For Jacky, on the Golden Ratio

This is of course phi = (1+sqrt(5))/2. If we take rational numbers of
the form a + b*phi we get something called an algebraic number field,
and if a and b are integers, we get its ring of algebraic integers;
these are rather closely analogous to the ordinary rational numbers
and ordinary integers, respectively.

We also have unique prime factorization. The primes fall into three
groups--the first are those of the form 5n+2 or 5n-2, such as
2,3,7,13 etc. These are inert--they are still primes for golden ratio
numbers. The second kind, of the form 5n+1 or 5n-1 (which we may also
call 10n+1 or 10n-1, since they are all odd) split--that is, factor
into two primes. For example 11 = (3+phi)(3-1/phi) = (3+phi)(4-phi),
and 3+phi and 4-phi are now the new primes. Finally, 5 is unique in
that it ramifies--sqrt(5) is now the new prime number. Phi itself is
called a unit, and has to be added to the factorization list also. If
we put it all together, the new 11-limit looks like

phi^a 2^b 3^c sqrt(5)^d 7^e (3+phi)^f (4-phi)^g,

where the exponents are integers.

We can do all the usual things with these new primes (and our unit,
phi) such as ets, MOS and so forth. Warning--I have no reason to
think this makes much sense musically! A good scale division for this
would be the 19 division, where we have approximations phi~2^13/19,
3~2^(30/19), sqrt(5)~2^(22/19), 7~2^(53/19), 3+phi~2^(42/19) and
4-phi~2^(24/19). If that isn't good enough, there's always the
111-division, which comes through like a champ. If you prefer, you
can also make phi and not 2 the exact interval, of course.