Tuning-Math Digests messages 9775 - 9799

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Message: 9775

Date: Tue, 03 Feb 2004 01:48:27

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

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 "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:
> > > > > 
> 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 *
> > 
> > Thanks Paul. Fascinating to look at, isn't it. So organic. Some 
> order,
> > some randomness. 
> > 
> > I think that planar temperaments are inherently less useful than
> > linear (which are less useful than equal).
> 
> I completely agree if you replace "less useful" with "more complex".
> 
> > This is mostly due to the
> > melodic dimension, which Herman mentions all the time, but we are
> > completely ignoring (except in so far as harmonic complexity implies
> > melodic complexity).
> 
> I disagree that it's about an ignored melodic dimension. Instead, 
> it's as I said before, these complexity values are not directly 
> comparable, because what's the length of an area? What's the area of 
> a volume.
> 
> > We are not measuring things like evenness and
> > transposability when deciding what is in and what is out. And that's
> > OK. We have to learn to crawl before we can walk.
> 
> Well, we're definitely agreed that a 7-limit planar temperament based 
> on a particular comma is quite a bit more complex than a 5-limit 
> linear temperament based on that same comma.
> 
> > But because planar are inherently less even and less transposable 
> than
> > linear I think there are only a very few interesting or useful 7-
> limit
> > planars.
> 
> Sure. I kind of figured the ragismic planar deserved to be in there, 
> but I wouldn't insist on it.
> 
> > Since you favour linear moats,
> 
> Where did you get that idea? Curved is fine too.

What range of exponents are acceptable to you? Isn't 1 near the
(geometric) middle of them?

> 
> > I suggest
> > 50/49
> > 49/48
> > 64/63
> > 81/80
> > 126/125
> > 225/224
> > 245/243
> 
> I definitely wouldn't want to throw out 28/27, 36/35 . . .

Gene, I hope you're happy I'm using slashes here. I agree there isn't
likely to be any confusion in this discussion since we're not talking
about individual pitches at all.

Why not. I have enough trouble wondering why anyone would use a
5-limit _linear_ temperament that was non-unique, 7-limit planar
stretches my credibility even further. Can you propose a scale or
finite tuning in these that you think might be useful as an
approximation of 7-limit JI?

Moat-wise, I can see my way to adding 36/35 and 128/125. That probably
gives the biggest moat possible (percentage-wise) particularly if you
use an exponent greater than 1. Unless you were to have one with an
exponent less than 1 (which I don't like) and go all the way up to
include 21/20 (which seems lidicrous to me).


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Message: 9776

Date: Tue, 03 Feb 2004 01:52:00

Subject: Re: Back to the 5-limit cutoff

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "David Bowen" <dmb0317@f...> wrote:
> Sorry for my delay in entering this discussion, but I'm a Digest
subscriber.
...

Hi David. It's good to hear from someone other than the usual
suspects. Thanks.


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Message: 9777

Date: Tue, 03 Feb 2004 01:58:33

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:

> > I definitely wouldn't want to throw out 28/27, 36/35 . . .
> 
> Gene, I hope you're happy I'm using slashes here. I agree there 
isn't
> likely to be any confusion in this discussion since we're not 
talking
> about individual pitches at all.
> 
> Why not. I have enough trouble wondering why anyone would use a
> 5-limit _linear_ temperament that was non-unique, 7-limit planar
> stretches my credibility even further. Can you propose a scale or
> finite tuning in these that you think might be useful as an
> approximation of 7-limit JI?

Not right now, must jet soon . . . This is Herman's department, or 
maybe Gene's . . .

> Moat-wise, I can see my way to adding 36/35 and 128/125. That 
probably
> gives the biggest moat possible (percentage-wise) particularly if 
you
> use an exponent greater than 1. Unless you were to have one with an
> exponent less than 1 (which I don't like)

Maybe you'll reconsider when you look at the ET graphs I just posted.

> and go all the way up to
> include 21/20 (which seems lidicrous to me).

It doesn't seem that lidicrous :) to me . . . Seriously, I think all 
kinds of novel effects could be obtained if 21/20 vanished, and if 
you used full 1:2:3:4:5:6:7:8:9:10 chords, there would certainly be 
no confusion over what the chords were 'representing' -- you might 
simply have to use the kinds of timbres that George and I were 
talking about . . . Maybe Herman would like to entertain us with some 
sort of example . . .


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Message: 9778

Date: Tue, 03 Feb 2004 03:31:10

Subject: TOP Equal Temperament graphs! (was: Re: Cross-check for TOP 5-limit 12-equal)

From: Dave Keenan

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

> I used this latter complexity measure to create these graphs:

Thanks for doing these Paul.

> Yahoo groups: /tuning_files/files/et3.gif *

I'm not familiar enough with 3-limit harmony (or rather ignoring
5-limit harmony) to comment on this, but I think I could be happy with
a straight line cutoff here.

> Yahoo groups: /tuning_files/files/et5.gif *

For this I'd go for a cutoff that just includes 15, 29, 46, 53, which
has a good enough straight-line moat, but admittedly it would be
widened slightly by using an exponent slightly less than 1.

> Yahoo groups: /tuning_files/files/et7.gif *

Here I assume you are referring to the difficulty of finding a moat
that includes both 12 and 72 and keeps out things like 58 and 39.

To me, this is just evidence that 72-ET would not be of much interest
as a 7-limit temperament (due to its complexity) if it wasn't for the
fact that it is a subdivision of 12-ET. So we could justify its
inclusion an an historical special case whether it was inside any moat
or not.

That's another dimension of usefulness that we're not considering --
12-ness.

> Yahoo groups: /tuning_files/files/et11.gif *

Here we can include 22, 31, 41, 46, and 72 with a straight line, but
admittedly it would be a somewhat wider moat if the exponent was made
slightly less than one.

Looking at these has disposed me more towards linear moats and less
towards quadratic ones, but only slightly toward powers slightly less
than one.

If I revisit the 5-limit linear temperament plot and look for good
straight (or near-straight) moats, I find there are none that would
include 2187/2048 that I could accept, because they would either mean
including too much dross at the high complexity end of things, or
would make 25/24 and 135/128 look far better than the marginal things
that they are.

But I could accept a straight line (or one with exponent slightly less
than 1) that excluded not only 2187/2048 and 3125/2916 but also
6561/6250 and 20480/19683, and included semisixths (78732/78125).

I'm guessing Gene would be happy with that too, since it looks more
like a log-flat badness cutoff with additional cutoffs on error and
complexity.


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Message: 9780

Date: Tue, 03 Feb 2004 04:45:01

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

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 "Paul Erlich" <perlich@a...> 
> wrote:
> 
> > > I definitely wouldn't want to throw out 28/27, 36/35 . . .
> > 
> > Gene, I hope you're happy I'm using slashes here. I agree there 
> isn't
> > likely to be any confusion in this discussion since we're not 
> talking
> > about individual pitches at all.
> > 
> > Why not. I have enough trouble wondering why anyone would use a
> > 5-limit _linear_ temperament that was non-unique, 7-limit planar
> > stretches my credibility even further. Can you propose a scale or
> > finite tuning in these that you think might be useful as an
> > approximation of 7-limit JI?
> 
> Not right now, must jet soon . . . This is Herman's department, or 
> maybe Gene's . . .
> 
> > Moat-wise, I can see my way to adding 36/35 and 128/125. That 
> probably
> > gives the biggest moat possible (percentage-wise) particularly if 
> you
> > use an exponent greater than 1. Unless you were to have one with an
> > exponent less than 1 (which I don't like)
> 
> Maybe you'll reconsider when you look at the ET graphs I just posted.
> 
> > and go all the way up to
> > include 21/20 (which seems lidicrous to me).
> 
> It doesn't seem that lidicrous :) to me . . . 

Definition of "lidicrous": so ludicrous that you can't type correctly. ;-)

> Seriously, I think all 
> kinds of novel effects could be obtained if 21/20 vanished, 

"All kinds of novel effects" is one thing and "approximating 7-limit
JI" is another.

> and if 
> you used full 1:2:3:4:5:6:7:8:9:10 chords, there would certainly be 
> no confusion over what the chords were 'representing' -- you might 
> simply have to use the kinds of timbres that George and I were 
> talking about . . . Maybe Herman would like to entertain us with some 
> sort of example . . .

It's the lack of counterexamples I'm more worried about.

I understand you claim that 12-ET is an approximation of JI for all
limits.

If the obtaining of relative consonance by using timbres of poorly
defined pitch in massive otonalities is a sufficient criterion for
temperament-hood (JI approximation) then please give me a non-trivial
planar tuning that _doesn't_ work like that. Otherwise we have a
reductio ad absurdum.



By the way, the TOP tuning of the 21/20 planar temperament has the
following errors in the primes (to the nearest cent).
2 +10 c
3 -15 c
5 +23 c
7 -27 c

So we have the following large errors in certain intervals
2:3  -25 c
7:10 +60 c

3:4  +35 c
5:7  -50 c

3:5  +38 c
4:7  -47 c

The approximations of 3:4 and 5:7 are the same interval, so are the
approximations of 3:5 and 4:7, and 2:3 is the same as 7:10.


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Message: 9783

Date: Wed, 04 Feb 2004 09:08:14

Subject: TOP Equal Temperament graphs! (was: Re: Cross-check for TOP 5-limit 12-equal)

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
> wrote:
> So 12-equal makes it in for you at 7 but not at 11?

12-equal doesn't really make it in as 7-limit for me personally, but I
was trying to keep you happy too.

> > Looking at these has disposed me more towards linear moats and less
> > towards quadratic ones,
> 
> So it worked!

Yes. And tomorrow someone might come up with something to convince me
of something different. :-)


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Message: 9784

Date: Wed, 04 Feb 2004 21:38:36

Subject: Re: Duals to ems optimization

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:
> 
> > It's not surprising given how Gene set it up: with the same 
weighting 
> > that gives equilateral triangles and tetrahedra in the 5-limit 
and 7-
> > limit lattices . . .
> 
> Actually, isosceles triangles.

Huh? You said this was the *unweighted* optimization scheme! Also, 
you never followed up on the loose ends of this thread, including the 
post "Attn: Gene" (not 2).

> The fifth gets a length of log(3) (or
> cents(3) or whatever log you are using) and the major and minor 
thirds
> have the same length, log(5).

You're seem to be contradicting yourself now -- in the post you're 
replying to, it said (your writing):

||5/4|| = ||7/4|| = ||11/8|| = sqrt(11).


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Message: 9786

Date: Wed, 04 Feb 2004 21:40:27

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:
> > --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
> > wrote:
> > > Perhaps we should limit such tests to otonalities having at 
most one
> > > note per prime (or odd) in the limit. e.g. If you can't make a
> > > convincing major triad then it aint 5-limit. And you can't use
> > > scale-spectrum timbres although you can use inharmonics that 
have no
> > > relation to the scale.
> > 
> > yes, mastuuuhhhhh . . . =(
> 
> It was just a suggestion. I wrote "perhaps we should" and "e.g.". 
> 
> What does "=(" mean?
> 
> I'm guessing you think it's a bad idea.

It's a picture of me succumbing to your authority.


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Message: 9787

Date: Wed, 04 Feb 2004 01:26:22

Subject: TOP Equal Temperament graphs! (was: Re: Cross-check for TOP 5-limit 12-equal)

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 "Paul Erlich" <perlich@a...> 
> > wrote:
> 
> > I used this latter complexity measure to create these graphs:
> 
> Thanks for doing these Paul.
> 
> > Yahoo groups: /tuning_files/files/et3.gif *
> 
> I'm not familiar enough with 3-limit harmony (or rather ignoring
> 5-limit harmony) to comment on this, but I think I could be happy 
with
> a straight line cutoff here.
> 
> > Yahoo groups: /tuning_files/files/et5.gif *
> 
> For this I'd go for a cutoff that just includes 15, 29, 46, 53, 
which
> has a good enough straight-line moat, but admittedly it would be
> widened slightly by using an exponent slightly less than 1.

Looks like a 'constellation' -- with 12 stars :(  :)

> Here I assume you are referring to the difficulty of finding a moat
> that includes both 12 and 72 and keeps out things like 58 and 39.

I also would have liked to see 43 and 50, but I suppose these are 
just 'footnoats' . . .

> > Yahoo groups: /tuning_files/files/et7.gif *
> 
> Here I assume you are referring to the difficulty of finding a moat
> that includes both 12 and 72 and keeps out things like 58 and 39.

Hmm . . . I see a rivulet, not a moat . . . 58 has gotten a lot more 
attention than 39 . . . but I was actually referring indirectly to 
how, in these graphs, the density of temperaments is not constant 
along each equicomplexity line, as in the comma graphs . . .

> > Yahoo groups: /tuning_files/files/et11.gif *
> 
> Here we can include 22, 31, 41, 46, and 72 with a straight line, but
> admittedly it would be a somewhat wider moat if the exponent was 
made
> slightly less than one.

So 12-equal makes it in for you at 7 but not at 11?

> Looking at these has disposed me more towards linear moats and less
> towards quadratic ones,

So it worked!


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Message: 9789

Date: Wed, 04 Feb 2004 21:50:11

Subject: Re: Paul's 32 again

From: Paul Erlich

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

> Here it is,

I put this list together in a fairly absurd manner from the 
information you gave, because it was the best I could do. I 
appreciate your great efforts to help out, but the most valuable 
thing you could do in this case would be to come up with a new list. 
The reason Dave and I wanted it in single-line format was so that we 
could graph it and make decisions based on it. Starting with the same 
32 defeats the whole purpose, I'm afraid.


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Message: 9792

Date: Wed, 04 Feb 2004 22:04:07

Subject: Re: Paul32 ordered by a beep-ennealimmal measure

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> Here are the same temperaments, ordered by error * complexity^(2.8).
> If the exponent was 2.7996... then ennealimma and beep would be the
> same, but why get fancy? I'm thinking an error cutoff of 15 and a
> badness cutoff of 4200 might work, looking at this; that would 
include
> schismic. More ruthlessly, we might try 3500. Really savage would be
> 3000; bye-bye miracle.

Strange; miracle was #3 according to log-flat, wasn't it?


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Message: 9799

Date: Wed, 04 Feb 2004 05:26:43

Subject: Re: Comma reduction?

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:
> 
> > > Are the 2 commas in the 7-limit always linearly independent?
> > 
> > Yes, they are never 'collinear'.
> 
> By definition of a 7-limit linear temperament.
> 
> > > How
> > > are they generated, (from wedgies OR matrices)?
> > 
> > You can pick them off the tree. We've been looking at some of 
> > the 'fruits' here.
> 
> Trees don't work for me. You can get them from direct comma searches
> or extract them out of temperaments, etc.
> 
> > > Also, was told
> > > that the complement of a wedge product in the 5-limit is the 
same
> > > as the cross-product, how does this work in the 7-limit?
> 
> The complement of a 2-val is a 2-monzo, and vice-versa, which just
> involves reordering.
> 
> ~<<||l1 l2 l3 l4 l5 l6|| = <<l6 -l5 l4 l3 -l2 l1||

Shouldn't that be

~||l1 l2 l3 l4 l5 l6>> = <<l6 -l5 l4 l3 -l2 l1||

or something?


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