maldv/crabcanon · Datasets at Hugging Face

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emotional and artistic expression. The telltale sign of a fugue is the way it begins: with a
single voice singing its theme. When it is done, then a second voice enters, either five
scale-notes up, or four down. Meanwhile the first voice goes on, singing the
"countersubject": a secondary theme, chosen to provide rhythmic, harmonic, and melodic
contrasts to the subject. Each of the voices enters in turn, singing the theme, often to the
accompaniment of the countersubject in some other voice, with the remaining voices
doing whatever fanciful things entered the composer's mind. When all the voices have
"arrived", then there are no rules. There are, to be sure, standard kinds of things to do-but
not so standard that one can merely compose a fugue by formula. The two fugues in the
Musical Offering are outstanding examples of fugues that could never have been
"composed by formula". There is something much deeper in them than mere fugality.

All in all, the Musical Offering represents one of Bach's supreme accomplishments in
counterpoint. It is itself one large intellectual fugue, in


Introduction: A Musico-Logical Offering


9



which many ideas and forms have been woven together, and in which playful double
meanings and subtle allusions are commonplace. And it is a very beautiful creation of the
human intellect which we can appreciate forever. (The entire work is wonderfully
described in the book f. S. Bach's Musical Offering, by H. T. David.)

An Endlessly Rising Canon

There is one canon in the Musical Offering which is particularly unusual. Labeled simply
"Canon per Tonos", it has three voices. The uppermost voice sings a variant of the Royal
Theme, while underneath it, two voices provide a canonic harmonization based on a
second theme. The lower of this pair sings its theme in C minor (which is the key of the
canon as a whole), and the upper of the pair sings the same theme displaced upwards in
pitch by an interval of a fifth. What makes this canon different from any other, however,
is that when it concludes-or, rather, seems to conclude-it is no longer in the key of C
minor, but now is in D minor. Somehow Bach has contrived to modulate (change keys)
right under the listener's nose. And it is so constructed that this "ending" ties smoothly
onto the beginning again; thus one can repeat the process and return in the key of E, only
to join again to the beginning. These successive modulations lead the ear to increasingly
remote provinces of tonality, so that after several of them, one would expect to be
hopelessly far away from the starting key. And yet magically, after exactly six such
modulations, the original key of C minor has been restored! All the voices are exactly one
octave higher than they were at the beginning, and here the piece may be broken off in a
musically agreeable way. Such, one imagines, was Bach's intention; but Bach indubitably
also relished the implication that this process could go on ad infinitum, which is perhaps
why he wrote in the margin "As the modulation rises, so may the King's Glory." To
emphasize its potentially infinite aspect, I like to call this the "Endlessly Rising Canon".

In this canon, Bach has given us our first example of the notion of Strange Loops. The
"Strange Loop" phenomenon occurs whenever, by moving upwards (or downwards)
through the levels of some hierarchical system, we unexpectedly find ourselves right
back where we started. (Here, the system is that of musical keys.) Sometimes I use the
term Tangled Hierarchy to describe a system in which a Strange Loop occurs. As we go
on, the theme of Strange Loops will recur again and again. Sometimes it will be hidden,
other times it will be out in the open; sometimes it will be right side up, other times it will
be upside down, or backwards. "Quaerendo invenietis" is my advice to the reader.


Escher

To my mind, the most beautiful and powerful visual realizations of this notion of Strange
Loops exist in the work of the Dutch graphic artist M. C. Escher, who lived from 1902 to
1972. Escher was the creator of some of the


Introduction: A Musico-Logical Offering


10




FIGURE 5. Waterfall, by M. C. Escher (lithograph, 1961).


most intellectually stimulating drawings of all time. Many of them have their origin in
paradox, illusion, or double-meaning. Mathematicians were among the first admirers of
Escher's drawings, and this is understandable because they often are based on
mathematical principles of symmetry or pattern ... But there is much more to a typical
Escher drawing than just symmetry or pattern; there is often an underlying idea, realized
in artistic form. And in pailicular, the Strange Loop is one of the most recurrent themes in
Escher's work. Look, for example, at the lithograph Waterfall (Fig. 5), and compare its
six-step endlessly falling loop with the six-step endlessly rising loop of the "Canon per
Tonos". The similarity of vision is


Introduction: A Musico-Logical Offering


11





FIGURE 6. Ascending and Descending, by M. C. Escher (lithograph, 1960).

emotional and artistic expression. The telltale sign of a fugue is the way it begins: with a

single voice singing its theme. When it is done, then a second voice enters, either five

scale-notes up, or four down. Meanwhile the first voice goes on, singing the

"countersubject": a secondary theme, chosen to provide rhythmic, harmonic, and melodic

contrasts to the subject. Each of the voices enters in turn, singing the theme, often to the

accompaniment of the countersubject in some other voice, with the remaining voices

doing whatever fanciful things entered the composer's mind. When all the voices have

"arrived", then there are no rules. There are, to be sure, standard kinds of things to do-but

not so standard that one can merely compose a fugue by formula. The two fugues in the

Musical Offering are outstanding examples of fugues that could never have been

"composed by formula". There is something much deeper in them than mere fugality.

All in all, the Musical Offering represents one of Bach's supreme accomplishments in

counterpoint. It is itself one large intellectual fugue, in

Introduction: A Musico-Logical Offering

9

which many ideas and forms have been woven together, and in which playful double

meanings and subtle allusions are commonplace. And it is a very beautiful creation of the

human intellect which we can appreciate forever. (The entire work is wonderfully

described in the book f. S. Bach's Musical Offering, by H. T. David.)

An Endlessly Rising Canon

There is one canon in the Musical Offering which is particularly unusual. Labeled simply

"Canon per Tonos", it has three voices. The uppermost voice sings a variant of the Royal

Theme, while underneath it, two voices provide a canonic harmonization based on a

second theme. The lower of this pair sings its theme in C minor (which is the key of the

canon as a whole), and the upper of the pair sings the same theme displaced upwards in

pitch by an interval of a fifth. What makes this canon different from any other, however,

is that when it concludes-or, rather, seems to conclude-it is no longer in the key of C

minor, but now is in D minor. Somehow Bach has contrived to modulate (change keys)

right under the listener's nose. And it is so constructed that this "ending" ties smoothly

onto the beginning again; thus one can repeat the process and return in the key of E, only

to join again to the beginning. These successive modulations lead the ear to increasingly

remote provinces of tonality, so that after several of them, one would expect to be

hopelessly far away from the starting key. And yet magically, after exactly six such

modulations, the original key of C minor has been restored! All the voices are exactly one

octave higher than they were at the beginning, and here the piece may be broken off in a

musically agreeable way. Such, one imagines, was Bach's intention; but Bach indubitably

also relished the implication that this process could go on ad infinitum, which is perhaps

why he wrote in the margin "As the modulation rises, so may the King's Glory." To

emphasize its potentially infinite aspect, I like to call this the "Endlessly Rising Canon".

In this canon, Bach has given us our first example of the notion of Strange Loops. The

"Strange Loop" phenomenon occurs whenever, by moving upwards (or downwards)

through the levels of some hierarchical system, we unexpectedly find ourselves right

back where we started. (Here, the system is that of musical keys.) Sometimes I use the

term Tangled Hierarchy to describe a system in which a Strange Loop occurs. As we go

on, the theme of Strange Loops will recur again and again. Sometimes it will be hidden,

other times it will be out in the open; sometimes it will be right side up, other times it will

be upside down, or backwards. "Quaerendo invenietis" is my advice to the reader.

Escher

To my mind, the most beautiful and powerful visual realizations of this notion of Strange

Loops exist in the work of the Dutch graphic artist M. C. Escher, who lived from 1902 to

1972. Escher was the creator of some of the

Introduction: A Musico-Logical Offering

10

FIGURE 5. Waterfall, by M. C. Escher (lithograph, 1961).

most intellectually stimulating drawings of all time. Many of them have their origin in

paradox, illusion, or double-meaning. Mathematicians were among the first admirers of

Escher's drawings, and this is understandable because they often are based on

mathematical principles of symmetry or pattern ... But there is much more to a typical

Escher drawing than just symmetry or pattern; there is often an underlying idea, realized

in artistic form. And in pailicular, the Strange Loop is one of the most recurrent themes in

Escher's work. Look, for example, at the lithograph Waterfall (Fig. 5), and compare its

six-step endlessly falling loop with the six-step endlessly rising loop of the "Canon per

Tonos". The similarity of vision is

Introduction: A Musico-Logical Offering

11

FIGURE 6. Ascending and Descending, by M. C. Escher (lithograph, 1960).