> On Tue, Mar 8, 2011 at 7:52 PM, Michael Gogins <michael.gogins@gmail.com> 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
>
>
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