Tuning-Math Digests messages 10025 - 10049

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Message: 10025

Date: Tue, 10 Feb 2004 02:40:43

Subject: Re: The same page

From: Carl Lumma

> Complexity is the Erlich magic L1 norm; if
> <<a1 a2 a3 a4 a5 a6|| is the wedgie, then complexity is
> |a1/p3|+|a2/p5|+|a3/p7|+|a4/p3p5|+|a5/p3p7|+|a6/p5p7|.
> Log complexity is log of this. Error is the distance from
> the JIP of the 7-limit TOP tuning for the temperament; log
> (complexity) and log(error) are logs of complexity and
> error, so defined.

What are p3, p5, etc.?

-Carl


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Message: 10026

Date: Tue, 10 Feb 2004 20:11:21

Subject: Re: The same page

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:
> > --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" 
> <gwsmith@s...> 
> > wrote:
> > > 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,

I'd like to understand this slight fiddling, and apply this 
understanding to the 7-limit linear case (and elsewhere).

> 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?

Not following.

> > > 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.

They're merely duals of one another, but why this definition? I can 
understand taking the wedge product of monzos much better than I 
understand the wedge product of vals.


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Message: 10027

Date: Tue, 10 Feb 2004 21:20:23

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

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 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".
> 
> 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.

This is a music *theory* paper, so presenting the bare minimum of 
math to actually *derive* our results is appropriate. True, only 
heavy theorists will probably want to reproduce the calculations. But 
we want to leave referees, at least, with fairly complete confidence 
that what we're doing is correct.


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Message: 10028

Date: Tue, 10 Feb 2004 22:51:06

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

From: Carl Lumma

>> >> Could you do me a favor and attempt to speak to me as a human being, 
>> >> and not deal with me like a chess opponent, trying to look several 
>> >> moves ahead so that you can defeat me?
>> >
>> >I washed out of the first round of the US correspondence championship.
>> >It's my brother who is the grandmaster.
>> 
>> Is he really a grandmaster?
>
>He's not an OTB grandmaster, he is a correspondence grandmaster and
>several times US champion.

Wow, cool!  I don't follow the correspondence scene, but I always get
a kick out of the name of the column in No Life magazine: The Check Is
In The Mail.
:)

-Carl


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Message: 10032

Date: Tue, 10 Feb 2004 20:15:01

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:
> > --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" 
> <gwsmith@s...> 
> > wrote:
> > > --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" 
<d.keenan@b...> 
> > > wrote:
> > > 
> > > > The error is minimax error in cents where the weighting is 
log_2
> > > (n*d)
> > > > for the ratio n/d in lowest terms.
> > 
> > The weighting is actually ONE OVER log2(n*d).
> 
> It's not either one when I'm doing it, to me log(n/d)/log(n*d) is 
> just a variant on epimericity.

I'm not following you, and I'm at a loss to understand why the above 
definition of TOP error is suddenly a problem for you.

> > > What in the world does this mean?
> > 
> > Just because he's off by a multiplicate inverse, you suddenly 
have 
> no 
> > idea what he's talking about?
> 
> No, and you aren't making much sense to me either; we seem to have 
> differing ideas of what the topic under discussion is. Maybe I'm 
not 
> tracking it, but I thought we were talking about TOP error and 
> complexity.

Yes; TOP error is always defined as above (though there are plenty of 
equivalent definitions, such as over the primes alone).
> 
> > > Do you mean TOP error for an equal 
> > > temperament,
> > 
> > Of course that's what he means.
> > 
> > > which is dual to the above?
> > 
> > Dual? How does duality come into play here?
> 
> The dual to Tenney distance is how the error is measured.

Sure, but Dave's talking about measuring the errors directly (which 
may be less useful for finding a solution, but may be more useful for 
understanding what the solution means).


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Message: 10034

Date: Tue, 10 Feb 2004 00:01:26

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

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >> I'm sorry it came across that way. But the fact is we had already
> >> thought about it and found it too extreme, not possible to match 
up
> >> with the historical data (vague though that is). Sorry we didn't
> >> spell that out.
> >
> >It would be nice if some attempt was made to bring the rest of us 
on 
> >board. I don't know what Carl or Graham think, but I have not been 
> >convinced.
> 
> My latest position is that I can live with log-flat badness with
> appropriate cutoffs.  The problem with anything more tricky

More tricky? Log-flat is tricky enough to be interesting for 
mathematicians and mind-boggling for musicians.

> is that
> we have no data.  Not vague historical data, actually no data.

Less data than in the log-flat case?

> 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. ;)


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Message: 10036

Date: Tue, 10 Feb 2004 12:18:09

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

From: Carl Lumma

>> >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".

That's all the more reason particulars like moats v. logflat don't
matter -- if I can fish, I can implement my own list.  But I still
don't think the hypothetical "musicians" you're battering Gene
with will be able to do it.

>> >> >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?

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.

>> My 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.

Which part?  Can the equation/coordinates that defines your fav. moat
be taken from a 5-limit plot and slapped onto a 7-limit one?

-Carl


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Message: 10037

Date: Tue, 10 Feb 2004 21:27:12

Subject: Re: !

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" 
<gwsmith@s...> 
> wrote:
> > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:
> > 
> > > Did either of you guys look at the loglog version of the moat-
of-
> 23 7-
> > > limit linear temperaments?
> > 
> > I have a plot with unlabled axes and a curved red line on it.
> > Obviously, since I don't know what is being plotted, I draw no 
> >conclusion.
> 
> I've already clarified this for you!!

Again: The horizontal axis, as always, is *complexity*. The vertical 
axis, as always, is *error*. We've already established we're on the 
same page on those. It's easy to see, by the tick marks, if either or 
both of the axes is scaled logarithmically. The red line is our 
proposed moat. And again, the 7-limit 'linear' temperaments are 
indexed as follows (I show the first three numbers in the val-wedgie, 
since you feel they are the most important):

1) 1 4 10
2) 2 -4 -4
3) 5 1 12 
4) 7 9 13 
5) 1 4 -2 
6) 3 0 -6 
7) 4 -3 2 
8) 2 8 1 
9) 6 5 3 
10) 1 9 -2 
11) 2 8 8 
12) 6 -7 -2
13) 6 10 10 
14) 7 -3 8 
15) 4 4 4 
16) 1 -8 -14 
17) 3 0 6 
18) 0 0 12 
19) 1 4 -9 
20) 0 5 0 
21) 3 12 -1 
22) 10 9 7 
23) 3 5 -6 
24) 9 5 -3 
25) 8 6 6 
26) 6 -2 -2 
27) 6 5 22 
28) 3 12 11 
29) 2 -9 -4 
30) 11 13 17 
31) 6 10 3 
32) 4 2 2


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Message: 10038

Date: Tue, 10 Feb 2004 00:09:02

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

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> > >Complexity is horizontal, error is vertical,
> > 
> > Aha.
> > 
> > >labels are the notes per octave of the ET.
> > 
> > How can error be in notes?
> 
> Sorry. I was referring to the labels on the points. i.e. each point 
is
> labelled with the n of the n-tET that it is.
> 
> The error is minimax error in cents where the weighting is log_2
(n*d)
> for the ratio n/d in lowest terms.
> 
> The complexity I'm not sure about. Paul?

This was explained in the post itself, though it's obviously giving 
something extremely close to the number of notes per octave.


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Message: 10041

Date: Tue, 10 Feb 2004 20:17:42

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 "Dave Keenan" <d.keenan@b...> 
> wrote:
> 
> > 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.
> 
> The ets Paul just got through plotting are lying more or less along 
> straight lines. I don't see any way to make a sensible moat unless 
> your line follows the lay of the land, so to speak.

Except you suddenly depart from the lay of the land in two places? 
Why this suddenness? What powerful psychological force operates at 
these two points?

> > Is there a simpler function of log(err) and log(comp) that gives
> > similar shaped curves in the region of interest?
> 
> Lines.

As I showed, the curve in question, on a log-log plot, looks clearly 
unlike a line.


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Message: 10042

Date: Tue, 10 Feb 2004 21:27:57

Subject: Re: !

From: Dave Keenan

--- 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.

> I'd say that if you planned to use any set of commas that generate 
> the ET's kernel (for chord-pump progressions, say), we're justified 
> to consider that you're planning to use the ET itself.

Fair enough.


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Message: 10043

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

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

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >> ... the dreaded error and complexity bounds.
> >
> >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.

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

Same reason as above.


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Message: 10045

Date: Tue, 10 Feb 2004 06:49:50

Subject: Re: !

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:
> > Why should we want to multiply instead of add?
> 
> Oh, for God's sake Paul-have you looked at your own plots? Did you 
> notice how straight the thing looks in loglog coordinates? Your plots 
> make it clear that loglog is the right approach. Look at them!

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.

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.


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Message: 10046

Date: Tue, 10 Feb 2004 16:11:27

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

From: Carl Lumma

>> I haven't seen any cluster analysis yet!
>
>It was principal components analysis, but the reasoning behind the 
>implementation was obscure.

I have no idea what you're talking about.

>> >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.
>
>OK, Carl, so everyone's been sorely underestimating the true 
>usefulness of 665-equal and 612-equal, yes?

No.

-Carl


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Message: 10047

Date: Tue, 10 Feb 2004 20:21:04

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

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >> 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.
> 
> I recall that Dave rejected the idea of a critical exponent.  But I
> didn't understand it until I coded it.  But anyway, it's no big 
deal.
> At the level I'd imagine this stuff being explained to a full-time
> musician, it wouldn't be any harder to explain than a moat.

I think the regular plot will be easier to explain than the log-log 
plot. After that, the moat will be quite easy to demonstrate. It 
seems you just grokked it yourself.

> >> or why we should
> >> want to add instead of multiply to get badness.
> >
> >Why should we want to multiply instead of add?
> 
> Gene multiplies logs, and you and Dave are adding them.
> Or so I thought...

No, Gene adds logs, while Dave and I add without taking logs.


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Message: 10049

Date: Tue, 10 Feb 2004 00:12:09

Subject: Re: Loglog

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" 
<gwsmith@s...> 
> wrote:
> > I checked the files I saved of the graphs being posted, and found 
> no 
> > loglog examples. I then went over to tuning-files, and found one 
> > example,
> 
> You missed quite a few then,

My apologies again, these used log of error, but not log of 
complexity. Using log of complexity crammed all the interesting stuff 
to the far left to the point of illegibility, in the cases I 
originally tried.


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