[Smt-talk] Two thoughts on Normal Form

Dmitri Tymoczko dmitri at Princeton.EDU
Wed Sep 12 07:51:59 PDT 2012

Sorry to be late with this.

On Sep 7, 2012, at 12:36 PM, Charles J. Smith wrote:

> So along these lines, what is the "normal order" of a 7th chord? 

Standard normal form for the V7 chord is (0, 3, 6, 8) and the geometrical normal form is (0, 2, 6, 9).  

Actually, from the geometrical point of view you could put the smallest interval in any position you want -- first, second, last, or even in the "wraparound" position from last to first.  First is an obvious but non-obligatory choice.  The important point is just that you minimize some consecutive interval, rather than some interval between nonconsecutive notes.

> Does it make any sense to think of functional-harmonic entities in these terms at all? 

Yes, I think so!  My own view is that we need lots of different sorts of approximations when doing music theory -- which we choose depends on what we want to know.  

A very simple example is when we ask "how does Bach resolve the V7 chord in the chorales?"  The music contains a very wide range of contrapuntal schemas -- you can have the third in the soprano, the third in the alto, etc., and you could be in many different keys, notes could be in different octaves.  By putting the voice leading into "normal form" you eliminate a lot of this variation, producing the basic schemas I described in my earlier email.  The normal form is a bookkeeping device that doesn't detract from or derogate the chord's dominant function.

> If not, why do we expend so much thought on a conceptualization without much in the way of historical roots (so to speak)? If so, shouldn't a functionally relevant normal order have some resonance in how we think of normal order in less tonal collections?

It's important to realize that normal form is in no way special.  It's just a conventional way of referring to an equivalence class (i.e. a large group of chords).  Intrinsically, the major triad set class is no better represented by [047] than by [038] or [059].  For lots of purposes (e.g. counting, or communicating with others) it's useful to just choose one.

Now, Charles and I have had an ongoing discussion about just what role the notion of "root" plays in classical harmony.  My own view is that you can describe a lot of the syntax of classical harmony without recourse to roots.  For instance, you can say "in F major, the chord containing pitch classes {C, E, G, Bb} often moves to the chord containing {F, A, C}," without invoking the notion of "root" at all.  To me it's an open question when you actually need roots for any deep theoretical reason.  (One answer, which I explore in Gollin and Rehding's Riemann volume, involves what I call "triadic extension symmetry," or the tendency of triads and sevenths to behave similarly when they are on the same root.)  This is a complicated and interesting subject which I want to think more about.


Dmitri Tymoczko
Professor of Music
310 Woolworth Center
Princeton, NJ 08544-1007
(609) 258-4255 (ph), (609) 258-6793 (fax)

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