The Scale Tree

In the latest issue of American Scientist (July-August 2000), there is
an article by Brian Hayes entitled "On the Teeth of Wheels" in which he
discusses the Stern-Brocot Tree, which turns out to be identical with
Wilson's Scale Tree and which may also be used to approximate
complicated ratios with smaller integers. Hayes mentioned a variant of
it last year (July -August 1999) in an article on "Vibonacci Numbers," a
form of the Fibonacci series in which the plus sign of the recursion
formula is randomly replaced by either a plus or a minus sign.

The original reference to the Stern-Brocot tree is Stern, M. A., 1858.
Ueber eine zahlentheoretische Funktion. Journal fuer die reine und
angewandte Mathematik. 55:193-220. This journal is sometimes called
Crelle's Journal after its first editor. Brocot, who applied the tree to
gear teeth ratios published in 1861. (Brocot, Achille. 1861. Calcul des
rouage par approximation, nouvelle me'thode. Revue chronome'trique.
Journal des horlogers, scientifique et pratique 3: 186-194. The Hayes's
article reference is formally Hayes, Brian. On The Teeth Of Wheels.
American Scientist 88: 296-300. The article also contains a picture of
the reconstructed Antikythera mechanism, a mechanical analog
astronomical computer about 2000 years old and found in sunken ship in
the Mediterranean sea. Derek de Solla Price was the one who figured out
what it was.

A secondary reference for the Stern-Brocot Tree is Concrete Mathematics,
A Foundation for Computer Science by Ronald L. Graham, Donald E. Knuth
and Oren Patashnik. 1989. Addison-Wesley Publishing Company, Reading MA.
This is probably much easier to find than the original journals.

--John

Daniel Wolf wrote,

<<A given number of scale tones will appear on the tree in every
relatively-prime set of sums. For example, one finds scales with 12
tones as the sum of 1+11 and 5+7. (The non-relatively-prime sums,
2+10=12, 3+9=12, and 4+8=12, generate deficient and redundant
solutions, so we can ignore them). This is useful because a given
scale's appearance in different places on the tree indicates
different -- and often surprisingly so -- ways of working.>>

Hi Daniel,

Shouldn't that say something like "a given number of scale tones will
appear on the tree in every set of sums that have no common divisor
greater than 1", as opposed to "a given number of scale tones will
appear on the tree in every relatively-prime set of sums"?

Also, you don't necessarily have to ignore sums with a common divisor.
You could scale the periodicity by the GCD and go on about the
business of the tree in pretty much the same ways that you mention.
Though you'd have to carry transformations across each segment of your
periodicity, etc.

By the way, I think posts like the ones your undertaking here are a
great idea and should hopefully spur more participation from a wider
percentage of the list members; especially those who might find a lot
of terminology and ideas flying around between a small group of people
much too quickly to get a grip on.

--Dan Stearns

I wrote,

<<Shouldn't that say something like [SNIP]>>

Whoops, for some reason my brain registered co-prime for
"relatively-prime", sorry about that.

--Dan Stearns