Davis, beehive, and the Hanson/kleismic temperament

In the Spring 1986 volume of Interval, there is an article by James
Davis on his beehive keyboard. This is a keyboard based on a tuning
with two generators, the horizontal generator hor = 3^(1/6), and the
vertical generator vert = sqrt(4/3). This leads to a mapping where
2 = hor^3 vert (exact), 3 = hor^6 (exact) and 5 ~ hor^8 vert (1.35
cents flat.) In other words, we have the mapping

[<3 6 8|, <1 0 1|]

This leads to the result that the kleisma is tempered out, so that the
beehive layout is a lattice for the r2 temperament which is variously
called hanson or kleismic.

I'd be interested to know when Hanson proposed hanson, because if it
was after 1986, Davis has priority.

If we look at where 7-limit approximations appear on this temperament,
we find that hor^2 vert^11 is 4.4 cents sharp. This is more accurate
and less complex than hor^13 vert^(-3), which is 5 cents sharp. The
first approximation would give the <<6 5 -31 -6 -66 -86|| temperament,
the second is catakleismic. Since Paul would pitch a fit if I
suggested "davis" as a name for the first temperament, I'm suggesting
"beehive" for it. The reasoning behind that is even though Davis
thought of the beehive keyboard as a 5-limit keyboard, the 7-limit
approximations are, after all, on it.

The 7-limit "beehive" temperament then would be the one with 5120/5103
and 15625/15552 as a TM basis, and the above wedgie. Does anyone know
James Davis? His input on this would be nice to have.

This article is courtesy of Jonathan Glaiser, who gave me a copy of
the back issue in question.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> In the Spring 1986 volume of Interval, there is an article by James
> Davis on his beehive keyboard. This is a keyboard based on a tuning
> with two generators, the horizontal generator hor = 3^(1/6), and the
> vertical generator vert = sqrt(4/3). This leads to a mapping where
> 2 = hor^3 vert (exact), 3 = hor^6 (exact) and 5 ~ hor^8 vert (1.35
> cents flat.)

It should be noted that this tuning is simply the minimax tuning.

>I'd be interested to know when Hanson proposed hanson, because if
>it was after 1986, Davis has priority.

According to a 1989 paper by Hanson, published in XH and available
here...

http://www.anaphoria.com/hanson.PDF

...Hanson came up with his 53-tone keyboard in 1942, but did not
realize that it could have been generated by minor thirds until
Erv pointed it out to him in 1978.

Hanson's keyboard is interesting. His L-shaped digitals allow two
degrees of freedom relative to other notes while holding a note.

-Carl

On 5/23/05, Gene Ward Smith <gwsmith@svpal.org> wrote:

> I'd be interested to know when Hanson proposed hanson, because if it
> was after 1986, Davis has priority.

Presumably the article published in Xenharmonikon 12, 1989:

http://xh.xentonic.org/tables-of-contents.html

in which case Davis has clear priority of publication. If you ask
Kraig, he might tell you that Hanson had the idea earlier -- but so
might Davis, of course.

Graham