Tuning-Math Digests messages 6075 - 6099

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Message: 6075

Date: Fri, 17 Jan 2003 18:11:39

Subject: Re: A common notation for JI and ETs

From: David C Keenan

Continuing down the ratio popularity list, of those we don't yet have a 
symbol for:

There are two 175 (5*5*7)commas of interest
175-diesis     512:525    43.41 c  //|    0.40 c schisma
175'-diesis 127575:131072 46.82 c  ./|)   exact, no symbol without 5'

two 245 (5*7*7) commas
245-comma      243:245    14.19 c  ~|(    0.54 c schisma
245'-diesis 524288:535815 37.65 c  /|~    0.40 c schisma

two 625 (5^4) commas
625-comma 4100625:4194304 39.11 c  (|(      0.20 and 1.04 c schismas
625-comma     625:648     62.57 c  '(|)     0.08 c sch, no sym w/o 5'

You remember that you were concerned about symbols for 23 and 24 degrees of 
494-ET if we eliminated the |\) and (/| symbols. Although I didn't notice 
at the time, we already had one for 23deg494 which is the 5:49'-diesis '|))

But we are left with the problem of 24deg494, or put another way: What 
should be the symbols for the apotome complements of |)) and '|)).

The apotome complement of 56.48 c |)) is 59.16 c. The only possible 
two-flag symbol for that is |\)

It seemed a lot tidier when we had (/| and |\) as complements, however this 
had a serious lateral confusability problem which we might now consider 
solved.

So 24deg494 would be .|\)

Not very nice to have to introduce 5' accents just for these two degrees, 
but there it is.
-- Dave Keenan
Brisbane, Australia
Dave Keenan's Home Page *


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Message: 6077

Date: Fri, 17 Jan 2003 08:26:46

Subject: Re: Nonoctave scales and linear temperaments

From: Graham Breed

Gene Ward Smith  wrote:

> The prime number theorem says pi(x)~x/(ln(x)-1).

Oh, I think I had that backwards.

>>Whereas combing equal temperaments only gives O(n**2) calculations, 
>>where n is the number of ETs you consider.  I find n=20 works well, 
>>requiring O(400) candidates. 
> 
> Once you take wedgies, you should have fewer candidates.

Maybe, but isn't taking wedgies as hard as finding the maps?  How do you 
avoid calculating a huge number of wedge products?

(Oh, n=20 gives exactly 190 candidates)


                         Graham


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Message: 6079

Date: Fri, 17 Jan 2003 08:31:53

Subject: Re: heuristic and straightness

From: Carl Lumma

>it shows that the 'expressibility' metric is not quite the same
>as the taxicab metric on the isosceles-triangular lattice. you
>need to use scalene triangles of a certain type, which i don't
>think is what you were thinking when you wrote "taxicab
>complexity" above.

That's true; I just count the rungs.  Any lattice that's
'topologically' (?) equivalent to the triangular lattice
will do.

>straightness applies to a set of unison vectors. different sets
>of unison vectors can define the same temperament. a temperament
>may look good on the basis of being defined by good unison
>vectors. but in fact you may end up with a terrible temperament
>if the unison vectors point in approximately the same direction.

Why would it be terrible?

>>Yahoo groups: /tuning-math/messages/2491?expand=1 *
//
>>Which column is the heuristic,
> 
>column V is proportional to the heuristic error, and Y is 
>proportional to the heuristic complexity.
> 
>>what are the other columns,
> 
>U is the rms error, W is the ratio of the two error measures. X
>is the complexity (weighted rms of generators-per-consonance at
>that point i believe), and Z is the ratio of the two complexity
>measures.
> 
>>and what are their values expected
>>to do (go down or up...)?
> 
>W and Z are expected to remain relatively constant.

Thanks again!  Now everything is clear.  Except how you
derrived the heuristics!

Seriously man, one expository blurb would save you from having
to do this for each person who's interested, and for me again
when I've forgotten it in 6 months.  :~)

-Carl


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Message: 6082

Date: Fri, 17 Jan 2003 00:13:49

Subject: Re: Nonoctave scales and linear temperaments

From: Carl Lumma

>>Can you find them at the end of a map-space search, and take
>>them out?
> 
>what do you want to take out?

Can I identify the duplicate temperaments?

>from what?

A search of all possible maps.

> what does "//" mean?

Cut.  I've been using it since I've been on these lists.

>correct. the untweakable generator has been tweaked. is that all?

Assuming 2:1 reduction makes me squirm in my chair, is all.
Plentiful near-2:1s should emerge from the search if the criteria
are right.

>>Thanks again.  So if reduction is necc., it means that a
>>temperament can be described by two different lists of commas,
>>right?
> 
>right, although for single-comma temperaments, only one choice
>leaves you without torsion.

Thanks yet again.

>>This means we'll have the same problem searching comma
>>space as we did map space.  So wedgies are our last hope.
>
>the problem i was pointing out with map space, i think, was
>that the arbitrariness of the set of generators means your
>complexity ranking (if it's just based on the numbers of the
>map) will be meaningless.

Oh.  Now I get it!  You're right.  But doesn't the same
problem occur with different commatic representations, when
defining complexity off the commas?

>>Are you saying a badness cutoff is not sufficient to give a
>>finite list of temperaments?
>
>exactly. in *every* complexity range you have about the same
>number of temperaments with log-flat badness lower than some
>cutoff -- and there are an infinite number of non-overlapping
>complexity ranges.

Oh.  I guess I need some examples, then, of most of the simple
temperaments that are garbage...

-Carl


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Message: 6087

Date: Fri, 17 Jan 2003 10:16:06

Subject: Re: heuristic and straightness

From: Graham Breed

--- wallyesterpaulrus wrote:

> straightness applies to a set of unison vectors. different sets of 
> unison vectors can define the same temperament. a temperament may 
> look good on the basis of being defined by good unison vectors. but 
> in fact you may end up with a terrible temperament if the unison 
> vectors point in approximately the same direction.

Then that's what you need to reduce the complexity of the search.  I
can't find a quantitative definition of "straightness" in the
archives.  What is it?  I presume it works for insufficient sets of
unison vectors.


                   Graham


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Message: 6089

Date: Fri, 17 Jan 2003 10:18:52

Subject: Re: heuristic and straightness

From: Carl Lumma

>>That's true; I just count the rungs.  Any lattice that's
>>'topologically' (?) equivalent to the triangular lattice
>>will do.
>
>you mean approximately?

?  I originally meant that counting the rungs between a dyad
and the origin would be approximately the same as the log of
the odd limit of the dyad...

         lnoddlimit  7limittaxicab  ratio
15:8     2.7             2          1.35
5:3      1.6             1          1.6
105:64   4.7             3          1.6
225:224  5.4             4          1.35

>>>but in fact you may end up with a terrible temperament
>>>if the unison vectors point in approximately the same
>>>direction.
>>
>>Why would it be terrible?
>
>heuristically speaking,
> 
>in most cases the *difference* between the unison vectors
>will be of similar or greater magnitude in terms of JI comma
>interval size as the unison vectors themselves, but since
>the angle is very small, this *difference* vector will be
>very short (i.e., low complexity). a comma of a given JI
>interval size will lead to much higher error if tempered out
>over much fewer consonant rungs (i.e., if it's very short)
>than if it's tempered out over more consonant rungs (i.e.,
>if it's long). therefore, you may end up with a temperament
>with much larger error than you would have expected given
>your original pair of unison vectors.

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?
If so, this means that badness is not fixed for a given
temperament...

Also, can I change the straightness by transposing pitches
by uvs?

Finally, is "commatic basis" an acceptable synonym for
"kernel"?

>i did post the derivation a while back, probably before i
>even used the word "heuristic", but if you search for
>"heuristic", i think you'll find a post that links to the
>derivation post.

I think I remember it coming out, but I couldn't find it
today in my searches.  I did try.

>i welcome suggestions or just make your own blurb, and
>let's put this on a webpage somewhere.

Maybe the original exposition can just be updated a bit, and
then monz or I could host it, certainly.

-C.


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Message: 6093

Date: Fri, 17 Jan 2003 10:26:40

Subject: Re: heuristic and straightness

From: Graham Breed

wallyesterpaulrus  wrote:

> insufficient? how can a set of unison vectors be insufficient?

One unison vector is insufficient to define a 7-limit linear 
temperament, two or three unison vectors are insufficient to define a 
21-limit linear temperament.  For an arbitrary search to be practicable, 
there has to be a way of rejecting sets of three unison vectors because 
you know they can't give a good linear temperament.  That would reduce 
the search to more like O(n**3) in the number of unison vectors instead 
of O(n**6).

                   Graham


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Message: 6097

Date: Sat, 18 Jan 2003 01:19:14

Subject: Re: heuristic and straightness

From: Carl Lumma

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

You said the temperament got worse, but the badness
remained constant.  -C.


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Message: 6099

Date: Sat, 18 Jan 2003 05:43:22

Subject: Re: heuristic and straightness

From: Carl Lumma

>>in most cases the *difference* between the unison vectors
>>will be of similar or greater magnitude in terms of JI comma
>>interval size as the unison vectors themselves, but since
>>the angle is very small, this *difference* vector will be
>>very short (i.e., low complexity). a comma of a given JI
>>interval size will lead to much higher error if tempered out
>>over much fewer consonant rungs (i.e., if it's very short)
>>than if it's tempered out over more consonant rungs (i.e.,
>>if it's long). therefore, you may end up with a temperament
>>with much larger error than you would have expected given
>>your original pair of unison vectors.
>
>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?
>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 [the temperament] become "terrible"?
>
>>if you change to a "straighter" pair of unison vectors,

Wait a minute -- straightness goes up or down with the angle
between the vectors?  I thought up.

>>one or both of them will have to be a lot shorter, thus less
>>distribution of error and a worse temperament.

I don't follow the 'shorter' bit.  The only thing I thought
straightness did was make the difference vector more complex.

>shortening the unison vectors makes the temperament worse, but
>in a given temperament, this would be counteracted by an
>increase in straighness, which makes the temperament better.

You lost me.

>overall, the measure must remain fixed for a given temperament,
>otherwise it's meaningless.

If by "the measure", you mean badness, I agree.

-Carl


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