[Smt-talk] Two thoughts on Normal Form

[Murphy, Scott Brandon]

Good to read another SMT-talk contribution from Dmitri.

>1. The first is that the standard definition of "normal form" is from a
>geometrical point of view a little awkward.  We train students to order
>pitch classes (in an ascending way, spanning less than an octave,
>transposing so that the first note is 0) so as to minimize the distance
>from the first pitch to the last.  Geometrically, the more natural thing
>to do is to order them (in an ascending way, spanning less than an
>octave, transposing so that the first note is 0), so as to minimize the
>distance from the first note to the SECOND note.
...
>Anyway, if you're bored with teaching standard "normal form" it might be
>interesting to explore these two different approaches.

I understand how this is more natural from a geometric perspective (as
well as from other perspectives, like lexicographical ordering, which is
also quite "natural," I suppose), but there are other perspectives from
which the traditional approach may still be more natural than what Dmitri
proposes.  For example, I'm thinking of a perspective in which normal form
in "chromatic semitonal space" is analogized with simple (i.e. solely
pitch-class based) tertian root finding in "diatonic third space": both
invoke a "most packed" total ordering of the pitch classes within a
certain cyclical space as, among other things, a means to an appellative
end.  I've found that students comfortable with simple root finding manage
pretty well both the cognitive leap to thinking of this process as
ordering "most packed" notes in diatonic third space, and, subsequently,
the cognitive leap that substitutes the circle of chromatic half steps for
the circle of diatonic thirds.

Perhaps this could provide yet another way to alleviate the aforementioned
boredom, unless you already use this pedagogy and are bored with it too.

Cheers,

-Scott

--
Scott Murphy
Associate Professor, Music Theory
University of Kansas School of Music
President, Music Theory Midwest
smurphy at ku.edu