Tuning-Math Digests messages 10200 - 10224

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Message: 10200

Date: Thu, 12 Feb 2004 04:15:37

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:
> 
> > If you mean, log-flat with no other cutoffs, then no. I don't think
> > this was ever on the table, even from Gene. There is an infinite
> > number in the increasing complexity direction. I understand this would
> > be a single straight line on the log-log plot parallel to the apparent
> > lower left "edge" of the populated region.
> 
> It would in particular be the line which is a lim sup for the slope,
> and hence containing an infinite supply of temperaments.
> 
> Limit superior and limit inferior - Wikipedia, the free encyclopedia *
> 
> > If you mean log-flat badness in conjunction with error and complexity
> > cutoffs then it can stay on the table if you like, but I don't know
> > how you can psychologically justify the corners.
> 
> Can you specifically cite when an obnoxious temperament turned up in a
> corner, and you couldn't get rid of it without losing something good?

No. I can't, but it may have happened and I wouldn't have known since
we only started plotting things very recently.

However, it may well be possible in some cases to find a wide enough
moat in the right (subjective) ballpark that can accomodate both a
smooth curve and a tri-linear (bad, comp, err) cutoff giving the same
list.

That's the beauty of moats.


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Message: 10201

Date: Thu, 12 Feb 2004 04:31:53

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

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >> >> Dave doesn't seem to want the macros which would
> >> >> be necessary for the scale-building stuff.
> >
> >To me, in the context of the current highly mathematical discussion,
> >this said to me that you think macros are necessary (i.e. you can't do
> >without them) for scale-building stuff.
> >
> >I think this is obviously wrong since you can show how to build a
> >scale using meantone which is not a macrotemperament. 
> >
> >But since I now learn that you apparently only meant "desirable"
> >rather than "necessary" in the strict logical sense,
> 
> Hate to nitpick now that we understand each other, but it has
> nothing to do with strict logic, but rather *what* one wants to
> do.  Try this again:
> 
> >>Do *what* without them?  Build any decent scale (the above sense)?
> >>Or run any kind of decent scale-building program (the sense in
> >>which I said "necessary")?

By "scale-building stuff" I assumed you mean "showing readers how to
take temperaments and build scales from them, complete with several
examples".

This corresponds closely to the second option above. I don't think our
difference of interpretation has anything to do with that. But the
words "any kind of decent" did not appear in your original statement.
If they had, there would have been no problem.

So continuing the nitpicking:

"necessary for <whatever>" does not mean "indispensable for any kind
of decent <whatever>". But I agree that "necessary for any kind of
decent <whatever>" would have been just as good as "desirable for
<whatever>".

But of course I still disagree with your opinion on this. 

> >you should note
> >that I long ago agreed to neutral thirds and pelogic being on the
> >5-limit list. Surely they are macro enough for your purposes.
> 
> Herman just got through posting on tuning how beep is a great
> temperament for scale-building.

It may be good for scale building, but it isn't a temperament in the
sense of approximation of JI. Herman agrees.


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Message: 10202

Date: Thu, 12 Feb 2004 02:31:59

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

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >> > >> It is well known that Dave, for example, is far more
> >> > >> micro-biased than I! 
> >> > >
> >> > >?
> >> > 
> >> > What's your question?
> >> 
> >> What does micro-biased mean, on what basis do you say this about you 
> >> vs. Dave, and what is its relevance here?
> >
> >I'd like to know what you mean by micro-biased. It may well be true,
> >but I'd like to know.
> 
> Of all the amazing things I've seen on these lists, the failure of
> both you and Paul to understand the meaning of "micro-biased" is
> possibly the most amazing.

You misjudge. It wasn't failure to understand, it was carefulness in
checking for possible misunderstandings, rather than immediately
telling someone they are wrong. Something that surely we'd all like to
see more of.

> 
> >At the moment I fell you should be calling me "centrally biased" or
> >some such.
> 
> Obviously you did understand it!

Yes, as it turns out (but might not have).

> >I don't want to include either the very high error low
> >complexity or very high complexity low error temperaments that a
> >log-flat cutoff alone would include.
> 
> Yes, you are apparently centrally biased.  You should like circles
> in that case.  :)

Yes, I do, so far. Haven't you read that?

For me there are three candidates on the table at the moment. log-log
circles or ellipses, log-log hyperbolae, and linear-linear
nearly-straight-lines.

I'm guessing that one can probably make any one of these fit within
any given moat. If so, a major reason to prefer one over another would
be the number of free parameters and the simplicity of the expression
for the cutoff relation in terms of error and complexity. 

What happens to the curve once it is free of the pack (sorry, reading
about too many antarctic sea voyages lately) doesn't much matter,
although I guess you could still argue psychological plausibility from
the curve's behaviour in regions that don't happen to be populated.


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Message: 10203

Date: Thu, 12 Feb 2004 05:47:00

Subject: Bye for a while

From: Dave Keenan

That's it from me for a while. I'm a tuning list addict. I don't know
how to moderate my list use, so I'm going cold turkey.

I'm happy that my position is now understood (but not agreed with) by
all concerned, re this temperament list cutoff business.

Sometimes I think it's all so unimportant, and no one but us is terribly
interested in it anyway. The idea of worrying about priority makes me
laugh.

Also, when you read stuff like this [off topic]

Life After the Oil Crash *

I feel like we're just fiddling while Rome burns.

You know how to email me.

Tamam Shud


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Message: 10205

Date: Thu, 12 Feb 2004 21:35:28

Subject: Re: Symmetrical complexity for 5 and 7 limit temperaments

From: Carl Lumma

>> > The symmetrical complexity for codimension one (5-limit linear,
>> > 7-limit planar) is the symmetrical lattice distance for the
>> > comma defining it.
>> 
>> Is this the same as the "n" in your "interval count" message,
>> then?  That was what I've been calling "taxicab distance to
>> the comma".
>
>No, it's the symmetrical Euclidean lattice distance.

Ok, let's take a look...

>To get to a note-class in what I called shell n^2 from the unison
>in 7-limit, you need at minimum n steps because a straight line
>path of 1-step intervals takes you only out to a distance n.

Oh yeah, here you're squaring n, even though the 7-limit is
3- or 4-dimensional.

So what the hell is a "step" in a "straight line path"?

>For 81/80 this is ceil(sqrt(13))=4 steps, and for
>2401/2400 it is ceil(sqrt(11))=4 steps also.

Where do 13 and 11 come from?

>In fact, both can be reached in four steps in only one way, up
>to commuitivity; we have
>
>81/80 = (6/5)(3/2)^3 (1/4)
>2401/2400 = (7/6)(7/5)^2(7/4) (1/4)

This sure looks like taxicab, but what are the "(1/4)" terms?

-Carl


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Message: 10206

Date: Thu, 12 Feb 2004 02:40:08

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

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >> Dave doesn't seem to want the macros which would
> >> be necessary for the scale-building stuff.
> >
> >What are macros?
> 
> Again, I'm amazed that this well-worn terminology isn't effective
> here.  AKA exos?

Again, just being careful since my current understanding did not agree
with them being necessary for scale building.

It wouldn't be the first time we both thought we understood the
meaning of a term and eventually discovered we were poles apart. (Damn
those Antarctic stories :-)

> >Why can't you do scale-building stuff without them?
> 
> I don't know that it can't, but they're certainly fertile for
> scale-building.

Carl, "necessary" means you can't do without them. Please be careful
about your use of hyperbole (as opposed to Gene's use of hyperbolas
:-), particularly since frustration is running high in all quarters at
present.


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Message: 10211

Date: Thu, 12 Feb 2004 00:52:06

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:
> 
> > I don't see how the fact that x/612ths of an octave is a fine way to
> > tune the ennealimmal generator has any bearing on the musical
> > usefulness of 612-ET _as_an_ET_. 
> 
> Try thinking like a computer composer, who could well desire to keep
> track of things more easily by scoreing in terms of reasonably small
> integers.

OK. I'll grant that that is a form of musical usefulness for 612-ET
but it is not derived directly from the n-limit properties of 612-ET
itself, but only indirectly from the properties of ennealimmal (which
are themselves of dubious utility). 

Even if we want to include _this_ kind of derived usefulness in our
considerations I think an ET should only inherit a tiny fraction of
the usefulness of an LT it supports.

The other way of approaching this (which I favour at present) is to
say that the support of LTs should have no such impact on the
inclusion or otherwise of an ET _as_an_ET_, because the ET will get
it's due in this regard when the LTs are listed since we would include
a column giving the ETs that well support each ET.


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Message: 10212

Date: Thu, 12 Feb 2004 01:51:25

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

From: Carl Lumma

>> >> Can you name the temperaments that fell outside of the top 20
>> >> on Gene's 114 list?
>> >
>> >Yes.
>> 
>> Eep!  Sorry, I meant the ones that you want that fell outside
>> Gene's top 20/114.
>
>Oh. Sorry. I just don't have any enthusiasm for working this out now.
>I just know that I like Paul's latest list (which I can't easily find)
>because it has the ones I want plus a few more that bring it up
>against a reasonable moat.

And which list is that?

-Carl


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Message: 10213

Date: Thu, 12 Feb 2004 02:54:02

Subject: Re: !

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >You know what a moat is right?
> 
> Obviously not!  :(
> 
> >You have the castle (the circle is its
> >outer bound) with people (temperaments) inside.
> 
> Then it's the same as a circle!

No. A circle is infinitesimally thin. A moat has real thickness. If
we're talking circular cutoffs then we'd say the moat is annular. The
circle is only one edge of the moat. Of course it doesn't have to be
circular, but continuing in that vein ...

You could draw the smallest circle that encloses all the lucky
temperaments and then you could draw another one outside that which is
the largest circle that still encloses the same ones and no others.
The space between them is the moat. You can then give a quantitative
measure of the size of the moat as the percentage difference between
the radii of the two circles.

The term "moat" came to mind because the temperaments sometimes look
like constellations and in the Niven and Pournelle books "The Mote in
Gods Eye" and "The Moat around Murcheson's Eye", "the Moat" is a vast
region of space with no stars.


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Message: 10214

Date: Thu, 12 Feb 2004 01:59:25

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

From: Carl Lumma

>So continuing the nitpicking:
>
>"necessary for <whatever>" does not mean "indispensable for any kind
>of decent <whatever>". But I agree that "necessary for any kind of
>decent <whatever>" would have been just as good as "desirable for
><whatever>".

I'm sorry, I can't parse this.

>But of course I still disagree with your opinion on this. 

I think you understand me, but ultimately I'm not sure since I
couldn't parse the above.

>> >you should note
>> >that I long ago agreed to neutral thirds and pelogic being on the
>> >5-limit list. Surely they are macro enough for your purposes.
>> 
>> Herman just got through posting on tuning how beep is a great
>> temperament for scale-building.
>
>It may be good for scale building, but it isn't a temperament in the
>sense of approximation of JI. Herman agrees.

But if you view it as a "temperament in the sense of 'approximating'
JI" it still works.  The point is that temperament, in whatever
sense, is useful for all sorts of reasons.

-Carl


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Message: 10216

Date: Thu, 12 Feb 2004 00:58:12

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

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> We have a choice -- derive badness from first principles or cook
> it from a survey of the tuning list, our personal tastes, etc.

What first principles of the human psychology of the musical use of
temperaments did you have in mind?


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Message: 10217

Date: Thu, 12 Feb 2004 03:00:50

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

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >> We have a choice -- derive badness from first principles or cook
> >> it from a survey of the tuning list, our personal tastes, etc.
> >
> >What first principles of the human psychology of the musical use of
> >temperaments did you have in mind?
> 
> Since I'm not aware of any, and since we don't have the means to
> experimentally determine any, I suggest using only mathematical
> first principles

But badness is clearly a psychological property, what have
mathematical first principles got to do with it?

> , or very simple ideas like...
> 
> () For a number of notes n, we would expect more dyads in the
> 7-limit than the 5-limit.
> 
> () I expect to find a new best comma after searching n notes
> in the 5-limit, n(something) notes in the 7-limit.

These sound reasonable, but I don't see how to use them to determine
psychologically reasonable cutoff for lists of the temperaments most
likely to be musically useful.

I think we have no choice but to "cook it from a survey of the tuning
list, our personal tastes, etc.". Some of us have been doing informal
surveys of these questions on the tuning list for a decade or more.


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Message: 10223

Date: Fri, 13 Feb 2004 13:32:08

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

From: Carl Lumma

>> Dragnabbit, we've already been through this.  I do *not* mean
>> *diatonics*, I mean *scales*, YOUR definition, pitches.
>
>A discrete set of notes?

Pitches, yes.

-Carl


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Message: 10224

Date: Fri, 13 Feb 2004 20:00:57

Subject: Re: loglog!

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >> >> >The complexity measures cannot be compared across different 
> >> >> >dimensionalities, any more than lengths can be compared with 
> >> >> >areas can be compared with volumes.
> >> >> 
> >> >> Not if it's number of notes, I guess.
> >> >
> >> >What's number of notes??
> >> 
> >> Complexity units.
> >
> >It's only that (or very nearly that) in the ET cases.
> 
> Your creepy complexity is giving notes, clearly.

Hmm . . .And what do you propose to use for the 5-limit linear and 7-
limit planar cases?

> >So it the below 
> >still relevant?
> 
> Yes!  It's a fundamental question about how to view complexity.
> I'd be most interested in your answer.

Again, I view complexity as a measure of length, area, volume . . . 
in the Tenney lattice with taxicab metric. We're measuring the size 
of the finite dimensions of the periodicity slice, periodicity tube, 
periodicity block . . . 

> >> >> I've suggested in the
> >> >> past adjusting for it, crudely, by dividing by pi(lim).
> >> >
> >> >Huh? What's that?
> >> 
> >> If we're counting dyads, I suppose higher limits ought to do
> >> better with constant notes.
> >> If we're counting complete chords,
> >> they ought to do worse.  Yes/no?

Still have no idea how to approach this questioning, or what the 
thinking behind it is . . .


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