Consistency generalized

I've been mulling over Paul Erlich's concept of consistency for a little
while now. When he first described it it really caught my attention,
because it dovetailed nicely with a concept I've used in my little
theoretical musings for years. The tunings I am interested in are what
you might call "level 2 consistent": not only are all intervals up to a
given limit represented consistently, but all combinations of them as
well. Thus, for example, at the 5-limit 7TET is consistent, but it is
not level 2 consistent because 5/4 is represented by two steps, but
25/16 is closer to five steps than four.

To generalize, an ET is level N consistent at the K-limit if all
approximations of K-limit intervals vary from just by no more than
1/(2N) of a step. This means that all combinations of up to N K-limit
intervals are represented consistently.

While Paul E. explores ETs which are level 1 consistent, I tend to be
more interested in level 2 consistency, which is a much more stringent
criterion. How much more? Well, I wrote a program to figure out
consistency levels at all limits up to 31 for all ETs up to 1200TET.
(Overkill, I know, but CPU cycles are cheap.) The output is available
at . ("." indicates
level 1 consistency.) Perusing it, one finds that the first ET with
level 2 consistency at the 7-limit is 31, at the 9-limit 41, at the
11-limit 72, at the 13-limit 270, and at the 15-limit 494. No ET below
1200 exhibits level 2 consistency at the 17-limit. Contrast this with
29 and 41 being level 1 consistent at the 15-limit, and 58 being level 1
consistent at the 17-limit.

--pH (manynote@library.wustl.edu or http://library.wustl.edu/~manynote)
O
/\ "Churchill? Can he run a hundred balls?"
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