Understanding wedgies

OK, I read some explanations of wedgies and now I understand how to get a wedgie from a description of a tuning, and why it's unique. What I don't understand is what the wedgie actually means, and how to use it.

A val is a linear function from the space of monzos to the integers. What, then, is a wedgie? What does it map to what?

Given a wedgie and a monzo, how can you get the monzo's mapping in that temperament in terms of periods and generators?

The first few components of a wedgie seem to signify the proportion of the number of generators used to get different primes. For example, meantone starts <<1,4,10... so the mapping of 5 uses 4 times as many generators as the mapping of 3 (which just happens to be the generator) and 7 uses 10 times as many. What do the rest of the components mean?

Will I understand all this after taking a course in linear algebra or differential geometry? =P

Keenan

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

> A val is a linear function from the space of monzos to the integers.
What, then,
> is a wedgie? What does it map to what?

By wedging, a multival of order n maps one of order m to one of order
m+n. By taking complements, you can get more types of mappings.

> Given a wedgie and a monzo, how can you get the monzo's mapping in that
> temperament in terms of periods and generators?

The most straightforward way is to first compute the period-generator
mapping, one way to do that is to start with a period and generator
expressed in terms of p-limit intervals. We can use Tvc = ~(~T^c),
where T is the wedgie and c is an interval; this maps to a val. By
doing this with the basis generator pair, you can solve for anything else.

For example, if T = <<1 4 10 4 13 12||, then |1 0 0 0> is 2 and
|-1 1 0 0> is 3/2. We have

Tv|1 0 0 0> = <0 -1 -4 -10|
Tv|-1 1 0 0> = <1 1 0 -3|

This is all the information you need to get a mapping, since the
mapping is [<1 1 0 -3|, <0 1 4 10|]. Moreover, any other of these
products with a 7-limit interval will be a linear combination of the
above two. If P is the period and G the generator, the mapping is just

[TvG, -TvP]

This requires us to use a period other than a a fraction of an octave,
of course. For instance, for ennealimmal, we could use 27/25 for the
period and 5/3 for the generator, and get the appropriate mapping. But
there are other ways of getting a mapping from the wedgie.

> The first few components of a wedgie seem to signify the proportion
of the
> number of generators used to get different primes. For example,
meantone starts
> <<1,4,10... so the mapping of 5 uses 4 times as many generators as
the mapping
> of 3 (which just happens to be the generator) and 7 uses 10 times as
many. What
> do the rest of the components mean?

The first few are the complexity of the other primes in terms of 2, or
equivalently, of 2 in terms of the other primes. That is, 3 is one
generator step when 2 is the period, 5 is four generator steps, 7 is
ten generator steps. The other numbers give the complexity of 5 in
terms of 3, 7 in terms of 3, and 7 in terms of 5. Sort of the
Bohlen-Pierce part of the wedgie.

> Will I understand all this after taking a course in linear algebra or
> differential geometry? =P

Possibly. Or it could turn out you understand those a lot better after
understanding this.

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

> [TvG, -TvP]

I should add, if TvG maps the octave to a negative number (for
instance, what you get from 4/3 rather than 3/2) then change the sign
of the mapping to [-TvG, TvP].

"Gene Ward Smith" <gwsmith@svpal.org> writes:

> The most straightforward way is to first compute the period-generator
> mapping, one way to do that is to start with a period and generator
> expressed in terms of p-limit intervals. We can use Tvc = ~(~T^c),
> where T is the wedgie and c is an interval; this maps to a val. By
> doing this with the basis generator pair, you can solve for anything else.
>
> For example, if T = <<1 4 10 4 13 12||, then |1 0 0 0> is 2 and
> |-1 1 0 0> is 3/2. We have

What leads you to use 2 and 3/2, as opposed to some other interval
pair?

> Tv|1 0 0 0> = <0 -1 -4 -10|
> Tv|-1 1 0 0> = <1 1 0 -3|
>
> This is all the information you need to get a mapping, since the
> mapping is [<1 1 0 -3|, <0 1 4 10|].

Definition of "mapping", please? I know, I know, *everybody* knows
this -- except I don't. What have you just computed here?

(Then there's the question of *how* you computed it. Apparently it
involves complementing a wedgie -- which I don't know how to do -- and
doing a wedge product of the wedgie complement with the interval --
which I don't know how to do -- and complementing the result -- which
I don't know how to do.)

> This requires us to use a period other than a a fraction of an octave,
> of course.

Oh, of course.

Why?

- Rich Holmes

--- In tuning-math@yahoogroups.com, Rich Holmes<rsholmes@m...> wrote:

> What leads you to use 2 and 3/2, as opposed to some other interval
> pair?

I'd have to know on other grounds it works. I can show how to get a
period and generator using no prior assumptions from the wedgie, however.

> > Tv|1 0 0 0> = <0 -1 -4 -10|
> > Tv|-1 1 0 0> = <1 1 0 -3|
> >
> > This is all the information you need to get a mapping, since the
> > mapping is [<1 1 0 -3|, <0 1 4 10|].
>
> Definition of "mapping", please? I know, I know, *everybody* knows
> this -- except I don't. What have you just computed here?

This is a pair of vals, which can be thought of as a 4x2 matrix. Given
a 5-limit interval, the first val maps to the number of octaves
needed, and the second to the number of generators. For instance,
7/5 is represented by |0 0 -1 1>, then
<1 1 0 -3|0 0 -1 1> = -3
<0 1 4 10|0 0 -1 1> = 6
so 7/5 is the tempered version of 2^(-3)*(3/2)^6 = 729/512

> (Then there's the question of *how* you computed it. Apparently it
> involves complementing a wedgie -- which I don't know how to do -- and
> doing a wedge product of the wedgie complement with the interval --
> which I don't know how to do -- and complementing the result -- which
> I don't know how to do.)

Would it help if someone put up some pseudocode? Or alternatively, can
you read Graham's Python script? Some suite of programs is really how
to do this.

> > This requires us to use a period other than a a fraction of an octave,
> > of course.
>
> Oh, of course.
>
> Why?

Definitional hogwash. If you allow Monzo's fractional monzos,
fractions of an octave work just fine. The period can be found as the
GCD of the first n wedgie coefficients, where n is the number of odd
primes in your tuning limit. If you call that number Q, then setting

P = |1/Q 0 0 ...0>

works as a "P" you can stick into [TvG, -TvP]

Whatever method you use, you are simply taking the first n wedgie
values, dividing out the GCD, sticking a zero in front, and that's the
generator part of the map; hence this part of the problem isn't a
concern. The *period* part of the map is <Q x1 x2 .. xp|, where the
x's have to be determined. They are constrained by the fact that the
octave and period parts of the map, wedged together, must give the
wedgie. Hence you can stick in indeterminates, solve the resulting
equation, find a particular solution giving integer values, and you
have a period and generator pair. Now you can reduce this to something
you like, such as the smallest generator.

"Gene Ward Smith" <gwsmith@svpal.org> writes:

> Would it help if someone put up some pseudocode? Or alternatively, can
> you read Graham's Python script? Some suite of programs is really how
> to do this.

Actually <http://66.98.148.43/~xenharmo/wedge.html> answers this
fairly well, once you get past the first paragraph. Between that,
Herman's page, and a few recent messages here, there's probably enough
explanation to go on, for a while.

It'd be nice if this were all in one introductory text, preferably in
terms digestable by musicians, though.

- Rich Holmes

Rich Holmes<rsholmes@mailbox.syr.edu> writes:

> Actually <http://66.98.148.43/~xenharmo/wedge.html> answers this
> fairly well, once you get past the first paragraph.

Hmm, it says there in the 5 limit

(a2 v2 + a3 v3 + a5 v5)/\(b2 v2 + b3 v3 + b5 v5) =

(a3 b5 - a5 b3) v3 /\ v5 + (a2 b5 - a5 b2) v2/\v5
+ (a2 b3 - a3 b2) v2/\v3

which is fine, but then I become baffled by

(a2 v2 + a3 v3 + a5 v5 + a7 v7)/\(b2 v2 + b3 v3 + b5 v5 + b7 v7) =

(a2 b3 - a3 b2) v5/\v7 + (a2 b5-a5 b2) v3/\v7 + (a2 b7-b2 a7) v3/\v5 +
(a3 b5 - a5 b3) v2/\v7 + (a3 b7-a7 b3) v2/\v5 + (a5 b7-b5 a7) v2/\v3

How do you end up with terms like (a2 b3 - a3 b2) v5/\v7 ? I would've
expected things like (a2 b3 - a3 b2) v2/\v3 , as in the 5 limit.

- Rich Holmes

--- In tuning-math@yahoogroups.com, Rich Holmes<rsholmes@m...> wrote:

> (a2 v2 + a3 v3 + a5 v5 + a7 v7)/\(b2 v2 + b3 v3 + b5 v5 + b7 v7) =
>
> (a2 b3 - a3 b2) v5/\v7 + (a2 b5-a5 b2) v3/\v7 + (a2 b7-b2 a7) v3/\v5 +
> (a3 b5 - a5 b3) v2/\v7 + (a3 b7-a7 b3) v2/\v5 + (a5 b7-b5 a7) v2/\v3
>
> How do you end up with terms like (a2 b3 - a3 b2) v5/\v7 ? I would've
> expected things like (a2 b3 - a3 b2) v2/\v3 , as in the 5 limit.

Sorry about that. :(

--- In tuning-math@yahoogroups.com, Rich Holmes<rsholmes@m...> wrote:
>
> which is fine, but then I become baffled by

I've cleaned up the wedge and wedgie pages a little; these things had
problems going back to the earliest versions, where I was trying to
make things overly simple.

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

> The first few components of a wedgie seem to signify the proportion
of the
> number of generators used to get different primes.

Yes -- *given* that your period is an octave (prime 2).

For example, meantone starts
> <<1,4,10... so the mapping of 5 uses 4 times as many generators as
the mapping
> of 3 (which just happens to be the generator) and 7 uses 10 times as
many. What
> do the rest of the components mean?

I'm sure someone answered this already, but it's basically the same
thing but using primes other than 2 as the period. Moreover, the
numbers you already looked (1, 4, and 10) at tell you how many
generators it takes to get 2 when 3, 5, or 7, respectively, is the
period.