Tuning-Math Digests messages 6202 - 6226

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Message: 6202

Date: Sun, 26 Jan 2003 22:36:27

Subject: Re: Graham's top 20, with standard vals

From: Carl Lumma

>>Say what?  I thought a val was the complement of a vector.
> 
>It's the dual of a vector, if by a vector you mean an interval
>in Monzo notation. Due to the magic of Poincare duality, you
>can wedge with either a vector or a val.

I may be dangerously close to understanding vals.  I've read
the definition in monz's dictionary.  Anybody care to give
an example?  Maybe Paul could shed some light on a layman's
definition.

-Carl


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Message: 6204

Date: Sun, 26 Jan 2003 22:37:35

Subject: Re: Graham's Top 20 13-limit temperaments

From: Graham Breed

Gene Ward Smith  wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" <clumma@y...> wrote:
> 
>>>i have access to a 2.4 GHz machine for running Matlab overnight or 
>>>for however long it takes. i'd be happy to try whatever algorithms 
>>>you wish to spell out.
>>
>>One page claims Matlab is implemented in C.  I seem to think Maple
>>is implemented in Maple, but I can't find that in the manual now.
>>I'd be surprised if either of them were faster than python, but
>>I could very well be wrong.
> 
> 
> Maple is in C also, but it isn't designed for speed. For instance, the float data type has a precision defined by "Digits",
and the int data type allows for ints as big as the machine can
handle.

This has come up on comp.lang.python.  People who've used both say
that 
Numeric Python is slightly faster than Matlab, although Matlab's
matrix 
operations are faster.  Also, if there's a native library for Matlab
but 
not Python then Matlab's much faster.

There are also people using Python to drive Matlab.

Probably Maple is similar.  As Python doesn't come with arbitrary 
precision floating point, Maple will be faster when you need it. 
Python 
does have arbitrary sized integers, and they now interact seemlessly 
with the normal integers.  So it looks comparable to Maple for what we

need.  However, the Numeric extensions use a Fortran library that only

works with floating point -- there aren't any routines to efficiently 
find the adjoint of an integer matrix.  There's also nothing for wedge

products.


                     Graham


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Message: 6209

Date: Sun, 26 Jan 2003 09:41:16

Subject: Re: Temperament finder update

From: Graham Breed

Carl Lumma  wrote:

> numpy or Numarray?

The one you get from "ppm install Numeric" in ActivePython.  I think 
that's numpy.


                      Graham


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Message: 6210

Date: Sun, 26 Jan 2003 10:41:22

Subject: Re: Graham's Top 20 13-limit temperaments

From: Graham Breed

wallyesterpaulrus  wrote:

> calculate the numerators and denominators here which came out in 
> scientific notation, making it impossible for yahoo to sort by 
> denominator:
> 
> Yahoo groups: /tuning/database? *
> method=reportRows&tbl=10&sortBy=4

" large limma", "0 3 -2", "27", "25", "133.237575", "beep", "4, 5, 9", "[1200, 268.056439]", "[[1,0] [2,-2] [3,-3]]", "35.609240" *

                     Graham


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Message: 6213

Date: Mon, 27 Jan 2003 09:17:34

Subject: Re: Graham's Top 20 13-limit temperaments

From: Carl Lumma

> better than any other complexity measure! cool, so you must
> *really* like the heuristic for complexity . . .

Apparently so.

> my best recollection, off the top of my head:
> 
> log-flat badness < 3500, rms error < 50 cents, geometric
> complexity < 104-151 (doesn't matter where you draw the
> line in this range).

Ok, but two small nits:

() Is that geometric complexity as Gene defines it?

() Being that badness is just a combination error and
complexity, why is it needed / how can it change the
bounds on the list?

-Carl


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Message: 6214

Date: Mon, 27 Jan 2003 12:34:21

Subject: Re: Calculating geometric complexity II

From: Graham Breed

Gene Ward Smith  wrote:
> Here are Maple routines which have the exact coefficients. They are not, of course, computationally effiecient, but it would be easy to
> calculate
the logarithms only once if that is a problem, though I havn't found
complexity calculations to be a bottleneck. These should be readily
translatable to Matlab, Python, or anything else.

How are you indexing your wedgies?  I use tuples, so that when 
multiplying 1-vectors,

z[i, j] = x[i,] + y[j,]

where x[0,] is the octave coefficient, x[1,] the 3:1 and so on.  Can
you 
provide conversion tables between your [i] and my [i,j]?

I think I've got the idea of vals as well.  A dual isn't the same as a

complement!  Using ^ for the wedge product, and ~ for the complement,
we 
have

h12^~comma = ~comma^h12 = {}

where "h12" is the val for 5-limit 12-equal, "comma" is the unison 
vector for 81:80 and {} is the empty wedgie.

Vals and unison vectors are both 1-vectors.  For duality, I'll have to

add a flag to each object.  So the complement operation also inverts
the 
flag.  A unison vector has the dual flag set to 0 and a val has the
dual 
flag set to 1.

To compute a wedge product, both dual flags has to agree.  So when you

ask to calculate h12^comma, the function can look at the two dual
flags, 
see they aren't the same, and take the complement of the second
element 
to make it so.  That means, asking for

h12^comma

means you get

h12^~comma

and it doesn't matter if you meant

~h12^comma

because the dual flag gets set again for the next step of the 
calculation.  And in this case the result is the same anyway.

You can also say that

h12^h7 == comma

because h12^h7 will have its dual flag set, and the comparison
function 
knows it really has to return

h12^h7 == ~comma

and the routine for calculating a linear temperament knows it needs to

start with a wedgie that has its dual flag cleared, and so if you feed

it h12^h7 it converts it to ~(h12^h7).

I still don't know how to store a multivector so that it's its own
dual 
which seems to be what you're doing.


                   Graham


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Message: 6217

Date: Mon, 27 Jan 2003 10:55:35

Subject: Re: Graham's Top 20 13-limit temperaments

From: Carl Lumma

>because otherwise you'd have a huge number of temperaments, and not 
>the same number in each complexity range. for example, imagine how 
>many possible temperaments there must be with rms error < 50 cents 
>and complexity between, say, 74 and 104. some huge number.

Right on.  -C.


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Message: 6221

Date: Mon, 27 Jan 2003 08:08:37

Subject: Re: A common notation for JI and ETs

From: monz

> From: <gdsecor@xxxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Monday, January 27, 2003 6:52 AM
> Subject: [tuning-math] Re: A common notation for JI and ETs
>
>
> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith 
> <genewardsmith@j...>" <genewardsmith@j...> wrote:
> > --- In tuning-math@xxxxxxxxxxx.xxxx "gdsecor <gdsecor@y...>" 
> <gdsecor@y...> wrote:
> > 
> > > ***** HEY IF ANYBODY ELSE OUT THERE IS READING THIS,
> > > HERE'S A QUESTION:  What other ETs above 494 besides
> > > 612 and 624 would you want to notate -- ones in which
> > > the 5' comma (a.k.a, historical 5-schisma, 32768:32805)
> > > is either a single degree of the ET or vanishes?
> > 
> > 665, 684, 730, 742 and 836.
> 
> Thanks, Gene.  I'll also add 653 to that list.
> 
> But we won't be able to notate 684, because the 5' comma
> vanishes and no other symbol in the sagittal notation
> would represent a single degree, either.
> 
> --George



how about 768? ... because it's the tuning resolution for a
number of popular electronic instruments.



-monz


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Message: 6222

Date: Mon, 27 Jan 2003 16:29:31

Subject: Re: Calculating geometric complexity II

From: Graham Breed

Gene Ward Smith  wrote:

> I'm not sure if we are speaking the same language, but I'm using lexicographical order; that is, z[0,1], z[0,2] .... z[0,n] would
> be
followed by z[1,2]...z[1,n] and so forth. This gives a linear
temperament wedgie as the product of two vals, and puts the 2-part,
which is related to the generators column of the period-generator
matrix, at the beginning.

Oh, that's good.  It should be the same as my invariant.  But are 
7-limit wedge products taken from vectors or vals?

I get 7-limit meantone as 21.97, 11-limit meantone as 31.72 and 
h12^h19^h22 in the 11-limit as 29.52.  The planar temperament with 
441:440 and 225:224 is 34.44.

> Are you still using empty wedgies for zero vectors? I hope this
isn't giving problems. In any case, the above is definitional; ~comma
is the
> 3-product such that h ^ ~comma = h(comma), so that we can identify 
> compliments with duals.

Empty wedgies are empty wedgies.  I haven't had any trouble with them.

> I'm simply being unsophisticated about it--I store the wedgies as
lists, and reverse the ordering when I compute from commas, etc. in
order to get the lists to be the same.

That sounds like taking the complement.  I thought you said you didn't

have to because you were using duality.  And how can you be sure that 
reversing the list will do the trick?  Some of the coefficients should

be negated if you aren't using a special ordering.


                     Graham


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Message: 6223

Date: Mon, 27 Jan 2003 16:38:22

Subject: Re: Calculating geometric complexity II

From: Graham Breed

Gene Ward Smith  wrote:

> I'm not sure if we are speaking the same language, but I'm using lexicographical order; that is, z[0,1], z[0,2] .... z[0,n] would
> be
followed by z[1,2]...z[1,n] and so forth. This gives a linear
temperament wedgie as the product of two vals, and puts the 2-part,
which is related to the generators column of the period-generator
matrix, at the beginning.

Oh, and I expect you're indexing from 1 as well.  In which case I get

7-limit meantone  23.76
11-limit meantone  23.85
h12^h19^h22  22.77
441:440 ^ 225:224  28.57


                        Graham


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