An eikosany lattice

All this business about who invented the eikosany has led me to wonder
about its properties, and whether or not there is something
particularly interesting about it. It has a high degree of symmetry
({5,7,11} symmetry) which suggested looking at it as possibly a ball
type scale under some metric or other. By setting a large number of
inequalities saying that the eikosany was within 1 of
[3/2,1/2,1/2,1/2] (the centroid) whereas other points were greater
than 1 from this, I ended up with a set of inequalities a quadratic
form defining such a metric had to satisfy, and by picking points in
the middle of the range of possibilites, finally ended up with

|| |a b c d e> ||_eik = sqrt(3*b^2 + 6b(c+d+e) + 10(c^2+d^2+e^2) +
9(cd + de + ec)

as a seminorm, which is a norm on 11-limit pitch classes, making the
11-limit into a lattice. The eikosany is the second ball around the
pitch class defined by |0 3/2 1/2 1/2 1/2>

Shells around |0 3/2 1/2 1/2 1/2>

First shell 3/sqrt(2) radius
{45, 63, 99, 105, 165, 231}

Second shell sqrt(15/2) radius
{15, 21, 27, 33, 35, 55, 77, 135, 189, 297, 315, 385, 495, 693}

Third shell sqrt(21/2) radius
{9, 81, 1155, 385/3}

Fourth shell sqrt(31/2) radius
{75, 147, 363, 693/5, 495/7, 315/11}

Shells around |0 0 0 0 0>

First shell zero radius
{1}

Second shell sqrt(3) radius
{3, 1/3}

Third shell sqrt(7) radius
{11/3, 5/3, 7/3, 3/7, 3/11, 3/5}

Fourth shell sqrt(10) radius
{5, 7, 11, 11/9, 9/5, 9/7, 1/5, 9/11, 5/9, 7/9, 1/7, 1/11}

The eikosany is the union of the first and second shells around
|0 3/2 1/2 1/2 1/2>; other scales can be obtained from other balls.

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

> || |a b c d e> ||_eik = sqrt(3*b^2 + 6b(c+d+e) + 10(c^2+d^2+e^2) +
> 9(cd + de + ec)

It occurred to me that I really should set the c^2+d^2+e^2 term
equal to the (cd+de+ec) term, since then in the no-threes context it
becomes the familiar lattice A3 of the 7-limit. I found the following
norm, which is a good deal simpler and has that nice property, and is
simple enough we may as well take it as the offical eikosany metric on
11-limit pitch classes.

|| |0 b c d e> ||_eik = sqrt(b^2 + 2b(c+d+e)+3(c^2+d^2+e^2+cd+ce+ec))

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

> || |0 b c d e> ||_eik = sqrt(b^2 + 2b(c+d+e)+3(c^2+d^2+e^2+cd+ce+ec))

Pondering this, it occurred to me to see if

|| |0 b c d e> || = sqrt(b^2+2b(c+d+e)+4(c^2+d^2+e^2+cd+de+ec))

works--and it does. It makes the eikosany into ball 3 rather than ball
2, however. The significance of this is that this metric is one I've
already proposed and discussed, where 5,7,9 and 11 are symmetrical.
Had I ever done a survey of scales centered on holes with this metric,
the eikosany would have turned up on it. This also tells us the
lattice in question can be viewed as two A4 lattices glued together,
which gives a better geometric picture.