maldv/crabcanon · Datasets at Hugging Face

text stringlengths 0 174
which sits a boy, relaxing in the heat, while two floors below him a woman-perhaps his
mother-gazes out of the window from her apartment which sits directly above a picture
gallery where a young man is standing, looking at a picture of a ship in the harbor of a
small town, perhaps a Maltese town-What!? We are back on the same level as we began,
though all logic dictates that we cannot be. Let us draw a diagram of what we see (Fig.

143) .

What this diagram shows is three kinds of "in-ness". The gallery is physically in the town
("inclusion"); the town is artistically in the picture ("depiction"); the picture is mentally
in the person ("representation"). Now while this diagram may seem satisfying, in fact it is
arbitrary, for the number of levels shown is quite arbitrary. Look below at another way of
representing the top half alone (Fig. 144).
inclusion

We have eliminated the "town" level; conceptually it was useful, but can just as well be
done without. Figure 144 looks just like the diagram for Drawing Hands: a Strange Loop
of two steps. The division markers are arbitrary, even if they seem natural to our minds.
This can be further accentuated by showing even more "collapsed" schematic diagrams of
Print Gallery , such as that in Figure 145.

inclusion + depiction

This exhibits the paradox of the picture in the starkest terms. Now-if the picture is "inside
itself', then is the young man also inside himself-. This question is answered in Figure
146.

inclusion + depiction + representation

Thus, we see the young man "inside himself, in a funny sense which is made up of
compounding three distinct senses of "in”.

This diagram reminds us of the Epimenides paradox with its one-step self¬
reference, while the two-step diagram resembles the sentence pair each of which refers to
the other. We cannot make the loop any tighter, but we can open it wider, by choosing to
insert any number of intermediate levels, such as "picture frame", "arcade", and
"building". If we do so, we will have many-step Strange Loops, whose diagrams are
isomorphic to those of Waterfall (Fig. 5) or Ascending and Descending (Fig. 6). The
number of levels is determined by what we feel is "natural", which may vary according to
context, purpose, or frame of mind. The Central Xmaps-Dog, Crab, Sloth, and Pipe-can
all be seen as involving three-step Strange Loops; alternatively, they can all be collapsed
into two- or one-step loops;, then again, they can be expanded out into multistage loops.
Where one perceives the levels is a matter of intuition and esthetic preference.

Now are we, the observers of Print Gallery, also sucked into ourselves by virtue
of looking at it? Not really. We manage to escape that particular vortex by being outside
of the system. And when we look at the picture, we see things which the young man can
certainly not see, such as Escher’s

Signature, "MCE", in the central "blemish". Though the blemish seems like a defect,
perhaps the defect lies in our expectations, for in fact Escher could not have completed
that portion of the picture without being inconsistent with the rules by which he was
drawing the picture. That center of the whorl is-and must be-incomplete. Escher could
have made it arbitrarily small, but he could not have gotten rid of it. Thus we, on the
outside, can know that Print Gallery is essentially incomplete-a fact which the young
man, on the inside, can never know. Escher has thus given a pictorial parable for Godel’s
Incompleteness Theorem. And that is why the strands of Godel and Escher are so deeply
interwoven in my book.

A Bach Vortex Where All Levels Cross

One cannot help being reminded, when one looks at the diagrams of Strange Loops, of
the Endlessly Rising Canon from the Musical Offering. A diagram of it would consist of
six steps, as is shown in Figure 147. It is too
bad that when it returns to C, it is an octave higher rather than at the exact original pitch.
Astonishingly enough, it is possible to arrange for it to return exactly to the starting pitch,
by using what are called Shepard tones, after the psychologist Roger Shepard, who
discovered the idea. The principle of a Shepard-tone scale is shown in Figure 14$. In
words, it is this: you play parallel scales in several different octave ranges. Each note is
weighted independently, and as the notes rise, the weights shift. You make the top
octave gradually fade out, while at the same time you are gradually bringing in the
bottom octave. Just at the moment you would ordinarily be one octave higher, the
weights have shifted precisely so as to reproduce the starting pitch ... Thus you can go
"up and up forever", never getting any higher! You can try it at your piano. It works even
better if the pitches can be synthesized accurately under computer control. Then the
illusion is bewilderingly strong.

This wonderful musical discovery allows the Endlessly Rising Canon to be played
in such a way that it joins back onto itself after going "up" an octave. This idea, which
Scott Kim and I conceived jointly, has been realized on tape, using a computer music
system. The effect is very subtle-but very real. It is quite interesting that Bach himself
was apparently aware, in some sense, of such scales, for in his music one can
occasionally find passages which roughly exploit the general principle of Shepard tones-
for instance, about halfway through the Fantasia from the Fantasia and Fugue in G Minor,
for organ.

In his book /. S. Bach's Musical Offering, Hans Theodore David writes:

Throughout the Musical Offering, the reader, performer, or listener is to search for
the Royal theme in all its forms. The entire work, therefore, is a ricercar in the
original, literal sense of the word.'

I think this is true; one cannot look deeply enough into the Musical Offering. There is
always more after one thinks one knows everything. For instance, towards the very end of
the Six-Part Ricercar, the one he declined to improvise, Bach slyly hid his own name,
split between two of the upper voices. Things are going on on many levels in the Musical
Offering. There are tricks with notes and letters; there are ingenious variations on the
King's Theme; there are original kinds of canons; there are extraordinarily complex
fugues; there is beauty and extreme depth of emotion; even an exultation in the many-
leveledness of the work comes through. The Musical Offering is a fugue of fugues, a

which sits a boy, relaxing in the heat, while two floors below him a woman-perhaps his

mother-gazes out of the window from her apartment which sits directly above a picture

gallery where a young man is standing, looking at a picture of a ship in the harbor of a

small town, perhaps a Maltese town-What!? We are back on the same level as we began,

though all logic dictates that we cannot be. Let us draw a diagram of what we see (Fig.

143) .

What this diagram shows is three kinds of "in-ness". The gallery is physically in the town

("inclusion"); the town is artistically in the picture ("depiction"); the picture is mentally

in the person ("representation"). Now while this diagram may seem satisfying, in fact it is

arbitrary, for the number of levels shown is quite arbitrary. Look below at another way of

representing the top half alone (Fig. 144).

inclusion

We have eliminated the "town" level; conceptually it was useful, but can just as well be

done without. Figure 144 looks just like the diagram for Drawing Hands: a Strange Loop

of two steps. The division markers are arbitrary, even if they seem natural to our minds.

This can be further accentuated by showing even more "collapsed" schematic diagrams of

Print Gallery , such as that in Figure 145.

inclusion + depiction

This exhibits the paradox of the picture in the starkest terms. Now-if the picture is "inside

itself', then is the young man also inside himself-. This question is answered in Figure

146.

inclusion + depiction + representation

Thus, we see the young man "inside himself, in a funny sense which is made up of

compounding three distinct senses of "in”.

This diagram reminds us of the Epimenides paradox with its one-step self¬

reference, while the two-step diagram resembles the sentence pair each of which refers to

the other. We cannot make the loop any tighter, but we can open it wider, by choosing to

insert any number of intermediate levels, such as "picture frame", "arcade", and

"building". If we do so, we will have many-step Strange Loops, whose diagrams are

isomorphic to those of Waterfall (Fig. 5) or Ascending and Descending (Fig. 6). The

number of levels is determined by what we feel is "natural", which may vary according to

context, purpose, or frame of mind. The Central Xmaps-Dog, Crab, Sloth, and Pipe-can

all be seen as involving three-step Strange Loops; alternatively, they can all be collapsed

into two- or one-step loops;, then again, they can be expanded out into multistage loops.

Where one perceives the levels is a matter of intuition and esthetic preference.

Now are we, the observers of Print Gallery, also sucked into ourselves by virtue

of looking at it? Not really. We manage to escape that particular vortex by being outside

of the system. And when we look at the picture, we see things which the young man can

certainly not see, such as Escher’s

Signature, "MCE", in the central "blemish". Though the blemish seems like a defect,

perhaps the defect lies in our expectations, for in fact Escher could not have completed

that portion of the picture without being inconsistent with the rules by which he was

drawing the picture. That center of the whorl is-and must be-incomplete. Escher could

have made it arbitrarily small, but he could not have gotten rid of it. Thus we, on the

outside, can know that Print Gallery is essentially incomplete-a fact which the young

man, on the inside, can never know. Escher has thus given a pictorial parable for Godel’s

Incompleteness Theorem. And that is why the strands of Godel and Escher are so deeply

interwoven in my book.

A Bach Vortex Where All Levels Cross

One cannot help being reminded, when one looks at the diagrams of Strange Loops, of

the Endlessly Rising Canon from the Musical Offering. A diagram of it would consist of

six steps, as is shown in Figure 147. It is too

bad that when it returns to C, it is an octave higher rather than at the exact original pitch.

Astonishingly enough, it is possible to arrange for it to return exactly to the starting pitch,

by using what are called Shepard tones, after the psychologist Roger Shepard, who

discovered the idea. The principle of a Shepard-tone scale is shown in Figure 14$. In

words, it is this: you play parallel scales in several different octave ranges. Each note is

weighted independently, and as the notes rise, the weights shift. You make the top

octave gradually fade out, while at the same time you are gradually bringing in the

bottom octave. Just at the moment you would ordinarily be one octave higher, the

weights have shifted precisely so as to reproduce the starting pitch ... Thus you can go

"up and up forever", never getting any higher! You can try it at your piano. It works even

better if the pitches can be synthesized accurately under computer control. Then the

illusion is bewilderingly strong.

This wonderful musical discovery allows the Endlessly Rising Canon to be played

in such a way that it joins back onto itself after going "up" an octave. This idea, which

Scott Kim and I conceived jointly, has been realized on tape, using a computer music

system. The effect is very subtle-but very real. It is quite interesting that Bach himself

was apparently aware, in some sense, of such scales, for in his music one can

occasionally find passages which roughly exploit the general principle of Shepard tones-

for instance, about halfway through the Fantasia from the Fantasia and Fugue in G Minor,

for organ.

In his book /. S. Bach's Musical Offering, Hans Theodore David writes:

Throughout the Musical Offering, the reader, performer, or listener is to search for

the Royal theme in all its forms. The entire work, therefore, is a ricercar in the

original, literal sense of the word.'

I think this is true; one cannot look deeply enough into the Musical Offering. There is

always more after one thinks one knows everything. For instance, towards the very end of

the Six-Part Ricercar, the one he declined to improvise, Bach slyly hid his own name,

split between two of the upper voices. Things are going on on many levels in the Musical

Offering. There are tricks with notes and letters; there are ingenious variations on the

King's Theme; there are original kinds of canons; there are extraordinarily complex

fugues; there is beauty and extreme depth of emotion; even an exultation in the many-

leveledness of the work comes through. The Musical Offering is a fugue of fugues, a