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🔗Gene Ward Smith <> <>

2/21/2003 8:25:10 PM

I was looking into why my attempt to find an analog to Miracle with
2401/2400 a comma and with a generator between 50/49 and 49/48 didn't
work, and found out something fascinating--such a thing is
algebraically impossible. However, we can find an analogue of sorts.

For a 7-limit wedgie to have 2401/2400 as a comma, it must be of the
form [4b - 2a, a, b, -4c - 5a, -2c - 5b, c] with a, b, c relatively
prime. This tells us, for instance, that a fifth always requires an
even number of generator steps to be represented, and so a fifth can
never be a generator for such a temperament. If we look at the number
of generator steps needed for 49/48 (or equivalently, 50/49), which
is the first component of the wedge of the above wedgie with the
Monzo representation of 49/48, we get that it is 2a-2b, so we can't
have what I was looking for, which is that this be 1. However, we
*can* get something with two generator steps, with a wedgie
[2n, n+2, n+1, -5n - 4c - 10, -5n - 2c - 5, c]. Such a temperament
will have a generator of 7/5; of course (7/5)^2 = 49/25 = 2 (49/50)
and so this has the required two-step relationship to 50/49 (and so

If we look for wedgies of low badness in this form, we find them for
n = -48, -15, -13 and most especially, n = 20. This gives us a good
microtemperament, which is the closest thing available as an analogue
to Miracle unless you are willing to count things like Ennealimmal,
with non-octave periods. The wedgie is
[40, 22, 21, -58, -79, -13], the mapping
[[1, 21, 13, 13], [0, -40, -22, -21]], 83/171 is a poptimal
generator, and it can be described as 68&171 in terms of standard
vals. The TM basis is [2401/2400, 48828125/48771072].