Even more miracles

In "More on miracles",
<url: http//groups.yahoo.com/groups/tuning/message/28290>, I
mentioned that the next miracle generator pops up at around 35.4
cents; let's work this out explicitly. The math works out more easily
if we use the units of a skhisma, 1/612 of an octave, so that 1 cent
= 0.51 sk, so these are what I use below.

In these units, the syntonic comma (81/80) is 11 sk, the Pythagorean
comma is 12 sk, and the skhisma is therefore the difference. We also
therefore have that 1/4 comma meantone temperament has a fifth which
is 11/4 = 2.75 sk flat. The major third is 197 sk, and so the
meantone is half of this, or 98.5 sk; it is the first miracle
generator.

The next miracle, with miracle generator, 2^(7/72), also works out
nicely in these units, being exactly 59.5 sk. Since 5/3 is 451 sk, if
we add this to 59.5 sk we get 510.5; half of this is the miracle
fourth of 255.25, 1.25 sk sharp from the just fourth at 254 sk. These
relationships, which are very nearly exact, illustrate why I like
skhismas better than cents as units; but it is even nicer for the
next miracle.

If we take as our miracle generator 2^(1/34)--one scale step of the
34-et--we have exactly 18 sk. In skhismas 3 is 970 sk; (970+18)/2 =
494 sk is therefore the miracle 7/4; it is 0.1 sk flat. If we
interate our miracle generator within 3 we have an analogue of
Blackjack; cutting the pattern off at an octave and taking a scale of
68 steps gives us a Blackjack-like scale consisting of two 34-ets 2
skhismas apart. If we iterate instead the miracle 7/4 within 3, we
can get, for instance, a scale with steps L of 10 sk and s of 8 sk,
which is 26 Ls followed by Lss, then 25 Ls followed by Lss for a
total of 108 divisions of 3; if we cut this off at an octave we have
26 Ls followed by Lss, then 6 Ls followed by L. It has many intervals
which are justly tuned for all practical purposes, but at 68 steps to
an octave its pretty well in tune anyway, of course.

I was tempted to call this "moron miracles" but I don't think it is
quite that useless; however we are definately pushing the envelope
here.