# 'Groups', abstract composition



## soundandfury (Jul 12, 2008)

Well, I keep being told that the classical style is outdated, so I decided to write something abstract and modern.
*'Groups'*, a study in abstract algebra.
Listen: http://jttlov.no-ip.org/music/groups.ogg
Score: http://jttlov.no-ip.org/music/groups.pdf

*The explanation*: it's all about maths, and specifically finite groups of order up to 6, expressed as subgroups of permutation groups.
First we have the trivial group, {e} (or Z₁), which is isomorphic to S₁ - in other words, it does nothing. This is represented by the bare C octaves which open the piece.
Next comes Z₂, the only group of order 2, which is isomorphic to S₂. It acts on the set {C,G}, in a fairly simple way.
There is also only one group of order 3, that is Z₃; it acts on {C,E,G}, again making the cyclic structure very obvious.
Of order 4 there are two groups, Z₄ and V₄ (the Klein 4-group, isomorphic to Z₂xZ₂). We take Z₄ first, cycling {C,E,G,Bb}. V₄ also fits into S₄, but in a different way; here it acts on the set {C,Eb,G,Bb} and consists of the swaps (C G) and (Eb Bb) and their product (note that the swaps are each a perfect fifth, therefore preserving a degree of harmonic structure).
There is only one group of order 5, Z₅, and it will fit in nothing smaller than S₅, whose rising cycles on {C,Eb,F,G,Bb} produce a feeling of increasing energy and approaching climax.
This climax is provided by the sudden force with which Z₆ is introduced. This is a special moment as Z₆ is the first group we've encountered which fits inside a symmetric group smaller than the left regular action provides: Z₆ fits inside S₅, here acting on {C,Eb,F,G,Bb}.
But there is another group of order 6, the dihedral group D₆ (the symmetries of a triangle) - and this group is isomorphic to S₃. This surprising compactness is expressed by the crunchy chords provided by acting on the set {C,D,Gb}.
And there we have all the groups of order ≤ 6, expressed in musical form as permutation subgroups; the piece is closed off by a resolution, these last two bars being the only ones not constrained by group structure. An awful lot of maths for 53 seconds of music!


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## Rasa (Apr 23, 2009)

O look, it comes with a manual


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## regressivetransphobe (May 16, 2011)

I'd be lying if I said I got the maths, but I can indeed hear the maths. So many maths.


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