Tuning-Math Digests messages 9878 - 9902

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Message: 9878

Date: Sun, 08 Feb 2004 21:36:15

Subject: 23 "pro-moated" 7-limit linear temps, L_1 complex.(was: Re: 126 7-limit linears)

From: Paul Erlich

> A bit more concavity still and we include
> 
> 45. Blackwood

Following what Dave did for the 5-limit and ET cases, I found that an 
exponent of 2/3 produces the desired moat, for example when

err^(2/3)/6.3+complexity^(2/3)/9.35 < 1.

Please look at the resulting graph:

Yahoo groups: /tuning_files/files/Erlich/7lin23.gif *

The temperaments in thie graph are identified by their ranking 
according to the badness measure implied above:

1. Huygens meantone
2. Pajara
3. Magic
4. Semisixths
5. Dominant Seventh
6. Tripletone
7. Negri
8. Hemifourths
9. Kleismic/Hanson
10. Superpythagorean
11. Injera
12. Miracle
13. Biporky
14. Orwell
15. Diminished
16. Schismic
17. Augmented
18. 1/12 oct. period, 25 cent generator (we discussed this years ago)
19. Flattone
20. Blackwood
21. Supermajor seconds
22. Nonkleismic
23. Porcupine

Here is the data for first three wedgie entries and implied badness, 
for the implied top 32:

            1            4           10      0.68784
            2           -4           -4      0.78033
            5            1           12      0.78742
            7            9           13      0.78759
            1            4           -2      0.82001
            3            0           -6      0.83995
            4           -3            2      0.86556
            2            8            1      0.87254
            6            5            3      0.87815
            1            9           -2      0.88068
            2            8            8      0.89638
            6           -7           -2      0.90191
            6           10           10       0.9041
            7           -3            8      0.91204
            4            4            4      0.91347
            1           -8          -14      0.91872
            3            0            6      0.93351
            0            0           12      0.93521
            1            4           -9      0.93554
            0            5            0       0.9488
            3           12           -1       0.9593
           10            9            7      0.95971
            3            5           -6      0.97257
            9            5           -3       1.0207
            8            6            6       1.0259
            6           -2           -2       1.0335
            6            5           22       1.0337
            3           12           11       1.0342
            2           -9           -4       1.0395
           11           13           17       1.0435
            6           10            3       1.0498
            4            2            2       1.0499


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Message: 9879

Date: Sun, 08 Feb 2004 21:39:35

Subject: Re: Jamesbond in 14-et

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:
> 
> > Did you see the horagram I posted?
> 
> You posted a lot of horagrams. So far I've not seen what you can do
> with them you can't do with a little algebra,

"A little algebra" to you is "a little forbidding" for most musicians.

> but maybe this horagram
> will make me see the point of it all. Which one is it?

jamesbond.bmp, zipped inside

Yahoo groups: /tuning_files/files/Erlich/sevenlimit.zip *


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Message: 9880

Date: Sun, 08 Feb 2004 21:44:38

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:
> > --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" 
<gwsmith@s...> 
> > wrote:
> > > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" 
<perlich@a...> 
> > wrote:
> > > 
> > > > > I
> > > > > think using log(err) and log(complexity) makes far more 
sense.
> > > > 
> > > > I don't think they make more sense practically. 
> > > 
> > > I think they probably will make more sense both practically and
> > > theoretically,
> > 
> > As I see it, no way.
> 
> You've taken me seriously enough to at least looked at it, or are 
you
> just blowing the issue off?

What, you didn't believe me when I said "countless hours"?

> >  Example: when you look at the graph with log
> > (err) as one of the axes, the indication is that JI is infinitely 
far 
> > away. This is ridiculous.
> 
> No it isn't. JI has *zero error*!

Yes, that's very different from "minus infinity error"!

> > The JI line should be right there, with 
> > some temperaments many times more distant from it than others. 
> > Otherwise, you're operating in the realm of hopelessly 
impractical 
> > abstraction.
> 
> It's what we've been doing, in effect, for the last few years, so 
this
> argument makes no sense at all to me. 

The only criterion for making sense is agreeing with habit? Why don't 
you actually take me seriously enough to at least look at it, instead 
of blowing the issue off?

> > > but you've been ignoring this issue. Are you going to
> > > think about it, at least?
> > 
> > Countless hours already spent thinking about it, and discussing 
it 
> > here.
> 
> The only one who seems to have thought about it is me.

Then you must not be reading our posts.

>I've been
> trying to get you to at least think about it, so far with no 
>success.

What would count as "thinking about it" to you? Agreeing with you?

> Do you care about convincing the rest of us that what you are doing
> makes a particle of sense, or is it going to be a committee of two?

Not only have we tried to be completely explicit and impartial in our 
logic, but you may note that Herman's guidelines had a very big 
influence on us. How big is the committee that thinks log-flat has 
any practical relevance?


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Message: 9881

Date: Sun, 08 Feb 2004 22:42:43

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

From: Carl Lumma

>> 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 is that
we have no data.  Not vague historical data, actually no data.  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.  But as long as Dave and Paul were having fun I
didn't want to say anything.  They have a way of coming up with neat
stuff, though so far their conversation has been impenetrable to me.

-Carl


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Message: 9883

Date: Sun, 08 Feb 2004 22:48:19

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

From: Carl Lumma

>At one stage Carl gave some good arguments why the cutoff might be
>as far from loglog as
>
>err^2 + k * comp^2 < x

Yes, I think I did say that, in multiplicitive form.

>And I went along with this until I saw the ET plots.

Ok, can you recommend a plot to look at, and what you saw that
changed your mind?  None of the plots I've seen have been labeled
nor made any sense to me.

>Perhaps this
>still makes sense as a badness measure for ranking temperaments, but
>not as a cutoff for what to include in an article. But I'm not even
>sure if that's a coherent suggestion.

Which suggestion?

-Carl


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Message: 9886

Date: Sun, 08 Feb 2004 23:56:24

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

From: Carl Lumma

>> >At one stage Carl gave some good arguments why the cutoff might be
>> >as far from loglog as
>> >
>> >err^2 + k * comp^2 < x
>> 
>> Yes, I think I did say that, in multiplicitive form.
>> 
>> >And I went along with this until I saw the ET plots.
>> 
>> Ok, can you recommend a plot to look at, and what you saw that
>> changed your mind?  None of the plots I've seen have been labeled
>> nor made any sense to me.
>
>Those Paul gave in 
>
>Yahoo groups: /tuning-math/message/9202 *
>
>Particularly the 5-limit one, which I assume most people have the
>greatest feel for.
>
>Complexity is horizontal, error is vertical,

Aha.

>labels are the notes per octave of the ET.

How can error be in notes?

>> >Perhaps this
>> >still makes sense as a badness measure for ranking temperaments, but
>> >not as a cutoff for what to include in an article. But I'm not even
>> >sure if that's a coherent suggestion.
>> 
>> Which suggestion?
>
>That something might make a good badness measure for ranking temps but
>not be good for determining a cutoff. I'd like to retract that now.

Ok.

-Carl


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Message: 9888

Date: Sun, 08 Feb 2004 22:46:38

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:
> 
> > What, you didn't believe me when I said "countless hours"?
> 
> I'll believe it when I see the log-log plots I've been trying to get
> you to do, with no success.

I've posted quite a few log-log plots, thanks very much. Seeing them 
is what made me realize that they assume JI is infinitely far away, 
and how absurd that is.

> > > The only one who seems to have thought about it is me.
> > 
> > Then you must not be reading our posts.
> 
> Which ones addressed this issue?

All the ones about 5-limit linear temperaments and about ETs in 
various limits.

> > >I've been
> > > trying to get you to at least think about it, so far with no 
> > >success.
> > 
> > What would count as "thinking about it" to you? Agreeing with you?
> 
> Some evidence you've actually considered it would be nice. A plot
> would be grand.

OK, what would you like me to plot that I haven't already plotted? 
I've looked at every single plot I've posted in both log-log and 
linear axes, just out of curiosity, but none of them change the fact 
that considering JI to be infinitely far away is like agreeing with 
Zeno that you can never traverse a room. Any measure of the "pain" of 
error will not predict this infinite distance, if it has any 
connection with the real world.

> Some attempt to theoretically justify what you two are
> doing would be appreciated.

Dave and I just recently shared some new theoretical insights, I 
thought. Carl sort of got hooked in too, on the "rectangular badness" 
issue. Why don't you take some time and re-read what you glossed 
over -- I've glossed over reading things before, but I think I've 
been pretty comprehensive since last month.


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Message: 9889

Date: Sun, 08 Feb 2004 22:53:10

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:

> A plot
> would be grand.

Here's the log-log version of the most recent plot:

groups.yahoo.com/group/tuning_files/files/Erlich/7lin23loglog.gif


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Message: 9891

Date: Sun, 08 Feb 2004 22:58:22

Subject: A post with pending questions

From: Paul Erlich

9052


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Message: 9896

Date: Mon, 09 Feb 2004 01:09:10

Subject: 23 "pro-moated" 7-limit linear temps, L_1 complex.(was: Re: 126 7-limit linears)

From: Dave Keenan

I wrote:
> This looks reasonable to me as a cutoff, although maybe still too
> many, ...

After more careful examination, I find this moat to be ideal. I can't
find one closer to the origin without leaving out temperaments I
really wouldn't want to leave out.

Well done.


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Message: 9898

Date: Mon, 09 Feb 2004 23:14:07

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

From: Carl Lumma

>> For musicians, I'd make the list 5 for each limit; 10 tops.  For
>> people reading a theory paper, 20 would be interesting.
>
>Ridiculous. I've *composed* in about that many temperaments.

You're not a professional musician, are you?

-Carl


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