Sounds of the Collatz Conjecture: Generating Music from the 3x + 1 Problem 

I love that 8421 ‘motif’ for all the strategy 1s, it sounds kinda like the game over theme for all the failed attempts to disprove the Collatz Conjecture

I can see why proving this is so enticing for mathematicians- the patterns start to jump out, which is red meat to a mathematician.

I love how the sequence starting on 27 using pitch classes has those funny little runs of little trills, particularly the modulo 12 system

everybody gangsta until a node hits 16

4:55 GameCube intro??

This sounds like how aliens would compose music

moving up and down the circle of fifths sounds pretty neat

I love the use of a marimba for the first few sequences.

I'm both a computer programmer and a musician so I find this very interesting. It's remarkably musical. I think your choice of sound is great. These sequences would make wonderful percussion solos in a large ensemble (orchestra, brass band, concert band etc) piece. I wish that I could compose. Thanks very much for this.

The sonification of such sequences has always been of interest to me. As someone with no sight, it is often possible to identify patterns using sonification. Your channel is great, and has encouraged me to find apython library which will enable me to sonify such patterns. Many thanks.

When I was in college I made a little box where you hit a button a certain number of times and it would play a MIDI sequence based on the hailstone sequence generated from the total number of button presses, then challenged my classmates to make the longest sequence they could as my midterm project. They hated it

I'm interested in what it sounds like if you just use the number as a frequency with the units chosen so that 1 is the lowest pitch that can be heard (usually about 20Hz). This makes halving the number musically meaningful because you go down exactly one octave. Multiplying by 3 and adding 1 is slightly more complicated. Multiplying by three is going up an octave plus a perfect fifth. Adding one makes it more than that, exactly how much depending on how high the number was. For example, 3→10 is going up an octave plus a major sixth. 5→16 is going up an octave plus a minor sixth. This system would allow numbers up to about 1,000 for someone with good hearing.

So this was just “intro to jazz”?

Beautiful. Just Beautiful.

another idea i came up with: using microtonal scale with logarythmic frequencies. difference between big numbers in your examples are much bigger than between small ones, so log scale can help with it. we already percieve freqeuncies logarithmically, so this scale will basically be smth like "next note is 10hz higher than the previous one" instead of 12EDO "next note is twelfth root of two times higher than the previous one"

Trying to prove this has driven men mad. No kidding.

Nice. It would be cool to try something like mod 144. Limits the range but uses more of the keyboard than just the pitch class.

I think the main problem with the directional/bounded one is that a number can’t go up two times in a row, so the note sequence generally trends downward.

6:46 “A nice feature of this strategy is that it confines the note to a finite range, so we can arbitrarily long sequences” The sequence: To infinity, and beyond!

I once did an experiment where I used the Collatz conjecture to determine the length of a section in 8th notes. It came out rather decent, rendered an accompanying animation with blender and sent that in for my composing class "exam" in music school a couple of years ago. Edit: had to look it up to be sure but... I started from 25, which gave a nice balance of number of numbers and not going absurdly high

Thank you, algorithms. New favorite channel right here.

I think the main reason it sounds more musical than a random sequence of notes is that every time it goes up it must go down on the next step, since if n is odd then 3n+1 is even. I'm guessing if you removed that regularity and used f(n) = n/2 if n is even, (3n+1)/2 if n is odd it would just sound random.

You should try this with a pentatonic scale. It would probably sound more sonorous and pleasant.

Amazing video. I really really like your channel content! Keep it up.A few days ago I asked chatgpt about possible analogs of circle of fifths in 3 dimensions, 4 dimensions or higher. And I got interesting resultings every time I asked. It would be amazing if you do an video about it. I will paste some interesting results here: Sure, here are a few potential higher-dimensional analogs of the circle of fifths: 1. **Sphere of Harmonic Relationships:** Imagine a three-dimensional sphere where each point represents a musical key, and the distance between points indicates the harmonic relationship between those keys. This model could incorporate not only fifths but also other intervals and harmonic concepts. 2. **Hypergraph of Musical Elements:** Visualize a hypergraph where each node represents a musical element (such as keys, chords, modes, etc.), and hyperedges represent relationships between multiple elements. This could capture complex relationships beyond just linear progressions. 3. **Tensor Representation:** Utilize a tensor framework where each dimension represents a different musical element (e.g., keys, chords, time signatures), allowing for a multidimensional representation of musical relationships. 4. **Topological Map:** Create a topological map where musical elements are represented as points on a surface, and the topology of the surface captures relationships between elements. This could provide a geometrically intuitive way to explore complex musical structures. Ok other results from chatgpt with slightly different question about hyperdimensional circle of fifths: : 1. **Multi-dimensional Pitch Space**: Some music theorists and composers have conceptualized pitch relationships in multi-dimensional spaces beyond the traditional linear or circular representations. These spaces can include dimensions for pitch height, duration, timbre, and other musical parameters. 2. **Spectral Analysis**: In the field of spectral music, composers like Gérard Grisey and Tristan Murail have explored the use of multi-dimensional pitch spaces based on the analysis of sound spectra. These spaces represent complex relationships between harmonic partials and timbral characteristics. 3. **Mathematical Models**: Mathematically inclined composers and theorists have investigated higher-dimensional spaces using concepts from topology, group theory, and other mathematical frameworks. These models aim to represent the intricate relationships between musical elements in more abstract and complex ways. 4. **Interactive Visualization Tools**: With advancements in digital technology, there have been efforts to develop interactive visualization tools for exploring hypertonal analogs and multi-dimensional pitch spaces. These tools often allow users to navigate through complex musical structures and relationships in real-time.

Bro the tritone major third one is so cool

strategy 3 sounds great

This sounds so metal 🤘

What a vibe!!

all positive integers will reach a power of two is another way of putting it i think

This sounds good 👌 😌

A sound that probably kept many a mathematicians awake at night scratching their heads down to the skull.

Incredible

Perhaps limit it to a particular key scale, major or minor instead of chromatic?

Yay a new algo video

new AlgoMotion video just dropped! Give me more combinations of random mathematical concepts with music theory!!!

imagine playing music at a concert but the sheet music is 8:41 or 9:56 :skull:

Wouldn't the most obvious strategy be to go up by a perfect fifth and down by an octave?

Strategy 4 sounds like a flashback sequence

I'd suggest precalculating the sequence and scaling the note intervals accordingly so they fit it into your 4 octave range without direction-reversal shenanigans!

Your very inspiring video made me think of the following new thought: given a set of Collatz sequences, when considering a new sequence, terminate the sequence as soon as it falls on a number which has appeared in any sequence already observed. Then the length of the sequence is measured only to that length. I wonder if this "Reduced Collatz Sequence" sheds any new light on things. If we have a function on the natural numbers that yields the "reduced length" of the corresponding Collatz sequence, then how quickly does the envelope of that function increase? ~~~~Arthur Ogawa

7:36 had cool sound to it

I was fascinated as to how quickly I adoped to expecting, musically, for the sequence to go to the value 1. Or, again musically, the root if you wish.

Love the 27

I can imagine further constraining it within a key. Maybe use another sequence to control the key or mode. You could basically generate infinite background music for a video game that changed constantly while always maintaining a specific mood.

You should try the mapping in "absolute" frequency. Frequencies get farther apart higher on the keys because they are ratios rather than absolute distances.

Wonderful. I am a mathemusician myself and I'd like to propose the following "bebop" strategy. Fix a note (e.g Bb) and an associated "Bebop scale": this is like the usual major scale, except there is an extra step (Ab) between G and A. This makes the scale of 8 notes and is useful in improvising to stay on the same chord when playing the scale linearly (try to go downward from Bb). When collatz go up, you would go two steps up on the scale, neglecting the extra step (this will result in upward arpeggios on II/V or I/IV/VI). When collatz go down, you would go one step down on the scale using the extra step, playing a downward scale. Then swing a bit the notes (even notes last 2 beats and odd notes last 1 beat) and set a trumpet like sound. Regarding boundedness, you can transpose 1 octave down or up the note your going to play if it goes out from the registry you fixed. I guess this would result in a very nice bebop like improvisation. If you want to rock, add a II/V/I jazz backtrack and you are ready to publish. I would like to listen to this veery much! Please make it!! A small mathematical note: when we only use the up/down information, it seems like we are using less information. However, froma mathematical pint of view, if you give me the story of a number trhougb Collatz (ups and downs), I am able to uniquely identify the starting number. So on balance we are using the same amount of information.

Alien jazz, neat

Strategy #4 forever

More musical than a lot of atonal modern music.

As an economist, all I can see is the long run underlying process of treasury interest rates haha

Would be interesting to mod 8 the numbers and apply them to the notes of scales. Thumbs up if you agree he should do this.

we making a frums song with this one 🔥🔥

0:01 sounds like a music from portal 2

4:54 Gamecube intro ahh beat

Obviously I'd heard of 3x+1 quite a few times but I would never have thought about this how much time do you spend finding these ideas ?

I'd like to hear one confined to a major scale!

this would make a SICK pokémon battle theme

Can you reverse into a sequence starting with 1 and branching where applicable (1 comes from 2 comes from 4 - 4 comes from 8 or 1). There are trails of doubling forever to inaudible, but reliably also these "one less and divide 3" drops down into more trails of doubling. Just seems interesting what chords the branches might make.

I could swear I've heard this melody in the Serial Experiments Lain soundtrack

Doing this on a chromatic scale just sounds like Harry Partch percussion runs… can we get this in some 46 note to the octave tuning (i can’t remember exactly how many notes he uses lol)?

6:35 has the beginning of the Buddy Holly riff

This almost sounds like the Jurassic Park sega genesis game music

I think it would be interesting to potentially use a bounded set of four or five octaves but for half steps exceeding the bounded range add additional notes by the same rules creating chords.

2:53 is like boss music, especially if you were to loop it

This feels almost like developing variations

Maybe I can use these for my ringtone

Okay!

I had a stroke listening to this

I think using the log of the hailstone numbers would be a good way to help limit the range but keep the shape of the sequence

Mesmerizing. First metods resembled some human composed music from 1960-1970 era wich was often used as musical ilutration to educational or documentary movies

John Petrucci: "Guys, i have an idea for the next album"

What about converting numbers to HZ and snapping to the nearest note (or for even more fun, just running with the micro tones)

Ahh, the good ol’ CS50 shorts exercise.

The last example reminds me a lot of "Tail of Benin" theme by Walter Smith III 😂

Use the frequency to encode long sequences.

This video have to has millions view

A lot it feels like something Jordan Rudess would have played at some point.

I did the sequence for the numbers "1234567890" repeated 9 times and it obeyed the conjecture

It would be interesting if you mapped the binary representation of the hailstone number to a set of notes eg 0001 -> {C4}, 1011 -> {C4, D4, F#4} etc, there should most definitely be a way to make it sound nice since there are some 'regularities' in collatz sequences in base 2.

I like strategy 5

you should try turning one of these into an actual good bit of music by messsing with the notes as little as possible (changing their length and adding in silent beats is perfectly cool tho)

This sounds like a 1970’s Miles Davis intro

What is your sheet music? Collatz conjecture

The ending one is very similar to the intro to Long Piece No. 3 Part 1 by Egg

I feel like there i a critical issue with some of these strategies: seems like we should impose an additional restriction, that 4-2-1-4 puts you back on the same note. it is, after all, a loop

Taking logarithm of the number would keep the sequence bound to sensible scale range quite easily. Of course it would need rounding however

27: "hold my beer"

Yo! Try using the harmonic series(or just multiplying a fundamental freq(preferably a low one) by the hailstone number) instead of the out of tune 12tet! And coming up with a strategy with that, like locking every note to an octave(2/1) or a 3/1. The subharmonic series have the opposite feeling to the harmonic series. its dividing a base freq by an integer, instead of multiplying.

Heres an idea: put shepherds tone version of each note so that limiting the range might sound smoother? idk im coming up with this idea at 3 am

I propose an alternate strategy: the note's pitch and duration should be equal to the pitch and duration at that index of beethoven's 5th symphony (using the modulus of the number of available notes)

"There Is A Fractal Named Collatz"

Rather than mapping the numbers to notes, why not map them to frequencies instead? Then, a number that's twice as large would map to a note exactly 1 octave higher - essentially a logarithmic mapping; you could go a few thousand numbers high and still be in a useable set of octaves. For example, number n ---> frequency of 20+n Hz. So 1 would be 21Hz (just above/below the lower limit of human hearing) and 20000 would be 20020Hz (somewhere around the upper limit)...

She was a shy nerdy girl obsessed with numbers; he, a bad boy musician! Watch them discover the meaning of music by composing with math, and on the way, they will find love! Coming soon to Disney Channel: Mathemusical!

Weird music good info 😂

I am reminded of the Ice Caverns song from final fantasy ix

Is it just me or anyone else noticed that the valley points (lowest) of the patterns are always prime numbers?

Strategy 3 sounds like gamecube intro

Mark my words, I will write a song out of this concept that doesn't sound like randomness

half the time i hear my gamecube starting up

What are the chances that yt recommended this to me because those numbers looked like osu hitcircles guys?