Whenever we talk about patterns,

manipulation of patterns and uses

of contrapuntal techniques to modify

a particular pattern, we always go

back to the work of Johann Sebastian Bach,

who was a master of composition and

manipulation - I mean in this particular

aspect of contrapuntal manipulation

of musical ideas - and in particular

there are some influential works that Bach

produced. One is the art of the fugue

and the other one is the musical offering

where he uses all these manipulation

techniques in an amazingly masterful way.

Now, the examples that I want to talk

about in this context are related to

the idea of a canon. For those of you who

are not familiar with what a canon is,

a canon is a piece of music that is based

on a single melodic line, that is played

or sung by a performer,

and then a second performer

enters with the same identical melodic

line but at a different time. So there is

a superposition of two identical lines,

with different starting points in time.

And the combination of these two identical

lines gives rise to harmonic and melodic

effects that generate

the composition itself.

If one is a student in countrapoint that

has to go through training in how to write

canons and canons of very different kinds,

this is not the place for me to go into

all these details, but there are some

very sophisticated canons that

one can write, and one is what is called

a crab canon.

A crab canon is a canon where there is

one melodic line that is sung or played

by one performer,

and the same melodic line is played

but retrograted by the other one.

So you have a single melody that one

performer plays left to right, and the

other one plays right to left at

the same time. And there is a famous

crab canon in the musical offering of

J. S. Bach that you can hear, that you can

listen to in the PowerPoint of this unit.

You can see actually the whole canon as it

is written. As you see, it is written only

in one line, and so the first performer,

the first voice, plays it normally, left

to right, and the second one plays

the score backwards. And this is actually

how these crab canons were written at

the time of J. S. Bach.

The second example is another canon

from the musical offering of J. S. Bach,

it is called the mirror canon. In a mirror

canon, we still have a single melodic

line, that is the principle line of

the canon, but then the response, or

the second performer enters at a

given point of the performance of

the first line, but instead of repeating

the same melodic content, it plays

the same motif, but inverted. So it's

an equilibrium between the original line

and the inverted line that start at

different times. Both a crab and a mirror

canon are demonstrated in the notebook,

where you can actually play with

the structure of it using very few lines

of code, so again it is a demonstration

of how these kind of algorithmic processes

give rise to very interesting

musical structures.