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Message: 10175 - Contents - Hide Contents

Date: Thu, 12 Feb 2004 20:33:28

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

From: Graham Breed

Dave Keenan wrote:


> I think that was _convex_ hull. That's just the smallest convex shape
> that encloses a set of points - a polygon with the "outermost" points
> as its vertices. I think it goes without saying that we would never
> exclude a temperament that was inside the convex hull of the included
> temperaments. Although not as formalised (yet), moats can be
> considered as taking the convex hull idea a little further, by
> including not only those _inside_ the convex hull but also those
> _close_ to the outside of it, and insisting that the hull is not only
> convex, but smooth.


Thanks!  I asked because they got mentioned in a paper relating to a 
project I was looking at and have now chosen.  What I'll be looking at 
needn't be convex but does have to be smooth.


>> - would k-means have anything to do with the clustering?
>> 
>> K-Mean Clustering Tutorial * [with cont.]  (Wayb.)
> 

> Sorry. Don't know anything about these.


Same project -- I was told to look them up.  Fuzzy k-means in fact, 
which will be the opposite of "moats".  I suppose you could segment 
temperaments into two categories, one "good" and the other "bad" but 
it'd need a peculiar geometry.  Alternatively, how about separate 
categories for "run of the mill", "way out" and "microtemperaments"?

Anyway, it may have something to do with clustering theory.


Happy de-tox! (if you haven't already left)


                Graham

Message: 10176 - Contents - Hide Contents

Date: Thu, 12 Feb 2004 01:11:12

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

From: Gene Ward Smith

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


> I'm starting to think that for you, "listen to me" means, "agree with
> everything I say", or at least "follow up every direction of
> investigation I suggest".


It means at least not to scoff at the idea of using lines without
first thinking about what they can do, not telling people that what
they are doing is useless and they should toddle off and compute
another list instead, and in general that you let someone outside of
the committee of two in on the conversation if they seem to want to help.


> What do we have to do to convince you we're listening?

 You'd convince
> us that you were listening to us if you didn't have to ask for reposts
> of information that's already been recently posted twice.


Paul posted a loglog graph, and I had trouble understanding it at
first--my eyesight is probably not as good as yours and I have trouble
seeing these things, and was relying mostly on what Paul said about
it--and it got me off onto the wrong foot, with the whole business of
trying to make elliptical regions. Then I forgot that is what the
graph, was, sorry--but it wasn't labeled when I looked at it again.


> Sorry. I'm getting hot under the collar again, and that's pretty bad
> because I'm not even wearing a shirt ..., but it's already 30 degrees
> Celsius here with extreme humidity, at 10 in the morning. :-)


Ouch. Try moving to California. :)

Message: 10177 - Contents - Hide Contents

Date: Thu, 12 Feb 2004 03:19:45

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

From: Dave Keenan

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

> [Dave Keenan wrote:]

>> 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.
> 

> Can we keep log-flat on the table for the moment?


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.

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.

Message: 10178 - Contents - Hide Contents

Date: Thu, 12 Feb 2004 20:57:37

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

From: Graham Breed

Dave Keenan wrote:


> I'm guessing ennealimmal is so complex and so close to 7-limit JI that
> most musicians would happily use the two interchangeably without
> noticing. Is that something we could test?


Ennealimmal isn't *that* complex.  By my measure, 27, so you get 18 
tetrads in the 45 note MOS.  That's within what's possible to fit to a 
keyboard, and more useful as a general notation system for computer 
music (it might be too strange for musicians to read, but who knows?)

It is very accurate -- to 0.2 cents.

IIRC, it tunes out the 2401:2400 comma.  So that comma pump I came up 
with already shows something that would work in ennealimmal but not 
7-limit JI.  And you get conceptual simplicity relative to the planar 
temperament, as well as more accuracy than miracle.


> 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_. One can't hear the difference between
> ennealimmal tuned as a subset of 612-ET and tuned with an
> ever-so-slightly different generator that is an irrational fraction of
> an octave.


Yes, the 7-limit minimax ennealimmal generator is 24.993/612 octaves, so 
you probably couldn't tell the difference.  As Gene said, it's a matter 
of simplicity and convenience.  It's much easier to think in integer 
steps than multiples of two generators.  And a computer could easily 
hold a 612 note octave tuning table, to allow for arbitrary modulations. 
  A lot of concepts relating to algorithmic composition work best with 
cyclic groups.

Quite indigestible for most of us, but not in loony territory yet.


                  Graham

Message: 10179 - Contents - Hide Contents

Date: Thu, 12 Feb 2004 03:26:07

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

From: Gene Ward Smith

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


>> But badness is clearly a psychological property,
> 

> No it isn't!  What evidence do we have that badness means anything
> musical at all?


Without a definition of "badness", we can't possibly have evidence for
what it means. Mostly, it's been a number used as an aid for a
decision proceedure. Log flat badness, occurring at the critical
exponent, could claim to be a "property". Keenan's psychological
badness is undefined.

Message: 10180 - Contents - Hide Contents

Date: Thu, 12 Feb 2004 22:27:27

Subject: Re: Rhombic dodecahedron scale

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" 


> Thanks Gene. This is neat. It will keep me busy for awhile.


If you want to learn more about the Jacobi theta function, you can 
read the Wikipedia article "Theta function", which must be a good one 
since I wrote it. :)

Message: 10181 - Contents - Hide Contents

Date: Thu, 12 Feb 2004 03:31:37

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

From: Gene Ward Smith

--- 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,... * [with cont.]  (Wayb.)


> 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?

Message: 10182 - Contents - Hide Contents

Date: Thu, 12 Feb 2004 22:42:29

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

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:

 And a computer could easily 

> hold a 612 note octave tuning table, to allow for arbitrary 
modulations. 


Most people would probably be satisfied with the tuning accuracy of 
171 notes. Joe would be satisfied with 72 notes, I suppose, given 
that he thinks that is JI. Last time I started to do an ennealimmal 
piece, I ended up with 171-et instead because the tuning sounded fine 
to me and you also temper out the schisma, which is convenient. This 
time I'm sticking to pure, top-tuned ennealimmal, but not because 171 
isn't actually already an acceptable way to tune it, at least to me.

Message: 10183 - Contents - Hide Contents

Date: Thu, 12 Feb 2004 01:17:20

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? Why can't you do scale-building stuff without them?


>>>>> 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?
> 

> Micro-biased means biased in favor of microtemperaments.


Then you are wrong. I'm not micro-biased. I'm biased in favour of
temperaments whose error and complexity are not enormously greater
than any that have ever been used before _as_approximations_of_JI_. In
my book, if you can't make a recognisably consonant complete otonal
n-limit chord in it, with no odentity duplicated at octaves, and
without using a custom made scale-based timbre, then it aint an
n-limit temperament (i.e. approximation of n-limit JI), useful though
it may be for other purposes.


> I've
> historically fought for macros vs. Dave.  But in general if it
> ever appears that I'm taking a side on any of these lists, please
> stop adn consider that I rarely do so -- I sometimes appear to
> do so if a position hasn't been _explored_ to my satisfaction.


And good on you for that. You and Paul forced me to nail down what I
object to about them, as above.

Message: 10184 - Contents - Hide Contents

Date: Thu, 12 Feb 2004 03:31:08

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

From: Dave Keenan

Carl, you 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, 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.

Message: 10185 - Contents - Hide Contents

Date: Thu, 12 Feb 2004 23:04:56

Subject: Interval-count complexity

From: Gene Ward Smith

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. Hence, for 
something in shell n, at a distance of sqrt(n), you need at minimum 
ceil(sqrt(n)) steps. For 81/80 this is ceil(sqrt(13))=4 steps, and for
2401/2400 it is ceil(sqrt(11))=4 steps also. 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)

Any comma pump for these commas will be related to some arragement of 
these steps. You could say from this that 81/80 and 2401/2400 are 
equally complex; however 2401/2400 involves all septimal intervals, 
and 81/80 all 5-limit intervals, and weighting the intervals will 
make it less complex. 4375/4374 is in shell 35, so
we need at least six steps to get to it; but in fact seven are 
required and again this is unique

4375/4374 = (7/6)(4/3)^2(5/3)^4 (1/16)

Weighting will bring down the relative complexity of this compared to 
2401/2400, since we have six 5-limit steps and only one septimal step.

Message: 10186 - Contents - Hide Contents

Date: Thu, 12 Feb 2004 01:25:56

Subject: Re: !

From: Dave Keenan

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

> Incidentally, I don't see the point of a moat vs. a circle, since
> the moat's 'hole' is apparently empty on your charts -- but I
> guess the moat is only meant for linear-linear, or?


There's no vs. between moats and circles. And moats are meant just as
much for log-log. 

The circle would ideally be placed so that it just touches two or
three of the included temperaments and has no temperament close to the
outside of it, i.e. so it is surrounded by a moat.

You know what a moat is right? You have the castle (the circle is its
outer bound) with people (temperaments) inside . Then you have the
moat surrounding that, with no people in it. Then you have the rest of
the world with the rest of the people in it.

Message: 10187 - Contents - Hide Contents

Date: Thu, 12 Feb 2004 23:38:16

Subject: A 9-limit diamond packing

From: Gene Ward Smith

If we start by considering 5,7, and 9 to be equal, we can put note-
classes of the form 9^a 5^b 7^c into a fcc lattice like the 7-limit 
lattice. If we take the centroid of 1,9,5,7, where 9 is represented by
[0 1 1], 5 by [1 0 1] and 7 by [1 1 0], we get ([0 0 0]+[0 1 1]+[1 0 
1]+[1 1 0])/4 = [1/2 1/2 1/2]. I propose placing 3 there. This would 
gives us a diamond packing, not a lattice--that is, 9-limit notes are 
arranged like the carbon atom of a diamond.

However, we have both otonalities and utonalities to deal with. If we 
scale everything up by two, we have 3 at [1 1 1] surrounded by 9, 5, 
and 7, and inverting this gives 1/3 at [-1 -1 -1] surrounded by 1/9 
1/5 and 1/7. Transposing both of these to the unison, we have the 
unison surrounded by two tetrahedra, which represent the 1:5:7:9 and 
1:1/5:1/7:1/9 tetrads respectively, and are represented by [-1 1 1], 
[1 -1 1], [1 1 -1] and [-1 -1 -1] for the major tetrad, with sum of 
coordinates congruent to 1 mod 4, and [1 -1 -1], [-1 1 -1], [-1 -1 1] 
and [1 1 1] a minor tetrad, with sum of coordinates congruent to -1 
mod 4. The fcc lattice we started with is now everything with even 
coordinates which sum to 0 mod 4, if we add everything with even 
coordinates which sum to 2 mod 4, we get the body-centered cubic 
lattice. We can do this by adding the minor tetrad around [1 1 1] to 
the major one. Now I've got a bcc lattice of notes, consisting of 
every triple with all even or all odd coodinates, but since 3*3=9 I 
have the same note represented more than once.

Message: 10188 - Contents - Hide Contents

Date: Thu, 12 Feb 2004 01:28:34

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

From: Dave Keenan

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


> 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 LT.


I had "ET" at the end there by mistake.

Message: 10189 - Contents - Hide Contents

Date: Thu, 12 Feb 2004 15:47:30

Subject: Re: !

From: Carl Lumma

At 04:11 PM 2/11/2004, you wrote:

>--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:

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

>>> 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.
>> 

>> It does? Look at the graph! How can you make 0 cents = zero when it's 
>> infinitely far away? And what about the position of the origin on the 
>> *complexity* axis??
>> 

>>> No origin; pfff.
>> 
>> piano-forte-forte-forte?
>> 

>> P.S. The relative scaling of the two axes is completely arbitrary, 
>> so, even if you actually selected an origin, the circle would produce 
>> different results for a different relative scaling.
>

>But an elliptical cutoff in log-log space could be made to work. You
>do have to choose nonzero values of error and complexity to represent
>zero pain (the center of the ellipse). But that's OK.


Glad you agree.


>So I'd also like to see if one of these elliptical-log beasties can be
>made to give the same list as Paul's red line on the 7-limit-LT
>log-log plot. How about it Carl?


I didn't follow how a circle became an ellipse.

Sorry all if I'm dropping correspondence of late.  I'm having a
nightmare of a time with my provider at the moment.

-Carl

Message: 10190 - Contents - Hide Contents

Date: Thu, 12 Feb 2004 16:24:32

Subject: Re: A 9-limit diamond packing

From: Carl Lumma

Can you, uh, draw it?

-Carl

Message: 10191 - Contents - Hide Contents

Date: Thu, 12 Feb 2004 00:11:26

Subject: Re: !

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:

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

>> 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.
> 

> It does? Look at the graph! How can you make 0 cents = zero when it's 
> infinitely far away? And what about the position of the origin on the 
> *complexity* axis??
> 

>> No origin; pfff.
> 
> piano-forte-forte-forte?
> 

> P.S. The relative scaling of the two axes is completely arbitrary, 
> so, even if you actually selected an origin, the circle would produce 
> different results for a different relative scaling.


But an elliptical cutoff in log-log space could be made to work. You
do have to choose nonzero values of error and complexity to represent
zero pain (the center of the ellipse). But that's OK. As Graham
pointed out, that's exactly what we do with loudness.

So I'd also like to see if one of these elliptical-log beasties can be
made to give the same list as Paul's red line on the 7-limit-LT
log-log plot. How about it Carl?

Message: 10192 - Contents - Hide Contents

Date: Thu, 12 Feb 2004 03:43:05

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,
> 

> No it isn't!  What evidence do we have that badness means anything
> musical at all?


So why call it "badness". Bad for whom? Bad for what? 

Humans and music, that's who and what.

We either want to _make_ badness mean something psychological or use a
different word for this thing we are trying to come up with to model
the psychology of musical usefulness of temperaments, at least in so
far as to produce a short-list.


>> what have mathematical first principles got to do with it?
> 

> What _don't_ they have to do with?  For folks into "digital physics"
> like me, nothing.


Sure. Everything may, _in principle_, be derivable from mathematics
but the intervening complexity of human neuro-physiology is such that
this is utterly irrelevant to what we are trying to do here.

Message: 10195 - Contents - Hide Contents

Date: Thu, 12 Feb 2004 04:00:26

Subject: Re: acceptace regions

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 log(error) and log(complexity) are already so far away from any
>> reasonable kind of pain(error) and pain(complexity) that I fear this
>> will just make it worse. You convinced us to humour you with the
>> log-log thing, but this seems to be going too far.
> 

> One posting you use the word "exciting" and in the next you aren't
> even willing to look at a plot, because doing that might be to go too
> far in humoring an obviously confused person. Which is it?


Wow! You're good at putting offensive words in people's mouths aren't
you? All I said was I don't thing it's worth the trouble of doing a u
v plot of it. I'd rather see it on log log and even better, linear
linear. But if you've done the plot, I'm happy to look at it.


>> OK. So long as its something close to Paul's list, otherwise we're
>> just wasting our time.
> 

> Does this mean you didn't even *look* at my lists??


Of course I looked at it, but I didn't line it up against Paul's and
check off the differences. It looks pretty close from memory, but it
will mean a lot more if I see the two cutoff lines plotted on the same
graph, preferably linear linear, but I'm happy to look at the others too.


>> We're not trying to _avoid_ subjectivity, we're trying to _model_ it.
> 

> The best model for total subjectivity is simply to pick any
> temperametns you like for any reason you like. We could try that.


I mean we are trying to model some kind of statistical average of
human subjectivity on the question of the musical usefulness of a
temperament.

It should be obvious that humans will agree with each other as to what
is a good temperament, infinitely better than any of them will agree
with a random temperament selector.


> 

>> And we're trying to  do so in such a way that 
>> (a) the model has only a small number of parameters, preferably no
>> more than 3.
> 

> Moving the origin of my uv stuff is two. Adding a hyperbola gives you
> three.


Good. I'm just waiting to see it plotted.


>> (c) we have agreement from as many people, as possible.
> 

> Has this last been attempted? It seems to me it's got to be whatever
> you and Paul want, lately.


It's what this whole thing is about. We're arguing. Trying to convince
each other. I've moved a considerable way towards accepting more
extreme errors and complexities than I would have thought acceptable
two years ago, thanks to the arguments of people on this list.
Sometimes the process is painful but we do seem like we might be
converging.

Message: 10196 - Contents - Hide Contents

Date: Thu, 12 Feb 2004 01:52:33

Subject: Re: acceptace regions

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:

>> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...>
wrote:

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

>>>> We might try in analyzing or plotting 7-limit linear
temperaments a 
>>>> transformation like this:
>>>> 
>>>> u = 4 - ln(complexity) - ln(error)
>>>> v = 12 - 4 ln(complexity) - ln(error)
>> 

>> In the terms Paul, Carl and I have been using, this is a cutoff
relation
>> 
>> max[ln(complexity)/4 + ln(error)/4, 
>> 4 * ln(complexity)/3 + ln(error)/12] < 1
> 

> Only if you choose to use it for one. It's a coordinate
> transformation, primarily. 
> 
> 

>> It seems we may be moving towards some kind of agreement. :-)
> 

> I've been *trying* to help you and Paul here, with all of this stuff
> about convex hulls and what not which I have been told is useless. I
> hope I am finally communicating *something*. This idea, by the way, is
> an old one but not much heed was paid to it by anyone, including me,
> and I proposed it. The new aspects are to use it as a coordinate
> transformation in a loglog context, and to draw hyperbolas if you want
> to smooth corners.
> 

>> OK. But I don't think it will help to do a u v plot.
> 
> Why not?


Well log(error) and log(complexity) are already so far away from any
reasonable kind of pain(error) and pain(complexity) that I fear this
will just make it worse. You convinced us to humour you with the
log-log thing, but this seems to be going too far.

What is the point of putting all the math-complexity of the cutoff
relation into the coordinate transformation just so the cutoff
relation looks simple. The math-complexity is still there.

> 
>  I'd prefer to see

>> it on the existing log log plot,
> 

> Why not both?


Sure. I'd like to see it on linear-linear too. But since I'm not doing
it, I figure I can't ask for too much.


>  and I'd really like to see if you can

>> come up with one of these hyperbolic-log beasties that gives the same
>> list as Pauls red curve. This is exciting. :-)
> 

> Probably that can be done, but what is special about Paul's red curve?
> I didn't like it, and in any case Paul tells us that the reason it
> drops off so fast at the end is not because he was trying to nuke
> ennealimmal, but because he ran out of things to plot. 


OK. So long as its something close to Paul's list, otherwise we're
just wasting our time.


> The massive
> subjectivity of it all is what I'm hoping to avoid.


I'm not sure why I'm not getting through with this, but you can't
avoid subjectivity if you're trying to make a list of things that are
likely to be of use to human musicians, not mathematician, not aliens
from outer space, not humpback whales, but humans.

We're not trying to _avoid_ subjectivity, we're trying to _model_ it.
And we're trying to  do so in such a way that 
(a) the model has only a small number of parameters, preferably no
more than 3.
(b) the list chosen is as insensitive as possible to the values of the
parameters
(c) we have agreement from as many people, as possible.

Message: 10197 - Contents - Hide Contents

Date: Thu, 12 Feb 2004 04:10:37

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

From: Dave Keenan

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

> [Dave wrote:]

>> Well yeah but we're probably within a factor of 2 of agreeing.
>> Another species could disagree with us by orders of magnitude.
> 

> This was addressed to "So Carl".  


I think that's a fairly universal english-speaking way of announcing
that you are about to ask the named person a question. What of it?


> Am I not human?


I have always assumed so. Are you telling me you're not? :-) What's
your point.


> I think we want roughly the same things.  


Wow. That would be great.


> Except I want to answer
> questions like those I just mentioned (which among other things
> investigate making complexity comparable across harmonic limit and
> dimensionality), and why Paul's creepy complexity gives the
> numbers it does, before continuing.


Go for it.


> And I maintain that a survey of the tuning list would be a
> cataclysmic scientific error.  But with reasonable pain axes,
> such as cents**2 and 2**notes, finding the widest reasonably-
> convex moat that encloses the desired number of temperaments
> for each case (limit, dimensionality) would seem to be a good
> idea and sufficient to eliminate the need for a survey.


Woah. Now we're talking!


>> But cutoffs utterly violate log-flat badness in the regions
>> outside of them.
> 

> I have no problem with smoothing the cutoff region.

Great!


>>> 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.

Message: 10198 - Contents - Hide Contents

Date: Thu, 12 Feb 2004 00:28:59

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:
>  a week starting next week (Tue to Fri).
> 
> 

>> Why can't you understand that the mathematical fact that temperaments
>> come out with a straight edge on a log-log plot has absolutely no
>> bearing on which of them will be found musically useful to humans.
> 

> Does this mean you are simply going to blow me off and ignore the hard
> work I have done trying to accomodate both you and Paul?

Absolutely not.


>> I can certainly understand Paul's impatience. 
> 

> Can you understand my impatience with the fact that neither you nor
> Paul seems willing even to listen to me?


For god's sake man, what _are_ you talking about?

I'm starting to think that for you, "listen to me" means, "agree with
everything I say", or at least "follow up every direction of
investigation I suggest".

What do we have to do to convince you we're listening? You'd convince
us that you were listening to us if you didn't have to ask for reposts
of information that's already been recently posted twice.

Sorry. I'm getting hot under the collar again, and that's pretty bad
because I'm not even wearing a shirt ..., but it's already 30 degrees
Celsius here with extreme humidity, at 10 in the morning. :-)

Message: 10199 - Contents - Hide Contents

Date: Thu, 12 Feb 2004 02:14:25

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

From: Dave Keenan

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

>> Humans seem to find a particular region of complexity and error
>> attractive and have a certain approximate function relating error and
>> complexity to usefulness. Extra-terrestrial music-makers (or humpback
>> whales) may find completely different regions attractive.
> 

> This seems to be the key statement of this thread.  I don't think
> this has been established.  If it had, I'd be all for it.  But it
> seems instead that whenever you cut out temperament T, somebody
> could come along and do something with T that would make you wish
> you hadn't have cut it.  Therefore it seems logical to use something
> that allows a comparison of temperaments in any range (like logflat).


So Carl. You  really think it's possible that some human musician
could find the temperament where 3/2 vanishes to be a useful
approximation of 5-limit JI (but hey at least the complexity is
0.001)?  And likewise for some temperament where the number of
generators to each prime is around a google (but hey at least the
error is 10^-99 cents)?


> Then no matter what T is, we can say...
> 
> "You could have used U, which is in the same range but better."
> 
> ...or...
> 
> "T's the best in that range.  Bravo!"
> 
> ...The worst that could happen would seem to be...
> 
> "T falls outside the range we established for our paper, sorry."
> 
> ...in which case the reader could perform his own analysis in the
> above way.  With a cooked acceptance region, however, the following
> could happen...
> 
> "Oh, T.  It didn't meet our guesses about human cognition, but YMMV."


I don't understand why you think log-flat is a magic bullet in this
regard. If you use log flat badness and include the same number of
temperaments as Paul and I and Gene are considering (around 20), then
exactly the same scenario is possible, only this time it will be
temperaments with moderate amounts of both error and complexity that
are omitted and the objecting musician won't be fictitious, he'll be
Herman Miller.

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