>> 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.
That doesn't seem to be true. There are lots of low-complexity
temperaments with arbitrarily high error. Wouldn't you have to bound
epimericity or something like that?
> Most are complete crap.
Agreed.
Message: 10053 - Contents - Hide Contents
Date: Tue, 10 Feb 2004 00:17:14
Subject: Re: The seven-limit lattices
From: Gene Ward Smith
--- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
> How is it that unit cube have 14 tones? Thanks
The cube is a cube of chords; it has 8 chords but 14 notes.
I've added a bunch, and there is still more that could be said about
all this.
Message: 10054 - Contents - Hide Contents
Date: Tue, 10 Feb 2004 04:00:13
Subject: Re: 23 "pro-moated" 7-limit linear temps
From: Paul Erlich
--- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma" <ekin@l...> wrote:
>>> is that
>>> we have no data. Not vague historical data, actually no data.
>>
>> Less data than in the log-flat case?
>
> log-flat is natural, in a way. And it should be one of the easier
> concepts around here to explain to musicians.
I don't recall even Dave understanding its derivation, let along any
full-time musicians.
> So far, you and Dave
> have not done any kind of job explaining "moats",
I thought we had, over and over again.
> or why we should
> want to add instead of multiply to get badness.
Why should we want to multiply instead of add?
> I notice that you're now saying that you don't want to use badness
> at all, which is what I said would be the logical extreme of a
> suggestion you made, and you argued against it!!
No, Carl, I was arguing against it being a logical extreme of, or
having any correlation in desirability with, that suggestion (which
was a hypothetical max(a*error, b*complexity) criterion).
>>> By putting all this energy into the list of temperaments, we're
>>> loosing 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.
>>
>> Seems to be a creepy coincidence, since it's an affine-geometrical
>> measure of area in the Tenney lattice, not something with units of
>> number of notes. But I'm not surprised that it gives more "notes"
>> for more complex temperaments, and fewer for less complex
>> temperaments. ;)
>
> I mean, it seems to favor DE scales,
Right, and that seems to me to be just a coincidence, but there are
many unanswered questions, for example in the post about 12-equal
complexity calculation.
Message: 10055 - Contents - Hide Contents
Date: Tue, 10 Feb 2004 06:58:04
Subject: Re: !
From: Gene Ward Smith
--- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...>
wrote:
> I don't much care how it's plotted, so long as we zoom in on the
> interesting bit. So, on these plots, what shape would you make a
> smooth curve that encloses only (or mostly) those ETs that musicians
> have actually found useful (or that you think are likely to be found
> useful) for approximating JI to the relevant limit? Having regard
for
> the difficulty caused by complexity as well as error.
Straight lines, or if you absolutely must, round out the straight
lines by making them algebraic curves of high degree.
> I wonder if, when you say that there is no particular problem with
> complexity you are thinking of cases where you may use a subset of
an
> ET, in the way that Joseph Pehrson is using a 21 note subset of 72-
ET.
> In that case you are really using a linear temperament, not the ET
> itself. I think the complexity of an ET should be considered as if
you
> planned to use _all_ its notes.
You are not thinking like a computer-based composer, and that is
likely to be an increasingly important consideration as time goes on.
Technology has a habit of getting both better and more available.
Around here Fry's sometimes sells new computers--good ones--for $99
as a loss leader. A cheap monitor, a pair of those cheap earphones
I've been hearing about on tuning, and some freeware and you're in
business, if only people like us will just tell you how.
Message: 10056 - Contents - Hide Contents
Date: Tue, 10 Feb 2004 20:26:44
Subject: Re: 23 "pro-moated" 7-limit linear temps
From: Paul Erlich
--- 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?
We don't care, since we're including *all* the systems with error and
complexity no worse than *any* of these systems, as well as miracle.
And that's quite a few!
> 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.
Except on lutes . . .
> 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,
Nasty, nasty us.
Message: 10057 - Contents - Hide Contents
Date: Tue, 10 Feb 2004 21:31:21
Subject: Re: 23 "pro-moated" 7-limit linear temps
From: Gene Ward Smith
--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:
> I think the regular plot will be easier to explain than the log-log
> plot.
Are you going to actually explain it, or just sweep that under the
rug? In other words, are you going to explain why what you are doing
makes sense? If you propose a real explanation, in the sense of
rational justification, good luck.
Message: 10058 - Contents - Hide Contents
Date: Tue, 10 Feb 2004 00:21:54
Subject: Re: 23 "pro-moated" 7-limit linear temps
From: Gene Ward Smith
--- 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?
Message: 10059 - Contents - Hide Contents
Date: Tue, 10 Feb 2004 04:01:15
Subject: Re: 23 "pro-moated" 7-limit linear temps
From: Paul Erlich
--- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma" <ekin@l...> wrote:
>>>> 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?
How can I illustrate infinity?
> It looks like the zero-error line is
> three dashes away on the above loglog plot. :)
Since you're smiliing, I'll assume you "got it".
Message: 10060 - Contents - Hide Contents
Date: Tue, 10 Feb 2004 07:06:25
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 "Paul Erlich" <perlich@a...>
> wrote:
>
>> But there's less to tweak -- we just find the thickest moat than
>> encloses the systems in the same ballpark as the ones we know we
>> definitely want to include. This seems a lot less arbitrary than
>> tweaking *three* parameters to satisfy one's sensibilities as best
> as
>> possible.
>
> Your plots make it clear you'd better trash the idea of doing moats
> in anything but loglog.
I don't have a problem with that. I still think the simplest curves
through moats that are in the right ballpark will be of the form
(err/k1)^p + (comp/k2)^p < x where p is 1 or slightly less than 1.
in terms of log(err) and log(comp) that's equivalent to
exp([log(err) - k1] * p) + exp([log(comp) - k2)] * p) < x
with a different choice of k1, k2 and x.
Is there a simpler function of log(err) and log(comp) that gives
similar shaped curves in the region of interest?
Message: 10061 - Contents - Hide Contents
Date: Tue, 10 Feb 2004 16:15:06
Subject: Re: loglog!
From: Carl Lumma
>> >or 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.
>
>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. I've suggested in the
past adjusting for it, crudely, by dividing by pi(lim). But
I think by fiddling with the variables in my scheme procedure
one could do better.
-Carl
Message: 10062 - Contents - Hide Contents
Date: Tue, 10 Feb 2004 20:29:46
Subject: Re: The seven-limit lattices
From: Paul Erlich
--- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith"
<gwsmith@s...>
> wrote:
>> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad"
>>
>
>>> How is it that unit cube have 14 tones? Thanks
>>
>> The cube is a cube of chords; it has 8 chords but 14 notes.
>>
>> I've added a bunch, and there is still more that could be said
> about
>> all this.
>
> Actually, I have a few questions. I've hopefully thought this
through
> so that my questions are good ones.
>
> 1. The 14 notes of the stellated hexany. Do these correspond to the
> sides of the cubeoctohedran? (Or the points on a D3 face centered
> lattice)...
>
> 2. Where in the cubeoctohedran are the tetrahedra and octahedra?
> Could you please give an example of a hexany...
Hi Paul, you may want to read this paper:
http://lumma.org/tuning/erlich/erlich-tFoT.pdf - Type Ok * [with cont.] (Wayb.)
through the pictures of the hexany and stellated hexany.
-Paul
P.S. Carl -- this used to come up in Google, but no longer does :(
Message: 10063 - Contents - Hide Contents
Date: Tue, 10 Feb 2004 21:33:26
Subject: Re: !
From: Paul Erlich
--- 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 "Dave Keenan" <d.keenan@b...>
>> wrote:
>>> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith"
<gwsmith@s...>
>>> wrote:
>>>> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich"
<perlich@a...>
>>>> wrote:
>
> Dave wrote:
>>> I don't much care how it's plotted, so long as we zoom in on the
>>> interesting bit. So, on these plots, what shape would you make a
>>> smooth curve that encloses only (or mostly) those ETs that
musicians
>>> have actually found useful (or that you think are likely to be
found
>>> useful) for approximating JI to the relevant limit? Having
regard
>> for
>>> the difficulty caused by complexity as well as error.
>>
>> Did either of you guys look at the loglog version of the moat-of-
23 7-
>> limit linear temperaments?
>
> Sure. I looked at it and agree with it just fine. That should be
> obvious since I agreed just fine with it on a linear-linear plot. I
> was asking Gene what shape _he_ thought it should be, and
particularly
> in regard to 5-limit ETs. He says "a straight line", so I think
we're
> doomed to disagree.
Yes, a straight line in loglog will either include a finite (?)
number of ultra-high-error temperaments, or an infinite number of
ultra-high-complexity temperaments. Unless you replace the straight
line with a "staple", which places far too much importance on the
arbitrary location of the corners.
Personally, I don't think we should even be looking at the loglog
plots, since the axes don't represent quantities even in the ballpark
of what can be considered 'pain'. View the lay of the land as you
find it, not after an ultra-clever mathematical transformation of it.
Message: 10064 - Contents - Hide Contents
Date: Tue, 10 Feb 2004 00:32:56
Subject: Re: 23 "pro-moated" 7-limit linear temps
From: Gene Ward Smith
--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> Aha, now I understand your objection. But wait, what's stopping
> this from being a rectangle? Is the badness bound giving the
> line AB? If so, it looks like a badness cutoff alone would give a
> finite region...
You don't need a finite region, just a finite number of temperaments
below the badness line. This is easily accomplished in loglog as
well; the difficulty is that people are likely to be unhappy with the
fact that both very high error, low complexity temperaments and very
low error, high complexity temperaments are likely to be included.
Since these are evil, and simply excluding them on the grounds we
don't want them is also evil, 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.
Message: 10065 - Contents - Hide Contents
Date: Tue, 10 Feb 2004 04:10:32
Subject: Re: 23 "pro-moated" 7-limit linear temps
From: Paul Erlich
--- 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.
>
> 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.
Thus it's great for a paper for mathematicians. Not for musicians.
>> 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!
>> 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.
From a purely mathematical standpoint only.
>> 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 factor infinity??
> to satisfy your
> sensibilities as best as possible.
But there's less to tweak -- we just find the thickest moat than
encloses the systems in the same ballpark as the ones we know we
definitely want to include. This seems a lot less arbitrary than
tweaking *three* parameters to satisfy one's sensibilities as best as
possible.
Message: 10066 - Contents - Hide Contents
Date: Tue, 10 Feb 2004 07:09:01
Subject: Re: The same page
From: Gene Ward Smith
--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...>
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith"
<gwsmith@s...>
> wrote:
>> Just to be sure we are on it, in terms of defintions of compexity
> and
>> error, here is my page.
>
> Where are ETs?
Forgot 'em, but you seem to have them figured out. Modulo some slight
fiddling if you must fiddle, complexity is n for the n-et, so log
complexity is log(n).
>> so log(complexity) is loglog(n*d). Error is
>> distance from the TOP tuning to the JIP, or in other words the
max
> of
>> the absolute values of the errors for 2, 3 and 5 in TOP tuning,
>> divided by log2(2), log2(3) and log2(5) respectively.
>
> It also can be expressed as log(n/d)/log(n*d) (*1200).
How can either log(n*d) or loglog(n*d) also be expressed as
epimericity, which this is very close to being?
>
>> Log(error) is
>> the log of this. Loglog plots compare loglog(n*d) with log(error).
>
> i.e., log(log(n*d)) with log(log(n/d)/log(n*d)).
>>
>> 7-limit linear
>>
>> Complexity is the Erlich magic L1 norm; if <<a1 a2 a3 a4 a5 a6||
is
>> the wedgie,
>
> val-wedgie, yes.
That's how "wedgie" is defined.
Message: 10067 - Contents - Hide Contents
Date: Tue, 10 Feb 2004 12:30:41
Subject: Re: The same page
From: Carl Lumma
>> >nybody 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>>
Great, so what happens when the monzos are commas being
tempered out?
A chart running over comma useful things would help our
endeavor tremendously.
>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).
Ok great. But what's all about this algebraic dual? Is this
something I can do to matrices, like complement and transpose?
Again, a chart of these things would be awesome.
-Carl
Message: 10068 - Contents - Hide Contents
Date: Tue, 10 Feb 2004 16:17:40
Subject: Re: 23 "pro-moated" 7-limit linear temps
From: Carl Lumma
>> >ssuming 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?
>
>We don't care, since we're including *all* the systems with error and
>complexity no worse than *any* of these systems, as well as miracle.
>And that's quite a few!
But you can still make the same kind of error.
-Carl
Message: 10069 - Contents - Hide Contents
Date: Tue, 10 Feb 2004 21:35:48
Subject: Re: Rhombic dodecahedron scale
From: Gene Ward Smith
--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
>> Here is a scale which arose when I was considering adding to the seven
>> limit lattices web page. 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?
>> 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.) But we don't seem to be using the same defintion of "dual".
>> 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,
>
> Ah, that would be the 'dual' operation I was thinking it above.
> I saw a graphic of this on site about Fuller once.
>
>> 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.
Message: 10070 - Contents - Hide Contents
Date: Tue, 10 Feb 2004 00:35:58
Subject: Re: 23 "pro-moated" 7-limit linear temps
From: Carl Lumma
>> >s that
>> we have no data. Not vague historical data, actually no data.
>
>Less data than in the log-flat case?
log-flat is natural, in a way. And it should be one of the easier
concepts around here to explain to musicians. So far, you and Dave
have not done any kind of job explaining "moats", or why we should
want to add instead of multiply to get badness.
I notice that you're now saying that you don't want to use badness
at all, which is what I said would be the logical extreme of a
suggestion you made, and you argued against it!!
>> By putting all this energy into the list of temperaments, we're
>> loosing 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.
>
>Seems to be a creepy coincidence, since it's an affine-geometrical
>measure of area in the Tenney lattice, not something with units of
>number of notes. But I'm not surprised that it gives more "notes"
>for more complex temperaments, and fewer for less complex
>temperaments. ;)
I mean, it seems to favor DE scales, but which ones and why?
-C.
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Message: 10071 - Contents - Hide Contents
Date: Tue, 10 Feb 2004 04:16:22
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:
>
>> Related how? Via log-flat badness? Meanwhile, error and
complexity
>> are related?
>
> If C is the complexity and E is the error for a 7-limit linear
> temperament which belongs to an infinite list of best examples,
Other than via log-flat badness?
> then
> E ~ k C^2. Taking the logs, log(E) ~ 2 log(C) + c. In other words,
we
> have a relationship; one can very roughly be estimated in terms of
> the other.
>> OK, I'll do more of them when I have a chance, but ultimately, I
>> don't think I want to force any musician to think about what log
>> (error) means or what log(complexity) means.
>
> You are going to explain TOP error and complexity, but this would
be
> too much math??
No, just too abstract a measure to wrap one's head around intuitively.
>>> High complexity really isn't such a big deal for some uses. JI
> can
>> be
>>> said to have infinite complexity in a sense (no amount of
fifths
>> and
>>> octaves will net you a pure major third, etc) which I think
shows
>>> Paul's worry about where it is on the graph is absurd,
>>
>> No, it shows the bullshit you're putting into my mouth is absurd,
> as
>> I agreed in a recent post.
>
> You go on and on about not finding the zero error line, though
> evidently not finding the infinite complexity line is not a
>problem.
No, it's not a problem, any more than not finding the infinite error
line is a problem.
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Message: 10072 - Contents - Hide Contents
Date: Tue, 10 Feb 2004 07:10:56
Subject: Re: Loglog
From: Dave Keenan
--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...>
> wrote:
>
>> 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?
>
> No, the good stuff lies along lines, making the whole moat business
> both much easier and far more logical--in case that matters to anyone.
On the loglog plot. The good stuff looks to me like a bite taken out
of the lower left side of the sheet of temperaments. Teeth marks and
all ;-).
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Message: 10073 - Contents - Hide Contents
Date: Tue, 10 Feb 2004 16:18:29
Subject: Re: The seven-limit lattices
From: Carl Lumma
>P.S. Carl -- this used to come up in Google, but no longer does :(
Dunno why that would be. I'll keep my feelers out on it.
-Carl
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Message: 10074 - Contents - Hide Contents
Date: Tue, 10 Feb 2004 20:34:23
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:
>
>>> Your plots make it clear you'd better trash the idea of doing
moats
>>> in anything but loglog.
>>
>> On the contrary.
>
> You did notice the approximately linear arragement, I presume?
Again: I've known this to be the case for years.
> Does
> that suggest anything to you?
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.
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