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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
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
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
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
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
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
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.
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
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
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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