1. Of course this tonal answer could not enter during the F# of the Subject. I suspect the point is relevant to the stretto character of the fugue.

Christoph Wolff points out that Friedrich Wilhelm Marpurg, soon after Bach's death, had already observed the features of this fugue so far discussed. In an essay entitled Analysen von Bachschen Fugenthemen, Fugen und Kanons (Berlin, 1753-54), Marpurg points out that "the leap down a fifth, which the octave above the tonic note makes here, was prohibited by older authorities, on the grounds that it renders the tonality uncertain" ("Der Sprung, den hier die Oktave der Hauptnote in die Unterquint thut, wurde bey den Alten verboten, weil er die Tonart ungewiss macht."). Some seven years later, in an essay entitled Themenbeantwortung und Durchfuehrung in einigen Fugen des Wohltemperierten Klaviers (Berlin, 1760), Marpurg repeats the above sentence and elaborates it with the following continuation: "Indeed, the Subject at issue here does not proclaim the key of D major, but much more G major; and one does not know where one is at home tonally, until the entrance of the Answer" ("In der That zeigt der hier vorhandene Fuehrer nicht die Tonart d dur, sondern vielmehr g dur an; und man merkt es erst bey dem Eintritt des Gefaehrten, wo man zu Hause ist."). Marpurg recognizes the problematic character of this phenomenon, but does not pursue it farther from a theoretical point of view, contenting himself with pedagogical advice: "Such exceptions to the rules can be ventured only by Masters, and beginners will do well to cleave to the rule that requires a fugue theme to indicate the key unambiguously" ("Dergleichen Ausnahmen von der Regel koennen nur von Meistern vorgenommen werden, und Anfaenger thun wohl bey der Regel zu bleiben, welche einen deutlichen und die Tonart gehoerig anzeigenden Fugensatz erfordert.").

The Marpurg passages are reproduced in Bach-Dokumente, ed. Bach-Archiv Leipzig, Volume III: Dokumente zum Nachwirken Johann Sebastian Bachs 1750-1800, ed. Hans-Joachim Schulze (Leipzig and Kassel, 1972). The quote from 1753 appears on page 28, and the quote from 1760 on page 156.

2. Since the Subject contains neither a C nor a C#, it contains no tritone-dyad. Related issues in musical cognition are discussed by Richmond Browne in "The Tonal Implications of the Diatonic Set," In Theory Only 5.6-7 (1981), 3-21; also by Helen Brown and David Butler in "Diatonic Trichords as Minimal Tonal Cue Cells," In Theory Only 5.6-7 (1981), 37-55.

3. "A tonality is expressed by the exclusive use of all its tones." Arnold Schoenberg, Structural Functions of Harmony (New York: W.W. Norton & Company, Inc., 1954), page 11. The sentence leads off a section entitled "Establishment of Tonality."

4. The phenomenon is suggestive as regards the end of the answer. Since the end of the subject in this fugue modulates from E minor to B minor, the end of the answer should normatively modulate back from B minor to E minor, requiring adjustment of the melody accordingly. Instead Bach gives a real answer, modulating from B minor to F# minor. One hears how the tones missing from the subject--E#, G#, and A--play a characteristic role over the second half of the answer, during the modulation to F# minor.

No doubt the chromatic closure is subordinate to other aspects of the real answer. Since the fugue has only two voices, the exposition is complete at the end of the answer, and Bach seems eager to move on tonally at once, rather than returning to E minor. Still, the chromatic closure has a certain effect, not least in that very connection.

5. Lewis Lockwood recalls an analysis course taught by Edward Lowinsky at Queens College, in which Lowinsky approached Book I as a whole by stressing the completely hexachordal nature of the subject for the first fugue in the Book, and the completely chromatic nature of the subject for the last fugue.

End of Footnotes