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Message: 2275 - Contents - Hide Contents Date: Fri, 07 Dec 2001 06:37:45 Subject: Re: The grooviest linear temperaments for 7-limit music From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> Yes -- I discussed the situation a few messages back. We use an > objective measure, and cut things off in a nice wide gap.You are thinking that gens^2 cents, and Ennealimmal as the shut-off point, would be a good plan?
Message: 2276 - Contents - Hide Contents Date: Fri, 07 Dec 2001 08:27:10 Subject: Re: The grooviest linear temperaments for 7-limit music From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:>> If with all quantities positive we have g^2 c < A and c > B, then >> 1/c < 1/B, and so g^2 < A/B and g < sqrt(A/B). However, it probably >> makes more sense to use g>=1, so that if g^2 c <= A then c <= A.> Are you saying that using g>=1 is enough to make this a closed search?All it does is put an upper limit on how far out of tune the worst cases can be, so we really need to bound c below or g above to get a finite search.
Message: 2277 - Contents - Hide Contents Date: Fri, 7 Dec 2001 21:14 +00 Subject: Re: More lists From: graham@xxxxxxxxxx.xx.xx paul@xxxxxxxxxxxxx.xxx (paulerlich) wrote:> --- In tuning-math@y..., graham@m... wrote:>> produce files using Dave Keenan's new figure of demerit. That is >> >> width**2 * math.exp((error/self.stdError*3)**2) >> I thought Dave Keenan wanted to use Gene's "step" measure. In > addition, I think it should be weighted to favor the simpler > consonances.Yes, but width is the analog for the way I'm calculating it.> Are you missing any "slippery" examples that don't come easily out of > two ETs?Yes, as always.> Since you're doing so much work to get the unison vectors, shouldn't > we be thinking about _starting_ with unison vectors?Yes, I've done that, and so has Gene.>> Another idea would be to take all the intervals between second- > order>> intervals below a certain size, and use them as unison vectors to > generate>> temperaments. I might try that. >> That should plug a lot of holes.I need to be able to take all combinations. So far, I can only do that for the 7-limit, where they're pairs. I'll have to think about the general case. It'll probably involve recursion. I'm also worried about the speed of this search, because there are going to be a lot more unison vector combinations that ET pairs for the higher limits. It may be more efficient to take different readings of inconsistent ETs. Graham
Message: 2278 - Contents - Hide Contents Date: Fri, 07 Dec 2001 06:42:45 Subject: Re: The grooviest linear temperaments for 7-limit music From: dkeenanuqnetau --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:>>> It's the only >>> measure that doesn't favor a certain range of acceptable values > for>>> error or for complexity. It only favors the best examples within >> each >>> range. >>>> What _objective_ reason is there, to choose it over gens^3 * cents > or>> gens^2.3785 * cents? >> Because those measures give an overall "slope" to the results, in > analogy to what the Farey series seeding does to harmonic entropy.What's objective about that? A certain slope may be _real_. i.e. humans on average may experience it that way, in which case the "flat" case will really be favouring one extreme. I understand what the slope is in the HE case, but what slope are you talking about re badness of linear temperament? Badness wrt what?
Message: 2279 - Contents - Hide Contents Date: Fri, 07 Dec 2001 08:33:25 Subject: Re: The grooviest linear temperaments for 7-limit music From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> How do you know you wouldn't be missing any good ones?You'd need bounds on what counted for good; I'll think about it.
Message: 2280 - Contents - Hide Contents Date: Fri, 7 Dec 2001 21:33 +00 Subject: Re: More lists From: graham@xxxxxxxxxx.xx.xx Dave Keenan wrote:> I note that Graham is using maximum width and (optimised) maximum > error where Gene is using rms width and (optimised) rms error. It will > be interesting to see if this alone makes much difference to the > rankings. I doubt it.Oh yes, I forgot to say before. Here's the difference RMS errors make in the 11-limit: 1 1 2 2 4 4 3 3 7 6 5 9 15 5 12 14 6 16 14 17 The left hand column is the minimax ranking in terms of the RMS one, and the other one is the other way round. So they mostly agree on the best ones, but disagree on the mediocre ones. To check my RMS optimization's working, is a 116.6722643 cent generator right for Miracle in the 11-limit? RMS error of 1.9732 cents. Graham
Message: 2281 - Contents - Hide Contents Date: Fri, 07 Dec 2001 00:01:24 Subject: Re: Wedge products From: genewardsmith --- In tuning-math@y..., graham@m... wrote:> What is this "wedge invariant" he keeps using, and how do you go from it > to get a list of unison vectors?First you order the basis so that a wedge product taken from two ets or two unison vectors will correspond: Yahoo groups: /tuning-math/message/1553 * [with cont.] Then you put the wedge product into a standard form, by (1) Dividing through by the gcd of the coefficients, and (2) Changing sign if need be, so that the 5-limit comma (or unison) 2^w[6] * 3^(-w[2])*5^w[1] where w is the wedgie, is greater than 1. If it equals 1, go on to the next invariant comma, which leaves out 5, and if that is 1 also to the one which leaves out 3. See Yahoo groups: /tuning-math/message/1555 * [with cont.] for the invariant commas. The result of this standardization is the wedge invariant, or wedgie, which uniquely determins the temperament.> I'm not sure how to implement these things in Python. The above should do for the 7-limit; in general is another matter.
Message: 2282 - Contents - Hide Contents Date: Fri, 07 Dec 2001 06:47:57 Subject: Re: The grooviest linear temperaments for 7-limit music From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:>> There's not much to the 5-limit--it basically is a mere comma > search,>> and that can be done expeditiously using a decent 5-limit notation.> A decent 5-limit notation?We could search (16/15)^a (25/24)^b (81/80)^c to start out with, and go to something more extreme if wanted.
Message: 2283 - Contents - Hide Contents Date: Fri, 07 Dec 2001 08:35:53 Subject: Re: The grooviest linear temperaments for 7-limit music From: dkeenanuqnetau --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote: >>> I may be able to answer that when someone explains what is flat > with>> respect to what. >> Paul did that.Not in any way that makes any sense to me. I don't think Pauk really understands it either and may be starting to realise that. I'm starting to wonder if there's a conspiracy here to make me think I'm going crazy. :-) Is anyone else getting this "gens^2 * cents is the only 'flat' metric" thing?> An analogy would be to use n^(4/3) cents when seaching > for 7-limit ets; this will give you a list which does not favor > either high or low numbers n,I'm sorry. This makes no sense to me either. _How_ would you use n^(4/3) cents? Can you prove this to me? Or better still just prove whatever it is you are trying to say about gens^2 * cents being a "flat" badness metric for linear temperaments.> I don't think you can have much of a theory about what a bunch of > cranky individualists might like, but I hope we could cut it off when > the difference could no longer be percieved. Can anyone hear the > difference between Ennealimmal and just?Well that is precisely a theory about humans, as opposed to say grasshoppers or rocks or computers. If you guys can't explain this to me, I don't think you've got much chance of getting published in a refereed journal. It doesn't involve anything beyond high school math.
Message: 2284 - Contents - Hide Contents Date: Fri, 07 Dec 2001 22:33:29 Subject: Re: More lists From: dkeenanuqnetau --- In tuning-math@y..., graham@m... wrote:> Oh yes, I forgot to say before. Here's the difference RMS errors make in > the 11-limit: ... > The left hand column is the minimax ranking in terms of the RMS one, and > the other one is the other way round. So they mostly agree on the best > ones, but disagree on the mediocre ones.Ok. Thanks. That was a good way of showing it.> To check my RMS optimization's working, is a 116.6722643 cent generator > right for Miracle in the 11-limit? RMS error of 1.9732 cents.I get 116.678 and 1.9017. Did you include the squared error for 1:3 twice? I think you should since it occurs twice in an 11-limit hexad, as both 1:3 and 3:9. So then you must divide by 15, not 14, to get the mean. Actually, I see that this doesn't explain our discrepancy.
Message: 2285 - Contents - Hide Contents Date: Fri, 07 Dec 2001 02:31:26 Subject: Re: The slippery six From: paulerlich --- In tuning-math@y..., graham@m... wrote:> Gene:>>> There are six 7-limit linear temperaments which are on the list of >> 66>>> obtained from pairs of commas which did not turn up on the list of >>> 505 obtained from pairs of ets. > > Paul:>> That's a good indication that Graham may have missed these too, since >> he also started from pairs of ETs . . . Graham? >> Yes, it looks like Gene's doing the same search as me, and so he's finding > the same weaknesses. I did point this one out before, so I'm not sure why > he isn't doing a different search, more in line with his thinking.Well, he is -- he _also_ started from unison vectors and found the slippery six (I bet there are simpler examples too). _My_ thinking would be to _only_ start from unison vectors, _not_ ETs. I gave a heuristic on _which_ unison vectors are most likely to help when they are part of a reduced basis.> So again, if you take each consistent ET and choose each possible > generator, you can get a list of linear temperaments that way. It gets > messy, because there's more than one mapping for each generator. In fact, > that sounds much like the very problem we're trying to solve in the first > place.Hmm . . . I'm not sure I see it that way.> If anybody wants to do some real work, this is something to look at. >I guess so, but it's not a route I would probably travel.> > Graham
Message: 2286 - Contents - Hide Contents Date: Fri, 07 Dec 2001 06:49:02 Subject: Re: The grooviest linear temperaments for 7-limit music From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: >>> Yes -- I discussed the situation a few messages back. We use an >> objective measure, and cut things off in a nice wide gap. >> You are thinking that gens^2 cents, and Ennealimmal as the shut-off > point, would be a good plan?Possibly, though since gens and cents are two dimensions, we really need a shuf-off _curve_, don't we?
Message: 2287 - Contents - Hide Contents Date: Fri, 07 Dec 2001 09:11:20 Subject: Re: The grooviest linear temperaments for 7-limit music From: genewardsmith --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:>> An analogy would be to use n^(4/3) cents when > seaching>> for 7-limit ets; this will give you a list which does not favor >> either high or low numbers n,> I'm sorry. This makes no sense to me either. _How_ would you use > n^(4/3) cents? Can you prove this to me?The argument for n^(4/3) is required in order to get the argument for gens^2 cents, so this is the place to start. The argument comes from the theory of simultaneous Diophantine approximation, where it is shown that there is a constant c, depending on d, such that for any d irrational numbers x1, x2, ... xd there will be an infinite number of solutions n to n^(1+1/d) |xi - pi/n| < c In the case of the 7-limit, we want to simultaneously approximate log2(3), log2(5) and log2(7), so d=3.> If you guys can't explain this to me, I don't think you've got much > chance of getting published in a refereed journal. It doesn't involve > anything beyond high school math.Explain what? Diophantine approximation, or why to use that theoretical basis, or what? *What* doesn't involve more than high school math? The theorem I mentioned isn't hard to prove but it does use Dirichlet's pidgeonhole principle, which is also not hard but which you probably did not learn in high school and which I would not propose to discuss in the pages of a music journal, given that I have reason to think that there is a limit to how much math they would find acceptable.
Message: 2289 - Contents - Hide Contents Date: Fri, 07 Dec 2001 02:36:36 Subject: Re: The grooviest linear temperaments for 7-limit music From: paulerlich --- In tuning-math@y..., graham@m... wrote:> Dan Stearns:>>> Of course it might help if I understood it all a bit better too! I >>> feel like I'm getting there though, I just wish Gene were a little >> bit>>> more generous with the narrative--either that or someone else >> besides>>> him were saying the same things slightly differently... that helps >> me >>> sometimes too. > > Paul Erlich:>> I think he's the only one who understands abstract algebra around >> here, so in a lot of cases, that isn't really possible, >> unfortunately . . . of course, I should study up on it, but I should >> also make more music, and get more sleep, and . . . >> Most of the results Gene's getting don't require anything I don't > understand. So I said all these things differently a few months ago. If > you want to catch up, try getting the source code from > <Automatically generated temperaments * [with cont.] (Wayb.)> and an interpreter and try > puzzling it out. I haven't had any feedback at all on readability, so I > don't know easy it'll be for a newbie. > > The method shouldn't be difficult for Dan to understand. You generate a > linear temperament from two equal temperaments.I _really hope_ that that's not what all or even most of Gene's narrative has been about!!> That's exactly like > finding an MOS on the scale tree, except you have to do it for all > consonant intervals instead of only the octave.This I don't see at all. Don't you mean "all fractions 1/N of an octave" rather than "all consonant intervals"?> The wedge products are more difficult, but I don't see them as being at > all important in this context.Well then, when Dan asks about what is going on here, and you come back saying you already understood it all a few months ago, you're actually making a very selective reply to Dan's question, aren't you?
Message: 2290 - Contents - Hide Contents Date: Fri, 07 Dec 2001 06:51:02 Subject: Re: The grooviest linear temperaments for 7-limit music From: genewardsmith --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:> I understand what the slope is in the HE case, but what slope are you > talking about re badness of linear temperament? Badness wrt what?What is the problem with a "flat" system and a cutoff? It doesn't commit to any particular theory about what humans are like and what they should want, and I think that's a good plan.
Message: 2291 - Contents - Hide Contents Date: Fri, 07 Dec 2001 09:48:55 Subject: Simultaneous Diophantine approximation in Hardy and Wright From: genewardsmith I think Paul has this, at any rate, and it's a useful thing to have. You can find a proof of the most basic version of the theorem on page 176 of the fourth edition, Theorem 200.
Message: 2292 - Contents - Hide Contents Date: Fri, 07 Dec 2001 02:45:07 Subject: Re: More lists From: paulerlich --- In tuning-math@y..., graham@m... wrote:> I've updated the script at <Automatically generated temperaments * [with cont.] (Wayb.)> to > produce files using Dave Keenan's new figure of demerit. That is > > width**2 * math.exp((error/self.stdError*3)**2)I thought Dave Keenan wanted to use Gene's "step" measure. In addition, I think it should be weighted to favor the simpler consonances.> > The stdError is from some complexity calculations we did before. I forget > what, but it's 17 cents. The results are at > > <3 4 5 7 8 9 10 12 15 16 18 19 22 23 25 26 27 2... * [with cont.] (Wayb.)> > <4 5 6 9 10 12 15 16 18 19 22 26 27 29 31 35 36... * [with cont.] (Wayb.)> > <5 12 19 22 26 27 29 31 41 46 50 53 58 60 68 70... * [with cont.] (Wayb.)> > <22 26 29 31 41 46 58 72 80 87 89 94 111 113 11... * [with cont.] (Wayb.)> > <26 29 41 46 58 72 80 87 94 111 113 121 130 149... * [with cont.] (Wayb.)> > <29 41 58 72 80 87 94 111 121 130 149 159 183 1... * [with cont.] (Wayb.)> > <58 72 80 94 111 121 149 159 183 217 253 282 30... * [with cont.] (Wayb.)> > <80 94 111 121 217 282 311 320 364 388 400 422 ... * [with cont.] (Wayb.)> > <94 111 217 282 311 364 388 400 422 436 460 525... * [with cont.] (Wayb.)> > > They seem to make good enough sense.Are you missing any "slippery" examples that don't come easily out of two ETs? Since you're doing so much work to get the unison vectors, shouldn't we be thinking about _starting_ with unison vectors?> > Another idea would be to take all the intervals between second- order > intervals below a certain size, and use them as unison vectors to generate > temperaments. I might try that.That should plug a lot of holes.
Message: 2293 - Contents - Hide Contents Date: Fri, 07 Dec 2001 06:52:39 Subject: Re: The grooviest linear temperaments for 7-limit music From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: >>>> There's not much to the 5-limit--it basically is a mere comma >> search,>>> and that can be done expeditiously using a decent 5-limit > notation. >>> A decent 5-limit notation? >> We could search (16/15)^a (25/24)^b (81/80)^c to start out with, and > go to something more extreme if wanted.More extreme? I'm not getting this.
Message: 2294 - Contents - Hide Contents Date: Fri, 07 Dec 2001 09:50:55 Subject: Re: The grooviest linear temperaments for 7-limit music From: dkeenanuqnetau --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote: >>>> An analogy would be to use n^(4/3) cents when >> seaching>>> for 7-limit ets; this will give you a list which does not favor >>> either high or low numbers n, >>> I'm sorry. This makes no sense to me either. _How_ would you use >> n^(4/3) cents? Can you prove this to me? >> The argument for n^(4/3) is required in order to get the argument for > gens^2 cents, so this is the place to start. The argument comes from > the theory of simultaneous Diophantine approximation,Oh damn. Ok forget about proving it to me. Just please try to get me to understand what it is you are saying. I just thought that getting you to prove it to me my be the easiest way for me to understand what it was I had asked you to prove. Apparently not. So ... What is n? What is a 7-limit et? How does one use n^(4/3) to get a list of them? How would one check to see whether the list favours high or low n.>> If you guys can't explain this to me, I don't think you've got much >> chance of getting published in a refereed journal. It doesn't > involve>> anything beyond high school math. >> Explain what? Diophantine approximation, or why to use that > theoretical basis, or what? *What* doesn't involve more than high > school math?Your (and Paul's) statements so far about badness metrics and flatness.> The theorem I mentioned isn't hard to prove but it does > use Dirichlet's pidgeonhole principle, which is also not hard but > which you probably did not learn in high school and which I would not > propose to discuss in the pages of a music journal, given that I have > reason to think that there is a limit to how much math they would > find acceptable. Agreed.But surely you can get me to understand what you actually mean by "flat" here. I may well be prepared to just believe the theorem as stated, if I can understand what it means. But no matter what you come up with I can't see how you can get past the fact that gens and cents are fundamentally incomensurable quantities, so somewhere there has to be a parameter that says how bad they are relative to each other. Currently you are saying that doubling gens is twice as bad as doubling cents. Why? What if 99% of humans don't experience it like that. And why should they both be treated logarithmically? k*log(gens) + log(cents) gives the same ranking as gens^2 * cents when k=2. Why not use k*gens + cents. e.g. if badness was simply gens + cents and you listed everything with badness not more than 30 then you don't need any additional cutoffs. You automatically eliminate anything with gens> 30 or cents > 30 (actually cents > 29 because gens can't go below 1).
Message: 2295 - Contents - Hide Contents Date: Fri, 07 Dec 2001 02:48:30 Subject: Re: The grooviest linear temperaments for 7-limit music From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., graham@m... wrote: >>> The wedge products are more difficult, but I don't see them as > being at>> all important in this context. Working with unison vectors is more >> trouble. >> If working with unison vectors is more trouble, why not wedge > products? The wedgie is good for the following reasons: > > (1) It is easy to compute, given a either pair of ets, a pair of > unison vectors, or a generator map. > > (2) It uniquely defines the temperament, so that temperaments > obtained by any method can be merged into one list. > > (3) It automatically eliminates torsion problems. > > (4) Given the wedgie, it is easy to compute assoicated ets, a > generating pair of unison vectors, or a generator map. Hence it is > easy to go from any one of these to any other. > > (5) By adding or subtracting wedgies we can produce new temperaments. > > Given all of that, I think you are missing a bet by dismissing them; > they could easily be incorporated into your code. > > I've got code for that at>> unison vectors is an outstanding problem that Gene might have > solved, but>> I haven't seen any source code yet. >> I don't know what good Maple code will do, but here it is: > > findcoms := proc(l) > local p,q,r,p1,q1,r1,s,u,v,w; > s := igcd(l[1], l[2], l[6]); > u := [l[6]/s, -l[2]/s, l[1]/s,0]; > v := [p,q,r,1]; > w := w7l(u,v); > s := isolve({l[1]-w[1],l[2]-w[2],l[3]-w[3],l[4]-w[4],l[5]-w[5],l[6]- w > [6]}); > s := subs(_N1=0,s); > p1 := subs(s,p); > q1 := subs(s,q); > r1 := subs(s,r); > v := 2^p1 * 3^q1 * 5^r1 * 7; > if v < 1 then v := 1/v fi; > w := 2^u[1] * 3^u[2] * 5^u[3]; > if w < 1 then w := 1/w fi; > [w, v] end: > > coms := proc(l) > local v; > v := findcoms(l); > com7(v[1],v[2]) end: > > "w7l" takes two vectors representing intervals, and computes the > wegdge product. "isolve" gives integer solutions to a linear > equation; I get an undeterminded varable "_N1" in this way which I > can set equal to any integer, so I set it to 0.The solutions represent?> The pair of unisons > returned in this way can be LLL reduced by the "com7" function, which > takes a pair of intervals and LLL reduces them.Why not TM-reduce them?
Message: 2296 - Contents - Hide Contents Date: Fri, 07 Dec 2001 06:51:43 Subject: Re: The grooviest linear temperaments for 7-limit music From: paulerlich --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:>> Because those measures give an overall "slope" to the results, in >> analogy to what the Farey series seeding does to harmonic entropy. >> What's objective about that? A certain slope may be _real_. i.e. > humans on average may experience it that way, in which case the "flat" > case will really be favouring one extreme.But I don't feel comfortable deciding that for anyone.> I understand what the slope is in the HE case, but what slope are you > talking about re badness of linear temperament? Badness wrt what?Both step and cent.
Message: 2297 - Contents - Hide Contents Date: Fri, 7 Dec 2001 11:30:56 Subject: Math modelization and musical perception From: Pierre Lamothe Here is my first message in MaMuPhi having goal to justify my math approach. I analyse in this part what is invariant within the sound parametric fields at the perceptive (rather than sensitive) viewpoint. I just evoked here a structural schema for the harmonic aspect. Before to add flesh, I have first to confront, in my next message (not before 2002), the perception phenomenology with all the scientific theories about consonance and dissonance. It will be easier after that to analyse, at relational viewpoint, the two types of topological structure implied in harmonic perception, and to study the natural (unconscious and conscious) strategies of musical expression in both creation and interpretation. Question de F. Nicolas (Introduction) * [with cont.] (Wayb.) Pierre [This message contained attachments]
Message: 2298 - Contents - Hide Contents Date: Fri, 07 Dec 2001 02:51:17 Subject: Re: A Geometric Algebra tutorial for Matlab From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> I found this at: > > GABLE: A Matlab Geometric Algebra Tutorial * [with cont.] (Wayb.) > > "This page contains the Matlab software and the PostScript and PDF > versions of a tutorial for learning Geometric Algebra. This tutorial > is aimed at the sophomore college level, although it may provide a > gentle introduction to anyone interested in the topic." > > The relevance of this is that the Clifford algebra of geometric > algebra is a beast which contains all the wedge products within it.Thanks so much. This may be especially useful to me, since I dropped Math 310 as a sophomore, and I use Matlab for everything.
Message: 2299 - Contents - Hide Contents Date: Fri, 07 Dec 2001 06:53:15 Subject: Re: The grooviest linear temperaments for 7-limit music From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> Possibly, though since gens and cents are two dimensions, we really > need a shuf-off _curve_, don't we?If we bound one of them and gens^2 cents, we've bound the other; that's what I'd do.
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