definition of a group

>From Eric's Treasure Trove of Mathematics (on line mathematical
encyclopedia)

A group G is defined as a set of finite or infinite objects or operators
(called ``elements'') that may be combined or ``multiplied'' to form
well-defined products which satisfy

1. If A and B are two elements, then the product AB is also a member
of the set.

2. The defined multiplication is associative, i.e. (AB)C A(BC)

3. There is a unit ``identity'' element I (a.k.a., 1 or E, the
latter deriving from the

German ``Einheit'' meaning unity) such that IA AI A for every
element in the set.

4. Therex must be an inverse or reciprocal of each element.

Eric's Treasure Trove
http://www.gps.caltech.edu/~eww/math/math.html

John Starrett

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Thanks, Jonathan (S), for reviewing the Southern California Microtonal
Group's latest concert.

I must confess that, regretably, I have to agree (from tapes Brian has
sent me) with your impression of the Southern California Microtonal Group's
improvs. And this isn't unique to them either. I think Neil is right that
many of us tend to ramble in our improvs. Much of it has little drama.
When each improv ends I have little understanding of why it should have
ended there rather than a few minutes earlier, or later. That's much of
the reason why I stick to planned composition. My own improvs aren't any
better.

I've been trying to interpret these sorts of works in a different light
though. An ambient-music kind of light. Perhaps it works better that way,
but I'll be honest with you: Few of those improvs compel me to give them
more than one listening to let me find that ethic. And I suppose it's
pretty obvious that the idea of an ambient-music concert is, by definition,
meaningless.

That line is a hard one to draw though. A few months ago, I picked up a
copy of Cultrane's "Interstellar Regions". Without a doubt I need to
develop a new music-listening ethic to understand it. Most of the
individual parts in each title of that CD don't seem to have a lot to do
with each other either. I'm certainly willing (when I have the time) to
give it several more good hard listens to try to find that required
music-listening ethic. Perhaps I ought to give the SCMG's improvs more
time too.

But I believe there a silver lining to the SCMG's works though.

I get the strong sense that the Group's members tend to cancel each
other out. Brian McLaren's and Jeff Stayton's own PRECOMPOSED music, I
think anyway, is pretty good stuff. Jonathan's work with vocal harmonics
is very entertaining, and he certainly got the audience boogying in El Paso
recently! And without a doubt, Bill Wesley has some really fantastic
instrument-building ideas. And even when they work together, I have heard
interesting timbres and textures in the tapes Brian has sent me, and I
think I can learn from them. But I sadly regret to say that I find the
high points in their improvs far too rare.

I personally think that the improvisors among us would be well served to
predesignate some more structure to their works. There's value for
improvisationalists to start each improv with:
1. A series of preintended and prepracticed underlying melodies and chord
progressions.
2. A predesignated overall impression of what feeling we're trying for
(fun,
lost in space, renaissancish sound, romantic, sad, frightening, or
whatever).
3. A high-level outline of how the music is to procede - where the high
points in the drama will be, and with what melodic and chordal
resources
they'll achieve that, where the loud and soft points will be, and so
forth.

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