Tuning-Math Digests messages 10551 - 10575

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Message: 10551

Date: Sun, 07 Mar 2004 09:44:25

Subject: Re: Hanzos

From: Graham Breed

Paul Erlich wrote (2nd March):

> Right, but that was a specific set of calculations, not a general 
> proof. You were just looking at 'linear' temperaments, if I recall 
> correctly, but torsion can afflict all types of temperaments. Plus it 
> seemed your method was far less elegant.

What was inelegant about the torsion finder?  I don't remember anything. 
  I thought it was easier than for the octave-specific case because you 
don't have to check for a common divisor in the left hand column because 
the left hand column isn't there.  So torsion should show up iff the 
adjoint matrix has a common divisor.

That certainly should work, because if it gives different results to the 
octave-specific case it means the left hand (octave-specific) column 
must miss a common divisor that the rest of the adjoint has.  For the 
simple case where the common divisor is 2, that means a power of each 
odd prime is equal to an odd number of octaves plus an even number of 
some comma.  Divide it through, and you get a non-square rational number 
to be the square of another rational number, and so there's a formula 
for the exact solution of the square root of a rational number!  This 
runs into problems with the fundamental theorem of arithmetic, so it 
can't be possible.

Anyway, I'm less likely to miss something in the tests than that proof. 
  And the general case doesn't matter, only the specific cases that 
refer to interesting temperaments.  I don't even think all of those 
matter -- if the overwhelming majority of cases of torsion are 
discovered and dealt with, that's good enough.  There are lots of sets 
of unison vectors that don't give sensible results, and there's always a 
set of unison vectors to give a specific temperament without torsion.

The linear temperament issue doesn't matter.  Torsion is a property of 
periodicity blocks, and it shouldn't matter how many chromatic unison 
vectors you choose (or which ones).  I think I checked the whole adjoint 
matrix anyway.  I thought we could prove that (for sensibly small unison 
vectors) the octave-equivalent adjoint is a subset of the 
octave-specific adjoint.

> I care, and I hope Gene does too. I'd like to see this revisited.

The problem is that you have to consider modulo arithmetic in the 
optimization problem.  Let's take quarter comma meantone as the example. 
  The rule is that four fifths have to add up to a 5:4.  In other words, 
the generator is a quarter of a 5:4.  Well, 5:4 is 386.3 cents so the 
generator must be 96.6 cents.  You can see this is wrong because 96.6 is 
absurdly inaccurate for a 3:2 of 702.0 cents.

There are four different results of division by 4 in modulo arithmetic. 
  So a quarter of 5:4 could be 96.6, 396.6, 696.6 or 996.6 cents. 
Obviously you choose the best one.

The old way of finding the minimax is to set the intervals between every 
consonant interval and every other consonant interval to be just.  Then 
take whichever of these tunings gives the best result.  The same should 
work for the octave equivalent case, so long as you take all the 
different generators that make each interval just.

The newer way is to trace the worst error function downhill until you 
get to the bottom.  I didn't expect this to work, but it does because 
the function only has one local minimum.  I think it's that bit less 
likely to work in the octave-equivalent case, but you could give it a try.

For the RMS, it gets more complicated because the total error can't be 
represented as a single quadratic function anymore.  Maybe your 
numerical packages can solve this.  The only way I can think is to find 
the minimax using the method above.  The problem is that it would take a 
lot longer, whereas currently this is a simple calculation.  Perhaps 
only solve for one consonance, and hope you get in the right basin of 
attraction.  Still, it can certainly be done.


                     Graham


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Message: 10552

Date: Sun, 07 Mar 2004 23:53:21

Subject: Re: Octave equivalent calculations (Was: Hanzos

From: Graham Breed

Gene Ward Smith wrote:

> That would depend on your definition of better; but the one with the
> TOP tuning closest to JI is the "standard" val, <21 33 48 59 73|. If
> we put that together with 41, we get an 11-limit temperament which
> adds 245/242 rather than 385/384 to 225/224 and 1029/1024, leading to
> a temperament with a half-secor generator. Incidentally, if we add 
> 243/242 to 225/224 and 1029/1024, what does your program give?

My definition is the closest to JI.  <21 33 48 59 72| has a worst 
11-limit error of 0.81 scale steps, but for <21 33 49 59 73| it's 0.92 
steps.  There isn't a standard val because the temperament is inconsistent.

I get this temperament from the nearest prime approximations of 21 and 
41 and also the unison vectors you gave:

3/62, 58.4 cent generator

basis:
(1.0, 0.048631497540919798)

mapping by period and generator:
[(1, 0), (1, 12), (3, -14), (3, -4), (4, -11)]

mapping by steps:
[(41, 21), (65, 33), (95, 49), (115, 59), (142, 73)]

highest interval width: 38
complexity measure: 38  (41 for smallest MOS)
highest error: 0.008391  (10.069 cents)
unique

Which looks like what you said, but not miracle.  You get miracle from 
that other set.

> At this point it is by no means clear your matrix methods are giving
> correct answers, and in any case you seem to be working a lot harder
> for them.

They're giving exactly the same results as the wedgies.


                 Graham


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Message: 10556

Date: Mon, 08 Mar 2004 18:31:13

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.

Does this lead to shells with cardinalities = rhombic dodecahedral 
numbers?


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Message: 10557

Date: Mon, 08 Mar 2004 20:15:51

Subject: Re: Between Hahn and Euclid

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:
> --- 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.
> 
> Does this lead to shells with cardinalities = rhombic dodecahedral 
> numbers?

Apparently the latter are customarily defined with respect to the 
cubic, not FCC, lattice :(

Rhombic Dodecahedral Number -- from MathWorld *


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Message: 10563

Date: Mon, 08 Mar 2004 09:04:11

Subject: Re: Octave equivalent calculations (Was: Hanzos

From: Graham Breed

Me:
>>They're giving exactly the same results as the wedgies.

Gene:
> They don't seem to be doing that. Are you saying you are getting
> results corresponding to my claims?

I get the results I say I get, whether using wedgies, matrices (for 
unison vectors) or the direct method for combining ETs.  You keep 
claiming I'm wrong, but don't say why.  Except for torsion, the 
arithmetic is identical for matrices and wedge products, so I don't see 
what difference it should make.


                     Graham


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Message: 10565

Date: Mon, 08 Mar 2004 03:29:42

Subject: inspiration for "jumping jacks"?

From: Carl Lumma

Jumping Champion -- from MathWorld *

-C.



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Message: 10566

Date: Mon, 08 Mar 2004 21:54:47

Subject: Re: Dual L1 norm deep hole scales

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:

> Ball 2 radius 6 14 notes
> [1, 21/20, 7/6, 6/5, 49/40, 5/4, 21/16, 7/5, 3/2, 8/5, 42/25, 
> 12/7, 7/4,9/5]

Just as I suspected. We've finally constructed the Stellated Hexany, 
aka Mandala!


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Message: 10567

Date: Mon, 08 Mar 2004 21:59:40

Subject: Re: Dual L1 norm deep hole scales

From: Paul Erlich

P.S. Why aren't they simply the "L1 deep hole" results? Why dual? I 
tend to think of the monzo space or lattice of notes itself as 
the 'standard' space, while the space of linear functionals on it 
(breeds) as its dual. Or am I misunderstanding?


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Message: 10574

Date: Mon, 08 Mar 2004 22:46:50

Subject: Re: Dual L1 norm deep hole scales

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:
> > --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" 
<gwsmith@s...> 
> > wrote:
> > 
> > > Ball 2 radius 6 14 notes
> > > [1, 21/20, 7/6, 6/5, 49/40, 5/4, 21/16, 7/5, 3/2, 8/5, 42/25, 
> > > 12/7, 7/4,9/5]
> > 
> > Just as I suspected. We've finally constructed the Stellated 
Hexany, 
> > aka Mandala!
> 
> Eh, I think that's turned up already. :)

Did it?


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