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The Geometry of Music
Posted by
kdawson
on Wed Mar 12, 2008 04:26 AM
from the fantasia-with-strings dept.
from the fantasia-with-strings dept.
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.
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Submission: The Geometry of Music by Anonymous Coward
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Hmmmm. (Score:4, Insightful)
Re: (Score:2, Insightful)
Re:Hmmmm. (Score:5, Insightful)
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ObligatoryJack Black quotation (Score:5, Funny)
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Re: (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
Re:Hmmmm. (Score:4, Funny)
Yeah, Science will print any crackpot theory...oh wait...dammit...I've conflated Slashdot and Science, again! Second time this week...
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Re: (Score:3, Interesting)
String theory is similar. The pattern of fundamental particles is baffling when we look at them from the perspective of
Related: (Score:2, Insightful)
http://www.spiritofmaat.com/archive/jan4/williams.htm [spiritofmaat.com]
The Naked Scientist (Score:4, Interesting)
Actually (Score:5, Interesting)
Re: (Score:3, Funny)
Re:Actually (Score:5, Insightful)
You're right. Plato did it in the Timaeus about 2500 years ago.
It's nice to see folks eschewing traditional Western culture and then 'discovering' things the same Western tradition developed over two millenia ago.
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Re:Actually (Score:5, Informative)
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Dirk Gently (Score:5, Funny)
Re:Dirk Gently (Score:5, Funny)
Yes.
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Re: (Score:3, Funny)
Yes.
No.
Re: (Score:2, Funny)
Yes and No.
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Re: (Score:3, Funny)
Re: (Score:3, Funny)
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one suggestion.. (Score:5, Interesting)
Also, as a composer myself, I'd like to be able to see what they look like
Seems to me (Score:4, Funny)
but this goes for any stream of information (Score:4, Insightful)
The difficulty usually comes when trying to describe a higher dimensional space in a system with *less* dimensions, the other way around is trivial.
Re:but this goes for any stream of information (Score:4, Insightful)
The article is light on mathematical details, but it seems that the achievement is that this space of points has been characterized in a useful way. The story is not that now it can be done with even more dimensions (which as you point out would be trivial). Rather, the story is that now this space of points has been characterized at all, and this description just so happened to require several or many dimensions.
Since this paper is the first ever on musical theory to be published in Science, which is a highly prestigious peer-reviewed journal, we can assume that the paper is saying something interesting within its field. Specifically, we can assume that this is not just a question of fitting some standard statistical model to some data points.
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Here comes the land rush (Score:2, Interesting)
the story is that now this space of points has been characterized at all, and this description just so happened to require several or many dimensions.
In the 19th century, the land area of Earth was pretty much completely explored and divided into parcels of real property. This characterization allows composers to explore the space of possible chord progressions, put them into works, and copyright the works. Once the interesting parts of this space have been filled with copyright claims, will the rest of the composers have to stop composing or risk infringing?
Re:Here comes the land rush (Score:4, Informative)
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Melodies are just as finite (Score:3, Insightful)
Re: (Score:2)
Since this paper is the first ever on musical theory to be published in Science, which is a highly prestigious peer-reviewed journal, we can assume that the paper is saying something interesting within its field.
I wouldn't reason that broadly from prestige, but then I score the utility of Wikipedia higher than most, and the utility of peer-review lower than most.
The difference in my view comes down to a different perception of what "utility" encompasses: I don't concede special prominence to the narrow utility of career advancement. No doubt I'll soon be called to testify in front of the "House Committee on Un-American Activities".
Listen to any background conversation at your local hot-tub or donut shop. Would t
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.
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Re: (Score:3, Informative)
At the very least, it would be foolish to take this as some kind of indication about the universe, i.e. this isn't an indication that string theory is correct, that the universe has more than 4 dimensions, or that music exists in "higher dimensions".
Being able to "represent" something in higher dimensional space just means it has more than 3 quantifiable features.
Re: (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 dimension
Does it work backwards too? (Score:2, Insightful)
Windowlicker (Score:5, Interesting)
Musical DNA (Score:4, Interesting)
Damn... (Score:2)
Applied theory (Score:4, Funny)
The Silmarillion: Music: Math For JRRT's Cosmos (Score:2)
http://en.wikipedia.org/wiki/Silmarillian [wikipedia.org]
Music was Tolkien's "math" for his world's creation.
I always thought it insightful that Tolkien utilized music/song as the vehicle whereby his cosmos was created. Melkor would later "bend" the song to his own and, thus, launch the epic and birth the foundations for the rest of the cosmology that lead to the LoTR.
oh give me some real news please! (Score:2)
b. it was dumb then: if you through in enough extra dimensions and presume a few "hidden" parameters, you could get a theory that would not only "explain" all sequences of notes ever written but explain my girlfriend's choices in shoes...as a function of every third word in speeches of a randomly selected political candidate.
How is this a surprise? (Score:3, Informative)
MOD PARENT UP. (Score:4, Informative)
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First Article on Music Theory they ever published? (Score:5, Informative)
Vol. 313. no. 5783, pp. 72 - 74
DOI: 10.1126/science.1126287
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
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
Pitch is Boring -- study rhythm (Score:3, Insightful)
Some Old, Some New (Score:3, Informative)
The use of the circle to described musical perceptions is not new. It's been used to describe among other things the "ascending/descending" illusion. However, the use of other topological/dimensional concepts is novel, and pretty damn awesome. I've studied musical perception and its physiology, and a circle is definitely insufficient. More dimensions are required, as the waveforms involved are never (as early as the ear, much less in neural processing) sine waves. A simple example is the fact that inclusion of noise improves reception. The ear itself introduces noise, quite possibly for this purpose. Another is the multimodal (ie. harmonics) nature of most musical instruments. For instance, look inside a piano. The "notes" have more than one string. Even a single string vibrates in a complex set of harmonic frequencies. Now consider that the complex harmonics alone can be used to recreate the missing fundamental (the "main") note in perception, and possibly even in the instrument. Many different multimodal waveforms can create the same result. That requires different approach paths to the solution, and that requires more dimensions.
Sadly, very few in the relevant psychological fields are prepared to understand and incorporate this theory into their work. I still can't find more than a handful that can understand nonlinear statistics above 2 dimensions, even though they often use them for such as fMRI (the vast majority team up with biophysicists who do understand it). When they do manage to grasp the concepts in TFA, or find enough people from a relevant field who do with whom they can work, the results will be damn interesting.
Re:Well... (Score:4, Funny)
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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.
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Sonny Bono owns you (Score:4, Insightful)
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Re:Watched the .movs (Score:4, Insightful)
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