46 augmented scales

I'm getting more scales out of this than I expected; below are 46
distinct augmented scales obtained by reducing the 53 Fokker blocks to
augmented. It would be interesting to know what these look like on a
rectangular lattice; I may try Maple on that. The top generators for
augmented are 399.020 for the period and 93.145 for the "generator",
which would be one way to convert these to cents.

{[0, 0], [0, 2], [-1, 2], [0, -1], [-1, 1], [-1, -2], [-1, -1], [-1,
0], [0, 3], [1, 1], [1, 0], [1, -3]}
{[0, 0], [-1, 2], [0, -1], [-1, 1], [-1, -2], [-1, -1], [-1, 0], [0,
3], [1, 1], [1, 0], [0, 1], [1, -2]}
{[0, 0], [-1, 2], [0, -1], [-1, -1], [0, -3], [1, 2], [-1, 0], [1, 1],
[1, 0], [1, -1], [0, 1], [1, -2]}
{[0, 0], [0, 2], [0, -1], [-1, 1], [-1, -2], [-2, 1], [-1, -1], [0,
-2], [-1, 0], [-2, 0], [1, -3], [1, -1]}
{[-1, 3], [0, 0], [-2, 1], [-2, 2], [-1, -1], [-1, -4], [0, -3], [0,
-2], [-1, 0], [1, -1], [0, 1], [1, -2]}
{[0, 0], [0, 2], [-1, 2], [0, -1], [-1, 1], [-2, 2], [-1, -1], [-1,
0], [1, 1], [1, 0], [1, -1], [0, 1]}
{[0, 0], [0, -1], [-1, 1], [-1, -1], [0, -2], [-1, 0], [1, 1], [1, 0],
[2, -2], [1, -1], [0, 1], [1, -2]}
{[0, 0], [0, -1], [-1, 1], [-1, -2], [-2, 1], [-1, -1], [0, -2], [-1,
0], [-2, 0], [1, -1], [0, 1], [1, -2]}
{[0, 0], [0, 2], [-1, 2], [0, -1], [-1, 1], [-1, -1], [1, 2], [-1, 0],
[1, 1], [1, 0], [1, -1], [0, 1]}
{[0, 0], [-1, 2], [0, -1], [-1, 1], [-1, -2], [-1, -1], [-1, 0], [0,
3], [1, 1], [1, 0], [1, -3], [1, -2]}
{[-1, 4], [-1, 2], [0, -1], [-1, -2], [-2, 1], [0, -3], [0, -5], [-1,
0], [0, 3], [1, 0], [0, 1], [1, -2]}
{[-1, 4], [-1, 2], [0, -1], [-1, -2], [-2, 1], [0, -3], [0, -5], [-1,
0], [0, 3], [0, 1], [1, -2], [2, -4]}

{[0, 0], [0, 2], [-1, 2], [0, -1], [0, -3], [-2, 3], [-1, 0], [1, 1],
[1, -3], [2, -2], [1, -1], [2, -4]}
{[-1, 3], [0, 0], [0, 2], [-1, 1], [-2, 2], [-1, -1], [0, -2], [-1,
0], [1, 1], [1, 0], [1, -1], [0, 1]}
{[2, -3], [0, 0], [0, -1], [-1, 1], [0, -2], [1, 2], [-1, 0], [0, 3],
[1, 0], [2, -1], [0, 1], [1, -2]}
{[0, 0], [0, 2], [-1, 2], [0, -1], [0, -3], [-2, 3], [-1, 0], [1, 1],
[1, 0], [2, -2], [1, -1], [0, 1]}
{[2, -3], [-1, 3], [-1, 1], [0, -2], [1, -4], [-2, 4], [1, 2], [0, 3],
[1, 0], [2, -1], [0, 1], [1, -2]}
{[-1, 4], [0, 2], [-1, 2], [0, -1], [0, -3], [-2, 3], [-1, 0], [1, 1],
[1, -3], [2, -2], [1, -1], [2, -4]}
{[-1, 4], [-1, 2], [0, -1], [-1, -2], [0, -3], [0, -5], [-3, 5], [-1,
0], [0, 3], [1, 0], [0, 1], [1, -2]}
{[2, -3], [-1, 3], [-1, 5], [0, 0], [0, 2], [-1, 1], [-2, 2], [-1,
-1], [0, -2], [0, -4], [1, 0], [2, -5]}
{[2, -3], [0, -1], [-1, 1], [0, -2], [1, -4], [-2, 4], [1, 2], [0, 3],
[1, 0], [2, -1], [0, 1], [1, -2]}
{[-1, 4], [0, 2], [-1, 2], [0, -1], [0, -3], [-2, 3], [-1, 0], [1, 1],
[2, -2], [1, -1], [0, 1], [2, -4]}
{[2, -3], [-1, 3], [-1, 5], [0, 0], [0, 2], [-1, 1], [-2, 2], [-1,
-1], [0, -2], [0, -4], [1, 0], [1, -1]}
{[-1, 4], [-1, 2], [0, -1], [-1, 1], [-2, 2], [-1, -1], [-1, -3], [0,
-4], [0, 3], [1, 0], [0, 1], [1, -2]}
{[-1, 4], [0, -1], [-1, 1], [-2, 2], [-1, -1], [0, -2], [-1, -3], [0,
-4], [0, 3], [1, 0], [0, 1], [1, -2]}
{[2, -3], [-1, 4], [-1, 2], [0, -1], [-1, -1], [1, 2], [0, -4], [0,
3], [1, 0], [0, 1], [-2, 5], [1, -2]}
{[2, -3], [-1, 3], [-1, 5], [0, 0], [-1, 1], [-2, 2], [-1, -1], [0,
-2], [0, -4], [1, 0], [1, -2], [2, -5]}
{[-1, 3], [0, 0], [0, 2], [-1, 1], [-2, 1], [-1, -1], [0, -2], [0,
-4], [-3, 4], [0, -6], [1, -3], [1, -1]}
{[0, 0], [0, 2], [0, -1], [-1, 1], [-1, -2], [-1, -1], [0, -2], [-1,
0], [0, 3], [1, 1], [1, 0], [0, 1]}
{[0, 0], [-1, 2], [0, -1], [-1, 1], [-2, 1], [-2, 2], [-1, -1], [-1,
0], [1, 0], [1, -3], [1, -1], [1, -2]}
{[-1, 3], [0, 0], [-1, 2], [-1, -2], [-1, -1], [0, -4], [0, 4], [0,
3], [1, 1], [1, -3], [-2, 5], [1, -2]}
{[-1, 3], [0, 0], [-1, 2], [-2, 2], [-1, -1], [0, -3], [0, -4], [0,
4], [1, 1], [0, 1], [1, -2], [2, -5]}
{[0, 0], [-1, 2], [0, -1], [0, -4], [0, 4], [0, 3], [1, 1], [1, -3],
[2, -2], [2, -1], [-2, 5], [1, -2]}
{[2, -3], [-1, 5], [0, 2], [0, -1], [-1, 1], [0, -2], [-2, 3], [1,
-4], [-1, 0], [1, 0], [2, -2], [1, -1]}
{[-1, 3], [0, 0], [-1, 2], [-2, 2], [-1, -1], [0, -3], [-1, 0], [0,
4], [1, 1], [0, 1], [1, -2], [2, -5]}
{[-1, 3], [0, 0], [-1, 2], [-1, -2], [-1, -1], [0, -4], [-1, 7], [0,
4], [1, 1], [1, -3], [-2, 5], [2, -6]}
{[-1, 3], [0, 0], [-1, -1], [0, -3], [1, 2], [0, -4], [-2, 6], [0, 4],
[0, 1], [0, 5], [1, -2], [2, -5]}
{[-1, 3], [-1, 2], [-1, 1], [-1, -2], [-2, 1], [-1, -1], [1, -4], [-1,
0], [-2, 0], [1, -3], [1, -1], [1, -2]}
{[0, 0], [-1, 2], [0, -1], [-1, 1], [-2, 2], [-2, 3], [-1, 0], [1, 1],
[1, 0], [1, -1], [0, 1], [1, -2]}
{[0, 0], [0, 2], [0, -1], [-2, 2], [0, -3], [0, -2], [-2, 3], [-1, 0],
[1, 1], [1, 0], [1, -1], [0, 1]}
{[0, 0], [0, -1], [-2, 1], [-2, 2], [0, -3], [0, -2], [-2, 3], [-1,
0], [1, 0], [1, -1], [0, 1], [1, -2]}
{[-1, 4], [0, 2], [2, -7], [-1, 1], [0, -2], [-2, 3], [-1, -3], [0,
-6], [1, -5], [-4, 8], [-1, 0], [1, -1]}
{[-1, 3], [-1, -1], [0, -3], [1, 2], [-1, 7], [-2, 6], [2, -8], [-1,
0], [0, 1], [0, 5], [1, -2], [2, -4]}
{[-1, 3], [0, 0], [-1, 2], [-1, -2], [-1, -1], [0, -4], [-1, 6], [0,
4], [1, 1], [1, -3], [-2, 5], [2, -5]}
{[0, 0], [-1, 2], [0, -1], [0, -4], [-1, 6], [-3, 7], [1, -7], [3,
-8], [0, 3], [1, 1], [1, -3], [2, -2]}
{[-1, 4], [0, 2], [0, -1], [-1, 1], [0, -4], [-2, 6], [1, 0], [1, -3],
[2, -2], [1, 3], [0, 5], [2, -5]}

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

> I'm getting more scales out of this than I expected; below are 46
> distinct augmented scales obtained by reducing the 53 Fokker blocks to
> augmented.

There are so many of these I decided to do a little sorting out. By
"diameter" I mean a sort of complexity measure, where the difference
between the largest and smallest values for each of the generators is
found and the maximum taken. Here are the scales with diameter less
than eight.

Diameter 3

{[0, 0], [0, 2], [-1, 2], [0, -1], [-1, 1], [-1, -1],
[-1, 0], [1, 1],[1, 0], [0, 1], [1, -1], [-2, 2]}

{[0, -2], [0, 0], [0, -1], [-1, 1], [-1, -1], [-1, 0],
[1, 1], [1, 0], [0, 1], [1, -2], [1, -1], [2, -2]}

{[0, -2], [-2,0], [0, 0], [0, -1], [-1, 1], [-1, -2],
[-1, -1], [-1, 0], [0, 1], [1, -2], [1, -1], [-2, 1]}

{[1, 2], [0, 0], [0, 2], [-1, 2], [0, -1], [-1, 1],
[-1, -1],[-1, 0], [1, 1], [1, 0], [0, 1], [1, -1]}

Diameter 5

{[0, 0], [-1, 2], [0, -1], [-1, 1], [-1, -2], [-1, -1],
[-1, 0], [0, 3], [1, 1], [1, 0], [0, 1], [1, -2]}

{[1, 2], [0, 0], [-1, 2], [0, -1], [-1, -1], [-1, 0],
[1, 1], [1, 0], [0, 1], [1, -2], [0, -3], [1, -1]}

{[0, -2], [-2,0], [0, 0], [0, 2], [0, -1], [-1, 1],
[-1, -2], [-1, -1], [-1, 0], [1, -3], [1, -1], [-2, 1]}

{[0, -2], [0, 0], [0, 2], [-1, 1], [-1, -1], [-1, 0],
[1, 1], [1, 0], [0, 1], [1, -1], [-1, 3], [-2, 2]}

{[0, -2], [0, 0], [0, 2], [0, -1],[-1, 1], [-1, -2],
[-1, -1], [-1, 0], [0, 3], [1, 1], [1, 0], [0, 1]}

{[0, 0], [-1, 2], [0, -1], [-1, 1], [-1, -1], [-1, 0],
[1, 0], [1, -3], [1, -2], [1,-1], [-2, 1], [-2, 2]}

{[0, 0], [-1, 2], [0, -1], [-1, 1], [-1, 0], [1, 1],
[1, 0], [0, 1], [1, -2], [1, -1], [-2, 2], [-2, 3]}

Diameter 6

{[0, 0], [0, 2], [-1, 2], [0, -1], [-1, 1], [-1, -2],
[-1, -1], [-1, 0], [0, 3], [1, 1], [1, 0], [1, -3]}

{[0, 0], [-1, 2], [0, -1], [-1, 1], [-1, -2], [-1, -1],
[-1, 0], [0, 3], [1, 1], [1, 0], [1, -3], [1, -2]}

{[0, -2], [1, 2], [0, 0], [0, -1], [-1, 1], [-1, 0],
[0, 3], [1, 0], [0, 1], [1, -2], [2,-3], [2, -1]}

{[0, 0], [0, 2], [-1, 2], [0, -1], [-1, 0], [1, 1],
[1, 0], [0,1], [0, -3], [1, -1], [2, -2], [-2, 3]}

{[0, -2], [0, 0], [0, 2], [0, -1], [-1, 0], [1, 1],
[1, 0], [0, 1], [0, -3], [1, -1], [-2, 2], [-2, 3]}

{[0, -2], [0, 0], [0, -1], [-1, 0], [1, 0], [0, 1], [1, -2], [0, -3],
[1, -1], [-2, 1],[-2, 2], [-2, 3]}}

Diameter 7

{[0, -2], [0, 0], [-1, -1], [-1, 0], [0, 1], [1, -2],
[0, -3], [1, -1],[-2, 1], [-1, 3], [-2, 2], [-1, -4]}

{[0, 0], [0, 2], [-1, 2], [0, -1], [-1,0], [1, 1],
[1, -3], [0, -3], [1, -1], [2, -2], [2, -4], [-2, 3]}

{[-2, 0], [-1, 2], [-1, 1], [-1, -2], [-1, -1], [-1, 0],
[1, -3], [1, -2], [1, -1], [-2,1], [-1, 3], [1, -4]}

>> I'm getting more scales out of this than I expected; below are 46
>> distinct augmented scales obtained by reducing the 53 Fokker blocks to
>> augmented.
>
>There are so many of these I decided to do a little sorting out. By
>"diameter" I mean a sort of complexity measure, where the difference
>between the largest and smallest values

In cents?

>for each of the generators is
>found and the maximum taken. Here are the scales with diameter less
>than eight.

Can you think of another term? Paul Hahn has used it in a graph-
theory sense...

http://library.wustl.edu/~manynote/music.html

-Carl

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

Sorry, I neglected to octave-reduce these. That should cut down on the
number.