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Message: 5227

Date: Wed, 5 Dec 2001 10:28 +00

Subject: Re: Top 20

From: graham@xxxxxxxxxx.xx.xx

Paul wrote:

> Graham's ... missed ennealimmal because . . . ?

It's too complex.  I get a complexity of 27, but 7-limit temperaments are 
capped at 18.  Also, I only consider the first 20 consistent ETs in that 
list, which goes up to 42 for the 7-limit, and you need 27 and 45 for 
ennealimmal.

Anyway, I have it now

3/8, 49.0 cent generator

basis:
(0.111111111111, 0.0408387831857)

mapping by period and generator:
[(9, 0), (15, -2), (22, -3), (26, -2)]

mapping by steps:
[(45, 27), (71, 43), (104, 63), (126, 76)]

unison vectors:
[[-5, -1, -2, 4], [-1, -7, 4, 1]]

highest interval width: 3
complexity measure: 27  (45 for smallest MOS)
highest error: 0.000170  (0.204 cents)
unique


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It isn't in my local copy, but I think that's out of date.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.


                        Graham


top of page bottom of page up down Message: 5228 Date: Wed, 05 Dec 2001 18:29:39 Subject: Re: Top 20 From: paulerlich --- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote: > Gene, > > As these are linear temperaments, could you also include the generator > and the period in your lists? > > thanks, > > --Dan Stearns Yes -- this would answer much of what went unanswered in my questions. Also, where's double-diatonic (14+12)? I wouldn't think that should be too much worse than paultone, but . . . can you show exactly how "step" is computed, with an example (no wedgies please)?
top of page bottom of page up down Message: 5229 Date: Wed, 05 Dec 2001 05:31:11 Subject: Re: Top 20 From: paulerlich --- In tuning-math@y..., "ideaofgod" <genewardsmith@j...> wrote: > --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: > > Gene, this is shaping up to be an immense contribution you're > making > > to tuning theory. > > Thanks. > > > > 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? > > 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. > > > 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 > > Well, I could but what's the rationale? Cubic growth is already > enough to give us a finite list; we don't need expondential growth. So what's the rationale for cubic growth as opposed to any other function that gives you > > 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)? > > There are far too many answers to this question. > <25/24,28/27,21/20,27/25> makes a nice basis for a notation, but > there are far too many of those also. > Would a list of ets help? How about just the usual details -- generator, mapping. > > > (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? > > 27 or 45 notes would be good--or even 72. 45 notes is just two more > than the Partch 43, and gives a large supply of essentially just > 7-limit harmonies. Not more than MIRACLE-41, though, does it? > > > > (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. > > From Graham's page I got the idea this was supposed to be 5-limit, > but in fact Keenan views it as 7-limit, so "kleismic" is the official > name. Is this Dave Keenan's chain-of-minor-thirds thingy? > > > (20) [1,4,-2,-16,6,4] <36/35,64/63> Looks suspiciously like 12- et > > > meantone. > > > > What's the generator? > > A sharp fifth, but otherwise it's like 12-et. I derived this several years ago, so I forget the cents value. 704? > > > Where is Huygens meantone in all this? > > Coming up soon, I'd guess. Should I keep on going? Please do, unless you think you may be missing some due to the limitations of your search.
top of page bottom of page up down Message: 5230 Date: Wed, 05 Dec 2001 19:26:31 Subject: Sorry Gene From: paulerlich Gene, for some reason the message that contains the questions I was referring to just got posted to the website now. Some sort of internet bottleneck, I suppose. So you can't be blamed for not having answered them!
top of page bottom of page up down Message: 5235 Date: Thu, 6 Dec 2001 20:48 +00 Subject: More lists From: graham@xxxxxxxxxx.xx.xx I've updated the script at <Automatically generated temperaments *> to produce files using Dave Keenan's new figure of demerit. That is width**2 * math.exp((error/self.stdError*3)**2) 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 28 29 31 34 35 37 39 41 *> <4 5 6 9 10 12 15 16 18 19 22 26 27 29 31 35 36 37 41 42 43 45 46 49 50 *> <5 12 19 22 26 27 29 31 41 46 50 53 58 60 68 70 72 77 80 84 87 89 91 94 99 *> <22 26 29 31 41 46 58 72 80 87 89 94 111 113 118 121 130 145 149 152 159 166 171 176 183 *> <26 29 41 46 58 72 80 87 94 111 113 121 130 149 159 166 171 183 190 198 212 217 224 241 253 *> <29 41 58 72 80 87 94 111 121 130 149 159 183 190 198 212 217 224 241 253 270 282 296 301 311 *> <58 72 80 94 111 121 149 159 183 217 253 282 301 311 320 364 388 400 414 422 436 441 460 494 525 *> <80 94 111 121 217 282 311 320 364 388 400 422 436 460 525 581 597 624 643 653 692 718 742 771 867 *> <94 111 217 282 311 364 388 400 422 436 460 525 581 597 624 643 653 718 742 771 867 908 935 997 1065 *> They seem to make good enough sense. I haven't taken the training wheels off completely, but loosened them as far as I did for the microtemperaments. The other files haven't been updated, and I'm not even calculating the MOS-rated list any more. I've also changed the program to print out equivalences between second-order ratios instead of unison vectors. That means the higher limits have a huge number of equivalences. For example, at the bottom of the 21-limit list there's an 11-limit unique temperament consistent with 111 and 282. It has a complexity of 174 and all intervals to within 2 cents of just. With something that complex, are there any second-order equivalences? Yes, lots. Including one interval that can be taken 11 different ways: 144:143 =~ 196:195 =~ 171:170 =~ 210:209 =~ 225:224 =~ 209:208 =~ 221:220 =~ 170:169 =~ 273:272 =~ 289:288 =~ 190:189 and that picked out of 197 lines of numerical vomit. I could clean it up, but I don't know if I should. If anybody thought the extended 21-limit was pretty, they can't have been paying attention. It should be possible to get some unison vectors without torsion from this list! If the temperament's second-order unique, I'll have to use the original method. Some 5-limit temperaments are, but they aren't a problem anyway. A few 7-limit temperaments are too, notably including shrutar. Ennealimmal for all its complexity has 49:40 =~ 60:49 50:49 =~ 49:48 One problem with calculating the unison vectors from these equivalences is I'd have to check they were linearly independent without using Numeric. Or move the generating function to vectors.py. But I don't know if I'll bother, because the equivalences are the important things anyway. 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. Oh yes. Seeing as a 7-limit microtemperament is now causing something of a storm, notice that the top 11-limit one is 26+46 (complexity of 30, errors within 2.5 cents). And the simplest with all errors below a cent is 118+152 (complexity of 74). Graham
top of page bottom of page up down Message: 5240 Date: Thu, 06 Dec 2001 06:35:18 Subject: Re: The grooviest linear temperaments for 7-limit music From: dkeenanuqnetau --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote: > --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: > > > I think you misunderstood Dave -- he wanted the *goodness* for the > > cents factor to be a Gaussian. > > I don't think penalizing a system for being good can possibly be > defended, so I'm at a loss here. I'm not sure who is confused about what. gaussian(x) = exp(-(x/k)^2) goodness = gaussian(cents_error) badness = 1/goodness = 1/exp(-(cents_error/k)^2) = exp((cents_error/k)^2) sinh might be fine too. I'm not familiar. The problems, as I see them, are (a) some temperaments that require ridiculously numbers of notes are near the top of the list only because they have errors of a fraction of a cent, but once it's less than about a cent, this should not be enough to redeeem them. And (b) some others with ridiculously large errors are near the top of the list only because they come out needing few notes. I think that the first can be fixed by applying a function to the cents error that treats all very small errors as being equal, and the latter might be fixed by dropping back from steps^3 to steps^2. -- Dave Keenan
top of page bottom of page up down Message: 5242 Date: Thu, 6 Dec 2001 11:16 +00 Subject: Re: The slippery six From: graham@xxxxxxxxxx.xx.xx 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. 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. If anybody wants to do some real work, this is something to look at. Graham
top of page bottom of page up down Message: 5243 Date: Thu, 6 Dec 2001 11:16 +00 Subject: Re: The slippery six From: graham@xxxxxxxxxx.xx.xx > By 46+34 I mean a particular system of generators in the 80-et, and > that is determined without reference to what the maps are. Graham > means by it the associated linear temperament, and that is *not* > determined without reference to the maps, and so is not strictly well- > defined. It is determined only mod 40 if you assume it should follow > the 46+34 of the 80-et. Gene, when I called you on this before you were definitely talking about temperaments. I wouldn't have mentioned it otherwise. Graham
top of page bottom of page up down Message: 5244 Date: Thu, 6 Dec 2001 11:16 +00 Subject: Re: The grooviest linear temperaments for 7-limit music From: graham@xxxxxxxxxx.xx.xx Paul wrote: > As for the other part, the dissonance measure . . . by doing it > Gene's way, we're going to end up with all the most interesting > temperaments for a wide variety of different ranges, from "you'll > never hear a beat" to "wafso-just" to "quasi-just" to "tempered" > to "needing adaptive tuning/timbring". Thus our top 30 or whatever > will have much of interest to all different schools of microtonal > composers. Oh, if you think one list can please everybody. I'd rather ask people what they want, and produce a short list that's likely to have their ideal temperament on it. That's why I keep up the .key and .micro files. Most importantly, why I release all the source code for a Free platform so that anybody can try out their own ideas. Nothing Gene's done so far couldn't have been done by modifying that code. Graham
top of page bottom of page up down Message: 5245 Date: Thu, 6 Dec 2001 11:16 +00 Subject: Re: The grooviest linear temperaments for 7-limit music From: graham@xxxxxxxxxx.xx.xx Dave Keenan wrote: > (b) some others with ridiculously large errors are near the top of the > list only because they come out needing few notes. > > I think that the first can be fixed by applying a function to the cents > error that treats all very small errors as being equal, and the latter > might be fixed by dropping back from steps^3 to steps^2. No, you get ridiculously large errors near the top with steps^2 as well. Graham
top of page bottom of page up down Message: 5246 Date: Thu, 06 Dec 2001 00:10:38 Subject: Re: The slippery six From: paulerlich --- In tuning-math@y..., "ideaofgod" <genewardsmith@j...> wrote: > 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. That's a good indication that Graham may have missed these too, since he also started from pairs of ETs . . . Graham? > They seem to be ones which are so > closely tied to one particular et that they don't show up by studying > pairs. Also for some reason there are two 9/7-systems on the list. > > (1) [6,10,10,-5,1,2] ets: 22 > > [0 2] > [3 1] > [5 1] > [5 2] > > a = 7.98567775 / 22 (~9/7) ; b = 1/2 You know, I was just going to ask you what happened to this one, as I remember it from the even earlier survey that you and Graham did, coming from my list of commas. > (5) [0,-12,-12,6,19,-19] > > [ 0 12] > [ 0 19] > [-1 28] > [-1 34] > > a = 23.40769169 cents; b = 100 cents > measure 9556 Oh yeah, this one again! > (6) [-2,4,-30,-81,42,11] ets: 46,80 > > [ 0 2] > [-1 4] > [ 2 3] > [-15 18] > > a = 33.01588032 / 80 (~4/3); b = 1/2 > measure 26079 So this _isn't_ 46+34??
top of page bottom of page up down Message: 5247 Date: Thu, 6 Dec 2001 11:16 +00 Subject: Re: The grooviest linear temperaments for 7-limit music From: graham@xxxxxxxxxx.xx.xx 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 *> 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. 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. 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. I've got code for that at <Unison vector to MOS script *>. Going from temperaments to unison vectors is an outstanding problem that Gene might have solved, but I haven't seen any source code yet. Graham
top of page bottom of page up down Message: 5248 Date: Thu, 6 Dec 2001 08:59:33 Subject: Re: The wedge invariant commas From: monz Hi Gene, > From: ideaofgod <genewardsmith@xxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Wednesday, December 05, 2001 5:08 PM > Subject: [tuning-math] Re: The wedge invariant commas > > > ... so I've messed > things up around here by introducing multilinear algebra, Baker's > theorem and what-not, as well as something I (and Pierre) saw as > relevant already, namely abelian groups (or Z-modules, as Pierre > prefers to call them), and quadratic forms in connection with > lattices. I'm having lots of trouble understanding what's been discussed on this list since you joined. But this bit of your post jumped out at me, and I thought you'd find this profitable: Mark Lindley & Ronald Turner-Smith. 1993. _Mathematical Models of Musical Scales: A New Approach_. Orpheus-Schriftenreihe zu Grundfragen der Musik vol. 66, Verlag für systematische Musikwissenschaft, Bonn-Bad Godesberg. Lindley, Mark and Ronald Turner-Smith. "An Algebraic Approach to Mathematical Models of Scales", Music Theory Online vol. 0 no. 3, June 1993. === === ============= ==== * Lindley/Turner-Smith view tuning systems as abelian groups. (see especially paragraph [5] of the latter article) love / peace / harmony ... -monz http://www.monz.org * "All roads lead to n^0" _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail Setup *
top of page bottom of page up down Message: 5249 Date: Thu, 06 Dec 2001 00:35:04 Subject: Re: The grooviest linear temperaments for 7-limit music From: paulerlich --- In tuning-math@y..., David C Keenan <d.keenan@u...> 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), What human-perception-or-cognition-based justification is there for using step^2 ??? > Yes. Once the deviation goes past about 20 cents it's irrelevant > how big it is, That's not true -- you're ignoring both adaptive tuning and adaptive timbring. >and a 0.1 cent deviation does not sound 10 times better than a 1.0 >cent deviation, it sounds about the same. In my own musical endeavors, this is true, but with all the strict-JI obsessed people out there, a 0.1 cent deviation may end up being 10 times more interesting than a 1.0 cent deviation. > I suggest this figure-of->demerit. > > step^2 [...] Again, what on earth does step^2 tell you about how composers and performers would rate a temperament? OK, step^2 is the number of possible dyads in the typical scale. Step^3 is the number of possible triads. Why is the former so much more "human-perception-or-cognition- based" to you than the latter? As for the other part, the dissonance measure . . . by doing it Gene's way, we're going to end up with all the most interesting temperaments for a wide variety of different ranges, from "you'll never hear a beat" to "wafso-just" to "quasi-just" to "tempered" to "needing adaptive tuning/timbring". Thus our top 30 or whatever will have much of interest to all different schools of microtonal composers.
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