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Experiment 1

Inverse ゛Contrapunctus a 3゛entirely as they are.

In the course of analysis on Contrapunctus a 3, I pointed out that the
voice the mirrored image is reflected to is different from the voice the
image is stated in when inversing a whole piece upside down.
Then I wondered why the voices have to be interchanged ? Searching for
the answer, we try this experiment; inverting the whole piece upside down
entirely as they are, by pivoting ゛h-c'゛ like Cp.12.

BGM is the result of the experiment.
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Soon, you may notice something is funny.



Melody in blue in the above music is derived from the inversion of the
thematic image in soprano from bar 8.
So, the mirrored image in bass has actually been moved one octave higher.
(Did you notice it already?)
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Compared with Cp.12, the image in this piece is set in high register.
Now, let's see the difference between the two pieces.

Bass
Tenor
Alto
Soprano
Whole
piece
Contrapunctus 12
C-e'
c-a'
f-e''
a-b''
C-b''
Contrapunctus a 3 C
F-d'
−
g-g''
e'-c'''
F-c'''

The concern about making a mirrored image is especially the register of
soprano because the lowest note in this experiment is five degree higher
than that of Cp.12. When the piece is inverted and soprano is replaced to
bass, the highest note will become ゛g゛, which is much lower than the
standard register of bass. And the lowest note ゛F ゛among the whole piece
of ゛Contrapunctus a 3゛ will become the highest ゛f ''゛ when making a
mirrored image. This is again fifth degree or so lower than the standard
register.
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Well, you may come to understand that the register gets one-sided toward
far down low when inverted. Then, how about shifting all the voices one
octave higher, like bass?

Well, this is practically wrong because keyboards those days were shorter
than now and the highest register was normally ゛c''' ゛, and even the
highest note ゛e ''゛of Contrapunctus inversus a 3 stands extraordinary,
therefore, lifting the highest note ゛f ''゛one octave higher is out of question.
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Additionally, the image in bass is designed quite suitably for bass, (weird to
say) in other wards, it sounds void as a melody. In this way, if this is
reflected to soprano, the music itself will surly not sound interesting.
BGM may sound strange to you.



The music above is just an example. Mostly like this except a few themes.
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That's all for now, but let's review the output we had from the experiment.

1. The register gets one-sided toward far down low if it is inverted upside
down like Cp.12.
2. The register gets too high if the inverted mirror image is lifted one
octave higher.
3. The image in bass has many void melodies so it is not suitable for
soprano.

For these reasons, Contrapunctus inversus a 3 is best to be as it is today,
and Bach might have composed it under a lot of calculations.
Now, we might as well checking the process.

1. Lifted the melody reflected to bass one octave higher.
2. Lifted the melody reflected to tenor two octaves higher
and made it be soprano.
3. Made the melody reflected to soprano be alto.

Let me tell you further that the melody in alto stays at its original position
after inversion like Cp.12.

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