Tuning-Math Digests messages 2202 - 2226

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Message: 2202

Date: Wed, 05 Dec 2001 19:55:02

Subject: Re: Top 20

From: paulerlich

--- In tuning-math@y..., "ideaofgod" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> I really had put the list out for a preliminary review, to get 
> feedback on whether the ordering seemed to make sense.

I really can't complain!

> Why don't I 
> work on it some more and see what I get?

Awesome!


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Message: 2203

Date: Wed, 05 Dec 2001 06:38:37

Subject: Re: Top 20

From: paulerlich

--- In tuning-math@y..., "ideaofgod" <genewardsmith@j...> wrote:

> I'm going to merge lists, and then expand by taking sums of wedge 
> invariants, but I need a decision on cut-offs. I am thinking the 
end 
> product would be additively closed--a list where any sum or 
> difference of two wedge invariants on the list was beyond the cut-
> off; but I have 173 in this list below 10000 already, so there's 
also 
> a question of how many of these we can handle.

Maybe Matlab would help. Do you have it? Can you write programs for 
it in matrix notation?

> > How did you decide on this criterion? Would you please try
> > 
> > Z^(step^(1/3)) cents
> 
> Well, I could but what's the rationale?

You said it sounded plausible that the amount of tempering associated 
with a unison vector was

(n-d)/(d*log(d))

which is

(n-d)/(2^length*length)

in the Tenney lattice. Now if a 3-d (my way) orthogonal block 
typically has "step" notes, then the tempering along each unison 
vector will typically involve a length of step^(1/3) . . . so this 
becomes
 
(n-d)/(2^(step^(1/3))*length)

Now if we say our 'badness measure' is proportional to amount of 
tempering times length, we have

badness = (n-d)/(2^(step^(1/3)))

Now in general, it seems that any worthwhile 7-limit temperament can 
be described with roughly orthogonal superparticular unison vectors 
(I kinda asked you about this sorta) . . . so it seems that we can say

n-d = 1

and make our goodness measure

2^(step^(1/3))

Is that some sloppy thinking or what (but shouldn't the exponential 
part be right)?


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Message: 2204

Date: Wed, 05 Dec 2001 19:56:49

Subject: Re: Top 20

From: paulerlich

--- In tuning-math@y..., "ideaofgod" <genewardsmith@j...> wrote:

> Just that it is a simple function with faster than quadratic 
growth, 
> but not a great deal faster. When in a polynomial growth situation, 
> one normally uses x^n for some expondent n which need not be an 
> integer.

step^3 measures the number of possible triads in the typical 
scale . . . so I guess it makes some sense . . .


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Message: 2207

Date: Wed, 05 Dec 2001 07:21:34

Subject: Re: Top 20

From: paulerlich

--- In tuning-math@y..., "ideaofgod" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> 
> > Now in general, it seems that any worthwhile 7-limit temperament 
> can 
> > be described with roughly orthogonal superparticular unison 
vectors 
> > (I kinda asked you about this sorta) . . . so it seems that we 
can 
> say
> 
> This is how you sneak in exponential growth, but is it plausible? 
The 
> TM reduced basis I get for a lot of good temperaments (eg. Miracle) 
> are not all superparticular.

2401:2400 and 225:224 are roughly orthogonal. So it _can_ . . . how 
about others?


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Message: 2208

Date: Wed, 5 Dec 2001 14:20:30

Subject: Re: The wedge invariant commas

From: monz

Can you guys please explain what you've been discussing
here for about the past two months?  I'm totally lost.


-monz


 



_________________________________________________________

Do You Yahoo!?

Get your free @yahoo.com address at Yahoo! Mail Setup *


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Message: 2209

Date: Wed, 05 Dec 2001 01:38:05

Subject: Re: List cut-off point

From: genewardsmith@xxxx.xxx

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning-math@y..., genewardsmith@j... wrote:
> > --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> > 
> > > Can you explain this sentence? I don't understand it at all.
> > 
> > It's simply conjecture on my part that the higher of a pair of 
twin 
> > primes should have a comparitively larger largest superparticular 
> > ratio associated to it than the lower,
> 
> Assuming this is true, can you explain the sentence?

The superparticular ratio commas are rather special ones, coming in 
more profusion than with other differences "a" in (b+a)/b, and so if 
there are expecially large ones, I would expect the associated 
temperaments to be especially good. I'd expect something more cooking 
in the 13-limit than the 11-limit, therefore.


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Message: 2211

Date: Wed, 05 Dec 2001 22:41:52

Subject: Re: The wedge invariant commas

From: paulerlich

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:

> Can you guys please explain what you've been discussing
> here for about the past two months?  I'm totally lost.
> 
> 
> -monz

Hi Monz,

There is little hope of having a full and rigorous understanding of 
everything Gene is doing without some serious undergraduate and 
graduate abstract algebra courses. Apparently, he himself didn't 
realize how many of the important mathematical concepts he was 
familiar with (torsion, multilinear algebra, . . .) actually could be 
important in music theory until he got here.

But basically, the whole field of periodicity blocks and regular 
temperaments seems to be on a much more solid mathematical foundation 
than before. This means that all kinds of difficult particular 
questions can be answered, deeper relationships between structures 
discerned, and comprehensive survey conducted (now being done for the 
linear temperament, octave-equivalent, 7-limit case).

Perhaps it would be best if you went back to the archives from when 
you last were active here, and tried to follow as much as you could 
from there, working your way to the present as slowly, and with as 
many questions, as you need to.

Good luck
-Paul


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Message: 2212

Date: Wed, 05 Dec 2001 01:45:00

Subject: Re: List cut-off point

From: Paul Erlich

--- In tuning-math@y..., genewardsmith@j... wrote:

> The superparticular ratio commas are rather special ones, coming in 
> more profusion than with other differences "a" in (b+a)/b, and so 
if 
> there are expecially large ones, I would expect the associated 
> temperaments to be especially good. I'd expect something more 
cooking 
> in the 13-limit than the 11-limit, therefore.

The jump from the longest 7-limit superparticular to the longest 11-
limit superparticular, you're saying, is not nearly as great as the 
jump from the longest 11-limit superparticular to the largest 13-
limit superparticular? I bet John Chalmers on the tuning list could 
immediately verify whether that's true. He might be interested to 
learn of a mathematical explanation of this fact.


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Message: 2213

Date: Wed, 05 Dec 2001 07:31:38

Subject: Re: Top 20

From: paulerlich

--- In tuning-math@y..., "ideaofgod" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> 
> > 2401:2400 and 225:224 are roughly orthogonal. So it _can_ . . . 
how 
> > about others?
> 
> I don't think you can make <2401/2400, 65625/65536> superparticular.
> What about <2401/2400, 3136/3125>?

If you can't, just think of (n-d) as an additional penalty for 
complexity. Length alone isn't much of a penalty -- it's sorta like 
step^(1/3)!


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Message: 2214

Date: Wed, 5 Dec 2001 23:26 +00

Subject: Re: Top 20

From: graham@xxxxxxxxxx.xx.xx

graham@xxxxxxxxxx.xx.xx () wrote:

> I'll add it to the catalog sometime.  It should be at the top of the 
> 7-limit microtemperaments at 
> <404 Not Found * Search for http://www.microtonal.co.uk/limit7.micro in Wayback Machine>.  It isn't in my local copy, 
> but I think that's out of date.  I'll have a look when I connect to 
> send this.

It was there.

I've added files with a .cubed suffix to show my version of the new figure 
of demerit (I don't do all this RMS stuff).  Doesn't look like an 
improvement to me, but I've still got the safety harness on.

If you want to play with the parameters, get the source code.  See, as 
usual, <Automatically generated temperaments *>.


                     Graham


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Message: 2215

Date: Wed, 05 Dec 2001 02:12:43

Subject: Re: List cut-off point

From: genewardsmith@xxxx.xxx

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> The jump from the longest 7-limit superparticular to the longest 11-
> limit superparticular, you're saying, is not nearly as great as the 
> jump from the longest 11-limit superparticular to the largest 13-
> limit superparticular? I bet John Chalmers on the tuning list could 
> immediately verify whether that's true. He might be interested to 
> learn of a mathematical explanation of this fact.

Yes, take the ratio log(T(superparticular))/log(T(prime)) and I'm 
guessing 7,13,19 stick out. 23 even more so--it is an isolate, with a 
distance of 4 to 19 and 6 to 29.


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Message: 2218

Date: Wed, 05 Dec 2001 02:08:59

Subject: Top 20

From: genewardsmith@xxxx.xxx

I started from 990 pairs of ets, from which I got 505 linear 7-limit 
temperaments. The top 20 in terms of step^3 cents turned out to be:

(1) [2,3,1,-6,4,0] <21/20,27/25>

(2) [1,-1,0,3,3,-4] <8/7,15/14>

(3) [0,2,2,-1,-3,3] <9/8,15/14>

(4) [4,2,2,-1,8,6] <25/24,49/48>

(5) [2,1,3,4,1,-3] <15/14,25/24>

(6) [2,1,-1,-5,7,-3] <21/20,25/24>

(7) [2,-1,1,5,4,-6] <15/14,35/32>

(8) [1,-1,1,5,1,-4] <7/6,16/15>

(9) [1,-1,-2,-2,6,-4] <16/15,21/20>

(10) [4,4,4,-2,5,-3] <36/35,50/49>

(11) [18,27,18,-34,22,1] <2401/2400,4375/4374> Ennealimmal

(12) [2,-2,1,8,4,-8] <16/15,49/48>

(13) [0,0,3,7,-5,0] <10/9,16/15>

(14) [6,5,3,-7,12,-6] <49/48,126/125> Pretty good for not having a 
name--"septimal kleismic" maybe?

(15) [0,5,0,-14,0,8] <28/27,49/48>

(16) [6,-7,-2,15,20,-25] <225/224,1029/1024> Miracle

(17) [2,-4,-4,2,12,-11] <50/49,64/63> Paultone

(18) [2,-2,-2,1,9,-8] <16/15,50/49>

(19) [10,9,7,-9,17,-9] <126/125,1728/1715> This one should have a 
name if it doesn't already. If I call it "nonkleismic" will that 
force someone to come up with a good one?

(20) [1,4,-2,-16,6,4] <36/35,64/63> Looks suspiciously like 12-et 
meantone.


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Message: 2219

Date: Wed, 05 Dec 2001 07:45:45

Subject: Re: Top 20

From: paulerlich

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> If you can't, just think of (n-d) as an additional penalty for 
> complexity. Length alone isn't much of a penalty -- it's sorta like 
> step^(1/3)!

Hey Gene -- something's wrong with my thinking here . . . note that 
the cents error _is_ the amount of tempering! So my criterion would 
be applied _without_ multiplying by the cents error . . . it would be 
a decent criterion with which to _constrain a search_, but definitely 
not for a final ranking . . .


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Message: 2221

Date: Wed, 05 Dec 2001 02:43:22

Subject: Re: List cut-off point

From: Paul Erlich

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> 
> > The jump from the longest 7-limit superparticular to the longest 
11-
> > limit superparticular, you're saying, is not nearly as great as 
the 
> > jump from the longest 11-limit superparticular to the largest 13-
> > limit superparticular? I bet John Chalmers on the tuning list 
could 
> > immediately verify whether that's true. He might be interested to 
> > learn of a mathematical explanation of this fact.
> 
> Yes, take the ratio log(T(superparticular))/log(T(prime)) and I'm 
> guessing 7,13,19 stick out. 23 even more so--it is an isolate, with 
a 
> distance of 4 to 19 and 6 to 29.

John Chalmers calculated all the superparticulars with numerator and 
denominator less than 10,000,000,000 (IIRC), for numerator and 
denominator up to 23. Can he verify this?


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Message: 2222

Date: Wed, 05 Dec 2001 08:00:09

Subject: Re: More temperaments

From: paulerlich

Could you take a look at my questions on the first 20?


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Message: 2223

Date: Wed, 05 Dec 2001 02:46:56

Subject: Re: List cut-off point

From: genewardsmith@xxxx.xxx

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> John Chalmers calculated all the superparticulars with numerator 
and 
> denominator less than 10,000,000,000 (IIRC), for numerator and 
> denominator up to 23. Can he verify this?

That would be a very useful thing to upload to the files area or 
stick on a web page.


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Message: 2224

Date: Wed, 05 Dec 2001 08:33:05

Subject: The time

From: paulerlich

Wow -- the time of day is actually coming up correctly on these 
messages! That means I better go to bed . . .


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