Complete form of Unfinished Fugue
A great secrecy in the art of fugue can be said this unfinished triple fugue.
Not a once, the common theme, employed to all the other pieces, is used in
this piece.
G.Nottebohm(1881) has clarified that the three themes in this fugue are
combinable with the common theme, in short, four-theme fugue is
simultaneously playable in terms of counterpoint. With this discovery, this
piece is widely accepted as a member of the art of fugue.
(it is still controversial though.)
I studied the structure of this piece, compared it with the other pieces, and
then approached its desirable figure, on condition that it is a four-theme
fugue.
Conclusion
1. Structure: My speculation is that the continuing part of the unfinished
fugue includes the exposition of the forth theme = common theme, and the
combination of the four themes.
2. Bar number: My speculation is that there must be 293 bars or so.
3. Exposition of the forth theme: My speculation is that the forth theme
is exposed in the order of soprano→alto→tenor→bass in terms of voice.
4. Resister limit: The unfinished fugue is strict to the regulation that it is
playable by two hands and each voice keeps a regular register.
5. Possibility to be a mirror fugue: it is impossible to be a mirror fugue
because if inverted upside down, it would generate a dissonant chord.
Now, let's see the details.
1. Structure of the piece
Four voices and four beat: It is discontinued at bar 239 in the manuscript.
The fugue with three themes can be divided as follows.
|
Bar |
Content |
Bar number |
|
|
|
|
1-115
|
Exposition and development of the first theme |
115
|
|
114※-193
|
Exposition of the second theme and
combination of two themes |
80
|
|
193-239<
|
Exposition of the third theme and
combination of three themes |
47<
|
※The second theme is exposed at the time of cadenza.
Easily imagined is that the continuing part must include the exposition of
the forth theme and the combination of the four themes.
2. Bar number
Another hidden structure of the unfinished fugue is the first theme never
fails to appear in bass such as the beginning, bar 61, bar 148, and bar 234,
keeping its figure as it is,
|
Bar |
Content |
Bar
number |
|
|
|
|
1-
|
Exposition of the first theme |
60
|
|
61-
|
Re-exposition of the first theme |
87
|
|
148-
|
Re-exposition of the first theme after
the exposition of the second theme |
86
|
|
234-
|
Re-exposition of the first theme after
the exposition of the third theme |
6<
|
Remarkably, the second part has almost the same bar number as the third
part. Bach tends to build a structure of each section symmetrically by
equating bar number like seen in Cp.8 and ゛Canon alla Ottava゛
|
Contrapunctus 8 |
|
Bar |
Content |
Bar
number |
|
|
|
|
1-38
|
Exposition of the first theme |
38
|
|
39-92
|
Combination of the first theme and the second |
54
|
|
93-146
|
Exposition of the third theme and the re-
exposition of the first theme and the second |
54
|
|
147-188
|
Combination of three themes |
42
|
|
Canon alla Ottava |
|
Bar |
Content |
Bar
number |
|
|
|
|
1-
|
Thematic exposition in the upper voice (tonal key) |
24
|
|
25-
|
Thematic exposition in the upper voice (dominant key） |
16
|
|
41-
|
Thematic exposition in the upper voice
(dominant key, inverted form) |
21
|
|
62-
|
Thematic exposition in the upper voice
(tonal key, inverted form, variant form) |
15
|
|
77-
|
Thematic exposition in the upper voice (tonal key) |
27
|
It suggests that the unfinished fugue might have been composed in
symmetric structure, thus, I supposed that the section 4 of the unfinished
fugue would be 60 bars or so, and would totally come to 293 bars or so.
3. Exposition of the forth theme - Order of voices
When the three themes of the unfinished fugue are exposed for the first
time, each appears in each voice in different order, which is common to the
three simple fugues out of the four.
|
Unfinished fugue |
Simple fugue |
Order of voices |
|
|
|
|
The first theme |
Contrapunctus 2 |
Bass→Tenor→Alto→Soprano |
|
The second theme |
Contrapunctus 1 |
Alto→Soprano→Bass→Tenor |
|
The third theme |
Contrapunctus 3 |
Tenor→Alto→Soprano→Bass |
|
Contrapunctus 4 |
Soprano→Alto→Tenor→Bass |
Taking this into account, I speculate that the exposition of the forth theme
issoprano→alto→tenor→bassorder, like Cp.4.
4. Resister limit
Like the other pieces, the unfinished fugue is playable by two hands, that is,
the register may not deviate from two hands range of playability.
And the pieces with four voices in the art of fugue observe a certain
register that each voice is vocally rational and feasible.
|
Bass
|
Tenor
|
Alto
|
Soprano
|
|
|
|
|
|
Pieces except
unfinished fugue |
C-g'
|
A-d''
|
e-a''
|
a-c'''
|
|
unfinished fugue |
C-d'
|
c-b'
|
f-g''
|
d'-c'''
|
In these circumstances, continuing part of the unfinished fugue may have
double rules; it should stay within the range of two hands, and each voice
should observe a certain register.
5. Possibility to be a mirror fugue
Bach's short biography referring tothe art of fuguereads ゛final fugue
including four themes with all voices inverted later ゛. Comprehending this
statement as the unfinished fugue, a theory that the unfinished fugue is a
mirror fugue has been prevailed.
As a result, it cannot be a mirror fugue.
You will be convinced it if you invert this whole piece upside down. Best
example to deny the possibility of a mirror fugue can be seen at ゛cadenza゛
prior to the exposition of the third theme. A strange chord of e-c'-e'-e''
will be born when you make this temporally pausing cadenza in dominant key
invert upside down. So, it is hard to believe that Bach dare utilize this
unusual chord even as a temporal pause. (see experiment 0)
For your information, there is another disagreeable theory that four themes
in the continuing part of the unfinished fugue are possibly invertible. My
denial to this is driven from the fact that no fugue with multiple themes are
invertible as far as we know.
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