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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
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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> Ennealimmal
You win! But somewhere out there, I wonder . . . What are some manageable MOSs of this?
> (12) [2,-2,1,8,4,-8] <16/15,49/48>
JI or Planar (ditto)
> > (13) [0,0,3,7,-5,0] <10/9,16/15> JI " > > (14) [6,5,3,-7,12,-6] <49/48,126/125> Pretty good for not having a > name--"septimal kleismic" maybe?
Please post details. Is this Dave Keenan's chain-of-minor-thirds thingy? It loses on tetrachordality.
> (15) [0,5,0,-14,0,8] <28/27,49/48>
JI or Planar
> > (16) [6,-7,-2,15,20,-25] <225/224,1029/1024> Miracle > > (17) [2,-4,-4,2,12,-11] <50/49,64/63> Paultone > > (18) [2,-2,-2,1,9,-8] <16/15,50/49>
JI or Planar
> > (19) [10,9,7,-9,17,-9] <126/125,1728/1715> This one should have a > name if it doesn't already. If I call it "nonkleismic" will that > force someone to come up with a good one?
Is this Graham's #1 7-limit? And he missed ennealimmal because . . . ?
> (20) [1,4,-2,-16,6,4] <36/35,64/63> Looks suspiciously like 12-et > meantone.
What's the generator? Where is Huygens meantone in all this?
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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
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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 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)?
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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.
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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!


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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


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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.
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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.
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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.
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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


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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.
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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.


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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.
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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.
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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
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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.


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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
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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
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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
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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
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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 = 1/2
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??
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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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