26et and meantone projections

The 26 division was mentioned again recently on the tuning list, and
here is another curious property for it. A comma perpendicular to
81/80 in the plane of symmetrical 5-limit note classes is 78125/73728.
This is
also a left eigenvector for the eigenvalue 1 of the projection for
7/26 comma meantone; in other words, 7/26 comma meantone leaves it fixed.
Moreover, {81/80, 78125/73728} is the 5-limit TM basis for 26-et. The
corresponding Fokker blocks have perpendicular sides, leading to more
or less rectangular blocks in the symmetrical plane. Perhaps I will
investigate these blocks; if nothing else, they'd look interesting in
Tonescape.

Of course it follows from all this that the complexity of 78125/73728
in meantone is 26, and that the size of it represents the degree of
flatness of the wolf for Meantone[26] in any given tuning. In
7/26-comma meantone, the size of 78125/73728 is just 78125/73728, or
100 cents, so we get a wolf "fifth" 100 cents flat.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> The 26 division was mentioned again recently on the tuning list, and
> here is another curious property for it. A comma perpendicular to
> 81/80 in the plane of symmetrical 5-limit note classes is 78125/73728.
> This is
> also a left eigenvector for the eigenvalue 1 of the projection for
> 7/26 comma meantone; in other words, 7/26 comma meantone leaves it
fixed.
> Moreover, {81/80, 78125/73728} is the 5-limit TM basis for 26-et. The
> corresponding Fokker blocks have perpendicular sides, leading to more
> or less rectangular blocks in the symmetrical plane. Perhaps I will
> investigate these blocks; if nothing else, they'd look interesting in
> Tonescape.
>
> Of course it follows from all this that the complexity of 78125/73728
> in meantone is 26, and that the size of it represents the degree of
> flatness of the wolf for Meantone[26] in any given tuning. In
> 7/26-comma meantone, the size of 78125/73728 is just 78125/73728, or
> 100 cents, so we get a wolf "fifth" 100 cents flat.

That's pretty flat! I found a way to superimpose tilted rectangles on
Paul's zoom diagrams (the kind that treats 3/2, 5/4 and 6/5 on equal
footing) My rectangles actually add the total (abs) error for 3/2 and
5/4, and let 6/5 be a passive consequence. It's vaguely interesting, 12-
et is on the fourth ring with 16 cents, along with 28, 19, 29, 37
In Ring 3: 22, 55, 43 In Ring 2: 31, 41, 34, 46 etc. Ring 1: 53, 65...
Oh never mind. I just meant to say that 26 is pretty bad, it goes to
show that bad temperaments can still have interesting properties. (For
sure, patterns in my study of Z-relations don't relate at all to
whether a temperament is good or not, even though 22 and 31 have some
neat properties.) I'm excited to see you applying eigenvectors to
tuning theory. Once I finish reviewing orthogonal projection of commas
I'm going to try to figure that out too.

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul_hjelmstad@a...> wrote:

> > Of course it follows from all this that the complexity of 78125/73728
> > in meantone is 26, and that the size of it represents the degree of
> > flatness of the wolf for Meantone[26] in any given tuning. In
> > 7/26-comma meantone, the size of 78125/73728 is just 78125/73728, or
> > 100 cents, so we get a wolf "fifth" 100 cents flat.
>
> That's pretty flat!

It has to be. While 7/26 comma has a fifth of 696.165 cents, 26-et
has a fifth of only 692.308 cents. Hence to complete the circle of
fifths, the wolf has to be flattened. Multiply the difference between
the fifths by 26, and you get 100.286 cents, which is exactly the size
of 78125/73728.

>I'm excited to see you applying eigenvectors to
> tuning theory. Once I finish reviewing orthogonal projection of commas
> I'm going to try to figure that out too.

You kind of inspired me to look at these matrices by asking about
projections. I'm still hoping to get something out of it beyond a new
point of view, but it is nice to see you can put the tuning data in a
package together with the temperament data, by which I mean the
algebraic features involving commas and vals of the temperament.