Math & Music: The Severed Alliance. Some Recent Academic Approaches (Do Not Read If You Hate Drums)

I recently came upon a bunch of interesting academic papers by a guy named Godfried T. Toussaint , including this one

"The Rhythm that Conquered the World: What Makes a “Good” Rhythm Good?"
http://www-cgrl.cs.mcgill.ca/~godfried/publications/Percussive-Notes-Web.pdf,

which use various metrics to analyze and catalog various rhythms in order to identify what characteristics make them durable and popular, especially the one rhythm known as the Bo Diddley Beat or the son clave, in addition to other names it might have. I posted links to this paper and some others on another thread and then somebody reposted on yet another but I kind of thought maybe this topic should have its own thread. Included in most of these papers are a bunch of cool looking symmetric diagrams that illustrate the rhythmic timeline or, as his book is called The Geometry of Musical Rhythm. I also came across a recent book by another guy named Dmitri Tymoczko who has being doing some research using mathematical techniques to analyze harmony, which features some similar circular diagrams, only in pitch space as he called it. It turns out that these two guys run seminars together and proof read each other's work, if not actually coauthoring papers together. Meanwhile last year in The Best Writing on Mathematics 2012 I read this article

"Can One Hear the Sound of a Theorem" by Rob Schneiderman
http://www.ams.org/notices/201107/rtx110700929p.pdf

In which the author, a mathematics researcher and practicing jazz pianist, argues something to the affect that most mathematics about music ends up amounting to so many blinking lights that give some sort of superficial illustration but don't really get at anything deep or profound. In particular, he criticizes Tymoczko's work, and there was brief exchange of letters between the two of them here: http://www.ams.org/notices/201110/rtx111001408p.pdf

The purpose of this thread is to discuss this set of ideas in particular, but also any other related stuff you may find interesting or relevant. My point of view starts from one of sympathy with Schneiderman, but I find the Toussaint stuff interesting, and the math starts out simple enough and the basic conclusions are kind of hard to argue with and not too overreaching. Tymoczko is a smart guy with all kind of kinds of credentials as an academic and a composer but seems to me that the math gets kind of overly complicated pretty quickly. OK, I'm running out of space in this tiny box, end of introductory remarks.

Seems to me that after skimming his book Tymoczko is trying to extend certain ideas of classical harmony to have a general theory of What Makes Music Tonal? - he is careful to avoid the Geirbait and notes that he also likes atonal music and is not making a value judgement. To this end he introduces the idea of "efficient voice leading" which takes places, ultimately in a four dimensional space- hence one section is entitled "Chopin's Tesseract." Most notably he analyzes certain famous passages in the work of Wagner that confounded classical theorists and says something to the effect that "yes, this voice leading is inefficient, but, subject to a certain constraint, it turns out to be efficient after all.

His project might be useful and he has some smart folks in his corner- glowing blurb from Vijay Iyer, anecdote about exchanging letters with Noam Elkies, but from this vantage point it seems that using a tessaract to illustrate voice leading is overkill, especially since there are already some other tools for this- the piano keyboard, standard musical notation and, as cryptic as it may seem, the guitar fretboard.

The rhythmic stuff in the other guy's work seems to me to have been less combed over over the centuries and appeals to some obvious points
1) A good rhythm is almost symmetric. We like the symmetry and the slight asymmetry makes for the spice
2) A good rhythm will be based on a cross-beat and will fake you out slightly about the time signature a little bit before reaffirming it, so the "winner" in the rhythm sweepstakes is the one that has its first three beats sounding like one is going to be in a treble meter, but then the thing turns out to be four.

Note that I don't think reading this stuff will make you a better musician, at least to first order. If you want to play Afro-Cuban music, it probably won't help you too much to study the mathematical properties of the clave. But it might help you in the second order, it might satisfy some mental itch to think about these things abstractly thereby freeing your mind so you can get down to the real business of memorizing the various interlocking rhythmic patterns that imply the clave.

https://www.youtube.com/watch?v=kn0pYn4h4n4

One thing Dmitri T is drawing on and then deviating from are the ideas of Heinrich Schenker, who thought that in 1722 a wrong turn was taken when Rameau published his Treatise on Harmony, creating the major/minor system and codifying what became the standard way of thinking about harmony for the next two centuries, overshadowing the contrapuntal ideas of Bach as exemplified in the first volume of The Well-Tempered Clavier, published that same year. I guess the tesseract is supposed to create a "big tent" in which all the harmonic ideas can live.

OK, I am typing on fumes, hoping someone who has some formal training in these things will show up and correct my errors.

Really I just created this thread because I couldn't get the title out of my head.

Just remembered it works better when the two terms are switched

But, getting back to the topic at hand, let's not forget the work of Junior Dad C.P.E. Bach

fuck drums shut this thread down

making a change.org petition about this how dare you

At last. A thoughtful respondent has arrived!

bo diddley beat = son clave seems a little superficial to me.

Well, there's more to both then just those five beats. There's everything else that goes with.

You coming to the ACPT this weekend, zachlyon?

right, and in context, the function of those five beats in the bo diddley beat and in the clave is completely different

i've abandoned the crossword thread ;_; xp

Title of the thread says math and music. Don't see the word philosophy, Hurting.

I mean the two are the same to the extent that those five beats are the same, once you abstract away everything else. The musical situations they appear in are quite different. Are you saying that to identify them at any level is useless, trivial, misleading, challops bait?

it just sounds a little
http://www.mnn.com/sites/default/files/user-59/hurricane-sandy-golden-spiral.jpg

DO U SEE?

Perhaps you would prefer it be called by the name al-thaqil al-awwal

i think the bo diddley beat definitely functions as a clave in the same way that son does (just in a much more overt, less subtle way). haven't read the article but i had a conversation with a drummer friend recently about this exactly.

i'm endlessly fascinated by how claves work in different music, especially the one bar clave (dotted quarter + dotted quarter + quarter, depending on how you're counting) and the ones that are basically five dotted 8th notes and can start either on the downbeat, the 'e' of 1, or the 'and' of one. i feel like people don't talk about these rhythms as claves like they do son or rhumba but it's so important to new orleans music, electronic music, brazilian drumming, everything.

Yeah, that first one, which is basically the baião or tumbao bass pattern to name two among its many uses. Before I looked at any of those papers I was thinking that it was the simplest thing you could do to count three against a duple meter. When you get to the next bar, you could keep the triple meter thing going and hit on the and of one, which is kind of an interesting exercise, but it is more interesting to either restart or "answer" the pattern. Or you could even do a Lalo Schifrin and take it into Mission Impossible territory with 5/4.

I like the circular diagrams because that's how I think about rhythms a lot of the time. And also it means you can render the "Bela Lugosi's Dead" rhythm as a pentagram.

I think I like that particular rhythm

o--o--o---o--o--

because it's basically the result of playing 5 beats over 4 quarter notes, but quantized to sixteenth notes. very symmetrical.

I guess that's exactly what that pdf says, now that I'm actually reading it.

Never thought about circles myself until I saw those diagrams but it seems like it might be useful

I guess I'm just not fond of the approach here of assuming some kind of genetically "good" rhythm based on the fact that a certain 3/2 pattern can be sussed out in various kinds of music. Is there data that this rhythm is actually more popular than other rhythms? Isn't it more likely the result of common ancestry?

Also it starts from a very western-centric way of reducing the rhythm to where it falls in a set of 4/4 bars. The "essence" of the beat is treated as the particular 3/2 pattern as it can be translated into western notation, not the accenting, the feel, etc. I think to a person who wasn't familiar with the concepts, a son rhythm and the bo diddley beat would not sound very similar.

Well, what about
Shave and a haircut

Six bits

"Shave and a haircut" isn't the same rhythm unless you put an accent over the "a" between and and haircut.

It's also a little like that Joseph Campbell Hero With a Thousand Faces stuff -- if you strip away all the details and fudge things a little, all these stories are the same!

which is a useful & legitimate way to look at music

not to mention fun. It's not like dogma is being prescribed here or anything.

Isn't it more likely the result of common ancestry?

totally, and of course the details are everything, but i'm also interested in thinking back to the earliest development of common rhythms like these...like, what makes a rhythm appealing and useful? why that rhythm in the first place?

I've been listening to Beethoven symphonies a lot lately actually and was thinking about how "square" (not necessarily in a bad way) the rhythms often are, and I was thinking about how unlike the rhythms of most of the world's other cultures these rhythms are, although they still tend to be a westerner's frame of reference for interpreting other culture's rhythms (even the fact that we think of a son clave as "3/2" is evidence of this.) I'm guessing there's some explanation that stems from the invention of the clock and from militarism (marching rhythms).

western classical music is very non-metronomic

There is a reason for that guy in the front with the baton. Gotta love that rubato.

Seriously. Been going to the opera again this season and there is a lot more rhythmic variety, even within one work, than one stereotypically thinks.

western classical music is very non-metronomic

― death and darkness and other night kinda shit (crüt), Monday, March 3, 2014 4:14 PM Bookmark Flag Post Permalink

I would say that's an inaccurate generalization as much as it would be a generalization to call it metronomic. However I was more talking about the fact that Beethoven is very downbeat heavy and not very "syncopated" (again, a western-centric word).

There is certainly "rhythmic variety" in western classical music, but it is variety that centers around a certain understanding of strong and weak beats, and deviation from that is generally understood as deviation with that as the implied center, at least up to a certain point in history.

Fair enough. I remember your man Han Bennink, whom I went to see at Tonic, but I believe you missed, saying something to the effect that in Europe drumming survived in music in the military, marching bands.Ah, yes

a long time ago someone told me about the book that Bob Moses wrote, and apparently it talks about how each beat of the measure has it's own emotion (especially in terms of feeling more or less stable, wanting to resolve, etc), the same as degrees of a scale. i never read it but i think about this all the time.

Yeah, me too, lately.

and here's a pdf (the book's out of print)! page 12.

I've had some interesting conversations about this kind of thing recently, perhaps I will provide you with some cherce nuggets

western-centric way of reducing the rhythm to where it falls in a set of 4/4 bars. The "essence" of the beat is treated as the particular 3/2 pattern as it can be translated into western notation

Also, I read somewhere recently, can't remember where, that in Africa, in the transcription of African drumming, they are aware, like everybody else, of the limits of Western musical notation, but it is still used. Guido d'Arezzo built to last.

"Inside a Master Drummer's Mind: A Quantitative Theory of Structures in African Music," by Willie Anku
http://www.redalyc.org/pdf/822/82201105.pdf

heh. i do like that i've never heard anyone say the word clave in new orleans, it just...is. aural tradition for sure.

Be careful what you whistle in Mexico: http://www.furious.com/perfect/bodiddleybeat.html

Hurting's initial approach reminds me of discussion long ago I watched between a relative and a friend, who are born exactly one day apart, in which, surprise, both of their most annoying qualities were at the forefront. My relative is one of those guys who learns some kind of fact a little after other people, like me, and then enlightens the savages with it, or his brother put it "educates another partygoer unto the magicks of science." My friend is one of those people who doesn't like to be pinned down or come clean, when he's going to make a decision tells you every reason why he shouldn't do it and then does it anyway, leaving you holding the bag in some kind of way, and finally, as is relevant to this discussion, in any aesthetic discussion is prone you make all the propositions and then shoot down what you say as being too limiting, missing the mark, too literal-minded, always ending up with the question "But don't you think there is something more?" What that more is however, he ain't saying (I may allow you to guess what professions these two ended up in)

Anyway, now that I've dispensed with these initial ad hominem advisories, the story was that the one guy said "did you now that all sound can be broken down into a series of overtones and that this completely describes what is going on?" The other guy got annoyed and said "But when Miles Davis is playing a melody or a solo is that really all there is, I don't think that can be the whole story?" First guy: "Yes, that's all there is to it." Second guy: "But don't you think there is something more?" Rinse and Repeat.

I was annoyed by the two of them because it seemed to me that basically they wanted to have an argument, they didn't want to see the merit in what the other guy was saying. Both ways of describing something can be valid if used appropriately, both can have explanatory power if applied correctly, one doesn't rule out the other one, unless you really feel you have to choose a side. Years later I asked the second guy, "Do you remember that conversation?" and, as Jordan said earlier he replied "I think about it all the time!"

More Songs About Chocolate and Girls Songs About Buildings and Food links from the other thread:

That guy has lots of papers with those kinds of diagrams.
A Mathematical Analysis of African, Brazilian, and Cuban Clave Rhythms: http://cgm.cs.mcgill.ca/~godfried/publications/clave.pdf
El Compas Flamenco: A Phylogenetic Analysis http://cgm.cs.mcgill.ca/~godfried/publications/winfield.pdf
The Euclidean Algorithm Generates Traditional Musical Rhythms http://archive.bridgesmathart.org/2005/bridges2005-47.pdf

― In Walked Sho-Bud (James Redd and the Blecchs), Friday, February 14, 2014 5:30 PM (2 weeks ago)

Did we link this paper yet? http://poisson.phc.unipi.it/~fidanza/matemusica/papers/Toussaint%20-%20The%20Geometry%20of%20Musical%20Rhythm.pdf

― In Walked Sho-Bud (James Redd and the Blecchs), Saturday, February 15, 2014 2:29 PM (2 weeks ago) Bookmark Flag Post Permalink

This review seems to get it just about right: http://mtosmt.org/issues/mto.13.19.2/mto.13.19.2.gotham.php

― In Walked Sho-Bud (James Redd and the Blecchs), Saturday, February 15, 2014 2:43 PM (2 weeks ago) Bookmark Flag Post Permalink

http://www.mtosmt.org/issues/mto.00.6.1/mto.00.6.1.anku_frames.html

― In Walked Sho-Bud (James Redd and the Blecchs), Sunday, February 16, 2014 10:21 AM (2 weeks ago)

that mind of the master drummer article is cool, I need to give that one some time

Hm. One of those links seems to break Zing

And now most have these have gone off the net because of crüt

what did I do!!!

Sorry, j/k. Didn't really think it was you, you just had the right vowel sound in your name to complete the expression.

Here is review of the Tymoczko book in case you are interested: http://www.mtosmt.org/issues/mto.11.17.3/mto.11.17.3.hook.html

Ah, it's been a while since I read anything in this vein! I'm both interested and sceptical.

anything in this vein

meaning "mathematical modelling of pitch", which used to be a major area of interest for me, not what Tymoczko himself is doing specifically, which is probably fairly distinctive.

Ah, it's been a while since I read anything in this vein! I'm both interested and sceptical.

― EveningStar (Sund4r), Tuesday, March 11, 2014 11:46 PM (Yesterday) Bookmark Flag Post Permalink

My understanding is most of the history of math as applied to music up to and even after equal temperament had to do with finding various rational numbers to divide up the scale pleasingly. I believe Galileo's father had a big success in this. Then every once in a while somebody throws some group theory at musical structures such as guy named Riemann -Hugo, not Bernhard, but I don't know too much about this

Actually the book Musimathics: The Matematical Foundations of Music, by Gareth Loy seems to be an interesting overview of a bunch of different topics, moving from equal-temperament and just intonation through the physics of sound, along with psychoacoustics and acoustics, finally ending up at various methods of composition, stochastic and aleatory.

Not a lot of posters on this thread but those that are are choice.

Anyway feel like the approach taken in that one book of building up a big theory and just plucking the low hanging fruit of the chord sequences from the theory tree is kind of not quite the most useful. More interested in simple ways to enumerate or generate the various facts related to the usual scale suspects.

Anyway the other thread reminded that I found an interesting if verging on eccentric book about
modal music that I can link to. I think one of the authors has a technical degree but I am tempted to say in this case it falls more under the rubric of polymath and music. By the other thread I mean worst music writing of course.

http://www.modalogy.net/

Feel bad for saying "verging on eccentric." Should have said "original" instead.

I want to post in this thread but want to read more Tymoczko first.

Loved the part in the MTO review of his book that said "if it seems that Tymoczko may overestimate his radicalism, it is not because he is doing so in the spirit of self-promotion, but because the bar for radicalism is set so low"

Schneidermann typed:

Three overlapping goals of music theory are to explain why music sounds the way it does, find good ways to listen to music, and describe how to create music.

I am 100% on board with #1. Of #3 I think "but this only describes how to create specific kinds of music, I mean, it's useful for film composers I guess? but otherwise no?" Of #2 I think "ohhhh I get it, you're an asshole!!"

Not Schneidermann specifically, I agree with that guy, I'm referring to that notion

Just read his section addressing that second point which was pretty otm.

Yeah I like that guy, I like this thread! I gotta read this Tymoczko book

Better you than me ;)

Seriously the part I read was kind of interesting but I didn't want to follow him all the way into Chopin's Tesseract. I think the Modalogy's Chromatic Cube concept might be more useful.

http://www.modalogy.net/ask.html

I've been listening to Beethoven symphonies a lot lately actually and was thinking about how "square" (not necessarily in a bad way) the rhythms often are, and I was thinking about how unlike the rhythms of most of the world's other cultures these rhythms are, although they still tend to be a westerner's frame of reference for interpreting other culture's rhythms (even the fact that we think of a son clave as "3/2" is evidence of this.) I'm guessing there's some explanation that stems from the invention of the clock and from militarism (marching rhythms).

― james franco tur(oll)ing test (Hurting 2), Monday, March 3, 2014 4:11 PM (4 months ago) Bookmark Flag Post Permalink

http://www.smithsonianmag.com/smart-news/was-beethovens-metronome-wrong-9140958/
http://www.ams.org/notices/201309/rnoti-p1146.pdf

https://www.youtube.com/watch?v=wvJAgrUBF4w

http://blogs.scientificamerican.com/but-seriously/2013/06/13/chladni-figures-amazing-resonance-experiment/

Which links to http://skullsinthestars.com/2013/05/02/physics-demonstrations-chladni-patterns/

Just was in contact with a guy who played on Dmitri T's record but he didn't have to read the book first.

http://www.independent.co.uk/arts-entertainment/classical/features/the-enduring-myth-of-music-and-maths-2307387.html

Exactly

http://docstudio.tvo.org/sites/default/files/imagecache/masterseries_620x350/The-Man-Who-Saved-Geometry%5B1%5D_0.jpg

^H.S.M or Harold Scott MacDonald "Donald" Coxeter, musically inclined mathematician, Bourbaki antagonist and King of Infinite Space.

http://www.amazon.com/Music-Mathematics-From-Pythagoras-Fractals/dp/0199298939
Know nothing about this book except I have enjoyed one of the editors writings on graph theory.

http://galileospendulum.org/2012/06/29/galileo-was-a-kickin-lute-player/

(Warning, considered posting on worst writing about music thread)

Could have sworn one dj on WKCR Bachfest was introduced as Terry Tao.

Right now, listening to Rob Schneiderman playing with his old buddy Brian Lynch's Unsung Heroes. Pretty good.

https://www.youtube.com/watch?v=9MzKx0fKg5o

Had this discussion the other day and it almost got a little heated.