Questions on Fugue No. 10
Well-Tempered Clavier, Book I
by Johann Sebastian Bach

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Before answering anything, click the following buttons to familiarize yourself  with the concept of compound melody as well as some additional Schenkerian terms.

Foreground: This subject contains a compound melody -- two melodies in one.  We'll use
Schenkerian graphing technique to sort out which melody is which.

Middleground: This view represents the most foreground diminutions in gray. These
diminutions, which consist of arpeggiations, neighbors, passing notes, and consonants skips
are represented as belonging to a main note (solid). Their purpose is to prolong the main notes.
Here we begin to see the outline of a compound melody.

Melody #1: Consists of a prolonged E which crosses over Melody #2 (at the straight line)
to arrive at its neighbor, D, a ninth below. The final D is prolonged by a prefixed consonant
skip from F#, which is itself prefixed by an upper neighbor, G.   Sing this line and the next,
then play the subject while listening for each of its melodies.

Melody #2: Descends from E to the dominant in a series of prefixed chromatic neighbors.
The dominant pitch, B, is prolonged by both its upper and lower neighbors.  While the contour
of the first two gray notes may look like passing tones, they have more connection with the

1. In Schenkerian analysis it is sometimes useful to "verticalize" melodic strings.  While this subject presents pitches one after the other, it implies harmonies that can be heard in simultaneity.  Where does the Middleground verticalize pitches that were heard one after the other in the Foreground?

at the beginning
upon arrival at the C#
at the diagonal unfolding of the tritone
at the end

2. Which statement is true of the Middleground?

It contains four lower neighbors and one consonant skip.
The diagonal "unfolding" of E/A# implies the dominant of em.
The diagonal "unfolding" of E/A# implies the dominant of bm.
The arpeggiation prolongs the dominant of em.

3. The subject's first lower neighbor (D#) could be heard as connected to the E that follows it, or the D-natural that follows the E. How do you hear the D#?

I hear it as connected to the following E.
I hear it as connected to the following D-natural.
I hear it as connected to both the E and D-natural.

4. How many of the following are true?

It is impossible to have a fugue with fewer than two voices.
A compound melody gives the illusion of two voices.
It would be impossible to write a fugue for a solo instrument like the flute.
This erstwhile two-voiced fugue could be heard as having three, or even four, voices.

none of the above
fewer than three of the above
more than two of the above

5. Make this fugue into a Möbius strip (download the pdf).  How many times must the playhead traverse the strip to "play" the fugue and counterfugue one time each?

once
twice
three times
four times

6. How many times was it necessary for you to open the playhead and re grip the strip for double counterpoint to occur?

never
once
twice

7. Watch the No Magic at All Möbius video, paying special attention to experiment no. 1. That you can draw a connecting line down the center of the strip, without lifting your pencil, proves that the plane of the Möbius strip has but one side.  How does this lesson apply to the E minor fugue?

The fugue and counterfugue are contrapuntally one.
Double counterpoint is like the continuous deformation of
lines in space that makes the Möbius strip "happen."
One of the above is true.
Both of the above are true.

8. Measures 1-19 and 20-38 of the fugue are contrapuntally identical, with the latter being the contrapuntal inversion of the former.  They differ by one interval, however, and that to return the fugue to the key in which it started.  That difference is found in a comparison of m. 29 with m. 10.  Which statement is true?

A rising 3rd becomes a rising 4th.
A rising 3rd becomes a falling 6th.
A falling 3rd becomes a rising 6th.
A falling 3rd becomes a rising 4th.

9. In answer to "Why is this fugue also a canon?," choose the best option.

(1) The high voice of mm. 1-4 & 11-12 are replicated, at the 4th below, by the low voice of mm. 3-6 & 13-14.
(2) The low voice of mm. 20-23 & 30-31 are replicated, at the 5th above, by the high voice of mm. 22-25 & 33-34.
Options 1 & 2 describe canonic episodes, neither of which qualifies the fugue, in sum, as a canon.
When taken together, options 1 & 2 make the whole fugue a canon.
Although the author wrote that the fugue is a canon, he was fudging.  The Coda
does not participate in any canonic imitation.

10. Is there a relationship between the fugue's compound melody (two in one) and the fugue/counterfugue structure (two fugues in one)? If so, might you logically infer that Bach intended for the relationship to exist and brought it into existence? If so, do you think it reasonable that one can discern the author's intentions through analysis?

(1) There is a relationship, but it could have been accidental or subconscious.
(2) My intuition is that Bach created the relationship, but agree with #1.
(3) I think one can learn about the creator by studying the creation, but don't have the intuition of #2.
(4) I agree with #3 but think this particular observation falls in the category of #1.

(5) This is a Möbius question.

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