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.

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

-monz

> From: "monz" <monz@attglobal.net>
> To: <tuning-math@yahoogroups.com>
> 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

----- Original Message -----
From: "monz" <monz@attglobal.net>
To: <tuning-math@yahoogroups.com>
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

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:

http://x31eq.com/meantone.htm#11lim

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

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> 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.

The first order of business is to decide which of the two extensions
of septimal meantone, both good, we are looking at--what I've called
"meanpop" or "meantone".

"Meantone" uses the map [1,4,10,18] whereas "meanpop" goes
[1,4,10,-13]. 18 - (-13) = 31 and these are the same for the 31-et,
which is one possible choice for the boundry between them.

The 30th row of the Farey sequence for poptimal q-comma meantones
goes 3/13 < 7/30 < 4/17 < 5/21 < 6/25 < 7/29.

The Farey sequence for the 250th row for rational (equal temperament)
poptimal meantones is 83/198 < 96/225 < 13/31

The range of poptimal brats is from 5.892 to 16.083.

The 40th row of the Farey sequence for meanpop goes
10/39 < 9/35 < 8/31.

The 250th row of the Farey sequence for the generator goes
73/174 < 60/143 < 47/112.

The range of poptimal brats is [infinity, -10.4789].

It would seem that the 31-et is the obvious choice for meantone, and
the 112-et, as I've mentioned before, works nicely for meanpop. If we
want good q-comma numbers, 3/13 or 4/17 would be fine for meantone,
and 10/39, 9/35 or 8/31 would have to do for meanpop. So far as brats
go, choosing infinity for meanpop and 6 for meantone would work.

All of these are minimal, in that the poptimal values at least include
the ones given, but might include more (especially under Paul's
definition.)

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> It would seem that the 31-et is the obvious choice for meantone, and
> the 112-et, as I've mentioned before, works nicely for meanpop. If we
> want good q-comma numbers, 3/13 or 4/17 would be fine for meantone,
> and 10/39, 9/35 or 8/31 would have to do for meanpop.

Duh. I forgot 1/4-comma for meanpop! Use that for meanpop, and
3/13-comma for meantone, and everything is kopacetic.

hi Graham,

> From: "Graham Breed" <graham@microtonal.co.uk>
> To: <tuning-math@yahoogroups.com>
> 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:

http://sonic-arts.org/dict/meantone-error/meantone-error.htm

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