Tuning-Math Digests messages 6606 - 6630

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Message: 6606

Date: Sun, 9 Mar 2003 15:07:32

Subject: good 11-limit meantones

From: monz

hey all,


i've found some interesting meantones which give
the lowest error-from-JI values thru the 11-limit:
11/48-, 8/35-, and 3/13-comma are three representative
examples.

can anyone (Gene?) give some data on these?
if it's been done before, please supply links.
thanks.




-monz


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Message: 6607

Date: Sun, 9 Mar 2003 15:10:59

Subject: Re: good 11-limit meantones

From: monz

> From: "monz" <monz@xxxxxxxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Sunday, March 09, 2003 3:07 PM
> Subject: [tuning-math] good 11-limit meantones
>
>
> i've found some interesting meantones which give
> the lowest error-from-JI values thru the 11-limit:
> 11/48-, 8/35-, and 3/13-comma are three representative
> examples.
> 
> can anyone (Gene?) give some data on these?
> if it's been done before, please supply links.
> thanks.



2/9-comma is also quite good in 11-limit, and in fact
is the one that got me started on this.



-monz


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Message: 6608

Date: Sun, 9 Mar 2003 16:07:11

Subject: Re: good 11-limit meantones

From: monz

----- Original Message ----- 
From: "monz" <monz@xxxxxxxxx.xxx>
To: <tuning-math@xxxxxxxxxxx.xxx>
Sent: Sunday, March 09, 2003 3:07 PM
Subject: [tuning-math] good 11-limit meantones


> hey all,
> 
> 
> i've found some interesting meantones which give
> the lowest error-from-JI values thru the 11-limit:
> 11/48-, 8/35-, and 3/13-comma are three representative
> examples.




as for equal-temperaments which fall in this range,
31edo is pretty darn good for a low-cardinality EDO,
generator 2^(18/31).

2^(79/136) is very good, 2^(97/167) better still,
and 2^(176/303) really fantastic.

feedback appreciated.



-monz


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Message: 6609

Date: Mon, 10 Mar 2003 08:56:18

Subject: Re: good 11-limit meantones

From: monz

hi Graham,


> From: "Graham Breed" <graham@xxxxxxxxxx.xx.xx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Monday, March 10, 2003 2:19 AM
> Subject: [tuning-math] Re: good 11-limit meantones
>
>
> monz wrote:
> 
> > 2^(79/136) is very good, 2^(97/167) better still,
> > and 2^(176/303) really fantastic.
> 
> What mapping are you using for 136?  


i made the generator 2^(79/136), and the mapping
of ratios to generators follows table i put at
the bottom of this webpage:

Definitions of tuning terms: meantone-from-JI error, (c) 2003 by Joe Monzo *

gen. ratio
-18  16/11   
-17  12/11   
-16  18/11
... 
-10   8/7
-9   12/7
-8   14/11
... 
-6   10/7
... 
-4    8/5
-3    6/5
... 
-1    4/3 
... 
+1    3/2 
... 
+3    5/3
+4    5/4 
... 
+6    7/5 
...
+8   11/7
+9    7/6 
+10   7/4 
... 
+16  11/9
+17  11/6 
+18  11/8
               

> The red line shows 3, so remember 9 is twice as far out.  Why aren't 
> ratios of 9 included in your applet?  I can see that 3/13-comma meantone 
> would be close to the minimax for the simpler mapping if you ignore 9:8, 
> 10:9 and 9:7.


oops ... i'm always thinking in terms of prime-numbers,
and including ratios according to prime-limit rather
than odd-limit.  i guess i should add ratios of 9,
but that will be a lot of work as i'll have to redo
every graph.



-monz


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Message: 6610

Date: Mon, 10 Mar 2003 10:19:39

Subject: Re: good 11-limit meantones

From: Graham Breed

monz wrote:

> 2^(79/136) is very good, 2^(97/167) better still,
> and 2^(176/303) really fantastic.

What mapping are you using for 136?  I can find two fairly good meantones

[136, 215, 316, 382, 470]
[136, 215, 316, 381, 470]

Both have a fifth of 79 steps, which matches your generator.  And both 
have a worst 11-limit error of 11.7 cents.  For 31-equal, this is only 
11.1 cents.

There are two relatively simple 11-limt meantone mappings.  This one, 
consistent with 31- and 43-equal:

   1   0
   2  -1
   4  -4
   7 -10
  11 -18

and this one

   1   0
   2  -1
   4  -4
   7 -10
  -2  13

consistent with 31 and 50.  The first one optimizes with a fourth of 
503.3 cents and a worst 11-limit error of 11.0 cents.  That's 
0.2437-comma meantone, you can find rational approximations I'm sure. 
The other one optimizes at 502.4 cents (exactly 1/4-comma by the looks 
of it) and a worst error of 10.8 cents.

My approximation graphs are here:

Meantone temperaments *

The red line shows 3, so remember 9 is twice as far out.  Why aren't 
ratios of 9 included in your applet?  I can see that 3/13-comma meantone 
would be close to the minimax for the simpler mapping if you ignore 9:8, 
10:9 and 9:7.


                          Graham


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Message: 6613

Date: Wed, 12 Mar 2003 09:23:03

Subject: Rothenberg giveaway

From: Carl Lumma

All;

I've got a spare copy of Rothenberg's three seminal, and very
mathy papers on scale theory.  The first person to write off-list
with a valid snail address and legible sentence saying why you're
interested, gets it.

-Carl


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Message: 6620

Date: Sat, 22 Mar 2003 18:43:19

Subject: Re: More on MOS/temperaments

From: Carl Lumma

Hmm, something's amiss.  Anybody else get this list sent to
them by e-mail?  I got msg. 6056 but not 6057-6067.  Just got
6068 & 9.

>we already started on this in a series of posts, in particular we 
>were looking at scales where the major tetrad and the minor tetrad 
>arise from the same pattern of scale steps.

!  Just when I was thinking, "when would I ever be interested
in this incomplete-chain extra-intervals stuff"...  Good looking
out, Paul!

More later.

-C.


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Message: 6621

Date: Sat, 22 Mar 2003 22:24:22

Subject: Re: this T[n] business

From: Carl Lumma

>>>So it seems my assertion is wrong; simple ratios don't tend
>>>to be bigger.
>>
>>that's the great thing about tenney complexity (as opposed to
>>farey, mann, etc.)!
>
>Ah, you've said that before, I think!

I've just verified this for the Farey series.

-Carl


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Message: 6622

Date: Sat, 22 Mar 2003 19:12:26

Subject: Re: More on MOS/temperaments

From: Carl Lumma

>>And recall, this is only one comma!
> 
>i'm not sure what you mean to imply by that -- and of course this 
>chroma is only defined plus or minus any arbitrary number of
>commatic unison vectors.

Maybe Gene's alluding to the planar and higher cases.

-Carl


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Message: 6623

Date: Sat, 22 Mar 2003 22:27:54

Subject: T[n] where n is small

From: Carl Lumma

Gene,

What about turning this on scales, n < 11?

I don't know how many lines of maple you do this with,
but if they're few you can post them here and I can
either translate to Scheme or run them in maple myself.

-Carl


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Message: 6624

Date: Sat, 22 Mar 2003 19:14:57

Subject: Re: 12 notes with 36/35 a chromatic comma

From: Carl Lumma

>Below I give the name, the wedgie, a TM reduced scale, major and
>minor tetrads going around this scale.

Bingo!

-C.


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