Math paper relevant to our interests - "set of differences of a given set"

http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.32.1933

If a scale is a set S, then the set of intervals of that scale is the set of differences of S, that is, {x - y | x, y in S}. I had been thinking that this construction was well understood, but apparently not.

In rank 2 it's obvious how to minimize the size of the set of differences - just make the points consecutive on the octave-equivalent lattice, yielding a MOS.

In rank >= 3 it's much more difficult. Carl had been saying that he knew how to minimize this over all free Fokker blocks / wakalixes, but I believe I have dispproved this with my recent counterexample. The linked paper discusses this question specifically, but I haven't had a chance to read it yet.

Keenan

--- In tuning-math@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:
>
> http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.32.1933
>
> If a scale is a set S, then the set of intervals of that scale is the set of differences of S, that is, {x - y | x, y in S}. I had been thinking that this construction was well understood, but apparently not.

http://xenharmonic.wikispaces.com/Diamonds

--- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
> --- In tuning-math@yahoogroups.com, "Keenan Pepper" <keenanpepper@> wrote:
> >
> > http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.32.1933
> >
> > If a scale is a set S, then the set of intervals of that scale is the set of differences of S, that is, {x - y | x, y in S}. I had been thinking that this construction was well understood, but apparently not.
>
> http://xenharmonic.wikispaces.com/Diamonds

So, how can you tell which (free) Fokker block has the smallest set of differences out of all the Fokker blocks with the same commas?

What is the minimum cardinality of the set of differences for any rank-3 scale with n notes? Rank r?

By "well understood", I mean it would be easy to answer questions like that.

Keenan

--- In tuning-math@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@> wrote:
> > --- In tuning-math@yahoogroups.com, "Keenan Pepper" <keenanpepper@> wrote:
> > >
> > > http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.32.1933
> > >
> > > If a scale is a set S, then the set of intervals of that scale is the set of differences of S, that is, {x - y | x, y in S}. I had been thinking that this construction was well understood, but apparently not.
> >
> > http://xenharmonic.wikispaces.com/Diamonds
>
> So, how can you tell which (free) Fokker block has the smallest set of differences out of all the Fokker blocks with the same commas?
>
> What is the minimum cardinality of the set of differences for any rank-3 scale with n notes? Rank r?
>
> By "well understood", I mean it would be easy to answer questions like that.
>
> Keenan
>
I'm reading it now --- could this be integrated with a study of interval vectors, Z-relations, etc. in ranks above 1, I wonder. (For example, would there be anything corresponding with a FLID from rank one (and finite) interval vectors?) FLIDs are also called Difference Sets...I think that's right anyway. This ties into projective planes, 13 points and 13 lines is a fun one.

pgh