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