Group Theory And Music

http://bookboon.com/en/textbooks/mathematics/an-introduction-to-group-theory

 

I came across this book randomly and I'm sure someone may be interested in this.  To be honest, I haven't read the first 3 chapters that relate to group theory, so I'm unsure if the concept is self-contained or if it assumes knowledge regarding abstract algebra already.  I did a quick read through the last chapter, where the music theory is presented.  If a person is interested in how group theory can be used to describe musical relations, then it's a fairly interesting and straight forward read.  I think, if someone here has read or studied abstract algebra and had a hard time understanding typical "concrete" examples, but understands music, then this book can be useful.

 

The topics presented in the book are rather basic when it comes to music, but it should be fairly obvious on how to extend the concepts to more complex problems.  If anyone else is familiar with the subject at say the typical undergraduate level care to weigh in and give their opinion on the matter that would be appreciated.  I may be interested in teaching a summer course on this, but I'm unsure if the book is suitable or the material really that interesting.

Thanks for posting this, I am taking Group Theory next year and have already seen some concepts.
I'd be interested in looking at maths and music for a future project at university, so I will have a look at this book sometime.

 I may be interested in teaching a summer course on this, but I'm unsure if the book is suitable or the material really that interesting.

Is there any nontrivial application to music in that book? I dunno. Not gonna read the whole thing just to find out though

Meh, but thanks.

Is there any nontrivial application to music in that book? I dunno. Not gonna read the whole thing just to find out though

Depends on how you define trivial.  If applying groups to musical theory is obvious, then no.  If the term isomorphism is new to you, then yes. 

Depends on how you define trivial.  If applying groups to musical theory is obvious, then no.  If the term isomorphism is new to you, then yes. 

Just read a book on group theory, then. Otherwise just go for the Mazzola.  

I'm utterly confused as to what you're trying to get at.  So i'll just eat a bagel.

I'm utterly confused as to what you're trying to get at.  So i'll just eat a bagel.

http://books.google.ca/books/about/The_Topos_of_Music.html?id=6I9U9-Rls8oC

 

The ne plus ultra of math-music. It's really hard to tell if it's BS or not, and I have a math degree

From what I can tell, the book is riddled with Topos theory.  A subfield of category theory that I'm not particularly strong and coupled with the fact  that i'm also not particularly intuitive on topology makes this book fairly hard for me to glance through.  So, I don't think I can decide how much of it is an acceptable use or a common use for said theories, especially regarding mathematics.  Nevertheless, it seems more like a reference than something to learn from and I'm sure the small group of people who can comprehend the music theory and mathematics will have divergent opinions on the merit.

I can't comment on this book because my math skills just aren't good enough to understand it, but I do know that there are plenty of simpler relationships between math and music that are often overlooked by composers that could potentially have a huge impact on their musical thought. I think that these should be focused on before we start trying to bring group theory into the equation. Books such as this one, that claim to shed light on some incredible new relationship between math and music, always reek of vanity publishing to me. 

I think you misunderstand the intent of the book.  It's a book to learn group theory with application to music theory.  I don't believe the author(s) claim that what they are doing is making new and insightful relationship, but rather attempting to show how a person can use group theory to formulate common properties within music.  Truth be told, it is already well established that music theory can be described by abstract algebra and combinatorial analysis.  The key word is describe, just as differential equations describe F = ma.  The relation is one thing, the ability to use it for your own purposes is another thing entirely, but first a person must be aware of the relation. 

I think you misunderstand the intent of the book.  It's a book to learn group theory with application to music theory.  I don't believe the author(s) claim that what they are doing is making new and insightful relationship, but rather attempting to show how a person can use group theory to formulate common properties within music.  Truth be told, it is already well established that music theory can be described by abstract algebra and combinatorial analysis.  The key word is describe, just as differential equations describe F = ma.  The relation is one thing, the ability to use it for your own purposes is another thing entirely, but first a person must be aware of the relation. 

 

But this is kind of my point. You can't really use mathematics to describe music on any meaningful level. The only thing you can do with it is label certain musical phenomena. This is especially true of the abstract algebra approach to music analysis. If you've ever read Allen Forte's The Structure of Atonal Music, you'll know what I'm talking about. He may be likely to say something along the lines of "Pitch class set A and pitch class set B are transpositionally related. Therefore B=T(A,11). Several pitch classes are present in both A and B. Therefore A and B contain the invariant subset C. C=·(A+B)." 

 

The observation may be true but it isn't really saying what the musical purpose of such a device is. It's the musical-mathematical equivalent of pointing to a chair and saying "that is a chair" but being unable to explain what the chair is used for. I'm sure this kind of analysis is useful to a small niche of people and I don't really want to knock it but you've posted this on a forum for composers and my honest opinion is that a book using group theory to attempt to describe musical phenomena is going to be of limited use to most composers in the same way that most mathematical descriptions of music are of limited use to most composers.

If that's your point, it is a fairly basic and naïve point.  So with that said, I'll choose to disengage from this line of dialog because it's apparent that you and I are approaching these ideas from different backgrounds.  I simply do find it meaningful to point a basic chair and call it a chair and rhen see what else is true about the chair by looking at the structure and hopefully finding more underlying structures, ie it has legs or chairs made out of wood are more common than chairs made out of plastic.  While, I can see why, on a purely musical creational aspect, that can be deemed as meaningless, I think on a more intellectual level, as in a level of knowing something for the sake of knowing it, it's rather meaningful.  Again, different perspectives. 

 

However, I also suspect you didn't actually read chapter four.  If the bijective relations to you was clearly obvious and the isomorphism are trivial consequences, then  my hat is off to you.  Keep in mind, this book is a MATH book that talks about how math can be used in music, NOT a music book that uses math. 

If that's your point, it is a fairly basic and naïve point.  So with that said, I'll choose to disengage from this line of dialog because it's apparent that you and I are approaching these ideas from different backgrounds.  I simply do find it meaningful to point a basic chair and call it a chair and rhen see what else is true about the chair by looking at the structure and hopefully finding more underlying structures, ie it has legs or chairs made out of wood are more common than chairs made out of plastic.  While, I can see why, on a purely musical creational aspect, that can be deemed as meaningless, I think on a more intellectual level, as in a level of knowing something for the sake of knowing it, it's rather meaningful.  Again, different perspectives. 

 

However, I also suspect you didn't actually read chapter four.  If the bijective relations to you was clearly obvious and the isomorphism are trivial consequences, then  my hat is off to you.  Keep in mind, this book is a MATH book that talks about how math can be used in music, NOT a music book that uses math. 

 

 

You asked if the material is interesting; I told you why it isn't interesting.

 

You also misunderstood my chair analogy or you consciously chose to confirm the point I was making with it.

I was hoping to hear from someone who actually studied math or at the very least could prove the FTA.  Clearly, this book presents no real value or of interest to a composer for music sake.  I was hoping that would be fairly obvious.

I was hoping to hear from someone who actually studied math or at the very least could prove the FTA.  Clearly, this book presents no real value or of interest to a composer for music sake.  I was hoping that would be fairly obvious.

 

 

Yet you posted it in the section of the forums entitled Composer's Headquarters...

I know how to do long division, does that count?

Oddly enough....Euclidean algorithm is rather useful in abstract algebra.

You're wasting your time on him, Ralph.

But this is kind of my point. You can't really use mathematics to describe music on any meaningful level. The only thing you can do with it is label certain musical phenomena. This is especially true of the abstract algebra approach to music analysis. If you've ever read Allen Forte's The Structure of Atonal Music, you'll know what I'm talking about. He may be likely to say something along the lines of "Pitch class set A and pitch class set B are transpositionally related. Therefore B=T(A,11). Several pitch classes are present in both A and B. Therefore A and B contain the invariant subset C. C=·(A+B)." 

 

The observation may be true but it isn't really saying what the musical purpose of such a device is. It's the musical-mathematical equivalent of pointing to a chair and saying "that is a chair" but being unable to explain what the chair is used for. I'm sure this kind of analysis is useful to a small niche of people and I don't really want to knock it but you've posted this on a forum for composers and my honest opinion is that a book using group theory to attempt to describe musical phenomena is going to be of limited use to most composers in the same way that most mathematical descriptions of music are of limited use to most composers.

 

Perhaps if you understood the theory better you would be able to infer musical purpose from it. Being familiar enough with pitch sets I can tell you that a set of information similar to the example you so effortlessly write off provides me with a wealth of knowledge concerning the musical purpose of said pitch sets, especially given a context in which to view it.

 

Of course you're going to somehow say I'm wrong, I didn't listen to what you had to say, or some nonsense and then continue to spout more nonsense. So go ahead.

Interesting, The mathetical side of music. I have a better ide: learn acustics(the study of sound)  I learned it and it was rather impressive but hard. Higher level stuff. Tunings and ect. True, this book may offer cool insights but it may not offer acoustics. :)

Perhaps if you understood the theory better you would be able to infer musical purpose from it. Being familiar enough with pitch sets I can tell you that a set of information similar to the example you so effortlessly write off provides me with a wealth of knowledge concerning the musical purpose of said pitch sets, especially given a context in which to view it.

 

Of course you're going to somehow say I'm wrong, I didn't listen to what you had to say, or some nonsense and then continue to spout more nonsense. So go ahead.

 

Fair enough, if you want to pretend that transpositional invariance is worth talking about in any composition then go ahead. I'm not going to lie to myself and pretend that it is.

 

The context I was referring to with that example was mm. 1-2 of the 2nd of Stravinsky's 3 pieces for String Quartet where there are 2 chords split across the barline that are both transpositions of pc 4-8, A being T=4 and B being T=3 (Hence the T=11 between them). While you were busy pretending that set theory is useful, the rest of us had already noted that A was a transposition of B and that the two chords had 2 notes in common and then wondered whether this may be something which is likely to recur throughout the movement and we did it without resorting to mathematics or pretentiousness.

 

 

In fact, looking at that example again, I could probably take the pretentiousness further and start talking about the interval class vector of the set being 200121 and how that means there will be a prevalence of either class 1, 4,5 or 6 intervals. Or I could look at the score for just a minute and see the clear harmonic focus on stacked 5ths/tritones. Which is the more useful observation? 

I think you're missing the point.  I don't believe the idea behind using mathematics to describe a pitch set is to add anything more meaningful to what may be in your arsenal of musical knowledge.  Clearly music students have been doing well enough at looking at notes on paper without group theory for a while now.  However, I do equate the use of this method to be the same as formulating physics in terms of the Lagrangian or Hamiltonian.  While these formularizations don't give you any new information per se, that doesn't mean that the new frameworks cannot be more useful.  For example, the Hamiltonian is rather useful in celestial mechanics.  Therefore, I believe, for someone who has the necessary background (which you don't), a person can use group theory and make meaningful statements that perhaps wouldn't be so obvious any other way and do so in a consist matter. 

 

So if your only real qualm is that an analysis of music can be done without mathematics, you'll have no argument from me.  I simply don't understand why you are opposed to other people finding merit in using math to study music.  Has it crossed your mind that perhaps the goal is not the same?  Perhaps using abstract algebra to study a musical structure is not so much about studying it for musical purposes but rather as an algebraic structure, and clearly one obvious benefit of this method is that a person doesn't even have to know music theory if they can recognize the structures, alas I feel that last part might fly above your head though. 

I think you're missing the point.  I don't believe the idea behind using mathematics to describe a pitch set is to add anything more meaningful to what may be in your arsenal of musical knowledge.  Clearly music students have been doing well enough at looking at notes on paper without group theory for a while now.  However, I do equate the use of this method to be the same as formulating physics in terms of the Lagrangian or Hamiltonian.  While these formularizations don't give you any new information per se, that doesn't mean that the new frameworks cannot be more useful.  For example, the Hamiltonian is rather useful in celestial mechanics.  Therefore, I believe, for someone who has the necessary background (which you don't), a person can use group theory and make meaningful statements that perhaps wouldn't be so obvious any other way and do so in a consist matter. 

 

So if your only real qualm is that an analysis of music can be done without mathematics, you'll have no argument from me.  I simply don't understand why you are opposed to other people finding merit in using math to study music.  Has it crossed your mind that perhaps the goal is not the same?  Perhaps using abstract algebra to study a musical structure is not so much about studying it for musical purposes but rather as an algebraic structure, and clearly one obvious benefit of this method is that a person doesn't even have to know music theory if they can recognize the structures, alas I feel that last part might fly above your head though. 

 

Look, I'm not taking anything away from the mathematical value of applying mathematical theories to music. I can believe that mathematicians may be able to find value in applying group theory to music and as you say, they may not even need any prior music theory knowledge to do so. In fact, I remember a math exercise I had to do once that involved time signatures but assumed no prior musical knowledge so I get your point, I really do. I'm just saying that it isn't of much, if any, practical use to composers. You've said it's not supposed to be. Fair enough. My last post wasn't aimed towards you; it was aimed towards someone who thinks you can use set theory to say something musically meaningful. I have tried to demonstrate that you can't. If anything, you would seem to agree with me judging by what you've just written so I don't see what the problem is, apart from the fact that I've practically derailed your thread, which I apologise for.

My last post wasn't aimed towards you; it was aimed towards someone who thinks you can use set theory to say something musically meaningful. I have tried to demonstrate you cant.

While the heyday of dodecaphony has long passed, it's not fair to assess the music in such a way, especially since it's not even possible to 'demonstrate' what is or isn't musical or meaningful since everyone has different views on the matter. Some of the most engaging pieces of music I've ever heard were constructed using set theory, pieces like Boulez's 1st Piano Sonata, Babbitt's More Melisma, Wuorinen's Chamber Concerto for Flute and Ten Players, Webern's Variations for Piano, et al.