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If Markov matrices were limited to a single state per seed then their
musical usefulness would be limited
(as I am sure you have realised from playing around with the example in
part 1).
Luckily, there are no such limits and seeds can in fact be any length
you like.
Let's look back again at the set of numbers we used in part 1.
{64,62,60,62,64,64,64,62,62,62,64,67,67,64,62,60,62,64,64,64,64,62,62,64,62,60}
To produce a Markov matrix of a depth of more than 1 (1 was the depth
for the matrix in the previous example) we use seeds of more than one
element.
Let us build a matrix based on a seed of two elements instead of one (i.e.
a Markov depth of 2). We calculate a matrix with a depth of 2 in the same
way
we calculate a matrix with a depth of 1. The only difference is that we
base all our calculations on any two numbers being followed by another
number.
Remember our seed is now 2. Output is always 1 regardless of the Markov
depth.
Our primary matrix now looks like this.
|
|
60 |
62 |
64 |
67 |
| 64,62 |
0.6 |
0.4 |
0.0 |
0.0 |
| 62,60 |
0.0 |
1.0 |
0.0 |
0.0 |
| 60,62 |
0.0 |
0.0 |
1.0 |
0.0 |
| 62,64 |
0.0 |
0.25 |
0.50 |
0.25 |
| 64,64 |
0.0 |
0.4 |
0.6 |
0.0 |
| 62,62 |
0.0 |
0.3333333 |
0.6666666 |
0.0 |
| 67,67 |
0.0 |
0.0 |
1.0 |
0.0 |
| 64,67 |
0.0 |
0.0 |
0.0 |
1.0 |
| 67,64 |
0.0 |
1.0 |
0.0 |
0.0 |
The obvious change is that the far left hand column now contains 2 numbers
(prior states) as the seed rather than 1.
Two more obvious changes have occurred. There are more rows than there
were previously and there are more 1.0 (100%) weighings than there
were before. These last two points seem obvious, but they point to the
most important feature of Markov chains; the ability to maintain local
context
and provide for more discriminatory, therefore musically coherent, output.
The output of the matrix maintains the 'character' of the data used to
construct it. Musically we might say it stores a sense of 'style'.
A Markov chain can be of any depth (any seed length) but there are a
number of important considerations when regarding depth.
As the depth of the Markov chain approaches the length of the phrase (data)
the novelty (randomness) of the output decreases.
Take the above matrix as an example. When we calculated the matrix for
a Markov chain of depth 1, then 25% of the rows had a 100%
(no randomness) chance of selecting a particular note. Now in our matrix
of depth 2 more than 50% of rows have a 100% chance of selecting
a particular note. This is the tradeoff with Markov chains. The guarantee
of a particular sequence introduces stability and thematic coherence
but also introduces the chance of a very similar and even identical arrangement.
The resulting melody might be too close to the original.
To test this matrix we are not going to write another engine but instead
make use of another of jMusic's in built classes.
This class is called PhraseMatrix and it will calculate a matrix and generate
a phrase of any length based on the calculated matrix.
It will also do this for you for any depth level and calculates matrices
for pitch, rhythm and volume (yes, of course you can create matrices for
characteristics other than pitch, for example rhythms value, dynamic,
or position in the bar).
Here is an example class which uses the PhraseMatrix.
Click here to view/download the source.
1: import jm.JMC;
2: import jm.music.data.*;
3: import jm.music.tools.*;
4: import jm.util.*;
6: public class MarkovTest implements JMC{
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