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The Geometry of Music

kdawson posted more than 6 years ago | from the fantasia-with-strings dept.

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.

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Hmmmm. (4, Insightful)

jd (1658) | more than 6 years ago | (#22726214)

Neanderthals had flutes and discovered the octave. If we are to assume music is linked to string theory, then the problem of where they all went is solved! They were the aliens all the time! (Seriously, the paper is interesting, but you can always describe a simple system with a complex one. I'd want solid evidence that this is the reduced form.)

Re:Hmmmm. (2, Insightful)

edittard (805475) | more than 6 years ago | (#22726238)

Neanderthals had flutes and discovered the octave.
If you're referring to the bone fragment (singuilar) with holes in, it's by no means proven that it was a flute, or even that the holes were man made.

Goatse (-1, Troll)

Anonymous Coward | more than 6 years ago | (#22726266)

Goatse [twofo.co.uk]

You nerds love it.

Re:Goatse (-1, Offtopic)

Anonymous Coward | more than 6 years ago | (#22726304)

didn't we find this guy a while ago?
Goatse's actualy name is Kirk Johnson or frank or something.

Re:Hmmmm. (5, Insightful)

espiesior (1254968) | more than 6 years ago | (#22726566)

The wording is quite misleading. Tymoczko used "string theory" math... i.e. Geometric Topology (the article tries to play with "orbifolds" - fancy manifolds). Doesn't mean that string theory and music theory are intrinsically related in the physical world (which they are for the obvious OTHER reason), but rather, they can be expressed by the same monsters in the world of mathematics.

ObligatoryJack Black quotation (4, Funny)

Potor (658520) | more than 6 years ago | (#22726918)

I don't need no instructions to know how to ROCK!

Re:ObligatoryJack Black quotation (1)

Nombre_de_Usuario (1253486) | more than 6 years ago | (#22727664)

nice nice. Rocking our with string theory.

Re:ObligatoryJack Black quotation (1)

GigaHurtsMyRobot (1143329) | more than 6 years ago | (#22728248)

You mean obligatory Carl from Aqua Teen quote.

Re:ObligatoryJack Black quotation (1)

Potor (658520) | more than 6 years ago | (#22728368)

well, i do stand corrected. living outside of north america for quite some time has left gaping holes in my knowledge of tv culture ... i guess JB got it from carl.

will check that show out - thanks

Re:Hmmmm. (2, Interesting)

Bombula (670389) | more than 6 years ago | (#22727090)

you can always describe a simple system with a complex one

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:Hmmmm. (3, Funny)

TapeCutter (624760) | more than 6 years ago | (#22727314)

"I'd want solid evidence"

Yeah, Science will print any crackpot theory...oh wait...dammit...I've conflated Slashdot and Science, again! Second time this week...

The Naked Scientist (4, Interesting)

DKlineburg (1074921) | more than 6 years ago | (#22726254)

The Naked Scientist [thenakedscientists.com] actully just had a Podcast [nakeddiscovery.com] [MP3 Link] about music and science. If you find music and science interesting, I think it is a good listen. Not quite on the string theory level, but non the less I think it is relivant.

overthinking thing (0)

Anonymous Coward | more than 6 years ago | (#22726262)

yes thats all very nice. I could also write 100 pages on the cultural significance of party pies and you know what? they'd still just be party pies.

Way to overthink things.

Actually (5, Interesting)

El Lobo (994537) | more than 6 years ago | (#22726272)

It's not the first time music has been represemted as mathematical equations, or even as a random events. Hell, even Bach experimented by throwing a pair of dices while composing some of his most popular baroque parts.

Re:Actually (1)

Patchw0rk F0g (663145) | more than 6 years ago | (#22726292)

It's not the first time music has been represemted as mathematical equations, or even as a random events. Hell, even Bach experimented by throwing a pair of dices while composing some of his most popular baroque parts.
And the two have what in common?

Mathematical equations are, by definition, not random. I don't see the correlation.

Re:Actually (4, Informative)

Yetihehe (971185) | more than 6 years ago | (#22726340)

Mathematical equations can be stochastic, they may have defined certain probabilities of happening. Stochastic L-Systems [wikipedia.org] are good for demonstrating outcomes of some stochastic equations (I'm telling it after weekend with L-system parser for school project).

Re:Actually (3, Funny)

El Yanqui (1111145) | more than 6 years ago | (#22726382)

Dice, shmice. More cowbell is all that's needed to solve the equation.

Re:Actually (4, Insightful)

baldass_newbie (136609) | more than 6 years ago | (#22726648)

It's not the first time music has been represemted as mathematical equations

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.

Re:Actually (1)

Gryle (933382) | more than 6 years ago | (#22727168)

I read somewhere that the Euler Beta-function that formed the foundation of a lot of string theory was originally used to describe the motion of violin strings or something along those lines. Go figure

Re:Actually (1)

xarak (458209) | more than 6 years ago | (#22727292)



That would be "pair of dice" or "pair of die".

Grammar, however, is sometimes random.

Re:Actually (1, Informative)

Anonymous Coward | more than 6 years ago | (#22728000)

It should be made clear that 1) it wasn't the J.S. Bach most folks know and love, but his kid, C.P.E. ... maybe. We know Mozart and some other guys did it. And 2) they invented musical dice games, not used them in their serious compositions. But yes, it's an interesting precedent to 20th-century people who did, like John Cage. See http://www.carousel-music.com/shooters.html [carousel-music.com] , and a page at http://links.jstor.org/sici?sici=0027-4224(197804)59%3A2%3C180%3ADMITEC%3E2.0.CO%3B2-X&size=LARGE&origin=JSTOR-enlargePage [jstor.org]

Dirk Gently (5, Funny)

freaknl (1194831) | more than 6 years ago | (#22726276)

Am I the only one who immediately thought of the computer scientist in Douglas Adams' Dirk Gently's Holistic Detective Agency?

Re:Dirk Gently (4, Funny)

Decameron81 (628548) | more than 6 years ago | (#22726282)

Am I the only one who immediately thought of the computer scientist in Douglas Adams' Dirk Gently's Holistic Detective Agency?


Yes.

Re:Dirk Gently (2, Funny)

Smordnys s'regrepsA (1160895) | more than 6 years ago | (#22726300)

Am I the only one who immediately thought of the computer scientist in Douglas Adams' Dirk Gently's Holistic Detective Agency?



Yes.


No.

Re:Dirk Gently (2, Funny)

SimonGhent (57578) | more than 6 years ago | (#22726356)

Am I the only one who immediately thought of the computer scientist in Douglas Adams' Dirk Gently's Holistic Detective Agency?


Yes.


No.


Yes and No.

Re:Dirk Gently (1)

jcuervo (715139) | more than 6 years ago | (#22726398)

Am I the only one who immediately thought of the computer scientist in Douglas Adams' Dirk Gently's Holistic Detective Agency?
Yes.
No.
Yes and No.
42.

Re:Dirk Gently (2, Funny)

Thanshin (1188877) | more than 6 years ago | (#22726442)

Am I the only one who immediately thought of the computer scientist in Douglas Adams' Dirk Gently's Holistic Detective Agency?
Yes.
No.
Yes and No.
42.

 
All the responses are wrong, including this one and excluding the next.

Re:Dirk Gently (2, Funny)

F34nor (321515) | more than 6 years ago | (#22726480)

Clean pressed Irish linen sheets every day.

Re:Dirk Gently (1)

UnHolier than ever (803328) | more than 6 years ago | (#22727080)

I suspect you cannot even begin to comprehend just how wrong you are. But you are. Trust me, you *are*.

Re:Dirk Gently (1)

Bastard of Subhumani (827601) | more than 6 years ago | (#22727366)

>>>> Am I the only one who immediately thought of the computer scientist in Douglas Adams'
>>>> Dirk Gently's Holistic Detective Agency?

>>> Yes.

>> No.

> Yes and No.

Yes OR no.

But we can't tell until we open the box.

Re:Dirk Gently (2, Funny)

Hillgiant (916436) | more than 6 years ago | (#22727714)

>>>> Am I the only one who immediately thought of the computer scientist in Douglas Adams'
>>>> Dirk Gently's Holistic Detective Agency?

>>> Yes.

>> No.

> Yes and No.

Yes OR no.
Possibly Maybe.
Probably No.

Re:Dirk Gently (1)

Arancaytar (966377) | more than 6 years ago | (#22726612)

No you're not. "Anthem" popped into my mind immediately! =)

no, but coincidentally... (-1, Offtopic)

Anonymous Coward | more than 6 years ago | (#22726620)

tymoczko's the name of my mother's doctor;-)

Re:no, but coincidentally... (-1, Offtopic)

Anonymous Coward | more than 6 years ago | (#22727242)

One hopes that her disease is not as unpronounceable as the doctor's name.

Re:Dirk Gently (0, Offtopic)

Eli Gottlieb (917758) | more than 6 years ago | (#22728130)

There's a computer-scientist in that book? I really need to find a copy; I've only read "Long, Dark Teatime of the Soul" and the "Hitchhiker's Guide" series.

one suggestion.. (4, Interesting)

unfunk (804468) | more than 6 years ago | (#22726298)

...instead of having to play some of my own compositions on my nonexistent MIDI keyboard (my only MIDI device is my guitar amp effects controller), or manually entering the chords one by one, how about giving us the option to directly open MIDI files? MIDI files can be found for just about every equally-tempered piece of music you can think of, and it would be very interesting to see what they "look" like.

Also, as a composer myself, I'd like to be able to see what they look like :)

Re:one suggestion.. (1, Informative)

Anonymous Coward | more than 6 years ago | (#22727114)

Have you considered getting a software sampler/synth/tracker, such as Cubase? Importing MIDI files into that gives you a pretty good visual representation of what they "look" like.

Re:one suggestion.. (1)

Anonymous Coward | more than 6 years ago | (#22727864)

There are tons of ways to edit MIDI files directly via GUI or API. My favorite is Cakewalk [cakewalk.com] , but there are plenty others out there. Cakewalk actually started out as a MIDI editor, and evolved into recording live audio much later. Their professional line for audio production is called SONAR, which I still use frequently. Their consumer line is called Music Creator.

Re:one suggestion.. (1)

TACD (514008) | more than 6 years ago | (#22728186)

I was hoping for this myself, and managed to find a solution (for Windows at least). Install the free MIDI loopback driver from http://www.nerds.de/en/loopbe1.html [nerds.de] and set that to be the output of your favourite MIDI player (a quick and easy possibility is http://notation.com/DownloadNotationPlayer.htm [notation.com] ) Easy! Seems to work well as long as the MIDI in question only has one track ;)

Seems to me (3, Funny)

Smordnys s'regrepsA (1160895) | more than 6 years ago | (#22726316)

Lots of people found out exactly this in the sixties.

...or, maybe it wasn't the music, but the copious amount of hallucinogens that were taking them to higher dimensions.

Watched the .movs (1)

800DeadCCs (996359) | more than 6 years ago | (#22726322)

Most people just use milkdrop.

Not to say it's not interesting, in a navel gazing sort of way,
mixing numbers from one system into another (mathematical reese's peanut butter cups?),
but would running an episode of american idol through it give goatse?

Re:Watched the .movs (3, Insightful)

Oktober Sunset (838224) | more than 6 years ago | (#22727286)

but would running an episode of american idol through it give goatse?
If it did, It would be an improvement.

Re:Watched the .movs (2, Funny)

carpe.cervisiam (900585) | more than 6 years ago | (#22727754)

Only if the camera was pointed at Simon for the whole episode.

Well... (1)

tenco (773732) | more than 6 years ago | (#22726326)

that only backs my thesis that european (tonal) 12-tone music is very primitive and constricted.

Re:Well... (4, Funny)

Yoozer (1055188) | more than 6 years ago | (#22726460)

Quit channeling Stockhausen ;).

but this goes for any stream of information (3, Insightful)

jacquesm (154384) | more than 6 years ago | (#22726348)

not to belittle the guys achievements, but isn't it so that any sequence of bits can be represented by any arbitrary higher dimensional space ?

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 (3, Insightful)

Bjarke Roune (107212) | more than 6 years ago | (#22726564)

It is often true that if you have some parametrized way for describing data, then you generally want as few parameters as possible. You definitely want fewer parameters than data points, so going to more parameters or dimensions is not an achievement, as you point out.

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.

Here comes the land rush (2, Interesting)

tepples (727027) | more than 6 years ago | (#22726686)

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 (3, Informative)

radicalskeptic (644346) | more than 6 years ago | (#22726912)

You can't copyright chord progressions, only melodies. Most famously, there have been dozens of jazz standards written that are based on the chord progression of Gershwin's "I've Got Rhythm". In fact, there's even a name for that form: "'Rhythm Changes".

Melodies are just as finite (2, Insightful)

tepples (727027) | more than 6 years ago | (#22728126)

You can't copyright chord progressions only melodies.
Even if this is true, melodies are just as finite [slashdot.org] .

Re:but this goes for any stream of information (1)

epine (68316) | more than 6 years ago | (#22726856)

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 the average opinion overheard be damaged or improved by a quick visit to the Wikipedia on the subject discussed? Wikipedia can locate Iraq on a map. Most Americans can't.

The American view seems to be if you're not getting paid to do so, why bother? Don't waste your time. The information you need is found in the authoritative literature of your profession. This system produces strong economic results, which goes a long ways toward paying for the rather bad political results corresponding to a blinkered electorate.

I'm just saying that how a person frames "utility" amounts to a value statement and that prestige and peer-review are relative to purpose. You don't need to study anthropology very long before you get a good look at cultural credence effects (eminently peer-reviewed) meanwhile overturned.

From the perspective of algorithmic complexity, a scientist ought to be compelled to believe *all* hypotheses that haven't yet been falsified (with an exponential weighting function diminishing likelihood as a function of expression length). But science tends to have a rather severe constructive bias, which is culturally enforced.

Read the Summers debate, these voices are the same people performing auspicious peer-review behind the ivory curtain. Is your confidence shaken? Mine was. Probably not so much by this link, but by the rest of what I read at the time. Note that on the surface they aren't even managing to debate the same point, but the undercurrent concerns career advancement.

http://www.edge.org/3rd_culture/debate05/debate05_index.html [edge.org]

Or this one, which amused me earlier this evening: Tall people earn more because they're smarter. [slate.com] But hey, if you don't like it, I'm sure it was peer-reviewed.

One thing you can count upon, no future Harvard president will be caught dead discussing this research in a public forum.

So, what one can infer from the prestige of the journal Science is that if this result is a sophisticated math bamboozle, it's at the most sophisticated end of the bamboozle spectrum. I'll give them that much.

I don't actually suspect this is a bamboozle. I wouldn't be at all surprised that a more compact expression of chordal music is possible in a higher dimensional space. I have trouble believing Bach could do what he did if his mind was manipulating the same representation as the rest of us. The extent to which Bach intuited this higher representation, if it in fact exists, would be hard to establish.

The metaphor I would use is self-organized quasi-periodic tiling. Bach seemed to sense the local rules which governed whether the pattern could be sustained and extended (mellifluously), though he might not have had a conscious grasp on Penrose tiling itself, or whatever its analog might be in contrapunctal composition.

Re:but this goes for any stream of information (1)

31eq (29480) | more than 6 years ago | (#22726604)

That's certainly a criticism of string theory, because it uses extra dimensions to try and explain what happens in the usual four dimensions. But Tymoczko uses exactly the number of dimensions you'd expect in order to model voice leading. He happened to end up with a geometry that's known from string theory, and has some interesting properties.

Re:but this goes for any stream of information (3, Interesting)

Sage Gaspar (688563) | more than 6 years ago | (#22727024)

He essentially came up with (or used someone else's) model for putting some sort of measure of distance on music, then studied its phase space, which is minimal in dimension.

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:but this goes for any stream of information (1)

Threni (635302) | more than 6 years ago | (#22727178)

Couldn't you take any piece of information - including a zip file containing all the books, music, films ever produced - and represent it as a single floating point number between 0 and 1? Couldn't you then make a mark on a piece of glass (or something - it would have to be a material which would let you make arbitrarily accurate marks) for storage? Did I dream this, or was it the subject of a science fiction book a long time ago?

Re:but this goes for any stream of information (2, Informative)

nine-times (778537) | more than 6 years ago | (#22727358)

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.

Does it work backwards too? (3, Insightful)

josgeluk (842109) | more than 6 years ago | (#22726360)

It would have been nice if the author had provided some examples of music that his model predicts. If I walk a circle in his four-dimensional space, what does it sound like?

Big Note Theory (0)

Anonymous Coward | more than 6 years ago | (#22726392)

This reminds me of the clever Big Note Theory Frank Zappa talked about.

Windowlicker (5, Interesting)

lobiusmoop (305328) | more than 6 years ago | (#22726406)

This reminds me of the Aphex Twin track Windowlicker [wikipedia.org] , which, when viewed via a spectrogram, shows hidden images - Richard D. James' face, and a spiral. This explains why the track sounds so weird in places - the music is being warped to generate the images.

Re:Windowlicker (0, Troll)

Anonymous Coward | more than 6 years ago | (#22726454)

It might also explain why Aphex Twin sucks ass. The guy isn't a musician, he's a novelty.

Re:Windowlicker (1)

spintriae (958955) | more than 6 years ago | (#22726622)

"Windowlicker" never sounds weird, unless by weird you mean awesome! It does indeed have a hidden spiral image, but the sound it's represented by only enters at the end when the track is over. The weird track that you're thinking of with the face image is the second one on the single. That one is not so much a song as a series of images respected by sound.

Re:Windowlicker (1)

Yoozer (1055188) | more than 6 years ago | (#22726780)

Actually, the image is being interpreted as a sonogram and converted back to music. Metasynth [metasynth.com] for the Mac (which is most likely what Richard D. James used) and Coagula [passagen.se] for the PC can do this. X is time, Y is frequency, and the color's intensity is the volume.

Re:Windowlicker (0)

Anonymous Coward | more than 6 years ago | (#22727064)

wrong track, polynomial C on the windowlicker EP has the images not the windowlicker track itself

Re:Windowlicker (1, Interesting)

Anonymous Coward | more than 6 years ago | (#22727192)

Details on the hidden "Devil Face" here:
http://www.bastwood.com/aphex.php [bastwood.com]

Musical DNA (4, Interesting)

dgreenbe (242142) | more than 6 years ago | (#22726414)

Musical DNA Software [musicaldnasoftware.com] is actually doing something useful with mathematical patterns generated from music. Check it out.

Re:Musical DNA (1)

adamziegler (1082701) | more than 6 years ago | (#22727570)

Wow... this looks very interesting. Thanks for posting the link.

Interesting, though limited. (1, Interesting)

Anonymous Coward | more than 6 years ago | (#22726422)

FTA: "Exactly how one style relates to another, however, has remained a mystery--except over one brief stretch of musical history. That, says Princeton University composer Dmitri Tymoczko, "is why, no matter where you go to school, you learn almost exclusively about classical music from about 1700 to 1900. It's kind of ridiculous.""

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. (5, Interesting)

sticks_us (150624) | more than 6 years ago | (#22726588)

The study of Music Theory is highly recommended (though I wouldn't recommend it for a career choice) for anyone of a technical nature who really wants to be challenged.

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.

Sonny Bono owns you (3, Insightful)

tepples (727027) | more than 6 years ago | (#22726706)

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).
"White" and "Europeans" might come from cultural bias, but the "dead" part comes from copyright, specifically the U.S. term extensions of 1976 and 1998. It's much more expensive for schools to provide copies of "Rhapsody in Blue" or any more recent work, so schools just pretend Gershwin's compositions never happened.

Re:Interesting, though limited. (1)

TRS80NT (695421) | more than 6 years ago | (#22727008)

"... there are about a zillion compositions that use this chord progression (I-IV-V)..."

Of course half of them are versions of Louie, Louie


Re:Interesting, though limited. (1)

Jerf (17166) | more than 6 years ago | (#22728546)

for example, the reason we concentrate on music by dead white europeans from 1700-1900 may include a cultural bias, not just technical
We concentrate on the dead white men because it so happens that dead white men wrote modern musical theory. There were other musical traditions, but the dead white men, in terms of this article, are the ones who stumbled upon the rich musical space that we now mostly occupy. (History shows there was a lot of resistance on this front at the time.) As is so often the case, being the first they had the most to map out. They were done before anybody else really got into the act.

It is worth pointing out that "white men" is a bit of an unfair characterization, as if it were some sort of exclusive enterprise; in the 18th and 19th century, having significant influences and contributions from everybody from England to Russia qualifies as a pretty diverse endeavor. It's not like the collaboration and communication of the modern world was in place.

Nowadays, nobody owns it. I've listened to music from all over the world, and everybody has freely appropriated the 12-tone equal temperament scale, and put their own spins on it. (One of my favorite pieces that is interesting this way is a Japanese cover of a Beatles song with clear modern European continental techno influences. Now that's culture. :) )

Damn... (1)

EReidJ (551124) | more than 6 years ago | (#22726440)

While I'm still interested in the paper, I was very excited for a moment because I thought it said "choral" music, not "chordal" music. Damn. (Check my sig.)

Choir (1)

tepples (727027) | more than 6 years ago | (#22726724)

I was very excited for a moment because I thought it said "choral" music, not "chordal" music.
Choral music since about the late 16th century is a subset of chordal music, so you're still in luck. (See also madrigals [wikipedia.org] .)

Applied theory (4, Funny)

Blighten (992637) | more than 6 years ago | (#22726446)

Finally there's a hard piece of work that demostrates the usefulness of String Theory.... oh wait.... it doesn't.

What a clever chap! (1, Funny)

Anonymous Coward | more than 6 years ago | (#22726500)

Everything in this guys work reeks of cleverness.

However, it's still some of the coldest work that I've ever come across.

The net result? A waste of time...

A clever waste of time tho!

Well done, let's all pat ourselves on the back.

Re:What a clever chap! (0)

Anonymous Coward | more than 6 years ago | (#22726848)

this is what happens when you've already been tenured and can sit and relax letting your mind wander freely in all directions. The greatest works of science have been produced like that. No graduate student or postdoc ever produced ground-breaking work (besides collaborative accidents or lucky guesses?...) so yes this guy has the luxury to connect dots and do whatever his mind desires to, for the rest of us... look there is a postdoc opening in Zimbabwe, it's a 1/2 + 1/2 year position subject to funding approval... you'd better produce 2-3 papers out of that position if you want your CV to survive the flood...

I'm a hug fan of music ... (1)

ta ma de (851887) | more than 6 years ago | (#22726526)

However, adding a bunch of adjustable parameters in order to get a good fit is not what I like about music.

*yawn* (1)

youthoftoday (975074) | more than 6 years ago | (#22726562)

The fact that it was the first musical paper in Science says more about Science, frankly. The application of computers to music for analysis and retrieval has been around since the 50s. Take a look in MIT's journal of computer music for example.

In other news: patterns have been found for the specification of common, re-usable designs in object oriented software...

The Silmarillion: Music: Math For JRRT's Cosmos (1)

Himring (646324) | more than 6 years ago | (#22726692)

The Silmarillion:
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.

Not all music... (1)

Samah (729132) | more than 6 years ago | (#22726740)

> And their chord progressions tend to be efficient, changing as few notes, by as little as possible, from one chord to the next.
If you want non-standard chord progressions, listen to anything by Trent Gardner [wikipedia.org] of Magellan [magnacarta.net] .

Oh Wow, Man (1)

that this is not und (1026860) | more than 6 years ago | (#22726762)

So this guy has proven that every piece of music can be converted into a Windows Media Player Light Show?

Any 'doze user could have demonstrated that with a default OEM install of Windows XP Home and his stack of Led Zeppelin CDs.

Failing to see what great about this... (1)

dcgrp (1096745) | more than 6 years ago | (#22726798)

As a musician myself, I am curious if anyone (musicians and non musicians included) are finding actual musical usefulness out of this thing? For me, it is nowhere near a replacement or even an aid to listening to sound and judging it only on what I hear. Chopin circle video and see that I've already visualized this particular piece in a similar manner. Maybe it would be useful to find the connections between more complex sounds that my ears cannot discern very well, but, I'm just not sure yet. Any thoughts?

Flash and Quicktime (0)

Anonymous Coward | more than 6 years ago | (#22726958)

A lovely combination. Movies don't open on Firefox for me, the Quicktime plugin crashes on IE, and then on Firefox when I copy the link over. Finally, can't save the videos to disk thanks to flash

Re:Flash and Quicktime (1)

presarioD (771260) | more than 6 years ago | (#22728234)

open the movie with firefox and the mplayer-plugin, let it run although you don't see anything after it buffers. Then locate the "Cache" directory of firefox in your local account and do:

find $HOME/.mozila/firefox/"your profile number here"/Cache -amin -2 -exec file {} \;

that will pinpoint where the quicktime movies are, then just copy them somewhere more permanent and open them with your favorite movie player like mplayer for example...it worked for me and felt the genuine rush of beating all these stupid restrictions on right-click-download moronic sites try to impose...

oh give me some real news please! (1)

museumpeace (735109) | more than 6 years ago | (#22726976)

a. this is last years story
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.

Boooring (2, Informative)

chord.wav (599850) | more than 6 years ago | (#22727118)

I've never seen such a boring visual representation of music! While it may be accurate, even MS Media Player Visuals are better!

I was expecting to be blown up with something like this:
Flight 404 on Vimeo [vimeo.com]

How is this a surprise? (2, Informative)

the eric conspiracy (20178) | more than 6 years ago | (#22727198)

Music and geometry have followed the same paths in western civilization since the days of Pythagoras.

FINALLY we have proof... (1)

clonan (64380) | more than 6 years ago | (#22727390)

that listening to that music DOES make you "square"!!!

Hanasmus (1)

Whiteox (919863) | more than 6 years ago | (#22727542)

Any discussion of music and science would be meaningless unless taken in the framework of Gurdjeiff's teachings on the Law of Octaves.
You are all hanassmus individuals!

First Article on Music Theory they ever published? (4, Informative)

Vreejack (68778) | more than 6 years ago | (#22727572)

This is emphatically NOT the first paper on music theory they have ever run. A cursory search turned up several other recent papers. I'm too busy reading Dmitri Tymoczko's report on "The Geometry of Musical Chords" to write any more ---Science 7 July 2006:
Vol. 313. no. 5783, pp. 72 - 74
DOI: 10.1126/science.1126287

common sense perhaps... (1)

neuromountain (1255052) | more than 6 years ago | (#22727752)

If you are dealing with an 88-key (or however many key) instrument and a ten-fingered human, one would think that music is a sequences trajectory of ten-dimensional subspaces in an 88-dimensional space. A rather binary one. It would be interesting to see how to model the interactions of multiple instruments with different dimensionalities.

Steve Coleman and M-Base (1)

frog_strat (852055) | more than 6 years ago | (#22727792)

I just watched a DVD by Steve Coleman and his work with the mathematical M-Base approach to composition. Near the end of the DVD is footage of the band performing live, reading the music off of LCD screens. A computer is writing the music. http://www.m-base.com/videos.html [m-base.com]

Well duh! (0)

Anonymous Coward | more than 6 years ago | (#22727846)

"chordal music can be represented in a higher-dimensional space."

Doesn't that apply to anything? I mean, while a circle can't be represented in one-dimensional space it can be represented in three-dimensional space and higher. Seems like that would apply to every facet of life.

eh? (1)

chucken (750893) | more than 6 years ago | (#22727910)

From synopsis: "[he has discovered that] chordal music can be represented in a higher-dimensional space". Eh? You can represent just about anything in "higher dimensional space". It's not a discovery. The patterns revealed in higher dimensional space, however, might be interesting....

Multidimensions are unnecessary (4, Interesting)

Ralph Spoilsport (673134) | more than 6 years ago | (#22727944)

You don't need "higher dimensions" to do this at all. In fact, it's insanely simple, and is governed by numbers. Like Boards of Canada said, Music is Math.

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? (2, Interesting)

Sean Cribbs (927082) | more than 6 years ago | (#22728004)

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.

Musicla Dice Game on Commodore 128 (1)

PHAEDRU5 (213667) | more than 6 years ago | (#22728568)

Back in the day, and I mean *back* in the day, Compute! magazine published an article about a dice game for generating minuets that was popular in Mozart's time. Pick two random start conditions, walk a set of states, et voila, a minuet.

I thought I had the actual issue, but I can't find it. Probably one of the documents fortunately lost in the floods of 1967, or somesuch.

Johannes Kepler (1)

alan_dershowitz (586542) | more than 6 years ago | (#22728574)

Kepler wrote "The Harmonies of the Worlds" in the mid-1600's, which detailed a supposed connection between math and geometry, music and physics (specifically, planetary motion.) I know a few very smart people who hold this book in high regard, but it's hard for me to tell if it's something really profound or just a bunch of bullshit. Point however is that people have been making geometrical representations of music for a long time, if I understand the issue correctly. Doing this with string theory is very interesting though.

Grr... (2, Insightful)

ExecutorElassus (1202245) | more than 6 years ago | (#22728592)

I realize this is probably whiny of me, but it would have been nice if he hadn't built his entire freaking page as a Flash object. Since I run Firefox3 on 64bit Linux, the only way to see swf content is through an ugly hack that rarely works. This is one case where it does not: I just get a big white page. Is there another link to the article?
/rant
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