Tuning-Math Digests messages 9975 - 9997

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Message: 9975

Date: Tue, 10 Feb 2004 01:00:08

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

From: Carl Lumma

>>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.

logflat is unique among badness functions I know of in that it does
not favor any region of complexity or error (thus it reveals
something about the natural distribution of temperaments) and has
zero free variables.

>Log-flat badness with cutoffs

The cutoffs are of course completely arbitrary, but can be easily
justified and explained in the context of a paper.

>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?

Ignoring the cutoffs, logflat does reveal something fundamental about
the distribution of temperaments.  Whether musically appropriate or
not (utterly unfalsifiable assumptions), it gives an unbiased view
of ennealimmal vs. meantone, etc.

>i.e. How should we decide what cutoffs to use on error, complexity
>and log-flat badness?

You can tweak them to satisfy your sensibilities as best as possible,
same as you're tweaking the moat to factor infinity to satisfy your
sensibilities as best as possible.

>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? 

Because nobody here or on the tuning list has the slightest clue
about what's musically useful.  Nobody has composed more than a few
ditties in any of these systems.

>> 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 *

It doesn't explain what the heck a moat is, for starters.

-Carl


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Message: 9977

Date: Tue, 10 Feb 2004 16:35:07

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

From: Carl Lumma

>> The rectangle enclosed by error and complexity bounds.  You answered
>> that the axes were infinitely far away, but the badness line AB
>> doesn't seem to be helping that.
>
>If you simply bound complexity alone, you get a finite number of
>temperaments. Most are complete crap.

Above I suggest a rectangle which bounds complexity and error, not
complexity alone.

In the circle suggestion I suggest a circle plus a complexity bound
is sufficient.

-Carl


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Message: 9978

Date: Tue, 10 Feb 2004 19:50:39

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

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >Thus it's great for a paper for mathematicians. Not for musicians.
> 
> The *contents* of the list is what's great for musicians, not
> how it was generated.

No; I agree with Graham that we should "teach a man to fish".

> >> >Log-flat badness with cutoffs
> >> 
> >> The cutoffs are of course completely arbitrary, but can be easily
> >> justified and explained in the context of a paper.
> >
> >But there are *three* of them!
> 
> ...still trying to understand why the rectangle doesn't enclose
> a finite number of temperaments...

Which rectangle?

> With moats it seems you're pretty-much able to hand pick the list,

No way, dude! The decision is virtually made for us. If you can find 
a wider moat in the vicinity, we'll adopt it.

> By thoughts are that in the 5-limit, we might reasonably have a
> chance of guessing a good list.  But beyond that, I would cry
> Judas if anyone here claimed they could hand-pick anything.  So,
> my question to you is: can a 5-limit moat be extrapolated upwards
> nicely?

Not sure what you mean by that.


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Message: 9980

Date: Tue, 10 Feb 2004 21:56:27

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

From: Graham Breed

Dave Keenan wrote:

> I disagree. It's just too hard for non-mathematicians. Unless by
> "fish" you mean "go to Graham's web site and use the temperament
> finder there" in which case I'm all for it! And this would let us not
> worry too much that we may have left some temperament out of the paper
> that someone someday may find useful.

That's roughly what I meant.  Of course, the temperament finder could 
always do with improving (even mathematicians have trouble understanding 
it!) and could do with a good user guide -- hence the "teach" part.  And 
I could really do with help with that.

We also need to give mathematicians the instructions for writing their 
own temperament finders for their own websites, or software packages, or 
idle amusement.

Either endeavour would be more worthy of my time than endless 
discussions about what temperaments to include on a list.  But, while 
I'm here:

- log-flat looks like a good place to start

- silence is negative infinity in decibels

- spherical projection!

- can somebody give a friendly explanation of complex hulls?

- would k-means have anything to do with the clustering?

K-Mean Clustering Tutorial *


                  Graham


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Message: 9982

Date: Tue, 10 Feb 2004 05:03:50

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:
> 
> > Well Paul and I see it as bringing it in closer touch with reality.
> 
> Convince us. Make a case. Show some loglog plots and prove they make 
> no sense. Explain why what you are doing does make sense. Is this an 
> unreasonable request?

I seem to have been doing nothing but that for the past two days. The
fact that you haven't recognised it as such says to me that we're
somehow talking past each other much worse than I thought. 

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) 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.

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.

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. Nor are they interested in a 5-limit temperament
where 6/5 is distributed. 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.

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".

The way to find a useful moat is to start with the temperaments you
know everyone will want included, and those that almost no one will
care about, and check out the space between the two.


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Message: 9984

Date: Tue, 10 Feb 2004 16:41:13

Subject: Re: Rhombic dodecahedron scale

From: Carl Lumma

>> >A Voronoi cell for a lattice is every point
>> >at least as close (closer, for an interior point) to a paricular
>> >vertex than to any other vertex. The Voronoi cells for the
>> >face-centered cubic lattice of 7-limit intervals is the rhombic
>> >dodecahedron
>> 
>> Something Fuller demonstrated, in his own tongue.
>
>Right. Fuller?

Buckminster.

>> >These
>> >fill the whole space, like a bee's honeycomb.
>> 
>> Isn't it also the dual to the FCC lattice (hmm, maybe dual isn't
>> the right word here...)
>
>The dual to the fcc lattice is the bcc lattice (body-centered cubic
>lattice.)

Indeed, sorry.

>> >The Delaunay celles of a
>> >lattice are the convex hulls of the lattice points closest to a
>> >Voronoi cell vertex; in this case we get tetrahedra and octahedra,
>> >which are the holes of the lattice, and are tetrads or hexanies.
>> >The six (+-1 0 0) verticies of the Voronoi cell
>> 
>> *The* Voronoi cell?  Which one do you mean?
>
>The one around the unison, (0 0 0). Others are merely translates.

Ah, yes.

-Carl


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Message: 9987

Date: Tue, 10 Feb 2004 21:53:17

Subject: Re: The same page

From: Paul Erlich

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

> > >In 4D (e.g., 7-limit), for linear temperaments the bival is dual 
to 
> > >the bimonzo, and both are referred to as the "wedgie" (though 
Gene 
> > >uses the bival form).
> 
> Both are referred to as the "wedgie" by whom?

For example, in the original post to Paul Hj. explaining Pascal's 
triangle. Clearly there, when there's only one val involved, the 
wedgie can only be a multimonzo, not a multival.

> > Ok great.  But what's all about this algebraic dual?  Is this
> > something I can do to matrices, like complement and transpose?
> 
> It's the complement.

Oh yeah -- sorry!


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Message: 9990

Date: Tue, 10 Feb 2004 19:56:01

Subject: Re: The same page

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >> > 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,

No, it's dual to the comma.

> so val ^ val must not be a val-wedgie.

Yes, I meant val ^ val.

> 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 ]  ->  ?

<val] ^ <val] -> <<bival||
[monzo> ^ [monzo> -> ||bimonzo>>

In 3D (e.g., 5-limit), for linear temperaments the bival is dual to 
the monzo, and for equal temperaments the bimonzo is dual to the val.

In 4D (e.g., 7-limit), for linear temperaments the bival is dual to 
the bimonzo, and both are referred to as the "wedgie" (though Gene 
uses the bival form).
 
> ...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?

No -- (5-limit linear) error. See the original message.


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Message: 9995

Date: Tue, 10 Feb 2004 07:39:38

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:


> Then first loglog plots I've seen were just now posted by Paul; they 
> make a *very* strong case for loglog, not at all to my surprise. It 
> would be interesting now to see linears.

Paul posted the linear versions a while ago. And I posted a link to
Paul's message introducing them, for Carl earlier today.

> > 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.
> 
> There's only one way to do it, which is to do it. I don't see why 
> this is any kind of argument. Something on the very edge of your 
> criterion is by definition marginal, whereever your margin lies. You 
> can try to avoid this by moats, but that's only going to take you so 
> far, and if you are not careful (and I've seen no signs of care) into 
> regions where the justification is dubious. If you want a list, why 
> not just pick your favorites and put them on it?

Because if someone plots them on a graph (whether log or linear),
along with some nearby ones we left out, then if the only way to draw
a line separating them is to have lots of zigs and zags in it, they
will have good reason to complain.

> > 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".
> 
> If the moat is gerrymandered, you get that question anyway, don't you?
> 

Sure. But the wider and smoother your moat, the easier you can be let
off the hook. :-)

Also "gerrymander" is a derogatory term and originally applied only to
electoral boundaries redefined to suit the encumbent.

We have no need to apologise for choosing boundaries that we know to
the best of our combined knowledge to only include the X most useful
temperaments. Indeed that's the whole idea.

We're never going to agree with everyone, but a good moat will lessen
the scope for disagreement.

> > The way to find a useful moat is to start with the temperaments you
> > know everyone will want included, and those that almost no one will
> > care about, and check out the space between the two.
> 
> Right. Then you put them on a loglog plot, and try to draw a straight 
> line between them, and find to your amazement that it works. 

No! I'm afraid I've tried, but I can find absolutely no way to make a
straight line work for this on a log log plot.

> Now you 
> only have the corners to worry about, and what you are doing is 
> easier to justify.

If it was a straight line, why would I have corners to worry about?

> Is this so bad? 

Now I've tried, and got the results I fully expected from my
experience of the kinds of things that happen when you go from
linearlinear to loglog.

> Why the opposition to even trying?

Because I was pretty sure from the above experience and having already
looked at it on both linear-linear and log-linear that it would be a
waste of time.

> When the response is "this isn't helping" my impression is that I am 
> not being listened to at all, hence I started shouting. Now I think I 
> may have gotten through a little, so let's talk.

Sure.


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Message: 9997

Date: Tue, 10 Feb 2004 21:05:12

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

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >> >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?

Carl,

I've been saying we have evidence regarding "5-limit ETs". The above
are all linear temperaments, not equal temperaments, and one is
strictly 7-limit. Also I'm not sure I'm parsing that sentence correctly.

I think you're asking how well we have explored any systems other than
those linear temperaments you mention. My answer is, "Not very far".
But at least you seem to agree that the systems you mention have been
somewhat explored, and so we have some evidence of their musical
usefulness. The same goes for several _equal_ temperaments.

> 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.

You've got the wrong end of the stick here, and are putting words in
my mouth. I never proposed using
failure-to-be-mentioned-on-the-tuning- lits as a reason to exclude a
temperament, that would be ridiculous. I only propose using those that
_have_ been mentioned (as useful), and general discussions on
desirable properties, as a starting point and then widening the circle
roughly equally in all directions from there.

> 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,

Even if that were true, it would not disqualify their past testimony.

> 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.

And why would you limit this information to those who have posted? We
have also heard about composers who never go near the tuning lists.
Darreg, Blackwood, Negri and Hanson come immediately to mind.


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