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Message: 2225 - Contents - Hide Contents 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 * [with cont.] (Wayb.) -- 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
Message: 2226 - Contents - Hide Contents 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> EnnealimmalYou 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?
Message: 2229 - Contents - Hide Contents 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 <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. Graham
Message: 2230 - Contents - Hide Contents 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 StearnsYes -- 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)?
Message: 2231 - Contents - Hide Contents 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.
Message: 2232 - Contents - Hide Contents 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!
Message: 2233 - Contents - Hide Contents Date: Wed, 05 Dec 2001 20:24:25 Subject: Superparticulars From: John Chalmers I looked for superparticular ratios whose prime factors were no larger than 23 and whose numerators were less than or equal to ten million (10 ^7) as that seemed the practical limit of my computer at thattime (1997, XH (17). My source was a paper by Bernd Streitberg and Klaus Balzer, 1988, The Sound of Mathematics, Proceedings of the 1988 International Computer Music Conference 158-165. They searched at the five limit to 10 ^12 and found only 10 (2/1, 3/2, 4/3, 5/4, 6/5, 9/8, 10/9, 16/15, 25/24 and 81/80). I found only 240 at the 23 limit. I've summarized the numbers at each prime limit and the cumulative totals below: Limit Number Total 2 1 1 3 3 4 5 6 10 7 13 23 11 17 40 13 28 68 17 40 108 19 58 166 23 74 240 I'm not sure what the question was about lengths (?) at each limit; I hope these data help answer it. --John
Message: 2234 - Contents - Hide Contents Date: Thu, 06 Dec 2001 05:11:47 Subject: Re: The slippery six From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> Hmm . . . you keep avoiding my whining about consistency (most > recently with regard to 21), and this would seem to be a good place > to bring it up again. You told Graham that something like 46+34 to > you would be _defined_ so that the 80 would come out right, not > necessarily the individual ETs. Now you seem to be contradicting > yourself. What gives?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.> And you brought up 80 when we were discussing ways of extending > diaschismic to 11-limit, if you recall . . . probably this same > mapping through the 7-limit.The point being, there was more than one sensible way to do it, which were the same in the 80-et but not as linear temperaments.
Message: 2235 - Contents - Hide Contents Date: Thu, 06 Dec 2001 20:01:26 Subject: Re: The slippery six From: genewardsmith --- In tuning-math@y..., graham@m... wrote:> Gene, when I called you on this before you were definitely talking about > temperaments. I wouldn't have mentioned it otherwise.I was talking about 34&46, not 34+46; the first is not well-defined, which was the point of my example.
Message: 2236 - Contents - Hide Contents Date: Thu, 06 Dec 2001 05:15:03 Subject: Re: Superparticulars From: genewardsmith --- In tuning-math@y..., John Chalmers <JHCHALMERS@U...> wrote:> I'm not sure what the question was about lengths (?) at > each limit; I hope these data help answer it.What's really best would be the data itself, and not a summary, but if that is too onerous the largest for each prime would be nice.
Message: 2237 - Contents - Hide Contents
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 * [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)
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. 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
Message: 2238 - Contents - Hide Contents Date: Thu, 06 Dec 2001 05:55:39 Subject: Re: The grooviest linear temperaments for 7-limit music From: genewardsmith --- 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.
Message: 2239 - Contents - Hide Contents Date: Thu, 06 Dec 2001 13:43:07 Subject: Re: Digest Number 185 From: John Chalmers Gene et al. Here's the whole article on superparticulars as an asci file. Ignore the first column of numbers, they were an internal check. The Number of 23-Prime-Limit Superparticular Ratios Less than 10,000,000 It has been conjectured that there are only 10 superparticular (epimore) ratios whose terms are factorable by 2, 3 and 5 1. These ratios are 2/1, 3/2, 4/3, 5/4, 6/5, 9/8, 10/9, 16/15, 25/24, and 81/80 2 and are well-known in music theory. Computer searches have verified this conjecture with numerators up to 1 x 1012, but a general proof is not known, though it has been claimed that one was known to the ancient Pythagoreans 3. I find this conjecture astonishing and have repeated the search on a Mac SE/30 up to 107 with a Microsoft QuickBASIC program I recently wrote4 . Out of curiosity, I decided to extend the search to include each prime limit up to 23 inclusive with numerators less than or equal to 107, which seems to be the practical limit for my program and system. I have found 240 ratios, the list of which I have appended below as Table 1. Needless to say, the largest intervals at each prime limit, including the recently discovered "ragisma," 4375/4374 5 have been exploited by musicians and theorists. John H. Chalmers Rancho Santa Fe, California January, 26, 1997 Table 1. Numbers of Superparticular Ratios Less Than 107 at the 23 Prime-Limit The number of New Ratios at each new prime limit is indicated as well as the Cumulative Total. The numbers in the first column are the order of generation numbers of the ratios as they are found by my program and are perhaps of less general interest. Ratios of 2 1 2 / 1 New Ratios = 1 Cumulative Total = 1 Ratios of 3 2 3 / 2 3 4 / 3 8 9 / 8 New Ratios = 3 Cumulative Total = 4 Ratios of 5 4 5 / 4 5 6 / 5 9 10 / 9 15 16 / 15 24 25 / 24 51 81 / 80 New Ratios = 6 Cumulative Total = 10 Ratios of 7 6 7 / 6 7 8 / 7 14 15 / 14 20 21 / 20 27 28 / 27 31 36 / 35 36 49 / 48 37 50 / 49 43 64 / 63 62 126 / 125 80 225 / 224 153 2401 / 2400 171 4375 / 4374 New Ratios = 13 Cumulative Total = 23 Ratios of 11 10 11 / 10 11 12 / 11 21 22 / 21 28 33 / 32 34 45 / 44 40 55 / 54 41 56 / 55 56 99 / 98 57 100 / 99 61 121 / 120 73 176 / 175 82 243 / 242 99 385 / 384 103 441 / 440 113 540 / 539 161 3025 / 3024 186 9801 / 9800 New Ratios = 17 Cumulative Total = 40 Ratios of 13 12 13 / 12 13 14 / 13 25 26 / 25 26 27 / 26 33 40 / 39 44 65 / 64 45 66 / 65 50 78 / 77 53 91 / 90 58 105 / 104 65 144 / 143 70 169 / 168 75 196 / 195 92 325 / 324 94 351 / 350 95 352 / 351 97 364 / 363 117 625 / 624 118 676 / 675 120 729 / 728 128 1001 / 1000 143 1716 / 1715 149 2080 / 2079 168 4096 / 4095 170 4225 / 4224 181 6656 / 6655 189 10648 / 10647 223 123201 / 123200 New Ratios = 28 Cumulative Total = 68 Ratios of 17 16 17 / 16 17 18 / 17 29 34 / 33 30 35 / 34 38 51 / 50 39 52 / 51 52 85 / 84 60 120 / 119 64 136 / 135 67 154 / 153 71 170 / 169 79 221 / 220 84 256 / 255 85 273 / 272 88 289 / 288 98 375 / 374 104 442 / 441 114 561 / 560 116 595 / 594 119 715 / 714 123 833 / 832 126 936 / 935 129 1089 / 1088 131 1156 / 1155 134 1225 / 1224 135 1275 / 1274 142 1701 / 1700 148 2058 / 2057 154 2431 / 2430 156 2500 / 2499 157 2601 / 2600 174 4914 / 4913 177 5832 / 5831 194 12376 / 12375 199 14400 / 14399 209 28561 / 28560 211 31213 / 31212 212 37180 / 37179 227 194481 / 194480 233 336141 / 336140 New Ratios = 40 Cumulative Total = 108 Ratios of 19 18 19 / 18 19 20 / 19 32 39 / 38 42 57 / 56 48 76 / 75 49 77 / 76 55 96 / 95 63 133 / 132 66 153 / 152 72 171 / 170 74 190 / 189 77 209 / 208 78 210 / 209 87 286 / 285 91 324 / 323 93 343 / 342 96 361 / 360 102 400 / 399 105 456 / 455 107 476 / 475 109 495 / 494 111 513 / 512 127 969 / 968 133 1216 / 1215 137 1331 / 1330 138 1445 / 1444 140 1521 / 1520 141 1540 / 1539 144 1729 / 1728 152 2376 / 2375 155 2432 / 2431 160 2926 / 2925 163 3136 / 3135 164 3250 / 3249 169 4200 / 4199 176 5776 / 5775 178 5929 / 5928 179 5985 / 5984 180 6175 / 6174 182 6860 / 6859 187 10241 / 10240 190 10830 / 10829 195 12636 / 12635 196 13377 / 13376 197 14080 / 14079 198 14365 / 14364 205 23409 / 23408 208 27456 / 27455 210 28900 / 28899 214 43681 / 43680 219 89376 / 89375 221 104976 / 104975 226 165376 / 165375 229 228096 / 228095 234 601426 / 601425 235 633556 / 633555 236 709632 / 709631 240 5909761 / 5909760 New Ratios = 58 Cumulative Total = 166 Ratios of 23 22 23 / 22 23 24 / 23 35 46 / 45 46 69 / 68 47 70 / 69 54 92 / 91 59 115 / 114 68 161 / 160 69 162 / 161 76 208 / 207 81 231 / 230 83 253 / 252 86 276 / 275 89 300 / 299 90 323 / 322 100 391 / 390 101 392 / 391 106 460 / 459 108 484 / 483 110 507 / 506 112 529 / 528 115 576 / 575 121 736 / 735 122 760 / 759 124 875 / 874 125 897 / 896 130 1105 / 1104 132 1197 / 1196 136 1288 / 1287 139 1496 / 1495 145 1863 / 1862 146 2024 / 2023 147 2025 / 2024 150 2185 / 2184 151 2300 / 2299 158 2646 / 2645 159 2737 / 2736 162 3060 / 3059 165 3381 / 3380 166 3520 / 3519 167 3888 / 3887 172 4693 / 4692 173 4761 / 4760 175 5083 / 5082 183 7866 / 7865 184 8281 / 8280 185 8625 / 8624 188 10626 / 10625 191 11271 / 11270 192 11662 / 11661 193 12168 / 12167 200 16929 / 16928 201 19551 / 19550 202 21505 / 21504 203 21736 / 21735 204 23276 / 23275 206 25025 / 25024 207 25921 / 25920 213 43264 / 43263 215 52326 / 52325 216 71875 / 71874 217 75141 / 75140 218 76545 / 76544 220 104329 / 104328 222 122452 / 122451 224 126225 / 126224 225 152881 / 152880 228 202125 / 202124 230 264385 / 264384 231 282625 / 282624 232 328510 / 328509 237 2023425 / 2023424 238 4096576 / 4096575 239 5142501 / 5142500 New Ratios = 74 Cumulative Total = 240 1 Streitberg, Bernd and Klaus Balzer. 1988. The Sound of Mathematics. Proceedings of the 1988 International Computer Music Conference: 158-165. 2 ibid. 3 ibid. 4 This program is an update of one I wrote in 1993 and mentioned in a letter to Ervin Wilson who persuaded me to write up my results for Xenharmonikon 17. 5 Personal communication, Erv Wilson.
Message: 2240 - Contents - Hide Contents Date: Thu, 06 Dec 2001 05:57:46 Subject: Re: The grooviest linear temperaments for 7-limit music From: genewardsmith --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:> Ok. Maybe I don't have good argument for that. Try > > step^3 * exp((cents/k)^2)This looks like hyper-exponential growth penalizing badness, not goodness.
Message: 2241 - Contents - Hide Contents Date: Thu, 06 Dec 2001 21:46:07 Subject: Re: The grooviest linear temperaments for 7-limit music From: genewardsmith --- 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 vector to MOS script * [with cont.] (Wayb.)>. Going from temperaments to > 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 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.
Message: 2242 - Contents - Hide Contents 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
Message: 2243 - Contents - Hide Contents Date: Thu, 06 Dec 2001 23:17:28 Subject: A Geometric Algebra tutorial for Matlab From: genewardsmith 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.
Message: 2244 - Contents - Hide Contents 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
Message: 2245 - Contents - Hide Contents 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
Message: 2246 - Contents - Hide Contents 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
Message: 2247 - Contents - Hide Contents 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
Message: 2248 - Contents - Hide Contents 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 = 1/2You 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 9556Oh 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 26079So this _isn't_ 46+34??
Message: 2249 - Contents - Hide Contents 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 * [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. 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 * [with cont.] (Wayb.)>. 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
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