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.