[Smt-talk] Two thoughts on Normal Form

[Dmitri Tymoczko]

Hi Everyone,

As we transition into a new semester, and the nth iterations of our "introduction to set theory" classes, I had a couple thoughts to share about the idea of a set-theoretical "normal form."  Please let me know (on or off the list) if these ideas spark any new pedagogical approaches.

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.
	The difference can be seen on Figure 3.8.6 (page 91) in "A Geometry of Music."  Standard normal form produces the kite-shaped region from 000 to 006 to 048 to 066 and back to 000.  (See Callender's 2004 paper, which describes this region.)  The alternative produces the triangular region from 000 to 00[12] to 048 and back to 000.
	There are several advantages of this latter method.  Not only is the resulting figure simpler (in fact, it is always a "simplex" or generalized triangle), but it is topologically speaking a "cone" -- it can be decomposed into layers each of which has the same basic structure, with the layers identified by the smallest interval in the set class.  (These layers are the horizontal lines on Figure 3.8.6.)  This decomposition turns out to be important -- for instance, it is the key to generalizing the familiar "M operation" to continuous set-class space.  Roughly speaking the "continuous M operation" simply moves you between analogous points on these layers.  This is essentially projective geometry in a musical context (with the perfectly even chord as the origin).
	Anyway, if you're bored with teaching standard "normal form" it might be interesting to explore these two different approaches.  It could lead to some interesting discussions about the role of convention in music theory, for example.

2. The second thought is that the notion of "normal form" can be extended in a pretty natural way to voice leadings.  Let's limit ourselves to transposition for the time being, leaving inversion out of it (*).  A voice leading can be represented as a collection of ordered pairs, with the first element a pitch class, and the second a path in PC space.  So, to represent "G stays fixed, D moves up by two semitones to E, while B moves up by semitone to C" we can write:

[(G, 0), (D, 2), (B, 1)]

Now, to put the voice leading into normal form, we simply order and transpose the first element of each pair so that they're in normal form, keeping each path in pitch class space attached to its appropriate partner.  (This is because transposition does not affect a path in PC space.)  We get (using standard "normal form" for the time being):

[(C, 0), (E, 1), (G, 2)] or in numbers [(0, 0), (4, 1), (7, 2)]

This can be pretty useful in tonal theory and pedagogy.  For instance, here are all the five most common upper-voice voice-leading normal forms for all the V7 -> I progression in Bach's chorales.

[0, -4], [3, -2], [6, -1]	362 occurrences
[0, -1], [2, 0], [6, 1]		120 occurrences
[0, 1], [3, -2], [6, -1] 	53 occurrences
[0, -2], [3, -1], [5, 0]	19 occurrences
[0, -4], [3, 2], [6, -1]	10 occurrences

Remember these are upper voices, to be supplemented with a 5-1 in the bass.  

In this case, "voice leading normal form" helps us reduce a wide range of contrapuntal practice to a few basic schemas.  The first, which is overwhelmingly the most popular, has the leading tone falling to the fifth.  The second has a doubled root in the V7, with notes moving in the expected way.  The third produces an incomplete I chord with tripled root.  The fourth has no leading tone (and occurs most often in eighth notes, following another inversion of V7.  The fifth produces a doubled third.

Anyway, these are both ideas I've found useful in my own work, but which don't seem to deserve an academic paper -- so in this sense they seem appropriate for the SMT-list.

Best wishes,
DT

(*) inversion multiplies a path in PC space by -1.

Dmitri Tymoczko
Professor of Music
310 Woolworth Center
Princeton, NJ 08544-1007
(609) 258-4255 (ph), (609) 258-6793 (fax)
http://dmitri.tymoczko.com