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The Geometry of Music
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.
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.)
It's a small world and it smells funny; I'd buy another if it wasn't for the money; Take back what I paid (SoM)
http://en.wikipedia.org/wiki/Sacred_geometry
http://www.spiritofmaat.com/archive/jan4/williams.htm
The Naked Scientist actully just had a Podcast [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.
Memory is deceptive because it is colored by today's events. - Albert Einstein
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.
It's time to realise that Abble's products are the biggest abomination these days. Just say NO to the dumb iAbble way!!
Am I the only one who immediately thought of the computer scientist in Douglas Adams' Dirk Gently's Holistic Detective Agency?
...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
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.
Just -1, Troll talking to another.
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?
that only backs my thesis that european (tonal) 12-tone music is very primitive and constricted.
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.
MP3 Search Engine
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?
This reminds me of the Aphex Twin track Windowlicker, 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.
"I bless every day that I continue to live, for every day is pure profit."
Musical DNA Software is actually doing something useful with mathematical patterns generated from music. Check it out.
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?)
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.)
modern choral music...
Finally there's a hard piece of work that demostrates the usefulness of String Theory.... oh wait.... it doesn't.
However, adding a bunch of adjustable parameters in order to get a good fit is not what I like about music.
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...
-1 not first post
The Silmarillion:
http://en.wikipedia.org/wiki/Silmarillian
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.
"All great things are simple & expressed in a single word: freedom, justice, honor, duty, mercy, hope." --Churchill
> 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 of Magellan.
Homonyms are fun!
You're driving your car, but they're riding their bikes there.
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.
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?
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 don't need no instructions to know how to ROCK!
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.
SLASHDOT: news for people who can't concentrate on work or have no life at all and got tired of yelling back at the TV.
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
Music and geometry have followed the same paths in western civilization since the days of Pythagoras.
http://aboutscotland.co.uk/harmony/prop.html
that listening to that music DOES make you "square"!!!
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!
Don't be apathetic. Procrastinate!
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
"Will future ages believe that such stupid bigotry ever existed!" -- Ivanhoe
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.
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
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....
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
Shoes for Industry. Shoes for the Dead.
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 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.
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...
Yam, yam, uga booga, yam, yam, yade, yade, uga booga, yam, yam, yade, yade
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.
668: Neighbour of the Beast
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.
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
My favorite exploration of musical spaces more complex than the familiar Equal Temperament visible/audible on a standard piano keyboard is James Tenney's "Harmonic Space". Tenney was one of the first to synthesize music, at Bell Labs, and collaborated with the foremost avant garde composers, like John Cage. Harmonic Space is a way of writing musical relationships that are then performed, but not simply as a script of which "notes" to play. Rather the space is described in which pitches are allowed, then performers can play them in various sequences (melodic), or explore different harmonic subspaces together, or indeed travel through the space according to a predefined path.
It's fascinating, possibly more accurate than "octave/fifths" models and probably more accurate than staff representations. And the music is really interesting, often beautiful, but also something other than beautiful while also compelling, and at least something new to the ear. And, as a space, to the mind.
--
make install -not war
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."
/. discussion.
"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, 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
Spoon not. Fork, or fork not. There is no spoon.
Music theory is miles deep in frequency analysis, throw this one on that slag heap. I do congratulate him on proving that pitch is boring though: since his chordal (i.e., pitch-based) analysis manages to lump vastly different musics together, ironically he's shown that the vast majority of what makes them different from each other must be something else.
Ha, just found out that you can read it on Google books! Not costly anymore at all.
Donald Duck in Mathmagic Land
Were that I say, pancakes?
There are both HTTP downloads and torrents. The sheet music to two of the songs is provided in PDF and Lilypond format, with the others to follow soon.
My music has the Creative Commons Attribution-ShareAlike license. You could really help me out if you shared my music over the Internet.
I'm also offering to send free CDs - autographed - to anyone anywhere in the world; just email your snail mail address to support@oggfrog.com
While I presently work as a programmer, I have been studying piano intensively for several years with the aim of one day enrolling in music school to study musical composition. I want to write symphonies!
Request your free CD of my piano music.
The claim this is the first paper in Science regarding music theory is wrong. There have been others, including some on music theory and the physiological basis of music perception.
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.
"I may be synthetic, but I'm not stupid." -- Bishop 341-B
I'll be impressed when a musical composition he wrote rises to the top of the charts.
chordal music can be represented in a higher-dimensional space
Just about anything can be represented in a "higher-dimensional space". And chordal music forms a low-dimensional subspace. So what?
Ordinary music can be described in six dimension, each correspinding to a string, which we will denote by E, A, G, D B and E'. By "bending" and "plucking" the space so constructed, one can obtain different notes, each defined by a string and a function called a "bend". A discrete version can be achieved by restricting these functions to a discreet subset of the strings, elements of which are called "frets". Thus, for one instance, an "A minor" chord is the defined by x in M^6, x = (0, 1, 3, 3, 2, 1), x -> f(x), where f is a "bending" map from the six-dimensional "string-space", to the physical space of sound.
The music I play with my guitar, by plucking various strings of different lengths and tensions can be described by math that describes everything in terms of strings of varying lengths and tensions? Say it isn't so!
Guns don't kill people; Physics kills people! - John Lithgow as Dick Solomon on Third Rock From The Sun
Actually music does exist between space and time. I tried to go Dmitiri's URL but it is serviced by a MAC OS X server so I couldn't access it. I did download a file at a similar site using a PC server and opened it with Windows Media Player. It was supposed to be the Chopin. The problem is the visualisation looks just like the typical ones that display themselves during a typical audio tracks play. I was not impressed. That is not a new discovery. I have devised geometric shapes to represent scales and intervals for years. I use them when I practice playing cycles, like the tune Giant Steps by Coltrane. There is a close connection between Geometry and Music. No Bull!