Tuning-Math Digests messages 9675 - 9699

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Message: 9675

Date: Mon, 02 Feb 2004 23:42:16

Subject: Re: finding a moat in 7-limit commas a bit tougher . . .

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:
> > Yahoo groups: /tuning_files/files/Erlich/planar.gif *
> 
> Paul,
> 
> Please do another one of these without the labels, so we have a 
chance
> of eyeballing the moats.

Yahoo groups: /tuning_files/files/Erlich/planar0.gif *



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Message: 9677

Date: Mon, 02 Feb 2004 00:32:43

Subject: Weighting (was: 114 7-limit temperaments

From: Graham Breed

> Observation One: The extent and intensity of the influence of a
> magnet is in inverse proportion to its ratio to 1.

Hmm, that's fairly impenetrable.  But it does say "extend" and "inverse 
proportion".

> "To be taken in conjunction with the following"
> 
> Observation Two: The intensity of the urge for resolution is in
> direct proportion to the proximity of the temporarily magnetized
> tone to the magnet.

So it's only about resolution?

Carl:
> ?  The more complex ones already have the highest entropy.  You mean
> they gain the most entropy from the mistuning?  I think Paul's saying
> the entropy gain is about constant per mistuning of either complex
> or simple putative ratios.

Oh no, the simple intervals gain the most entropy.  That's Paul's 
argument for them being well tuned.  After a while, the complex 
intervals stop gaining entropy altogether, and even start losing it.  At 
that point I'd say they should be ignored altogether, rather than 
included with a weighting that ensures they can never be important. 
Some of the temperaments being bandied around here must get way beyond 
that point.  Actually, any non-unique temperament will be a problem.

What I meant is that, because the simple intervals have the least 
entropy to start with, they still have the least after mistuning, 
although they're gaining it more rapidly.

Carl:
> I was thinking about this last night before I passed out.  If you
> tally the number of each dyad at every beat in a piece of music and
> average, I think you'd find the most common dyads are octaves, to be
> followed by fifths and so on.  Thus if consonance really *does*
> deteriorate at the same rate for all ratios as Paul claims, one
> would place less mistuning on the simple ratios because they occur
> more often.  This is, I believe, what TOP does.

It depends on the music, of course.  My decimal counterpoint tends to 
use 4:6:7 a lot because it's simple, and not much of 6:5.  So tuning for 
such pieces would be different to TOP, which assumes a different pattern 
of intervals.

This would make more sense for evaluating complexity, although I'm not 
sure how you can write a piece of music without knowing what temperament 
you want it in.  Why temper at all in that situation?  But if you have 
some idea of the intervals you like, perhaps with a body of music in JI 
to count them from, you could find a temperament that makes them all 
nicely in tune and easy to find.


                  Graham


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Message: 9678

Date: Mon, 02 Feb 2004 04:00:09

Subject: Re: 114 7-limit temperaments

From: Paul Erlich

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

> >> Thus if consonance really *does*
> >> deteriorate at the same rate for all ratios as Paul claims,
> >
> >Where did I claim that?
> 
> In your decatonic paper you say the consonance deteriorates
> 'at least as fast', and opt to go sans weighting, IIRC.

Yes, the mathematics underlying harmonic entropy makes it clear that 
simpler ratios have more "room" around them, but when you actually 
calculate harmonic entropy itself, you end up finding that this 
doesn't translate into less sensitivity to mistuning. The paper is 
pre-harmonic entropy (it was invented too late) . . .


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Message: 9679

Date: Mon, 02 Feb 2004 05:32:08

Subject: Re: Back to the 5-limit cutoff

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >> >> >> >I'm arguing that, along this particular line of thinking, 
> >> >> >> >complexity does one thing to music, and error another, but
> >> >> >> >there's no urgent reason more of one should limit your
> >> >> >> >tolerance for the other . . .
> >> >> >> 
> >> >> >> Taking this to its logical extreme, wouldn't we abandon
> >> >> >> badness alltogether?
> >> >>
> >> >> >No, it would just become 'rectangular', as Dave noted.
> >> >> 
> >> >> I didn't follow that.
> >> >
> >> >Your badness function would become max(a*complexity, b*error),
> >> >thus having rectangular contours.
> >> 
> >> More of one can here influence the tolerance for the other.
> >
> >Not true.
> 
> Actually what are a and b?

Constants.

> But Yes, true.  Increasing my tolerance for complexity 
simultaneously
> increases my tolerance for error, since this is Max().

I have no idea why you say that. However, when I said "more of one", 
I didn't mean "more tolerance for one", I simply meant "higher values 
of one".


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Message: 9680

Date: Mon, 02 Feb 2004 08:55:58

Subject: Re: Back to the 5-limit cutoff

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >It's what you said yesterday (I think).
> >
> >At some point (1 cent, 0.5 cent?) the error is so low and the
> >complexity so high, that any further reduction in error is 
irrelevant
> >and will not cause you to allow any further complexity. So it 
should
> >be straight down to the complexity axis from there.
> 
> Picking a single point is hard.  It should be asymptotic.

Surely you don't mean asymptotic here, since asymptotic 
means "getting closer and closer to a line but never reaching it 
except in the limit of infinitely distance from the origin", right?

Asymptote -- from MathWorld *

Unless you're talking about log-flat badness, in which case you're 
not really responding to Dave's comment at all . . .


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Message: 9681

Date: Mon, 02 Feb 2004 23:59:25

Subject: Re: finding a moat in 7-limit commas a bit tougher . . .

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:
> > Yahoo groups: /tuning_files/files/Erlich/planar.gif *
> 
> And could you please multiply the vertical axis numbers by 1200. I'm
> getting tired of doing this mentally all the time, to make them mean
> something.

I re-uploaded

Yahoo groups: /tuning_files/files/Erlich/planar0.gif *

for you.


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Message: 9682

Date: Mon, 02 Feb 2004 01:02:00

Subject: Re: Back to the 5-limit cutoff

From: Carl Lumma

>Even if you accept this (which I don't), wouldn't it merely tell you 
>that the power should be *at least 2* or something, rather than 
>*exactly 2*?

Yes.  I was playing with things like comp**5(err**2) back in the day.
But I may have been missing out on the value of adding...

-Carl


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Message: 9684

Date: Mon, 02 Feb 2004 04:10:54

Subject: Re: Weighting

From: Paul Erlich

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

> > Anyway, Partch is saying you can create a dissonance by using a
> > complex interval that's close in size to a simple one.  I 
translate
> > his Observations into the present context thus...
> > 
> > 'The size (in cents) of the 'field of attraction' of an interval
> > is proportional to the size of the numbers in the ratio, and
> > the dissonance (as opposed to discordance) becomes *greater* as
> > it gets closer to the magnet.'
> 
> Since I don't know what he, or you, mean by a "magnet" I can only
> comment on the first part of this purported translation. And I find
> that it is utterly foreign to my experience, and I think yours. Did
> you accidentally drop an "inversely".

Yes.

> i.e. we can safely assume that
> Partch is only considering ratios in othe superset of all his JI
> scales, so things like 201:301 do not arise.

Yes.

> i.e. he's ignoring
> TOLERANCE and only considering COMPLEXITY.

Yes.

> So surely he means that as
> the numbers in the ratio get larger, the width of the field of
> attraction gets smaller.

Yes.

> To me, that's an argument for why TOP isn't necessarily what you 
>want.

Why, if this only addresses complexity and ignores tolerance? Partch 
isn't expressing his views on tolerance/mistuning here.

And while the Farey or whatever series that are used to calculate 
harmonic entropy follow this same observation if one equates "field 
of attraction" with "interval between it and adjacent ratios", the 
harmonic entropy that comes out of this shows that simpler ratios are 
most sensitive to mistuning, precisely because their great consonance 
arises from this very remoteness from neighbors, a unique property 
that rapidly subsides as one shifts away from the correct tuning.


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Message: 9685

Date: Mon, 02 Feb 2004 05:33:29

Subject: Re: The true top 32 in log-flat?

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >> >> >> >> > TOP generators [1201.698520, 504.1341314]
> >> >> >> 
> >> >> >> So how are these generators being chosen?  Hermite?
> >> >> >
> >> >> >No, just assume octave repetition, find the period (easy)
> >> >> >and then the unique generator that is between 0 and 1/2
> >> >> >period.
> >> >> >
> >> >> >> I confess
> >> >> >> I don't know how to 'refactor' a generator basis.
> >> >> >
> >> >> >What do you have in mind?
> >> >> 
> >> >> Isn't it possible to find alternate generator pairs that give
> >> >> the same temperament when carried out to infinity?
> >> >
> >> >Yup! You can assume tritave-equivalence instead of octave-
> >> >equivalence, for one thing . . .
> >> 
> >> And can doing so change the DES series?
> >
> >Well of course . . . can you think of any octave-repeating DESs
> >that are also tritave-repeating?
> 
> Right, so when trying to explain a creepy coincidence between
> complexity and DES cardinalities, might not we take this into
> account?

Sure . . . some of the ones that 'don't work' may be working for 
tritave-DESs rather than octave-DESs, is that what you were thinking?


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Message: 9686

Date: Mon, 02 Feb 2004 01:07:08

Subject: Re: Back to the 5-limit cutoff

From: Carl Lumma

>> >It's what you said yesterday (I think).
>> >
>> >At some point (1 cent, 0.5 cent?) the error is so low and the
>> >complexity so high, that any further reduction in error is
>> >irrelevant and will not cause you to allow any further complexity.
>> >So it should be straight down to the complexity axis from there.
>> 
>> Picking a single point is hard.  It should be asymptotic.
>
>Surely you don't mean asymptotic here, since asymptotic 
>means "getting closer and closer to a line but never reaching it 
>except in the limit of infinitely distance from the origin", right?
>
>Asymptote -- from MathWorld *

That's right.

>Unless you're talking about log-flat badness, in which case you're 
>not really responding to Dave's comment at all . . .

No, I was talking about what happens to error's contribution to
badness as it approaches zero.

-Carl


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Message: 9687

Date: Mon, 02 Feb 2004 02:27:18

Subject: Re: Back to the 5-limit cutoff

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >I'm arguing that, along this particular line of thinking, 
complexity 
> >does one thing to music, and error another, but there's no urgent 
> >reason more of one should limit your tolerance for the other . . .
> 
> Taking this to its logical extreme, wouldn't we abandon badness
> alltogether?
> 
> -Carl

No, it would just become 'rectangular', as Dave noted.


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Message: 9688

Date: Mon, 02 Feb 2004 04:18:15

Subject: Re: Back to the 5-limit cutoff

From: Paul Erlich

--- 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 "Paul Erlich" <perlich@a...> 
> wrote:
> > > You should know how to calculate them by now: log(n/d)*log(n*d) 
> > > and log(n*d) respectively.
> > 
> > You mean 
> > 
> > log(n/d)/log(n*d)
> > 
> > where n:d is the comma that vanishes.
> > 
> > I prefer these scalings
> > 
> > complexity = lg2(n*d)
> > 
> > error = comma_size_in_cents / complexity
> >       = 1200 * log(n/d) / log(n*d)
> > 
> > My favourite cutoff for 5-limit temperaments is now.
> > 
> > (error/8.13)^2 + (complexity/30.01)^2 < 1
> > 
> > This has an 8.5% moat, in the sense that we must go out to 
> > 
> > (error/8.13)^2 + (complexity/30.01)^2 < 1.085
> > 
> > before we will include another temperament (semisixths).
> > 
> > Note that I haven't called it a "badness" function, but rather a
> > "cutoff" function. So there's no need to see it as competing with
> > log-flat badness. What it is competing with is log-flat badness 
plus
> > cutoffs on error and complexity (or epimericity).
> > 
> > Yes it's arbitrary, but at least it's not capricious, thanks to 
the
> > existence of a reasonable-sized moat around it.
> > 
> > It includes the following 17 temperaments.
> 
> is this in order of (error/8.13)^2 + (complexity/30.01)^2 ?
> 
> > meantone	80:81
> > augmented	125:128
> > porcupine	243:250
> > diaschismic	2025:2048
> > diminished	625:648
> > magic	3072:3125
> > blackwood	243:256
> > kleismic	15552:15625
> > pelogic	128:135
> > 6561/6250	6250:6561
> > quartafifths (tetracot)	19683:20000
> > negri	16384:16875
> > 2187/2048	2048:2187
> > neutral thirds (dicot)	24:25
> > superpythag	19683:20480
> > schismic	32768:32805
> > 3125/2916	2916:3125
> > 
> > Does this leave out anybody's "must-have"s?
> > 
> > Or include anybody's "no-way!"s?
> 
> I suspect you could find a better moat if you included semisixths 
> too -- but you might need to hit at least one axis at less than a 
90-
> degree angle. Then again, you might not.

Also try including semisixths *and* wuerschmidt -- for a list of 19 --
 particularly if you're willing to try a straighter curve.


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Message: 9689

Date: Mon, 02 Feb 2004 05:35:51

Subject: Re: 7-limit horagrams

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >> >> >> Beautiful!  I take it the green lines are proper scales?
> >> >> >
> >> >> >Guess again (it's easy)!
> >> >> 
> >> >> Obviously not easy enough if we've had to exchange three
> >> >> messages about it.
> >> >
> >> >Then you can't actually be looking at the horagrams ;)
> >> 
> >> Why not just explain things rather than riddling your users?
> >
> >Because I'm trying to encourage some looking.
> 
> I've tested several possibilities about what the green could mean,
> and your continued refusal to simply provide the answer is assinine,
> with a double s.

I'm ssorry.

Green-black-green-black-green-black-green-black-green-black-green-
black . . .

Wasn't that your idea in the first place?


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Message: 9690

Date: Mon, 02 Feb 2004 01:11:46

Subject: Re: Back to the 5-limit cutoff

From: Carl Lumma

>> If I have a certain expectation of max error and a separate
>> expectation of max complexity, but I can't measure them directly,
>> I have to use Dave's formula, I wind up with more of whatever I
>> happened to expect less of.
>
>More of whatever you happened to expect less of? What do you mean? 
>Can you explain with an example?

If I'm bounding a list of temperaments with Dave's formula only,
and I desire that error not exceed 10 cents rms and complexity not
exceed 20 notes (and a and b somehow put cents and notes into the
same units), what bound on Dave's formula should I use?  If I pick
10 I won't see the larger temperaments I want, and if I pick 20
I'll see the less accurate temperaments I don't want.

>> Dave's function is thus a badness
>> function, since it represents both error and complexity.
>
>A badness function has to take error and complexity as inputs, and 
>give a number as output.

That's why the notion of badness is incompatible with the logical
extreme of your suggestion.

-C.


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Message: 9691

Date: Mon, 02 Feb 2004 02:31:55

Subject: Re: The true top 32 in log-flat?

From: Paul Erlich

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

> >> > TOP generators [1201.698520, 504.1341314]
> 
> So how are these generators being chosen?  Hermite?

No, just assume octave repetition, find the period (easy) and then 
the unique generator that is between 0 and 1/2 period.

> I confess
> I don't know how to 'refactor' a generator basis.

What do you have in mind?


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Message: 9692

Date: Mon, 02 Feb 2004 16:07:56

Subject: Re: Back to the 5-limit cutoff

From: Carl Lumma

>By the way, if you use 81/80 instead of 80:81, you are not going to 
>be inconsistent with that other fellow who uses 81:80 for the exact 
>same ratio. You will aslo be specifying an actual number. Numbers are 
>nice. This whole obsession with colons makes me want to give the 
>topic a colostomy. I have read no justification for it which makes 
>any sense to me.

There's a history in the literature of using ratios to notate pitches.
Normally around here we use them to notate intervals, but confusion
between the two has caused tragic misunderstandings and more than a
few flame wars.  So we adopted colon notation for intervals.  I have
no idea what the idea behind putting the smaller number first is,
and I don't approve of it.

-Carl


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Message: 9693

Date: Mon, 02 Feb 2004 02:32:30

Subject: Re: 7-limit horagrams

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma" <ekin@l...> wrote:
> Beautiful!  I take it the green lines are proper scales?
> 
> -C.

Guess again (it's easy)!


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Message: 9694

Date: Mon, 02 Feb 2004 05:38:56

Subject: Re: Back to the 5-limit cutoff

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:
> Also try including semisixths *and* wuerschmidt -- for a list of 19 --
>  particularly if you're willing to try a straighter curve.

No. There's no way to get a better moat by adding wuerschmidt. It's
too close to aristoxenean, and if you also add aristoxenean it's too
close to ... etc.


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Message: 9695

Date: Mon, 02 Feb 2004 09:12:17

Subject: Re: Back to the 5-limit cutoff

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >> >It's what you said yesterday (I think).
> >> >
> >> >At some point (1 cent, 0.5 cent?) the error is so low and the
> >> >complexity so high, that any further reduction in error is
> >> >irrelevant and will not cause you to allow any further 
complexity.
> >> >So it should be straight down to the complexity axis from there.
> >> 
> >> Picking a single point is hard.  It should be asymptotic.
> >
> >Surely you don't mean asymptotic here, since asymptotic 
> >means "getting closer and closer to a line but never reaching it 
> >except in the limit of infinitely distance from the origin", right?
> >
> >Asymptote -- from MathWorld *
> 
> That's right.

But without the "infinite distance" part?

> >Unless you're talking about log-flat badness, in which case you're 
> >not really responding to Dave's comment at all . . .
> 
> No, I was talking about what happens to error's contribution to
> badness as it approaches zero.

I've always seen 'asymptote' defined as in the diagrams, with 
something approaching infinity, not a finite limit. But OK.


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Message: 9696

Date: Mon, 02 Feb 2004 02:56:20

Subject: Re: 114 7-limit temperaments

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:
> Paul Erlich wrote:
> 
> > I don't think that's quite what Partch says. Manuel, at least, 
has 
> > always insisted that simpler ratios need to be tuned more 
accurately, 
> > and harmonic entropy and all the other discordance functions I've 
> > seen show that the increase in discordance for a given amount of 
> > mistuning is greatest for the simplest intervals.
> 
> Did you ever track down what Partch said?

Can't find my copy of Genesis!

> Harmonic entropy can obviously be used to prove whatever you like.  
It 
> also shows that the troughs get narrower the more complex the 
limit, so 
> it takes a smaller mistuning before the putative ratio becomes 
irrelevant.

Yes, this is what Partch and the mathematics that underlies harmonic 
entropy say.

> It also shows that, if all intervals are equally mistuned, the more 
> complex ones will have the highest entropy.

They had the highest entropy to begin with, and will get less on the 
margin.

> So they're the ones for 
> which the mistuning is most problematic,
> and where you should start for 
> optimization.

I've offered some arguments against this here, but 13:8 vs. 14:13 
example below seems to make it a bit moot . . .

> > Such distinctions may be important for *scales*, but for 
> > temperaments, I'm perfectly happy not to have to worry about 
them. 
> > Any reasons I shouldn't be?
> 
> You're using temperaments to construct scales, aren't you?

Not necessarily -- they can be used directly to construct music, 
mapped say to a MicroZone or a Z-Board.

http://www.starrlabs.com/keyboards.html *

> If you don't 
> want more than 18 notes in your scale, miracle is a contender in 
the 
> 7-limit but not the 9-limit.  And if you don't want errors more 
than 6 
> cents, you can use meantone in the 7-limit but not the 9-limit. 

What if you don't assume total octave-equivalence?

> There's 
> no point in using intervals that are uselessly complex or 
inaccurate so 
> you need to know whether you want the wider 9-limit when choosing 
the 
> temperament.

In the Tenney-lattice view of harmony, 'limit' and chord structure is 
a more fluid concept.

> > Tenney weighting can be conceived of in other ways than you're 
> > conceiving of it. For example, if you're looking at 13-limit, it 
> > suffices to minimize the maximum weighted error of {13:8, 13:9, 
> > 13:10, 13:11, 13:12, 14:13} or any such lattice-spanning set of 
> > intervals. Here the weights are all very close (13:8 gets 1.12 
times 
> > the weight of 14:13), *all* the ratios are ratios of 13 so 
simpler 
> > intervals are not directly weighted *at all*, and yet the TOP 
result 
> > will still be the same as if you just used the primes. I think 
TOP is 
> > far more robust than you're giving it credit for.
> 
> It's really an average over all odd-limit minimaxes.  And the 
higher you 
> get probably the less difference it makes -- but then the harder 
the 
> consonances will be to hear anyway.  For the special case of 7 vs 9 
> limit, which is the most important, it seems to make quite a 
difference.

Any examples?

> Oh, yes, I think the 9-limit calculation can be done by giving 3 a 
> weight of a half.

Which calculation are you referring to, exactly?

> That places 9 on an equal footing with 5 and 7, and I 
> think it works better than vaguely talking about the number of 
> consonances.

Number of consonances?

> After all, how do you share a comma between 3:2 and 9:8?

I'm not sure why you're asking this at this point, or what it 
means . . .

> I still don't know how the 15-limit would work.

?shrug?

> I'm expecting the limit of this calculation as the odd limit tends 
to 
> infinity will be the same as this Kees metric.

Can you clarify which calculation and which Kees metric you're 
talking about?

> And as the integer limit 
> goes to infinity, it'll probably give the Tenney metric.

I haven't the foggiest idea what you mean.

All I can say at this point is that n*d seems to be to be a better 
criterion to 'limit' than n (integer limit).

> But as the 
> integers don't get much beyond 10, infinity isn't really an 
important 
> consideration.

I wish I knew what it would be important for . . .

> Not that it does much harm either, because the minimax always 
depends on 
> the most complex intervals, which will have roughly equal 
weighting. 
> The same as octave specific metrics give roughly the same results 
as 
> odd-limit style octave equivalent ones if you allow for octave 
stretching.

I still remain unclear on what you were doing with your octave-
equivalent TOP stuff. Gene ended up interested in the topic later but 
you missed each other. I rediscovered your 'worst comma in 12-equal' 
when playing around with "orthogonalization" and now figure I must 
have misunderstood your code. You weren't searching an infinite 
number of commas, but just three, right?


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Message: 9697

Date: Mon, 02 Feb 2004 09:14:44

Subject: Re: Back to the 5-limit cutoff

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >> If I have a certain expectation of max error and a separate
> >> expectation of max complexity, but I can't measure them directly,
> >> I have to use Dave's formula, I wind up with more of whatever I
> >> happened to expect less of.
> >
> >More of whatever you happened to expect less of? What do you mean? 
> >Can you explain with an example?
> 
> If I'm bounding a list of temperaments with Dave's formula only,
> and I desire that error not exceed 10 cents rms and complexity not
> exceed 20 notes (and a and b somehow put cents and notes into the
> same units), what bound on Dave's formula should I use?

You'd pick a and b such that max(cents/10,complexity/20) < 1.

> >> Dave's function is thus a badness
> >> function, since it represents both error and complexity.
> >
> >A badness function has to take error and complexity as inputs, and 
> >give a number as output.
> 
> That's why the notion of badness is incompatible with the logical
> extreme of your suggestion.

Why? Max(cents/10,complexity/20) gives a number as output.


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Message: 9698

Date: Mon, 02 Feb 2004 01:24:22

Subject: Re: Back to the 5-limit cutoff

From: Carl Lumma

>> >> >I'm arguing that, along this particular line of thinking, 
>> >> >complexity does one thing to music, and error another, but
>> >> >there's no urgent reason more of one should limit your
>> >> >tolerance for the other . . .
//
>> 
>> If I'm bounding a list of temperaments with Dave's formula only,
>> and I desire that error not exceed 10 cents rms and complexity not
>> exceed 20 notes (and a and b somehow put cents and notes into the
>> same units), what bound on Dave's formula should I use?
>
>You'd pick a and b such that max(cents/10,complexity/20) < 1.

Ok, I walked into that one by giving fixed bounds on what I wanted.
But re. your original suggestion (above), for any fixed version of
the formula, more of one *increases* my tolerance for the other.

-Carl


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Message: 9699

Date: Mon, 02 Feb 2004 03:01:49

Subject: Re: The true top 32 in log-flat?

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:
> > There's something VERY CREEPY about my complexity values. I'm 
going 
> > to have to accept this as *the* correct scaling for complexity 
(I'm 
> > already convinced this is the correct formulation too, i.e. L_1 
> norm, 
> > for the time being) . . .
> 
> That's great, Paul. So what's the scaling?

I'm using your formula from

Yahoo groups: /tuning-math/message/8806 *

but instead of "max", I'm using "sum" . . .


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