supernatural superparticulars?

/tuning/topicId_31641.html#31663

> >Curiously, J Gill
>
> No -- but what seems to be the case very often, is that when one
> comes up with such a scale in the form of a periodicity block, one
> has quite a few arbitrary choices to make as to which version of a
> particular scale degree one wants (the different versions differing
> by a unison vector), and then _one such set_ of arbitrary choices
> does lead to a scale with superparticular step sizes.

Hi Paul...

Well, that's pretty *mysterious* isn't it? Why does that happen that
the superparticular step sizes result? Is it just the way the system
is set up. Spooky stuff! (If we can't believe in "magic primes"
that surely is something a little weird... yes?)

Joseph

--- In tuning-math@y..., "jpehrson2" <jpehrson@r...> wrote:
>
> /tuning/topicId_31641.html#31663
>
>
> > >Curiously, J Gill
> >
> > No -- but what seems to be the case very often, is that when one
> > comes up with such a scale in the form of a periodicity block,
one
> > has quite a few arbitrary choices to make as to which version of
a
> > particular scale degree one wants (the different versions
differing
> > by a unison vector), and then _one such set_ of arbitrary choices
> > does lead to a scale with superparticular step sizes.
>
> Hi Paul...
>
> Well, that's pretty *mysterious* isn't it? Why does that happen
that
> the superparticular step sizes result?

Usually they don't "result", but very often you can arbitrarily
choose to use them.

> Is it just the way the system
> is set up.

Well, superparticulars are not favored _by design_, if that's what
you mean. You should ask Kraig Grady about superparticular step sizes
too.

> Spooky stuff! (If we can't believe in "magic primes"
> that surely is something a little weird... yes?)

One of the themes on the tuning-math list (busier than ever) is
superparticulars . . . for example, we found that the graph of
ET "goodness" has "waves" in it, and the most prominent visible wave
by far for 7-limit rises and falls every 1664 ETs . . . and the
superparticular ratio that Graham made famous in his Blackjack
progression, 2401:2400, fits 1663.9 times in an octave.

Basically, the magic of superparticulars is that they're the smallest
possible intervals for a given complexity (or distance on the
lattice). Which is kind of obvious if you think about it. If the two
numbers in a ratio differ by 1, how can the interval be any smaller
without increasing the two numbers in the ratio?

Things fizzle out, though, once you get past the point where there
are no superparticulars left. Think 5-limit; you can look at the page

http://www.kees.cc/tuning/s235.html

and in the second column from the right, you'll see a bunch of
ratios. Certain rows have the first few columns in parentheses, but
those that don't correspond to the SMALLEST unison vectors for some
given DISTANCE LIMIT in the lattice, or ODD-LIMIT, or INTEGER-LIMIT
if you prefer. , you can see, are all superparticular, until you
reach the last two, your old friends 25:24 and 81:80. Then there are
no more superparticulars possible in the 5-limit.

Now look at

http://www.kees.cc/tuning/s2357.html

and again, all the non-parenthesized entries are superparticular,
until you reach the last two, 2401:2400 and 4375:4374. The latter is
the last superparticular possible in the 7-limit.