source file: mills2.txt
Date: Fri, 20 Dec 1996 00:23:48 -0800

Subject: Triadic or tetradic infill?

From: Daniel Wolf <106232.3266@compuserve.com>

I have spent a bit of time trying to figure out why there seems to be a
preference for a triadic (4-5-6-8) over a tetradic (4-5-6-7-8) octave
infill in tonal musics. Perhaps this was not exactly the right question,
and it should rather be: why is there a preference for an infill that
leaves a gap at top (i.e. 6-8 is left unfilled by 7)? Could there be an
explaination that does not go into supposed numerological properties of the
number 7? 
 
The best explaination of the infill question I have come up with is that
the infill is heirarchic, following the harmonic series: an octave infill
adds the fifth (2-3-4) and the next infill in of that fifth, not of the
octave (4-5-6-8), and the next infill (theoretically) would be of the third
(8-9-10-12-16)(which can be heard as a stable chord).

It is perhaps trivial to note that these infills are freshman sums.

There seems in this procedure to be a kind of equilibrium building, in that
the infill of the lowest interval counteracts the harmonic series effect of
increasingly smaller intervals at top. Perhaps this spacing equilibrium can
partially account for the prefered use of triads in the resolving position
of tonal cadences.

This procedure inverted appears to be effective for subharmonic sonorities
as well.

Now, if we would prefer to use 12th equivence instead of octave
equivalence, how might our infill procedure change? One immediate
possibility comes to mind: maintain the regular harmonic series and the
infill procedure will be very similar to the octave version 1-3, 1-2-3,
2-3-4-6, 4-5-6-8-12.  

And for kicks, here is a version for 7/4 equivalence: 4-7, 8-11-14,
16-19-22-28, 32-35-38-44-56

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