Oops, semiring

Sorry, a quick error: these would actually be a semiRING, not a
semifield. In general, chords have no transpositional inverse.

Also, I asked the MathOverflow people, who told me that this is just a
generalization of the group ring to monoids, and that it's called a
monoid semiring. That's a rather nice way of looking at it. So the
semiring of multichords over a group G of intervals is just N[G], and
the semiring of chords is B[G], where B is the 2-element idempotent
monoid. Rather simple.

-Mike

On Thu, Feb 21, 2013 at 7:02 AM, Mike Battaglia <battaglia01@gmail.com> wrote:
> A while ago, we were discussing an approach to forming scales which
> treated them as words over an alphabet; we'd talked about DL distance
> and the like. This effectively treats scales as elements in the free
> monoid G* over the underlying set of G, some abelian group of musical
> intervals. This is a neat idea, and it would be even neater if we
> could somehow find some algebraic structure that lets G and G* coexist
> as a single set with two binary operations which are related in some
> way. Unfortunately, I can't figure out how to do it.
>
> However, there really is a nice algebraic structure lurking about if
> you stop worrying about the ordering of elements in the scale. If you
> instead look at the free -commutative- monoid over G, then you get
> something which can naturally be interpreted as a monoid of chords or
> of unordered scales, where notes can be repeated more than once. This
> monoid can effectively be interpreted as the set of finite multisets
> of elements in G. I'll call the operation where you combine two
> intervals into a chord "aggregation."
>
> You can take this monoid to be the underlying monoid of a semifield,
> and then treat ordinary interval composition, e.g. the group operation
> of G, as semifield multiplication. This works out quite neatly:
> composition then distributes over aggregation, so you can do things
> like transpose a chord by an interval. More generally, you can
> transpose one chord by another, where transposition works out to be
> the Minkowski sum of the multisets; this isn't quite as useful as
> transposing a chord by an interval, but it certainly doesn't hurt.
> Music set theorists call this "set multiplication"
> (http://en.wikipedia.org/wiki/Multiplication_(music)) for some reason.
>
> An even nicer trick is to force the monoid to become idempotent, at
> which point you get a monoid of finite sets in G, with duplicate notes
> prohibited. The resulting semifield also then has the structure of a
> "dioid," which is the name that Wikipedia claims has been given to an
> idempotent semiring. All of the above still applies, except now
> there's no duplicate notes allowed. I'll call this "the semifield of
> chords," and the thing before this "the semifield of multichords,"
> where a multichord is a chord that can contain duplicate notes.
>
> I'm not sure if this makes anything easier to compute, but I find it
> interesting to note that the musical operations of "interval
> aggregation into a chord" and "interval composition" naturally give
> you the structure of a semifield like that. Forget about knowing what
> a semifield "looks like," now you even know what it sounds like!
>
> More interestingly, this semifield seems to be a rather natural
> construction associated to any monoid, of which our original group G
> is an example, so I'm rather curious if it has a name.
>
> -Mike