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Message: 4050 - Contents - Hide Contents

Date: Thu, 7 Mar 2002 14:41 +00

Subject: Re: listing linear temperaments

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <a663hl+fj4c@xxxxxxx.xxx>
Me:


>> All temperaments will contain multiple ETs, it's only a question of 
>> the algorithm being general enough to find them.  They aren't always 
>> consistent.  

Gene:


> An algorithm which will find some of them, consistent or not, is to 
> wedge with commas until you get an et.



It's not finding the ETs from the LT that's difficult.  You can walk the 
scale tree, and one day I will.  No, the difficult bit is finding the LT 
in the first place, and a good pair or ETs is an efficient way of doing 
that.  Checking all mappings of each individual ET may be better in some 
cases.

Another way I forgot to mention is to take all combinations of a set of 
unison vectors.  I've implemented this now, and I find it to be very slow 
when it goes beyond the 11-limit.  It may be improvable, but I think 
checking all versions of inconsistent ETs will be much more productive.


                      Graham
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Message: 4051 - Contents - Hide Contents

Date: Thu, 7 Mar 2002 17:00:54

Subject: Re: some output from Graham's cgi

From: manuel.op.de.coul@xxxxxxxxxxx.xxx

Thanks, I've put them in the archive.
Graham wrote:


> Which ones do you think the best are? 



I haven't taken a close look, I'll leave that to
you, Gene and Paul to decide.



>Are there any plans to add this functionality to Scala?



You mean generating these scales? That's possible now.
But if I get the files ready I don't need to worry about
a good file name, description, and number of generator
steps up and down. You always want to start the cycle on 1/1?
Or maybe a good convention is to have D somewhere in the
middle, if it's octave based.

Manuel
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Message: 4052 - Contents - Hide Contents

Date: Thu, 07 Mar 2002 09:22:48

Subject: Re: Some 58-et reduced bases

From: genewardsmith

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:


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


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


>>> 13-limit: <126/125, 144/143, 176/175, 196/195, 354/363>
>> 


>> <126/125, 144/143, 176/175, 196/195, 364/363>
> 


> i tried the fokker pb of these and it has five exceedingly small 
> steps -- 1375:1372s and 3025:3024s. so a reduction to 53 is implied 
> (as in cassandra, etc.). is there a better basis for holding up 58 in 
> this regard?



Here's another possibility, which I obtained by trying to minimize
Tenney height:

[1, 56/55, 36/35, 26/25, 22/21, 16/15, 14/13, 12/11, 11/10, 10/9, 
9/8,
8/7, 15/13, 7/6, 13/11, 6/5, 40/33, 11/9, 26/21, 5/4, 14/11, 9/7,
13/10, 21/16, 4/3, 27/20, 15/11, 11/8, 7/5, 45/32, 10/7, 13/9, 22/15,
40/27, 3/2, 32/21, 20/13, 14/9, 11/7, 8/5, 21/13, 13/8, 33/20, 5/3,
22/13, 12/7, 26/15, 7/4, 16/9, 9/5, 20/11, 11/6, 13/7, 15/8, 21/11,
25/13, 35/18, 55/28]

The smallest interval here is between 7/5 and 45/32; this could no
doubt be smoothed out more than this, but it seems clear that we can
get something decent in 58 tones, 13-limit JI which is epimorphic.
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Message: 4053 - Contents - Hide Contents

Date: Thu, 7 Mar 2002 17:34 +00

Subject: Re: some output from Graham's cgi

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <OFB9F45FC8.B7945257-ONC1256B75.00573332@xxxxxx.xxxxxxxxx.xx>
Manuel wrote:



> Thanks, I've put them in the archive.



You mean the Pajaras?



> Graham wrote:


>> Which ones do you think the best are? 
> 


> I haven't taken a close look, I'll leave that to
> you, Gene and Paul to decide.



I presume you have some kind of miracles already.  I have Scala files on 
my website.  They're linked to from 
<Miracle Temperament Home Page * [with cont.]  (Wayb.)>.  Currently in 72-equal, although 
I'd prefer them with the 11:8-just generator of 116.755 cents.

I'd rather you include a template for the keyboard mapping, so that it can 
be filled with any generator size.  But I don't think that'll work, as the 
mapping files aren't geared for open-ended tunings.  Perhaps you could 
make it a mode of 31, 41 and 72, but that still doesn't solve the general 
problem.

Also 58 from multiple-29 should be there.  I haven't tuned it up, so I'll 
have to assume minimax is best.  That's two lots of 29-equal 25 cents 
apart.



>> Are there any plans to add this functionality to Scala?
> 


> You mean generating these scales? That's possible now.



I was thinking of the searches and optimizations.  Is the documentation 
online?  Ah yes, <Scala help * [with cont.]  (Wayb.)>.  Well, 
BISTEP and CALCULATE/LEASTSQUARE seem to do the optimization bit.



> But if I get the files ready I don't need to worry about
> a good file name, description, and number of generator
> steps up and down. You always want to start the cycle on 1/1?
> Or maybe a good convention is to have D somewhere in the
> middle, if it's octave based.



It doesn't matter where you start the cycle, or exactly what size 
generator you choose.  I'm counting up from the root by default.  For 
people who have Scala, simple instructions for generating the scales are 
probably better than static files, so that they can play around with these 
things.  Although for people who don't have Scala, the lists of cents 
might be useful.

I've counted 95 files in my copy of the archive including the word 
'meantone'.  Do you want that many for every temperament we come up with?


                  Graham
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Message: 4054 - Contents - Hide Contents

Date: Thu, 07 Mar 2002 11:18:28

Subject: Filling the idea pool with Tenney reduced scales

From: genewardsmith

This business of finding the Tenney-reduced epimorphic scale for
various prime limits and numbers of steps looks like another useful
project. Here is 11-limit, 41-et:

[1, 56/55, 28/27, 21/20, 15/14, 12/11, 10/9, 9/8, 8/7, 7/6, 25/21, 
6/5, 11/9, 5/4, 14/11, 9/7, 21/16, 4/3, 15/11, 11/8, 7/5, 10/7, 16/11,
22/15, 3/2, 32/21, 14/9, 11/7, 8/5, 18/11, 5/3, 27/16, 12/7, 7/4,
16/9, 9/5, 11/6, 15/8, 21/11, 27/14, 49/25]

By way of comparison, here is Genesis Minus:

[1, 81/80, 33/32, 21/20, 16/15, 12/11, 10/9, 9/8, 8/7, 7/6, 32/27, 
6/5, 11/9, 5/4, 14/11, 9/7, 21/16, 4/3, 27/20, 11/8, 7/5, 10/7, 16/11,
40/27, 3/2, 32/21, 14/9, 11/7, 8/5, 18/11, 5/3, 27/16, 12/7, 7/4,
16/9, 9/5, 11/6, 15/8, 40/21, 64/33, 160/81]
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Message: 4055 - Contents - Hide Contents

Date: Thu, 07 Mar 2002 11:27:37

Subject: Re: omigawd

From: genewardsmith

--- In tuning-math@y..., Carl Lumma <carl@l...> wrote:


> There is sooo much stuff in the idea pool here... will we
> drown?  I hope somebody is on top of it all.
> 
> Capstone temperaments... there is only one per limit, right?
> 
> 5 = meantone
> 7 = ennealimmal
> 11 = hemiennealimmal
> 13 = It's too late at night to worry about 13.

> Gene, I have your top 20 for steps^3, but not steps^2 (you
> once said you had a list of 505 here...).



I do, but it's not a complete list like my 5-limit list.
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Message: 4056 - Contents - Hide Contents

Date: Thu, 07 Mar 2002 13:41:03

Subject: Re: Filling the idea pool with Tenney reduced scales

From: paulerlich

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


> This business of finding the Tenney-reduced epimorphic scale for 


various prime limits and numbers of steps looks like another useful 
project. Here is 11-limit, 41-et:


> 
> [1, 56/55, 28/27, 21/20, 15/14, 12/11, 10/9, 9/8, 8/7, 7/6, 25/21, 
> 6/5, 11/9, 5/4, 14/11, 9/7, 21/16, 4/3, 15/11, 11/8, 7/5, 10/7, 


16/11, 22/15, 3/2, 32/21, 14/9, 11/7, 8/5, 18/11, 5/3, 27/16, 12/7, 
7/4, 16/9, 9/5, 11/6, 15/8, 21/11, 27/14, 49/25]


> 
> By way of comparison, here is Genesis Minus:
> 
> [1, 81/80, 33/32, 21/20, 16/15, 12/11, 10/9, 9/8, 8/7, 7/6, 32/27, 
> 6/5, 11/9, 5/4, 14/11, 9/7, 21/16, 4/3, 27/20, 11/8, 7/5, 10/7, 


16/11, 40/27, 3/2, 32/21, 14/9, 11/7, 8/5, 18/11, 5/3, 27/16, 12/7, 
7/4, 16/9, 9/5, 11/6, 15/8, 40/21, 64/33, 160/81]

how about an evangelina example: 19-limit, 22-tone where 1216:1215 
and 57:56 are unison vectors?
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Message: 4057 - Contents - Hide Contents

Date: Fri, 08 Mar 2002 22:51:02

Subject: Re: 32 best 5-limit linear temperaments redux

From: dkeenanuqnetau

I don't think we are going to make any progress on this unless we can 
get beyond a badness measure that says the best 5-limit temperament is 
one that takes 49 generators before we get a single fifth (because it 
has such teensy weensy errors).

This badness measure also says that meantone is only 7th best (or 
thereabouts) and thinks that a temperament whose perfect fifth is 758 
cents and whose major third is 442 cents is only slightly worse than 
meantone (because it only needs 2 generators to get one of these 
supposed 1:3:5 chords).

Does anyone really believe this stuff?
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Message: 4058 - Contents - Hide Contents

Date: Fri, 8 Mar 2002 16:39:44

Subject: Yahoo has gone yahoo

From: monz

apologies to everyone for the delay in receiving
my posts for the last few days -- they've finally
just been sent successfully.

the problem was that after the crash that disabled
the Yahoo groups, they also changed their policy
for regular email (and my main account is at Yahoo),
so that it requires "authentication" (whatever that
means).  

my messages have been getting backed up in my emailer
since Wednesday because i didn't know about the change
in procedure until just now.  everything seems to be
fine now.



-monz


 






_________________________________________________________

Do You Yahoo!?

Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.]  (Wayb.)
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Message: 4059 - Contents - Hide Contents

Date: Fri, 08 Mar 2002 22:56:48

Subject: Re: Tenney reduced 22-et epimorphic scales

From: paulerlich

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


> 5-limit:
> 
> 1, 25/24, 16/15, 10/9, 9/8, 32/27, 6/5, 5/4, 32/25, 4/3, 25/18, 
> 45/32, 36/25, 3/2, 25/16, 8/5, 5/3, 27/16, 16/9, 9/5, 15/8, 48/25
> 
> 7-limit:
> 
> 1, 25/24, 15/14, 10/9, 8/7, 7/6, 6/5, 5/4, 9/7, 4/3, 25/18, 7/5, 
> 35/24, 3/2, 14/9, 8/5, 5/3, 12/7, 7/4, 9/5, 15/8, 27/14
> 
> 11-limit:
> 
> 1, 22/21, 15/14, 10/9, 8/7, 7/6, 6/5, 5/4, 9/7, 4/3, 11/8, 7/5, 
> 16/11, 3/2, 11/7, 8/5, 5/3, 12/7, 7/4, 9/5, 15/8, 21/11



it might make more sense to reduce by odd-limit, not tenney height, 
because these represent octave-repeating scales. 27/14 is not truly 
simpler than 48/25 in this context.
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Message: 4060 - Contents - Hide Contents

Date: Fri, 08 Mar 2002 08:09:28

Subject: Re: Filling the idea pool with Tenney reduced scales

From: genewardsmith

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:



> how about an evangelina example: 19-limit, 22-tone where 1216:1215 
> and 57:56 are unison vectors?



This is a bizarre request--1216/1215 comes out a comma if you map
19 to 94, and 57/56 if you map 19 to 93. To get them both to be commas, you need to screw the mapping of 7.
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Message: 4061 - Contents - Hide Contents

Date: Fri, 08 Mar 2002 09:35:25

Subject: 648/625 in the kernel

From: genewardsmith

Here are some 7-limit linear temperaments with 648/625 in the kernel:


wedgie   [4, 4, 4, -2, 5, -3]

map   [[0, 1, 1, 1], [4, 5, 8, 10]]

generators   385.6982078   300

bad   153.0959406   rms   19.13699259   g   2.828427124



wedgie   [0, 0, 4, 9, -6, 0]

map   [[0, 0, 0, -1], [4, 6, 9, 12]]

generators   293.9303312   300

bad   507.2211586   rms   63.40264486   g   2.828427124



wedgie   [8, 8, 4, -13, 16, -6]

map   [[0, -2, -2, -1], [4, 8, 11, 12]]

generators   254.9108878   300

bad   517.3802185   rms   17.63796199   g   5.416025604



wedgie   [12, 12, 8, -15, 21, -9]

map   [[0, -3, -3, -2], [4, 12, 15, 15]]

generators   568.6218810   300

bad   639.9052435   rms   9.998519430   g   8.



wedgie   [0, 0, 12, 28, -19, 0]

map   [[0, 0, 0, -1], [12, 19, 28, 36]]

generators   227.2636655   100

bad   708.5378231   rms   9.840803104   g   8.485281372



wedgie   [4, 4, -8, -30, 24, -3]

map   [[0, 1, 1, -2], [4, 5, 8, 14]]

generators   408.4361510   300

bad   867.7149165   rms   13.55804557   g   8.



wedgie   [4, 4, 16, 26, -14, -3]

map   [[0, 1, 1, 4], [4, 5, 8, 6]]

generators   391.9922675   300

bad   878.5571468   rms   9.151636944   g   9.797958972



wedgie   [16, 16, 12, -17, 26, -12]

map   [[0, -4, -4, -3], [4, 14, 17, 17]]

generators   576.5076868   300

bad   1045.372749   rms   9.116622806   g   10.70825227



wedgie   [8, 8, -8, -41, 35, -6]

map   [[0, 1, 1, -1], [8, 10, 16, 25]]

generators   388.2433283   150

bad   1184.484300   rms   10.09503666   g   10.83205120



wedgie   [12, 12, -4, -43, 40, -9]

map   [[0, 3, 3, -1], [4, 4, 7, 12]]

generators   231.3421146   300

bad   1228.221946   rms   9.031043720   g   11.66190379



wedgie   [4, 4, 12, 17, -8, -3]

map   [[0, 1, 1, 3], [4, 5, 8, 7]]

generators   425.9743487   300

bad   1246.939230   rms   24.61064270   g   7.118052167



wedgie   [8, 8, -4, -32, 29, -6]

map   [[0, 2, 2, -1], [4, 5, 8, 12]]

generators   205.9097122   300

bad   1270.415456   rms   17.64465912   g   8.485281372



wedgie   [8, 8, 16, 15, -3, -6]

map   [[0, -1, -1, -2], [8, 16, 22, 29]]

generators   490.5870474   150

bad   1314.171075   rms   15.40044228   g   9.237604309



wedgie   [8, 8, 20, 24, -9, -6]

map   [[0, 2, 2, 5], [4, 5, 8, 8]]

generators   193.5062010   300

bad   1426.736685   rms   10.49071092   g   11.66190379
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Message: 4062 - Contents - Hide Contents

Date: Fri, 08 Mar 2002 10:44:38

Subject: Tenney reduced 22-et epimorphic scales

From: genewardsmith

5-limit:

1, 25/24, 16/15, 10/9, 9/8, 32/27, 6/5, 5/4, 32/25, 4/3, 25/18, 
45/32, 36/25, 3/2, 25/16, 8/5, 5/3, 27/16, 16/9, 9/5, 15/8, 48/25

7-limit:

1, 25/24, 15/14, 10/9, 8/7, 7/6, 6/5, 5/4, 9/7, 4/3, 25/18, 7/5, 
35/24, 3/2, 14/9, 8/5, 5/3, 12/7, 7/4, 9/5, 15/8, 27/14

11-limit:

1, 22/21, 15/14, 10/9, 8/7, 7/6, 6/5, 5/4, 9/7, 4/3, 11/8, 7/5, 
16/11, 3/2, 11/7, 8/5, 5/3, 12/7, 7/4, 9/5, 15/8, 21/11
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Message: 4063 - Contents - Hide Contents

Date: Sat, 09 Mar 2002 13:37:40

Subject: Re: 32 best 5-limit linear temperaments redux

From: Carl Lumma


>What you're saying is that the search was too broad, it seems. That
>could be rectified if there was general agreement it is so by the
>simple expedient of leaving off the extremes.



I for one do not understand steps*cents for linear temperaments.
Shouldn't it be g*cents?

-Carl
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Message: 4064 - Contents - Hide Contents

Date: Sat, 09 Mar 2002 13:34:45

Subject: Re: omigawd

From: Carl Lumma


>> >here is sooo much stuff in the idea pool here... will we
>> drown?  I hope somebody is on top of it all.
>


>bless you carl for taking an interest. i mean that with all sincerity!

Thanks, Paul.

-Carl
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Message: 4065 - Contents - Hide Contents

Date: Sat, 09 Mar 2002 00:39:30

Subject: Re: 32 best 5-limit linear temperaments redux

From: genewardsmith

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



> I don't think we are going to make any progress on this unless we can 
> get beyond a badness measure that says the best 5-limit temperament is 
> one that takes 49 generators before we get a single fifth (because it 
> has such teensy weensy errors).



This is just like saying we should not regard 2460 as a super-good 
5-limit
scale because its errors are so small that it could make no practical
difference if they were larger, and that given the choice between 53
tones and 2460, 53 seems much more practical. This misses the point,
which is that 2460 is very, very good compared to other things *in its
size range*. If you compare wildly different values of "g", you are
getting into apples and elephants.



> This badness measure also says that meantone is only 7th best (or 
> thereabouts) and thinks that a temperament whose perfect fifth is
758 
> cents and whose major third is 442 cents is only slightly worse than

> meantone (because it only needs 2 generators to get one of these 
> supposed 1:3:5 chords).

> Does anyone really believe this stuff?



Paul has pointed out that ultra-funky scales may have more
possibilities than is at first apparent. Again, why compare apples
with e coli?
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Message: 4066 - Contents - Hide Contents

Date: Sat, 09 Mar 2002 10:05:16

Subject: Re: 32 best 5-limit linear temperaments redux

From: dkeenanuqnetau

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


> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> 
> You didn't give a list of e coli, a list of apples 


>> and a list of elephants, but only the "32 best 5-limit linear 
>> temperaments".
> 


> Is it the subject line you object to?



No it's the badness measure. I actually _want_ a badness measure that 
compares _all_ 5-limit temperaments irrespective of their e-coli-ness 
or elephant-ness, but actually takes into account that e-coli and 
elephants are inherently less interesting or useful than apples (but 
in a continuous manner not a discrete one as the analogy of e-coli, 
apples and elephants would suggest).

For 5-limit I'm currently using:

badness = wtd_rms_gens*EXP((rms_error/7.4_cents)^0.5)

Where wtd_rms_gens are weighted by log of odd limit.

I think this approaches your badness in the limit where the power (0.5 
above) goes to zero and the 7.4 cents does something else (I forget 
what).
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Message: 4067 - Contents - Hide Contents

Date: Sat, 09 Mar 2002 03:15:29

Subject: Seven and eight note Tenney reduced scales

From: genewardsmith

7 notes

5-limit: 1, 9/8, 5/4, 4/3, 3/2, 5/3, 9/5
7-limit: 1, 8/7, 5/4, 4/3, 3/2, 5/3, 7/4
11-limit: same as 7-limit

8 notes

5-limit 1, 10/9, 6/5, 4/3, 25/18, 3/2, 5/3, 9/5

h8 7-limit (not epimorphic):

1, 7/6, 6/5, 4/3, 9/7, 3/2, 5/3, 9/5

h8 + v7 7-limit (epimorphic):

1, 8/7, 6/5, 4/3, 10/7, 3/2, 5/3, 7/4

Here's a 7-free block for 2,3,5,11 h8:

1, 12/11, 6/5, 4/3, 11/8, 3/2, 18/11, 11/6

This vigorously 11-limit scale was toned down to

1, 10/9, 6/5, 4/3, 11/8, 3/2, 5/3, 9/5

by Tenney reduction.
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Message: 4068 - Contents - Hide Contents

Date: Sat, 09 Mar 2002 23:17:01

Subject: Re: 32 best 5-limit linear temperaments redux

From: dkeenanuqnetau

--- In tuning-math@y..., Carl Lumma <carl@l...> wrote:


>> What you're saying is that the search was too broad, it seems. That
>> could be rectified if there was general agreement it is so by the
>> simple expedient of leaving off the extremes.
> 


> I for one do not understand steps*cents for linear temperaments.
> Shouldn't it be g*cents?



Yes Gene is using gens*cents for linear temperaments, or rather gens^n 
* cents in general (different n>=1 for different limits). I don't like 
abbreviating the number of generators to "g". That's being 
unnecessarily obscure.
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Message: 4069 - Contents - Hide Contents

Date: Sat, 09 Mar 2002 07:56:19

Subject: Re: Tenney reduced 22-et epimorphic scales

From: genewardsmith

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:



> it might make more sense to reduce by odd-limit, not tenney height, 
> because these represent octave-repeating scales. 27/14 is not truly 
> simpler than 48/25 in this context.



I
wanted something which would be better at breaking ties, but it occurs
to me that tenney height itself could be the tie-breaker, so maybe I
should do that instead. Any other comments about this project?
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Message: 4070 - Contents - Hide Contents

Date: Sat, 09 Mar 2002 18:11:57

Subject: Re: 32 best 5-limit linear temperaments redux

From: Carl Lumma


>> > for one do not understand steps*cents for linear temperaments.
>> Shouldn't it be g*cents?
>


>Yes Gene is using gens*cents for linear temperaments, or rather gens^n 
>* cents in general (different n>=1 for different limits). I don't like 
>abbreviating the number of generators to "g". That's being 
>unnecessarily obscure.



Okay, thanks, this does answer my question -- he's using the number of
gens in one instance of the map, as opposed to the number in the et
that provides a near-optimal generator size, or something.

Though my particular suggestion was not only this; g is the rms of
gens in a map, or something.  I'm still not clear exactly how it's
calculated, or if it's different from what Graham calls complexity.
For the record, my preferred complexity measure is...

(/ (- (max map) (min map)) (card map))

...but anything that levels the field for different limits is
fine, and Gene's already been using g, so... 

-Carl
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Message: 4071 - Contents - Hide Contents

Date: Sat, 09 Mar 2002 09:21:39

Subject: Re: 32 best 5-limit linear temperaments redux

From: dkeenanuqnetau

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


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


>> I don't think we are going to make any progress on this unless we 
can 
>> get beyond a badness measure that says the best 5-limit 
temperament is 
>> one that takes 49 generators before we get a single fifth (because 
it 
>> has such teensy weensy errors).
> 


> This is just like saying we should not regard 2460 as a super-good 
> 5-limit scale because its errors are so small that it could make no 


practical difference if they were larger, and that given the choice 
between 53 tones and 2460, 53 seems much more practical. This misses 
the point, which is that 2460 is very, very good compared to other 
things *in its size range*. If you compare wildly different values of 
"g", you are getting into apples and elephants.


> 


>> This badness measure also says that meantone is only 7th best (or 
>> thereabouts) and thinks that a temperament whose perfect fifth is 
758 
>> cents and whose major third is 442 cents is only slightly worse 
than 
>> meantone (because it only needs 2 generators to get one of these 
>> supposed 1:3:5 chords).
> 
>> Does anyone really believe this stuff?
> 


> Paul has pointed out that ultra-funky scales may have more 


possibilities than is at first apparent. Again, why compare apples 
with e coli?

Why indeed? But that's exactly what you're doing. You only gave one 
list, in which a single badness metric compares all the temperamants 
against each other. You didn't give a list of e coli, a list of apples 
and a list of elephants, but only the "32 best 5-limit linear 
temperaments".
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Message: 4072 - Contents - Hide Contents

Date: Sat, 09 Mar 2002 09:39:32

Subject: Re: Tenney reduced 22-et epimorphic scales

From: genewardsmith

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

I wasn't too happy with the odd-limit results--here they are for
8 tones:

5-limit

1, 32/27, 9/8, 4/3, 81/64, 3/2, 16/9, 27/16

7-limit, h8

1, 7/6, 8/7, 4/3, 49/32, 3/2, 7/4, 12/7

7-limit, h8+v7

1, 8/7, 9/8, 4/3, 21/16, 3/2, 16/9, 7/4

None of these are epimorphic, and none use any 5s. I'll try the symmetric octave-class lattice distance next.
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Message: 4073 - Contents - Hide Contents

Date: Sat, 09 Mar 2002 09:41:24

Subject: Re: 32 best 5-limit linear temperaments redux

From: genewardsmith

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

You didn't give a list of e coli, a list of apples 


> and a list of elephants, but only the "32 best 5-limit linear 
> temperaments".



Is it the subject line you object to?
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Message: 4074 - Contents - Hide Contents

Date: Sat, 09 Mar 2002 20:15:21

Subject: Re: 32 best 5-limit linear temperaments redux

From: Carl Lumma


>> >kay, thanks, this does answer my question -- he's using the number of
>> gens in one instance of the map, as opposed to the number in the et
>> that provides a near-optimal generator size, or something.
>


>What's the difference?



Meantone has a very compact 5-limit map.  You only need 4 gens.  In
listening tests I've preferred a generator close to that of 69-et,
though the rms optimum is closer to 31-et IIRC.  In either case, why
should we penalize meantone because it takes 31 or 69 gens to yield
an et with the optimum generator?



>> Though my particular suggestion was not only this; g is the rms of
>> gens in a map, or something.  I'm still not clear exactly how it's
>> calculated, or if it's different from what Graham calls complexity.
>


>It's a different measure of complexity.



What is different from what??

Once again, I'll list my preferred map complexity measure in unambiguous
mathematical notation.  Why don't you and Graham give yours for the
record, so Dave can tell us which one he likes best?

Carl's preferred map complexity measure:
(/ (- (max map) (min map)) (card map))

Gene's preferred map complexity measure:
________________________________________

Graham's preferred map complexity measure:
________________________________________


-Carl
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