flexible mapping of meantones to PBs

Paul, I finally understand your objections to what I've been
saying!

Here's another example I just found:

(This is one of the examples in Fokker 1968, "Selections from the
Harmonic Lattice of Perfect Fifths and Major Thirds Containing
12, 19, 22, 31, 41 or 53 Notes".)

unison-vector matrix =

[ 4 -1]
[-1 5]

determinant = | 19 |

periodicity-block coordinates:

3^x 5^y ~cents

-1 -2 925.4175714
0 -2 427.3725723
1 -2 1129.327573
2 -2 631.282574
-1 -1 111.7312853
0 -1 813.6862861
1 -1 315.641287
2 -1 1017.596288
-1 0 498.0449991
0 0 0
1 0 701.9550009
-2 1 182.4037121
-1 1 884.358713
0 1 386.3137139
1 1 1088.268715
-2 2 568.717426
-1 2 70.67242686
0 2 772.6274277
1 2 274.5824286

Ah!... actually now I see what's happening.

The meantones most commonly associated with this periodicity-block
would be 1/3-comma and 19-EDO. I'm not latticing EDOs on this
spreadsheet, so we'll just stick with the fraction-of-a-comma type.

1/3-comma does indeed split the periodicity-block exactly in half,
just not along an axis I expected, as it doesn't follow the same
angle as either of the unison-vectors.

The meantone I found by eye to split it according to the same angle
as the unison-vector [-1 5] is 16/61-comma.

But I think now I understand what you've been getting at, Paul.

In the 1/3-comma chain,

closest JI
generator coordinate

+1 ( 1 0) - 1/3-comma
+2 (-2 1) + 1/3-comma
+3 (-1 1) exactly
+4 ( 0 1) - 1/3-comma
+5 ( 1 1) - 2/3-comma
+6 (-2 2) exactly
+6 (-1 2) - 1/3-comma
+6 ( 0 2) - 2/3-comma
+6 ( 1 2) - 1 comma
etc.

In my mapping done by eye, everything would be the same up
to +4 generator. Then I'd set +5 generator equal to
(-3 2) - 1/3-comma, rather than (1 1) - 2/3-comma, since
it's closer. And so on.

But then we end up with +6 generator mapped to (1 2) - 1 comma
instead of to exactly (-3 3), which is what I would get.

But *it doesn't matter which periodicity-block contains the
closest-approach ratio, because they're all equivalent!* Right?

Got it now. Whew!

It doesn't matter which fraction-of-a-comma meantone I lattice
within a periodicity-block -- they'll *all* split the block
exactly symmetrically in half. Only the angles and resulting
areas differ.

-monz

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--- In tuning-math@y..., "monz" <joemonz@y...> wrote:

> Paul, I finally understand your objections to what I've been
> saying!

I hope so . . . we've been "stuck" on the very same set of issues for
several years now.
>
>
> Here's another example I just found:
>
> (This is one of the examples in Fokker 1968, "Selections from the
> Harmonic Lattice of Perfect Fifths and Major Thirds Containing
> 12, 19, 22, 31, 41 or 53 Notes".)
>
>
>
> unison-vector matrix =
>
> [ 4 -1]
> [-1 5]
>
>
> determinant = | 19 |
>
>
> periodicity-block coordinates:
>
>
> 3^x 5^y ~cents
>
> -1 -2 925.4175714
> 0 -2 427.3725723
> 1 -2 1129.327573
> 2 -2 631.282574
> -1 -1 111.7312853
> 0 -1 813.6862861
> 1 -1 315.641287
> 2 -1 1017.596288
> -1 0 498.0449991
> 0 0 0
> 1 0 701.9550009
> -2 1 182.4037121
> -1 1 884.358713
> 0 1 386.3137139
> 1 1 1088.268715
> -2 2 568.717426
> -1 2 70.67242686
> 0 2 772.6274277
> 1 2 274.5824286
>
>
> Ah!... actually now I see what's happening.
>
> The meantones most commonly associated with this periodicity-block
> would be 1/3-comma and 19-EDO. I'm not latticing EDOs on this
> spreadsheet, so we'll just stick with the fraction-of-a-comma type.
>
> 1/3-comma does indeed split the periodicity-block exactly in half,
> just not along an axis I expected, as it doesn't follow the same
> angle as either of the unison-vectors.
>
> The meantone I found by eye to split it according to the same angle
> as the unison-vector [-1 5] is 16/61-comma.
>
>
> But I think now I understand what you've been getting at, Paul.
>
> In the 1/3-comma chain,
>
> closest JI
> generator coordinate
>
> +1 ( 1 0) - 1/3-comma
> +2 (-2 1) + 1/3-comma
> +3 (-1 1) exactly
> +4 ( 0 1) - 1/3-comma
> +5 ( 1 1) - 2/3-comma
> +6 (-2 2) exactly
> +6 (-1 2) - 1/3-comma
> +6 ( 0 2) - 2/3-comma
> +6 ( 1 2) - 1 comma
> etc.
>
> In my mapping done by eye, everything would be the same up
> to +4 generator. Then I'd set +5 generator equal to
> (-3 2) - 1/3-comma, rather than (1 1) - 2/3-comma, since
> it's closer. And so on.
>
> But then we end up with +6 generator mapped to (1 2) - 1 comma
> instead of to exactly (-3 3), which is what I would get.
>
> But *it doesn't matter which periodicity-block contains the
> closest-approach ratio, because they're all equivalent!* Right?

Sure, but not sure what you did above to decide that.

>
> Got it now. Whew!
>
>
> It doesn't matter which fraction-of-a-comma meantone I lattice
> within a periodicity-block -- they'll *all* split the block
> exactly symmetrically in half. Only the angles and resulting
> areas differ.

Well, this has nothing to do with any of my objections to what you've
been saying. You could perfectly well be interested, for some strange
musical contrivance or pure mathematical curiosity, in the meantone
that "splits" the "periodicity block" along an "axis" parallel to one
of the unison vectors, in the way you've been diagramming things, and
I'd have no problem helping you do so.