Tuning-Math Digests messages 11225 - 11249

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Message: 11225

Date: Sat, 03 Jul 2004 18:30:31

Subject: Re: NOT tuning

From: Carl Lumma

>> >Meantone
>> >
>> >5-limit: 698.0187 (43, 55, 98, 153, 251, 404)
>> >
>> >7-limit: 697.6458 (31, 43, 934, 977; 0.0286 cents flatter than 43)
>> 
>> Hmm... I dunno, this seems a bit far from the old-style rms
>> optimum.
>> 
>> -Carl
>
>Carl, when Graham investigated this same question here a few months 
>ago, he concluded that pure-octaves TOP would be a uniform stretching 
>or compression of TOP,

That seems obvious for ETs....

>except where TOP already had pure octaves, in 
>which case it would actually change!

That's impossible given the criterion of NOT.

Maybe I don't comprehend you.

-Carl


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Message: 11226

Date: Sat, 03 Jul 2004 20:25:09

Subject: Re: from linear to equal

From: Carl Lumma

>> 9-limit should also be considered when you're going "poptimal".
>
>True enough. Alas, even though we have the same wedgie, commas and
>tuning map, the poptimal range need not even overlap. Orwell is a
>typical example--there seems to be no overlap from 7 to 9, and none
>between 11 and 9, but the others are OK. So, 5 and 9 overlap, and
>have 43/190 as a common generator, but 7 and 9, no.

This is AWESOME.  Seriously, if you had come to me in a past
life and asked me to imagine the most heinously interesting
thing ever, for torturing curious folks in purgatory or
something, I wouldn't have come up with the half of this
temperaments thing.

How annoying, that there doesn't seem to be any really good way
to famlify the temperaments.

By the way, Gene, how does poptimal relate to TOP?  If the
commas dictate the TOP tuning, is there necessarily a generator/
period pair that give it?

And if there is, is it not specified exactly by the TOP tuning,
for a given map?

Maybe what I'm asking is, "could you walk me through the
functions you'd call to find a generator/period pair for a
7-limit linear temperament?".  I can look in your code.
Though I guess what I have doesn't cover TOP...

-Carl


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Message: 11227

Date: Sat, 03 Jul 2004 20:27:10

Subject: Re: NOT tuning

From: Carl Lumma

>> >except where TOP already had pure octaves, in 
>> >which case it would actually change!
>> 
>> That's impossible given the criterion of NOT.
>> 
>> Maybe I don't comprehend you.
>
>Some examples of this method of tuning would be nice, and
>a definition even better.

Which method?  Graham's?  I think he gave examples.

Graham, what's a good word to search for?  I know I have that
post.  I think I replied to it.

-Carl



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Message: 11230

Date: Sun, 04 Jul 2004 23:23:23

Subject: Re: A chart of syntonic comma temperaments

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Herman Miller <hmiller@I...> 
wrote:
> http://www.io.com/~hmiller/png/syntonic.png *
> 
> This is a chart of 7-limit temperaments that temper out the 
syntonic 
> comma 81;80. The horizontal axis is deviation from 3:1 and the 
vertical 
> axis is the deviation from 7:1. This time I limited the list to 7-
limit 
> consistent ET's.

What you call Mothra, I call Cynder, since it's basically the same as 
the Slendric or Wonder temperament, but with 5 thrown into the primal 
mix.

What you call Hemifourths, I call Semaphore.


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Message: 11235

Date: Sun, 04 Jul 2004 10:14:22

Subject: Re: from linear to equal

From: Carl Lumma

>> Isn't TOP a minimax (p=inf.) method?  Oh, but in the definition above
>> you restrict poptimal to octaves. . .  .
>
>TOP is a weighted method, not restricted, which can be regarded as
>minimax applied to just the primes.

Oh, and why should I believe that p=inf gives minimax again?

-Carl


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Message: 11236

Date: Sun, 04 Jul 2004 23:35:12

Subject: Re: bimonzos, and naming tunings (was: Gene's mail server))

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:
> 
> > > > Are you expecting people to read the comma values 
> > > > off of the bimonzo?
> > > 
> > > No. But as long as we're on the subject here, it might
> > > be worth reviewing here for list memmbers how you do that.
> > > Not in the paper.
> 
> > yes, please do review it!
> 
> If you have a bimonzo ||a1 a2 a3 a4 a5 a6>> then you can read off 
the
> odd comma of the temperament (the comma which is a ratio of odd
> integers) by taking out the common factor if needed of a1, a2, a3 to
> get b1, b2, b3, and then the comma is 3^b1 5^b2 7^b3, or its
> reciprocal if you need to make it bigger than 1. 
> 
> In general, however, reading the commas from a bimonzo is no easier
> than reading them from a bival, and in fact probably harder,

Why harder? Can you show this?

> and I
> don't think this makes much of a reason for using bimonzos. I really
> would like to know why Paul insists on this so stubbornly.

I welcome constructive suggestions for making the paper go bival, 
without adding to its math-heaviness.

> If you have a bival <<a1 a2 a3 a4 a5 a6||, then
> 
> 2^a4 3^(-a2) 5^a1 gives the 5-limit comma.

You don't need to remove common factors?

> 2^(a5) 3^(-a3) 7^(a1) gives the {2,3,7}-comma; that's 2 to the power
> of the (3,7) coefficient, 3 to minus the power of the (2,7)
> coefficient, and 7 to the power of the (2,3) coefficient;

How is this easier than the bimonzo case?


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Message: 11240

Date: Sun, 04 Jul 2004 00:27:37

Subject: Re: from linear to equal

From: Carl Lumma

>> By the way, Gene, how does poptimal relate to TOP? 
>
>Not very well, apparently.
>
> If the
>> commas dictate the TOP tuning, is there necessarily a
>> generator/period pair that give it?
>
>You've lost me.

I meant, for a given TOP-tuned linear temperament, does it not
stand to reason that there is at least one generator/period
pair (in cents) that produces scales in said tuning?

>> Maybe what I'm asking is, "could you walk me through the
>> functions you'd call to find a generator/period pair for a
>> 7-limit linear temperament?".  I can look in your code.
>> Though I guess what I have doesn't cover TOP...
>
>Do you mean in terms of cents,

Yes.

>or a generator period pair in terms of
>p-limit intervals which temper to the generator and period whatever
>tuning you use?

You lost me here; sounds interesting but I don't think I meant this.

>In terms of cents, the easy ones to find are TOP, NOT,
>rms, and minimax, but each of these is different; rms involves least
>squares, and the rest I set up as a linear programming problem and
>solve using Maple's simplex method implementation.

Ok.  I know roughly what this means.  But it'd still be nice to
know what data you feed into which processes.  You need a map at
some point, I'd think, so you can specify whether to find, say,
fifths or fourths for meantone... I'm trying to build a picture
of what kinds of things I need to know to get what kinds of
answers out.

-Carl


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Message: 11241

Date: Sun, 04 Jul 2004 11:21:49

Subject: Re: from linear to equal

From: Carl Lumma

>> >TOP is a weighted method, not restricted, which can be regarded as
>> >minimax applied to just the primes.
>> 
>> Oh, and why should I believe that p=inf gives minimax again?
>
>If you have a>b, then a^p+b^p is dominated by a as p goes to infinity,
>since (b/a)^p --> 0. Hence (|a|^p+|b|^p)^(1/p) --> max(|a|, |b|) as 
>p --> infinity. (If a and b are of the same size, the
>doubling makes no difference, since 2^(1/p)-->1)

Of course.  Thanks.

-Carl


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Message: 11242

Date: Sun, 04 Jul 2004 23:35:54

Subject: Re: from linear to equal

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:
> 
> > > Har. You think that is bad, try this: two different 11-limit 
linear
> > > temperaments are the meantone variants meantone or meanpop 
(sharing
> > > the same TOP tuning with the 7-limit temperament) and huygens 
> > (sharing
> > > the same NOT tuning with the 7-limit temperament.)
> > 
> > Isn't that a ridiculous name for an 11-limit temperament?
> 
> You'd maybe prefer Fokker?

Did Fokker have a particular route along the circle of fifths that he 
preferred to get 11?


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Message: 11245

Date: Sun, 04 Jul 2004 23:40:22

Subject: Re: from linear to equal

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:
> 
> > > This is quite an interesting approach. What makes poptimal 
> > >generators 
> > > good?
> > 
> > Not much, IMHO -- the "true" value of p in any situation will be 
some 
> > number, not an infinite range of numbers.
> 
> What in the world does this mean? What allegedly "true" value?

If you're using this p-norm model in the first place, it's probably 
because you think it's true for some value of p. If you run over all 
possible 'p's, you're violating the assumptions of any such model.

> > > And why can't p be 1?
> > 
> > My graphs show p going even slightly below 1, and I think this is 
> > more than appropriate when you look at the kinds of discordance 
> > curves Bill Sethares predicts and George Secor prefers. Very 
sharp 
> > spikes at the simple ratios.
> 
> If you go below 1 your don't even get a corresponding metric, but 
you
> can go as far as 1 and have a metric. Let's at least keep the 
triangle
> inequality, please. 

The behavior below 1 reflects the meaningful result that temperaments 
do not improve on JI tunings there. It's helpful to think of the 
bigger picture.

> As for 1, I think a lot of people would find the supposedly optimal
> tunings not really very optimal in some cases.

And yet there is a significant bunch of composers who refuse to 
temper, clinging to their JI scales. Might they be modelled too? (no 
offense to them, of course.)


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Message: 11246

Date: Sun, 04 Jul 2004 00:34:01

Subject: Re: from linear to equal

From: Carl Lumma

>> For pure octave tunings, a system I sometimes use is to close at a
>> "poptimal" generator. A generator is "poptimal" for a certain set of
>> octave-eqivalent consonances if there is some exponent p, 2 <= p <=
>> infinity, such that the sum of the pth powers of the absolute value
>> of the errors over the set of consonances is minimal.

I guess I never understood how poptimal is different than 'all
the error functions ever advocated here'.

>> A different naming convention than using TOP tuning would be to give
>> the same name iff the poptimal ranges intersect. This isn't very
>> convenient in practice, due to the difficulty of computing the
>> poptimal range, but clearly it leads to quite different results.
>> Miracle, for instance, has the same TOP tuning in the 5, 7 and 11
>> limits, but while the 5 and 7 limit poptimal ranges intersect,
>> the 5 and 11 or 7 and 11 ranges apparently do not, though as I say 
>> computing these is a pain, so I may have the range too small. In
>> any case, miracle closes at 175 in the 5 and 7 limits, and at 401
>> in the 11-limit.

Isn't TOP a minimax (p=inf.) method?  Oh, but in the definition above
you restrict poptimal to octaves. . .  .

-Carl


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