[Smt-talk] "Core syntax" and 6/4 chords etc.

[Dmitri Tymoczko]

I sense that this discussion is exhausting itself, and I don't want to  
overburden people's inboxes any more than I've done.  However, I do  
want to illustrate the fact that music-theoretical differences run  
very deep.  Olli clearly has little use for my approach, which aspires  
to be empirical and which derives from traditional harmonic theory.   
He has every right to feel this way, and as I said to him privately, I  
don't think either of us will argue the other out of their views.   
What I do hope for is some kind of clear statement of the points of  
our disagreement, which I think could be useful not just to us, but to  
others.

The problem, though, is that the discussion can go on for ever: for  
each argument that one of us finds definitive, the other will have a  
counterargument.

> FIRST a very basic note on "core syntax" and 6/4 chords. This point  
> seems self-evident and I have already made it, but since it almost  
> seems to have been forgotten at parts of this discussion, pardon me  
> for repeating it this context:
>
> Dissonance treatment, if anything, can be described as part of the  
> "core syntax" of conventional tonality (an aspect it inherits from  
> the preceding modal era). Dissonances resolve by step, and in  
> "strict style" they are also approached by step or by suspension.  
> Since bass-related fourths are (usually) dissonant, it follows that  
> they require stepwise voice-leading, which can be described as  
> passing, neighboring, or suspension (or incompletely neighboring/ 
> appoggiatura, in freer style). Hence these categories are crucial  
> simply for relating 6/4 usages with more general principles of  
> dissonance treatment, principles that have enormous descriptive  
> power for centuries of Western art music.

In the standard IV6->I6/4->ii6 pattern, you have something like (A, C,  
F)->(G, C, E)->(F, A, D).  Here the fourth (G, C) moves by a  
combination of step and leap to (F, A).  (See the transition of Mozart  
symphony #40, first movement, for example.)  It's not clear how this  
relates to or derives from Renaissance-era norms of dissonance  
treatment -- it certainly wouldn't be allowed in modal second-species  
counterpoint.  In fact, the voice leading is part of what makes me  
think that this chord should be considered a tonal idiom, one that  
depends crucially on the harmonies involved.

It is true that one rarely leaps into and out of a 6/4 chord, and that  
this is connected to the fact that the fourth is dissonant.  I have no  
objection to someone using the terms "neighboring"/"passing" to record  
how the bass is approached.  Though as others have mentioned, I would  
probably recommend distinguishing the "pedal 6/4" (where the bass  
doesn't move) from the "neighboring 6/4" where the bass does move.   
Furthermore, I would reiterate that one needs to explain that only a  
small number of these patterns actually occur: progressions such as  
(G, B, D, G)->(A, A, D, F)->(G, B, D, G), which involve beautiful  
neighboring motion in all voices, are almost unheard-of.

> SECOND, a comment on Dmitri's view that if we restrict ourselves to  
> quarter-note chords in Bach chorales, we "find many fewer passing  
> 6/4 chords than Olli does, and they almost all conform to the  
> standard type," i.e., between IV6 and II6/5. Actually, the share of  
> "non-standard" 6/4s in my little sample does not diminish  
> significantly, if we make the rhythmic restriction Dmitri suggests.  
> My sample of 42 chorales included 15 passing 6/4s, 8 of which were  
> "standard" (including cases in which II6/5 is chromaticized to  
> become V6/5 of V). Of these 15 occurrences, 7 are quarter-note  
> chords. Of these 7, 4 are "standard" and 3 "non-standard." Hence the  
> share of "non-standard" occurrences is almost half of the  
> occurrences (7/15 or 3/7), whether or not we count eight-note chords  
> or not. (For those interested in the "non-standard" occurrences,  
> I'll attach a list below.)

> – Chorale 11 (Jesu, nun sei geprieset), m. 17. Between C: V and d:  
> II4/3 (perhaps possible to interpret as between IVn and II4/3 in d).
> (The same chorale also includes a 6/4 in m. 19 between VI and II6/5,  
> but since this only involves a passing motion in only one voice, I  
> didn't include it in my count.)
> – Chorale 21 (Herzlich tut mich verlangen), m 3. Between VII°6/5 of  
> V and V6/5 of V.
> – Chorale 21, m. 5. Between V4/2 and VII6.

One of your quarter note examples is in 3/4 time.  In chorale style,  
this is a tricky time signature -- the second quarter often contains  
clearly nonharmonic structures, rather like the weak eighths of  
chorales.  The second of the three examples is a chromaticized version  
of the standard pattern.  The third is genuinely unusual; it suggests  
what in my book I call a "third substitution," whereby a 5/3 chord is  
replaced by a 6/3 over the same bass (or vice versa).  This sort of  
substitution is characteristic of Bach's harmonic practice, leading to  
some very marvelous progressions (see the Ab major fugue, WTC 2),  
though the appearance of the 6/4 here is exceptional.  Here, we find a  
6/3 being replaced by the 6/4.

So, given this one genuine nonstandard quarter-note passing 6/4 in 42  
chorales, that gives us an expectation of about 9 such rascals in the  
entire collection.  That sounds about right to me.  The question is:  
how much class time (or how many textbook pages) do you want to spend  
on a progression that occurs nine times in the entire chorales?

> FOURTH, a comment on another of Dmitri's statements:
>
> "It's also important to remember that Bach chorales have a lot of  
> genuinely weird stuff in them, stuff you don't find in Mozart or  
> elsewhere.  The same chorale [#14] has vi-iii and V-iii  
> progressions  (m. 2, m. 11-12)."
>
> I was astonished to see vi–iii characterized as "genuinely weird."  
> Actually, there is no vi–iii in the measures Dmitri mentions, but I  
> assume he refers to that in m. 1, after the initial ubbeat (please  
> correct me if I am wrong). Here the iii occurs on a weak beat  
> between strong-beat vi and IV supporting a passing 7^ in an 8^–7^–6^  
> top-voice progression. To me, this is anything but weird; it is one  
> of the most characteristic functions of iii. ("vi–iii" combinations  
> also occur in the extremely common sequence by descending thirds I– 
> V, vi–iii, IV etc; I wonder whether sequences were excluded from  
> Dmitri's Mozart counts?)

I agree with Fetis, who suggested that sequential tonality obeys  
different norms from non-sequential tonality: one finds root-position  
diminished triads, V-iii progressions, and so on.  What is unusual  
here is the appearance of vi-iii in a nonsequential context.  This, I  
believe, is quite rare.  There aren't any instances of the progression  
in the Mozart sonatas, for example; Mozart almost always uses I6 in  
place of the iii.

Again, I am acutely conscious of the fact that I've taken more than my  
share of your pixels.  Thanks to everyone for a stimulating  
discussion, and for helping me sharpen my views.  I promise you that,  
despite my interest in statistics, I am more interested in music, and  
in learning how this style works.  I am also interested in the odd and  
unusual progressions, and love collecting interesting-but-unusual  
harmonies.

DT

Dmitri Tymoczko
Associate Professor of Music
310 Woolworth Center
Princeton, NJ 08544-1007
(609) 258-4255 (ph), (609) 258-6793 (fax)
http://music.princeton.edu/~dmitri