The Magic of Superparticulars

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.