Tuning-Math Digests messages 6050 - 6074

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Message: 6050

Date: Fri, 17 Jan 2003 10:32:34

Subject: Re: heuristic and straightness

From: Carl Lumma

> > Right, the difference vector has to vanish, too.  Ok.  What
> > I don't get is, for a given temperament, can I change the
> > straightness by changing the unison vector representation?
> 
> yes.

Ok.

> > If so, this means that badness is not fixed for a given
> > temperament...
> 
> that's not true. since both the defining unison vectors *and*
> the straightness change, the badness can (and will) remain
> constant.

Then how can it become "terrible"?
 
> > Also, can I change the straightness by transposing pitches
> > by uvs?
> 
> this is meaningless, as we're talking about temperaments, not 
> irregular finite periodicity blocks. we're talking either equal 
> temperaments or infinite regular tunings.

Straightness certainly sounds like it can be defined on an
untempered block.

> > Finally, is "commatic basis" an acceptable synonym for
> > "kernel"?
> 
> no. every vector that can be generated from the basis belongs
> to the kernel, but not every set of n members of the kernel
> (even if they're linearly independent) is a basis (since you may
> get torsion).

So is it possible to have more than one kernel representation
for a temperament?

-Carl


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Message: 6051

Date: Fri, 17 Jan 2003 11:38:47

Subject: Re: heuristic and straightness

From: Graham Breed

Gene Ward Smith  wrote:

> One method which might come to the same thing as "straightness" in
effect is to take two commas, and combine to get a codimension 2
wedgie. Produce a list of these by taking the best (here you run into
geometric badness) and then wedge these with another comma, and so
forth.

So is "geometric badness" simplicity or complexity of the exterior
element?

                      Graham


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Message: 6052

Date: Fri, 17 Jan 2003 01:41:31

Subject: Re: Nonoctave scales and linear temperaments

From: Carl Lumma

[I wrote...]
>Perhaps we could enforce "validity", and maybe also Kees'
>'complexity validity'.

I guess I should have called that 'expressibility validity'.

-Carl


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Message: 6053

Date: Fri, 17 Jan 2003 10:33:29

Subject: Re: heuristic and straightness

From: Graham Breed

wallyesterpaulrus  wrote:

> then you need a 3-d generalization of "straightness". i bet if i went 
> and learned grassmann algebra i'd be able to get a better grasp on 
> all this.

Yes, probably.  Vectors are straighter the smaller the area of the 
parallelogram they describe.  The wedge product is a generalisation of 
area, as it tends towards the determinant.  So hopefully we can do 
something with the intermediate wedge products.  But I don't know what :(


                            Graham


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Message: 6054

Date: Fri, 17 Jan 2003 15:16:52

Subject: Re: [tuning] margo, manuel -- was it kirnberger?

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

Paul asked:
>was it kirnberger who proposed foreshortening each fifth by a schisma
>to approximate 12-equal? a chain of 12 such fifths would fail to
>close on itself by a mere "atom", or .015361 cents . . .

That I don't know, but he was at least the first to publish a temperament
with a fifth flattened by a schisma, in the 1780's. Then Prinz did so
around 1810.
Found this on Nigel Taylor's tuning page *

Manuel


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Message: 6055

Date: Fri, 17 Jan 2003 01:48:49

Subject: heuristic and straightness

From: Carl Lumma

> > >it's nice when there's only one comma. then the log of the
> > >numbers in the comma (say, the log of the odd limit) is an
> > >excellent estimate of complexity (it's what i call the
> > >heuristic complexity).
> > 
> > That's what I call taxicab complexity, I think.
> 
> not quite. for one thing, read this:
> 
> lattice orientation *
> 
> including the link to my observations.

All I get from this is that it depends whether one uses a
triangular or rectangular lattice.  I must be missing
something...

-----

> > >if there's more than one comma being tempered out, we need
> > >a notion of the "angle" between the commas . . .
> > 
> > Please explain.
>
> search for "straightess" in these archives . . .

For a given block, notes can be transposed by unison vectors.
This changes the shape of the block.  Does it change its
straightness?

Don't follow why the less straight blocks are supposed to
be less interesting.

-----

Here's what I found on the heuristic.  Last time I asked,
you referred me to this message:

Yahoo groups: /tuning-math/messages/2491?expand=1 *

Which I can't follow at all.  Which column is the heuristic,
what are the other columns, and what are their values expected
to do (go down or up...)?

I also found this blurb:

>the heuristics are only formulated for the one-unison-vector
>case (e.g., 5-limit linear temperaments), and no one has bothered
>to figure out the metric that makes it work exactly (though it
>seems like a tractable math problem). but they do seem to work
>within a factor of two for the current "step" and "cent"
>functions. "step" is approximately proportional to log(d),
>and "cent" is approximately proportional to (n-d)/(d*log(d)).

Why are they called "step" and "cent"?  How were they derrived?

-Carl


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Message: 6057

Date: Fri, 17 Jan 2003 18:38:30

Subject: Re: heuristic and straightness

From: Carl Lumma

> > > that's not true. since both the defining unison vectors *and*
> > > the straightness change, the badness can (and will) remain
> > > constant.
> > 
> > Then how can it become "terrible"?
> 
> if you change to a "straighter" pair of unison vectors, one or
> both of them will have to be a lot shorter, thus less distribution
> of error and a worse temperament.

I was trying to point out that badness here has failed to reflect
your opinion of the temperament.

>>>>Also, can I change the straightness by transposing pitches
>>>>by uvs?
>>> 
>>>this is meaningless, as we're talking about temperaments, not 
>>>irregular finite periodicity blocks. we're talking either equal 
>>>temperaments or infinite regular tunings.
>>
>>Straightness certainly sounds like it can be defined on an
>>untempered block.
>
>well, that's not the context in which it's been discussed, and
>thus not the context into which your questions about it were posed.

That's true, but I couldn't see making a separate post.  I'm
trying to understand your concept of straightness.

>no. the kernel is an infinite set of vectors, and is unique to the 
>temperament.

Whew.  Got it!

-C.


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Message: 6071

Date: Fri, 17 Jan 2003 11:14:25

Subject: Re: heuristic and straightness

From: Graham Breed

wallyesterpaulrus  wrote:

> so the main difference is that you're using a euclidean metric (for 
> geometric complexity), while i'm using a taxicab one (for heuristic 
> complexity).

I take it you're adding either the moduli or squares of the coefficients 
of the exterior element (wedge product of vector)?  Well, that's easy 
enough.  And the more complex the better, because that covers the 
complexity of the original unison vectors and their straightness.  If 
we've already chosen unison vectors that are small pitch intervals, do 
we have a badness measure?


                       Graham


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Message: 6074

Date: Fri, 17 Jan 2003 07:53:09

Subject: Re: Nonoctave scales and linear temperaments

From: Carl Lumma

>>"The results of a search of all possible maps is bound to return
>>pairs, trios, etc. of maps that represent the same temperament.
>>Can we find them in the mess of results?"  Was all I was asking!
> 
>yes, since you can calculate the wedgie of each. but clearly this
>is a terrible way to go about the search.

Well, well by now I of course agree.

>how would you even delimit it?

I won't ask...

>>Perhaps we could enforce "validity",
> 
> ?

That's Gene's name for a concept you said was equivalent to
the condition that all steps of a block be larger than its
unison vectors.

>>and maybe also Kees' 'complexity validity'.
> 
> ??

See later message.

-Carl


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