Solution
to Canon
1. a 2 cancrizans (crab)
Solution
for the first puzzle canon is in the form of:
A
dialogue between two women, Euterpe, named after the ancient Greek muse
of music, who also invented the flute,
Cecilia:
All
right. Let’s see what you have discovered.
How
many voices are there?
Look
at the title:
Euterpe:
”a 2” in the title indicates
that there are two voices in a canonical relationship with each other.
Cecilia:
How
many clefs did you find? Did you look at the entire piece, also at the end? Double-check
this before you proceed.
Are
you are ready to proceed?
Euterpe:
There is a C-clef at the beginning, and a C-clef at the end of the piece. This
also indicates that there are two canonical voices.
And
since there is no [start
here] or
any other symbols, I think the answer is that the canon has two voices.
Cecilia:
How should the follower be played? What is the relationship between the leader
and the follower?
Here’s
a hint: Look at the position of the second clef. Do this again before you
proceed.
Are
you are ready to proceed?
Euterpe:
Look! The C-clef, and the three flats at the end of the piece are facing
backwards, a mirror image of the clef and flats at the beginning!
Cecilia:
What
could this indicate? Think about this before you proceed.
Are
you are ready to proceed?
Euterpe:
Why, this must mean that the follower is to be played backwards!
Cecilia:
Look
at the title:
Euterpe:
Well, Bach refers to a “Crab” in the title – Come to think of it, a crab
looks the same if it crawls forwards, or backwards!
The
transformation principle, then, is backwards directionality.
Cecilia:
When does the follower begin?
Is
there a [start
here symbol]
?
Euterpe:
No. Since there is no [start
here symbol]
symbol, this must mean that both voices start simultaneously.
Therefore,
the solution must be that there are two voices, which start simultaneously,
and the follower is performed backwards – it begins at the end and proceeds
through the notes from the end to the beginning.
And
another thing I have noticed is that the canon is comprised of 8½
measures of the Royal theme, followed by 9½ measures of counterpoint, so that,
at all times, both are played, simultaneously, but in opposite directions --
either the Royal theme forwards, together with the counterpoint backwards, or
the Royal theme backwards, together with the counterpoint forwards.
Geometrical
transformation:
Make
two exact geometric forms, such as triangles, with very different angles, for
example:
Place
them on top of each other. Call the bottom one the leader, and the top one the
follower. Then, rotate the follower so that it ends up being backwards, but
beginning and ending at the same place as the leader, so you now see
geometrically, the transformation of backward directionality, which Bach uses
for canon 1.
Do
this first yourself before you watch the animation.
The
transformation occurs in the physical process of rotation. You can also think
about symetrical “right-“ and “left-handed” molecules.
During
the performance of the canon, the triangles are gradually rotated,
simultaneously, where each one is travelling toward the beginning position of
the other.
Then,
when the parts switch places, where the follower plays forwards, and the
leader backwards, the voices rotate again, towards their original positions.
See
the following animation with the parts played twice, switching parts:
Performance:
Play
or sing the first voice as written.
You can listen to the written voice here
Play
or sing the last measure backwards, that is, “c-e-g-c’”
Play
or sing the Royal theme backwards, starting from the “c” in measure 9.
Listen here to the Royal theme backwards
Play
or sing the whole piece backwards.
Listen here to the whole canon backwards.
Play
or sing the whole piece, simultaneously, with one type of voice forwards, and
another type backwards (different singing voices: soprano, tenor, etc, and/or
different instruments).
Do
that again, and then, immediately switch parts.
Listen here to the canon played twice -- first forwards, and then backwards.
Invent
your own musical example:
Invent
a simple four-note motive.
Play
or sing your motive backwards.
Try
playing your motive both forwards and backwards, simultaneously.
Notice
that this results in the two voices forming either consonant or dissonant
intervals between each other. This gives you a sense of how dissonances are
created, not arbitrarily, but as a result of transforming the motive you have
chosen, and playing it in its new form simultaneously with the original motive.
This
is a window through which you can see how musical problems are created, which
then beg to be resolved. We will see this process again, and again, as we
proceed to discover other kinds of transformation principles. This, then,
becomes a source of creative tension and development for composers of
classical music.
Perform
it with two different type of voices, and have the voices switch directions
afterwards.
Try
it again with another simple motive.
Study
how musical paradoxes are created and resolved
Now,
investigate how the use of this “backwards” transformation principle
creates musical development – lays the basis for a unified unfolding of a
musical idea, held together by an unending musical tension, caused by the
process of leading up to, and eruption of dissonant intervals between the
voices, and their resolution, not just once, but like an intense, pulsating,
forward moving drive.
Investigate
the intervallic relationships formed in the first half of this crab canon –
where the leader moves forward, and the follower moves backward – where the
directionality is dominated by the chromatic descent of the second part of the
theme.
How
does this change when the parts are switched? When the leader moves backwards
and the follower forwards? Here, as the above-mentioned second part of the
theme moves backwards, it actually ends up being flipped upside down, rising
chromatically upwards, and forming different intervals with the
opposite-moving counterpoint voice than in the first part. How do the
different intervals formed affect the music?
Do
you notice any ambiguity in the traditional notions of consonance and
dissonance? Are there some intervals which ought to sound consonant, like a
major third, which, actually, in certain places, sound dissonant, and
sometimes traditionally dissonant intervals, like the “Lydian” interval
comprised of three whole tones, sometimes, actually sound more like
resolutions after more dissonant sounding consonances?
Now,
compare this intense musical development in miniature, to the beginning,
2-voiced section of the 3-voiced Ricarcar. (measures ----).
