[Smt-talk] Nature and Labeling of the Cadential Six-Four

Dmitri Tymoczko dmitri at Princeton.EDU
Tue Feb 14 06:46:00 PST 2012

For fun, I checked some of Dimitar's statement's against my databases of RN analyses (Bach chorales and Mozart sonatas).

> Thus students are told that III connects very well to VI, but nobody mentions that VI-III is just as good.

In the Bach chorales, there are more iii-vi progressions as vi-iii (39 vs. 24, for a ratio of 1.625), but both are pretty rare; about 0.3% of all progressions (about 10,000 diatonic major-key progressions).  (Also, I'm pretty sure that a fair number of the vi-iii progressions appear in Pachelbel-sequence contexts.)  In the Mozart sonatas (leaving aside sequential contexts), there are no vi-iii progressions and just two iii-vi progressions (out of more than 13,000 progressions).  This fits with my sense that the best simple description of classical syntax is that iii is pretty much not used; it is rare in Bach, very, very rare in Haydn, Mozart, and Beethoven (outside of sequential contexts), and gradually stages a modest comeback in the 19th-century.

(NB: all stats major-mode only for simplicity).

A nice way to think about this is that tonal harmony involves some pretty strong "zeroth-order" (in the Markov model sense: brute probabilities about which chords appear) constraints: the chord on scale-degree 3 is usually a first-inversion triad, not a root-position triad, while the chord on ^5 is usually root position, rather than first-inversion (of course, there's always that cadential 6/4!).  Also, the chord on ^1 is usually a root-position, rather than first inversion triad.  These constraints have the effect of dramatically increasing the prevalence of tonic and dominant while suppressing all inversions of iii and the first inversion of vi.  

Of course, there are also "first-order constraints" about how the harmonies follow each other, but these "zeroth-order constraints" are pretty important.

> II6 connects perfectly not only to I6/4 (actually, the cases in classical music of II6 to I6/4 may be more than IV to I6/4; we cannot count them anyway) but also to I, in a plagal relationship.

In the Bach chorales and Mozart piano sonatas there are basically no instances of ii6->I.  (IIRC, there may be one potential instance in the chorales, but the more natural reading involves a funny passing tone; I can dig it up if anyone cares.)  Interestingly, there are just a handful of instances of ii6->I6/4 in Bach chorales (he prefers ii6/5 or IV, and in any case doesn't use I6/4 that much) as compared to about 275 instances of ii6->I6/4 in the Mozart sonatas. 

> III connects V4/3 perfectly, and this device is used in modulations; when your common chord happens to be III in the new key, V4/3 makes an excellent modulating chord to fully enter the new key.

I have just 4 Bach chorale instances of iii->V4/3 and none in Mozart.

Now of course, there are a variety of things you can say about this sort of statistical information.  For instance, you can say "well, all these progressions sound good to me, and that's what is important."  Then you are basically describing (or teaching) your own personal idiolect.  This is most useful if you are, yourself, an expert functionally-tonal composer; if you're not, it is a somewhat dicier proposition.

My own preference is to try to teach and theorize about what happens in actually existing repertoire.  So I tell my students that the iii chord is very rare in classical music, that ii6->I is virtually unknown.

> But the worst thing of all is that the tests in such books are designed in such a manner that they do not give you room or felxibility to point out some of the situations above! As a result, teachers who go by the book will take points off for any answer which is not fully compatible with that chart! A terrible example of thinking in a box! Clever teachers will devise their own tests on that topic.
> Also, many less common progressions do a good job such as V between two subdominants, i-v6-VI; i-v6-iv6 and others. If a skillful person knows how to voice-lead, any chord could connect any. In the beginning, however, our students must move along the typical situations, but even those are not comprehensively examined in Kostka/Payne and many other books.

Kostka and Payne's description of chord grammar is, in my opinion, the best of all the standard textbooks; it's pretty close to the one I offer in GOM chapter 7.  The biggest problem is that they claim vi->V is rare, when it is in fact extremely common.


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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