source file: m1451.txt
Date: Fri, 19 Jun 1998 02:45:24 +0000

Subject: Re: Commas and Consistency

From: kollos@cavehill.dnet.co.uk (Jonathan Walker)

After a long absence from the list due to the pressing need to complete
a long-overdue thesis, I noticed the current discussion of
syntonic-comma equivalents in ETs. I thought the following might be of
use to anyone interested in the topic:

If the best approximations of 3/2 and 5/4 are consistent in n-ET, for
any integer n, then

      4[1/2 + nlog_2 3/2] mod n == [1/2 + nlog_2 5/4]

but if the two sides are unequal then a syntonic-comma equivalent is
present in that n-ET. (The square brackets denote the "integer value"
function, i.e. "[x]" is the greatest integer less than any given real
number "x".)

If a syntonic-comma equivalent is present, then the size of the comma in
n-ET steps is, obviously enough:

      abs ( 4[1/2 + nlog_2 3/2] mod n - [1/2 + nlog_2 5/4] )

Anyone who has read this far will no doubt be able to adapt the above to
cover compatiblity between powers of any number of ratios. For example:

      -2[1/2 + nlog_2 3/2] mod n == [1/2 + nlog_2 7/4]

for compatibility between the best approximations in n-ET to 16/9 and
the septimal seventh.

Since I haven't seen any similar formulations elsewhere, I trust these
are originals; but if they've appeared on the list before, do tell me
who else obtained them.

Jonathan Walker