This was very interesting to me even though i only play the piano and do not compose! i am printing the article for my music teacher.

I hate to say it, but despite the best efforts of some in Music Theory, there has yet to be anything approaching a "Theory of Everything."  One big clue to even the most unseasoned eye is when Tymoczko makes an offhand comment about African music (as if all African music is similar).  Without data, your theory doesn't stand.

Interestingly, I studied that same Chopin etude in undergrad (it is a popular one due to its short length and somewhat puzzling chords).  Our conclusion was that the chords arrived, as did all harmony in Western Music BTW, through voice leading.  That is to say, each line doesn't stray far from its previous note.  That sounds remarkably like "when you plot the chord movement in geometric space, you can see Chopin is moving along very short lines, staying primarily within one region."

To me, what Tymoczko has discovered is simply voice leading.  He may have come up with a novel method of mathematically describing the logic behind voice leading, but this is no different than the work that other theorists like David Lewin have done.

What burns me is that not only do papers like this get routinely accepted at conferences like SMT and ICMC, but also that this is the work that makes mainstream news.  Sheesh.

In the eyes of a musician, this article states the obvious.

I do, however, resent the implication that a computer could ever possibly create anything that measures up to Beethoven or Bach.

Re: "The idea of expressing music geometrically goes back centuries."

Quite a few centuries, actually. It goes back to Pythagoras, the Father of Numbers, 582 BC – 507 BC.    

But who's counting ?

Yes, I hope the post makes clear that this scheme cannot yield new, as-yet-unheard melodies ... or produce computer-controlled music as good as Bach or Beethoven. One could argue that the same thing was said about chess ... that no computer could possibly play chess as well as a human (which we now know is not the case). But I think music requires a different sort of intelligence that is harder for computers to reproduce. It's not just a case of working through the largest number of possibilities to come up with an optimal decision tree. That would be a great topic for further discussion: Is there something about music (or poetry, or art...) that computers can't intrinsically match, or would you side with Ray Kurzweil and say that it's just a matter of time before artificial intelligence produces masterpieces?

Here's some more food for thought:

http://www.aaai.org/AITopics/html/music.html

http://www.kurzweilai.net/brain/frame.html?startThought=Music 

I, not Tymoczko, am the author of the paper to be presented at the ICMC. I hope I will succeed not only in presenting the rudiments of Tymoczko's work, but also in showing some ways it can be used for algorithmic composition.

As long as the limits of artificial intelligence are not clear, surely it is premature to decide whether, or not, a computer will ever compose music on the level of Bach or Beethoven. My personal opinion is that not only human thought, but Nature in general, cannot be completely simulated by computer software. But I can't prove this, and I'm not sure it can ever be proved.

However, as far as I am concerned, this is completely beside the point. I use software not to replace my musical intelligence, but to amplify and augment it. After working for years with algorithmic composition I am quite sure that I, at least, am making music with it that I would never be able to make without it.

I think Tymoczko's work will really expand what we can do with harmony and counterpoint in algorithmic composition. And in my opinion, this will be precisely BECAUSE the ideas are simple yet cover a lot of ground.

I've revised the post slightly to make it crystal-clear that Gogins, not Tymoczko, is the author of the ICMC research (the original language could be read ambiguously on that point). Dmitri didn't tell me who was working on that research, but I'm glad that Michael has weighed in. Lots of information from his Wiki page:

http://www.ruccas.org/index.php?Michael%20Gogins

Including a link to an MP3 of the sample algorithmic music:

http://ruccas.org/pub/Gogins/samplepiece.mp3

Many thanks to Michael for shedding more light on the subject.

You are on the right track discovering the universal structural metaphors of existance. Eventually 50 years from now, computers will be able to compose truely great music based on emotional experience and complex creative interpretation. Those who do not understand this have not the vision of where computers are heading. These are natural progressions of science, man and computer.

"I do, however, resent the implication that a computer could ever possibly create anything that measures up to Beethoven or Bach."

Well, the way I read the response to that question, no such thing was implied.

"That's not going to happen," he said. "This isn't going to take anyone from being a mediocre composer to a brilliant composer. But it might help you get from being a beginning composer to a pretty good composer. ... We're going to be able to instruct computers to produce musical results. At the very least, we won't ask computers to do impossible things."

I think the response made clear that mathematical evaluation of music cannot give a composer talent, but can only hone skills.  There is quite a difference between the two.

This, to me, is just another means to appreciate music...a means to view it through its harmony with string theory...not unlike plotting chords on a circle graph.  Just interesting.  It doesn't replace the pure joy of music, but it does augment it.  Nor is it made useless by any similar observations.

The most important relevance of this information is that we can discover more about our relationship to the universe.  Harmonics has for centuries been seen in association with God, angels, and other spiritual representations.  The question is why?  What is so important about harmony in the cosmos that we have associated it in that way?  
There are some incredibly smart people who have theorized and then computer generated fractals aka; fractal geometry. This shows us a glimpse at the natural process of cellular growth, crystal growth, and other natural phenomenon, which is shown to be complex and highly detailed mathematical symmetry.  
I am simply over joyed that there are people like Mr. Gogins, Mr. Tymoczko, and those in the string theory camp who are exploring the subject.  I for one anticipate all of this knowledge, and theory with great enthusiasm!        

Because I have considerably less expertise in musical theory, limited to "I like that" or I don't like that", I can't comment on the article.  I did want to drop you a note about a link on Defense Tech.  Seems there was an article about a wild and crazy theory concerning the warming of the Earth - in 1932!  Check that one out, Alan.

Thought was the molecular interaction which ultimately
reproduced the sounds of nature. The human species interacted with crude simulations of the natural world as to relate to our place through the musical expression. The great masters as we call them are just a guage the the musical venues we have explored thus far. We have gone beyond the earth into space. We will venture beyond the masters of yesterday if only to nuture our instinctual nature to precieve the universe spreading before us. Think of the composite melody and rythimic beat of just one galaxy. It is all numbers in a cadence we have yet to understand.

Did they consult Richard MacDuff and use the Anthem software (WayForward Technologies II) to map this out? Sounds a bit like the fundamental interconnectedness of all things..... Douglas Adams was a genius.

There was actually an interesting Star Trek Voyager episode related to this topic.  While visiting a very advanced race (who never heard music) the emergency EMH doctor was overheard singing opera to himself, which absolutely enthralled one of the Aliens who – as he was listening – heard underlying mathematical equations as much as the music itself.  The doctor was soon invited to sing for the entire planet and they even constructed an entire opera house for him.  They also invited him to stay.  He really did give grand – heartfelt performances.  But Voyager had to leave and the doctor was going, too.  So, this advanced culture created their own look-alike doctor to sing opera in his absence.  They claimed he would be even better than the original doctor.  The stage was set for this new doctor’s opening performance - done in honor of the departing doctor.  But as he began, something was decidedly wrong and different.  The new doctor was singing in mathematical tones, which sounded more like the ending of Close Encounters of the Third Kind than anything we’d recognize.  It was a very cold performance as well.  Voyager’s doctor was completely dismayed – especially with how the crowd seemed to love this new doctor way more than his performances.  Had this alien race gained nothing from his efforts?  He confronted the creator of this new synthesized doctor and passionately insisted that music was much more than equations, it was an expression of passion and raw emotion as well – something that obviously did not sink in…  

I certainly hope that computers will gain consciousness and perhaps even show emotions one day!  I think this would be a great achievement!  How we ourselves view music may be entirely unique to the way in which other intelligent beings (and perhaps other intelligent computers) would.  I guess the question would be, can we give a computer our own unique love for music – perhaps by inputting all past music?  Maybe!

This is like trying to find recurring patterns in pi. The theory might hold in the short term, but is completely clueless in the long term

The above arguments about human vs. computer music miss the point of art entirely.  It is not about saying one is better than the other.  Art is about presenting something for others to experience.  It is up to every individual to determine what they like and don't like.  I don't care if everyone says Bach was a genius, if I don't like it, I don't like it, so who cares.

These arrogant commentators on what is considered art, are the ones who need to chill out and just sit back and let art exist on its own merits, regardless of the source.

Guess what - we may find out one day that we humans are really just pre-programmed biological computers that are built by someone else in the universe.  We might be nothing more than fancy Commodore 64s.

I marvel at the postioning of these previous comments. It truly is a expression of music and interpretation. A measurement of varied intellects. Bravo to all.
As we advance technologically the inquiring mind strives for answers. Computers allow us to delve deeper.
This inspirational theroy can breed a whole new understanding towards music and should be positively observed regardless of personal beliefs!
The mere implications of a formula that can better inform the composer or audience of the techincal structure can be a valuble tool.
If we can now discern  basic composition of a score and measure the results of the impact on the listener we can truly bridge a gap towards universal connection and DNA  of music.
As a movement fills the room of people, emotional responses are created. It would be beautiful to coincide this structural formula and marry it to the other aspects of a show.  
This may seem a deep thought but the since music is an expression of energy with great intrigue should we embrace and analyze new compositional techniques. It may only fortify our impact on all.

lorenzo-Ikatchers

It might help people having a hard time understanding String Theory to watch the movie, "The Fifth Element".  Watch and listen to the blue woman singing opera. I hope that science and art can work together before it's to late! But, if everthing is just negative and positive vibrations it will never be too late.  Our past and future could be just one of our creators records going around in circles, skipping off track once and awhile...Or a alien dog's sneeze for that matter!  Love life and take time to smell the roses.  Sometimes the smallest things in life are the most important.

Interesting. It would be cool if somehow, one day, computers could compose music as amazing a Bach or Chopin or some of the other great classical composers, but using modern age instruments, such as synthesizer effects and nonacoustic instruments and such (like the music on the new version of Peter Pan - not the animated one. That movie had some really cool music because it was amazingly detailed, yet, a lot of the instruments were digital or effects off that could be derived using an accoustic instrument). I guess those would be called "New Age" instruments, I'm not sure....Anyways, a very interesting article. I couldn't view the QuickTime videos though, so I think I'm missing a lot.

It is interesting that while this topic was hot, the discussion was an age old one about the tension between humans and their technology. What I find most useful about the work is the new space in which we can visualize music. The musical score, as does the MIDI sequencer, graphs the pitch but separates in physical space by timbre, and it is unable to show pitch class relationships. Tymoczko's graph is able to overcome the problem of visualizing multiple pitch class relationships. It is at least a new and potentially useful way to visually represent music in order to gain a deeper understanding of its structure (over time). Until now our visual representions have been limited to the Cartesian plane, the circle of chromatic pitches, and the circle of 5ths, (along with some very creative but long disused visualiztions of the Medieval, Renaissance, and Baroque eras). As some have stated, it may not show us anything new; but it does allow us to see what we know more clearly.

Alan, for a completely different theory of music, you might be interested to look at http://whatismusic.info, which explains my super-stimulus theory of music - that music is a super-stimulus for the perception of musicality, where "musicality" is actually a hitherto unknown aspect of speech perception, which corresponds to the perception of some aspect of the internal mental state of the speaker (probably the speaker's current level of "consciousness" in relation to the significance and profundity of what they are saying).

According to my theory, all the different aspects of music arise from maximisation of one simple function of patterns of neural activity, with different aspects corresponding to maximisation of the same function across different parts of the brain which represent different aspects of information about speech (especially speech prosody and rhythm, but probably other aspects as well).

My theory is the first theory which potentially explains *all* aspects of music as they occur across *all* genres (but this ultimately depends on being able to describe the representations of information in all relevant cortical maps). At the same time the theory provides an account of music perception as an adaptation, because the perception of mental state assists in the comprehension of the significance of speech, and this theory of adaptation is not circular, i.e. it does not explain music in terms of music perception and music perception in terms of music.

hi

im a 19 years old student of Persian Gulf University of Bushehr,Iran
we are doing a research about geometry & music.

as a matter of fact we used the (the geometry of music) article in our research.but we couldnt understand what orbifold is or even its definition.

now we need a brief and explicit explanation about it.

thanks alot.

Hi, Roham: In the Web page I linked to, an orbifold is defined thusly:

"The orbifold of a symmetric surface is obtained by regarding as identical all the points of a surface that look alike."

As I understand it, you can use this example: Let's say you have a pattern that you can break up into identical symmetrical tiles, like a patterned bathroom tile floor (or a solid-colored floor, for that matter). The orbifold would be a geometrical description of that tile. The single tile serves as the "definition" for the elements of a whole symmetry group. That is, the entire bathroom floor.

In the same way, the structure of a musical piece could be described by the symmetrical use of a smaller element ... the orbifold. In the musical case, it's more a question of mathematics describing the chord progression than a geometrical pattern.

I hope mathematicians (or musicians) will feel free to correct or extend this explanation.

It seems to me that most of the comments here are just slightly off the mark in that music should be taken to mean the mathematics of musical harmonics, not the mathematics of any particular song. In physics any existing complex physical structure, atom, rock planet, etc may be taken to correspond to a particular combination of notes, a song or melody, but the physics, referring to the mathematical structures that underlie the existence of matter and its characteristics, correspond to tuning theory; specifically to the basic theory of proportions that follows from non-equal temperament tunings (so that would be pretty much anything pre-Bach).

In purely mathematical terms harmonics theory is a self-referential fractal geometry that is mathematically very similar to many concepts found in modern physics.

Pythagoras, by the way, was not the first to discover the relationship between geometry and mathematical harmonics. Most of this material can be found as far back a Sumeria and Babylon, which is where Pythagoras claimed to have learned it. Both Pythagoras and the Sumerians/Babylonians where working with tuning theory, not musical composition.

Just some observations.  I've arranged from scratch 126 public domain melodies for easy piano and posted the music free of charge to my website at easybyte.org.

Chords on my music editor (with perfect intonation) can sound different, sometimes a lot, when played on my electronic keyboard where all the notes are  slightly out of tune (tempered).  In particular, how  notes decay and faint harmonics interact have an enormous influence on the final esthetic quality.

Example, Beethoven's Moonlight Sonata first movement.  The right hand notes HAVE to decay like a guitar, otherwise the minor second chords sound unpleasantly dissonant.  I've heard it transcribed for organ, and it sounds awful.

If anyone has heard orchestral music transcribed for MIDI, you know MIDI just can't duplicate string sounds.  A string section has lots of instruments, all playing not exactly in tune, with vibrato, that creates a very complex signature sound.

One final caveat.  Chords are not played in isolation in music, and what sounds good depends on the preceding chords, chords that follow, tempo, instrumentation, etc.

So mapping chords in hyperspace might be interesting, but has little utility when creating actual music.

(And "music" in this sense is something that people genuinely prefer to listen, not the avant garde "noise" that people experiment with in music theory.  Avant garde is about as entertaining as listening to static on a mis-tuned radio or watching snow on a television set.)

Here we're up against the difficulty in communicating mathematics to laymen.  Most people don't see the value in creating a geometric structure to represent something, though they might be perfectly willing to applaud some new prediction or tangible result based on the structure.  The result is often no more interesting than the structure, to someone with a moderate appreciation of higher mathematics, especially higher geometry.

I'm particularly sorry to read the dismay that is clearly present in responses from musicians, which manifests itself as scorn.  While Messrs Timoczko and Goggins probably have no intention of stealing the few opportunities that working composers still enjoy today, they understandably feel threatened.  Musicians have always been jealous of their territories, feeling (often with justification) that only they understand what they do.  However, knowing that Pythagoras investigated the mathematics of music is not a basis for saying that another approach might be more fruitful.  Trying to keep mathematicians out of music is like trying to keep moths from the flame: if there is mathematics there, that's where mathematicians will want to go, like Sir Edmund Hillary!

This is not to say that this particular representation of a piece of music, and the conclusions drawn from the instances, will be particularly interesting.  I look forward to reading the article, but I'm reacting to the reactions above, before I forget!

I think I'm in love! I have wandered over the web, reading news and comments in many locations, but never, until now, have I come across a truly intelligent and fascinating group of comments, sent by obviously intelligent humans. Language is my metier, but I am also a musician and an artist. I have taught English, sign language, piano music and art and also engage in those disciplines with energy and compulsion. After my 21 year-old daughter suffered a damaging stroke, I was fascinated by the terrible damage it inflicted on her memory -- she lost her English but remembered German (she was fluent in it) and sign language (also fluent in that, having learned it from me). As she worked in rehab to regain her abilities in math and language, I worked with her to help her find ways back to the understandings she previously held. What worked best in her recovery? You guessed it -- music. Music indeed has charms, not only to "sooth the savage beast" but also to heal the damaged human. Music is the language of the universe, as is mathematics. Thank you for letting me eavesdrop on the music you all have made.

I'm interested in music savants and their take on the mathematical idea that, paraphrasing, "...life is but a song.".  The idea that the smallest pieces of which the parts of the universe are made (One can think of them as something like strings)vibrate...different vibration equals different wave or particle or beyond both and neither. Collections of differently vibrating strings are what literally matter...actual creation of the music or the mathematics of matter.  So, if anyone knows a musical savant, I'd like to ask her or him some questions about the mathematical equation that proves the unity of the micro and macrocosmic worlds.

The major issue with this research and theory is that it is based on "sound" and the mathematical equivilency of vibrations to numbers. Sound is physical. Music is metaphysical. Sound is just the vehicle or body of Music, not the soul or even its essence. Any geometric representation of "music" must be metaphysically based, not physically evolved.

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