Robotics Reference
In-Depth Information
Rest
C
D
E
F
G
A
B
Rest
0.38
0.17
0.10
0.10
0.06
0.13
0.03
0.02
C
0.36
0.23
0.13
0.07
0.02
0.10
0.03
0.07
D
0.26
0.20
0.21
0.19
0.03
0.06
0.01
0.05
E
0.22
0.15
0.18
0.16
0.16
0.12
0.01
0.00
F
0.15
0.00
0.14
0.35
0.14
0.20
0.01
0.01
G
0.29
0.14
0.00
0.16
0.06
0.26
0.08
0.00
A
0.17
0.05
0.07
0.00
0.02
0.36
0.15
0.17
B
0.18
0.30
0.12
0.01
0.01
0.08
0.21
0.08
Figure 37.
Pinkerton's first order Markov chain probabilities
when a group of notes are sounded simultaneously. The eighteenth-
century French composer Jean-Philippe Rameau carried out a similar sta-
tistical analysis of harmony as the basis for much of his extensive writings
on the subject.
The Markov chains employed by Pinkerton are called
first order
Markov chains because they rely only on the one previous note to select
which row of the table should be examined when selecting the next note.
First order chains are the most appropriate ones to use when the amount
of source data is insubstantial. But when there is a large amount of source
data available as there is today,
20
then it is possible to delve more deeply
into the source material and create Markov chain databases that are even
closer to the original style. If, for example, we had data for how often
each of the sequences of notes C, rest; C, C; C, D; C, E; C, F; C, G; C,
A; and C, B; is followed by a rest, by an A, by a B, etc., then whenever
there is a two-note sequence starting with a C, we would know the fre-
quency of occurrence of the note after that two-note sequence. Such data
creates what is called a
second order
Markov chain. And the same method
can go even deeper, giving rise to a third order Markov chain in which
sequences of three successive notes each have associated with them a list
ofthefrequenciesforthefourthnotebeingarest,anA,aB,....Byusing
a third or fourth order Markov chain, based on all the compositions of
a well-known composer, it is possible to create new works in which that
composer's style is recognisable. Imagine being able to generate literally
thousands of tunes in the style of Lennon-McCartney, or anyone else you
wish.
20
For example, enormous corpora of written English and vast numbers of songs and other musical
works available in MIDI and other digital formats.
