All set up

Dear Members of the tuning digest,

I was considering recent comments as made to the alternative tuning digest
about the relationships of set theory to tuning.

It made more sence, the more I thought about it, that is, if two numbers are
NOT relative primes, then it is likely that they will share at least some,
but perhaps not always all notes in common.

Relative primes in equal temperament terms will not be subsets or supersets
of each other.

Take for example 24ET, and 12ET, the 12 ET will be a subset of 24 ET.

In other words, ALL NOTES THAT 12ET HAS, 24 WILL HAVE.

18ET, and 12ET will share some notes in common but NOT all notes in common.

Two relative prime temperaments that are not, themselves primes, share no
notes in common.

In the case of 12ET and 18Et, what would the related Venn diagram look like?

I know that in the case of 24ET and 12ET, one circle would be inside
another, drawn here as hexes:

----------------
/ --------- \
/ / \ \
/ / 12ET \ \
/ \ / \
\ --------- /
\ 24 ET /
\ /
------------------

12 and 24 share all common multiples: 2, 4, 6, 12, and 1 (naturally)....

where as the relationship between 18ET and 12ET can be shown:

-------------
/ \
/ 18ET \
/ \
\ -------- /
\ / /\
\ / / \
---------- /
\ 12ET /
----------

.....where 1, 2, 3, 6, are common multiples with 18 and 12, but 9 isn;t a
multiple shared.

Two temperaments like 9, and 4, or 9 and 17, compared, show no overlap, and
can be shown:

---------- ----------
/ \ / \
/ \ / \
/ 9ET \ / 17ET \
\ / \ /
\ / \ /
\ / \ /
---------- ---------

This assummes, of course, "crispness" of sets, but what intrigues me, is the
relationship between sets, as equal temperaments, which are, at least,
PARTIALLY, and ALMOST related, like close equal temperaments, such as 12Et
and 13ET, or perhaps even 15Et and 32ET.

Is there an algorithm for acessing the relative amount of surface area
overlapped, assummeing that we use circles?

I thought that it would just be a matter of finding out the ratio (in other
words, the difference) between-each- and-every-note as to compared to
each-and-every-other-note in our temperaments, thus getting something liek
12ET and 17Et we get 12*17=204 calculations.

Am I on the right track here?

----Sarn.