This is more complex than pentatonic noodling, but I agree with other posts that this sounds the way it does mostly because of the choices made by the interpreter. The crucial choice was that every pixel of sierpinski’s triangle is a note in a major scale. All sierpinskis triangle encodes is the ordinal rank of the notes, everything else is added by the choice of how to translate it into the musical space. If the mapping were to a 12-tone chromatic scale and the result were somehow tonal with interesting harmonies that would be really remarkable. Even then though the 12 tone scale is an interpretation of tonality from western music.
That said even if it’s not a sign of the hand of god it’s still pretty cool…
> Even then though the 12 tone scale is an interpretation of tonality from western music
I hear this a lot, but this is a pretty uncharitable description of the 12-tone scale. The 12-tone scale is pretty easy to derive from first principles - the only non-mathematically-"obvious" part is the weird behavior of the multiplicative group generated by 2^(1/12) approximating lots of low-entropy rational numbers. The rest of it (identifying integer powers of 2, group properties of C12, etc) are relatively "obvious" to justify mathematically/physically. It's not just "what western people are used to" or something like that.
Doesn't this just get you the octaves, which are by no means unique to the 12-tone scale?
> group properties of C12
Can you expand on this? "Group" in what sense? C12?
I'm genuinely intrigued!
In any case, I don't think I disagree per se. But from what I can tell, the 12 tone scale only "falls out" if we apply two constraints: (1) roughly 10 tones in an octave, (2) steps are roughly equidistant (I'm _not_ alluding to equal temperament tuning here). Besides those constraints, why does, say, 16:15 get a tone (m2) but not 7:6?
So the question becomes: are those constraints justified?
EDIT: Partially answered my own question by consulting this diagram [1]: 16:15 has a lower "prime limit" than 7:6.
Isomorphic to the integers mod 12 (Z/Z12) - this is just a set with 12 things that wraps around. Things like the "circle of fifths" falls naturally out of the behavior of this group (7 is a generator of C12, and adding 7 in this group corresponds to multiplying by 2^(7/12) \approx 3/2).
Thanks for this comment. I mostly agree with you. A music theory professor used to lazily say that "nature says so" when asked why a major 3rd sounds better than an augmented 4th (i.e. tritone). Lacking a better explanation I assumed this had to do with consonant chords having notes whose wavelength had a lesser common multiple, but am not entirely sure. Can you provide some citations or resources describing the mathematical justification for a 12-tone basis for tonal music in more detail? I know there was a group theory element to some of Schoenberg's compositions but aside from that I have not been exposed much to this idea.
- Sound is vibrations produced by some object resonating and transmitted through the air. In the real world, things that resonate tend to have multiple resonant frequencies, and these resonant frequencies tend to be integer multiples of each other (due to physics). This sequence of integer multiples is called the harmonic series.
- Therefore, our ears are built to recognize the harmonic series, and to try to fit incoming sounds to the harmonic series in order to quickly and accurately identify the source of a sound (and isolate different sound sources).
- If and only if the least-common-multiple of a set of pitches is small, these pitches share many harmonics in common. If we hear these pitches together, our ears tend to recognize them as one blended tone, rather than as separate notes.
- Sounds that blend together are called “consonant”, and sounds that don’t are called “dissonant”.
As far as tuning systems go: frequencies are real numbers, but for practical reasons we would like to choose a discrete set of frequencies to work with when making music (unless you can build me a piano with an uncountably infinite number of keys). So, we choose notes that harmonize reasonably well together. But if you sit down and try to construct a tuning system by starting with one frequency and multiplying it by simple integer ratios, you’ll find that you do not obtain a finite closed group, which means you’ll have to make approximations somewhere. For instance, start with a frequency of 100 Hz and go up three perfect fifths - each is a 3:2 ratio so you end up at a 27:8 ratio, or 337.5 Hz. Then, start at the same base frequency and go up a major sixth (5:3) and then an octave (2:1), making an 10:3 ratio, or 333.3 Hz. These sound so close to each other that they are functionally the same note (you can listen to them on a website like https://onlinetonegenerator.com/), but no matter whether we choose to put one frequency or the other on our keyboard, some intervals will always be just a little off, so we have to make a compromise.
12-TET is the most common such compromise; it divides the octave into 12 equally-spaced notes (on a geometric series). Why 12? Because it gives us very close approximations of all the intervals used in Western music, without having so many notes that it would become unwieldy. Equal temperament has the advantage that the same interval sounds exactly the same everywhere — the 5th from F to C has the exact same ratio as the 5th from G to D, so you can play the same piece of music in different keys and all the intervals will sound exactly the same. It also guarantees that octaves (the most consonant interval) are tuned exactly and never approximated.
> Unfretted string ensembles, which can adjust the tuning of all notes except for open strings, and vocal groups, who have no mechanical tuning limitations, sometimes use a tuning much closer to just intonation for acoustic reasons. Other instruments, such as some wind, keyboard, and fretted instruments, often only approximate equal temperament, where technical limitations prevent exact tunings. Some wind instruments that can easily and spontaneously bend their tone, most notably trombones, use tuning similar to string ensembles and vocal groups.
Another choice made by the interpreter is not aligning any of the sides of the triangle vertically. If you took the piano roll and flipped each triangle 90 degrees to the right, you'd get regularly occurring note spam / lots of "chords" with several notes all half-steps away from each other, and it would sound very discordant.
There is deeper maths behind 12-tone octave than is implied by "western music".
It also encapsulates nearly all other scale systems found in the world.
The fact that 4/3, 5/4 and 6/5 are approximately 2^(1/12) apart, and 3/2 is twice as far as that from 4/3, pretty much made every other musical theory system irrelevant.
Reminds me of the infinite polyrhythm from the last summer of math exposition[0], though this has the more frequent notes in the base rather than in the treble. How pleasant the "Musification" is depends quite a lot on your choice of map.
Nice. Shameless plug, but I once did something like this with a Sierpinski triangle back in the day:
https://www.youtube.com/watch?v=72FmQAfQv1E
When you start to think about it, there are really a lot of different ways you could turn the Sierpinski triangle into sound. They generally have something interesting about them since the fractal pattern has a nice balance of same but different. I would personally recommend trying to make sound from a fractal as a good exercise to do if you have a Jupiter (or Colab) notebook handy.
This is pretty cool, even if by good it just means consonant. I happen to use fractals a fair bit in my own musical experiments, and I do it with a specific synth module called a fractal sequencer from Bloom. It takes a scale, a sequence length, a root sequence, and then various depths of mutations and iterates them. There is something about the way our minds select and sort information from noise that it's just complex enough to entrain the listeners attention where it seems predictable but it isn't, and it has just enough texture to create satisfying tension and release. My use of it has been pretty coarse and I just record live sessions without any composition, but as I get more competent with it, I could see some formal fractal techniques emerging.
Technically it's noise, but 1/f or recursive noise, and consonant instead of the dissonant noise we think of when we hear random music.
This reminded me of this way back page on the internet, from the time where phone ring tones could only be Midi files. (Sony Ericsson T610). I used "rabbit.mid" on that page as my ring tone, which gave my phone a very unique sound.
That page is but one of many different ways of mapping functions/fractals to sounds + music, while trying to build something artistic + aesthetically pleasing to the ear.
Sounds like a canon, as expected, just a bit fast.
But why just one scale? Put on the natural major scale, minor pentatonic scale, something like mixolydian, the blues scale, etc. They all have a chance to sound reasonably good, but with different flavors.
All 4 comments so far are middlebrow dismissals. Disapponting, HN.
It sounds pretty cool to me. It's not a fully formed musical piece in its own right, but did you expect drawing a fractal with note positions on a sequencer to sound like Beethoven's Fifth?
It's an curious little idea that produced an interesting sound, and unlike all the "this is what a black hole sounds like" nonsense it isn't presented as anything more than that. I expect with a little imagination the general idea could be developed into some interesting music, but the post doesn't even claim that.
Walk up to a nearest piano, hold down the pedal, and start playing on the black keys. You will find it completely impossible to play something that sounds bad. Much of what you play will even sound good. Someone ignorant of the piano might think you're a skilled pianist. But make no mistake: it is the tuning that is doing all the work.
I have nothing against this post, so long as people recognize this is an interpretation of the fractal designed to make it sound good. Any random input would also sound harmonious if mapped to the same scale.
It doesn't just has tone, it has rhythm too. Everything is based on powers of two, it will give you a nice 4:4 time signature at any scale. It is the same idea as for hitting the black keys randomly: you can't get wrong with powers of two.
(edit: note that in this case, it is a property of the fractal itself, not just the way it is interpreted)
This comment is so confidently ignorant of rhythm and melody, lol
Pay attention to the counterpoint (one voice rising, one voice falling) when one big triangle ends and the one above/below it starts.
Music is harmony, melody, rhythm, and dynamics. Dynamics means variations in softness/loudness, but none applied here, and someone else has noted the rhythmic component.
I wonder, with the right choice of mapping/filters, how much variation you can admit on the input side while still ending up with something that sounds basically okay at the end.
Context: I've been thinking about this recently whenever something to do with machine-written music comes up. My best attempt to describe what music is to me is something like "the patterned construction and resolution of rhythmic and tonal dissonance" — i.e. start neutral, create something that feels slightly uncomfortable, but hint through the pattern how it might be resolved to make you comfortable again. Kinda like storytelling, come to think of it.
So what if instead of a series of Sierpinski triangles, you used hailstone sequences, or anything else that contains interesting complexity and eventually returns to the origin. Could you mix and match different filters ("reject anything that doesn't fit this equality") and maps ("natural numbers to pentatonic scale", "natural numbers to commonly used chord progression"), and then apply them recursively? Could you end up with something that people would interpret as "real music", or would it always still sound like obviously computer generated "not-quite-music"?
I know that. I play the piano. I think I even qualify as a "fairly experienced singer and pianist" [1].
As others have pointed out, there is structure here beyond the fact that the tones are constrained in a way that results in harmonious intervals. This isn't random wind-chimes. Specifically:
* The mind tends to hear lines in the music, associating notes with particular preceding notes to hear melodies embedded in the polyphony. These lines move together in interesting and (to me) pleasant sounding ways, some moving together at fixed intervals and some moving in contrary motion. The avoidance of consecutive fifths and octaves in counterpoint is to prevent lines seeming to blend together into block chords, but here that happens a lot so the lines don't "live" for long. They briefly pop out of the harmony and merge back. There are many ways to hear that depending on what you focus on, so there's an ambiguity/interplay between hearing separate lines and hearing chords that tickles my brain pleasantly.
* The fractal is built step by step (in two different ways if you listen to the whole thing), which creates a rising/falling/growing pattern that builds a pleasing tension and release. It also provides a rhythmic framework and emphasises the self-similarity, as the previous pattern is repeated while further similar copies are added over the top.
Sierpinski's gasket isn't a picture. It's an abstract mathematical object. The image and the sound are both imperfect physical realisations of that platonic ideal and are directly related to each other by a very simple mapping. You could certainly argue that in the sound you are "hearing Sierpinski's gasket" to the same extent that you are "seeing Sierpinski's gasket" when you view the image. You'd be on far stronger ground than people who take physical phenomena such as black holes that in no sense "sound like" anything, arbitrarily map some aspect of them to a waveform and claim the result is the "sound of" the original phenomenon.
But the post doesn't make any such claim. It just points out that you can use the fractal to make something that both looks and sounds cool. I agree.
It’s because all five notes are adjacent to each other on the circle of fifths. Notes that are a fifth apart (adjacent on the circle of fifths) are consonant when played together. Limiting it to 5 notes makes it so that no two notes would sound too dissonant together. The more colors you try to use from the 12 chromatic notes, the greater potential for clashing colors. (The circle of fifths is kind of like a color wheel in that colors on opposite sides of the circle are maximally dissimilar to each other, so limiting yourself to 5 consecutive colors on the wheel means no colors are opposite each other)
I accidentally generated the Sierpinski Triangle when I was debugging a k-tree’s XOR space (I think I was using XOR with the distance function but I’d have to review the test case to be more exact). I wanted to visualize what was happening to the values and generated a bitmap of the coordinates translated to XOR space — it created this fractal. I might have some specifics wrong (it was a while ago) but the fractal popped out and I found it very interesting.
Seeing it used to generate music is also interesting. I enjoyed it as a curiosity and it reminded me of my own encounter with the fractal. Not sure where the hate comes from, it is really a fascinating sound.
If it was expanded into something musical, I would no doubt enjoy where that music takes me.
The title is truly misleading. Yes, natural scales sound good and we already know that. Gradually thickening an arpeggio or sequence makes it interesting for some time.
This is not what we expect when we see such a title. I'd except maybe some interesting microtonal or timbral effects. Even though microtonality rarely sounds 'good' since it's way too far from natural overtone scale.
It makes things sound inoffensive, but not necessarily good. This actually sounds good. It'd be a lovely generic arpeggio pattern in addition to the standard up/down/alternating patterns.
Great technical review. I appreciate the insight and thoroughness from someone who obviously knows their craft extremely well. The nuance in which you broke down how "[i]t sounds less than bad" left me incredulous, and then you completely rendered any type of retort useless by affirming that it was "... not good". There's just no countering that.
Spot on. I will never trust a musical review on fractal musical composition by anyone other than fairly experienced guitarists and/or pianists ever again.
My review was spot on. It’s subjectively not terrible but you wouldn’t use it as the basis of a composition.
Perhaps a school bell. But it would get annoying as fuck very quickly. Actually perhaps sticking it on repeat in a torture cell would be a good place for it. It’s not bad enough or good enough to be likeable.
>"Would it be possible to share one of your own compositions"
One does not have to be composer to judge music. Also it is very subjective. In my opinion the piece is a short phrase that does not cause immediate allergic reaction. But it is not a composition.
Some math translated to notes and producing reasonable harmony is not really that exciting imo.
That said even if it’s not a sign of the hand of god it’s still pretty cool…