Temperaments and their relationships to JI Unison Vectors Periodicity blocks

I have had some offline interactions with Paul Erlich who has kindly
educated me regarding certain aspects of unison vectors and
periodicity blocks in JI and my curiosity about the possibility of
analyzing or even creating temperaments using these conceptual tools.
This discussion has generated some hypotheses and questions for me
that I think should be shared with the online community here.

Question: SINCE -

1) to define a periodicity block in an n-dimensional JI lattice you
need n unison vectors,

AND IF

2) having defined a periodicity block you wish to close the gap and
construct a closed cycle temperament based on that block (as the
Pythagorean comma, a unison vector in one-dimensional JI, is closed
to construct 12-tET),

THEN

aren't the unison vectors in each dimension different from each other?

THEREFORE

how can we close more than one of them, simply letting the others
fall where they will as a consequence?

OR

is it possible to find common unison vector sizes in the lattice that
represent vertices of all n dimensions of the lattice with respect to
the origin at 1/1 and use that vector size as the "fulcrum" from
which to construct a temperament?

This is a mathematical and not a historical question. I am not
concerned here with how any temperament actually arose historically.
For example, 31-tET is a close xenharmonic approximation of 1/4-comma
meantone. 31-tET clearly has an evolutionary history, but can be
derived mathematically from a periodicity block involving a cycle of
31 pure M3s.

This comma is greater by far than the syntonic comma, but gets
divided by 31 in equally tempering the error rather than only four,
so the 3rds come out much purer than the fifths. So in this case we
choose the 3rds not simply because of the size of the unison vector,
but by this together with how many 3rds are in the cycle that will
divide it.

It seems that research could be done (if it hasn't been already)
using some such approach to investigate various temperaments, both
open (meantone) and closed (31-tET).

Any comments? Thank you in advance, all contributors!

Sincerely,

Bob

--- In tuning@y..., BobWendell@t... wrote:

> Question: SINCE -
>
> 1) to define a periodicity block in an n-dimensional JI lattice you
> need n unison vectors,
>
> AND IF
>
> 2) having defined a periodicity block you wish to close the gap and
> construct a closed cycle temperament based on that block (as the
> Pythagorean comma, a unison vector in one-dimensional JI, is closed
> to construct 12-tET),

So by "close the gap" you mean to temper out a particular unison
vector, or all of them . . . ?
>
> THEN
>
> aren't the unison vectors in each dimension different from each
other?

I'm confused by this question. Different unison vectors are different
from one another, obviously . . . so what are you asking?

> THEREFORE
>
> how can we close more than one of them, simply letting the others
> fall where they will as a consequence?

Well, closing all of them leads to an ET, as you seem to understand.
Closing not all of them leads to some other scale . . . For example,
let's say we're dealing with the diatonic scale, which is the
periodicity block defined by the two unison vectors 25:24 (the
chromatic semitone) and 81:80 (the syntonic comma). Shrinking 81:80
to nothing, while preserving the size of 25:24, leads to a diatonic
scale in 2/7-comma meantone temperament.

In fact, a few months ago I posted my Hypothesis, which states that
if you temper out all but one of the unison vectors of a Fokker
periodicity block, you end up with an MOS scale. We're discussing
this Hypothesis on tuning-math@yahoogroups.com.
>
> OR
>
> is it possible to find common unison vector sizes in the lattice
that
> represent vertices of all n dimensions of the lattice with respect
to
> the origin at 1/1 and use that vector size as the "fulcrum" from
> which to construct a temperament?

Any unison vector will represent a vertex in the lattice when
reckoned with respect to the origin at 1/1. I don't know what you
mean by "of all n dimesions" or by "fulcrum". A temperament can be
constructed by tempering out anywhere from 1 to n unison vectors. If
you temper out n (and do it uniformly), you have an ET. If you temper
out n-1, you have a linear temperament. If you temper out n-2, you
have a planar temperament (Dave Keenan has created some examples of
those).

I direct any future correspondence on this topic (which I warmly
encourage) to

tuning-math@yahoogroups.com

. . . the mathematical nature of this topic is uncomfortable for some
people on this list, and the tuning-math group was created explicitly
to alleviate this discomfort.

Sounds like you may be inventing some interesting new temperaments in
the future, Bob!