Reply to Paul Erlich

Paul: Your recent post on the extent to which various ET's
articulate the harmonic series is an interesting and potentially
useful set of data. The data are very relevant to the process of
mapping JI constructs in ET's.

To expand a bit, 22-tet expresses the 4:5:6:7 dominant 7th chord as
7:6:5:4 degrees of 22-tet and thus distinguishing each interval of a
second or 3rd (8/7, 7/6, 6/5 and 5/4) with a unique span roughly
proportional to its JI width. Fifteen-tet distinguishes the 16/15, 10/9
and 9/8 as 1, 2 and 3 degrees though it wildly distorts the the
differences between the intervals. Still the major scale is recognizable
as a "nearly just" scale (As "nearly just" is an oxymoron in English,
I proposed the term "dicaeoid," or "dikaioid," which means the same
thing and sounds more "academic," if nothing else.)

William Stoney did something similar with function called HarFac in
the paper below in which he scooped Erv and me. Back in the late 60's
we had no idea where we could publish an article on computer generated
tuning tables and the analysis of various ET's. (We didn't include a
HarFac, just the raw error of each harmonic and subharmonic interval,
alas.)

Stoney, W. "Theoretical Possibilities for Equally Tempered Musical
Systems", pp. 162-171 in The Computer and Music, H.B. Lincoln ed.,
Cornell University Press, Ithaca, 1970.


--John


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