Omnitetrachordal scales can be arbitrary rank

I'm trying to understand the exact relationship between omnitetrachordal scales (which ought to have a better name, I agree...) and distributionally even scales. I want to have a "fundamental theorem of omnitetrachordal scales" that completely characterizes them, with a bunch of strong if-and-only-if type statements.

I had previously conjectured that omnitetrachordal scales are always rank <= 3 (and Mike Battaglia had conjectured that they're related to rank-3 Fokker blocks), but this is false. Here's a counterexample (that also happens to have a 2.3.5.13 chord in it):

1/1 13/12 10/9 9/8 39/32 5/4 3/2 13/8 5/3 2/1

which has the following steps:

13/12 40/39 81/80 13/12 40/39 6/5 13/12 40/39 6/5

This is omnitetrachordal with 4/3 as the omnitetrachordality interval (OI), but it's rank 4.

Similarly, you can concoct an omnitetrachordal scale of arbitrary rank by combining shifted copies of {1/1, 9/8, 3/2}, where all of the shifts are smaller than 9/8, and all mutually incommensurate.

A necessary, but not sufficient, condition for a scale to be omnitetrachordal is for it to be a union of shifted copies of a specific MOS generated by the OI, with duplicate notes removed. The usual version with 4/3 as the OI means the scale has to be a union of shifted copies of {1/1, 9/8, 3/2}. This is easily proved from the definition of omnitetrachordality by considering that either {1/1, 9/8, 3/2} or {1/1, 4/3, 3/2} or {1/1, 4/3, 16/9} has to be present in each mode of the scale.

I'm looking for some restriction on the shifted copies that would make that both necessary and sufficient. I think the best approach is to think of it as a union of MOSes of the OI, which can possibly be different sizes. For example, this 8-note scale:

1/1 10/9 9/8 5/4 81/64 3/2 5/3 27/16 2/1

is omnitetrachordal even though it is impossible to express it as a union of shifted copies of the *same* MOS. It has to be pythagorean[5] + pythagorean[3], and the three notes of the smaller MOS have to fall between the three small intervals of the larger MOS...

Not a theorem yet, but so close I can taste it.

Keenan

I note that Pythagorean[5] is itself a self-shifted version of
Pythagorean[3]. This sort of trickery kept turning up when I was trying to
work out the same thing earlier.

I conjecture that an omnitetrachordal scale is -always- expressible as a
shifted copy of some MOS of the omnitetrachordality interval. This implies
a particular lattice construction that is a bunch of copies of a line,
convolved with some shifting kernel. (You can think in terms of convolving
indicator functions if you like, but don't forget to take sign() of the
result!)

I think there will be some condition of being in monotonic order placed on
the kernel, with respect to some weird val in 4/3-equivalent space. And
there should be some additional condition handling if the kernel has a
4/3-component too, which I'm not sure of. Maybe ensuring that it forms an
MOS with respect to 2/1.

-Mike

On Dec 24, 2011, at 1:31 PM, Keenan Pepper <keenanpepper@gmail.com> wrote:

I'm trying to understand the exact relationship between omnitetrachordal
scales (which ought to have a better name, I agree...) and distributionally
even scales. I want to have a "fundamental theorem of omnitetrachordal
scales" that completely characterizes them, with a bunch of strong
if-and-only-if type statements.

I had previously conjectured that omnitetrachordal scales are always rank
<= 3 (and Mike Battaglia had conjectured that they're related to rank-3
Fokker blocks), but this is false. Here's a counterexample (that also
happens to have a 2.3.5.13 chord in it):

1/1 13/12 10/9 9/8 39/32 5/4 3/2 13/8 5/3 2/1

which has the following steps:

13/12 40/39 81/80 13/12 40/39 6/5 13/12 40/39 6/5

This is omnitetrachordal with 4/3 as the omnitetrachordality interval (OI),
but it's rank 4.

Similarly, you can concoct an omnitetrachordal scale of arbitrary rank by
combining shifted copies of {1/1, 9/8, 3/2}, where all of the shifts are
smaller than 9/8, and all mutually incommensurate.

A necessary, but not sufficient, condition for a scale to be
omnitetrachordal is for it to be a union of shifted copies of a specific
MOS generated by the OI, with duplicate notes removed. The usual version
with 4/3 as the OI means the scale has to be a union of shifted copies of
{1/1, 9/8, 3/2}. This is easily proved from the definition of
omnitetrachordality by considering that either {1/1, 9/8, 3/2} or {1/1,
4/3, 3/2} or {1/1, 4/3, 16/9} has to be present in each mode of the scale.

I'm looking for some restriction on the shifted copies that would make that
both necessary and sufficient. I think the best approach is to think of it
as a union of MOSes of the OI, which can possibly be different sizes. For
example, this 8-note scale:

1/1 10/9 9/8 5/4 81/64 3/2 5/3 27/16 2/1

is omnitetrachordal even though it is impossible to express it as a union
of shifted copies of the *same* MOS. It has to be pythagorean[5] +
pythagorean[3], and the three notes of the smaller MOS have to fall between
the three small intervals of the larger MOS...

Not a theorem yet, but so close I can taste it.

Keenan