[Smt-talk] BELGIAN +6

Ildar Khannanov solfeggio7 at yahoo.com
Sun Nov 27 13:56:09 PST 2011

Dear Dmitri,
I apologize for miscommunication: I am with you on that Subdominant is Subdominant and cannot substitute for Dominant. You, in your turn, misread my statement concerning the alteration of the root of the  V/V chord. I wrote not about lowered scale step two, but about raised scale step two, and not in the main Dominant, but in the Dominant of Dominant. In the latter, you should agree, raised scale step two raises the root of the chord and creates a new situation: viidim7 of VI. So, I assume that alterations of scale steps cannot change the function of the chord. It does not matter what is raised or lowered in Gr+6. Raised sc step two is not an indication of its functional definition.
Derivation and substitution are two most important concepts of tonal music theory before Schenker. Derivattion imples hierarchy of chords. All the variety of chords cane be derived from a few fundamental functions. These functions produce logic and syntax of harmonic progression. Imagine: T-S-D-T, a full functional cycle. The Subdominant there participates in creation of tonal syntax, logic and meaning, but does not violate it. Another case: T-D/D-D (T). In this case, the Dominant of Dominant serves entirely different function: it produces the digression into the Dominant. It emphasizes the Dominant and deemphases Tonic. In the body of a harmonic progression it makes sense. A composer emphasizes Dominant temporarily and there is enough time to recoup and restaure the rule of Tonic. In the end, at the PAC, it does not make sense to emphasize the Dominant. 
The so-called German Augmented Sixth chord appears most commonly before the Perect Authentic Cadence, before the terminal Dominant, in a model harmonic progression, which is achttaktige Periodeform (Riemann). This metric period is a true Ursatz, a model which combines metric, formal, harmonic and rhetorical structures in one. The placement of Gr+6 in this model is at the end of Measure 6, before the PAC in mm. 7 and 8. Now, I wanted to know, why whould a composer introduce the emphasis on the Dominant in such an inapropriate place, where we should wrap things up and emphasize Tonic?
Yes, Beethoven liked V/V. This was, however, an exception from the rule. Look through 20 preparations of the PAC in music of Beethoven and you will find the preponderance of IV, ii6, ii6/5 etc. It makes sense to exacerbate the Dominantness before the Half Cadence, like, say, in the first phrase of Pathetique. But it does not make any sense to use V/V or viidim/V before the Cadential 6/4 - Dominant 5/3 - Tonic.
Our discussion of these two possibilities places us right smack in the middle of the century-long discussion of this same issue between Leningrad and Moscow schools of theory. They were fighting over this issue quite seriously. Yuri Tjulin suggested that there are two types of modes, diatonic and alterational. Major mode is diatonic, minor--alerational by definition. Alterational modes DERIVE from diatonic. By the way, I do not agree with your definition of "semitonal alteration" as a guarantee of the Dominant function. The Leading Tone in major is not a product of semitonal alteration, it is a diatonic scale step. Semitonal alterations do not exist in a vacuum. They exist in the context of tonality, or systematic relationship of tonal functions. Not all semitonal alterations created the situation of a leading tone. Thus raised scale step 4 can be a part of V/V as well as something else. I would even suggest this as the next step in your marvelous work
 on tonal geometry. You could depart from the idea of adjacency (which is abstract and unmusical) and separate geometires "in the key" from the "outside of the key."
Each function can be altered, or embellished. There are altearation of the chords of the Dominant function, and there are alterations of the chord of the Subdominant function (sorry that I have to recite the pages from the 1937 Russian textbook!). Subdominant can be altered and you will hear Neapolitan sixth chord, German Augmented Sixth chord, Double Subdominant with added fourth, Rachmaninoff Subdominant, etc.
Subdominant is a selfstanding independent function. The fact that it is placed between Tonic and Dominant in the tonal syntax should not diminish its functional weight. M. Catel knew the significance of the Subdominant. He suggested to modulate into la quinte en dessous, which means that he perceived it as sous-dominant just as Rameau suggested, a fifth below tonic. If it was just as scale step 4, it should be placed between sc st 3 and sc st 4. Textbooks do not offer extensive theoretical reasoning, but what is left for reading between the lines is the actual tradition.

 From: Dmitri Tymoczko <dmitri at princeton.edu>
To: smt smt-talk <smt-talk at societymusictheory.org> 
Sent: Friday, November 25, 2011 12:14 PM
Subject: Re: [Smt-talk] BELGIAN +6
Hi Ildar,

> Your hearing of the three national chords as derived from the V/V seems very intriguing to me. Indeed, is there not an Fa# in each of them? My position, nevertheless, of considering them a part of subdominant function is based upon the fact that these adjacencies (Fa# to Sol, Re# to Mi) are the parts of different keys, not the given Do major. They belong to different geometries and different physics. There is the geometry of tonic (TSDT) and other neighboring spaces with their own tension-resolution patterns. For being a Dominant function, the chord must have a leading tone in a given key, and that is what exactly missing in German, French and Italian chords. So, I should say that French, German and Italian chords in the given key serve the role of Subdominant because the original chords from which they are derived are IV and ii, and because all three do not have the Leading Tone in the given key. The fact that the altered chords have tendencies to
 leave a given geometry do
es not change their function in the original space of TSDT. And the V/V is not their only tendency, if to be precise.

My difficulty with this is the notion of "derived from" which strikes me as ambiguous.  The German 6/5 can be "derived" in multiple ways:

    Way 1: take viio6/5/V and lower the bass by semitone.
    Way 2: take iv6/5 and raise the root by semitone.

You seem to think that Way 2 is the right one, but I can't follow you in that.  To me, they're both equally true, which means that "derived from" can't help us determine function.

The other difficulty I have is the idea that semitonal alteration doesn't change chord function.  This I just don't believe.  For example, IV is a subdominant, but #ivo is a viio/V chord, an applied dominant.  Your argument seems to work just as well for applied dominants as for augmented sixths -- for you, viio/V should be a subdominant, not a "dominant of the dominant" chord.  I can't follow you here -- if the term "function" is to be useful for me, then there has to be a difference between subdominant and dominant of the dominant.

>  You observed the placement of the German chord in variations. Strange choice! I completely agree with you that we have to observe the placement of a chord in order to analyze its structural meaning. However, it seems more logical to me to find the most common placement of the German chord, in preparation of the terminal dominant or a cadence. I make my final definition of it s Subdominant because it is placed in the 6th metric measure of the metric period. Or, like in Mozart concerti and sonatas, right before the Cadenza, resolving into the dominant in the main key.

Just to be clear: my claim was that composers regularly use augmented sixths in passages where, in earlier versions of the same music, they had V/V (or vice versa), whereas you don't find the same intersubstitution of IV (or ii) and the augmented sixths.  In all honesty, this seems pretty decisive to me -- about as strong an argument as you're ever going to get in music theory.  It strongly suggests that composers conceive V/V and the augmented sixth as "basically the same" or "analogous" whereas they didn't feel that way about the augmented sixths and IV or ii.

My intuition is that V/V before V is about as common as augmented sixths are in the same spots -- and that the applied dominants appear in basically the same places, including before the I6/4 that leads to a cadenza.  (David Lewin used to emphasize to me that V/V was one of Beethoven's favorite chords.)  I'd need to look at statistics to back this up, but offhand that seems right.

>  There has been an objection earlier, that the three named chords cannot be the Subdominant because they do not have Fa, but Fa# instead. But, by the same logic, they cannot be the chords of the V/V because instead of Re the German chord has Re#.

Except that dominant chords have ^b2 all the time -- in late Beethoven, Schubert, Chopin, etc.  The only issue here is that the augmented sixths appeared first as V/V chords in minor, where the ^b6 is given by the key signature.  To me, that's not at all mysterious.  The obvious thought is that V/V in minor has voice leading awkwardnesses, with ^#6 moving down to ^5, contravening standard minor-mode practice.  Sometimes you even have ^b6->^#6->^5, which is quite awkward.  So composers just started retaining the key signature's ^b6 in the applied dominant V/V, producing the augmented sixth.

Best wishes,

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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