Tuning-Math Digests messages 9925 - 9949

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Message: 9925

Date: Mon, 09 Feb 2004 22:31:05

Subject: Re: The same page

From: Carl Lumma

>log2(3), log2(5), etc.

Thanks.  I think that's the same as Gene was using before, then.

-Carl


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Message: 9926

Date: Mon, 09 Feb 2004 23:49:59

Subject: 23 "pro-moated" 7-limit linear temps, L_1 complex.(was: Re: 126 7-limit linears)

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
wrote:

> Yes. That's a good point (about e.g. 5-limit JI having infinite
> complexity as a linear temperament), but obviously there's another
> point of view available where 5-limit JI has finite complexity as a
> planar temperament.

But there are no other 5-limit planar temperaments to compare it to, 
so this is irrelevant.

> Psychologically it would seem that there is some point in the
> complexity of low-error 5-limit linear temperaments where one would
> rather have the planar complexity of 5-limit JI than bother with the
> linear complexity of a temperament.

Rather, it seems to me, one wouldn't care. Certainly the linear 
temperament can never become *more* complex than a planar -- it can 
merely become *equally* complex for all intents and purposes.

> I suggest that occurs somewhere
> between the complexities of schismic and the least complex 
temperament
> with error less than schismic.

I suggest much more complex temperaments belong in a math paper, not 
a music paper.


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Message: 9927

Date: Mon, 09 Feb 2004 22:38:33

Subject: Re: The same page

From: Carl Lumma

>> > 5-limit, comma = n/d
>> > 
>> > Complexity is log2(n*d),
>> 
>> Yes, but this can also be expressed in other ways, for example if
>> 
>> <<a1 a2 a3||
>> 
>> is the val-wedgie (dual to the comma),

I thought val ^ val -> comma, so val ^ val must not be a val-wedgie.
What's a val-wedgie?

Anybody have a handy asci 'units' table for popular wedge products
in ket notation?  ie,

[ val >   ^ [ val >    ->  [[ wedgie >>
< monzo ] ^ < monzo ]  ->  ?

...etc.

>> > Error is the distance from the JIP of the 7-limit TOP 
>> > tuning for the temperament;
>> 
>> Or same as 5-limit linear error but with an additional term for 7.

What's linear error?

-Carl


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Message: 9928

Date: Mon, 09 Feb 2004 23:54:26

Subject: Re: Beep isn't useless....

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Herman Miller <hmiller@I...> 
wrote:

> but finding something that 
> doesn't at the same time exploit the meantone comma is going to be 
> tricky.

Yes, it's quite rare. Even the Canon has some rough spots where, for 
example, scale degree 2 appears over the IV chord and vi chord.


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Message: 9930

Date: Mon, 09 Feb 2004 23:56:53

Subject: 23 "pro-moated" 7-limit linear temps, L_1 complex.(was: Re: 126 7-limit linears)

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
wrote:

> I'm hoping paul can easily replot those ET plots loglog.

When I do so, at least keep in mind that rather than log(complexity), 
2^complexity has actually been proposed as a criterion (i.e., by 
Fokker), and that error^2, at least, has gotten much attention as a 
measure of pain, while log(error) has gotten none.


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Message: 9931

Date: Mon, 09 Feb 2004 22:44:01

Subject: Re: 23 "pro-moated" 7-limit linear temps

From: Carl Lumma

>We're trying to come up with some reasonable way to decide on which
>temperaments of each type to include in a paper on temperaments, given
>that space is always limited. We want to include those few (maybe only
>about 20 of each type)

For musicians, I'd make the list 5 for each limit; 10 tops.  For
people reading a theory paper, 20 would be interesting.

>which we feel are most likely to actually be
>found useful by musicians, and we want to be able to answer questions
>of the kind: "since you included this and this, then why didn't you
>included this". So Gene may have a point when he talks about cluster
>analysis, I just don't find his applications of it so far to be
>producing useful results.

I haven't seen any cluster analysis yet!

>Our starting point (but _only_ a starting point) is the knowledge
>we've built up, over many years spent on the tuning list, regarding
>what people find musically useful, with 5-limit ETs having had the
>greatest coverage.

You're gravely mistaken about the pertinence of this 'data source'.
Even worse than culling intervals from the Scala archive.

>It may be an objective mathematical fact that log-flat badness gives
>uniform distribution, but you don't need a multiple-choice survey to
>know it is a psychological fact that musicians aren't terribly
>interested in availing themselves of the full resources of 4276-ET
>()or whatever it was.

So far this can be addressed with a complexity bound.

>So we add complexity and error cutoffs which
>utterly violate log-flat badness in their region of application (so
>why  violate log-flat badness elsewhere and make the transition to
>non-violatedness as smooth as possible.

?

>Corners in the cutoff line are bad because there are too many ways for
>a temperament to be close to the outside of a corner.

Agreed.

>A moat is a wide and straight (or smoothly curved) band of white space
>on the complexity-error chart, surrounding your included temperaments.
>It is good to have a moat so that you can answer questions like "since
>you included this and this, then why didn't you included this", by at
>least offering that "it's a long way from any of the included
>temperaments, on an error complexity plot".

Okay, now I have a definition of moat.  How do they compare to Gene's
"acceptance regions"?

-Carl


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Message: 9932

Date: Mon, 09 Feb 2004 22:51:29

Subject: loglog!

From: Carl Lumma

Yahoo groups: /tuning-math/files/Paul/et5loglog.gif *

Ok, easy!  No moat needed, at least for ETs.  Just draw a
circle around the origin and grow the radius until it would
include something that exceeds a single bound -- a "TOP
notes per 1200 cents" bound.  For ETs at least.  Choose a
bound according to sensibilities in the 5-limit, round it
to the nearest ten, and use it for all limits.

-Carl


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Message: 9933

Date: Mon, 09 Feb 2004 00:40:53

Subject: 23 "pro-moated" 7-limit linear temps, L_1 complex.(was: Re: 126 7-limit linears)

From: Dave Keenan

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

> Some evidence you've actually considered it would be nice. A plot
> would be grand. Some attempt to theoretically justify what you two are
> doing would be appreciated.

I'm not sure what "it" is that you think we haven't considered. If
it's log-flat badness then that seems to have been the only such
measure being considered on this list for the past several years,
despite the objections (from psychology) that I thought I spelled out
in great detail when it was first mooted.

And by "theoretically justify" do you mean justify purely from
mathematical considerations? I believe that to be futile. It
eventually needs to be grounded in human psychology, both perceptual
and cognitive.

I understand you're still in favour of log-flat cutoffs which can be
written in the form

log(err) + k * log(complexity) < x

Paul and I have been considering those of the form 

err^p + k * comp^p < x

which can be made to look a lot like the previous one when 0<p<0.5.

Paul and I have not so much been trying to theoretically justify, but
rather empirically determine, appropriate values for p, admittedly
based on some pretty sketchy and anecdotal evidence. But that's all we
have.

By far the greatest body of evidence, about which temperaments people
consider musically interesting or useful, relates to equal
temperaments, particularly at the 5-limit.

And we find that what works best is a value of p that's slightly less
than one, i.e. the cutoff functions that we construct based on our
knowledge of which ETs have been popular historically, are somewhere
between log and linear, but much closer to linear.

Since you and Paul seem to have done a marvelous job of giving us
error and complexity measures that generalise from equal temps to
linear temps and beyond, then it seems likely that the general shape
of equal-interest contours we find for equal temps will be repeated
for higher dimensions. I suppose you could say this is the theoretical
part of the justification.

But rather than trying to come up with precise values for p and the
scaling constants for cutoffs, we are looking for what we call
"moats". These are places where moderate changes in these constants
will make no difference to which temperaments are included. They would
ideally look like a band of whitespace on the graph shaped like a pair
of back-to-back horns (something I hadn't realised before). In other
words it doesn't matter so much if a moat has a narrow waist. What is
most important is that it is wide near the axes.

But we can't just use any old moat. There are bound to be some very
wide moats that are unusable because they bear no resemblance to an
equal-interest cutoff.

The idea is that they should agree with the subjective cutoff
functions (implicit or otherwise) of as many different people as possible.


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Message: 9934

Date: Mon, 09 Feb 2004 00:57:22

Subject: 23 "pro-moated" 7-limit linear temps, L_1 complex.(was: Re: 126 7-limit linears)

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:
> > A bit more concavity still and we include
> > 
> > 45. Blackwood
> 
> Following what Dave did for the 5-limit and ET cases, I found that an 
> exponent of 2/3 produces the desired moat, for example when
> 
> err^(2/3)/6.3+complexity^(2/3)/9.35 < 1.

I prefer to put the scaling constants inside the exponentiation like this

(err/15.8)^(2/3) + (complexity/28.6)^(2/3) < 1.

Then you can see at a glance what maximum error and complexity are
allowed by this cutoff. For similar reasons I prefer to show the chart
with both axes starting from zero.

> Please look at the resulting graph:
> 
> Yahoo groups: /tuning_files/files/Erlich/7lin23.gif *
> 
> The temperaments in thie graph are identified by their ranking 
> according to the badness measure implied above:
> 
> 1. Huygens meantone
> 2. Pajara
> 3. Magic
> 4. Semisixths
> 5. Dominant Seventh
> 6. Tripletone
> 7. Negri
> 8. Hemifourths
> 9. Kleismic/Hanson
> 10. Superpythagorean
> 11. Injera
> 12. Miracle
> 13. Biporky
> 14. Orwell
> 15. Diminished
> 16. Schismic
> 17. Augmented
> 18. 1/12 oct. period, 25 cent generator (we discussed this years ago)
> 19. Flattone
> 20. Blackwood
> 21. Supermajor seconds
> 22. Nonkleismic
> 23. Porcupine

This looks reasonable to me as a cutoff, although maybe still too
many, but making a badness measure out of it may be going too far.



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Message: 9935

Date: Tue, 10 Feb 2004 00:14:08

Subject: Re: 23 "pro-moated" 7-limit linear temps

From: Carl Lumma

>> >Our starting point (but _only_ a starting point) is the knowledge
>> >we've built up, over many years spent on the tuning list, regarding
>> >what people find musically useful, with 5-limit ETs having had the
>> >greatest coverage.
>> 
>> You're gravely mistaken about the pertinence of this 'data source'.
>> Even worse than culling intervals from the Scala archive.
>
>How do you know this?

Assuming a system is never exhausted, how close do you think we've
come to where schismic, meantone, dominant 7ths, augmented, and
diminshed are today with any other system?

If you had gone to apply your program in Bach's time, would you have
included augmented and diminished?  "Oh, nobody's ever expressed
interest about them on a particular mailing list with about enough
aggregate musical talent to dimly light a pantry, so they must not be
worth mentioning."  It is said the musicians of Bach's time did not
accept the errors of 12-tET.

5-limit ETs being shown musically useful on the tuning list?
Exactly what music are you thinking of?  We're fortunate to have
had some great musicians working with new systems -- Haverstick,
Catler, Hobbs, Grady -- but we've chased all of them off the list,
and only Haverstick could be said to have worked in a "5-limit ET"
(and it's a stretch).  We've got Miller, Smith and Pehrson left,
with the promising Erlich and monz stuck in theory and/or 12-tET
land.  We're so far from any kind of form that would allow us to
make statements about musical utility that it's laughable.

-Carl


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Message: 9936

Date: Tue, 10 Feb 2004 00:43:33

Subject: Re: 23 "pro-moated" 7-limit linear temps

From: Carl Lumma

>>>My objection was not to limits on them per se, but to acceptance
>>>regions shaped like this (on a log-log plot).
>>>
>>>err
>>>|
>>>|   (a)
>>>|---\
>>>|    \
>>>|     \
>>>|      \ (b)
>>>|      |
>>>|      |
>>>------------ comp
>>>
>>>as opposed to a smooth curve that rounds off those corners marked
>>>(a) and (b).
>>
>>Aha, now I understand your objection.  But wait, what's stopping
>>this from being a rectangle?  Is the badness bound giving the
>>line AB?
>
>Yes.
>
>>If so, it looks like a badness cutoff alone would give a
>>finite region...
>
>No, because the zero-error line is infinitely far away on a loglog 
>plot.

Can you illustrate this?  It looks like the zero-error line is
three dashes away on the above loglog plot.  :)

>>>It turns out that the simplest way to round off those corners
>>>is to do the following on a linear-linear plot.
>> >
>> >err
>> >|
>> >|
>> >|\
>> >| \
>> >|  \
>> >|   \
>> >|    \
>> >------------ comp
>> 
>> Why not this on a loglog plot?
>
>Same reason as above.

-Carl


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Message: 9937

Date: Tue, 10 Feb 2004 04:18:36

Subject: 23 "pro-moated" 7-limit linear temps, L_1 complex.(was: Re: 126 7-limit linears)

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 "Dave Keenan" <d.keenan@b...> 
> > wrote:
> > 
> > > I'm hoping paul can easily replot those ET plots loglog.
> > 
> > When I do so, at least keep in mind that rather than log
> (complexity), 
> > 2^complexity has actually been proposed as a criterion (i.e., by 
> > Fokker), and that error^2, at least, has gotten much attention as 
a 
> > measure of pain, while log(error) has gotten none.
> 
> That it's gotten none is what I'm complaining about.

No one creates a psychological model where one of the response 
variables goes to minus infinity!

> That 
> 2^complexity has been discussed bores me to tears, unless you can 
> explain *why*.

One reason might be because, for an ET, the number of possible chords 
goes as 2^complexity.


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Message: 9939

Date: Tue, 10 Feb 2004 16:20:12

Subject: Re: 23 "pro-moated" 7-limit linear temps

From: Carl Lumma

>Years ago, when you first made be aware of this fact, I was seduced 
>by it, to Dave's dismay. Did you forget? Now, I'm thinking about it 
>from a musician's point of view. Simply put, music based on 
>constructs requiring large numbers of pitches doesn't seem to be able 
>to cohere in the way almost all the world's music does. Of all 
>people, I'm suprised Carl is now throwing his investigations along 
>these lines by the wayside.

I'm not.  It is well known that Dave, for example, is far more
micro-biased than I!  I'm just exploring possibilities.

-Carl


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Message: 9940

Date: Tue, 10 Feb 2004 20:36:47

Subject: Re: !

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:
> 
> > > Your plots 
> > > make it clear that loglog is the right approach. Look at them!
> > 
> > Geez, you must really be thinking like a mathematician and not a 
> > musician.
> 
> A musician is going to look at these plots, see that they show a
> slantwise arrangement of ets, and conclude circles are the way to
> analyze them,

I wasn't one of those who brought up or discussed circles, but I 
certainly wouldn't want to seduce musicians with a plot that is not 
likely to correspond with musically meaningful pain measures -- not 
by a long shot!


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Message: 9942

Date: Tue, 10 Feb 2004 00:44:19

Subject: Re: 23 "pro-moated" 7-limit linear temps

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> My latest position is that I can live with log-flat badness with
> appropriate cutoffs.  The problem with anything more tricky is that
> we have no data. Not vague historical data, actually no data.

Three questions regarding this statement.

1. Why is log-flat badness with cutoffs (on error and complexity) less
tricky than the cutoff functions Paul and I have been looking at.

Log-flat badness with cutoffs looks like this

max(err/k1, comp/k2, err * comp^k3) < x

or equivalently this (with different choices of k1, k2 and x)

max(err/k1, comp/k2, log(err) + k3*log(comp)) < x

where k3 is the number of primes divided by the number of primes less
the number of degrees of freedom.

This has two discontinuities in the cutoff curve.

Is that less tricky than the single straight line

err/k1 + comp/k2 < x  ?

or the slightly curved line 

(err/k1)^(2/3) + comp^(2/3) < x  ?

If so, why?

2. Assuming for the moment that we have no data, why isn't that just
as much of a problem for log-flat badness with e&c cutoffs as for any
other proposed cutoff relation? i.e. How should we decide what cutoffs
to use on error, complexity and log-flat badness?

3. Why don't discussions of the value of various temperaments in the
archives of the tuning list constitute data on this, or at least
evidence? 

I assume "evidence" is what you mean by "data" here. It's what I
meant. If, by "data", you mean something already organised as lists of
relevant numbers then I agree we don't have it, but what could
possibly be meant by "vague historical" lists of numbers.


> By
> putting all this energy into the list of temperaments, we're loosing
> touch with reality.

Well Paul and I see it as bringing it in closer touch with reality.

> Rather than worry about what is and isn't on
> the list, I'd like to figure out why Paul's creepy complexity gives
> the numbers it does.

Well sure. That would be a good thing to do. But I don't have a handle
on it. I think that's Paul and Gene's department. I'm happy just to
take it as evidence that Paul has hit on a very good complexity
measure and we should use it.

> But as long as Dave and Paul were having fun I
> didn't want to say anything.  They have a way of coming up with neat
> stuff, though so far their conversation has been impenetrable to me.

Thanks and sorry. Did this one help?

Yahoo groups: /tuning-math/message/9330 *


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Message: 9943

Date: Tue, 10 Feb 2004 04:20:25

Subject: Re: Loglog

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:
> 
> > My apologies again, these used log of error, but not log of 
> > complexity. Using log of complexity crammed all the interesting 
> stuff 
> > to the far left to the point of illegibility, in the cases I 
> > originally tried.
> 
> Sounds like a reason to get rid of most of your points, which are 
> gumming up the works anyway, and look at the good stuff.

I actually meant the right, not the left -- but this isn't so much of 
a problem for the loglog graph I made for you before and for the 
current batch, is it?


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Message: 9944

Date: Tue, 10 Feb 2004 16:22:50

Subject: Re: !

From: Carl Lumma

>> A musician is going to look at these plots, see that they show a
>> slantwise arrangement of ets, and conclude circles are the way to
>> analyze them,
>
>I wasn't one of those who brought up or discussed circles, but I 
>certainly wouldn't want to seduce musicians with a plot that is not 
>likely to correspond with musically meaningful pain measures -- not 
>by a long shot!

The circle rocks, dude.  It penalizes temperaments equally for trading
too much of their error for complexity, or complexity for error.  Look
at the plots, and the first things you hit are 19, 12, and 53.  And
22 in the 7-limit.  Further, my suggestion that 1cents = zero should
satisfy Dave's micro fears.  Or make 0 cents = zero.  It works either
way.  No origin; pfff.

-Carl


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Message: 9946

Date: Tue, 10 Feb 2004 20:39:14

Subject: Re: 23 "pro-moated" 7-limit linear temps

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:
> 
> > No way, dude! The decision is virtually made for us. 
> 
> Prove it. Give log-log plots for your proposed moats,

I already did that for one case, pointed it out twice, and asked for 
your comments.

> It's possible we could come to some kind of consensus
> if you would attempt to treat people with something better than the
> contempt you have shown lately.

I can take your attitude in no other way, unless you either ignored 
completely or have an abominably low level of respect for the 
discussions Dave and I posted on the topic.

Let's start over. If I'm willing to tolerate a certain level of 
error, and a certain level of complexity, why wouldn't I be willing 
to tolerate both together?


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Message: 9948

Date: Tue, 10 Feb 2004 00:44:15

Subject: Re: 23 "pro-moated" 7-limit linear temps

From: Carl Lumma

>we are left with trying to cook up some 
>scheme which doesn't look as if we are simply cooking up some scheme 
>to get rid of them. If this isn't basically just a shell game, I 
>think the thing should be defined in a way where the definition 
gives 
>us the list, and not the list the definition. Some kind of cluster 
>analysis or something.

Agreed completely, but let's hear Paul and Dave out.  They may
already have something!

-C.


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Message: 9949

Date: Tue, 10 Feb 2004 04:28:02

Subject: Re: 23 "pro-moated" 7-limit linear temps

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
> wrote:
> 
> > The error is minimax error in cents where the weighting is log_2
> (n*d)
> > for the ratio n/d in lowest terms.
> 
> What in the world does this mean? Do you mean TOP error for an equal 
> temperament, which is dual to the above?

Yes, I should have said the weights were 1/log_2(n*d) or that the
errors were _divided_ by the weights I gave.

Yes. I mean TOP error but didn't want to assume all readers would know
what that meant.


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