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Message: 4325 - Contents - Hide Contents

Date: Tue, 19 Mar 2002 08:42:02

Subject: Re: Weighting complexity (was: 32 best 5-limit linear temperaments)

From: genewardsmith

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> So Gene, > > How's the 5-limit list coming along.
Here's what I did for weighting: The rms weights could be regarded as factors of 3^(-1/2) times each of the errors of 3,5, and 5/3; when squared this becomes dividing the sum of squares by three. The new proposed weights were 1/ln(3), 1/ln(5) and 1/ln(5), suggesting we adjust things by a factor of 3*3^(-1/2)/(ln(3)^-2 + 2 ln(5)^-2). That's the origin of the mysterious weight factor. What do you think should be done?
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Message: 4327 - Contents - Hide Contents

Date: Wed, 20 Mar 2002 19:56:47

Subject: Re: A common notation for JI and ETs

From: gdsecor

--- In tuning-math@y..., manuel.op.de.coul@e... wrote:
> [Secor]
>>> That's something that I don't like about the Sims notation -- down >>> arrows used in conjunction with sharps, and up arrows with flats. > > [Keenan]
>> I think Manuel exempts sharps and flats from this criticism. >
> Yes indeed, for example, Eb/ is always the nearest tone to 6/5 > as E\ is always nearest to 5/4. > > Manuel
My objection is to alterations used in conjunction with sharps and flats that alter in the opposite direction of the sharp or flat by something approaching half of a sharp or flat. For example, 3/7 or 4/7 is much clearer than 1 minus 4/7 or 1 minus 3/7, but I would not object to 1 minus 2/7 (instead of 5/7). I have been dealing with this issue in evaluating ways to notate ratios such as 11/9, 16/13, 39/32, and 27/22, and I am persuaded that the alterations for these should not be in the opposite direction from an associated sharp or flat. In other words, relative to C, I would prefer to see these as varieties of E-semiflat rather than E- flat with varieties of semisharps. But (for other intervals) something no larger than 2 Didymus commas (~43 cents or ~3/8 apotome) altering in the opposite direction would be okay with me. --George
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Message: 4328 - Contents - Hide Contents

Date: Wed, 20 Mar 2002 20:40:34

Subject: Re: A common notation for JI and ETs

From: gdsecor

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:
>> The comma is 4095:4096 (~0.423 cents). ... > >> It does not vanish in 19, 27, >> 50, 58, 72, 149, 159, or 198, and this seems to be due to >> inconsistencies in those ET's. >
> No. Every one of these (except I didn't check 198) is > {1,3,5,7,9,13}-consistent. >
>> For those systems under 100 in which >> it does not vanish, I don't think that ratios of 13 will be used to >> define their notation, so this should not be a problem. >
> Well Gene and I are already using the 13-comma (1024:1053) to notate > 27-tET, and it looks like it would be pretty useful in 50-tET too. > There are many others under 100-tET where 4095:4096 doesn't vanish. > 37-tET is another such, where I was planning to use the 13-comma for > notation. 37-tET is {1,3,5,7,13} consistent.
I spent some time wrestling with 27-ET last night, and it proved to be a formidable opponent that severely limited my options. There is one approach that allows me to do it justice (using 13 -- what else is there?) that also takes the following into account: Yahoo groups: /tuning-math/message/3768 * [with cont.] With this it looks as if I am going to be stopping at the 17 limit, with intervals measurable in degrees of 183-ET. Once I have made a final decision regarding the symbols, I hope to have something to show you in about a week or so. --George
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Message: 4329 - Contents - Hide Contents

Date: Wed, 20 Mar 2002 03:34:36

Subject: Re: A common notation for JI and ETs

From: genewardsmith

--- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:

> Good work. Did you (or can you) prove that the smallest on each of John's > lists is the smallest there is?
Baker's theorem would give an effective bound, but not a good one. Proving that the seemingly smallest one is in fact the smallest one looks to me, as a number theorist, to be a difficult problem in number theory, though certainly not hard in the 3-limit. Maybe I should try showing 81/80 is the smallest in the 5-limit.
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Message: 4330 - Contents - Hide Contents

Date: Wed, 20 Mar 2002 21:38:04

Subject: Re: Gould Article

From: paulerlich

--- In tuning-math@y..., Mark Gould <mark.gould@a...> wrote:

> It appears that I need to join Tuning Math, as for some unknown reason, > pleople have mentioned a certain article.
unknown reason . . . well, you asked on the tuning list for reactions to your article, and since most of the generalizing-diatonicity work goes on over here, it's not too surprising that it first got mentioned here. if you're interested in generalizing diatonicity to higher harmonic limits, you're in the right place. hope you will participate and be patient during the weeks/months/years it often takes people to learn one another's languages.
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Message: 4331 - Contents - Hide Contents

Date: Wed, 20 Mar 2002 04:50:04

Subject: Re: A common notation for JI and ETs

From: dkeenanuqnetau

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote: >
>> Good work. Did you (or can you) prove that the smallest on each of John's >> lists is the smallest there is? >
> Baker's theorem would give an effective bound, but not a good one. > Proving that the seemingly smallest one is in fact the smallest one > looks to me, as a number theorist, to be a difficult problem in
number theory, though certainly not hard in the 3-limit. Maybe I should try showing 81/80 is the smallest in the 5-limit.
>
If it looks hard, forget it. I'm sure you've got better things to do with your time.
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Message: 4332 - Contents - Hide Contents

Date: Wed, 20 Mar 2002 07:01:15

Subject: Re: A common notation for JI and ETs

From: genewardsmith

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> If it looks hard, forget it. I'm sure you've got better things to do > with your time.
If I *could* prove it, it might be enough for a paper.
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Message: 4333 - Contents - Hide Contents

Date: Thu, 21 Mar 2002 22:51:59

Subject: Re: Hemithirds no 3s

From: Herman Miller

On Thu, 21 Mar 2002 08:47:09 -0000, "genewardsmith"
<genewardsmith@xxxx.xxx> wrote:

>If anyone wants to experiment with a well-in-tune, 12 or 13 or even 6 note scale, they might try septimal hemithirds, with 3136/3125 >as its comma. The deal is that the third (of course) takes two steps, and the 7/4 five steps. A comparable no 5s system would be no 5s >miracle, with 1029/1024 as a comma, I suppose.
Hmm, it seems to be compatible with 37-ET as well. I'm really going to have to start playing around with 37, if I can find the time; it looks like it has some good commas.
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Message: 4334 - Contents - Hide Contents

Date: Thu, 21 Mar 2002 23:09:29

Subject: Re: 25 best weighted generator steps 5-limit temperaments

From: Herman Miller

On Thu, 21 Mar 2002 20:47:00 -0000, "paulerlich" <paul@xxxxxxxxxxxxx.xxx>
wrote:

>--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote: >> 16875/16384 >>
>> map [[0, -4, 3], [1, 2, 2]] >> >> generators 126.2382718 1200 >> >> badness 716.2095928 rms 5.942562596 weighted_g 4.939565964 >
>someone tell me why i should care about this one.
It could have melodic uses in 19-ET, dividing major thirds into three equal parts. A 9-note scale is a pretty useful size for a basic scale. C Db D# E F F# G Ab A# B C
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Message: 4335 - Contents - Hide Contents

Date: Thu, 21 Mar 2002 00:43:32

Subject: Re: Gould Article

From: dkeenanuqnetau

--- In tuning-math@y..., Mark Gould <mark.gould@a...> wrote:
> It appears that I need to join Tuning Math, as for some unknown reason, > pleople have mentioned a certain article.
Paul, I think Mark is being facetious here. :-)
> Please feel free to lambast/berate/vilify etc, for any inaccuracies in the > theory within. Hi Mark,
It seems we must first berate you for failing to remove the quote of an entire tuning-math digest from the end of your post. :-) Regards, -- Dave Keenan
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Message: 4336 - Contents - Hide Contents

Date: Thu, 21 Mar 2002 01:20:16

Subject: Re: A common notation for JI and ETs

From: dkeenanuqnetau

--- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:
> My objection is to alterations used in conjunction with sharps and > flats that alter in the opposite direction of the sharp or flat by > something approaching half of a sharp or flat. For example, 3/7 or > 4/7 is much clearer than 1 minus 4/7 or 1 minus 3/7, but I would not > object to 1 minus 2/7 (instead of 5/7).
Aha! I'm glad you've clarified that. I assume then that you would just barely allow 1 minus 1/3, since that's closer to 2/7 than 3/7.
> I have been dealing with this issue in evaluating ways to notate > ratios such as 11/9, 16/13, 39/32, and 27/22, and I am persuaded that > the alterations for these should not be in the opposite direction > from an associated sharp or flat. In other words, relative to C, I > would prefer to see these as varieties of E-semiflat rather than E- > flat with varieties of semisharps. But (for other intervals) > something no larger than 2 Didymus commas (~43 cents or ~3/8 apotome) > altering in the opposite direction would be okay with me.
I totally agree, when the 242:243 or 507:512 vanishes (as in many ETs), but I don't see how it is possible when notating strict ratios. If two of the above come out as E] and E}, then the other two must be Eb{ and Eb[, where ] and } represent an increase, and { and [ a decrease, by the 11 and 13 comma respectively. Also which ever have no sharp or flat from F,C or G _will_ have a sharp or flat from A, E or B and vice versa.
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Message: 4337 - Contents - Hide Contents

Date: Thu, 21 Mar 2002 02:05:08

Subject: Re: A common notation for JI and ETs

From: dkeenanuqnetau

--- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:
> I spent some time wrestling with 27-ET last night, and it proved to > be a formidable opponent that severely limited my options. There is > one approach that allows me to do it justice (using 13 -- what else > is there?) ...
The only other option I could see was to notate it as every fourth note of 108-ET (= 9*12-ET), using a trinary notation where the 5-comma is one step, the 7-comma is 3 steps, and the apotome is 9 steps, but that would be to deny that it has a (just barely) usable fifth of its own.
> With this it looks as if I am going to be stopping at the 17 limit,
This might be made to work for ETs, but not JI. The 16:19:24 minor triad has a following.
> with intervals measurable in degrees of 183-ET.
I don't understand how this can work.
> Once I have made a > final decision regarding the symbols, I hope to have something to > show you in about a week or so.
I'm more interested in the sematics than the symbols at this stage. I wouldn't spend too much time on the symbols yet. I expect serious problems with the semantics.
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Message: 4338 - Contents - Hide Contents

Date: Thu, 21 Mar 2002 05:14:12

Subject: 25 best weighted generator steps 5-limit temperaments

From: genewardsmith

Yet another "best" list. This one uses weights of 1/ln(3), 1/ln(5) and
1/ln(5) times the steps to 3, 5, and 5/3. To get it to more or less correspond to the previous g, I adjusted by a factor of

(1/sqrt(3) + 1/sqrt(3) + 1/sqrt(3)) / (sqrt(1/ln(3) + 2 sqrt(1/ln(5))

This isn't the only adjustment you might think of, so your milage may vary. I did a search for badness (adjusted) less than 750, rms less
than 35 (as suggested by Dave) and adjusted g less than 20. Here's the
result:


135/128

map   [[0, -1, 3], [1, 2, 1]]

generators   522.8623453   1200

badness   347.2957065   rms   18.07773298   weighted_g   2.678254330


256/243

map   [[0, 0, 1], [5, 8, 10]]

generators   395.3362123   240

badness   612.7594631   rms   12.75974156   weighted_g   3.634818387


25/24

map   [[0, 2, 1], [1, 1, 2]]

generators   350.9775007   1200

badness   134.9517606   rms   28.85189698   weighted_g   1.672379084


648/625

map   [[0, -1, -1], [4, 8, 11]]

generators   505.8656425   300

badness   536.5364242   rms   11.06006024   weighted_g   3.647096372


16875/16384

map   [[0, -4, 3], [1, 2, 2]]

generators   126.2382718   1200

badness   716.2095928   rms   5.942562596   weighted_g   4.939565964


250/243

map   [[0, -3, -5], [1, 2, 3]]

generators   162.9960265   1200

badness   363.8269146   rms   7.975800816   weighted_g   3.573058894


128/125

map   [[0, -1, 0], [3, 6, 7]]

generators   491.2018553   400

badness   198.0597092   rms   9.677665980   weighted_g   2.735322279


3125/3072

map   [[0, 5, 1], [1, 0, 2]]

generators   379.9679494   1200

badness   368.6051706   rms   4.569472316   weighted_g   4.320809550


20000/19683

map   [[0, 4, 9], [1, 1, 1]]

generators   176.2822703   1200

badness   565.4935338   rms   2.504205191   weighted_g   6.089559954


81/80

map   [[0, -1, -4], [1, 2, 4]]

generators   503.8351546   1200

badness   81.02783490   rms   4.217730124   weighted_g   2.678254330


2048/2025

map   [[0, -1, 2], [2, 4, 3]]

generators   494.5534684   600

badness   167.3579339   rms   2.612822498   weighted_g   4.001094414


78732/78125

map   [[0, 7, 9], [1, -1, -1]]

generators   442.9792974   1200

badness   412.2839142   rms   1.157498146   weighted_g   7.088571030


393216/390625

map   [[0, 8, 1], [1, -1, 2]]

generators   387.8196732   1200

badness   373.3878885   rms   1.071950166   weighted_g   7.036043951


2109375/2097152

map   [[0, 7, -3], [1, 0, 3]]

generators   271.5895996   1200

badness   340.7350960   rms   .8004099292   weighted_g   7.522602841


4294967296/4271484375

map   [[0, -9, 7], [1, 2, 2]]

generators   55.27549315   1200

badness   687.5758107   rms   .4831084292   weighted_g   11.24843211


15625/15552

map   [[0, 6, 5], [1, 0, 1]]

generators   317.0796754   1200

badness   146.7501955   rms   1.029625097   weighted_g   5.223559222


1600000/1594323

map   [[0, -5, -13], [1, 3, 6]]

generators   339.5088258   1200

badness   252.5491335   rms   .3831037874   weighted_g   8.703150208


1224440064/1220703125

map   [[0, -13, -14], [1, 5, 6]]

generators   315.2509133   1200

badness   497.4863561   rms   .2766026501   weighted_g   12.16115842


10485760000/10460353203

map   [[0, 4, 21], [1, 0, -6]]

generators   475.5422335   1200

badness   441.3527601   rms   .1537673823   weighted_g   14.21152037


6115295232/6103515625

map   [[0, 7, 3], [2, -3, 2]]

generators   528.8539366   600

badness   313.0746567   rms   .1940181460   weighted_g   11.72920349


19073486328125/19042491875328

map   [[0, 1, 1], [19, 24, 38]]

generators   386.2396202   1200/19

badness   544.7739924   rms   .1047837215   weighted_g   17.32370777


32805/32768

map   [[0, -1, 8], [1, 2, -1]]

generators   498.2724869   1200

badness   39.20134011   rms   .1616904714   weighted_g   6.235512615


274877906944/274658203125

map   [[0, -15, 2], [1, 4, 2]]

generators   193.1996149   1200

badness   178.5465796   rms   .6082244804e-1   weighted_g   14.31844579


7629394531250/7625597484987

map   [[0, -2, -3], [9, 19, 28]]

generators   315.6754868   400/3

badness   203.0321497   rms   .2559261582e-1   weighted_g   19.94420419


9010162353515625/9007199254740992

map   [[0, -8, 5], [2, 9, 1]]

generators   437.2581077   600

badness   116.1747349   rms   .1772520822e-1   weighted_g   18.71429401


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Message: 4339 - Contents - Hide Contents

Date: Thu, 21 Mar 2002 08:47:09

Subject: Hemithirds no 3s

From: genewardsmith

If anyone wants to experiment with a well-in-tune, 12 or 13 or even 6 note scale, they might try septimal hemithirds, with 3136/3125
as its comma. The deal is that the third (of course) takes two steps, and the 7/4 five steps. A comparable no 5s system would be no 5s
miracle, with 1029/1024 as a comma, I suppose.


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Message: 4340 - Contents - Hide Contents

Date: Thu, 21 Mar 2002 12:31 +0

Subject: Re: Gould Article

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <B8BE8615.3860%mark.gould@xxxxxxx.xx.xx>
Mark Gould wrote:

> It appears that I need to join Tuning Math, as for some unknown reason, > pleople have mentioned a certain article.
It's an article on tuning and mathematics, isn't it? That suggests it's fairly relevant.
> Please feel free to lambast/berate/vilify etc, for any inaccuracies in > the > theory within.
As you're being negative, I'll suggest your use of "generated scale" is wrong, or at least inconsistent with Carey & Clampitt. You say something like generated scales will only have two step sizes, which is the extra criterion that makes a generated scale MOS or WF. I don't understand either of the second criteria you use for choosing diatonics. (That is the second numbered point on both the lists.) Can you clarify them? Graham
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Message: 4341 - Contents - Hide Contents

Date: Thu, 21 Mar 2002 20:47:00

Subject: Re: 25 best weighted generator steps 5-limit temperaments

From: paulerlich

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> Yet another "best" list. This one uses weights of 1/ln(3), 1/ln(5) and > 1/ln(5) times the steps to 3, 5, and 5/3. To get it to more or less
correspond to the previous g, I adjusted by a factor of
> > (1/sqrt(3) + 1/sqrt(3) + 1/sqrt(3)) / (sqrt(1/ln(3) + 2 sqrt(1/ln (5))
thanks for doing this, gene.
> This isn't the only adjustment you might think of, so your milage
may vary. I did a search for badness (adjusted) less than 750, rms less
> than 35 (as suggested by Dave) and adjusted g less than 20. Here's the > result: > > 256/243 > > map [[0, 0, 1], [5, 8, 10]] > > generators 395.3362123 240 > > badness 612.7594631 rms 12.75974156 weighted_g 3.634818387
blackwood's decatonic system. this is pretty essential, really. i can't believe we forgot all about it all this time. i'd like to see it in the paper.
> 16875/16384 > > map [[0, -4, 3], [1, 2, 2]] > > generators 126.2382718 1200 > > badness 716.2095928 rms 5.942562596 weighted_g 4.939565964
someone tell me why i should care about this one.
> 4294967296/4271484375 > > map [[0, -9, 7], [1, 2, 2]] > > generators 55.27549315 1200 > > badness 687.5758107 rms .4831084292 weighted_g 11.24843211
ditto. perhaps a weighted badness cutoff of 650, then.
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Message: 4342 - Contents - Hide Contents

Date: Thu, 21 Mar 2002 20:48:01

Subject: Re: Hemithirds no 3s

From: paulerlich

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:

> If anyone wants to experiment with a well-in-tune, 12 or 13 or even
6 note scale, they might try septimal hemithirds, with 3136/3125
> as its comma. The deal is that the third (of course) takes two
steps, and the 7/4 five steps. A comparable no 5s system would be no 5s
> miracle, with 1029/1024 as a comma, I suppose.
do we call the latter 'slendric'?
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Message: 4343 - Contents - Hide Contents

Date: Fri, 22 Mar 2002 02:14:10

Subject: Re: 25 best weighted generator steps 5-limit temperaments

From: dkeenanuqnetau

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> Yet another "best" list. This one uses weights of 1/ln(3), 1/ln(5) and > 1/ln(5) times the steps to 3, 5, and 5/3. Thanks Gene. > To get it to more or less
correspond to the previous g, I adjusted by a factor of
> > (1/sqrt(3) + 1/sqrt(3) + 1/sqrt(3)) / (sqrt(1/ln(3) + 2 sqrt(1/ln(5)) > > This isn't the only adjustment you might think of, so your milage may vary.
Indeed. I see no justification for a weighted rms (or any kind of average or mean) that can produce a result which is _outside_ the range of its inputs. Feeding it 1, 1, 1 should give you 1 no matter what the weights are.
>I did a search for badness (adjusted) less than 750, rms less > than 35 (as suggested by Dave) and adjusted g less than 20. Here's the > result:
The fact that you uncovered for the first time that quintuple thirds temperament (256/243), that Paul tells us has actually been used by Blackwood, is I think justification enough for using Paul's weighting of the numbers of generators to favour fifths. Your list contains all the 5-limit temperaments that I think are of some interest (except of course it is missing the degenerates). However there are five temperaments in your list that I think are of no interest whatsoever. If you were to reduce your complexity cutoff so it is between that of parakleismic and 10485760000/10460353203 we would agree on a best 20. In your terms the g value would need to be between 12.2 and 14.2. In terms of the true weighted rms complexity it needs to be between 11.7 and 13.5.
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Message: 4344 - Contents - Hide Contents

Date: Fri, 22 Mar 2002 21:56:02

Subject: Re: A group "best" listt

From: Herman Miller

On Fri, 22 Mar 2002 06:07:25 -0000, "genewardsmith"
<genewardsmith@xxxx.xxx> wrote:

>This was done in a way exactly comparable to the most recent 5-limit >search.
A suggestion: it might be easier to recognize some of the unison vectors if you supply the prime factorization in addition to the ratio. Here's one I didn't recognize right away:
>531441/524288
But expressed as 2^-19 * 3^12, it's clear that this is the familiar Pythagorean comma. As many times as I've seen "531441/524288", it somehow never stuck in my memory.
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Message: 4345 - Contents - Hide Contents

Date: Fri, 22 Mar 2002 03:15:59

Subject: Re: 25 best weighted generator steps 5-limit temperaments

From: dkeenanuqnetau

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> blackwood's decatonic system. this is pretty essential, really. i > can't believe we forgot all about it all this time. i'd like to see > it in the paper.
So would I. According to my badness it rates better than septathirds (4294967296/4271484375) or parakleismic (1224440064/1220703125).
> >> 16875/16384 >>
>> map [[0, -4, 3], [1, 2, 2]] >> >> generators 126.2382718 1200 >> >> badness 716.2095928 rms 5.942562596 weighted_g 4.939565964 >
> someone tell me why i should care about this one.
I'm calling this tertiathirds (was quadrafourths). By my badness it is better than: diminished, pelogic, twin-tertiatenths (semisuper), quintuple-thirds (Blackwood's decatonic) and parakleismic.
>> 4294967296/4271484375 >> >> map [[0, -9, 7], [1, 2, 2]] >> >> generators 55.27549315 1200 >> >> badness 687.5758107 rms .4831084292 weighted_g 11.24843211 > > ditto.
I'm calling this septathirds. By my badness it is marginally better than parakleismic. They both have rms error less than 0.5 c but tertiathirds needs one less generator (in weghted rms terms).
> perhaps a weighted badness cutoff of 650, then.
That won't work for me. The only way Gene and I can agree, when he is using sharp cutoffs and I'm using gentle rolloffs, is if there doesn't happen to be anything near the 3 upper/outer corners of Gene's "acceptance brick". I'm visualising here points in a 3D space where the horizontal dimensions are error and complexity and the vertical dimension is Gene's badness. There are very few choices for Gene's badness, error and complexity cutoffs, that agree with a single badness cutoff for me. Gene's already got badness and error cutoffs that work for me, all he needs to do now is lower his complexity cutoff from 20 to 14 (or 13 if he fixes up his weighted rms calculation). And "best 20" is a nice round number. Here they are by name, in order of weighted complexity. neutral thirds meantone pelogic augmented semiminorthirds (porcupine) quintuple thirds (Blackwood's decatonic) diminished diaschismic magic tertiathirds (quadrafourths) kleismic quartafifths (minimal diesic) schismic wuerschmidt semisixths (tiny diesic) subminor thirds (orwell) quintelevenths (AMT) septathirds (4294967296/4271484375) twin tertiatenths (semisuper) parakleismic (1224440064/1220703125) The fact that no-one's written any music in, or even heard of the last four (or is it the last six?) as 5-limit temperaments until recently, should indicate that we have gone far enough in the direction of increased complexity. As usual, you can experiment with agreement between Gene's and my lists, updated with Gene's latest temperaments, in http://uq.net.au/~zzdkeena/Music/5LimitTemp.xls.zip - Type Ok * [with cont.] (Wayb.)
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Message: 4346 - Contents - Hide Contents

Date: Fri, 22 Mar 2002 04:08:48

Subject: Re: 25 best weighted generator steps 5-limit temperaments

From: paulerlich

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
>> Yet another "best" list. This one uses weights of 1/ln(3), 1/ln (5) > and
>> 1/ln(5) times the steps to 3, 5, and 5/3. > > > Thanks Gene. >
>> To get it to more or less
> correspond to the previous g, I adjusted by a factor of >>
>> (1/sqrt(3) + 1/sqrt(3) + 1/sqrt(3)) / (sqrt(1/ln(3) + 2 > sqrt(1/ln(5)) >>
>> This isn't the only adjustment you might think of, so your milage > may vary. >
> Indeed. I see no justification for a weighted rms (or any kind of > average or mean) that can produce a result which is _outside_ the > range of its inputs. Feeding it 1, 1, 1 should give you 1 no matter > what the weights are. >
>> I did a search for badness (adjusted) less than 750, rms > less
>> than 35 (as suggested by Dave) and adjusted g less than 20. Here's > the >> result: >
> The fact that you uncovered for the first time that quintuple thirds > temperament (256/243), that Paul tells us has actually been used by > Blackwood, is I think justification enough for using Paul's weighting > of the numbers of generators to favour fifths. > > Your list contains all the 5-limit temperaments that I think are of > some interest (except of course it is missing the degenerates). > However there are five temperaments in your list that I think are of > no interest whatsoever. If you were to reduce your complexity cutoff > so it is between that of parakleismic and 10485760000/10460353203 we > would agree on a best 20. In your terms the g value would need to be > between 12.2 and 14.2. In terms of the true weighted rms complexity it > needs to be between 11.7 and 13.5.
a lower complexity seems desirable now but we'll regret it next year when we're working on the 19-limit.
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Message: 4347 - Contents - Hide Contents

Date: Fri, 22 Mar 2002 04:13:25

Subject: Re: 25 best weighted generator steps 5-limit temperaments

From: paulerlich

--- In tuning-math@y..., Herman Miller <hmiller@I...> wrote:
> On Thu, 21 Mar 2002 20:47:00 -0000, "paulerlich" <paul@s...> > wrote: >
>> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote: >>> 16875/16384 >>>
>>> map [[0, -4, 3], [1, 2, 2]] >>> >>> generators 126.2382718 1200 >>> >>> badness 716.2095928 rms 5.942562596 weighted_g 4.939565964 >>
>> someone tell me why i should care about this one. >
> It could have melodic uses in 19-ET, dividing major thirds into three equal > parts. A 9-note scale is a pretty useful size for a basic scale. > > C Db D# E F F# G Ab A# B C
dave and you both? ok, then i think we should keep this in the paper - - it's john negri's system in 19-equal (looks a little better in the 7-limit).
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Message: 4348 - Contents - Hide Contents

Date: Fri, 22 Mar 2002 05:43:17

Subject: Re: 25 best weighted generator steps 5-limit temperaments

From: dkeenanuqnetau

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> a lower complexity seems desirable now but we'll regret it next year > when we're working on the 19-limit.
I have always assumed we would allow higher complexities for higher primes. We might well be looking at stuff with up to 25 rms generators for 19-limit. But 13 is plenty for 5-limit.
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Message: 4349 - Contents - Hide Contents

Date: Fri, 22 Mar 2002 06:00:41

Subject: Re: 25 best weighted generator steps 5-limit temperaments

From: dkeenanuqnetau

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning-math@y..., Herman Miller <hmiller@I...> wrote:
>> On Thu, 21 Mar 2002 20:47:00 -0000, "paulerlich" <paul@s...> >> wrote:
>>> someone tell me why i should care about this one. [tertiathirds] >>
>> It could have melodic uses in 19-ET, dividing major thirds into > three equal
>> parts. A 9-note scale is a pretty useful size for a basic scale. >> >> C Db D# E F F# G Ab A# B C >
> dave and you both? ok, then i think we should keep this in the paper - > - it's john negri's system in 19-equal (looks a little better in the > 7-limit).
If you keep this one (tertiathirds) you're also gonna have to keep the other one you asked about, septathirds (4294967296/4271484375), since septathirds is better than tertiathirds by Gene's badness. But I agree that it's extremely boring melodically, being essentially 22-tET.
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