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Music Media Science

The Geometry of Music 170

Posted by kdawson
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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The Geometry of Music

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  • Hmmmm. (Score:4, Insightful)

    by jd (1658) <imipak@noSPam.yahoo.com> on Wednesday March 12, 2008 @05:31AM (#22726214) Homepage Journal
    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: (Score:2, Insightful)

      by edittard (805475)

      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.
    • Re:Hmmmm. (Score:5, Insightful)

      by espiesior (1254968) on Wednesday March 12, 2008 @07:13AM (#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.
      • by eh2o (471262)
        The string-theoretic meaning of "orbifold" is not the same as its meaning in geometric topology, and according to Wikipedia has "no geometric interpretation". I'm not an expert on the subject but at first glance I'm fairly certain that Tymoczko's orbifold is in the general sense, not the string-theoretic one. Oops?
    • by Potor (658520) <.farker1. .at. .gmail.com.> on Wednesday March 12, 2008 @08:34AM (#22726918) Journal
      I don't need no instructions to know how to ROCK!
    • Re: (Score:3, Interesting)

      by Bombula (670389)
      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

    • Re:Hmmmm. (Score:4, Funny)

      by TapeCutter (624760) on Wednesday March 12, 2008 @09:26AM (#22727314) Journal
      "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...
    • Re: (Score:3, Interesting)

      by ceoyoyo (59147)
      It looks like his system doesn't necessarily describe music in a conceptually simple way, but it does seem to represent the problem in a way that explains why some chord progressions work and others don't. In that way it's much simpler. Many problems work that way: if you look at them in the right higher dimensional space you see definite patterns that aren't obvious in other topologies.

      String theory is similar. The pattern of fundamental particles is baffling when we look at them from the perspective of
  • Related: (Score:2, Insightful)

    by Anonymous Coward
  • The Naked Scientist (Score:4, Interesting)

    by DKlineburg (1074921) on Wednesday March 12, 2008 @05:41AM (#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.
  • Actually (Score:5, Interesting)

    by El Lobo (994537) on Wednesday March 12, 2008 @05:47AM (#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.
    • 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: (Score:3, Funny)

      by El Yanqui (1111145)
      Dice, shmice. More cowbell is all that's needed to solve the equation.
    • Re:Actually (Score:5, Insightful)

      by baldass_newbie (136609) on Wednesday March 12, 2008 @07:39AM (#22726648) Homepage Journal

      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.
      • by ceoyoyo (59147)
        Plato and his ilk are the very foundation of "western culture."

        Besides, the comparison is like the one someone made the other day between some discoveries in quantum mechanics and buddhism. Sure, buddhists believe everything is connected, but that doesn't mean they discovered quantum entanglement.

        There have been lots of attempts to describe music mathematically, from Plato to neural networks. That doesn't mean Plato had it all figured out and we're just now rediscovering it.
      • by mblase (200735)
        It's nice to see folks eschewing traditional Western culture and then 'discovering' things the same Western tradition developed over two millenia ago.

        The Greeks two millennia ago had developed a theory of higher-dimensional space?
    • by Gryle (933382)
      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
  • Dirk Gently (Score:5, Funny)

    by freaknl (1194831) on Wednesday March 12, 2008 @05:49AM (#22726276)
    Am I the only one who immediately thought of the computer scientist in Douglas Adams' Dirk Gently's Holistic Detective Agency?
    • by Decameron81 (628548) on Wednesday March 12, 2008 @05:50AM (#22726282)

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


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



        Yes.


        No.
        • Re: (Score:2, Funny)

          by SimonGhent (57578)

          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.
          • by jcuervo (715139)

            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: (Score:3, Funny)

              by Thanshin (1188877)

              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.
          • >>>> 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: (Score:2, Funny)

              by Hillgiant (916436)

              >>>> 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.
    • No you're not. "Anthem" popped into my mind immediately! =)
    • by ozbird (127571)
      Am I the only one who immediately thought of the computer scientist in Douglas Adams' Dirk Gently's Holistic Detective Agency?

      No, but only because you looked like you knew where you were going with this.

      Follow up question: Am I the only one that thinks that the research was a waste of time, because the majority of "mainsteam" (*cough*RIAA*cough) music could be adequately described in only one dimension?
  • one suggestion.. (Score:5, Interesting)

    by unfunk (804468) on Wednesday March 12, 2008 @05:57AM (#22726298) Journal
    ...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 :)
    • by TACD (514008)
      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 ;)
    • by expro (597113)

      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.

      Sadly, only for those in 12ET supported well by Midi. It would be nice to see how this math facilitate more-intelligent translation into alternative ET scales. I always wanted to try 53 ET, myself.

  • Seems to me (Score:4, Funny)

    by Smordnys s'regrepsA (1160895) on Wednesday March 12, 2008 @06:02AM (#22726316) Journal
    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.
  • 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?
  • by tenco (773732)
    that only backs my thesis that european (tonal) 12-tone music is very primitive and constricted.
    • Re:Well... (Score:4, Funny)

      by Yoozer (1055188) on Wednesday March 12, 2008 @06:39AM (#22726460) Homepage
      Quit channeling Stockhausen ;).
    • that only backs my thesis that european (tonal) 12-tone music is very primitive and constricted.

      In terms of the entire gamut of audible sound, yes.

      In terms of audible sounds considered to have enough aesthetic value to be 'musical', tonal music in a 12-pitch-per-octave tuning system comprises a large percentage of that space.

      The realm of what is 'musical' is expanding over time, though. Yesterday's terrible noise is today's progressive music, and tomorrow's standard repertoire.
  • by jacquesm (154384) <j&ww,com> on Wednesday March 12, 2008 @06:11AM (#22726348) Homepage
    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.
    • by Bjarke Roune (107212) on Wednesday March 12, 2008 @07:12AM (#22726564) Homepage
      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.
      • by tepples (727027)

        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?

      • by epine (68316)

        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

    • by 31eq (29480)
      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.
    • by Sage Gaspar (688563) on Wednesday March 12, 2008 @08:52AM (#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: (Score:3, Informative)

      by nine-times (778537)

      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)

      by ceoyoyo (59147)
      Yes, but this isn't just a higher dimensional space. It's a higher dimensional space with a particular topology, and that topology makes the relationship between chords in a chord progression behave in certain recognizable ways.

      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
  • 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?
  • Windowlicker (Score:5, Interesting)

    by lobiusmoop (305328) on Wednesday March 12, 2008 @06:27AM (#22726406) Homepage
    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.
  • Musical DNA (Score:4, Interesting)

    by dgreenbe (242142) on Wednesday March 12, 2008 @06:29AM (#22726414)
    Musical DNA Software [musicaldnasoftware.com] is actually doing something useful with mathematical patterns generated from music. Check it out.
  • by Anonymous Coward
    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 conc
    • by sticks_us (150624) on Wednesday March 12, 2008 @07:21AM (#22726588) Homepage
      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.

      • by tepples (727027) <tepples@nOSpAM.gmail.com> on Wednesday March 12, 2008 @07:54AM (#22726706) Homepage Journal

        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.
      • by Jerf (17166)

        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, bein

        • by sticks_us (150624)
          I'd love to hear that Beatles cover. Got a link?

          The old "melting pot" metaphor is really cool when you look at how different cultures appropriate, and adopt, music
          from other regions. Most of my favorite music (Blues, Reggae, Jazz) can be traced to this kind of activity.

          Recently, for example, I've been digging Nyankol Mathaing (here's [theage.com.au]
          one link I just found). Awesome mash-up of euro-techno styles with microtonal, groovy Sudanese music.
  • 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.)
  • by Blighten (992637) on Wednesday March 12, 2008 @06:36AM (#22726446) Homepage
    Finally there's a hard piece of work that demostrates the usefulness of String Theory.... oh wait.... it doesn't.
    • Re: (Score:2, Funny)

      by LeadSongDog (1120683)

      Finally there's a hard piece of work that demostrates the usefulness of String Theory.... oh wait.... it doesn't.
      I'm pretty sure that string theory has managed to feed a number of math-geek's babies. You tell me if that's useful.
  • 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...
  • 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.

  • 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 (Score:2, Informative)

    by chord.wav (599850)
    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]
  • by the eric conspiracy (20178) on Wednesday March 12, 2008 @09:12AM (#22727198)
    Music and geometry have followed the same paths in western civilization since the days of Pythagoras.

  • that listening to that music DOES make you "square"!!!
  • by Vreejack (68778) on Wednesday March 12, 2008 @09:55AM (#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
  • by Ralph Spoilsport (673134) on Wednesday March 12, 2008 @10:37AM (#22727944) Journal
    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

    • by ceoyoyo (59147)
      And you end up with a very large number of pieces of crap.

      The magic is in predicting which points in that very large parameter space describe something you'd want to listen to.

      Now, your wardialing approach to copyright, that's a little too insightful. You do know the RIAA reads this site, right?
  • Riemann anyone? (Score:2, Interesting)

    by Sean Cribbs (927082)

    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,

  • 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.
  • 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 i
  • Grr... (Score:2, Insightful)

    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 [wikipedia.org] visible/audible on a standard piano keyboard is James Tenney's "Harmonic Space" [google.com]. 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 per
  • by chocolatetrumpet (73058) <slashdot&jonathanfilbert,com> on Wednesday March 12, 2008 @12:10PM (#22729050) Homepage Journal
    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."

    "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 /. discussion.
    • by ceoyoyo (59147)
      Maybe it's another American thing. I didn't take music in college, but I played in the wind ensemble so I had friends who did. They used to complain about one of their professors who was into metal and acid rock and would draw from the genre for his classes.
  • by kov (262834) on Wednesday March 12, 2008 @12:12PM (#22729062)
    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.
    • by ceoyoyo (59147)
      Percussionist? No, wait, no pitch... so drummer?

      Rhythm gets boring pretty fast by itself too. Combine rhythm and pitch and you've got something though.

      Even drummers are rather fond of things like different sized toms, bongos, etc. Those would be examples of... oh yeah, pitch.
  • Donald Duck in Mathmagic Land
  • My album Geometric Visions [geometricvisions.com] is inspired by geometry; one of the pieces is called Recursion. It is minimalist instrumental piano.

    There are both HTTP downloads and torrents [geometricvisions.com]. 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 [geometricvisions.com] - autographed - to anyone anywhere in the world; just

  • Some Old, Some New (Score:3, Informative)

    by DynaSoar (714234) on Wednesday March 12, 2008 @05:50PM (#22733124) Journal
    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.
  • by nguy (1207026)
    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 -
  • 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!

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