Order of the pieces
Up until now, many theories have arisen as for the order of pieces included
in the art of fugue, and the order varies depending on the music and
performers.
G. Butler's (1983) convinced one theory in his study of the ordering. He
looked into the art of fugue's first edition and confirmed that the numbering
had been changed after Bach's death. Then he reconstructed the original
ordering before its alteration.
※Note that the fist edition might have been supervised by Bach himself.
My study is base on the order by G. Butler's reconstruction.
Conclusion
The following chart is the result of my research. Now, please see the style
in the left column so that you will find the top (simple) and the bottom
(canon) make one pair, and then the upper center (3 themes and 2
themes) and the lower center (3 themes and mirror) make another pair
in terms of the number of bar. So the whole cycle is built in symmetric
structure. The alternation such as extension of a piece and insertion of a
new piece was thought to be made intentionally for the purpose of ideal
structure at the time of first edition's publication.
|
Style |
G. Butler's reconstruction |
Bar number
|
|
Simple |
Contrapunctus 1 |
78
|
372
|
|
Contrapunctus 2 |
84
|
|
Contrapunctus 3 |
72
|
|
Contrapunctus 4 |
138
|
|
Inversion |
Contrapunctus 5 |
90
|
230
|
Contrapunctus 6
a 4 in Stylo Francese |
79
|
Contrapunctus 7
a 4 per Augment et Diminut |
61
|
|
3 themes |
Contrapunctus 8 a 3 |
188
|
438
|
|
2 themes |
Contrapunctus 9 a 4
alla Duodecima |
130
|
250
|
Contrapunctus 10 a 4
alla Decima |
120
|
|
3 themes |
Contrapunctus 11 a 4 |
184
|
438
|
|
Mirror |
Contrapunctus inversus 12 a 4
/Contrapunctus inversus a 4 |
56
／56
|
254
|
Contrapunctus a 3
/Contrapunctus inversus a 3 |
71
／71
|
3 themes
（unfinished） |
Fuga a 3 Soggetti |
233<
(239<)
|
233<
(239<)
|
|
Canon※ |
Canon alla Ottava |
103
|
372
|
Canon alla Decima
Contrapunto alla Terza |
82
|
Canon alla Duodecima
in Contrapunto alla Quinta |
78
|
Canon per Augmentationen
in Contrario Motu |
109
|
|
※Some canons have a repeat but, repeated bar is not counted here. |
The detail is as follows.
1. Bar number and grouping of the pieces
The pieces can be divided and categorized into some groups, depending on
their musical styles. The column on the left shows each style while the
right shows the total number of bar on each group. Then you may realize
that the total bar number of two groups on both ends and another two
groups in center are respectively equivalent.
Symmetric structure by bar number is applicable to some of pieces in the
art of fugue, also unfinished fugue's complete form. Then I suppose that
this formula might have applied to the whole cycle of this music as well.
2. Alteration made at the first edition's publication
The alternation at the first edition's publication proves that the symmetric
structure was designed on purpose. Now I would like to remark the altered
points below at every group.
|
Style |
Altered point
|
Bar number
|
|
Simple |
Doubling bar number （Contrapunctus1〜3）
Extension（Contrapunctus1〜3）
New addition（Contrapunctus4） |
372
|
|
Inversion |
|
230
|
|
3 themes |
|
188
|
438
|
|
2 themes |
Doubling bar number（Contrapunctus9&10）
Extension（Contrapunctus10） |
250
|
|
3 themes |
|
184
|
438
|
|
Mirror |
|
254
|
3 themes
(unfinished) |
New addition |
239<
|
|
Canon |
Arranged work（Canon per Augmentationen）
New addition（Canon alla Decima & Duodecima) |
372
|
In this way, I believe that alternation such as extension or new addition was
made over in each group of ゛Simple, 2 themes and Canon゛, adjusting for
the accordance of bar number. Besides, the total bar number of two 3
theme fugues other than the unfinished fugue sums up to 372, which is
equal to the number of ゛Simple and Canon゛group. This fact needs further
examination.
3. Placement of the unfinished fugue
The study so far may jeopardize the placement of the unfinished fugue
because this piece, one of the set of ゛Inversion゛group is unbalanced the
counterpart with the total bar number of 230, already exceeding 230 even
in its incomplete fugue. Considering its structure without the common
theme and with this asymmetry for the cycle of the art of fugue, the
unfinished fugue might not be the member of the cycle.
This is just my presumption but Bach was discouraged from completing it
because of this excessive bar number. It is believed that composing,
publication, and extension of pieces and addition of new pieces, all these
were simultaneously ongoing at that time, then halfway built unfinished
fugue could not match with the finally created symmetric structure.
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