Minimal genera for equal temperaments

A nice way of denoting the elements of an equal temperament in some
prime limit is by way of an Euler genus, or rotated Euler genus, which
covers the temperament. An example of this would be the duodene for 12
in the 5-limit, but other genera cover it exactly as well.

For 19 in the 5-limit, the 3^3 5^4 genus covers it, with one pair of
notes differing by 16875/16384, the negri comma.

1, 16875/16384, 135/128, 1125/1024, 9/8, 75/64, 625/512, 5/4, 675/512,
5625/4096, 45/32, 375/256, 3/2, 25/16, 3375/2048, 27/16, 225/128,
1875/1024, 15/8, 125/64

If we rotate this by sending 3 to 16/5 and 5 to 24/5, then we get a
pair of notes differing by a comma:

1, 128/125, 648/625, 82944/78125, 3456/3125, 144/125, 6/5, 768/625,
32/25, 20736/15625, 864/625, 36/25, 4608/3125, 192/125, 8/5,
5184/3125, 216/125, 27648/15625, 1152/625, 48/25

The 3^6 5^2 genus, rotated by sending 3-->10/3 and 5-->16/3 gives the
following:

1, 250/243, 3125/2916, 10/9, 125/108, 15625/13122, 100/81, 625/486,
4/3, 15625/11664, 25/18, 3125/2187, 40/27, 125/81, 3125/1944, 5/3,
1250/729, 16/9, 15625/8748, 50/27, 625/324

This has two pairs of notes separated by a kleisma.

As well as a notation scheme, this sort of thing can lead to
interesting tempered scales as I mentioned before. Of course it isn't
much of a secret that 19 notes tempered by the kleisma, syntonic
comma, or negri comma is interesting but the idea still works for
higher prime limits. The obvious necessary condition is that if N is
the et, and if 3^e3 5^e5 ... p^ep defines the genus, then we must have
(e3+1)(e5+1) ... (ep+1) >= N.

The square 3^4 5^4 = 50625 works for 22, with three diaschisma pairs
tempered out. For 31, 3^3 5^7 with 32 notes, one pair being separated
by a semicomma, works nicely. There is also 3^10 5^2 rotated by
3-->10/3 and 5-->16/3 with 33 notes, emphasizing the relation of a
fifth, with two pairs separated by |9 9 -10>, and 3^6 5^4 rotated by
3-->16/5 5-->24/5 with 35 notes, and four pairs separated by a comma.