a pristine orderly mountain range, but the prime version was ugly looking. I looked to see if I still had a copy of it, but no, I'd have to do it over again.
Every time the commentery asked "can you hear..." about some aspect of the sound I was thinking "no, because I've only had a couple of seconds and you keep talikg over it". Unlike the video editor we've not heard the isolated sound pre-edit to be able to be reminded of the sound of each variant from a clip of a couple of seconds... It's worth the editor bearing in mind that viewers will be hearing these interesting sounds for the first time. Hopefully some more extended clips in the "making of..." video mentioned in the description.
The primes are inherently more interesting. Pi is just a bog standard irrational number, there's nothing special about it and the only reason it gets the recognition it does is because people with no understanding of math think its irrationality, which it shares with almost all other numbers, is some unique and magical property.
You do only have 19.980 possible distinct usable sounds (although anything above 19.000Hz is nigh unhearable), that's if you're not counting cent differences which are hard to spot without a good ear.
You are only counting frequency (and just the fundamental). You can make a note, for example 440hz, with so many different timbre, adding more tan 19.980 possible sounds.
@@boncoderz1430 I started studying for a Bachelor in software engineering this year. Music and maths are just hobbies of mine. I play piano, transcribe and produce music in my freetime and studied 2 semesters of pure maths but I quit the latter.
Fun Fact: N's Theme from Pokémon Black and White is composed entirely of Prime numbers. Junuchi Masuda, the composer, thought it would be fitting because N is incredibly smart
There’s a lot of interesting musical references in Black and White. N’s full name is Natural Harmonia Gropius, and his father is Ghetsis Harmonia Gropius, pronounced as “G-cis” in the original Japanese translation. N represents a natural chord, while his father represents the chord of G and C#, which is the tritone, and “the devil in music”.
8:56 - 9:13 This is unnecessarily funny, lol. The calming voice behind the rapidly increasing speed of percussions and afterwards cutting instantly is what gets me.
As someone who is a musician, enjoys coding, and holds a math degree, this was a very enjoyable video and has opened my mind to adding various aspects of primes and other mathematical concepts to my music. Thank you for creating and sharing this! I am in your debt.
This is incredibly cool. When you mentioned trying a faster tempo, it made me think what if we increased the tempo steadily as the song is progresses. Primes on the small end would one by one turn into continuous rising tones that would eventually fade away as they passed out of the audible range, leaving space to hear the larger primes. And we could set an "equalizer" to make them fade soon if we wanted. We could also have the prime-pitch coding change in a steady manner, so that increasingly large primes would pass in and out of the audible pitch range and the natural tempo range at the same time. Using this moving window of pitch and tempo might prevent the song as a whole from blowing up or fading out.
Yes i would mike to hear that, we could state it as "the notes that makes one [row] when the row [x] times lower (in the graph) makes some constant rythm" the "x" have to be bog enough so that when larges tempo are heard as such, the row x time uphead is heard as a tone
The Sieve of Eratosthenes algorithm stops at 10 (square root of 100) in your example. The remaining numbers are all prime. Think about it: Any factor above 10 would have been found by a factor below 10 (for numbers up to 100). This is why the sieve is so powerful. If we were to find all the primes to 10,000 we only have to do the sieve to 100. Primes to 1,000,000, only do the sieve to 1,000. Etc.
That's absolutely true --- probably should have mentioned it! Of course, if the idea is more of an infinite sieve, rather than one that stops at a certain point, then that's a different story, and that's kind of what I was ultimately going for
@@pauselab5569 @pause lab yes, but not as much. Imaging having to seive a million times, when you could also do it for 1,000 only. That is litteraly over-powered.
Believe it or not but this optimization barely matters. To understand this, think about sieving the multiples of 2. You need to cross off half the numbers in your list. Now think about sieving the multiples of 101. This only takes 1/50th the work. This is where the real power lies in the sieve of eratosthenes. We find that sieving all multiples of the numbers 2,3,…,k in the numbers up to N takes (1/2+1/3+…+1/k)N=Nlogk time. All your optimization does is use k=N^1/2 rather than k=N. That only yields an optimization of 1/2. But it gets even better. We only sieve multiples of primes. This yields Nloglogk time instead. So as k gets larger, the optimization factor tends to 1 and becomes unnoticeable, let alone significant. There are even optimizations to the sieve of eratosthenes to achieve linear running time btw.
I'd like to hear the undertones matched with the primes themselves and not the n-th prime. Maybe it wouldn't add much, but it might reveal some structure tonally as well as rhythmically. There are some impressive implications in the math of the harmonic series if you can build intuition around it. You could hypothetically use the log scale structure of the harmonic series to teach times tables with ear training. The kids would likely make weird errors around octaves and powers of two...
Yeah, that's an interesting point. Often in just intonation we talk about tunings having different prime limits (e.g. 5-limit, 7-limit), meaning we only allow frequency ratios that break down into prime factors below that limit. I guess in this video, I was more focused on the rhythmic aspect than the tuning, and by going with the nth prime I avoided getting too low to fast. I like your idea about times tables and ear training!
@@Anonymous-df8it wow, that would get incredibly high tempo, incredibly quickly. It would be a fun experiment! Considering the distance between primes grows logarithmically it would accelerate almost exponentially to make that distance perceived as linear. Now I'm curious how long you could play it before you exceed your bitrate!
@@lexinwonderland5741 Well, the more frequent polyrhythms (2,3,5 etc.) will eventually exceed the barrier of rhythm to pitch (20 Hz), so you could replace those with a sine wave. Similarly, when they become inaudible (20 kHz), you can stop playing them.
@@Anonymous-df8it I like how you think, friend. I made a version up to the 15th prime with overtones (4 octaves) and now you've got me wanting to play around even more haha. The bitrate question is still there with the increasing speed, but regardless this sounds like a fun weekend project!
Thanks so much! I just looked you up, and your channel is wonderful. (Watched your video on knots, which I've always been curious to know more about.) If you ever want to collaborate on a mathematical sonification of some sort, I'd definitely be interested!
5:31 Numbers 1-37 looped would sound beautiful. It's interesting to see the hear the microtonality as it leads your ear to the tonic, which is very common in middle eastern music. For example, listen to how to 31 is played how it leads your ear to 37. So pretty! 3, 13, 37, 87 numbers are on the tonic (stable root note)
the note frequencies vs primes rhythms are offset by one. so 31 -> 37 is actually 30 -> 36, which is 6->5 once you take out the common factors of 2 and 3, a minor third descent.
I think it is so nice when someone combines math with art ... It makes the whole concept much more understandable. What a great, inspiring playing around that is! Thank you so much. A true inspiration.
It would be interesting to hear it where it speeds up logarithmically to keep the introduction of new primes fairly consistent while also fading out the volume of old primes over time to clear out the noise. Would the sound of it stay loosely consistent while also morphing in an organic feeling way? You could reuse the same sounds after a certain point since they would fade to nothing, so it should be able to be done forever.
Sometimes I make generative modular synth music... so its all like sequenced and logic/math based. I love playing with primes. I'm making a modular EP now ^~^ Edit: You've given me insight by pointing out that prime polyrhythms have gaps at unassociated prime numbers. And repeat and the square of the first unused prime. ^~^ THANK YOU BUDDY
@@droughdough Polymeter: Tracks that play in different meters, de-synchronizing themselves from each other (e.g a 5/4 time and a 4/4 time playing together) Polyrhythm: Subdivisions that fit within the same bar and whose accents always start on the downbeat of a bar (e.g triplets playing against eighth notes both in 4/4 time)
@@harry_dum7721 to be fair, there's a very natural correspondence which is to take the least common denominator, call that one bar, and only play beat 1 of each part. That then gives a polyrhythm corresponding to the meter. For example 2+3+5 corresponds to 6:10:15. In general the product of the polyrhythm and the meter giving it is that denominator.
@@droughdough If you were to write out this music, it would all be 8th notes, they are just at different pitches. With polyrhythms that's not the case, as each different instrument/voice would have to be written as a different tuplet (like a triplet, or quintuplet). Here's the easiest way I can put it: Polymeter means there are multiple meters (time signatures) happening simultaneously, but we keep the tempo and the note divisions constant. This means the different instruments do NOT start together at the beginning of each measure, and instead it might take a few bars for them to get back together. As an example, imagine a piano playing 3 8th note pattern played on top of a guitar playing a 4 8th note pattern. The patterns are different lengths but the same speed. Polyrhythm means it's all the same time signature, but the different instruments are playing different speeds (or tuplets). In other words, within the space of one measure, one instrument might play 4 notes while the other instrument plays 5 notes, but they always start at back together at the beginning of each measure. The patterns are all the same length but different speeds but the same length
Honestly, the first 2 seconds of the polyrhythm sound cool, like a forest in the rain, but the type of rain where there's still sun beams entering through the cracks in the clouds
I watch both alot of math and alot of music theory videos and I absolutely love when TH-cam recognizes the intersection between the two and recommends videos like this one. Thank you for creating this awesome video.
I'm super curious just how terrible (or not?) it would sound to invert the entire rhythmic scheme and interpret 2 as 1/2 note, 3 as 1/3 note , 5 as 1/5th note etc, or 2BMP, 3BMP, 5BPM, same thing, just different abuses of notation. There's the downside that you have to decide how deep to take your recursion ahead of time and then rescale your playback speed to make sense for how deep you went, rather than just adding in parts until it's a mess and stopping, but has the potentially interesting advantage that you can go arbitrarily deep and still have all the cycles line up in a finite amount of time. Also, it would actually be a polyrhythm instead of a polymeter ;-) (for the record of course polymeter>>polyrhythm, I'm just pedantically teasing)
If you play this rhythm fast enough, it becomes a sound. An overtone series with just the primes resembles a lot a clarinet - except there is the 2nd partiaö, and the 1st missing (as well as all non-primes).
@@torydavis10 But you have to represent something infinitessimally close to the start while not having to run the video until the heat death of the universe self-corrects because the really big primes don't have to be represented before the heat death of the universe and would be represented at its own pace. While for this the really big primes would have to be represented incredibly quickly. With that said, it would be really interesting to try this with the first n primes (with n being a finite number)
@@marcevanstein I intend to control only the pitch. A vertical helical blade (probably) on top will capture the wind and spin a vertically stacked set of cams spaced according to the first handful (out of the presumed infinite number) of primes. The cams will then actuate hammers that strike tubular bells surrounding the structure.
Many years ago I had much the same idea, and made a song out of the first 17 primes running through their rhythm for a few minutes. I then had a musician friend choose samples for the soundscape. The end result was less about the rhythm of the primes, and more an evolving soundscape with a difficult-to-define rhythmic pace, but it was a fun project!
This is really cool! Considering there's sort of an upper and lower bound at which rhythm breaks down as far as human perception is concerned (I think the lower bound is around 33 bpm), you can choose to simply work with a finite number of primes and create a lot of different arrangements.
Thank you for explaining the threeness I hear, when I've caught it in my own experiments I thought it was simple bias. Funny thing, when the primes were speeded up with percussion in one of the last samples I perceived an almost horizontal and equal three-based pattern which simply kept emerging in a new timbral space, which seems independent of remainder of three rule. I also imagine I hear an accelerating clave, sometimes swapping polarity between 5:3 and 3:5 which I guess is a consequence of remainder distribution too. Excellent video!
People who are good at math are usually good at music. By using our sense of hearing, we are learning about prime numbers. When I was learning about prime numbers in school, I thought they seemed like awkward lonely numbers. I have since discovered that they are more valuable than I thought. This reminds me of people who seem dull and useless on the surface with hidden genius and talent underneath.
For a few years now I've been planning to make an album of electronic music using ideas from maths (I'm a maths grad working in software, but music is one of my passions). I've recently finished putting together my studio in a new home, and I'm beginning to work on the ideas, so this is truly inspiring for me. Thank you!
At the risk of sounding like my far distant teenage self - this is so cool. It combines several of my favorite things: math, music, design, and color. I may become addicted to watching it over and over. Thanks for making my day!
Woa! Reminds of something Sevish would compose. Great video! c: Side note: There's a cool lecture by Adam Neely where he showed that speeding up polyrhythms until the beats are like frequencies would result in intervals! so a 3,4,5 polyrhythm would make something like a major triad when sped up incredibly fast
I was looking through the comments for someone else who thought of this. I want to hear this sequence represented as a tone. Play the "2" rhythm at some audible frequency (>20Hz). I presume the resulting sound would start as a recognizable pitch but dissolve into noise fairly quickly.
I experimented for a bit with using the squares of primes as tempo markings and coming up with metric/tempo modulations to attempt to create seamless transitions. One thing I wish I could do is a synchronization experiment with a prime number of mechanical metronomes set to prime BPMs. I'd like to see what BPM they synchronize to.
Have you considered making a mapping that is calculated modulo some frequency? so that if a prime's frequency would be below a certain threshold, the value "wraps around" back to the difference between the threshold and the frequency? This would allow you to play your music indefinitely without going so low that the value is inaudible.
I actually really like this visual representation of successive primes, because it demonstrates a trend, if not a pattern. If there were a pattern, we'd be able to codify it into an equation to find the next prime. But our insane pattern recognition abilities see this and automatically say "dude see??? There's totally a pattern!!"
Wow, as far as hear and see, I wonder what it would sound like when playing the full prime-(factor) spectrum, i.e. at beat 102 the triad (2,3,17) in equal volume distribution 1:1:1. Would it be harmonic or catastrophic for the ear? At wich speed?
Returning here 2 years after I first saw the video, I realised just now how much I have learnt. I am so grateful for your channel, everytNice tutorialng
Wow, I just stumbled upon this video, and that's amazing! I always loved bridges between music and maths! Would it be possible to have a long version of this song on your channel?
Good idea --- I'll try to put one together! I was thinking of maybe also including a downloadable link to a midi file, in case people wanted to play with it
I love this thank you so much for uploading. Absolutely beautiful. I’m not good with written numbers but I love music. Content like this helps bridge my understanding and is so valuable to me.
Ok but what if we used other sequences of numbers and a different mapping algorithm? What about Fibonacci/Lucas numbers? triangulars, factorials, Van Eck? Good video though! I'm not complaining, I was just being curious
Fibonacci's tend to the golden ratio, which is the *most* irrational number there is (phi = 1 + 1/(1 + 1/(1 + 1/(1 +...... and that would be maximally dissonant.
What program did you use to make this? Because I'm doing a whole lot of CTL+C CTL+V in Cakewalk to make my variations. My theory has been that each prime number (P) follows the same rule. Starting at a high note, every time (P) reaches a new factor (^n), it plays the highest note. The tones go down the scale as (P^n-x) where n-x is greater than 0. This results in lower primes kind of crawling off of the song the longer it goes. If we follow this approach using the A minor pentatonic scale; P^n-0 = A7 P^n-1 = G7 P^n-2 = E7 P^n-3 = D7 P^n-4 = C7 P^n-5 = A6 ... So the first time you hear the note A6 is at number 66, because 64=2^(6) and 66=2^(6-5)*11^(1-0). I ran this with the numbers 2,3,5,7, & 11 and got this: drive.google.com/file/d/1-YIUfIMMKZDjd3G_cjUr_0c9ng_OkfgB/view?usp=drivesdk The half way mark is wicked satisfying 😩
@@christopherrice891 I might not understand your question, but I'm afraid I'm the wrong guy to ask. I only have a passing knowledge of number theory, no working knowledge.
its misleading to call this a polyrhythm, especially when you are letting it go to infinity this is more accurately a polymeter: different sized cycles on the same size subdivision. A polyrhythm is different sized subdivisions with one shared cycle length. Any finite polymeter is also a polyrhythm, but the cycle length of that polyrhythm is the least common multiple of all the parts; in the 2:3:5:7 example towards the beginning of this video for instance, you didn't even get close to playing a full cycle of the polyrhythm, as that'd be 210 subdivisions. So, what appears to be the 2 of the infinite polyrhythm is actually the 2 of the polymeter; the rhythm that has 2 evenly spaced hits over the full cycle length is... well, infinite. you're not technically wrong to call it an "infinite polyrhythm" in the sense that the cycle length is infinite, but I am sure this has made some people think that polyrhythm is polymeter I find polymeter often more musically useful, and I don't want to detract from the cool math here, but polyrhythm just isnt the right musical term
I really like all the different scales and tempi you used for the prime numbers. I couldn't help thinking this could easily be a theoretical example to demonstrate synaethesia. Although there are some more prevalent forms of synaesthesia, practically any combination of sensory substitution is possible. I like to imagine hearing colours or numbers, tasting or smelling colors, feeling the texture of temperature etc. Most people goosebump or shiver in response to a sudden or prolonged decrease in temperature. These responses can also be elicted by shock, fear, horror, awe and other emotions which may also be evoked by touch, sound and music etc. Whilst it is still a response to temperature change, I occasionally goosebump and shiver when going from a relatively warm to very hot environment. Usually this occurs when going from outdoors, on a day warm enough in direct sunlight for me to enjoy being in the shade or maybe find a cool breeze, then getting into a car with no air conditioning that has been in full sun for a while. I also fall into the minority of people with a photic sneeze reflex; we really aren't all wired the same.
I’ve used prime number polymeters quite frequently recently but never took a step back to think about some inherent properties of them. Thanks for the video!
I am neither a mathmetician nor a musician but this video was still incredibly interesting to me as an artist. This reminds me of the idea that everything in the end is just math.
For those of you wondering how this was made, check out the making-of video: th-cam.com/video/GzrTmukxUxA/w-d-xo.html
you didnt make an inverted harmonic series, you made the normal harmonic series with a scalar lol
Awesome video!
Can you help with a similar music project?
Planets like Saturn have rings of asteroids. They remind me of a vinyl record or CD. If we had a high-res image of Saturn, could it set to music?
For those of you!
Quite a few years ago I made an array of primes and then using bitwise math triggered midi notes in the same key as bits 3 through 12 or so turned on in the integers as the array was cycled through, the bits acting like the pegs in a music box. It played an interesting song. I also painted a picture with the bits of primes in a strip. Low bits at the top. If the bit were on I drew a small line on the screen down to about 24 bits or so. The ribbon of primes was ghastly looking, bizarre, never repeating. With the same algorithm if you painted all integers the strip looks like a pristine orderly mountain range, but the prime version was ugly looking. I looked to see if I still had a copy of it, but no, I'd have to do it over again.
If you think about it, all polyrhythms are just this polyrhythm with channels muted
@@joe_z Cool so say we do that.
See also: en.wikipedia.org/wiki/Euclidean_rhythm
the Schillinger system is worth a look … his theory of rhythm is quite relevant here …
if you think about it a bed is the same as a bathtub only without a bed and with a bathtub and in the bathroom
@@shum8104 It's literally a waterbed.
i'd love some no-commentary videos of this just playing for like 10 minutes, or maybe 1 hour, with different mappings
yeah, that would be great, just like the sorting algorithm sound videos made by someone else
Same
th-cam.com/video/EsO9COuLPIE/w-d-xo.html
Every time the commentery asked "can you hear..." about some aspect of the sound I was thinking "no, because I've only had a couple of seconds and you keep talikg over it". Unlike the video editor we've not heard the isolated sound pre-edit to be able to be reminded of the sound of each variant from a clip of a couple of seconds... It's worth the editor bearing in mind that viewers will be hearing these interesting sounds for the first time. Hopefully some more extended clips in the "making of..." video mentioned in the description.
if anyone's still looking for a no-commentary video of this, it's here:
th-cam.com/video/M48319x1Kg4/w-d-xo.html
This is waaayyy more interesting musically than all the (way too many) uninteresting pi mappings when pi day was at the top of its fad
Thanks so much! Yeah... I have a similar reaction to those, since statistically the digits of pi aren't too different from a random number generator.
The only good one I've seen was one where the tempo is π/4.
The primes are inherently more interesting. Pi is just a bog standard irrational number, there's nothing special about it and the only reason it gets the recognition it does is because people with no understanding of math think its irrationality, which it shares with almost all other numbers, is some unique and magical property.
@@Oneiroclast I do agree with you, but pi is still important nonetheless, it's just that a lot of people think it's important for the wrong reasons.
@@Oneiroclast If you were as good at math as you’re implying then you would know that Pi is indeed a special number…
"One of the problems with infinity is, that you do rather tend to run out of percussion sounds" 🤣
Impossible
Das true Mon because I'm a drummer and I run out sometimes...so frustrating!🫤🪘
Sounds like something Douglas Adams might've written had he been a music critic.
You do only have 19.980 possible distinct usable sounds (although anything above 19.000Hz is nigh unhearable), that's if you're not counting cent differences which are hard to spot without a good ear.
You are only counting frequency (and just the fundamental). You can make a note, for example 440hz, with so many different timbre, adding more tan 19.980 possible sounds.
As a musician and math enthusiast this video was very interesting and entertaining.
Might as well be my favorite so far.
@@boncoderz1430 I started studying for a Bachelor in software engineering this year.
Music and maths are just hobbies of mine. I play piano, transcribe and produce music in my freetime and studied 2 semesters of pure maths but I quit the latter.
Fun Fact: N's Theme from Pokémon Black and White is composed entirely of Prime numbers. Junuchi Masuda, the composer, thought it would be fitting because N is incredibly smart
There’s a lot of interesting musical references in Black and White. N’s full name is Natural Harmonia Gropius, and his father is Ghetsis Harmonia Gropius, pronounced as “G-cis” in the original Japanese translation. N represents a natural chord, while his father represents the chord of G and C#, which is the tritone, and “the devil in music”.
can you make a sort of 10 hour thing of just this, this is so cool and kind of calming to listen to tbh
seconded, this is so nice to listen to. i'm not sure if i like a scale or the (inverted) harmonic series better but i want MORE
Count me in.
Im gonna try to make it, wish me luck
Try some G. F. Haas or Xenakis' Rebonds
@@phoenizboiisawesome Any luck?
8:56 - 9:13 This is unnecessarily funny, lol. The calming voice behind the rapidly increasing speed of percussions and afterwards cutting instantly is what gets me.
"can you hear the cycles of the larger primes come into focus now?"
*sounds of a drum set crashing down the stairs*
@@faland0069 😂
As someone who is a musician, enjoys coding, and holds a math degree, this was a very enjoyable video and has opened my mind to adding various aspects of primes and other mathematical concepts to my music. Thank you for creating and sharing this! I am in your debt.
4:19 Chopin Prelude Op. 28, No. 10 in c# minor
True! Thanks for labeling it in case anyone was wondering
@@marcevanstein I love that you used that prelude! It’s a little less known, but it’s one of my favorites
I figured it was Chopin! Just couldn't figure out what piece. Thanks!
takes hat off to sir
Thanks you!!!!
This is incredibly cool. When you mentioned trying a faster tempo, it made me think what if we increased the tempo steadily as the song is progresses. Primes on the small end would one by one turn into continuous rising tones that would eventually fade away as they passed out of the audible range, leaving space to hear the larger primes. And we could set an "equalizer" to make them fade soon if we wanted. We could also have the prime-pitch coding change in a steady manner, so that increasingly large primes would pass in and out of the audible pitch range and the natural tempo range at the same time. Using this moving window of pitch and tempo might prevent the song as a whole from blowing up or fading out.
hell ya
Someone made that under a different comment, but ill post the link here as well m.th-cam.com/video/sdhpyBGP1xI/w-d-xo.html
That would be extremely hard to run for longer than a minute probably
Yes i would mike to hear that, we could state it as "the notes that makes one [row] when the row [x] times lower (in the graph) makes some constant rythm" the "x" have to be bog enough so that when larges tempo are heard as such, the row x time uphead is heard as a tone
This idea is ass.
I'm screenshotting it, and I'll be back once I've done what you said
9:05 "Can you hear the cycles of the larger primes now?" [Drummer falls downstairs]
The Sieve of Eratosthenes algorithm stops at 10 (square root of 100) in your example. The remaining numbers are all prime. Think about it: Any factor above 10 would have been found by a factor below 10 (for numbers up to 100). This is why the sieve is so powerful. If we were to find all the primes to 10,000 we only have to do the sieve to 100. Primes to 1,000,000, only do the sieve to 1,000. Etc.
That's absolutely true --- probably should have mentioned it! Of course, if the idea is more of an infinite sieve, rather than one that stops at a certain point, then that's a different story, and that's kind of what I was ultimately going for
Cool
Yes but it’s still way too much…
@@pauselab5569 @pause lab yes, but not as much. Imaging having to seive a million times, when you could also do it for 1,000 only.
That is litteraly over-powered.
Believe it or not but this optimization barely matters. To understand this, think about sieving the multiples of 2. You need to cross off half the numbers in your list. Now think about sieving the multiples of 101. This only takes 1/50th the work. This is where the real power lies in the sieve of eratosthenes.
We find that sieving all multiples of the numbers 2,3,…,k in the numbers up to N takes (1/2+1/3+…+1/k)N=Nlogk time. All your optimization does is use k=N^1/2 rather than k=N. That only yields an optimization of 1/2.
But it gets even better. We only sieve multiples of primes. This yields Nloglogk time instead. So as k gets larger, the optimization factor tends to 1 and becomes unnoticeable, let alone significant.
There are even optimizations to the sieve of eratosthenes to achieve linear running time btw.
I'd like to hear the undertones matched with the primes themselves and not the n-th prime. Maybe it wouldn't add much, but it might reveal some structure tonally as well as rhythmically. There are some impressive implications in the math of the harmonic series if you can build intuition around it. You could hypothetically use the log scale structure of the harmonic series to teach times tables with ear training. The kids would likely make weird errors around octaves and powers of two...
Yeah, that's an interesting point. Often in just intonation we talk about tunings having different prime limits (e.g. 5-limit, 7-limit), meaning we only allow frequency ratios that break down into prime factors below that limit. I guess in this video, I was more focused on the rhythmic aspect than the tuning, and by going with the nth prime I avoided getting too low to fast.
I like your idea about times tables and ear training!
@@marcevanstein What about varying the tempo between each prime to make all of the primes sound equidistant
@@Anonymous-df8it wow, that would get incredibly high tempo, incredibly quickly. It would be a fun experiment! Considering the distance between primes grows logarithmically it would accelerate almost exponentially to make that distance perceived as linear. Now I'm curious how long you could play it before you exceed your bitrate!
@@lexinwonderland5741 Well, the more frequent polyrhythms (2,3,5 etc.) will eventually exceed the barrier of rhythm to pitch (20 Hz), so you could replace those with a sine wave. Similarly, when they become inaudible (20 kHz), you can stop playing them.
@@Anonymous-df8it I like how you think, friend. I made a version up to the 15th prime with overtones (4 octaves) and now you've got me wanting to play around even more haha. The bitrate question is still there with the increasing speed, but regardless this sounds like a fun weekend project!
This video was a delight, thank you!
Thanks so much! I just looked you up, and your channel is wonderful. (Watched your video on knots, which I've always been curious to know more about.) If you ever want to collaborate on a mathematical sonification of some sort, I'd definitely be interested!
as a percussionist, programmer, and math student, this is ❤
:-) Do you have any videos of yourself playing percussion?
8:57 sounds like a percussion band falling down a staircase
5:31 Numbers 1-37 looped would sound beautiful. It's interesting to see the hear the microtonality as it leads your ear to the tonic, which is very common in middle eastern music.
For example, listen to how to 31 is played how it leads your ear to 37. So pretty!
3, 13, 37, 87 numbers are on the tonic (stable root note)
the note frequencies vs primes rhythms are offset by one. so 31 -> 37 is actually 30 -> 36, which is 6->5 once you take out the common factors of 2 and 3, a minor third descent.
I think it is so nice when someone combines math with art ...
It makes the whole concept much more understandable.
What a great, inspiring playing around that is!
Thank you so much. A true inspiration.
It would be interesting to hear it where it speeds up logarithmically to keep the introduction of new primes fairly consistent while also fading out the volume of old primes over time to clear out the noise. Would the sound of it stay loosely consistent while also morphing in an organic feeling way?
You could reuse the same sounds after a certain point since they would fade to nothing, so it should be able to be done forever.
This is a *very* interesting idea! Kind of like a Shepard tone, but for prime rhythms
Extra cursed shepherd tones
dude I'm absolutely stealing your idea it's amazing
@@bonbondojoe1522 Don't forget to share with us!
@@bonbondojoe1522 I request an update
The 2-3-5-7-11-13 is truly astounding, the emerging patterns are mesmerizing
Sometimes I make generative modular synth music... so its all like sequenced and logic/math based.
I love playing with primes.
I'm making a modular EP now ^~^
Edit: You've given me insight by pointing out that prime polyrhythms have gaps at unassociated prime numbers. And repeat and the square of the first unused prime. ^~^ THANK YOU BUDDY
Technically, these aren't polyrhythms, but polymeters. It's not one bar with different amounts of beats, it's a fixed pulse, but the bar lengths vary
Can you explain further?
@@droughdough
Polymeter: Tracks that play in different meters, de-synchronizing themselves from each other (e.g a 5/4 time and a 4/4 time playing together)
Polyrhythm: Subdivisions that fit within the same bar and whose accents always start on the downbeat of a bar (e.g triplets playing against eighth notes both in 4/4 time)
@@harry_dum7721 to be fair, there's a very natural correspondence which is to take the least common denominator, call that one bar, and only play beat 1 of each part. That then gives a polyrhythm corresponding to the meter. For example 2+3+5 corresponds to 6:10:15. In general the product of the polyrhythm and the meter giving it is that denominator.
This is exactly the type of comment I'd expect on a video like this.
@@droughdough If you were to write out this music, it would all be 8th notes, they are just at different pitches. With polyrhythms that's not the case, as each different instrument/voice would have to be written as a different tuplet (like a triplet, or quintuplet). Here's the easiest way I can put it:
Polymeter means there are multiple meters (time signatures) happening simultaneously, but we keep the tempo and the note divisions constant. This means the different instruments do NOT start together at the beginning of each measure, and instead it might take a few bars for them to get back together. As an example, imagine a piano playing 3 8th note pattern played on top of a guitar playing a 4 8th note pattern. The patterns are different lengths but the same speed.
Polyrhythm means it's all the same time signature, but the different instruments are playing different speeds (or tuplets). In other words, within the space of one measure, one instrument might play 4 notes while the other instrument plays 5 notes, but they always start at back together at the beginning of each measure. The patterns are all the same length but different speeds but the same length
h e a r i n g math properties of primes is amazing. Thanks for this experience! #peer_review
I've listened to so much microtonal music that I just plain enjoy this polyrhythm
Honestly, the first 2 seconds of the polyrhythm sound cool, like a forest in the rain, but the type of rain where there's still sun beams entering through the cracks in the clouds
Very poetic
.
I watch both alot of math and alot of music theory videos and I absolutely love when TH-cam recognizes the intersection between the two and recommends videos like this one. Thank you for creating this awesome video.
I'm super curious just how terrible (or not?) it would sound to invert the entire rhythmic scheme and interpret 2 as 1/2 note, 3 as 1/3 note , 5 as 1/5th note etc, or 2BMP, 3BMP, 5BPM, same thing, just different abuses of notation. There's the downside that you have to decide how deep to take your recursion ahead of time and then rescale your playback speed to make sense for how deep you went, rather than just adding in parts until it's a mess and stopping, but has the potentially interesting advantage that you can go arbitrarily deep and still have all the cycles line up in a finite amount of time. Also, it would actually be a polyrhythm instead of a polymeter ;-) (for the record of course polymeter>>polyrhythm, I'm just pedantically teasing)
If you play this rhythm fast enough, it becomes a sound. An overtone series with just the primes resembles a lot a clarinet - except there is the 2nd partiaö, and the 1st missing (as well as all non-primes).
not necessarily the same thing because then you'd just get infinite per unit time because of so many primes?
and this video did not run until the heat death of the universe, so what?
@@burkhardstackelberg1203 I never would have guessed that would sound like a clarinet, but somehow it makes sense.
@@torydavis10 But you have to represent something infinitessimally close to the start
while not having to run the video until the heat death of the universe self-corrects because the really big primes don't have to be represented before the heat death of the universe and would be represented at its own pace.
While for this the really big primes would have to be represented incredibly quickly.
With that said, it would be really interesting to try this with the first n primes (with n being a finite number)
So marvelous. You've inspired me to make a wind chime that does this for as many primes as I can design into it.
Ooh, sounds very cool. How do you control the wind chime rhythmically? Or is it in pitch?
@@marcevanstein I intend to control only the pitch. A vertical helical blade (probably) on top will capture the wind and spin a vertically stacked set of cams spaced according to the first handful (out of the presumed infinite number) of primes. The cams will then actuate hammers that strike tubular bells surrounding the structure.
Many years ago I had much the same idea, and made a song out of the first 17 primes running through their rhythm for a few minutes. I then had a musician friend choose samples for the soundscape. The end result was less about the rhythm of the primes, and more an evolving soundscape with a difficult-to-define rhythmic pace, but it was a fun project!
This is really cool! Considering there's sort of an upper and lower bound at which rhythm breaks down as far as human perception is concerned (I think the lower bound is around 33 bpm), you can choose to simply work with a finite number of primes and create a lot of different arrangements.
Thank you for explaining the threeness I hear, when I've caught it in my own experiments I thought it was
simple bias. Funny thing, when the primes were speeded up with percussion in one of the last samples
I perceived an almost horizontal and equal three-based pattern which simply kept emerging in a new
timbral space, which seems independent of remainder of three rule. I also imagine I hear an accelerating
clave, sometimes swapping polarity between 5:3 and 3:5 which I guess is a consequence of remainder
distribution too.
Excellent video!
This is a really, really creative proof of Euclid's Lemma.
I'd love to turn this prime polyrhythms section into a full piece of music! Feels kinda like a 7/8 or 11/8 measure, Love it!
People who are good at math are usually good at music. By using our sense of hearing, we are learning about prime numbers. When I was learning about prime numbers in school, I thought they seemed like awkward lonely numbers. I have since discovered that they are more valuable than I thought. This reminds me of people who seem dull and useless on the surface with hidden genius and talent underneath.
For a few years now I've been planning to make an album of electronic music using ideas from maths (I'm a maths grad working in software, but music is one of my passions). I've recently finished putting together my studio in a new home, and I'm beginning to work on the ideas, so this is truly inspiring for me. Thank you!
You may enjoy reading Haskell school of music, if not already familiar
@@herdenq Thanks for the tip! I'm aware of Haskell but have never used it. I've added the book to my shopping list :)
@@macronencer Sweet! Just subscribed~ I look forward to a potential update
This is a wonderful way to learn about primes. Lovely idea to put math to music.
Come on man, we need the speedcore and extratone version! I wanna hear the extratone of a 1000bpm 2:3:5:7 polyrhythm!
Very well done. Videos like this are refreshing to see on a site rife with silliness.
Mapping notes to primes gaps would be neat. Especially you won't run out of notes quickly
This is an *excellent* idea -- I love it! I think I'll probably try it
it starts to sound eerie in a beautiful way as it goes
This has successfully explained to me how a prime sieve works.
I remember seeing you somewhere
Pentonic Scale + Prime Mapping with reverse harmonics = HEAVENLY MUSIC
This is the rhythm of the primes
The primes
Oh yeah
The rhythm of the primes
This is the rhythm of my life
My life
Oh yeah
The rhythm of the primes
Gee, sorry, I made the same joke two days later
8 months later and I thought I was a genius for coming up with this too. 😅
At the risk of sounding like my far distant teenage self - this is so cool. It combines several of my favorite things: math, music, design, and color. I may become addicted to watching it over and over. Thanks for making my day!
This is way more interesting musically than most contemporary music….
I love how you can hear new twin or close primes when you hear two new drums in close succession
This truly is some Prime music
This is so very cool! Thank you for this great information and well presented. Fascinating
Woa! Reminds of something Sevish would compose. Great video! c:
Side note: There's a cool lecture by Adam Neely where he showed that speeding up polyrhythms until the beats are like frequencies would result in intervals! so a 3,4,5 polyrhythm would make something like a major triad when sped up incredibly fast
I was looking through the comments for someone else who thought of this. I want to hear this sequence represented as a tone. Play the "2" rhythm at some audible frequency (>20Hz). I presume the resulting sound would start as a recognizable pitch but dissolve into noise fairly quickly.
I have to say I found this absolutely fascinating. Sound with numbers genius.
I experimented for a bit with using the squares of primes as tempo markings and coming up with metric/tempo modulations to attempt to create seamless transitions. One thing I wish I could do is a synchronization experiment with a prime number of mechanical metronomes set to prime BPMs. I'd like to see what BPM they synchronize to.
I cannot describe how satisfying it was to watch this. Bravo.
Have you considered making a mapping that is calculated modulo some frequency? so that if a prime's frequency would be below a certain threshold, the value "wraps around" back to the difference between the threshold and the frequency? This would allow you to play your music indefinitely without going so low that the value is inaudible.
Reminds me of a similar video years ago mapping the C-major pentatonic scale to the digits of pi, plus some backing music
Sounds microtonal, I adore this.
It is microtonal! The inverted harmonic series is interesting though, because it still has a lot of pure intervals
after browsing through so many channels. Yours is by far the best. The explaining thod is so great and detailed even complex stuff is
This was so fascinating! I wonder if you would try speeding up the sequence even further to the point where the frequency ratios would build a chord?
one hand very eerie and unearthly sounding but on the other hand very interesting and unique sound
I actually really like this visual representation of successive primes, because it demonstrates a trend, if not a pattern. If there were a pattern, we'd be able to codify it into an equation to find the next prime. But our insane pattern recognition abilities see this and automatically say "dude see??? There's totally a pattern!!"
Congratulations! You discovered 90's industrial and trip hop music.
Wow, as far as hear and see, I wonder what it would sound like when playing the full prime-(factor) spectrum, i.e. at beat 102 the triad (2,3,17) in equal volume distribution 1:1:1. Would it be harmonic or catastrophic for the ear? At wich speed?
I have no words to describe how much I love this. Thank you so much!!!
Very cool video ! Thanks
Returning here 2 years after I first saw the video, I realised just now how much I have learnt. I am so grateful for your channel, everytNice tutorialng
Wow, I just stumbled upon this video, and that's amazing! I always loved bridges between music and maths!
Would it be possible to have a long version of this song on your channel?
Good idea --- I'll try to put one together! I was thinking of maybe also including a downloadable link to a midi file, in case people wanted to play with it
@@marcevanstein That sounds great, I won't miss that!
TNice tutorials is one of the best intro soft softs I've ever seen. The entire basic worksoftow with no B.S.!
I always wanted to hear base 12 PI mapped into chromatic scale.
I love this thank you so much for uploading. Absolutely beautiful.
I’m not good with written numbers but I love music. Content like this helps bridge my understanding and is so valuable to me.
5:23 This part is hilarious
This was rad. Really well done on a fascinating topic. As a software developer, physicist, and musician this was a very fun exploration of primes.
Ok but what if we used other sequences of numbers and a different mapping algorithm? What about Fibonacci/Lucas numbers? triangulars, factorials, Van Eck?
Good video though! I'm not complaining, I was just being curious
Fibonacci's tend to the golden ratio, which is the *most* irrational number there is (phi = 1 + 1/(1 + 1/(1 + 1/(1 +......
and that would be maximally dissonant.
@@DrDeuteron It depends on how its sound is represented, and the mapping algorithm. But I guess you're pretty much right
Beautiful video, and strangely pleasing to the year. Well done!
4:26 math jumpscare
I used to do this with a friend in 2009. Cute to see it replicated.
What program did you use to make this? Because I'm doing a whole lot of CTL+C CTL+V in Cakewalk to make my variations.
My theory has been that each prime number (P) follows the same rule. Starting at a high note, every time (P) reaches a new factor (^n), it plays the highest note. The tones go down the scale as (P^n-x) where n-x is greater than 0.
This results in lower primes kind of crawling off of the song the longer it goes. If we follow this approach using the A minor pentatonic scale;
P^n-0 = A7
P^n-1 = G7
P^n-2 = E7
P^n-3 = D7
P^n-4 = C7
P^n-5 = A6
...
So the first time you hear the note A6 is at number 66, because 64=2^(6) and 66=2^(6-5)*11^(1-0).
I ran this with the numbers 2,3,5,7, & 11 and got this: drive.google.com/file/d/1-YIUfIMMKZDjd3G_cjUr_0c9ng_OkfgB/view?usp=drivesdk
The half way mark is wicked satisfying 😩
I need help writing out the Math patterns for the prime numbers. May i please have your assistance doing this?
@@christopherrice891 I might not understand your question, but I'm afraid I'm the wrong guy to ask. I only have a passing knowledge of number theory, no working knowledge.
My man's dedication is over the top!
0:10 no never who does that
Two through sixteen goes incredibly hard.
its misleading to call this a polyrhythm, especially when you are letting it go to infinity
this is more accurately a polymeter: different sized cycles on the same size subdivision. A polyrhythm is different sized subdivisions with one shared cycle length.
Any finite polymeter is also a polyrhythm, but the cycle length of that polyrhythm is the least common multiple of all the parts; in the 2:3:5:7 example towards the beginning of this video for instance, you didn't even get close to playing a full cycle of the polyrhythm, as that'd be 210 subdivisions.
So, what appears to be the 2 of the infinite polyrhythm is actually the 2 of the polymeter; the rhythm that has 2 evenly spaced hits over the full cycle length is... well, infinite.
you're not technically wrong to call it an "infinite polyrhythm" in the sense that the cycle length is infinite, but I am sure this has made some people think that polyrhythm is polymeter
I find polymeter often more musically useful, and I don't want to detract from the cool math here, but polyrhythm just isnt the right musical term
I really like all the different scales and tempi you used for the prime numbers. I couldn't help thinking this could easily be a theoretical example to demonstrate synaethesia. Although there are some more prevalent forms of synaesthesia, practically any combination of sensory substitution is possible. I like to imagine hearing colours or numbers, tasting or smelling colors, feeling the texture of temperature etc.
Most people goosebump or shiver in response to a sudden or prolonged decrease in temperature. These responses can also be elicted by shock, fear, horror, awe and other emotions which may also be evoked by touch, sound and music etc. Whilst it is still a response to temperature change, I occasionally goosebump and shiver when going from a relatively warm to very hot environment. Usually this occurs when going from outdoors, on a day warm enough in direct sunlight for me to enjoy being in the shade or maybe find a cool breeze, then getting into a car with no air conditioning that has been in full sun for a while. I also fall into the minority of people with a photic sneeze reflex; we really aren't all wired the same.
5:09 8000 Hz / 12 does not equal 333 Hz;
He meant 4000Hz
This was fascinating, Marc. Thank you!
This is exceptional. The long mapping looks like an ocean at perspective.
I’ve used prime number polymeters quite frequently recently but never took a step back to think about some inherent properties of them. Thanks for the video!
Man I’m glad I found this video. I was already playing this symphony in my head. D4# and A5# as 2 and 3.
🌊 📂
I have to teach the prime numbers to my students. So thanks for letting me know about this Erathostenes method!
I like the way those rhythms sound!
Okay... this was my favorite of all the 3b1b vids
On this type of content I find your kind of voice the most convincing.
Im going to think about this video for a long time, really gave me lots of ideas
Really nice job. I did so stuff with MIDI/DAW/soft banks wayyy back in like 99/00 in college and was ok at it. I bought soft soft last
This is amazing, thank you. It almost looks like a perspective drawing
I am neither a mathmetician nor a musician but this video was still incredibly interesting to me as an artist. This reminds me of the idea that everything in the end is just math.
TNice tutorials tutorial is so useful,I tried tons of other tutorials but tNice tutorials was the best one
I love this!! Thank you for the clear explanations and for showing us the outcome! Nice!
Such a wonderous project!
This got me way too excited considering it is almost 1:30 a.m.! Thank you, this was beautiful!
yeah we all need the full audio of the thing just running to infinity now thx, this is the internet