links (was RE: harmonic entropy)

I still can't find anything on the 'simplicity' of mediants. But here's
an interesting link...

http://www.roanoke.infi.net/~bowerman/stern.html

...if you follow the link at the top, "3D Stern-Brocot Trees", there's
an interesting representation, not unlike one recently proposed by Keenan
Pepper.

And at the bottom of this page...

http://www.cut-the-knot.com/blue/Stern.html

...we see mention of our friend Pierre, and a discusison of his discovery
that the sum of the reciprocals of the Tenny complexities of the ratios in
a row of the Stern-Brocot tree sum to 1 -- which he mentioned in a recent
post. Wow.

-Carl

--- In tuning@egroups.com, Carl Lumma <CLUMMA@N...> wrote:

http://www.egroups.com/message/tuning/13069

> I still can't find anything on the 'simplicity' of mediants. But
here's an interesting link...
>
> http://www.roanoke.infi.net/~bowerman/stern.html
>
> ...if you follow the link at the top, "3D Stern-Brocot Trees",
there's an interesting representation, not unlike one recently
proposed by Keenan Pepper.
>

Thanks so much, Carl, for this excellent post. It's the first
glimmer of understanding (ahem) I have yet had about the Stern-Brocot
trees and the mediant process...

Also, fascinating visual depicting Keenan Peppers's mathematical
proposal... only this time using columns of a temple, rather than
trees.

I assume everybody has gone over to this link to see this. But, if
you haven't... don't miss it!!
___________ ___ __ _ _
Joseph Pehrson

>Also, fascinating visual depicting Keenan Peppers's mathematical
>proposal... only this time using columns of a temple, rather than
>trees.

That, and the columns represent points on the Stern-Brocot tree, instead
of points on the co-ordinate plane of the natural numbers.

-Carl

Joseph wrote,

>>Also, fascinating visual depicting Keenan Peppers's mathematical
>>proposal... only this time using columns of a temple, rather than
>>trees.

Carl wrote,

>That, and the columns represent points on the Stern-Brocot tree, instead
>of points on the co-ordinate plane of the natural numbers.

Carl, Joseph is absolutely right, and the colored-in points are the trees
you can see from the origin. Since the points on the Stern-Brocot tree are
the ratios in lowest terms, and those are the points you can see from the
origin, you're both right.

--- In tuning@egroups.com, Carl Lumma <CLUMMA@N...> wrote:

http://www.egroups.com/message/tuning/13139

> >Also, fascinating visual depicting Keenan Peppers's mathematical
> >proposal... only this time using columns of a temple, rather than
> >trees.
>
> That, and the columns represent points on the Stern-Brocot tree,
instead
> of points on the co-ordinate plane of the natural numbers.
>
> -Carl

Yes. I got that! Like they used to say, "groovy!" (or was it "gravy")

____________ ___ __ _
Joseph Pehrson

>>>Also, fascinating visual depicting Keenan Peppers's mathematical
>>>proposal... only this time using columns of a temple, rather than
>>>trees.
>>>
>>That, and the columns represent points on the Stern-Brocot tree, instead
>>of points on the co-ordinate plane of the natural numbers.
>
>Carl, Joseph is absolutely right, and the colored-in points are the trees
>you can see from the origin. Since the points on the Stern-Brocot tree are
>the ratios in lowest terms, and those are the points you can see from the
>origin, you're both right.

Right- but there is still a square grid, with trees on every point in
Pepper's version, while each unique ratio has only one column in the web
verison.

-Carl