Tuning-Math Archive Section 3: 2202

Message: 2202 - Contents - Hide Contents 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!

Message: 2203 - Contents - Hide Contents 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)?

Message: 2204 - Contents - Hide Contents 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 . . .

Message: 2207 - Contents - Hide Contents 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?

Message: 2208 - Contents - Hide Contents 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 - The best web-based email! * [with cont.] (Wayb.)

Message: 2209 - Contents - Hide Contents 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.

Message: 2211 - Contents - Hide Contents 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

Message: 2212 - Contents - Hide Contents 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.

Message: 2213 - Contents - Hide Contents 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)!

Message: 2214 - Contents - Hide Contents 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 > <4 5 6 9 10 12 15 16 18 19 22 26 27 29 31 35 36... * [with cont.] (Wayb.)>. 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 * [with cont.] (Wayb.)>. Graham

Message: 2215 - Contents - Hide Contents 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.

Message: 2218 - Contents - Hide Contents 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.

Message: 2219 - Contents - Hide Contents 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 . . .

Message: 2221 - Contents - Hide Contents 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?

Message: 2223 - Contents - Hide Contents 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.

Message: 2224 - Contents - Hide Contents 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 . . .