Interpreting a Sierpinski Triangle Fractal as musical notes sounds good
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That said even if it’s not a sign of the hand of god it’s still pretty cool…
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.
- https://en.wikipedia.org/wiki/Harmony#Perception
- https://en.wikipedia.org/wiki/Harmonic_series_(music)
- https://en.wikipedia.org/wiki/Consonance_and_dissonance
The short version is:
- 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.
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.
[1] https://en.m.wikipedia.org/wiki/File:Equal_Temper_w_limits.s...
> Can you expand on this? "Group" in what sense? C12?
Have a look here:
Correct, I was just referring to the convention that you use the same letter for f and 2f.
> Can you expand on this? "Group" in what sense? C12?
The cyclic group with 12 elements https://en.m.wikipedia.org/wiki/Cyclic_group
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).
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.
Complex Example : https://www.youtube.com/watch?v=ZdDT3hj7L2M
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.
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.
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.
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.
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.
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"?
(edit: note that in this case, it is a property of the fractal itself, not just the way it is interpreted)
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.
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.
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.
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.
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.
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.