Random interesting thing: intervals are a group, chords are a semifield

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

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>

This would be a lot clearer if you gave an example. What for instance, is

(1) [2,3,5} + [1,3,3,7]

(2) [2,3,5} * [1,3,3,7]

On Thu, Feb 21, 2013 at 2:39 PM, genewardsmith <genewardsmith@sbcglobal.net>
wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia wrote:
> >
>
> This would be a lot clearer if you gave an example. What for instance, is
>
> (1) [2,3,5} + [1,3,3,7]
>
> (2) [2,3,5} * [1,3,3,7]

I don't know what this notation means. Assuming those are both
supposed to be multisets, then {2,3,5} + {1,3,3,7} = {1,2,3,3,3,5,7},
and {2,3,5} * {1,3,3,7} =
{2*1,2*3,2*3,2*7,3*1,3*3,3*3,3*7,5*1,5*3,5*3,5*7} =
{2,6,6,14,3,9,9,21,5,15,15,35}.

More usefully, {1/1,5/4,3/2} * {3/2} = {3/2,15/8,9/4}, which has the
interpretation of being a transposed chord.

-Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

Thanks; you had me thoroughly confused by the semifield thing.