maldv/crabcanon · Datasets at Hugging Face

text stringlengths 0 174
meaning of the word; "ricercar" was, in fact, the original name for the musical form now
known as "fugue". By Bach's time, the word "fugue" (or fuga, in Latin and Italian) had
become standard, but the term "ricercar" had survived, and now designated an erudite
kind of fugue, perhaps too austerely intellectual for the common ear. A similar usage
survives in English today: the word "recherche" means, literally, "sought out", but carries
the same kind of implication, namely of esoteric or highbrow cleverness.

The trio sonata forms a delightful relief from the austerity of the fugues and canons,
because it is very melodious and sweet, almost dance-


introduction: A Musico-Logical Offering


7


able. Nevertheless, it too is based largely on the King's theme, chromatic and austere as it
is. It is rather miraculous that Bach could use such a theme to make so pleasing an
interlude.

The ten canons in the Musical Offering are among the most sophisticated canons Bach
ever wrote. However, curiously enough, Bach himself never wrote them out in full. This
was deliberate. They were posed as puzzles to King Frederick. It was a familiar musical
game of the day to give a single theme, together with some more or less tricky hints, and
to let the canon based on that theme be "discovered" by someone else. In order to know
how this is possible, you must understand a few facts about canons.

Canons and Fugues

The idea of a canon is that one single theme is played against itself. This is done by
having "copies" of the theme played by the various participating voices. But there are
means' ways to do this. The most straightforward of all canons is the round, such as
"Three Blind Mice", "Row, Row, Row Your Boat", or " Frere Jacques". Here, the theme
enters in the first voice and, after a fixed time-delay, a "copy" of it enters, in precisely the
same key. After the same fixed time-delay in the second voice, the third voice enters
carrying the theme, and so on. Most themes will not harmonize with themselves in this
way. In order for a theme to work as a canon theme, each of its notes must be able to
serve in a dual (or triple, or quadruple) role: it must firstly be part of a melody, and
secondly it must be part of a harmonization of the same melody. When there are three
canonical voices, for instance, each note of the theme must act in two distinct harmonic
ways, as well as melodically. Thus, each note in a canon has more than one musical
meaning; the listener's ear and brain automatically figure out the appropriate meaning, by
referring to context.

There are more complicated sorts of canons, of course. The first escalation in
complexity comes when the "copies" of the theme are staggered not only in time, but also
in pitch; thus, the first voice might sing the theme starting on C, and the second voice,
overlapping with the first voice, might sing the identical theme starting five notes higher,
on G. A third voice, starting on the D yet five notes higher, might overlap with the first
two, and so on. The next escalation in complexity comes when the speeds of the different
voices are not equal; thus, the second voice might sing twice as quickly, or twice as
slowly, as the first voice. The former is called diminution, the latter augmentation (since
the theme seems to shrink or to expand).

We are not yet done! The next stage of complexity in canon construction is to invert the
theme, which means to make a melody which jumps down wherever the original theme
jumps up, and by exactly the same number of semitones. This is a rather weird melodic
transformation, but when one has heard many themes inverted, it begins to seem quite
natural. Bach was especially fond of inversions, and they show up often in his work-and
the Musical Offering is no exception. (For a simple example of


Introduction: A Musico-Logical Offering


8



inversion, try the tune "Good King Wenceslas". When the original and its inversion are
sung together, starting an octave apart and staggered with a time-delay of two beats, a
pleasing canon results.) Finally, the most esoteric of "copies" is the retrograde copy-
where the theme is played backwards in time. A canon which uses this trick is
affectionately known as a crab canon, because of the peculiarities of crab locomotion.
Bach included a crab canon in the Musical Offering, needless to say. Notice that every
type of "copy" preserves all the information in the original theme, in the sense that the
theme is fully recoverable from any of the copies. Such an information preserving
transformation is often called an isomorphism, and we will have much traffic with
isomorphisms in this book.

Sometimes it is desirable to relax the tightness of the canon form. One way is to allow
slight departures from perfect copying, for the sake of more fluid harmony. Also, some
canons have "free" voices-voices which do not employ the canon's theme, but which
simply harmonize agreeably with the voices that are in canon with each other.

Each of the canons in the Musical Offering has for its theme a different variant of the
King's Theme, and all the devices described above for making canons intricate are
exploited to the hilt; in fact, they are occasionally combined. Thus, one three-voice canon
is labeled "Canon per Augmentationem, contrario Motu"; its middle voice is free (in fact,
it sings the Royal Theme), while the other two dance canonically above and below it,
using the devices of augmentation and inversion. Another bears simply the cryptic label
"Quaerendo invenietis" ("By seeking, you will discover"). All of the canon puzzles have
been solved. The canonical solutions were given by one of Bach's pupils, Johann Philipp
Kirnberger. But one might still wonder whether there are more solutions to seek!

I should also explain briefly what a fugue is. A fugue is like a canon, in that it is
usually based on one theme which gets played in different voices and different keys, and
occasionally at different speeds or upside down or backwards. However, the notion of
fugue is much less rigid than that of canon, and consequently it allows for more

meaning of the word; "ricercar" was, in fact, the original name for the musical form now

known as "fugue". By Bach's time, the word "fugue" (or fuga, in Latin and Italian) had

become standard, but the term "ricercar" had survived, and now designated an erudite

kind of fugue, perhaps too austerely intellectual for the common ear. A similar usage

survives in English today: the word "recherche" means, literally, "sought out", but carries

the same kind of implication, namely of esoteric or highbrow cleverness.

The trio sonata forms a delightful relief from the austerity of the fugues and canons,

because it is very melodious and sweet, almost dance-

introduction: A Musico-Logical Offering

7

able. Nevertheless, it too is based largely on the King's theme, chromatic and austere as it

is. It is rather miraculous that Bach could use such a theme to make so pleasing an

interlude.

The ten canons in the Musical Offering are among the most sophisticated canons Bach

ever wrote. However, curiously enough, Bach himself never wrote them out in full. This

was deliberate. They were posed as puzzles to King Frederick. It was a familiar musical

game of the day to give a single theme, together with some more or less tricky hints, and

to let the canon based on that theme be "discovered" by someone else. In order to know

how this is possible, you must understand a few facts about canons.

Canons and Fugues

The idea of a canon is that one single theme is played against itself. This is done by

having "copies" of the theme played by the various participating voices. But there are

means' ways to do this. The most straightforward of all canons is the round, such as

"Three Blind Mice", "Row, Row, Row Your Boat", or " Frere Jacques". Here, the theme

enters in the first voice and, after a fixed time-delay, a "copy" of it enters, in precisely the

same key. After the same fixed time-delay in the second voice, the third voice enters

carrying the theme, and so on. Most themes will not harmonize with themselves in this

way. In order for a theme to work as a canon theme, each of its notes must be able to

serve in a dual (or triple, or quadruple) role: it must firstly be part of a melody, and

secondly it must be part of a harmonization of the same melody. When there are three

canonical voices, for instance, each note of the theme must act in two distinct harmonic

ways, as well as melodically. Thus, each note in a canon has more than one musical

meaning; the listener's ear and brain automatically figure out the appropriate meaning, by

referring to context.

There are more complicated sorts of canons, of course. The first escalation in

complexity comes when the "copies" of the theme are staggered not only in time, but also

in pitch; thus, the first voice might sing the theme starting on C, and the second voice,

overlapping with the first voice, might sing the identical theme starting five notes higher,

on G. A third voice, starting on the D yet five notes higher, might overlap with the first

two, and so on. The next escalation in complexity comes when the speeds of the different

voices are not equal; thus, the second voice might sing twice as quickly, or twice as

slowly, as the first voice. The former is called diminution, the latter augmentation (since

the theme seems to shrink or to expand).

We are not yet done! The next stage of complexity in canon construction is to invert the

theme, which means to make a melody which jumps down wherever the original theme

jumps up, and by exactly the same number of semitones. This is a rather weird melodic

transformation, but when one has heard many themes inverted, it begins to seem quite

natural. Bach was especially fond of inversions, and they show up often in his work-and

the Musical Offering is no exception. (For a simple example of

Introduction: A Musico-Logical Offering

8

inversion, try the tune "Good King Wenceslas". When the original and its inversion are

sung together, starting an octave apart and staggered with a time-delay of two beats, a

pleasing canon results.) Finally, the most esoteric of "copies" is the retrograde copy-

where the theme is played backwards in time. A canon which uses this trick is

affectionately known as a crab canon, because of the peculiarities of crab locomotion.

Bach included a crab canon in the Musical Offering, needless to say. Notice that every

type of "copy" preserves all the information in the original theme, in the sense that the

theme is fully recoverable from any of the copies. Such an information preserving

transformation is often called an isomorphism, and we will have much traffic with

isomorphisms in this book.

Sometimes it is desirable to relax the tightness of the canon form. One way is to allow

slight departures from perfect copying, for the sake of more fluid harmony. Also, some

canons have "free" voices-voices which do not employ the canon's theme, but which

simply harmonize agreeably with the voices that are in canon with each other.

Each of the canons in the Musical Offering has for its theme a different variant of the

King's Theme, and all the devices described above for making canons intricate are

exploited to the hilt; in fact, they are occasionally combined. Thus, one three-voice canon

is labeled "Canon per Augmentationem, contrario Motu"; its middle voice is free (in fact,

it sings the Royal Theme), while the other two dance canonically above and below it,

using the devices of augmentation and inversion. Another bears simply the cryptic label

"Quaerendo invenietis" ("By seeking, you will discover"). All of the canon puzzles have

been solved. The canonical solutions were given by one of Bach's pupils, Johann Philipp

Kirnberger. But one might still wonder whether there are more solutions to seek!

I should also explain briefly what a fugue is. A fugue is like a canon, in that it is

usually based on one theme which gets played in different voices and different keys, and

occasionally at different speeds or upside down or backwards. However, the notion of

fugue is much less rigid than that of canon, and consequently it allows for more