Tuning-Math Digests messages 8575 - 8599

This is an Opt In Archive . We would like to hear from you if you want your posts included. For the contact address see About this archive. All posts are copyright (c).

Contents Hide Contents S 9

Previous Next

8000 8050 8100 8150 8200 8250 8300 8350 8400 8450 8500 8550 8600 8650 8700 8750 8800 8850 8900 8950

8550 - 8575 -



top of page bottom of page down


Message: 8575

Date: Wed, 26 Nov 2003 21:53:11

Subject: Re: Finding Generators to Primes etc

From: Graham Breed

Paul Erlich wrote:

> Isn't he *assuming* that the fifth is the generator here? Sorry, i'm 
> having trouble following his reasoning . . .

Yes, but that page is out of date, and wrong anyway for octave 
equivalent vectors.  But it's the best that's currently "published". 
I've explained the modern way several times on this list, but as you've 
obviously forgotten I'll try again.

You form a matrix with the octave (1 0 0 ...) at the top, then a 
chromatic unison vector (it doesn't matter which) and below them the 
commas.  Take the adjoint (the inverse multiplied by the determinant). 
The left hand column is an equal temperament mapping -- describing the 
periodicity block corresponding to the chromatic UV.  The gcd of the 
next column is the number of periods to the octave, and when you divide 
through by that GCD you get the generator mapping.

The wedge product version I pretty much showed in a recent bra/ket post.


                        Graham


top of page bottom of page up down


Message: 8576

Date: Wed, 26 Nov 2003 21:54:40

Subject: Re: Finding Generators to Primes etc

From: Graham Breed

> if there's only one comma, is that the wedgie?

Yes.


top of page bottom of page up down


Message: 8577

Date: Wed, 26 Nov 2003 22:09:14

Subject: Re: Finding Generators to Primes etc

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:
> Paul Erlich wrote:
> 
> > Isn't he *assuming* that the fifth is the generator here? Sorry, 
i'm 
> > having trouble following his reasoning . . .
> 
> Yes, but that page is out of date, and wrong anyway for octave 
> equivalent vectors.  But it's the best that's 
currently "published". 
> I've explained the modern way several times on this list, but as 
you've 
> obviously forgotten I'll try again.

If I don't have a direct understanding of how something works, I 
won't retain it. Sorry.

> You form a matrix with the octave (1 0 0 ...) at the top, then a 
> chromatic unison vector (it doesn't matter which)

Is this one of those cases where you're saying chromatic unison 
vector but don't really mean it?

Anyway, thanks, and I hope you'll update your pages.


top of page bottom of page up down


Message: 8578

Date: Wed, 26 Nov 2003 15:52:54

Subject: Re: Finding the wedge product?

From: Carl Lumma

>Now we list them in alphabetical (also numerical) order of index
>inside the correct number of brackets.

"Lexigraphic order", no?

More later; I can't think in this hick country!  (Montana)

-Carl


top of page bottom of page up down


Message: 8579

Date: Wed, 26 Nov 2003 09:06:27

Subject: Re: Finding Generators to Primes etc

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" 
> <paul.hjelmstad@u...> wrote:
> As I was saying, for each prime, use the mapping common to both 5/12 
> and 8/19, which is
> 
> prime 2 = 1 period
> prime 3 = 2 periods - 1 generator
> prime 5 = 4 periods - 4 generators
> 
> > And then is rms applied after
> > that?
> 
> yes, you want to 'solve' the above system of equations for the 
> generator, to minimize your desired error function.

Paul H,

I hope you've installed the optional Solver Add-in for Excel.


top of page bottom of page up down


Message: 8580

Date: Wed, 26 Nov 2003 09:28:58

Subject: Re: Finding the wedge product?

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> So, can we get a version of your Gospel with this rolled in?

Lets first take the simplest case worth considering. The wedge product
of two 3-limit (2D) vectors.

[a1 a2> ^ [b1 b2>

The procedure is to first list every product of a coefficient from A
with a coefficient from B, i.e. their ordinary scalar products. So
with 2 coefficients in each there will be 2x2 = 4 products to
consider, a1*b1, a1*b2, a2*b1, a2*b2.

As you calculate each product, combine the indices of the two
coefficients to make a compound index for it. It is important to keep
the indices in their original order at this stage. So we have

product index
a1*b1 11
a1*b2 12
a2*b1 21
a2*b2 22

There are certain rules about what to to with each product now,
depending on its compound index. There are 3 possibilities:

1. If the indexes have a digit in common then ignore it. Just throw
the product away. So we throw away a1*b1 and a2*b2.

2. Otherwise if the digits in the compound index are already in
alphabetical order, do nothing. So a1*b2 is just fine as it is.

3. Otherwise if they are not in alphabetical order, then put them in
alphabetical order. But first, look at each digit of the compound
index in turn, and count how many larger digits are to the left of it.
Add up all these left-and-larger counts as you go, or just keep
counting so their counts accumulate. If the result is an odd number
then negate the product, otherwise leave it as it was. Consider the
index 21. There are zero larger digits to the left of the 2 (because
there are _no_ digits to the left of it), and there is one larger
digit to the left of the 1, namely the 2. So the total of the
left-and-larger counts is 1, an odd number. So a2*b1 becomes -a2*b1.
We now have

product index
 a1*b2 12
-a2*b1 21


Now find any products that have the same index and add them together.
So we have only 

product       index
a1*b2 - a2*b1 12.

Now list all these sums in alphabetical order of their indices, inside
as many brackets as the sum of the number of brackets in the two
arguments, and pointing in the same direction. The wedge
product is only defined for values having their brackets pointing the
same way.

So our answer is

[[a1*b2-a2*b1>>

Now lets try something more messy. A 7-limit (4D) vector wedged with a
7-limit bivector. This might represent combining a third comma with
two that have already been combined, as an intermediate result on the
way to finding the ET mapping where these all vanish.

[a1 a2 a3 a4> ^ [[b12 b13 b14 b23 b24 b34>>

We first make the list of products of all pairs, with their compound
indices.

product index
a1*b12 112
a1*b13 113
a1*b14 114
a1*b23 123
a1*b24 124
a1*b34 134
a2*b12 212
a2*b13 213
a2*b14 214
a2*b23 223
a2*b24 224
a2*b34 234
a3*b12 312
a3*b13 313
a3*b14 314
a3*b23 323
a3*b24 324
a3*b34 334
a4*b12 412
a4*b13 413
a4*b14 414
a4*b23 423
a4*b24 424
a4*b34 434

Now we get rid of all those with two digits the same. Of course once
you've got the idea, you wouldn't even bother writing them down in the
first place. This leaves.

product index left-and-larger count
a1*b23 123
a1*b24 124
a1*b34 134
a2*b13 213 1
a2*b14 214 1
a2*b34 234
a3*b12 312 2
a3*b14 314 1
a3*b24 324 1
a4*b12 412 2
a4*b13 413 2
a4*b23 423 2

And we do the left-and-larger counts on the indices that aren't
already in alphabetical order (shown above), and negate the product if
this is odd. And we end up with:

product index
a1*b23 123
a1*b24 124
a1*b34 134
-a2*b13 123
-a2*b14 124
a2*b34 234
a3*b12 123
-a3*b14 134
-a3*b24 234
a4*b12 124
a4*b13 134
a4*b23 234

Now we sum the products having the same index.

product                  index
a1*b23 + a3*b12 - a2*b13 123
a1*b24 - a2*b14 + a4*b12 124
a1*b34 - a3*b14 + a4*b13 134
a2*b34 - a3*b24 + a4*b23 234

Now we list them in alphabetical (also numerical) order of index
inside the correct number of brackets.

[[[a1*b23+a3*b12-a2*b13 a1*b24-a2*b14+a4*b12 a1*b34-a3*b14+a4*b13
a2*b34-a3*b24+a4*b23>>>

Voila!


top of page bottom of page up down


Message: 8581

Date: Wed, 26 Nov 2003 10:52:18

Subject: Re: Finding Generators to Primes etc

From: monz

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" 
> <paul.hjelmstad@u...> wrote:
> >
> > Hmm. Sorry, let me approach it this way. in 12&19,
> > you have 5/12 for the one and 8/19 for the other.
> > How do you come up with one raw generator-to-prime mapping.
> 
> As I was saying, for each prime, use the mapping 
> common to both 5/12 and 8/19, which is
> 
> prime 2 = 1 period
> prime 3 = 2 periods - 1 generator
> prime 5 = 4 periods - 4 generators


i'm not understanding a lot of this ... but i am curious
about this: why are you overshooting the prime with the
periods and then subtracting generators (instead of
coming as close under the prime as you can with the periods,
then adding generators)?

the latter is the way i've always thought of prime-mapping.
is there some special reason to do it "backwards" like this?


-monz


top of page bottom of page up down


Message: 8582

Date: Thu, 27 Nov 2003 01:37:22

Subject: Re: Finding the wedge product?

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >Now we list them in alphabetical (also numerical) order of index
> >inside the correct number of brackets.
> 
> "Lexigraphic order", no?

Yes. Although I think it's "lexicographic".


top of page bottom of page up down


Message: 8583

Date: Thu, 27 Nov 2003 01:45:54

Subject: Re: Finding Generators to Primes etc

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:
> 
> > Now *I* need to figure out how to get generators from commas!
> 
> My personal approach is to turn everything into wedgies, and then 
> derive everything *from* wedgies.

OK. Now it's time for me to wrestle with the term "wedgie". It this a
synonym for "multivector" (both contravariant and covariant)?


top of page bottom of page up down


Message: 8595

Date: Mon, 01 Dec 2003 22:31:31

Subject: Re: Enumerating pitch class sets algebraically

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx jon wild <wild@f...> wrote:
> > 
> > Gene - pc-sets are usually enumerated using Polya's method. IIRC 
> there are
> > 351 in 12-tet.
> 
> Thanks, I'll research that. I don't see how there can be 351; 361 
> should be a minimum.

352 is the number I remember -- maybe that includes the empty set. 
This got discussed on the music theory yahoogroup (from which I've 
since unsubscribed due to anti-semitic and anti-german postings).


top of page bottom of page up down


Message: 8599

Date: Mon, 01 Dec 2003 22:51:49

Subject: Re: Enumerating pitch class sets algebraically

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:
> 
> > 352 is the number I remember -- maybe that includes the empty 
set. 
> > This got discussed on the music theory yahoogroup (from which 
I've 
> > since unsubscribed due to anti-semitic and anti-german postings).
> 
> Flavell acting up?

Yeah, that was the name. Are you familiar with it from elsewhere?


top of page bottom of page up

Previous Next

8000 8050 8100 8150 8200 8250 8300 8350 8400 8450 8500 8550 8600 8650 8700 8750 8800 8850 8900 8950

8550 - 8575 -

top of page