Tuning-Math Digests messages 5950 - 5974

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Message: 5950

Date: Mon, 13 Jan 2003 22:50:59

Subject: Re: Nonoctave scales and linear temperaments

From: Carl Lumma

>>I thought of that, but I thought also that as long as one always
>>uses the same set of targets across temperaments, one is ok.
>>Whaddya think?
//
>why would keeping the same set of targets help? complexity
>shouldn't be this arbitrary!

Because complexity is comparitive.  The idea of a complete
otonal chord is not less arbitrary.

>>>multiply the generator span of the otonal (or utonal) n-ad by the
>>>number of periods per octave.
>>
>>That's what I thought.  How does this compare to the taxicab
>>approach?  Say, for Pajara.
>
>i'm unclear on what taxicab approach you mean. be patient with me,
>i know this would be easier in person. but i have to go now.

Oh, sure, dude.  You still abroad?  I've got to go now, too.
Just count the number of generators it takes, on the shortest
route in a rect. lattice, to get to the approx. of the target
interval.

>i don't think anyone's been doing weighted error on this list.

Oh, crap.  What is it you've been pushing, then?
Weighted complexity?

>but if you did, you'd minimize
>
>f(w3*error(3),w5*error(5),w5*error(5:3))
>
>where f is either RMS or MAD or MAX or whatever, and w3 is your 
>weight on ratios of 3, and w5 is your weight on ratios of 5.

Thanks.  So my f is +, where you tend to use RMS.  And I've got
complexity in the w's, which is a mistake, as I'll post about
shortly...

-Carl


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Message: 5954

Date: Tue, 14 Jan 2003 16:33:43

Subject: Re: Notating Pajara

From: Graham Breed

Kalle Aho  wrote:

> I get 709.363 for 7-limit and 708.128 for 9-limit. 

But I agree with Gene's 706.843 for the 9-limit pajara minimax.  It 
looks like he accidentally copied that for the 7-limit.


                          Graham


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Message: 5965

Date: Tue, 14 Jan 2003 01:46:51

Subject: Re: Nonoctave scales and linear temperaments

From: Carl Lumma

>All of which being why I came up with geometric complexity,
>which is invariant with respect to choice of generators and
>does not have this problem.

Unfortunately, you may be the only one on this list that
understands geometric complexity.  :(

-Carl


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Message: 5973

Date: Tue, 14 Jan 2003 11:51:41

Subject: Re: Notating Pajara

From: manuel.op.de.coul@xxxxxxxxxxx.xxx

Gene wrote:
>minimax 706.8431431

Could this be wrong? I have 709.363 in the scale archive.

Manuel


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Message: 5974

Date: Tue, 14 Jan 2003 22:38:49

Subject: Re: Nonoctave scales and linear temperaments

From: Carl Lumma

>>Because complexity is comparitive.  The idea of a complete
>>otonal chord is not less arbitrary.
>
>right, but whether a particular mapping is more complex
>than another shouldn't be this arbitrary!

I'm lost.  If you agree with that, then what's arbitrary?

>>Oh, crap.  What is it you've been pushing, then?
>>Weighted complexity?
> 
>that's been much more common, yes.

Ok.  Here's my latest thinking, as promised.

Ideally we'd base everything on complete n-ads, with
harmonic entropy.  Since that's not available, we'll look
at dyadic breakdowns.

If you use the concept of odd limits, and your best way
of measuring the error of an n-ad is to break it down
into dyads, you're basically saying that ratio containing
n is much different than any ratio containing at most n-2.
Thus, I suspect that my sum of abs-errors for each odd
identity up to the limit would make sense despite the fact
that for dyads like 5:3 the errors may cancel.

If we throw out odd-limit, however, we might be better off.
If there were a weighting that followed Tenney limit but
was steep enough to make near-perfect 2:1s a fact of life
and anything much beyond the 17-limit go away, we could
have individually-weighted errors and 'limit infinity'.

We should be able to search map space and assign generator
values from scratch.  Pure 2:1 generators should definitely
not be assumed.  Instead, we might use the appearence of
many near-octave generators as evidence the weighting is
right.

As far as my combining error and complexity before optimizing
generators, that was wrong.  Moreover, combining them at all
is not for me.  I'm not bound to ask, "What's the 'best' temp.
in size range x?".  Rather, I might ask, "What's the most
accurate temperament in complexity range x?".  Which is just
a sort on all possible temperaments, first by complexity, then
by accuracy.  Which is how I set up Dave's 5-limit spreadsheet
after endlessly trying exponents in the badness calc. without
being able to get a sensicle ranking.

As for which complexity to use, we have the question of how to
define a map at 'limit infinity'. . .  In the meantime, what
about standard n-limit complexity?

() Gene's geometric complexity sounds interesting (assuming
it's limit-specific...).

() The number of notes of the temperament needed to get all of
the n-limit dyads.

() The taxicab complexity of the n-limit commas on the harmonic
lattice.  Or something that measured how much smaller the
average harmonic structure would be in the temperament than in
JI.  This sort of formulation is probably best, and may in fact
be what Gene's geometric complexity does...

>>>f(w3*error(3),w5*error(5),w5*error(5:3))
>>>
>>>where f is either RMS or MAD or MAX or whatever, and w3 is
>>>your weight on ratios of 3, and w5 is your weight on ratios
>>>of 5.
>> 
>>Thanks.  So my f is +,
>
>are you sure? aren't there absolute values, in which case it's
>equivalent to MAD? (or p=1, which gene doesn't want to consider)

Yep, you're right.  Though its choice was just an expedient
here, and it would be the MAD of just the identities, not of
all the dyads in the limit.

Last time we tested these things for all-the-dyads-in-a-chord,
I believe I preferred RMS.  Which is not to say that MAD
shouldn't be included in the poptimal series.

-Carl


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