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Message: 4275 - Contents - Hide Contents Date: Thu, 14 Mar 2002 19:15:43 Subject: Ferguson WHO??? (Re: [tuning] Re: "wolf" pack, rat pack [Isacoff])) From: genewardsmith --- In tuning-math@y..., "Orphon Soul, Inc." <tuning@o...> wrote:> I searched tuning AND tuning math and I didn't find either Ferguson or > Forcade. What exactly is this algorithm?It's an integer relation finding algorithm. There is even an integer relation finding web site now where you can email and get the relation (using PSLQ, I think.) Here's mathworld on it: Ferguson-Forcade Algorithm -- from MathWorld * [with cont.] I've not found these useful for searches, but maybe I should consider the matter; however brute force seems to suffice.

Message: 4277 - Contents - Hide Contents Date: Thu, 14 Mar 2002 20:12:51 Subject: Re: Ferguson found From: genewardsmith --- In tuning-math@y..., "Orphon Soul, Inc." <tuning@o...> wrote:> Thanks for the link to Harmonic Entropy, that was from a couple weeks before > I joined the online tuning world. I'm trying to find my way around the > weblinks and so far other than this 26 page .pdf file from NASA that's > taking forever to load I haven't been able to get hold of a digestible > definition as yet.That's got to be Bailey. :)> Is anyone here familiar enough with the PLSQ to sum it up in a few dozen > words and/or ascii?It's an integer relation algorithm.

Message: 4278 - Contents - Hide Contents Date: Thu, 14 Mar 2002 23:22:35 Subject: Re: Systematic naming of new temperaments (was: amt) From: Herman Miller On Wed, 13 Mar 2002 08:39:23 -0000, "dkeenanuqnetau" <d.keenan@xx.xxx.xx> wrote:>> When was that discussed? I was going over 7-limit planars on tuning,>and I recall someone saying that 126/125 had been looked at, but I >thought starling was a scale! >>> > >It is. Maybe I was the first to apply the term to the temperament the >scale is in. In referring to Genes recent 8-noter as Starling-8.It's referred to as "Starling temperament" on the Warped Canon page, although of course the canon doesn't take advantage of the 7-limit commas in this case. I believe I also referred to it as a temperament in some old posts to the tuning list. I described it as essentially what we'd now call a planar temperament with (approximate) major third and minor third generators, having 126/125 as a unison vector. This would have been sometime in 1999, but I don't have the exact reference. The prototypical Starling scale had a 312 cent minor third and a 388 cent major third. A later variation had slightly tempered octaves. But I'm starting to think that the exact size of the thirds isn't the most definitive feature of the tuning. Those were just the sizes of the geenrators that optimized a particular set of 7-limit intervals I was interested in at the time. -- see my music page ---> ---<The Music Page * [with cont.] (Wayb.)>-- hmiller (Herman Miller) "If all Printers were determin'd not to print any @io.com email password: thing till they were sure it would offend no body, \ "Subject: teamouse" / there would be very little printed." -Ben Franklin

Message: 4279 - Contents - Hide Contents Date: Thu, 14 Mar 2002 21:39:35 Subject: Starling temperament mapping From: David C Keenan Here's an opportunity to describe a planar temperament mapping in the canonical form I've just proposed in 'Standardising temperament mappings'. Starling temperament discovered and named by Herman Miller, 1999 Target [lg(2) lg(3) lg(5) lg(7)] Generators (cents) [1200 498.0 107.1] Mapping [1 2 2 1] [0 -1 1 5] [0 0 -1 -3] Which means for example that lg(7) ~= 1*1200 + 5*498.0 + -3*107.1 Can someone please check that that agrees with the fact that the 125:126 (2^1 * 3^2 * 5^-3 * 7^1) is distributed, and remind me how you do that? Gene, Can you please give your proposed canonical form if it differs from this, and explain the conventions involved. Regards, -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)

Message: 4280 - Contents - Hide Contents Date: Thu, 14 Mar 2002 00:37:11 Subject: Periods and generators From: genewardsmith My previous example with a title like this hasn't shown up, and now I'm trying to follow it up anyway. Nuts! I was going to add to a previous example of 720 as a period for meantone one of 16/15 as meantone. Assuming the posting does finally show up, continuewith this point: I want 16/15 to be my period, so I take 16/15^81/80 = h5. Now one solution to h5^g = 81/80 is g=h7, which gives us [h5,h7] as our map. We now have a "period" of about 16/15, inside of which is a "generator" of about 25/24. But a more sensible way of looking at it is that 16/15 and 25/24 are both generators, and we do 5-limit meantone by means of a two-dimensional generator lattice.

Message: 4281 - Contents - Hide Contents Date: Thu, 14 Mar 2002 02:33:27 Subject: Re: Systematic naming of new temperaments (was: amt) From: dkeenanuqnetau --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:>> --- In tuning-math@y..., Carl Lumma <carl@l...> wrote:>>> How do you differentiate two different mappings with the same >>> generator? >>>> I just say 5-limit whatever versus 7-limit whatever, and so on. >> that won't necessarily do the trick. > > look at the fourth and fifth entries here: > > 4 5 6 9 10 12 15 16 18 19 22 26 27 29 31 35 36... * [with cont.] (Wayb.)You'll have to do better than that. 270.8 cent generator, [-2, -3, -1] mapping, 45 cent minimum MA error The generator here must function as 7:8, 6:7 and 5:6. Look at the minimum error. As a 7-limit temperament it is garbage. But if forced to find a meaningful name for it I would call it "7:8 = 5:6" or "supermajor second minor thirds" or "diminished third minor thirds", after Gene's naming of "fourth thirds". 271.3 cent generator, [7, -3, 8] mapping, 4.5 cent minimum MA error This is of course 7-limit subminor thirds or 7-limit orwell. I must point out that Gene invented the "generator as fraction of consonant interval" nomenclature, with "quadrafourths". I'm just arguing that we should (a) extend its use, (b) refine the prefixes to more clearly indicate fractions as opposed to multiples, (c) use latin consistently (not mix it with greek), and (d) apply it always to the interval that is made up of the fewest generators (and doesn't contain any periods except as whole octaves). Regardsing (c), I wouldn't mind if they were consistently greek except I feel the latin ones work better regarding criterion (b) (fractions versus multiples). Regarding (d): In the case where there is more than one consonant interval which is made up of the minimum number of generators, then the largest and smallest in size should both be mentioned, as above. This was Gene's invention too, with fourth thirds. I also think that when a temperament _does_ extend to being good at a higher limit, fewer generators map to a consonant interval at the higher limit, then it fine to use the higher limit name prefixed by the limit. So 5-limit Orwell can also be called 5-limit subminor thirds and we can forget about calling it tertiminorsixths. Kleismic and parakleismic is a better example of the problem you were setting up for me. I approve of these names, and para (presumably meaning "near to") might generally be used for the badder of the two since para can also mean "improper". So the horrible 270.8 cent temperament above could also be para-orwell or para-subminor-thirds. However I remember Gene using cata- meaning "down". Its opposite is ana- (up). I quite like these two, and they (or para) would remove the temptation to use semi to mean "almost", and leave it to mean "half", as in semitone or semicircle. So the horrible 270.8 cent temperament above could also be cata-orwell or cata-subminor-thirds. And parakleismic could be catakleismic. Gene, What was it that you used cata- for?

Message: 4282 - Contents - Hide Contents Date: Thu, 14 Mar 2002 03:17:59 Subject: Re: Systematic naming of new temperaments (was: amt) From: genewardsmith --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:> Gene, What was it that you used cata- for?It had a generator in the 7 and 11 limit a whisker below the kleismic 5-limit generator, but I really shouldn't have called it that, since it is a whisker *above* the h19^h15 = [6,5,3,-7,12,-6] 7-limit generator, which I was taking to be the 7-limit kleismic. So it's a misnomer. The other reason is that it sounds like "cataclysmic"--ha ha. Get it? I don't either. :)

Message: 4283 - Contents - Hide Contents Date: Thu, 14 Mar 2002 04:30:07 Subject: Re: Weighting complexity (was: 32 best 5-limit linear temperaments) From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: >>> what do you mean (who and what limits)? >> I don't want to do another search, this time weighted, unless you >tell me what you want to look at.i think dave needs to answer this question (or repeat his answer).

Message: 4284 - Contents - Hide Contents Date: Thu, 14 Mar 2002 04:37:46 Subject: Re: Weighting complexity (was: 32 best 5-limit linear temperaments) From: paulerlich --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:>> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: >>>>> what do you mean (who and what limits)? >>>> I don't want to do another search, this time weighted, unless you >> tell me what you want to look at. >> i think dave needs to answer this question (or repeat his answer).well, failing that, just do it with the same bounds as before, or still more inclusive bounds if you prefer, and with the applied badness measure given for each entry. after that's done, we should strive for consensus and move on to the 7-limit.

Message: 4285 - Contents - Hide Contents Date: Thu, 14 Mar 2002 06:37:14 Subject: Re: Weighting complexity (was: 32 best 5-limit linear temperaments) From: dkeenanuqnetau --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:>> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:>>> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: >>>>>>> what do you mean (who and what limits)? >>>>>> I don't want to do another search, this time weighted, unless you >>> tell me what you want to look at. >>>> i think dave needs to answer this question (or repeat his answer).Yes I have already posted them, and they are also in my spreadsheet. But here they are again. 35 cents rms, 12 weighted-rms generators (or 10 for the shorter list) and max 625 for Gene's badness (using weighted complexity).

Message: 4286 - Contents - Hide Contents Date: Thu, 14 Mar 2002 09:54:28 Subject: Re: Weighting complexity (was: 32 best 5-limit linear temperaments) From: genewardsmith --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:> Yes I have already posted them, and they are also in my spreadsheet. > But here they are again. 35 cents rms, 12 weighted-rms generators (or > 10 for the shorter list) and max 625 for Gene's badness (using > weighted complexity).Here's some examples of weighted badness, using weighted complexity. Can you check your limits in terms of this? 81/80: 103.257 15625/15552: 187.010 250/243: 463.641 78732/78125: 525.371

Message: 4287 - Contents - Hide Contents Date: Thu, 14 Mar 2002 13:29:51 Subject: Re: some output from Graham's cgi From: manuel.op.de.coul@xxxxxxxxxxx.xxx Graham, When I enter these numbers in the linear temperament finder 13 28 12.0 25 10 0.5 the first one is: 13-limit 31/75, 495.6 cent generator basis: (1.0, 0.41300690788) mapping by period and generator: [(1, 0), (2, -1), (11, -21), (9, -15), (8, -11), (7, -8)] mapping by steps: [(46, 29), (73, 46), (107, 67), (129, 81), (159, 100), (170, 107)] highest interval width: 21 complexity measure: 21 (29 for smallest MOS) highest error: 0.009422 (11.306 cents) But when I do an RMS approximation I'm able to get a lower highest error of 10.868 cents (for 13/9), with a generator of 495.708356 cents. Did you exclude some consonant intervals? With many consonant intervals to approximate, minimax makes more sense than RMS I think. (29 for smallest MOS) This means 29 tones in the smallest MOS, doesn't it? Manuel

Message: 4288 - Contents - Hide Contents Date: Thu, 14 Mar 2002 13:33:36 Subject: Re: Ekmelische Musik From: manuel.op.de.coul@xxxxxxxxxxx.xxx Aaron, Thanks, I wasn't aware that it referred to the harmonic series.>Anyone know what's going on over there nowadays?I don't; they still give courses. In 1999 a CD titled "Ekmelische Musik" came out. Manuel

Message: 4289 - Contents - Hide Contents Date: Thu, 14 Mar 2002 13:52 +0 Subject: Re: some output from Graham's cgi From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <OFC5246248.AC822B6F-ONC1256B7B.00535BCC@xxxxxx.xxxxxxxxx.xx> Manuel wrote:> But when I do an RMS approximation I'm able to get a lower > highest error of 10.868 cents (for 13/9), with a generator of > 495.708356 cents. > Did you exclude some consonant intervals?No, I only exclude intervals that don't depend on the generator. I'm not sure if that's right or not, but it doesn't matter here because the period is an octave.> With many consonant intervals to approximate, minimax makes more sense > than RMS I think.Yes, but the RMS optimum is more efficient to calculate.> (29 for smallest MOS) > > This means 29 tones in the smallest MOS, doesn't it?Yes, the smallest one that contains a complete 13-limit otonality. Graham

Message: 4290 - Contents - Hide Contents Date: Thu, 14 Mar 2002 15:24:56 Subject: Re: some output from Graham's cgi From: manuel.op.de.coul@xxxxxxxxxxx.xxx>No, I only exclude intervals that don't depend on the generator.O! So you didn't include 7/5 in the given example? Wouldn't you, generally speaking, get better results if you did?>I'm not sure if that's right or not, but it doesn't matter here >because the period is an octave.I don't understand why it doesn't matter? Manuel

Message: 4291 - Contents - Hide Contents Date: Thu, 14 Mar 2002 14:59 +0 Subject: Re: some output from Graham's cgi From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <OFBF3E9787.74F450EC-ONC1256B7C.004EAB50@xxxxxx.xxxxxxxxx.xx> Manuel wrote:> O! So you didn't include 7/5 in the given example? Wouldn't > you, generally speaking, get better results if you did?What? No, 7:5 is 20 generator steps. It would be excluded in Twintone, where it's always a half-octave. On thinking about it more, this doesn't affect the optimum because it's only removing a constant from the RMS-by-generator-size function that's being minimized. So I was right to exclude such intervals. I don't know why we're getting different results.>> I'm not sure if that's right or not, but it doesn't matter here >> because the period is an octave. >> I don't understand why it doesn't matter?An interval would have to be equal to a unison to be excluded, which is much less likely than being equal to some division of an octave. Graham

Message: 4292 - Contents - Hide Contents Date: Thu, 14 Mar 2002 17:43:16 Subject: Re: some output from Graham's cgi From: manuel.op.de.coul@xxxxxxxxxxx.xxx Graham wrote:>I don't know why we're getting different results.Found the problem, I was using a wrong mapping somewhere. I confirmed your result using all intervals. Anyway, the minimax solution is 495.662963 cents. Highest error 10.595435 cents. Manuel

Message: 4294 - Contents - Hide Contents Date: Fri, 15 Mar 2002 16:01:09 Subject: Re: Standardising temperament mappings From: dkeenanuqnetau --- In tuning-math@y..., graham@m... wrote:> In-Reply-To: <a6t29j+cdq0@e...> > dkeenanuqnetau wrote: >>> I don't understand what you mean by "more robust". Whatever >> conventions we settle on to give a unique form will allow an >> octave-based temperament to be identified by the mapping-by-generators >> and the number of periods to an octave. Why would my proposal of >> generator less than half period, fail to do this. >> The generator might get larger, in which case it would be incorrectly > described. If you set the first non-zero element of the generator mapping > to be positive, the mapping is uniquely described regardless of the size > of the generator.You seem to be talking about what your program is doing internally. I don't care about that. I'm talking about what we present to people, irrespective of how it is generated. The important thing to me, is what will be most immediately meaningful or useful to musicians or composers. If your optimiser comes out with a generator that is between 0.5 and 1 period you can always subtract it from the period, or if it is between 1 and 1.5 periods you can subtract a period from it etc., and adjust the matrix accordingly. But the questions is, "Is this the most meaningful generator?". It could be argued for example in Pajara/Paultone that a generator which is a fourth or fifth may be more meaningful than the one of 108 cents. But it isn't clear to me how you would generalise that to cases where there isn't a fourth or fifth among the possible generator values. It just looks too messy to me. However Gene seems like he might be favouring a fourth when one exists in such cases. What rule are you using there Gene?> The one for generating linear temperaments from equal temperaments does. > But the one for generating linear temperaments from unison vectors > doesn't. Some generators are a lot bigger than the period because they > become larger on optimization. So I'll have to change that program to > adjust the mapping after optimization. OK.>> I assume you understood I meant "smallest with GCD(row) = 1", so no >> degeneracy. >> Um what if there's contorsion?I'm still confused about that. ...>So I won't be changing the library's internal > format, but can change the display the CGI uses. Sure.But don't bother changing anything yet. One point is to have the same format from both you and Gene, and Gene is still thinking about it.

Message: 4295 - Contents - Hide Contents Date: Fri, 15 Mar 2002 16:23:46 Subject: Re: Wedge product definitions from mathworld From: dkeenanuqnetau --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> Here is exterior algebra: > > http://mathworld.wolfram.com/ExteriorAlgebra.html * [with cont.] > > Here is wedge product: > > http://mathworld.wolfram.com/WedgeProduct.html * [with cont.] > > This could and should be made a lot simpler for musical purposes. Gene,If you have attempted that somewhere, could you please point me to it. The mathworld definition makes no sense to me whatsoever, since it is recursive and I can't find where the recursion bottoms out. Can you please give a definition that only applies to matrices or vectors (whatever makes sense for temperaments) and not general k-forms, and that eventually bottoms out into ordinary scalar multiplication *? Does it take two vectors and give another vector or what?

Message: 4296 - Contents - Hide Contents Date: Fri, 15 Mar 2002 16:42 +0 Subject: Re: Wedge product definitions from mathworld From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <a6t76i+vkbe@xxxxxxx.xxx> dkeenanuqnetau wrote:> The mathworld definition makes no sense to me whatsoever, since it is > recursive and I can't find where the recursion bottoms out. Can you > please give a definition that only applies to matrices or vectors > (whatever makes sense for temperaments) and not general k-forms, and > that eventually bottoms out into ordinary scalar multiplication *?My Python source code includes an implementation of wedge products. Whether that's easier to understand than the mathworld link I don't know. Their description looks clear enough, but not the source code. For mine, I think you need the __setitem__ and __getitem__ methods of Wedgable, as well as the obvious __mul__ which does the product.> Does it take two vectors and give another vector or what?It can take vectors, implemented as a list of real (or maybe even complex, but scalar) coefficients of a set of unison vectors. The wedge product of a pair of such vectors will be a list of coefficients of a combination of unison vectors. So if you start with 3-D vectors defined on i, j and k the wedge product will be defined on ij, ik and jk. My Wedgable class is to implement this generalisation of vectors, and the product of two wedgables is always another wedgable. Graham

Message: 4297 - Contents - Hide Contents Date: Fri, 15 Mar 2002 03:13:23 Subject: Re: Standardising temperament mappings From: genewardsmith --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:> Gene and Graham are using different ways of presenting both the basis > vector for a temperament and its mapping matrix.I wish I had checked what Graham did before coding, but I am inclined to think I want a different standard mapping anyway, since I have what seems like a good, general method convention suitable for planar and higher temperaments also.> I see the following issues which need to be addressed in the following order. > 1. Standardising the basis vectorWhat's the basis vector?

Message: 4298 - Contents - Hide Contents Date: Fri, 15 Mar 2002 17:05 +0 Subject: Re: Wedge product definitions from mathworld From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <memo.707336@xxx.xxxxxxxxx.xx.xx> I wrote:> My Python source code includes an implementation of wedge products. > Whether that's easier to understand than the mathworld link I don't > know. Their description looks clear enough, but not the source code. > For mine, I think you need the __setitem__ and __getitem__ methods of > Wedgable, as well as the obvious __mul__ which does the product.Oops, sorry. __mul__ is for multiplication by a scalar. You want the standalone wedgeProduct function.

Message: 4299 - Contents - Hide Contents Date: Fri, 15 Mar 2002 03:26:30 Subject: Re: Standardising temperament mappings From: dkeenanuqnetau --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote: >>> Gene and Graham are using different ways of presenting both the basis >> vector for a temperament and its mapping matrix. >> I wish I had checked what Graham did before coding, but I aminclined to think I want a different standard mapping anyway, since I have what seems like a good, general method convention suitable for planar and higher temperaments also.>>> I see the following issues which need to be addressed in the following order. >> 1. Standardising the basis vector >> What's the basis vector?The list of generators. Sorry.

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