[Smt-talk] Sequences

Dmitri Tymoczko dmitri at Princeton.EDU
Tue Mar 10 09:11:33 PDT 2009

Hi Dave,

Always good to hear from you, even when we disagree.

> Rather than make neo-Pistonian lists of root motions, I suggest  
> Matthew Brown’s book, Explaining Tonality, chapter 3 (indeed,  
> hello) as a good place to start for trying to understand tonal  
> sequences.  Steve Laitz’s textbook is also a place to find a useful  
> compendium of such phenomena.

I think I share your aversion to "neo-Pistonian lists," though  
perhaps we understand the term differently.  Recently I've been  
really interested in questions and models that unite the harmonic and  
contrapuntal domains.  (e.g. Q: "Which diatonic triads can be linked  
by single-step voice leading."  A: "Triads whose roots are a third  
apart.")  By dealing with both at once, we can sometimes come up with  
new insights.  For instance, I find it interesting that most of our  
examples of minor-third sequences do not use the more efficient  
stepwise descending voice leading, but instead ascending parallel  
voice leading; one reason for this may be that the stepwise sequence  
cycles through all its inversions in turn, leading to 6/4 chords.  I  
also think it's cool when voice-leading facts can help explain  
unusual root progressions -- as in "Hey Joe," perhaps.

> I haven’t counted them all yet, but check out Coltrane’s “Giant  
> Steps” for 7th chords in M3 cycles (albeit mediated by V-Is,  ii-V- 
> Is) and Messiaen’s Quartet for the end of Time V for triads in  
> minor 3rd cycles.

Good examples!  A few points:

	1. Definitely, there are some examples of M3-related dominant  
sevenths and m3-related triads.  My claim is statistical, though.
	2. In the twentieth century, dramatic harmonic juxtapositions appear  
frequently, often without efficient voice leading.  In nineteenth- 
century music, "odd" harmonic shifts very often exploit efficient  
voice leading.  This is one interesting difference between, say,  
Schubert and Prokofiev: both love juxtaposing M3-related chords, but  
Schubert more often emphasizes the smooth voice leading between them.
	3. As you probably know, "Giant Steps" never juxtaposes M3-related  
chords directly; instead, it juxtaposes m3- (or tritone-) related  
seventh chords: for example, Bmaj7 progresses directly to D7, which  
can perhaps be interpreted using the semitonal voice leading (B, D#,  
F#, A#)->(C, D, F#, A).  (Note that the bass plays chord-roots -- the  
"Giant Steps" of the title -- and doesn't participate in the  
schema.)  This is an example of what I call the "minor-third  
system."  If you actually look at Coltrane's original solo, you  
periodically see echoes of this voice leading in his playing.
	It's worth comparing this piece to the modulating E-minor passage in  
the exposition of the first movement of Beethoven's second piano  
sonata (Op. 2, no. 2, I, mm. 58ff).  There, m3-related dominant  
chords (diminished sevenths or V7 chords) give rise to minor third  
related tonics.  Here, you find a similar juxtaposition of m3-related  
sevenths, but since they're both dominants, you get different root  
relations among the tonics  If you just look at the tonic chords in  
these two passages, you miss out on the interesting relationships  
between the two sorts of modulatory procedure.


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

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