Survey II

<2401/2400, 3136/3125>

ets: 31, 68, 99, 130

Map:

[ 1 2]
[-1 14]
[ 2 6]
[ 2 9]

Adjusted map:

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

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

Related systems: 68+31 and 99+31

Errors:

3: .604
5: 1.506
7: .724

Comparison to 99:

3: 1.075
5: 1.565
7: .871

<5120/5103, 1728/1715>

ets: 53, 58, 111

Map:

[3 -1]
[4 0]
[3 6]
[9 -4]

Generators a = .3963765938 ~ 21/16; b = .189129781 ~ 8/7

Adjusted map:

[ 0 1]
[ 4 0]
[21 -6]
[-3 4]

Generators a and 1. Clearly there is something to be said for the
original set of generators, which are approximately 44/111 and 21/111.
We have more precisely a = 43.99780191 / 111.

Errors:

3: .75
5: 2.88
7: 4.15

Comparion with 111-et:

3: .65
5: 2.38
7: 4.22

This is in effect the 58+53 system of the 111-et; it is also related
to 53+5.

<64/63, 50/49>

ets: 10, 12, 22

Map:

[ 0 2]
[-1 4]
[ 2 3]
[ 2 4]

Generators a = .4093213919 = 9.005070622 / 22 (~4/3) and b = 1/2.

This is the Paultone system, which retains two familiar comma
relationships from the 12 et, while not being meantone.

Errors:

3: 6.86
5: -3.94
7: 13.55

Comparison with 22-et:

3: 7.14
5: -4.50
7: 12.99

The system is in effect 12+10 in the 22-et.

<4000/3969, 245/243>

ets: 27, 41, 68, 109

Map:

[1 4]
[1 -4]
[1 -14]
[2 -3]

Adjusted map:

[ 0 1]
[ 8 1]
[18 1]
[11 2]

Generators: a = .0734545064 (~21/20) = 4.994906384 / 68

This is closely allied to the 27+41 system of the 68 et, but the
tuning is somewhat improved. It is also close to the 68+41 system of
the 109-et.

Errors:

3: 3.21
5: 0.30
7: 0.77

Comparison to the 68-et:

3: 3.93
5: 1.92
7: 1.76

--- In tuning-math@y..., genewardsmith@j... wrote:
> <4000/3969, 245/243>
>
> ets: 27, 41, 68, 109

I forgot to check these for Minkowski reduction. The rest are
Minkowski reduced, but this one reduces to <2401/2400, 245/243>

--- In tuning-math@y..., genewardsmith@j... wrote:
> <2401/2400, 3136/3125>
>
> ets: 31, 68, 99, 130
>
> Map:
>
> [ 1 2]
> [-1 14]
> [ 2 6]
> [ 2 9]
>
> Adjusted map:
>
> [ 0 1]
> [16 -1]
> [ 2 2]
> [ 5 2]

Can you explain what "map" and "adjusted map" mean? What about the
mapping from generators to primes (which seems most important of all)?

> <5120/5103, 1728/1715>
>
> ets: 53, 58, 111
>
> Map:
>
> [3 -1]
> [4 0]
> [3 6]
> [9 -4]
>
> Generators a = .3963765938 ~ 21/16; b = .189129781 ~ 8/7

Hmm . . . I thought one of the generators had to be 1/N octaves, N
integer. So what's going on here?
>
> <4000/3969, 245/243>
>
> ets: 27, 41, 68, 109
>
> Map:
>
> [1 4]
> [1 -4]
> [1 -14]
> [2 -3]
>
> Adjusted map:
>
> [ 0 1]
> [ 8 1]
> [18 1]
> [11 2]
>
> Generators: a = .0734545064 (~21/20) = 4.994906384 / 68
>
> This is closely allied to the 27+41 system of the 68 et, but the
> tuning is somewhat improved. It is also close to the 68+41 system
of
> the 109-et.
>
> Errors:
>
> 3: 3.21
> 5: 0.30
> 7: 0.77
>
> Comparison to the 68-et:
>
> 3: 3.93
> 5: 1.92
> 7: 1.76

It's not clear to me that the tuning's improved. 5/3, 7/3, and 7/5
are all better in 68-tET.

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., genewardsmith@j... wrote:
> > <4000/3969, 245/243>
> >
> > ets: 27, 41, 68, 109
>
> I forgot to check these for Minkowski reduction. The rest are
> Minkowski reduced, but this one reduces to <2401/2400, 245/243>

Can you give the whole list, that you did LLL-reduced, Minkowski
reduced instead? How about the ETs?

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

> Can you explain what "map" and "adjusted map" mean? What about the
> mapping from generators to primes (which seems most important of
all)?

"Map" is what I get by LLL reducing the dual group to the two basis
intervals. "Adjusted map" takes linear combinations of the columns of
this matrix to get a generator/interval of repetition system. The
columns are vals in prime notation, and hence are maps to primes.

> > <5120/5103, 1728/1715>
> >
> > ets: 53, 58, 111
> >
> > Map:
> >
> > [3 -1]
> > [4 0]
> > [3 6]
> > [9 -4]
> >
> > Generators a = .3963765938 ~ 21/16; b = .189129781 ~ 8/7

> Hmm . . . I thought one of the generators had to be 1/N octaves, N
> integer. So what's going on here?

It's the difference between "map" and "adjusted map". This one seemed
interesting to look at, as well as the adjusted one.

> > Errors:
> >
> > 3: 3.21
> > 5: 0.30
> > 7: 0.77
> > Comparison to the 68-et:
> >
> > 3: 3.93
> > 5: 1.92
> > 7: 1.76

> It's not clear to me that the tuning's improved. 5/3, 7/3, and 7/5
> are all better in 68-tET.

Well, of course it's improved in a least-squares sense at any rate,
but the close-to-exact values of the 5 and 7 would be good in wide
interval contexts with sustained harmony, whereas the difference
between 5/3 7/3 and 7/5 does not seem so interesting. In fact,
however, this is another system which seems as if one might just as
well leave it in an et.

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
>
> > Can you explain what "map" and "adjusted map" mean? What about
the
> > mapping from generators to primes (which seems most important of
> all)?
>
> "Map" is what I get by LLL reducing the dual group to the two basis
> intervals.

No comprendo.

> "Adjusted map" takes linear combinations of the columns of
> this matrix to get a generator/interval of repetition system. The
> columns are vals in prime notation, and hence are maps to primes.

Oh, so the first column is the coefficient on a, and the second is
the coefficient on b?

> > Hmm . . . I thought one of the generators had to be 1/N octaves,
N
> > integer. So what's going on here?
>
> It's the difference between "map" and "adjusted map".

I'd like to understand this better.
>
> Well, of course it's improved in a least-squares sense at any rate,
> but the close-to-exact values of the 5 and 7 would be good in wide
> interval contexts with sustained harmony, whereas the difference
> between 5/3 7/3 and 7/5 does not seem so interesting.

Hmm . . . wide interval contexts? You mean bass-soprano dyadic
counterpoint??

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

> > "Map" is what I get by LLL reducing the dual group to the two
basis
> > intervals.

> No comprendo.

Usually this means I LLL reduce two of the ets I get in my list of
ets, and the ets in question are ones which have both basis elements
in the kernel. The intersection of the two kernels is the set of
intervals generated by the two basis elements, which is a group of
rank two. Dual to it is the group of rank two consisting of integral
combinations of two ets. Lattice basis reduction reduces this from a
two et basis to a basis of two vals which tell me the number of
generator steps for a prime--they are generator vals, you might say.
If we use a Tenney metric on intervals, we induce a dual metric on
the vals, which is a sup norm (maximum of absolute values) with an
adjustment of dividing the 3 row by log2(3), etc.

Is this something we need to try to explain? I think if I could
explain it to you (which I suspect this isn't quite doing yet) then
we should be able to get the point across, but how theoretical should
things get?

> > "Adjusted map" takes linear combinations of the columns of
> > this matrix to get a generator/interval of repetition system. The
> > columns are vals in prime notation, and hence are maps to primes.
>
> Oh, so the first column is the coefficient on a, and the second is
> the coefficient on b?

Yes, to get to each prime in turn. The first row tells you how many
a's and how many b's to get a 2, and so forth.

> > Well, of course it's improved in a least-squares sense at any
rate,
> > but the close-to-exact values of the 5 and 7 would be good in
wide
> > interval contexts with sustained harmony, whereas the difference
> > between 5/3 7/3 and 7/5 does not seem so interesting.
>
> Hmm . . . wide interval contexts? You mean bass-soprano dyadic
> counterpoint??

That's where this might be noticed. It's pretty small differences we
are discussing.

--- In tuning-math@y..., genewardsmith@j... wrote:
> <2401/2400, 3136/3125>
>
> ets: 31, 68, 99, 130
>
> Map:
>
> [ 1 2]
> [-1 14]
> [ 2 6]
> [ 2 9]
>
> Adjusted map:
>
> [ 0 1]
> [16 -1]
> [ 2 2]
> [ 5 2]
>
> Generators a = .1615916143 = 15.99756982 / 99; b = 1
>
> Related systems: 68+31 and 99+31
>
> Errors:
>
> 3: .604
> 5: 1.506
> 7: .724

Complexity 16, max. error 1.506

This was #1 on Graham's 7-limit list, right?

> <4000/3969, 245/243>
>
> ets: 27, 41, 68, 109
>
> Map:
>
> [1 4]
> [1 -4]
> [1 -14]
> [2 -3]
>
> Adjusted map:
>
> [ 0 1]
> [ 8 1]
> [18 1]
> [11 2]
>
> Generators: a = .0734545064 (~21/20) = 4.994906384 / 68
>
> This is closely allied to the 27+41 system of the 68 et, but the
> tuning is somewhat improved. It is also close to the 68+41 system
of
> the 109-et.
>
> Errors:
>
> 3: 3.21
> 5: 0.30
> 7: 0.77

Complexity 18, max. err. 3.21¢