I've received my copy of Dmitri Tymoczko's new book, _A Geometry of
Music: Harmony and Counterpoint in the Extended Common Practice_
(Oxford and New York: Oxford University Press, 2011).

In my view, this is a very important book in music theory, and I
highly recommend it to any computer music person with any sort of
interest in anything like chords, scales, and voice-leading.

I find that the work of Tymoczko (and his colleagues Clifton
Callender, Rachel Hall, Adrian Childs, etc.) in elaborating a
geometric theory of chords and voice-leadings facilitates generative
composition in ways that other formal or mathematical approaches to
music theory, such as the generative grammar of Jackendoff and
Lerdahl, simply do not (at least, not for me). I have composed a
number of works based on generating movements in chord spaces, either
directly moving chords as points in multi-dimensional spaces, or
applying neo-Riemannian operations such as the Generalized Contextual
Group of Fiore and Satyendra as implemented in such chord spaces, and
I presented a paper on this work of mine at the 2006 ICMC.

The great thing about the geometric approach is its ability to greatly
simplify and easily automate operations upon pitches and chords,
across various scales of musical structure, and including recursive
operations. It helps a great deal that the mathematics involved is not
very complicated once some quite basic principles of group theory and
quotient spaces are assimilated -- no need to learn category theory or
differential geometry! The required principles are presented with
exemplary clarity in Tymoczko's book. Furthermore, operations can be
efficiently implemented (for example, finding a well-formed
voice-leading by selecting the shortest of multiple paths through
multi-octave chord space is of O(log N) complexity and can be
implemented in a page or so of Python, whereas voice-leading by
formalizing the rules of _Gradus ad Parnassum_ is I guess of O(C^N)
complexity and takes multiple pages of code).

Just as Hiller and Isaacson, and Xenakis' sieves, opened up the use of
stochastic processes for algorithmic composition on various scales of
musical structure, and thus brought considerable variety and power
into atonal algorithmic composition, so operations in music spaces of
varying degrees of abstraction open up a good chunk of the resources
of tonality and extended tonality for algorithmic composition.

-- 
Michael Gogins
Irreducible Productions
http://www.michael-gogins.com
Michael dot Gogins at gmail dot com

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Thanks Michael for the recommendation and review!  I'll be ordering a
copy soon!  (And thanks Dave for mentioning the Supercollider book!)

On Tue, Mar 8, 2011 at 8:52 PM, Michael Gogins  wrote:
> I've received my copy of Dmitri Tymoczko's new book, _A Geometry of
> Music: Harmony and Counterpoint in the Extended Common Practice_
> (Oxford and New York: Oxford University Press, 2011).
>
> In my view, this is a very important book in music theory, and I
> highly recommend it to any computer music person with any sort of
> interest in anything like chords, scales, and voice-leading.
>
> I find that the work of Tymoczko (and his colleagues Clifton
> Callender, Rachel Hall, Adrian Childs, etc.) in elaborating a
> geometric theory of chords and voice-leadings facilitates generative
> composition in ways that other formal or mathematical approaches to
> music theory, such as the generative grammar of Jackendoff and
> Lerdahl, simply do not (at least, not for me). I have composed a
> number of works based on generating movements in chord spaces, either
> directly moving chords as points in multi-dimensional spaces, or
> applying neo-Riemannian operations such as the Generalized Contextual
> Group of Fiore and Satyendra as implemented in such chord spaces, and
> I presented a paper on this work of mine at the 2006 ICMC.
>
> The great thing about the geometric approach is its ability to greatly
> simplify and easily automate operations upon pitches and chords,
> across various scales of musical structure, and including recursive
> operations. It helps a great deal that the mathematics involved is not
> very complicated once some quite basic principles of group theory and
> quotient spaces are assimilated -- no need to learn category theory or
> differential geometry! The required principles are presented with
> exemplary clarity in Tymoczko's book. Furthermore, operations can be
> efficiently implemented (for example, finding a well-formed
> voice-leading by selecting the shortest of multiple paths through
> multi-octave chord space is of O(log N) complexity and can be
> implemented in a page or so of Python, whereas voice-leading by
> formalizing the rules of _Gradus ad Parnassum_ is I guess of O(C^N)
> complexity and takes multiple pages of code).
>
> Just as Hiller and Isaacson, and Xenakis' sieves, opened up the use of
> stochastic processes for algorithmic composition on various scales of
> musical structure, and thus brought considerable variety and power
> into atonal algorithmic composition, so operations in music spaces of
> varying degrees of abstraction open up a good chunk of the resources
> of tonality and extended tonality for algorithmic composition.
>
> --
> Michael Gogins
> Irreducible Productions
> http://www.michael-gogins.com
> Michael dot Gogins at gmail dot com
>
>
> Send bugs reports to the Sourceforge bug tracker
>            https://sourceforge.net/tracker/?group_id=81968&atid=564599
> Discussions of bugs and features can be posted here
> To unsubscribe, send email sympa@lists.bath.ac.uk with body "unsubscribe csound"
>
>


Send bugs reports to the Sourceforge bug tracker
            https://sourceforge.net/tracker/?group_id=81968&atid=564599
Discussions of bugs and features can be posted here
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I pretty much agree with Anthony.

First I think we need to back up a bit. The kind of music I make is
like the music of John Cage or some of the minimalists or Iannis
Xenakis. I don't hear the score in my head and then write it down. I
don't imagine the details in advance. I invent a process that I
believe to be musical, and sometimes it turns out that it actually is
musical!

My personal motive in using geometric music theory to compose is (a)
to save myself work and (b) to develop algorithms that create
structures that I could not have imagined, that nevertheless have a
much better than random chance of being what I would call "musically
well-formed" or, in other words, listenable.

Let me expand on that a bit -- I've always had pretty good luck
getting fractals or recursive functions or whatnot to generate music
that is sometimes quite satisfying to me and, it seems, occasionally
to others as well. But I've been weaned and raised on classical music
and jazz, and it always bugged me that my stuff was either atonal, or
had some hint of tonality that happened more or less by accident and I
somehow managed to tweak it into recognizable cadences and modulations
and such.

Here we need to back up yet again -- I have nothing against tonality
as such, there is music by atonal composers that I truly love and that
I believe is great music that will stand the test time of time.
Furthermore, there is 'acousmatic' music to which the question of
tonality or atonality is more or less irrelevant, and there is some of
that I love just as much. No, it's just that out of my own pieces, the
stuff I like best is the stuff with some tonality. (I think that's
because those pieces tend to have less monotony over longer periods of
time.)

It has never worked for me to manually push the algorithmically
generated notes around till they made that kind of sense. It has only
worked for me if the algorithm actually generates all the notes. So
the Tymoczko stuff appeals to me because it promises to make that much
easier.

And it actually does!

In other words, I'm interested in whether a music theory can motivate
itself to write in some style in an academic sense, certainly -- but
not in a practical sense. No, what I want is a geometric playground,
where, when my processes start running around and playing with each
other, this noises that happen are more listenable because they're
circulating in the right parts of chord space.

Regards,
Mike

On Wed, Mar 9, 2011 at 12:23 PM, Anthony Palomba  wrote:
> Hello Aaron,
>
> Actually I think you might be misunderstanding what this book is proposing.
> It does not
> claim to capture in essence of the composition of a musical language. It
> is a technique that
> allows one to represent harmonic relationships as a geometric space.
>
> No one is saying that it makes any creative decisions for you. Just as a
> painter using
> cartesian canvas is free to make creative decisions that they want to.
> Is the art created by the use of perspective suddenly "less art"?
> It does provide concepts like the "horizon" or "vanishing point", but it the
> end it is I who fill that spaces with meaningful expression.
>
> For me, what this technique boils down to is that it allows me the ability
> to study the structure of a musical grammar. It allows me to study
> the relationship between pitch spaces, and experiment with mapping complex
> behaviors on to modulations and progressions. These new behaviors form the
> foundation of a new musical grammar that I can express myself in.
>
> It is true that the masters did a great deal of this in their heads, which
> is
> quite humbling to me. But I think we are heading into territory where we
> would be developing musical grammars that are perhaps beyond what the
> unassisted human mind might be able to conceive of.
>
>
>
>
> Anthony
>
>



-- 
Michael Gogins
Irreducible Productions
http://www.michael-gogins.com
Michael dot Gogins at gmail dot com


Send bugs reports to the Sourceforge bug tracker
            https://sourceforge.net/tracker/?group_id=81968&atid=564599
Discussions of bugs and features can be posted here
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What you describe in terms of moving from to chord is a subset of what
Tymoczko and colleagues have done -- in terms of results. I would say,
a fairly restricted subset. It is not at all a subset conceptually. I
think this would become clearer to you if you read the book or the
papers which formed it. My personal Web site (see below) has my own
intro to this stuff, if you don't have ready access to the book or
papers.

The main thing about geometric music theory is that each chord is a
single point in a multi-dimensional space, in which each dimension
represents the pitch of one of the voices of the chord. A chord is not
a thing with so many voices - it is a single point. Furthermore, the
chord space is continuous, not discrete. Then, various degrees of
abstraction can be introduced by equating octaves, equating
inversions, equating different orders of pitches, and so on. This is
highly non-trivial and encodes in a concise and easy to see way a
great deal of information about what is happening in voice-leading and
harmony.

There's a Python app in the Csound examples directory that displays
chord space for trichords, and allows the user to play chords using
Csound.

Regards,
Mike

On Wed, Mar 9, 2011 at 3:11 PM, Aaron Krister Johnson
 wrote:
> Anthony and Michael,
>
> What you say makes total sense.
>
> I'm interested to see what you come up with musically from being in such a
> playground.
>
> I assume the algorithms would be extensible to any abstract tonal space
> (e.g. any arbitrary EDO tuning or Just Intontational fabric), b/c Tymoczko's
> work has been mentioned on the yahoo tuning list--Gene Ward Smith has
> structured things on such ideas. To me, the results weren't as satisfying as
> I think they might be b/c I think the timbral choices were bad/static, and I
> think there wasn't an apparent larger compositional 'will' at work, for lack
> of a better term....it seemed to wander aimlessly through an interval/chord
> space. Very cool for a short duration, then it becomes numbing.
>
> Some years ago, I wrote some Python MIDI code that would randomly pick a
> fifth, fourth, major or minor third to move as a root note from the current
> tonal root note, and calculate the smoothest voice-leading transition (for
> example, by using common tones) to the new chord. It was more a
> proof-of-concept for fun than a compositional tool, but it did suggest some
> flexible modulatory schemes. One could easily extend this concept into more
> far-flung regions of the overtone series (but beware that they may or may
> not sound as good as basic 5-limit harmony shifts!)
>
> My point is to question how close this concept is to what Tymoczko's basic
> concept is. If it's a shared essence, I would argue that such understanding
> is hundreds of years old, and not really new with Tymoczko, although he may
> have put a firm mathematical formality around it.
> At least on a superficial level, it sound like applying algorithms to moving
> about a triadic lattice structure, for example.
>
> AKJ
>
> On Wed, Mar 9, 2011 at 11:54 AM, Michael Gogins 
> wrote:
>>
>> I pretty much agree with Anthony.
>>
>> First I think we need to back up a bit. The kind of music I make is
>> like the music of John Cage or some of the minimalists or Iannis
>> Xenakis. I don't hear the score in my head and then write it down. I
>> don't imagine the details in advance. I invent a process that I
>> believe to be musical, and sometimes it turns out that it actually is
>> musical!
>>
>> My personal motive in using geometric music theory to compose is (a)
>> to save myself work and (b) to develop algorithms that create
>> structures that I could not have imagined, that nevertheless have a
>> much better than random chance of being what I would call "musically
>> well-formed" or, in other words, listenable.
>>
>> Let me expand on that a bit -- I've always had pretty good luck
>> getting fractals or recursive functions or whatnot to generate music
>> that is sometimes quite satisfying to me and, it seems, occasionally
>> to others as well. But I've been weaned and raised on classical music
>> and jazz, and it always bugged me that my stuff was either atonal, or
>> had some hint of tonality that happened more or less by accident and I
>> somehow managed to tweak it into recognizable cadences and modulations
>> and such.
>>
>> Here we need to back up yet again -- I have nothing against tonality
>> as such, there is music by atonal composers that I truly love and that
>> I believe is great music that will stand the test time of time.
>> Furthermore, there is 'acousmatic' music to which the question of
>> tonality or atonality is more or less irrelevant, and there is some of
>> that I love just as much. No, it's just that out of my own pieces, the
>> stuff I like best is the stuff with some tonality. (I think that's
>> because those pieces tend to have less monotony over longer periods of
>> time.)
>>
>> It has never worked for me to manually push the algorithmically
>> generated notes around till they made that kind of sense. It has only
>> worked for me if the algorithm actually generates all the notes. So
>> the Tymoczko stuff appeals to me because it promises to make that much
>> easier.
>>
>> And it actually does!
>>
>> In other words, I'm interested in whether a music theory can motivate
>> itself to write in some style in an academic sense, certainly -- but
>> not in a practical sense. No, what I want is a geometric playground,
>> where, when my processes start running around and playing with each
>> other, this noises that happen are more listenable because they're
>> circulating in the right parts of chord space.
>>
>> Regards,
>> Mike
>>
>> On Wed, Mar 9, 2011 at 12:23 PM, Anthony Palomba 
>> wrote:
>> > Hello Aaron,
>> >
>> > Actually I think you might be misunderstanding what this book is
>> > proposing.
>> > It does not
>> > claim to capture in essence of the composition of a musical language. It
>> > is a technique that
>> > allows one to represent harmonic relationships as a geometric space.
>> >
>> > No one is saying that it makes any creative decisions for you. Just as a
>> > painter using
>> > cartesian canvas is free to make creative decisions that they want to.
>> > Is the art created by the use of perspective suddenly "less art"?
>> > It does provide concepts like the "horizon" or "vanishing point", but it
>> > the
>> > end it is I who fill that spaces with meaningful expression.
>> >
>> > For me, what this technique boils down to is that it allows me the
>> > ability
>> > to study the structure of a musical grammar. It allows me to study
>> > the relationship between pitch spaces, and experiment with mapping
>> > complex
>> > behaviors on to modulations and progressions. These new behaviors form
>> > the
>> > foundation of a new musical grammar that I can express myself in.
>> >
>> > It is true that the masters did a great deal of this in their heads,
>> > which
>> > is
>> > quite humbling to me. But I think we are heading into territory where we
>> > would be developing musical grammars that are perhaps beyond what the
>> > unassisted human mind might be able to conceive of.
>> >
>> >
>> >
>> >
>> > Anthony
>> >
>> >
>>
>>
>>
>> --
>> Michael Gogins
>> Irreducible Productions
>> http://www.michael-gogins.com
>> Michael dot Gogins at gmail dot com
>>
>>
>> Send bugs reports to the Sourceforge bug tracker
>>            https://sourceforge.net/tracker/?group_id=81968&atid=564599
>> Discussions of bugs and features can be posted here
>> To unsubscribe, send email sympa@lists.bath.ac.uk with body "unsubscribe
>> csound"
>>
>
>
>
> --
> Aaron Krister Johnson
> http://www.akjmusic.com
> http://www.untwelve.org
>
>



-- 
Michael Gogins
Irreducible Productions
http://www.michael-gogins.com
Michael dot Gogins at gmail dot com


Send bugs reports to the Sourceforge bug tracker
            https://sourceforge.net/tracker/?group_id=81968&atid=564599
Discussions of bugs and features can be posted here
To unsubscribe, send email sympa@lists.bath.ac.uk with body "unsubscribe csound"