old diamonds

;Going over stuff that I might put on the web, I realized this unfinished
;work wasn't going to make it. I publish it here in case anybody finds it
;interesting...

It all starts with a group of things. Imagine a group called Moe. Say there
are x things in it. Imagine picking the things up and putting them in your
hand. There are obviously x different hands if you pick up only one of them
at a time. But how many different ways are there to hold two of them at a
time? Three at a time? The answers can be found in Pascal's triangle, by
reading across the row corresponding to x.

Now imagine a set of all the possible hands from Moe. This is called the
power set of Moe. How many things does it have in it? It makes sense to add
up all the numbers in the corresponding row of Pascal's triangle. It so
happens that this number is 2^x. If you've ever played a valved brass
instrument, you may have noticed that there are 8 = 2^3 possible ways to
valve the horn, at least on three-valve instruments.

Wilson has called the power set a 0/x thru x/x CPS, or "grand slam". He
pointed out that it contains all of the CPS's makable by taking any number
of things at a time from any subset of the source set. For example, the 2|6
[1 3 5 7 9 11] eikosany contains a 2|3 and a 3|3 dekany for each of the six
5-member subsets of [1 3 5 7 9 11]. In this way, each new row of Pascal's
triangle can be seen as restricting the number of possible source sets for
the previous rows.

What Wilson didn't point out is that the power set also contains the
diamond of every scale that can be built by chaining intervals from the
source set. In the case of [1 3 5 7] one such scale would be 1/1, 5/4,
105/64, 15/8. In the case of [1/1 3/2 5/3 7/5] one such scale would be 1/1,
5/4, 3/2, 7/4.
What is a diamond? It is a list of all the intervals in a scale. In other
words, it's what you get when you rotate the scale's interior intervals
around a common frequency -- what you get if you start each mode of the
scale on 1/1.

-C.

Carl Lumma wrote,

>Wilson has called the power set a 0/x thru x/x CPS, or "grand slam".

This is also known as the Euler genus with the same factors.