The Geometry of Music 170
An anonymous reader notes a Time.com profile of Princeton University music theorist Dmitri Tymoczko, who has applied some string-theory math to the study of music and found that all possible chordal music can be represented in a higher-dimensional space. His research was published last year in Science — it was the first paper on music theory they ever ran. The paper and background material, including movies, can be viewed at Tymoczko's site.
The Naked Scientist (Score:4, Interesting)
Actually (Score:5, Interesting)
one suggestion.. (Score:5, Interesting)
Also, as a composer myself, I'd like to be able to see what they look like
Windowlicker (Score:5, Interesting)
Musical DNA (Score:4, Interesting)
Interesting, though limited. (Score:1, Interesting)
The innovation in music over the last hundred years has not been about the notes you play, but the harmonic content of new sounds and their expression.
If you ignore that and concentrate on the chords, then much music (like blues and a lot of rock) becomes identical in analysis. (It's all 1-4-5 so it's all the same, right?)
Re:Interesting, though limited. (Score:5, Interesting)
Beyond the simple technicalities of measure-by-measure analysis (what notes combine to find what chord? what notes form a pattern to yield what scale?) the body of known music as a whole forms a massive network of associations and references in the form of quotes, parody, mimesis, etc...it's almost as if music comments about other music.
This network, combined with various social and cultural studies, really provides a rich field of exploration (for example, the reason we concentrate on music by dead white europeans from 1700-1900 may include a cultural bias, not just technical).
The professional, academic fields of Music Theory, History, and Ethnomusicology are only now beginning to broaden the discussion, having been stuck in the early 1900s (I've known professors of music who will say, without irony, that there's nothing worth discussing since ca. 1915).
So, on your I-IV-V comment, it's true that there are about a zillion compositions that use this chord progression, so an interesting question would be "what makes each composition different in its use of this repetitive structure?"
The answers are always interesting, and can include discussions of different genres, barely-perceptible rhythmic features borrowed from other cultures, sound textures, audio effects, and on and on.
Fun times.
Here comes the land rush (Score:2, Interesting)
Re:but this goes for any stream of information (Score:4, Interesting)
For example consider the space of all oriented lines through the origin in three dimensional space. If you think about it you can identify them uniquely with the points on the sphere (the one they pass through "on the way out") and if you consider their "distance" from each other to be the differences between the angles of departure from the origin you will generate the standard topology on the sphere. Now consider unoriented lines. You can start with the sphere again, but then you identify points on opposite sides with each other because it doesn't matter what direction you're going. This is RP^2, 2-dimensional real projective space, which is a lot different from your plain old sphere and represents a minimal parametrization of unoriented lines.
Re:Hmmmm. (Score:3, Interesting)
I suppose it depends on how you define complexity. If we assume that none of the 'dimensions' of music are infinite - ie pieces are not infinitely long, there are not an infinite number of instruments playing at once, there are not an infinite number of audible tones, etc - then musical space is, well, pretty darn finite as far as the math is concerned. There are, after all, only 12 notes spread across 12 or so audible octaves. Even if we do not limit music to the 12 discrete notes of Western scales, there are still a very finite number of frequencies discernable to the human ear.
So while the number of possible combinations and permutations within musical space is very large, it is certainly finite and clearly definable. The fact that string theory might be able to do this is nifty, though I have my doubts about how well it's really working from this paper.
Re:Windowlicker (Score:-1, Interesting)
http://www.bastwood.com/aphex.php [bastwood.com]
Multidimensions are unnecessary (Score:5, Interesting)
It works like this: you use an algorithm that puts together in a very orderly fashion every possible note combination. Think of this as Serialism gone buttfuck crazy. If your system has only one note, and only one duration, then it can be represented in binary: 1 = note, 0 = silence. You can arbitrarily limit the duration (set definition) in question. So, let's say it's 8 measures.
So, every possible combination of 1 and zero becomes a number in this system, and so every melody can be identified.
Now, just multiply pitches, give it a number, and you get melody - 1,6,21,4,55, etc. Then you establish a simple number as your base "speed" (say, 120) and you can calculate the fastest possible repetition of a sound before it buzzes into a sound itself (something over 20 beats per second, so let's say 64th notes) and you then establish that as your "Planck" note duration. You then establish the number of possible pitches (the MIDI 128 will do for now) and then it's on to harmony.
Harmony (harmonies, triads, and chords, clusters, etc.) is simply melody stacked on top of itself. So, you then put some upper limit on the number of "voices" you wish to consider. An orchestra has 80+ voices, so let's make it a nice number like 100. So, you then take one melody.
So, now we have to calculate all the possible (128) pitches and silences for 8 measures for one melody. That gives you a number. Then you calculate it for each voice in sequence, and that gives you another number. Keep calculating. You will end up with a VERY large number of numbers, but you will be able to calculate EVERY POSSIBLE melody, harmony, triad and chord, in EVERY POSSIBLE rhythm within the parameters of your system (which, at 64th notes at 120bpm with a range of 128 notes, is REALLY FREAKIN' HUGE).
Except for primes, all numbers are the products of two smaller numbers greater than 1, so, one could then arrive at an equation of simple numbers arranged in additions and multiplications that would provide the given number to express a given piece of music. In fact, it would, in essence, express ALL music, as a given song would consist of a number expressing 8 measures, which is then followed by another number expressing 8 measures, etc. It's completely linear.
So, the first 8 bars might be [(a+b+c)(df)+g] which is then followed by [h(ij)+(kl)] which describes the next 8 measures, etc.
The computer would do the calculations themselves on demand. And this is where the EVIL FUN begins:
What you do is with this system, ANY piece of notated music could be fed into the computer, and it would then "find" that music inside the system, and ALL SONGWRITERS would have to PAY royalties on the music the computer has generated.
"Buh buh buh I'm an artist and I wrote this song. It goes Gm / Gm7 / A / D / G for eight bars and then..." Buh buh bullshit buddy: you song is located RIGHT HERE in my MASTER MUSIC PLAN. It's number consists of 10^42 digits and starts with "234895230498000345600045345" and ends with "3489000234502340523065023045604004506340" See? Right there.... Now PAY UP MOTHERFUCKER...
"buh buh buh..."
"ALL YOUR SONGS ARE BELONG TO ME!!!! now PAY UP!!!! I make the RIAA look like a bunch of GIRL SCOUTS!!! PAY UP!!! NOW!!!!"
See? We don't need "multidimensional systems" to describe music - it can be done linearly. And it can make the guy who builds this damn thing filthy fucking rich.
RS
Riemann anyone? (Score:2, Interesting)
The only remarkable thing about this man's research (at least what I can tell from the superficial article) is that he got published in Science. Music theory scholars study all kinds of mathematical models with strong resemblances to his multi-dimensional lattices. There's a whole branch of music theory [wikipedia.org] devoted to graphical, parsimonious chordal analysis and derivatives thereof.
Neo-Riemannian theory centers around a triangularly-tiled toroidal space (usually represented as a flat plane) in which chords, represented as whole triangles, typically move one vertex at a time, flipping across the space along adjacent sides.
Music Education Needs His Help! (Score:3, Interesting)
"Kind of ridiculous?" It's abhorrent. Think about all of the musical innovation that has happened since 1900. It's off the collegiate music curriculum. Try doing that in the field of engineering or medicine and see how the public reacts. But since it's just music, it's OK. We can all thank the NASM [arts-accredit.org], the organization through which most music schools are accredited, for keeping us, figuratively, in the dark ages.
The public usually thinks of high standards as forcing everyone to do equally well. Unfortunately, they often result in everyone doing equally poorly; there are only so many hours available in a day, and so many credit hours available towards a degree. We need more diversity in music education, especially in higher ed. Perhaps Dmitri Tymoczko's work will help.
And now, back to your regularly scheduled
Re:Hmmmm. (Score:3, Interesting)
String theory is similar. The pattern of fundamental particles is baffling when we look at them from the perspective of real space. When you look at them in higher dimensional spaces with particular topologies, certain features become obvious, a consequence of the geometry. Certain other patterns (in string theory) may become consequences given better mathematical tools for analyzing the geometry.
11-dimensional string theory may not represent reality, but the patterns it uncovers suggest that it might be saying something fundamental. The same goes for this guy's geometric theory of music. The Neanderthals discovered that certain patterns of frequencies sound good (the scale). Others discovered that some of those notes sound good (or not) played together (chords). Others discovered that particular chords played in sequence sound good.
The scale is explained by basic physics. Chords themselves are explained by fairly simple relationships. Chord progressions are more complex, but look like they might be explained by relationships in higher dimensional geometry.
Re:but this goes for any stream of information (Score:3, Interesting)
If you take an old game of Asteroids, you might be confused about how the spaceship can fly off one side of the screen and suddenly reappear on the other side. But if you lift that 2D plane up into three dimensions and roll it up, connecting the sides, you've got a higher dimensional representation with a particular restricted topology that makes it perfectly obvious where the spaceship will show up if it flies "off the edge."
Re:Windowlicker (Score:-1, Interesting)
http://hem.passagen.se/rasmuse/Coagula.htm [passagen.se]
Granulab is fun to play with as well (same site, granular synthesis engine).