Re: [tuning-math] Digest Number 1005

(Mainly for Paul Hj.): Paul, to put it informally, the reason the EIS
sequence seems to collapse an interval vector and the interval vector of
that chord's complement is because that sequence has to do with
two-colourings of beads on a necklace. We're "two-colouring" the notes of
an ET, where the colours are "present" and "absent". If the colours are
black and white, then you can see more easily how a set and its complement
are naturally considered the same set.

Someone else mentioned the Polya enumeration theory, which we talked about
here a while ago - you can count set-classes with it, but you don't seem
to be able to count distinct interval vectors with it (because as Paul
says, it's not easy to find a transform that maps Z-related sets to one
another).

I'm really hoping that Gene or someone else might look at Rick Cohn's
recent article in Music Theory Online, where he develops a very simple
formula for enumerating tetrachords in universes of even cardinality. It
looks as though the formula should be generalisable to other cardinalities
of chord, and if someone here could see how, that would be a great result.
Haven't got the url handy but if you look for the article and can't find
it, let me know.

Best -Jon

--- In tuning-math@yahoogroups.com, jon wild <wild@f...> wrote:
>
> (Mainly for Paul Hj.): Paul, to put it informally, the reason the
EIS
> sequence seems to collapse an interval vector and the interval
vector of
> that chord's complement is because that sequence has to do with
> two-colourings of beads on a necklace. We're "two-colouring" the
notes of
> an ET, where the colours are "present" and "absent". If the colours
are
> black and white, then you can see more easily how a set and its
complement
> are naturally considered the same set.
>
> Someone else mentioned the Polya enumeration theory, which we
talked about
> here a while ago - you can count set-classes with it, but you don't
seem
> to be able to count distinct interval vectors with it (because as
Paul
> says, it's not easy to find a transform that maps Z-related sets to
one
> another).
>
> I'm really hoping that Gene or someone else might look at Rick
Cohn's
> recent article in Music Theory Online, where he develops a very
simple
> formula for enumerating tetrachords in universes of even
cardinality. It
> looks as though the formula should be generalisable to other
cardinalities
> of chord, and if someone here could see how, that would be a great
result.
> Haven't got the url handy but if you look for the article and can't
find
> it, let me know.
>
> Best -Jon

Thanks Jon. Also, see my table in files section of Intvec(n,m) counts.
I am up to 19. Today I hope to get to 22. (Whenever you get a chance,
I would love to analyze 23- and higher...)

--- In tuning-math@yahoogroups.com, jon wild <wild@f...> wrote:
>
> (Mainly for Paul Hj.): Paul, to put it informally, the reason the
EIS
> sequence seems to collapse an interval vector and the interval
vector of
> that chord's complement is because that sequence has to do with
> two-colourings of beads on a necklace. We're "two-colouring" the
notes of
> an ET, where the colours are "present" and "absent". If the colours
are
> black and white, then you can see more easily how a set and its
complement
> are naturally considered the same set.
>
> Someone else mentioned the Polya enumeration theory, which we
talked about
> here a while ago - you can count set-classes with it, but you don't
seem
> to be able to count distinct interval vectors with it (because as
Paul
> says, it's not easy to find a transform that maps Z-related sets to
one
> another).
>
> I'm really hoping that Gene or someone else might look at Rick
Cohn's
> recent article in Music Theory Online, where he develops a very
simple
> formula for enumerating tetrachords in universes of even
cardinality. It
> looks as though the formula should be generalisable to other
cardinalities
> of chord, and if someone here could see how, that would be a great
result.
> Haven't got the url handy but if you look for the article and can't
find
> it, let me know.
>
> Best -Jon

Mostly for Jon: I (myself) have calculated what I believe is Intvec
(4n,n) to a high degree (I will post the spreadsheet in my Files
section). This is based on the belief of a simple transformation for
the Z-relations, which holds true from n=1 to n=6 for certain. The
Intvec count is in the last column. My listing only shows 4n-
cardinality. I'll look for the MTO article...

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
> --- In tuning-math@yahoogroups.com, jon wild <wild@f...> wrote:
> >
> Mostly for Jon: I (myself) have calculated what I believe is Intvec
> (4n,n) to a high degree (I will post the spreadsheet in my Files
> section).

Oops I meant (4n,4)