Genus bridges

A 5-limit genus bridge for the prime q is a comma of the form
2^a 3^b 5^c q^(-1) for which b>=0, c>=0; and which is allowed to be
less than 1. Higher limit genus bridges are defined in the obvious
manner. The idea is that if c is a p-limit genus bridge for the prime
q, then q*c is a p-limit interval approximately equal to q, and such
that 3^b 5^c is an integer. Hence, genus(3^b 5^c) is defined, and
contains an approximate q.

The classic example of a 5-limit genus bridge is 225/224, which allows
genus(225) to be regarded as containing a 7. Another 5-limit genus
bridge is 5625/5632, which puts an approximate 11 into genus(5625).
Another bridge to 7 is given by |-22 1 10 -1>, which is very heavy on
the 5, and to 11 by |-24 10 5 0 -1>, which allows us to put an equal
emphasis on 9 as on 5.

Genus bridges as defined above are not, of course, the only way to
temper in some higher limit harmony. Genus(3^7 5^5), for instance,
contains {625, 1875, 2187, 3125, 5625}, which is a 4575/4374 version
of a pentad. We may also start with a non-contiguous set of primes;
6144/6125 is a genus bridge from {5,7} to 3, and 5120/5103 is a genus
bridge from {3,7} to 5; also for instance 2401/2400 is a xenharmonic
bridge from {5,7} intervals to 3.

We can therefore get a count of the number of 6144/6125 tetrads in
genus n from d(n/6125), and of 5120/5103 tetrads and pentads from
d(n/5103).