Starling example

Here's a matrix of 3 et columns giving "starling":

[[31, 49, 72, 87], [46, 73, 107, 129], [50, 79, 116, 140]]

Here's a unimodular transformation matrix:

[[-15, -4, 13], [-14, 4, 5], [16, -1, -9]]

Here's the final result, the Hermite normal form for starling:

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

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> Here's a matrix of 3 et columns giving "starling":
>
> [[31, 49, 72, 87], [46, 73, 107, 129], [50, 79, 116, 140]]
>
> Here's a unimodular transformation matrix:
>
> [[-15, -4, 13], [-14, 4, 5], [16, -1, -9]]
>
> Here's the final result, the Hermite normal form for starling:
>
> [[1, 0, 0, -1], [0, 1, 0, -2], [0, 0, 1, 3]]

And what are the 3 generators implied by that?

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> > Here's the final result, the Hermite normal form for starling:
> >
> > [[1, 0, 0, -1], [0, 1, 0, -2], [0, 0, 1, 3]]
>
> And what are the 3 generators implied by that?

From the fact that the top part of the matrix is the 3x3 identity it follows that the generators are what Hermite form produces if it can--generators approximating 2,3, and 5. If we assume pure octaves and take 7-limit rms values, we get

a = 1200.000
b = 1899.984
c = 2789.270

as generators, giving us an approximate 7 of 3367.841 cents. The fifth is the 12-et fifth, but of course the third and 7 are not close to 12-et. 31, 46 and 77 give fairly decent but not more than that versions of starling.