back to list

Meantone-superpyth notation

🔗Gene Ward Smith <gwsmith@svpal.org>

5/15/2005 3:24:28 PM

I've mentioned before that if you combine meantoen and superpyth, you
end up with a notation for the whole 7-limit. One way to notate that
would be by an ordered pair in standard notation; hence for example
(C,C) would be 1, (E,D#) would be 5/4, and so forth. Equivalently,
you can take an ordered pair of 3-limit intervals, so that
(81/64, 19683/16384) is 5/4. It is important that the pair is ordered;
(D#,E) is not 5/4, but 1594323/1310720. The ratio between these would
be notated (256/243,243/256), which is sort of cool.

If Q is a 3-limit interval, it is notated (Q,Q). Hence the 3-limit has
the property that every note Q in it belongs to the interval (Q,Q). It
is less obvious but also true that if R is a 5-limit note, notated in
this system by (P,Q), then R is contained in the interval from P to Q.
Hence for example we have 19683/16384 < 5/4 < 81/64, but also
19683/16384 < 1594323/1310720 < 81/64. This is no longer true for
7-limit intervals, the easiest counterexample being 7/6, which is
notated (19683/16384,32/27), but 7/6 falls below that interval.
Similarly, (32/27,19683/16384) is 2187/1792, lying above the interval.