changeset 53:2f783c4c3562

Added to intro of sec 4, moved some redundant material out of 4A.

Binary file draft.pdf has changed

--- a/draft.tex	Fri Mar 16 12:15:36 2012 +0000
+++ b/draft.tex	Fri Mar 16 13:02:43 2012 +0000
@@ -616,6 +616,7 @@
 
 
 	\subsection{First and higher order Markov chains}
+	\label{s:markov}
 	First order Markov chains are the simplest non-trivial models to which information 
 	dynamics methods can be applied. In \cite{AbdallahPlumbley2009} we derived
 	expressions for all the information measures described in \secrf{surprise-info-seq} for
@@ -796,6 +797,60 @@
 \section{Information dynamics as compositional aid}
 \label{s:composition}
 
+The use of stochastic processes in music composition has been widespread for
+decades---for instance Iannis Xenakis applied probabilistic mathematical models
+to the creation of musical materials\cite{Xenakis:1992ul}. While such processes
+can drive the \emph{generative} phase of the creative process, information dynamics
+can serve as a novel framework for a \emph{selective} phase, by 
+providing a set of criteria to be used in judging which of the 
+generated materials
+are of value. This alternation of generative and selective phases as been
+noted by art theorist Margaret Boden \cite{Boden1990}.
+
+Information-dynamic criteria can also be used as \emph{constraints} on the
+generative processes, for example, by specifying a certain temporal profile
+of suprisingness and uncertainty the composer wishes to induce in the listener
+as the piece unfolds.
+%stochastic and algorithmic processes: ; outputs can be filtered to match a set of
+%criteria defined in terms of information-dynamical characteristics, such as
+%predictability vs unpredictability
+%s model, this criteria thus becoming a means of interfacing with the generative processes.  
+
+The tools of information dynamics provide a way to constrain and select musical
+materials at the level of patterns of expectation, implication, uncertainty, and predictability.
+In particular, the behaviour of the predictive information rate (PIR) defined in 
+\secrf{process-info} make it interesting from a compositional point of view. The definition 
+of the PIR is such that it is low both for extremely regular processes, such as constant
+or periodic sequences, \emph{and} low for extremely random processes, where each symbol
+is chosen independently of the others, in a kind of `white noise'. In the former case,
+the pattern, once established, is completely predictable and therefore there is no
+\emph{new} information in subsequent observations. In the latter case, the randomness
+and independence of all elements of the sequence means that, though potentially surprising,
+each observation carries no information about the ones to come.
+
+Processes with high PIR maintain a certain kind of balance between
+predictability and unpredictability in such a way that the observer must continually
+pay attention to each new observation as it occurs in order to make the best
+possible predictions about the evolution of the seqeunce. This balance between predictability
+and unpredictability is reminiscent of the inverted `U' shape of the Wundt curve (see \figrf{wundt}), 
+which summarises the observations of Wundt that the greatest aesthetic value in art
+is to be found at intermediate levels of disorder, where there is a balance between
+`order' and `chaos'. 
+
+Using the methods of \secrf{markov}, we found \cite{AbdallahPlumbley2009}
+a similar shape when plotting entropy rate againt PIR---this is visible in the
+upper envelope of the scatter plot in \figrf{mtriscat}, which is a 3-D scatter plot of 
+three of the information measures discussed in \secrf{process-info} for several thousand 
+first-order Markov chain transition matrices generated by a random sampling method. 
+The coordinates of the `information space' are entropy rate ($h_\mu$), redundancy ($\rho_\mu$), and
+predictive information rate ($b_\mu$). The points along the 'redundancy' axis correspond
+to periodic Markov chains. Those along the `entropy' produce uncorrelated sequences
+with no temporal structure. Processes with high PIR are to be found at intermediate
+levels of entropy and redundancy.
+These observations led us to construct the `Melody Triangle' as a graphical interface
+for exploring the melodic patterns generated by each of the Markov chains represented
+as points in \figrf{mtriscat}.
+
   \begin{fig}{wundt}
     \raisebox{-4em}{\colfig[0.43]{wundt}}
  %  {\ \shortstack{{\Large$\longrightarrow$}\\ {\scriptsize\emph{exposure}}}\ }
@@ -808,31 +863,6 @@
     }
   \end{fig}
  
-The use of stochastic processes in music composition has been widespread for
-decades---for instance Iannis Xenakis applied probabilistic mathematical models
-to the creation of musical materials\cite{Xenakis:1992ul}.  Information dynamics
-can serve as a novel framework for the exploration of the possibilities of
-stochastic and algorithmic processes; outputs can be filtered to match a set of
-criteria defined in terms of information-dynamical characteristics, such as
-predictability vs unpredictability
-%s model, this criteria thus becoming a means of interfacing with the generative processes.  
-This allows a composer to explore musical possibilities at the high and abstract level of
-expectation, randomness and predictability.
-In particular, the behaviour of the predictive information rate (PIR) defined in 
-\secrf{process-info} make it interesting from a compositional point of view. The definition 
-of the PIR is such that it is low both for extremely regular processes, such as constant
-or periodic sequences, \emph{and} low for extremely random processes, where each symbol
-is chosen independently of the others, in a kind of `white noise'. In the former case,
-the pattern, once established, is completely predictable and therefore there is no
-\emph{new} information in subsequent observations. In the latter case, all the observations
-are random and independent, and hence unpredictable, but also not informative about
-any other observation. Processes with high PIR maintain a certain kind of balance between
-predictability and unpredictability in such a way that the observer must be continually
-paying attention to each new observation as it occurs in order to make the best
-possible predictions about the evolution of the seqeunce. This balance between predictability
-and unpredictability is reminiscent of the Wundt curve (see \figrf{wundt}), which 
-summarises the observations of Wundt [ref].. [etc \dots].
-
 
 %It is possible to apply information dynamics to the generation of content, such as to the composition of musical materials. 
 
@@ -844,6 +874,18 @@
 
  \subsection{The Melody Triangle}  
 
+The Melody Triangle is an exploratory interface for the discovery of melodic
+content, where the input---positions within a triangle---directly map to information
+theoretic measures of the output.  The measures---entropy rate, redundancy and
+predictive information rate---form a criteria with which to filter the output
+of the stochastic processes used to generate sequences of notes.  These measures
+address notions of expectation and surprise in music, and as such the Melody
+Triangle is a means of interfacing with a generative process in terms of the
+predictability of its output.
+
+The triangle is `populated' with first order Markov chain transition
+matrices as illustrated in \figrf{mtriscat}.
+
  \begin{fig}{mtriscat}
 	\colfig{mtriscat}
 	\caption{The population of transition matrices distributed along three axes of 
@@ -856,23 +898,6 @@
 	not visible in this plot, it is largely hollow in the middle.}
 \end{fig}
 
-The Melody Triangle is an exploratory interface for the discovery of melodic
-content, where the input---positions within a triangle---directly map to information
-theoretic measures of the output.  The measures---entropy rate, redundancy and
-predictive information rate---form a criteria with which to filter the output
-of the stochastic processes used to generate sequences of notes.  These measures
-address notions of expectation and surprise in music, and as such the Melody
-Triangle is a means of interfacing with a generative process in terms of the
-predictability of its output.
-
-The triangle is `populated' with possible parameter values for melody generators.
-These are plotted in a 3D information space of $\rho_\mu$ (redundancy), $h_\mu$ (entropy rate) and
-$b_\mu$ (predictive information rate), as defined in \secrf{process-info}.
- In our case we generated thousands of transition matrices, representing first-order
- Markov chains, by a random sampling method.  In figure \figrf{mtriscat} we
- see a representation of how these matrices are distributed in the 3D information
- space; each one of these points corresponds to a transition matrix.
-
 The distribution of transition matrices plotted in this space 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