Survey IX

<225/224, 1029/1024> reduced

Wedge invariant: [6,-7,-2,15,20,-25] length = 36.592

Ets: 31,41,72

Map:

[ 0 1]
[ 6 1]
[-7 3]
[-2 3]

Generators: a = .09714412267 = 6.994376832 / 72 (secor); b = 1

Errors and 72-et

3: -2.517 -1.955
5: -2.324 -2.980
7: -1.972 -2.159

Measures: 11.3213 sc; 78.2771 s2c

This is, of course, the 7-limit version of Miracle.

<2401/2400, 6144/6125>

Minkowski reduction: <2401/2400, 3136/3125>

Wedge invariant: [16,2,5,6,37,-34] length = 53.3479

Ets: 31,68,99,130

Map:

[ 0 1]
[16 -1]
[ 2 2]
[ 5 2]

Generators: a = .1615916143 = 15.99756982 / 99; b = 1

Errors and 99 et:

3: 0.604 1.075
5: 1.506 1.565
7: 0.724 0.871

Measures: 8.8323 sc; 89.1168 s2c

This beats Miracle in terms of step-cents, but not with step^2 cents.
The fact that so much 5 and 7 is available is the reason why. I was a
little disturbed to find that not only was this not Minkowski
reduced, it was also not even LLL-reduced. How did it get on my list?
Have I done this one already? :(

In any case, this is a very interesting system.

<2401/2400,5120/5103> reduced

Wedge invariant: [2,25,13,-40,-15,35] length = 62.0322

Ets: 41,58,99,140,239

Map:

[ 0 1]
[ 2 1]
[15 -1]
[13 -5]

Generators: a = .2928926789 (~49/40) = 70.00135026 / 239; b = 1

Errors and 239 et

3: .987 .974
5: .467 .297
7: .300 .212

Measures: 7.3682 sc; 92.7798 s2c

Another valuable system, particularly if you are fond of 7/5.

<1029/1024,126/125>

Wedge invariant: [9,5,-3,-21,30,-13] length = 40.3113

Ets: 15,16,31,46,77

Map:

[ 0 1]
[ 9 1]
[ 5 2]
[-3 3]

Generators: a = .06475590616 = 2.007433091 / 31; b = 1

Errors and 31 et

3: -2.59 -5.19
5: 2.22 0.78
7: -1.95 -1.08

Measures: 22.068 sc; 158.834 s2c

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

> <2401/2400, 6144/6125>
>
> Minkowski reduction: <2401/2400, 3136/3125>
>
> Wedge invariant: [16,2,5,6,37,-34] length = 53.3479
>
> Ets: 31,68,99,130
>
> Map:
>
> [ 0 1]
> [16 -1]
> [ 2 2]
> [ 5 2]
>
> Generators: a = .1615916143 = 15.99756982 / 99; b = 1
>
> Errors and 99 et:
>
> 3: 0.604 1.075
> 5: 1.506 1.565
> 7: 0.724 0.871
>
> Measures: 8.8323 sc; 89.1168 s2c
>
> This beats Miracle in terms of step-cents, but not with step^2
cents.
> The fact that so much 5 and 7 is available is the reason why. I was
a
> little disturbed to find that not only was this not Minkowski
> reduced, it was also not even LLL-reduced. How did it get on my
list?
> Have I done this one already? :(

Yes, you did do this one already. I asked/pointed out that it was the
same as Graham's #1 on http://x31eq.com/limit7.txt -- now
here it is again. Somehow, it made it into your survey _twice_.

> <2401/2400,5120/5103> reduced
>
> Wedge invariant: [2,25,13,-40,-15,35] length = 62.0322
>
> Ets: 41,58,99,140,239
>
> Map:
>
> [ 0 1]
> [ 2 1]
> [15 -1]
> [13 -5]
>
> Generators: a = .2928926789 (~49/40) = 70.00135026 / 239; b = 1
>
> Errors and 239 et
>
> 3: .987 .974
> 5: .467 .297
> 7: .300 .212
>
> Measures: 7.3682 sc; 92.7798 s2c
>
> Another valuable system, particularly if you are fond of 7/5.

Graham -- shouldn't this be on your list? Its complexity is only 15,
yet it has lower errors than your #1. Perhaps this shows a limitation
of your method?

>
>
> <1029/1024,126/125>
>
> Wedge invariant: [9,5,-3,-21,30,-13] length = 40.3113
>
> Ets: 15,16,31,46,77
>
> Map:
>
> [ 0 1]
> [ 9 1]
> [ 5 2]
> [-3 3]
>
> Generators: a = .06475590616 = 2.007433091 / 31; b = 1
>
> Errors and 31 et
>
> 3: -2.59 -5.19
> 5: 2.22 0.78
> 7: -1.95 -1.08
>
> Measures: 22.068 sc; 158.834 s2c

I've played with this system -- it extends pretty well into the 11-
limit, right? How would you describe this in terms of an ET+ET
genesis?

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

> > <2401/2400,5120/5103> reduced
> >
> > Wedge invariant: [2,25,13,-40,-15,35] length = 62.0322
> >
> > Ets: 41,58,99,140,239
> >
> > Map:
> >
> > [ 0 1]
> > [ 2 1]
> > [15 -1]
> > [13 -5]

> Graham -- shouldn't this be on your list? Its complexity is only
15,
> yet it has lower errors than your #1. Perhaps this shows a
limitation
> of your method?

It shows a limitation of my brain which you've probably already
noticed. If you look at the wedge invatiant, you'll see the "15"
should be "25", so it isn't nearly as good as I thought by looking at
my typo.

> >
> >
> > <1029/1024,126/125>
> >
> > Wedge invariant: [9,5,-3,-21,30,-13] length = 40.3113
> >
> > Ets: 15,16,31,46,77
> >
> > Map:
> >
> > [ 0 1]
> > [ 9 1]
> > [ 5 2]
> > [-3 3]

> I've played with this system -- it extends pretty well into the 11-
> limit, right? How would you describe this in terms of an ET+ET
> genesis?

It extends nicely to the 11-limit, with [9,5,-3,7] being the
generator map. For et plus et, all you need to do is see if two ets
on the et list add to a third. It's done pretty well by 16+15 in the
31-et, and also by 31+15=46, 46+31=77.

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

> > > <1029/1024,126/125>
> > >
> > > Wedge invariant: [9,5,-3,-21,30,-13] length = 40.3113
> > >
> > > Ets: 15,16,31,46,77
> > >
> > > Map:
> > >
> > > [ 0 1]
> > > [ 9 1]
> > > [ 5 2]
> > > [-3 3]
>
> > I've played with this system -- it extends pretty well into the
11-
> > limit, right? How would you describe this in terms of an ET+ET
> > genesis?
>
> It extends nicely to the 11-limit, with [9,5,-3,7] being the
> generator map.

Yes -- I thought those 11s would fall right in there!

> For et plus et, all you need to do is see if two ets
> on the et list add to a third. It's done pretty well by 16+15 in
the
> 31-et,

That's where I found it, on my guitar.

Graham, where does this one come in on your rankings?

> Graham, where does this one come in on your rankings?

11 in the 7-limit, 17 in the 11-limit.

--- In tuning-math@y..., graham@m... wrote:

> > Graham, where does this one come in on your rankings?
>
> 11 in the 7-limit,

Cool! I knew it had to be comparable to the ones in your top 10!