comparison draft.tex @ 17:e47aaea2ac28

Added images

  The measures are the entropy rate, redundancy and predictive information rate of the random process used to generate the sequence of notes. 
 These are all related to the predictability of the the sequence and as such address the notions of expectation and surprise in the perception of music.\emph{self-plagiarised}
  	
 Before the Melody Triangle can used, it has to be ÔpopulatedÕ with possible parameter values for the melody generators.
 These are then plotted in a 3d statistical space of redundancy, entropy rate and predictive information rate. 
-In our case we generated thousands of transition matrixes, representing first-order Markov chains, by a random sampling method. In figure x we see a representation of how these matrixes are distributed in the 3d statistical space; each one of these points corresponds to a transition matrix.\emph{self-plagiarised}
+In our case we generated thousands of transition matrixes, representing first-order Markov chains, by a random sampling method. In figure \ref{InfoDynEngine} we see a representation of how these matrixes are distributed in the 3d statistical space; each one of these points corresponds to a transition matrix.\emph{self-plagiarised}
+
+\begin{figure}
+\centering
+\includegraphics[width=0.5\textwidth]{MatrixDistribution.png}
+\caption{The population of transition matrixes distributed along three axes of redundancy, entropy rate and predictive information rate.  Note how the distribution makes a curved triangle-like plane floating in 3d space.  \label{InfoDynEngine}}
+\end{figure}
+ 
 	
 When we look at the distribution of transition matrixes plotted in this space, we see that it forms an arch shape that is fairly thin. 
 It thus becomes a reasonable approximation to pretend that it is just a sheet in two dimensions; and so we stretch out this curved arc into a flat triangle. 
 It is this triangular sheet that is our ÔMelody TriangleÕ and forms the interface by which the system is controlled. \emph{self-plagiarised}
 	
 When the Melody Triangle is used, regardless of whether it is as a screen based system, or as an interactive installation, it involves a mapping to this statistical space. 
 When the user, through the interface, selects a position within the triangle, the corresponding transition matrix is returned. 
-Figure x shows how the triangle maps to different measures of redundancy, entropy rate and predictive information rate.\emph{self-plagiarised}
-	
+Figure \ref{TheTriangle} shows how the triangle maps to different measures of redundancy, entropy rate and predictive information rate.\emph{self-plagiarised}
+ \begin{figure}
+\centering
+\includegraphics[width=0.5\textwidth]{TheTriangle.pdf}
+\caption{The Melody Triangle\label{TheTriangle}}
+\end{figure}	
 Each corner corresponds to three different extremes of predictability and unpredictability, which could be loosely characterised as ÔperiodicityÕ, ÔnoiseÕ and ÔrepetitionÕ. 
 Melodies from the ÔnoiseÕ corner have no discernible pattern; they have high entropy rate, low predictive information rate and low redundancy. 
 These melodies are essentially totally random. 
 A melody along the ÔperiodicityÕ to ÔrepetitionÕ edge are all deterministic loops that get shorter as we approach the ÔrepetitionÕ corner, until it becomes just one repeating note. 
 It is the areas in between the extremes that provide the more ÔinterestingÕ melodies.