Comma sequences, flags, and Schubert cycles

We can define a comma sequence more generally than one characterizing
a family relationship as simply a sequence of commas c1, c2, ... cn
where the commas are independent--ie, the matrix of monzos is of rank
n. Such a sequence of commas define a sequence of subspaces; c1
defines the line through the origin containing c1, which we may call
V1, c2 and c1 define a plane through the origin, V2, and so forth.
Each Vi is of dimension i, and contains the previous and is contained
in the subsequent subspace of the sequence V1 < V2 < ... < Vn. Such a
sequence of subspaces mathematicians, in their jargon-laden way, call
a flag:

http://en.wikipedia.org/wiki/Flag_%28mathematics%29

http://mathworld.wolfram.com/VectorSpaceFlag.html

Given a flag, we can ask for the corresponding Schubert variety (cell,
cycle, these are closely related ideas.) This means we find every
subspace of a given dimension, the dimension of whose intersections
with the subspaces of the flag are fixed.

http://mathworld.wolfram.com/SchubertVariety.html

http://en.wikipedia.org/wiki/Grassmannian

None of this is going to make a particle of sense without an example,
so here is one. If we take the TM basis for 31-et in the 7-limit, we
get 81/80, 126/125, 1029/1024 in order of Tenney height. This comma
sequence determines a corresponding flag, the promo defined by 81/80,
or the 81/80-planar 7-limit temperament, the plane defined by
{81/80, 126/125}, which is 7-limit meantone, and the space spanned by
{81/80, 126/125, 1029/1024}, which is the kernel of <31 49 72 87|.

Now I'm simply going to take the example given on page 197 of
Griffiths and Harris, "Principles of Algebraic Geometry" of "Schubert
cycles" (meaning they are intended to represent something in
"cohomology", the meaning of which we will ignore.) We have the
following Schubert varieties of planes through the origin in 4-space,
which means projective lines in 3-space, which we can identify with
wedgies (6 dimensional), which are really points in projective
5-space, which lie on a projective variety of dimension 4. This
confusing mess I should try to sort out better some time, but for now
simply note we are talking about 7-limit linear temperaments!

We have:

(1) All (7-limit, linear) temperaments sending some comma of 7-limit
meantone to the unison--that is, something of the form
(81/80)^i (126/125)^j is in the kernel. Example: miracle.

(2) Temperaments supported by 31-equal--which is to say, with a kernel
contained in the kernel of the 7-limit 31-et val. Example: valentine.

(3) Temperaments containing 81/80. Example: godzilla.

(4) Temperaments containing 81/80 and supported by 31-et. Example: mothra.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> We have:
>
> (1) All (7-limit, linear) temperaments sending some comma of 7-limit
> meantone to the unison--that is, something of the form
> (81/80)^i (126/125)^j is in the kernel. Example: miracle.
>
> (2) Temperaments supported by 31-equal--which is to say, with a kernel
> contained in the kernel of the 7-limit 31-et val. Example: valentine.
>
> (3) Temperaments containing 81/80. Example: godzilla.
>
> (4) Temperaments containing 81/80 and supported by 31-et. Example:
mothra.

This is the usual way of proceeding, with the idea of applying it all
to dark and sinister things called the cohomology of grassmannians and
the Schubert Calculus. For our purposes, it seems to be better to
refine this by settling boundry issues, which expands the above
classification of 7-limit temperaments by means of what we might call
the 7-limit TM flag for 31-et to the following:

1. Not supported by 31, common comma with 31 is not in the kernel of
meantone. Examples are blackwood, with common comma 1029/1024, and
hemififths, with common comma 2401/2400.

2. Not supported by 31, common comma with 31 is in the kernel of
meantone but is not 81/80. Examples are magic, pajara and garibaldi,
with common comma 225/224, and sensi, with common comma 126/125.

3. Not supported by 31, common comma with 31 is 81/80. Examples are
injera and godzilla.

4. Supported by 31, common comma with meantone is not 81/80. Examples
are miracle and orwell, with common comma with meantone of 225/224,
and hemiwuerschmidt, with common comma with meantone of 3136/3125.

5. Supported by 31, common comma with meantone is 81/80, but is not
meantone. Examples are mothra (supermajor seconds) semififths and squares.

6. Meantone. Obviously, the only example of this is meantone.