Hexanies into hexagons

Using breed (2401/2400) tempering, we can represent 7-limit note
classes by a two-dimensional lattice. The generators for this are 10/7
and a neutral third which can be written either 49/40 or 60/49. Here
is a portion of the lattice, taken as rectangular, with 49/40
vertically and
10/7 horizontally; because of the usual Yahoo massacre of such
diagrams you should read this so that the spaces appear.

72/49 -- 21/20 -- 3/2 -- 15/14 -- 49/32
| | | | |
6/5 -- 12/7 -- 49/40 -- 7/4 -- 5/4
| | | | |
49/25 -- 7/5 -- 1 -- 10/7 -- 50/49

Note that hexanies, in this picture, now become hexagons, and any
hexagon surrounding an interval of 10/7 is in fact a hexany; the two
notes in the center are not a part of the hexany, though in fact
adding them to make an eight-note scale is an interesting possibility.

Gene Ward Smith wrote:
> Using breed (2401/2400) tempering, we can represent 7-limit note
> classes by a two-dimensional lattice. The generators for this are 10/7
> and a neutral third which can be written either 49/40 or 60/49. Here
> is a portion of the lattice, taken as rectangular, with 49/40
> vertically and
> 10/7 horizontally; because of the usual Yahoo massacre of such
> diagrams you should read this so that the spaces appear.
> > 72/49 -- 21/20 -- 3/2 -- 15/14 -- 49/32
> | | | | |
> 6/5 -- 12/7 -- 49/40 -- 7/4 -- 5/4
> | | | | |
> 49/25 -- 7/5 -- 1 -- 10/7 -- 50/49
> > Note that hexanies, in this picture, now become hexagons, and any
> hexagon surrounding an interval of 10/7 is in fact a hexany; the two
> notes in the center are not a part of the hexany, though in fact
> adding them to make an eight-note scale is an interesting possibility.

I'm surprised this hasn't been noticed before, since this resembles the usual 2D projection of the 7-limit lattice (not counting reflection).

5/4
/|\
/7/4\
/ / \ \
1/1-----3/2

It's a fortunate coincidence that 2401/2400 ends up directly above 1/1, so that we only need 4 layers of the lattice if we're willing to ignore it or temper it out. It's convenient for tables and spreadsheets that this fits into a 2D grid. But then I've been seeing this as just a crude 2-dimensional representation of an actually 3D structure, and didn't realize that it could also be a representation of a 2401/2400 planar temperament. That's nice!

So the representation of hexanies as hexagons follows logically from the fact that the projection of an octahedron onto a plane is a hexagon.

> I'm surprised this hasn't been noticed before, since this resembles the
> usual 2D projection of the 7-limit lattice (not counting reflection).
>
> 5/4
> /|\
> /7/4\
> / / \ \
> 1/1-----3/2

I was surprised too, back in 2001 or whenever I noticed them, hehe

> It's a fortunate coincidence that 2401/2400 ends up directly above 1/1,
> so that we only need 4 layers of the lattice if we're willing to ignore
> it or temper it out. It's convenient for tables and spreadsheets that
> this fits into a 2D grid. But then I've been seeing this as just a crude
> 2-dimensional representation of an actually 3D structure, and didn't
> realize that it could also be a representation of a 2401/2400 planar
> temperament. That's nice!

There are other ways of doing it as well. As my website's down, try
the archive:

http://tinyurl.com/8ukmo

All of them can be a 3-D temperament (I see you still call them planar
temperaments...) and for that matter any 3-D temperament corresponds
to more than one 2-D lattice (with octave equivalence).

> So the representation of hexanies as hexagons follows logically from the
> fact that the projection of an octahedron onto a plane is a hexagon.

Also that there are 6 notes in a hexany!

Graham