[Smt-talk] stretto calculation

[Olli Väisälä]

Dear Stephen and List,

To Prof. Schubert's list of articles I would add:

Lynden De Young, Stretto by Direct and Contrary Motion in Three  
Fugues by J. S. Bach, Indiana Theory Review Vol. 18/2 (1997).

If I recall correctly, this article describes the special technique  
evident in the B-flat minor Fugue (WTC II). Basically the technique  
can be described as follows:  The theme is, for the most part,  
harmonizable by the regular alternation of I and VII°7 in half notes.  
Consequently, (1) if you transpose the theme by a seventh, it is  
harmonizable by the alternation of VII° and "VI7," the latter of  
which is good enough to approximate I since the "root G" is  
metrically weak (cf. A in bar 2 with G in bar 28). Owing to the  
similar regular alternation in both cases, the original and the  
transposition work together in stretto if you shift the latter by a  
half note. (2) If you invert the theme by using the ^3 as axis of  
symmetry, you get another version with alternating I and VII°7, and,  
by transposing this at the lower seventh (upper ninth), II and I7, of  
which the II is can work as equivalent to VII°7. These features  
account for all the strettos in this Fugue (except that the end of  
the theme does not quite follow the alternating pattern).

Another of De Young's examples was the D-minor Fugue from WTC I. In  
this case, too, you can harmonize the theme with alternating I and VII 
°7, but the rhythm is now quarter-note + half-note. For this reason,  
the stretto won't work in the upper seventh, but will work in the  
octave, with a dotted half note as the temporal interval. In this  
case, too, you can incorporate the inversion to the strettos by using  
the ^3 as axis of symmetry, since this maps I to I and VII°7 to VII° 
7. This accounts for most but not all strettos in this Fugue.

In principle, the B-flat fugue subject also works with the stretto at  
the octave, if the temporal interval is two beats. (I just tried it,  
it sounds OK.) However, Bach did not use this, perhaps because it is  
too confusing metrically.

However, the I–VII°7-alternation technique certainly accounts for  
only a minority of Bach's strettos (and works less well in major). I,  
too, mostly recommend trial-and-error for my students.

Olli Väisälä
ovaisala at siba.fi





> I have always tested (and taught) the suitability of a subject for  
> stretto
> by trial and error, using subject and answer forms (including  
> inversions)
> at various pitch and time intervals. Yet it seems to me that there  
> must be
> a more dependable and rigorous method of composing and, especially,
> evaluating a subject a subject to maximize its "strettobility." One  
> way is
> to make it sequential, but there must be others. I noticed at the SMT
> counterpoint session that Robert Gauldin's comments on the Bb minor  
> fugue
> (WTC2, Eric Wen's talk) seemed to come out of a fairly well worked out
> stretto methodology, but it doesn't seem to be set forth in his  
> 18th-c.
> cpt. book, where the comments on stretto are fairly sketchy.
> Schubert&Neidhofer also appear to recommend a trial and error  
> approach,
> but have a short section about making first species reductions of  
> subject
> and answers to facilitate the discovery process.
>
> Does anyone have any thoughts on this?
>
> Thanks,
>
> Stephen Slottow
> University of North Texas