Fun with product words

One way to approach this is via adding the steps of two different scales. Example:

Roulette[7]: [5, 5, 5, 5, 5, 5, 1]
Meantone[7]: [5, 5, 3, 5, 5, 5, 3]
(Meantone+Roulette)/2: [5, 5, 4, 5, 5, 5, 2]

Modes:
[5, 5, 5, 5, 5, 1, 5]=>[5, 5, 4, 5, 5, 3, 4]
[5, 5, 5, 5, 1, 5, 5]=>[5, 5, 4, 5, 3, 5, 4]
[5, 5, 5, 1, 5, 5, 5]=>[5, 5, 4, 3, 5, 5, 4]
[5, 5, 1, 5, 5, 5, 5]=>[5, 5, 2, 5, 5, 5, 4]
[5, 1, 5, 5, 5, 5, 5]=>[5, 3, 4, 5, 5, 5, 4]
[1, 5, 5, 5, 5, 5, 5]=>[3, 5, 4, 5, 5, 5, 4]

On Mon, Nov 14, 2011 at 9:33 PM, genewardsmith
<genewardsmith@sbcglobal.net> wrote:
>
> One way to approach this is via adding the steps of two different scales. Example:
>
> Roulette[7]: [5, 5, 5, 5, 5, 5, 1]
> Meantone[7]: [5, 5, 3, 5, 5, 5, 3]
> (Meantone+Roulette)/2: [5, 5, 4, 5, 5, 5, 2]
>
> Modes:
> [5, 5, 5, 5, 5, 1, 5]=>[5, 5, 4, 5, 5, 3, 4]
> [5, 5, 5, 5, 1, 5, 5]=>[5, 5, 4, 5, 3, 5, 4]
> [5, 5, 5, 1, 5, 5, 5]=>[5, 5, 4, 3, 5, 5, 4]
> [5, 5, 1, 5, 5, 5, 5]=>[5, 5, 2, 5, 5, 5, 4]
> [5, 1, 5, 5, 5, 5, 5]=>[5, 3, 4, 5, 5, 5, 4]
> [1, 5, 5, 5, 5, 5, 5]=>[3, 5, 4, 5, 5, 5, 4]

Right, and then if you change meantone[7] around, there are 49
combinations in total.

Replacing roulette with tetracot makes it easy to see how this relates
to the Fokker lattice; then you can do some neat stuff with Fokker
blocks too. For example, if you change the mode of meantone[7] that
you're multiplying by, and you change it by moving one more meantone
generator "up," you shift the resultant Fokker block over in the 81/80
direction such that one of the points falls off the edge and is
replaced by another point 20000/19683 away. And if you change the
tetracot mode by moving one tetracot generator "up," you end up doing
the reverse of that. Keenan has a concept called "virtual generator"
for rank-3 scales which might prove useful here, but I'll leave it to
him to explain.

-Mike