Are music and mathematics aesthetically AND inherently compatible? Inspired by a musical rendering of pi.
This video is aesthetically pleasing to me but I can’t help but be skeptical. Assuming any arbitrary mathematical sequence that is arbitrarily truncated and correlated, might this also not be true?
I’m probably slightly more briefed in math than music, but let’s assume the average doper skilled in neither.
I have this vague notion that western music is based on an arbitrary exponential/logarithmic scale also based on the arbitrary division of eight, also based arbitrarily on some specific base note.
I’m willing to be corrected.
There’s a whole bunch of skeptically based questions I want to ask. But the combinations are too overwhelming. So let’s stick with…
Maybe it’s a chicken and the egg problem - is music theory based on some arbitrary commonly agreed on math system… or does math correspond to some arbitrarily agreed on easy to replicate music system?
OK let’s get back to the core of the question…
Assuming any arbitrarily useful music notation system, will mathematical constants always be aesthetically beautiful?
I don’t understand that “core question”, given that I don’t find e or i particularly beautiful, but the theoretical link between math and music goes at least as far back as Aristotle; the practical link is older. IIRC, he described mathematically an equation which was already being applied practically: musical instruments were already made in a certain way because “that’s what you do so it sounds good” (panpipes and hand harps with specific reed/stringth length relationships), he wrote that relationship in mathematical terms. I think his work also describes the concept of “harmonics” - but this is all from barely-remembered lessons taken some 30 years ago, so take it with a huge chunk of salt.
Honestly, I’m surprised the project in the OP-linked video worked out so well (I was expecting a cacaphonous mess like the “mathy” stuff they were into when I was a music composition major).
But a good composer can use the right chords to cover over a multitude of discordant sins ;).
(But the chords he used, apparently, were also in some way constrained by the decimal expansion of pi, though I wasn’t clear in exactly what way–still, a nice feat of engineering. I wonder how many other ways he tried to interpret the expansion til he hit on this way…)
The “musical rendering of pi” is basically just noodling about within the c-major scale, so it’s no big surprise that the outcome sounds like “real” music. Any mathematical constant will do this, I suppose, if you just translate the figures directly into tones that are all within the same scale. It all depends on how you “translate” from a series of numbers into “tones”. A “translation” of a random number (or a mathematical constant, if you like) into wavelenghts by multiplying each figure with 675 and using the result as the number of hz to play will probably not result in anything musically meaningful.
The notes of the scale are not distributed randomly. They align with different proportional fractions of string lenght (easily discernible when playing guitar, for instance). I.e. an octave represents the fraction 2/1 (string length is halved), the “fourth” is 4/3, the “fifth” is 3/2, etc.
I don’t think that either system evolved from the other, but the realization that what the ear percieves as harmonic intervals corresponds to quite simple mathematical properties of string lenght have been seen as a significant sign to the underlying “harmony of nature” since the Pythagoreans onward. I am not sure that the question of why this is the case, can even be answered.
As an experiment, I tried basically the same thing with the constant e and within the space of half an hour or so got something which is fairly workable–given a day’s work I think it could be made to sound downright pleasant. (IMO it’s almost there already). I think you can use this link to listen to it. (Please ignore the name–it’s just what my daughter exclaimed in reaction to hearing (a slower version of) it.)
Next, I’ll try it with a randomy generated string of 16 digits. If that works out, we’ve got further evidence that it’s a trick and doesn’t mean there’s anything special, musically, about constants like these.
The effect is due more to the constraints placed on the project by its remaining entirely in one key. That would make (I predict) the great majority of random digit strings “workable” as melodies like this.
Meant to add in edit: the choice of e turns out to be fortuitous in a sense–the translation from decimal expansion directly to corresponding steps on a scale turned out to yield immediately a very traditionally structured, very I-IV-V-ish melody.
“The circle of fifths illustrates that chords are also assigned numbers.”
I find this to be misleading, if not exactly false. A less misleading way to put it would be “The circle of fifths is an illustration of the way we assign numbers to chords.” My phrasing makes it clear that the assignment is something we do. The way he worded it makes it sound like the assignment is in some sense purely natural.
Also–in the song the dude totally cheats. The song is based on the first 31 digits of pi, but you’ll note that his melody is actually 32 notes long. Where did the extra note come from? If you watch the numbers on the screen carefully you’ll note that the extra note comes at the very end, and it’s got nothing to do with pi, and it’s exactly the note he needs to make for a musically natural transition back into the beginning of the sequence.
The digits of pi are, in a sense, apparently random, so I’d call music based on them music based on randomness, not on mathematics.
As Panurge noted, there are simple mathematical ratios between notes that sound harmonius together. For example, if one string is 1/2 the length of another, the notes at which they vibrate are an octave apart. Here’s one site I found on the connection between math and music; I’m sure there are others.
Anecdotally, I know a number (no pun intended) of people who are gifted at both math and music. Pure math and pure music are both abstract, and deal with patterns and abstract relationships. And they’re both subjects at which child prdigies and young geniuses often turn up, perhaps because they don’t have extensive knowledge of the world or of human nature or a deep background of facts as prerequisites.
If you’re asking for the specific underlying mathmatics related to music, this thread may belong in GQ.
While I don’t have the deep math knowledge to articulate what is going on, I can say this:
With all of the digital modeling going on these days in music - where you can have a guitar in Standard Tuning, but the pickup can grab the tones and present a set of tones in a completely different open-chord tuning (so, in effect, you can play a guitar in Standard tuning as if it was in Open G or DADGAD tuning - very weird if the amp’s volume is so low that you can hear the “unplugged” Standard tones with the amplified open tuning tones) - this is clearly an example where there is enough inherent compatibility that the math can be used to digitally manipulate the sound before it is transduced back from signal into sound…
Beyond that, as a musician, it is hard not to listen to Bach without reveling in the sheer mathematical beauty of his intersecting harmonic lines.
I’m not quite sure what y’all have been getting on up until now, and I don’t have the time to really give it a think-through, but if you just want an example of a song that sounds really neat and uses some interesting mathematical concepts, check out The Fibonacci sequence in Tool’s Lateralus.
Is the beauty mathematical or musical? Both, perhaps. Bach was certainly very well aware of various mathematical properties of the different harmonic relations. I think they are definitely related, but as I said above, I can’t articulate how. Pure mathematics definitely has an aesthetical side as well - I believe that some mathematicians even consider this a primary feature of pure mathematics. Perhaps we are hard-wired into appreciating certain relations, ratios, etc. and these relations then pop up where ever we apply this sense of beauty, i.e. music, architecture, mathematics, etc.?
Two small anecdotes: a) The library where I work used a systematic catalogue many years ago. Music was placed under “Mathematics, Applied”. b) The classical music traditions in India are very closely connected with various cosmological, philosophical and scientific concepts. So much so, that music can actually be considered a way, perhaps even a shortcut, of gaining insight into the “highest truth/concept” i.e. brahman. Even the earliest musical theoreticians in India, Matanga, Bharata, the author of the Dattilam, etc. were aware of the intrisical parallels between music and mathematics.