Tuning-Math Digests messages 10526 - 10550

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Message: 10526

Date: Fri, 05 Mar 2004 21:12:11

Subject: Re: Hahn norm formula

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:
> 
> > > Despite its apparent theoretical interest, Crystal Ball Two 
should
> > > suffice as a name,
> > 
> > Paul Hahn might have called it the 7-limit radius 2 scale or 
perhaps 
> > more likely, the Level 2 7-limit Diamond.
> 
> Since apparently it is called a cystal ball in some nonmusical
> connection already, it seems to me better to stick with that, on 
your
> own principle of established names.

In nonmusical connections?


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Message: 10527

Date: Fri, 05 Mar 2004 13:21:37

Subject: Re: Hahn norm formula

From: Carl Lumma

Can someone briefly explain shell and hull, and the difference?
The mathworld definitions are, as usual, obtuse, and it isn't
clear which shell definition is in use here.  And has anyone
noticed how mathworld is slowly becoming a Mathematica help file?

My guess (based on the "generalization of an annulus" definition
of shell) is that in 3-D, a hull is a surface while a shell is a
volume.  Is this correct?

-Carl


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Message: 10528

Date: Fri, 05 Mar 2004 21:25:31

Subject: Re: Hahn norm formula

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> Can someone briefly explain shell and hull, and the difference?

Gene uses "shell" to mean the set of notes at a given "distance" from 
the origin. I know what a "convex hull" is but don't know what "hull" 
in general means, if anything.


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Message: 10536

Date: Sat, 06 Mar 2004 23:44:11

Subject: Re: Canonical generators for 7-limit planar temperaments

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
> wrote:
> > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" 
> > <paul.hjelmstad@u...> wrote: 
> >  
> > > > I gave a semidefinite form in terms of 3, 5 and 9; you feed 3/2 
> > and 
> > > > 9/7 into that, and get a definite form in terms of those two. 
> > >  
> > > Okay. I have to figure out how you killed off "c", and how "b" 
> is  
> > > based on 9/7... 
> > > on 9/7, and so forth 
> >  
> > The new a and b are different from the old a and b; I killed off c 
> > because I never fed a c in in the first place.
> 
> True, I should know better. I just need to figure out how it is 
> *calculated*. I certainly don't expect you to crunch numbers for me...
> that I can do myself (hopefully) once I understand things 
> theoretically. Here's a more theoretical question: Why does doing
> an orthogonal projection make the comma vanish? 

I think I can jump in with the answer here -- because it makes all
comma-separated pairs of notes overlap exactly, thus representing the
comma by a zero length in the lattice.


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Message: 10537

Date: Sat, 06 Mar 2004 23:46:12

Subject: Re: Between Hahn and Euclid

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
> wrote:
> 
>  If we put p=1 into it, I wonder if that is
> > useful for anything? Paul liked the L1 error; this would be the
> > corresponding norm on note classes.
> 
> I looked at the p=1 norm around the unison. The first two shells
> correspond to the first two shells of Euclidean, leading to this
> 19-note scale once again:
> 
> [1, 21/20, 15/14, 8/7, 7/6, 6/5, 5/4, 4/3, 48/35, 7/5, 10/7, 35/24,
> 3/2, 8/5, 5/3, 12/7, 7/4, 28/15, 40/21]
> 
> The next shell, with 36 elements, is the union of the third and fourth
> Euclidean shells, and the one after that, with 24 elements, is the
> Euclidean fifth shell. Beyond this point the shells don't correspond.

Cool. What if you center around a deep hole? Thanks.


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Message: 10545

Date: Sun, 07 Mar 2004 15:00:58

Subject: Re: Hanzos

From: Graham Breed

Gene Ward Smith wrote:

> There's nothing more complicated about it really if you do it in exact
> analogy to TOP; the question then is what are you taking to be the
> analog of the Tenney norm?

You can do it the old pre-TOP way by taking the number of n-limit 
intervals to the comma.  Or use the Kees metric, being like the Tenney 
one, but only considering the larger odd number in the ratio.  Either 
way, it could well give the right tuning (depending on what you're doing 
for multiple commas) but you still have to be able to extract the right 
generator from it.


                  Graham


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Message: 10546

Date: Sun, 07 Mar 2004 15:50:43

Subject: Re: Hanzos

From: Graham Breed

Gene Ward Smith wrote:

> I think this would be clearer with some examples. Let's say you have
> 21 and 41. How do you get miracle out of the pair of them by your
> method? Then the same question for 1029/1024 and 16875/16807.

Even with the octave specific method, 21 and 41 don't give miracle, but:

3/62, 58.4 cent generator

basis:
(1.0, 0.048647720621243257)

mapping by period and generator:
[(1, 0), (1, 12), (1, 27), (3, -4), (2, 30)]

mapping by steps:
[(41, 21), (65, 33), (95, 48), (115, 59), (142, 72)]

highest interval width: 34
complexity measure: 34  (41 for smallest MOS)
highest error: 0.008440  (10.128 cents)

1029/1024 and 16875/16807 confuse the program, which suggests that 
whatever I was doing before won't work here.  Hmm.  Anyway, using 3:1 as 
the chroma gives
[[  1,   0,   0,   0],
  [  0,   1,   0,   0],
  [-10,   1,   0,   3],
  [  0,   3,   4,  -5]]

and an adjoint of

[[ 12,   0,   0,   0],
  [ 24,  12,   0,   0],
  [ 22, -14,   5,   3],
  [ 32,  -4,   4,   0]]

So there is some spurious torsion in the second column.  I wonder how I 
got rid of it before.  The way the algorithm actually works, it 
discovers that a period of 6 will work before it tries the period of 3 
that it thinks will work.

Well, if you need to find torsion with octave-equivalent vectors, you 
can always work out the periodicity block I suppose.


                 Graham


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Message: 10548

Date: Sun, 07 Mar 2004 22:01:28

Subject: Octave equivalent calculations (Was: Hanzos

From: Graham Breed

Gene Ward Smith wrote:

> The two vals <21 33 49 59| and <41 65 95 115| should lead to miracle;
> if they don't something is wrong. The same goes in the 11-limit for
> <21 33 49 59 72| and <41 65 95 115 142|.

Why should they?  They never have before.

> This is the two vals <41 65 95 115 142| and <21 33 48 59 72|, which
> leads to the question why the second val for 21-equal; it doesn't seem
> like first choice. The temperament in question, with TM basis {100/99,
> 245/243, 1029/1024}, is associated firmly to 41 at any rate.

That val is for the best equal temperament, isn't it?  Can you find a 
better one?

> I think you are making a good case for the claim the best plan is to
> use wedgies.

Wedgies give identical results to matrices, so what difference does it make?

                 Graham


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