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
@@No1rated_salesman1997 1 2 4 8 16 32 64 And 2^n Will be very short And also 5 is gangsta too On the end almost every number has 5 and after 5 16 8 4 2 1
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.
That was what I was thinking too! What's also interesting is how every key has different properties (e.g. some have better perfect fifths, others have better perfect fourths etc. depending on their prime factorization). Another interesting thing would be to go the other way (i.e. linearly increasing period; the first one would be ~50 microseconds), which also has the same variety in keys, whilst also having a simple physical interpretation- linearly increasing lengths of the vibrating object (e.g., multiples of Tv/4 and Tv/2 for closed and open pipes respectively)
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.
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!
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"
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.
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
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.
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.
What's cool about this is that, since the Collatz sequences can never end up on a multiple of 3 (unless you start on a multiple of 3 but then the sequence moves away from 3's after the first odd number), the pitch class strategy will never use the notes C, D sharp, F sharp, or A. This means that the melodies using this strategy will always be in the C sharp/D octatonic scale!
@@official-obama it trivializes the harmonic aspect by defining the variation to be harmonious. What's the point of tying it to that sequence at all if you're just going to do this?
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
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.
This is called algorithmic or procedural music. You might be interested in the work of Nicklas Nygren, a game developer who specializes in algorithmic music. Two of his games, Uurnog and Ynglet, have entirely procedural soundtracks.
You should try the mapping in "absolute" frequency. Frequencies get farther apart higher on the keys because they are ratios rather than absolute distances.
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)...
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.
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
How about use pitch class + Each digit has Each octave 2 is D in 1st(lowest) octave, 20 is D2 + C#1 2024 is D4 + C#3 + D2 + E1 like that. you can use UP to 7 digit and remove 2 black note if you want
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)
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.
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.
Are you talking about playing two at the same time or averaging the current number (eg because idk 3*+1 for the third one in lists 4,12,58 and 4,57,23 is 58 and 23) then modulating it to 127
Great video! One small suggestion though: the volume difference between your voice and the notes was quite large, maybe try balancing them a bit better. Apart from that, this video was really cool, well-made and informative! keep up the great work, you've earned yourself a subscriber :3
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)?
