Re: pairwise well-formed scales

About Keenan's "triple wakalixes":

In case you're interested, I found that pairwise well-formed scales are, with exactly one exception, constructed as two chains of a single generator such that the lengths of the chains differ by one. For example the JI diatonic is two chains of 5ths, of lengths 4 (F-C-G-D) and 3 (A-E-B). You can always specify a lattice where a pwwf scale looks like the JI diatonic as it appears on the usual lattice of fifths and thirds; for example a pwwf scale of 11 notes will look like a chain of 6 and a chain of 5, and if we choose the right lattice we can display it symmetrically like:

* * * * *
* * * * * *

I can prove this for each length of scale separately, by an exhaustive process, but there's probably a nice proof available for the general case.

The one exception is the 7-note template abacaba, unique among pairwise well-formed scales for being inversionally symmetrical and for having unique multiplicities of scale step. This one needs three chains of the same generator, which you can spread out like a hexagon on the appropriate lattice:

* *
* * *
* *

As a chain of 2 generators, g and h, it will always be expressible as g-h-g-g-h-g. I mentioned this to Gene earlier, not sure if it was on the list or off.

Jon Wild

On Mon, Oct 17, 2011 at 5:28 PM, Jon Wild <wild@music.mcgill.ca> wrote:
>
> About Keenan's "triple wakalixes":
>
> In case you're interested, I found that pairwise well-formed scales are,
> with exactly one exception, constructed as two chains of a single
> generator such that the lengths of the chains differ by one. For example
> the JI diatonic is two chains of 5ths, of lengths 4 (F-C-G-D) and 3
> (A-E-B). You can always specify a lattice where a pwwf scale looks like
> the JI diatonic as it appears on the usual lattice of fifths and thirds;
> for example a pwwf scale of 11 notes will look like a chain of 6 and a
> chain of 5, and if we choose the right lattice we can display it
> symmetrically like:

We did go over this before, although I remember Keenan also had a
counterexample for aabcb, which was a plus sign. I'm not sure how that
resolved itself.

It's also true that triple wakalixes look like this on the lattice:

http://www.sciencephoto.com/image/78248/large/C0013846-Uniform_tiling_pattern-SPL.jpg

A trihexagonal tiling of the lattice will be a triple wakalix iff
there are no points in the triangles. Then it's possible to turn the
hexagon into a parallelogram three separate ways, by grouping it with
the triangles adjacent to any of the 3 sets of parallel lines in the
hexagon.

Keenan had a link posted to this, but I'm not sure where it went. Where is it?

But anyway, what you say is interesting, because if a triple wakalix
that's 101 notes wide fits in as a hexagon 51 and 50 notes long, it's
difficult for me to figure out how exactly that'd turn into a
trihexagonal tiling, but I assume it would.

-Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Mon, Oct 17, 2011 at 5:28 PM, Jon Wild <wild@...> wrote:
> >
> > About Keenan's "triple wakalixes":
> >
> > In case you're interested, I found that pairwise well-formed scales are,
> > with exactly one exception, constructed as two chains of a single
> > generator such that the lengths of the chains differ by one. For example
> > the JI diatonic is two chains of 5ths, of lengths 4 (F-C-G-D) and 3
> > (A-E-B). You can always specify a lattice where a pwwf scale looks like
> > the JI diatonic as it appears on the usual lattice of fifths and thirds;
> > for example a pwwf scale of 11 notes will look like a chain of 6 and a
> > chain of 5, and if we choose the right lattice we can display it
> > symmetrically like:
>
> We did go over this before, although I remember Keenan also had a
> counterexample for aabcb, which was a plus sign. I'm not sure how that
> resolved itself.

The aabcb case is unique, because it is max-variety-3, but it is not a triple wakalix because if you temper out the difference between b and c, you get aabbb, which is not a MOS.

So my findings are identical to Jon Wild's.

> It's also true that triple wakalixes look like this on the lattice:
>
> http://www.sciencephoto.com/image/78248/large/C0013846-Uniform_tiling_pattern-SPL.jpg
>
> A trihexagonal tiling of the lattice will be a triple wakalix iff
> there are no points in the triangles. Then it's possible to turn the
> hexagon into a parallelogram three separate ways, by grouping it with
> the triangles adjacent to any of the 3 sets of parallel lines in the
> hexagon.
>
> Keenan had a link posted to this, but I'm not sure where it went. Where is it?

Sorry, I'll upload this again.

> But anyway, what you say is interesting, because if a triple wakalix
> that's 101 notes wide fits in as a hexagon 51 and 50 notes long, it's
> difficult for me to figure out how exactly that'd turn into a
> trihexagonal tiling, but I assume it would.

I'll make a picture of this too.

Keenan

Mike wrote:

> We did go over this before, although I remember Keenan also had a > counterexample for aabcb, which was a plus sign. I'm not sure how that > resolved itself.

This isn't pairwise well-formed, since conflating b and c gives aaxxx, which is not well-formed. (It *is* quasi-pairwise well-formed though.)

Jon

On Mon, Oct 17, 2011 at 8:04 PM, Jon Wild <wild@music.mcgill.ca> wrote:
>
> Mike wrote:
>
> > We did go over this before, although I remember Keenan also had a
> > counterexample for aabcb, which was a plus sign. I'm not sure how that
> > resolved itself.
>
> This isn't pairwise well-formed, since conflating b and c gives aaxxx,
> which is not well-formed. (It *is* quasi-pairwise well-formed though.)

Yeah, what it was is that aabcb is the only max-variety 3 scale that's
not pairwise well-formed. I still need to work my way through the
latest wave of papers.

-Mike

--- In tuning-math@yahoogroups.com, Jon Wild <wild@...> wrote:

> As a chain of 2 generators, g and h, it will always be expressible as
> g-h-g-g-h-g. I mentioned this to Gene earlier, not sure if it was on the
> list or off.

I'm way behind; I'm not sure what list you mean or what might be on it.

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

> A trihexagonal tiling of the lattice will be a triple wakalix iff
> there are no points in the triangles.

Could someone give an actual definition of a wakalix?

Gene wrote:

>Could someone give an actual definition of a wakalix?

I'll leave it for Keenan to clarify the definitions but I'd
like to suggest we assume Zabka is right for now and drop
terms like "wakalix". And please, don't call the other things
Lumma blocks.

-Carl

On Mon, Oct 17, 2011 at 9:16 PM, genewardsmith
<genewardsmith@sbcglobal.net> wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > A trihexagonal tiling of the lattice will be a triple wakalix iff
> > there are no points in the triangles.
>
> Could someone give an actual definition of a wakalix?

A wakalix is a Fokker block in which the vertices don't belong to the
lattice, but their lengths do.

-Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Mon, Oct 17, 2011 at 9:16 PM, genewardsmith
> <genewardsmith@...> wrote:
> >
> > --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
> >
> > > A trihexagonal tiling of the lattice will be a triple wakalix iff
> > > there are no points in the triangles.
> >
> > Could someone give an actual definition of a wakalix?
>
> A wakalix is a Fokker block in which the vertices don't belong to the
> lattice, but their lengths do.

"Wakalix" is identical to "(free) Fokker block", so let's stop using it.

"Triple wakalix" is identical to "pairwise well-formed scale", so let's stop using that too.

Keenan

--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:
>
> Gene wrote:
>
> >Could someone give an actual definition of a wakalix?
>
> I'll leave it for Keenan to clarify the definitions but I'd
> like to suggest we assume Zabka is right for now and drop
> terms like "wakalix". And please, don't call the other things
> Lumma blocks.

Ok, so instead of "Lumma block" we should say "strictly comma-delimited GTS", because those are identical.

Keenan

--- In tuning-math@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:

> Ok, so instead of "Lumma block" we should say "strictly comma-delimited GTS", because those are identical.

What's a "GTS", and why must we use yet another acronym? I think Fokker has priority anyway.

--- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning-math@yahoogroups.com, "Keenan Pepper" <keenanpepper@> wrote:
>
> > Ok, so instead of "Lumma block" we should say "strictly comma-delimited GTS", because those are identical.
>
> What's a "GTS", and why must we use yet another acronym? I think Fokker has priority anyway.

And if he doesn't certainly other people have been using strictly delimited blocks for many years, whereas surveys of the non-strict blocks have been sitting around in my folder in the files section for years.

--- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:

> And if he doesn't certainly other people have been using strictly delimited blocks for many years, whereas surveys of the non-strict blocks have been sitting around in my folder in the files section for years.
>

What about "Fokker block" and "strictly delimited Fokker block"?

--- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
> What about "Fokker block" and "strictly delimited Fokker block"?

That's fine with me. "Lumma block" = "strictly comma-delimited Fokker block".

Keenan

>What about "Fokker block" and "strictly delimited Fokker block"?

Sounds good.

>"Lumma block" = "strictly comma-delimited Fokker block"

"Lumma block", by the way, has been used before to mean something
entirely different (a 7-limit block with 225/224).

-Carl