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Message: 5200 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: 5201 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: 5202 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: 5205 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: 5206 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: 5207 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: 5209 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: 5210 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: 5211 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: 5212 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 *>. 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: 5213 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: 5216 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: 5217 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: 5219 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: 5220 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: 5221 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: 5222 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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Message: 5223 Date: Wed, 05 Dec 2001 16:19:26 Subject: Re: The grooviest linear temperaments for 7-limit music From: David C Keenan I haven't read any of the messages about this in tuning-math. I'm purely responding to Paul's summary and subsequent responses by Paul and Gene on the tuning list. --- In tuning@y..., "paulerlich" <paul@s...> wrote: > --- In tuning@y..., "dkeenanuqnetau" <D.KEENAN@U...> wrote: > > Thanks for this summary Paul, but ... > > You mean you haven't been on tuning-math@y... ? Get thee > hence :) > > > > He proposed a 'badness' measure defined as > > > > > > step^3 cent > > > > > > where step is a measure of the typical number of notes in a scale > > for > > > this temperament (given any desired degree of harmonic depth), > > > > What the heck does that mean? > > step is the RMS of the numbers of generators required to get to each > ratio of the tonality diamond from the 1/1, I think. This is good. More comprehensive than what Graham and I were using. > > How does he justify cubing it? > --- In tuning@y..., "ideaofgod" <genewardsmith@j...> wrote: > An order of growth estimate shows there should be an infinite list > for step^2, but not neccesarily for anything higher, and looking far > out makes it clear step^3 gives a finite list. What this means, of > course, is that in some sense step^2 is the right way to measure > goodness. Yes! Only squared, not cubed. > Step^3 weighs the small systems more heavily, and that is > why we see so many of them to start with. I believe the way to fix this is not to go to step^3 (I don't think there's any human-perception-or-cognition-based justification for doing that), but instead to correct the raw cents to some kind of dissonance or justness measure (more on this below). > > > and > > > cent is a measure of the deviation from JI 'consonances' in cents. > > > > Yes but which measure of deviation? minimum maximum absolute or > > minimum root mean squared or something else? > > RMS Fine. > > How does he justify not applying a human sensory correction to this? > > A human sensory correction? Yes. Once the deviation goes past about 20 cents it's irrelevant how big it is, and a 0.1 cent deviation does not sound 10 times better than a 1.0 cent deviation, it sounds about the same. I suggest this figure-of-demerit. step^2 * exp((cents/k)^2), where k is somewhere between 5 and 15 cents I think this will give a ranking of temperaments that corresponds more to how composers or performers would rank them. -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * -- A country which has dangled the sword of nuclear holocaust over the world for half a century and claims that someone else invented terrorism is a country out of touch with reality. --John K. Stoner
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Message: 5224 Date: Wed, 05 Dec 2001 03:17:38 Subject: Re: Top 20 From: Paul Erlich Gene, this is shaping up to be an immense contribution you're making to tuning theory. > I started from 990 pairs of ets, from which I got 505 linear 7- limit > temperaments. You'll also try starting from an expanded list of UVs, correct? The top 20 in terms of step^3 cents How did you decide on this criterion? Would you please try Z^(step^(1/3)) cents where you're free to pick Z to be 2 or e or whatever. turned out to be: wedgie univectors > (1) [2,3,1,-6,4,0] <21/20,27/25> JI block (what simple UVs complete a TMR (TM-reduced) basis for this)? > (2) [1,-1,0,3,3,-4] <8/7,15/14> JI block (ditto) > (3) [0,2,2,-1,-3,3] <9/8,15/14> JI (ditto) > (4) [4,2,2,-1,8,6] <25/24,49/48> JI or Planar (ditto) > (5) [2,1,3,4,1,-3] <15/14,25/24> JI (ditto) > (6) [2,1,-1,-5,7,-3] <21/20,25/24> JI " > (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> JI or Planar " > (11) [18,27,18,-34,22,1] <2401/2400,4375/4374> Ennealimmal You win! But somewhere out there, I wonder . . . What are some manageable MOSs of this? > (12) [2,-2,1,8,4,-8] <16/15,49/48> JI or Planar (ditto) > > (13) [0,0,3,7,-5,0] <10/9,16/15> JI " > > (14) [6,5,3,-7,12,-6] <49/48,126/125> Pretty good for not having a > name--"septimal kleismic" maybe? Please post details. Is this Dave Keenan's chain-of-minor-thirds thingy? It loses on tetrachordality. > (15) [0,5,0,-14,0,8] <28/27,49/48> JI or Planar > > (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> JI or Planar > > (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? Is this Graham's #1 7-limit? And he missed ennealimmal because . . . ? > (20) [1,4,-2,-16,6,4] <36/35,64/63> Looks suspiciously like 12-et > meantone. What's the generator? Where is Huygens meantone in all this?
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