Wedgies and generators

Suppose that W is a bival wedgie for a linear temperament. Then we can
take a product with a monzo u which maps to a val via W$u = ~(~W ^ u).
If u is a generator for the temperament, the val will be a
corresponding mapping to primes, up to sign. If u and v are a pair of
generators generating the temperament, then [+-W$v, -+W$u] will be the
mapping to primes for the pair u,v; we can choose signs in order to
normalize but we should make one a + and the other a minus if u and v
are both positive.

For instance, suppose u = 15/14 and v = 7/5, and W is the wedgie for
pajara. Then W$(15/14) = <2 3 5 6| in the 7-limit, <2 3 5 6 8| in the
11-limit. -W$(7/5) = <0 1 -2 -2| in the 7-limit, <0 1 -2 -2 -6| in the
11-limit. Putting them together gives a mapping for pajara; we can
choose other values for u (16/15, 21/20, etc ) or v (10/7, 99/70 etc)
and get the same result so long as they are equivalent under mapping
to pajara.

In particular, v=2 can be a period, in which case if we take the
mapping [W$g, -W$2] for the generator g and 2, we get a mapping for
generator g and period 2. For 7-limit meantone, for instance, we get

g=3; [W$3, -W$2] = [<1 0 -4 -13|, <0 1 4 10|]

g=3/2; [W$(3/2), -W$2] = [<1 1 0 -3|, <0 1 4 10|]

g=4/3; [W$(4/3), -W$2] = [<-1 -2 -4 -7|, <0 1 4 10|]

The mapping for 4/3 would normally be normalized by taking minus of
it, [<1 2 4 7|, <0 -1 -4 -10|].

If 2 is a generator, the second part of the mapping, -W$2, can be read
off directly from the wedgie by taking the first pi(p)-1 values of the
wedgie, where p is the prime limit. If 2 is not a generator, it will
be n times a generator, where n is the number of periods to the
octave. Hence the second part of the mapping can again be read easily
from the wedgie; n is the gcd of the first pi(p)-1 elements, and the
mapping val is those elements divided through by n. Since we are
normally concered with period-generator types of generators, the bival
wedgie has a clear advantage over the multimonzo alternative, in
giving the information we most want immediately and up front.

The other elements of the wedgie can be thought of in various ways;
for instance the first pi(p)-1 are from all the elements past the zero
in -W$2, and that pattern continues, the next pi(p)-2 are the elements
past the zero in -W$3, and so forth. Here's meantone:

-W$2: <0 1 4 10|
-W$3: <-1 0 4 13|
-W$5: <-4 -4 0 12|
-W$7: <-10 -13 -12 0|

Put it all together and you get an antisymmetric matrix, the upper
right corner of which is the wedgie. I think Herman was the one who
suggested writing it in the form

<<1 4 10
4 13
12||

which has a certain logic.

The elements are also related to commas for the temperament with three
prime factors; pi(p) choose 3 of these can be read off the wedgie;
since these are commas rather than vals we can do this a little easier
with the multimonzo form of the wedgie (not worrying about the signs
of the exponents), but since we don't normally want or need to do this
I don't see it as an important consideration. The val side and its
mappings is more immediate. Of course, since only two vals are
involved it is also easier to compute the wedgie, beyond the 7-limit,
from vals in the first place.

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

> Put it all together and you get an antisymmetric matrix, the upper
> right corner of which is the wedgie. I think Herman was the one who
> suggested writing it in the form
>
> <<1 4 10
> 4 13
> 12||

When I asked you, some time ago, about these triangles of numbers and
their great similarity to the upper triangle of certain matrix of
vanishing commas, you had no response.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
>
> > Put it all together and you get an antisymmetric matrix, the upper
> > right corner of which is the wedgie. I think Herman was the one who
> > suggested writing it in the form
> >
> > <<1 4 10
> > 4 13
> > 12||
>
> When I asked you, some time ago, about these triangles of numbers and
> their great similarity to the upper triangle of certain matrix of
> vanishing commas, you had no response.

Sorry. I did discuss it fairly extensively when I first introduced the
wedge product, if I recall correctly, in connection with the commutator.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> > --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> > wrote:
> >
> > > Put it all together and you get an antisymmetric matrix, the
upper
> > > right corner of which is the wedgie. I think Herman was the one
who
> > > suggested writing it in the form
> > >
> > > <<1 4 10
> > > 4 13
> > > 12||
> >
> > When I asked you, some time ago, about these triangles of numbers
and
> > their great similarity to the upper triangle of certain matrix of
> > vanishing commas, you had no response.
>
> Sorry. I did discuss it fairly extensively when I first introduced
the
> wedge product, if I recall correctly, in connection with the
commutator.

If you were teaching a class, I would have failed out a long time ago.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> If you were teaching a class, I would have failed out a long time
ago.

If I were teaching a class, I would be obligated to remember someone
asked me a question, and not forget about it in the press of other
considerations, or because I'd need to think about it to give an
answer.

Anyway, I don't recall the question.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> [to Gene]
> If you were teaching a class, I would have failed out a
> long time ago.

imagine the trouble i'm having keeping up with Gene's work.

:(

-monz