Getting from Wedgies to Generators

One feature of tuning theory discourse that I haven't thus far been able to figure out by myself is the use of bival wedgies to describe rank-2 moments of symmetry that are compatible with a given equal temperament. I'd greatly appreciate it if some of you could explain this at a basic level & help me understand.

For instance this 94edo page:

http://xenharmonic.wikispaces.com/94edo

includes this excerpt:

Below are some 23-limit temperaments supported by 94et. [snip]

46&94 <<8 30 -18 -4 -28 8 -24 2 ... ||
68&94 <<20 28 2 -10 24 20 34 52 ... ||
53&94 <<1 -8 -14 23 20 -46 -3 -35 ... ||
41&94 <<1 -8 -14 23 20 48 -3 -35 ... ||
135&94 <<1 -8 -14 23 20 48 -3 59 ... ||
130&94 <<6 -48 10 -50 26 6 -18 -22 ... ||
58&94 <<6 46 10 44 26 6 -18 -22 ... ||
50&94 <<24 -4 40 -12 10 24 22 6 ... ||
72&94 <<12 -2 20 -6 52 12 -36 -44 ... ||
94 solo <<12 -2 20 -6 -42 12 -36 -44 ... ||
80&94 <<18 44 30 38 -16 18 40 28 ... ||

Okay, me again. Now tell me if I'm wrong here. A wedgie represents any number as factorized into prime powers. Between the wedge brackets & the vertical brackets is a list of exponents, in order, of the primes within some prime limit. Raise each prime to its corresponding exponent, then multiply the resulting prime powers together, & you obtain the represented number. Have I gone wrong yet?

Well, the few wedgies in that list that I've bothered to check all seem to represent very small numbers. Does that mean they are commas? Is each the comma (or product of commas?) whose tempering out defines a temperament? If not, please correct me.

If so, how do you or I get from there, the defining comma value, to the size of the generator? In the excerpt shown above, looking at, say, 94 solo, what size generator in 94edo scale degrees is implied or encoded by the wedgie? How do I figure that out also for other temperaments without having to ask every time?

Thanks for your patience with my ignorance & errors, & thanks in advance for all efforts to cure me of such ills.

Cheers!
==
MLV aka Manu Phonic

--- In tuning@yahoogroups.com, "manuphonic" <manuphonic@...> wrote:
>
> One feature of tuning theory discourse that I haven't thus far been able to figure out by myself is the use of bival wedgies to describe rank-2 moments of symmetry that are compatible with a given equal temperament. I'd greatly appreciate it if some of you could explain this at a basic level & help me understand.

Wedgies are really hard to understand. They're not intuitive.

They don't describe MOSes at all, they only describe infinite temperaments. For example, <<1 4 10 4 13 12|| describes meantone as a whole, not any of its MOSes (pentatonic, diatonic, chromatic, etc.).

They're the perfect mathematical abstraction to represent any regular temperament uniquely, but they are not very practically useful. If you can ever get a "reduced mapping" rather than a wedgie, that's much easier to understand if you're not a math wizard.

> For instance this 94edo page:
>
> http://xenharmonic.wikispaces.com/94edo
>
> includes this excerpt:
>
> Below are some 23-limit temperaments supported by 94et. [snip]
>
> 46&94 <<8 30 -18 -4 -28 8 -24 2 ... ||
> 68&94 <<20 28 2 -10 24 20 34 52 ... ||
> 53&94 <<1 -8 -14 23 20 -46 -3 -35 ... ||
> 41&94 <<1 -8 -14 23 20 48 -3 -35 ... ||
> 135&94 <<1 -8 -14 23 20 48 -3 59 ... ||
> 130&94 <<6 -48 10 -50 26 6 -18 -22 ... ||
> 58&94 <<6 46 10 44 26 6 -18 -22 ... ||
> 50&94 <<24 -4 40 -12 10 24 22 6 ... ||
> 72&94 <<12 -2 20 -6 52 12 -36 -44 ... ||
> 94 solo <<12 -2 20 -6 -42 12 -36 -44 ... ||
> 80&94 <<18 44 30 38 -16 18 40 28 ... ||
>
> Okay, me again. Now tell me if I'm wrong here. A wedgie represents any number as factorized into prime powers. Between the wedge brackets & the vertical brackets is a list of exponents, in order, of the primes within some prime limit. Raise each prime to its corresponding exponent, then multiply the resulting prime powers together, & you obtain the represented number. Have I gone wrong yet?

No, this is quite wrong. The thing you're describing is a monzo, not a wedgie. The numbers in a wedgie do not correspond to single primes, but *pairs* of primes.

The examples above are not complete wedgies, hence the '...'s. The full monzos would have 36 entries, because there are 9 primes up to and including 23, so there are 9*8/2 = 36 possible pairs of those 9 primes.

The reason it only gives the first 8 is because those are all the pairs where one of the primes in the pair is 2. This is sort of all you need to understand the temperament; if you want to you can think of the first number as the number of generators you need to get to 3, the second as the number you need to get to 5, and so on. Then all the rest of the numbers after the ones involving the prime 2 are just confusing junk.

(Of course, they actually are meaningful, not just junk. I think the best explanation is here if you're interested:
http://xenharmonic.wikispaces.com/Wedgies+and+Multivals
Relevant stuff is in the third paragraph.)

> Well, the few wedgies in that list that I've bothered to check all seem to represent very small numbers. Does that mean they are commas? Is each the comma (or product of commas?) whose tempering out defines a temperament? If not, please correct me.

This is also wrong. These are 29-limit rank-2 temperaments, which means a hell of a lot of commas are all being tempered out. 29-limit JI is rank-9, so if you only tempered out 1 comma you would get a rank-8 system, not rank-2.

> If so, how do you or I get from there, the defining comma value, to the size of the generator? In the excerpt shown above, looking at, say, 94 solo, what size generator in 94edo scale degrees is implied or encoded by the wedgie? How do I figure that out also for other temperaments without having to ask every time?

I actually don't know the answer to this question off the top of my head. Maybe someone like Gene will step in and say how to find a generator given a rank-2 wedgie.

> Thanks for your patience with my ignorance & errors, & thanks in advance for all efforts to cure me of such ills.

No problem!

Keenan

Keenan, I cannot tell you how much I appreciate your elucidating explanation.
==
MLV aka Manu Phonic

--- In tuning@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:
>
> --- In tuning@yahoogroups.com, "manuphonic" <manuphonic@> wrote:
> >
> > One feature of tuning theory discourse that I haven't thus far been able to figure out by myself is the use of bival wedgies to describe rank-2 moments of symmetry that are compatible with a given equal temperament. I'd greatly appreciate it if some of you could explain this at a basic level & help me understand.
>
> Wedgies are really hard to understand. They're not intuitive.
>
> They don't describe MOSes at all, they only describe infinite temperaments. For example, <<1 4 10 4 13 12|| describes meantone as a whole, not any of its MOSes (pentatonic, diatonic, chromatic, etc.).
>
> They're the perfect mathematical abstraction to represent any regular temperament uniquely, but they are not very practically useful. If you can ever get a "reduced mapping" rather than a wedgie, that's much easier to understand if you're not a math wizard.
>
> > For instance this 94edo page:
> >
> > http://xenharmonic.wikispaces.com/94edo
> >
> > includes this excerpt:
> >
> > Below are some 23-limit temperaments supported by 94et. [snip]
> >
> > 46&94 <<8 30 -18 -4 -28 8 -24 2 ... ||
> > 68&94 <<20 28 2 -10 24 20 34 52 ... ||
> > 53&94 <<1 -8 -14 23 20 -46 -3 -35 ... ||
> > 41&94 <<1 -8 -14 23 20 48 -3 -35 ... ||
> > 135&94 <<1 -8 -14 23 20 48 -3 59 ... ||
> > 130&94 <<6 -48 10 -50 26 6 -18 -22 ... ||
> > 58&94 <<6 46 10 44 26 6 -18 -22 ... ||
> > 50&94 <<24 -4 40 -12 10 24 22 6 ... ||
> > 72&94 <<12 -2 20 -6 52 12 -36 -44 ... ||
> > 94 solo <<12 -2 20 -6 -42 12 -36 -44 ... ||
> > 80&94 <<18 44 30 38 -16 18 40 28 ... ||
> >
> > Okay, me again. Now tell me if I'm wrong here. A wedgie represents any number as factorized into prime powers. Between the wedge brackets & the vertical brackets is a list of exponents, in order, of the primes within some prime limit. Raise each prime to its corresponding exponent, then multiply the resulting prime powers together, & you obtain the represented number. Have I gone wrong yet?
>
> No, this is quite wrong. The thing you're describing is a monzo, not a wedgie. The numbers in a wedgie do not correspond to single primes, but *pairs* of primes.
>
> The examples above are not complete wedgies, hence the '...'s. The full monzos would have 36 entries, because there are 9 primes up to and including 23, so there are 9*8/2 = 36 possible pairs of those 9 primes.
>
> The reason it only gives the first 8 is because those are all the pairs where one of the primes in the pair is 2. This is sort of all you need to understand the temperament; if you want to you can think of the first number as the number of generators you need to get to 3, the second as the number you need to get to 5, and so on. Then all the rest of the numbers after the ones involving the prime 2 are just confusing junk.
>
> (Of course, they actually are meaningful, not just junk. I think the best explanation is here if you're interested:
> http://xenharmonic.wikispaces.com/Wedgies+and+Multivals
> Relevant stuff is in the third paragraph.)
>
> > Well, the few wedgies in that list that I've bothered to check all seem to represent very small numbers. Does that mean they are commas? Is each the comma (or product of commas?) whose tempering out defines a temperament? If not, please correct me.
>
> This is also wrong. These are 29-limit rank-2 temperaments, which means a hell of a lot of commas are all being tempered out. 29-limit JI is rank-9, so if you only tempered out 1 comma you would get a rank-8 system, not rank-2.
>
> > If so, how do you or I get from there, the defining comma value, to the size of the generator? In the excerpt shown above, looking at, say, 94 solo, what size generator in 94edo scale degrees is implied or encoded by the wedgie? How do I figure that out also for other temperaments without having to ask every time?
>
> I actually don't know the answer to this question off the top of my head. Maybe someone like Gene will step in and say how to find a generator given a rank-2 wedgie.
>
> > Thanks for your patience with my ignorance & errors, & thanks in advance for all efforts to cure me of such ills.
>
> No problem!
>
> Keenan
>

--- In tuning@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:
>
> --- In tuning@yahoogroups.com, "manuphonic" <manuphonic@> wrote:

[snip]

> > For instance this 94edo page:
> >
> > http://xenharmonic.wikispaces.com/94edo
> >
> > includes this excerpt:
> >
> > Below are some 23-limit temperaments supported by 94et. [snip]
> >
> > 46&94 <<8 30 -18 -4 -28 8 -24 2 ... ||
> > 68&94 <<20 28 2 -10 24 20 34 52 ... ||
> > 53&94 <<1 -8 -14 23 20 -46 -3 -35 ... ||
> > 41&94 <<1 -8 -14 23 20 48 -3 -35 ... ||
> > 135&94 <<1 -8 -14 23 20 48 -3 59 ... ||
> > 130&94 <<6 -48 10 -50 26 6 -18 -22 ... ||
> > 58&94 <<6 46 10 44 26 6 -18 -22 ... ||
> > 50&94 <<24 -4 40 -12 10 24 22 6 ... ||
> > 72&94 <<12 -2 20 -6 52 12 -36 -44 ... ||
> > 94 solo <<12 -2 20 -6 -42 12 -36 -44 ... ||
> > 80&94 <<18 44 30 38 -16 18 40 28 ... ||

[snip]

> > In the excerpt shown above, looking at, say, 94 solo, what size generator in 94edo scale degrees is implied or encoded by the wedgie? How do I figure that out also for other temperaments without having to ask every time?
>
> I actually don't know the answer to this question off the top of my head. Maybe someone like Gene will step in and say how to find a generator given a rank-2 wedgie.

What about it, Gene? Anyone?

Thanks!
==
MLV aka Manu Phonic

"manuphonic" <manuphonic@...> wrote:

> > I actually don't know the answer to this question off
> > the top of my head. Maybe someone like Gene will step
> > in and say how to find a generator given a rank-2
> > wedgie.
>
> What about it, Gene? Anyone?

Eugh! Right. What you can do is take the first d numbers
where d is the number of primes you're dealing with. Then
take d-1 numbers, add a zero to the front, and put them in
the second row. If it's a rank 3 temperament, take another
d-2 numbers, but two zeros in front, and you have the third
row. And so on up to the rank you're dealing with. Then
you have a mapping, that may have torsion, but that you can
feed into an optimal tuning algorithm.

That'll be on the Xenwiki somewhere, and also here:

http://x31eq.com/te.pdf

Graham

"manuphonic" <manuphonic@...> wrote:

> What about it, Gene? Anyone?

Oh, wait, what I just said won't work because there are
other entries that are guaranteed to be zero. And the
octave-equivalent part comes first.

You can rearrange it somehow to get a mapping, anyway.
Sometimes.

For rank 2, the wedgie is really the triangular part of a
matrix defined as

W[i,j] = M[1,i]*M[2,j] - M[2,i]*M[1,j]

That's what you need.

Graham

--- In tuning@yahoogroups.com, "manuphonic" <manuphonic@...> wrote:

> > I actually don't know the answer to this question off the top of my head. Maybe someone like Gene will step in and say how to find a generator given a rank-2 wedgie.
>
> What about it, Gene? Anyone?

It can get complicated, but start with this:

http://xenharmonic.wikispaces.com/Abstract+regular+temperament

This has a lot about about transforming from one way of representing a regular temperament to another; for instance, from wedgies to val lists.

http://xenharmonic.wikispaces.com/Transversal+generators

This shows how to get generators which correspond with val lists.

Some time ago, I left some of my comments on this in message # 100994, IIRC.

Petr