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\def\firstauthor{Henrik Ekeus}
\def\secondauthor{Samer A. Abdallah}
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\def\fourthauthor{Peter W. McOwan}

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   {\firstauthor, \secondauthor, \thirdauthor, \fourthauthor}  {Queen Mary University of London \\  Centre for Digital Music \\ School of Electronic Engineering and Computer Science\\%
     {\tt \href{mailto:samer.abdallah@eecs.qmul.ac.uk}{\{samer.abdallah,hekeus\}@eecs.qmul.ac.uk}}}

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\begin{abstract}
The Melody Triangle is an interface for the discovery of melodic materials, where the inputÑpositions within a triangleÑdirectly map to information theoretic properties of the output.  The measures are the entropy rate, redundancy and \emph{predictive information rate}\cite{Abdallah:2009p4089} of the random process used to generate the sequence of notes. These are all related to the \emph{predictability} of the the sequence and as such address the notions of expectation and surprise in the perception of music. 

We describe some of the relevant ideas from information dynamics, how the Melody Triangle is defined in terms of these, and describe two physical incarnations of the Melody Triangle. The first is a multi-user installation where collaboration in a performative setting provides a playful yet informative way to explore expectation and surprise in music.  The second is a screen based interface where the Melody Triangle becomes a compositional tool for the generation of musical textures; the user's control  at the abstract level of randomness and predictability. Finally we outline a pilot study where the screen-based interface was used under experimental conditions to determine how the three measures of predictive information rate, entropy and redundancy might relate to musical preference.  	
\end{abstract}
%

\section{Information Dynamics}\label{sec:Information_dynamics}
Music involves patterns in time.  When listening to music we continually build and re-evaluate expectations of what is to come next.  Composers commonly, consciously or not, play with this process by setting up expectations which may, or may not be fulfilled, manipulating the expectations of the listener and inducing surprise or not as the music progresses 
%and surprise in the listener has been articulated by music theorist Meyer
\cite{Meyer:1967,Narmour:1977}.  Central to this is the idea that music is not a static object, presented as a whole, 
%as the grammatical analysis of Lerdahl and Jackendoff \cite{Lerdahl:1983} might imply, 
but as a phenomenon that `unfolds' and is experienced \emph{in time}.   
 
Information dynamics\cite{Abdallah:2009p4089} considers several different kinds of predictability in musical patterns, how these might be quantified using the tools of information theory, 
%human listeners might perceive these, 
and how they shape or affect the listening experience.  Central to this is the idea that listeners maintain a dynamically evolving statistical model that enables them to make predictions about how a piece of music will continue.  They do this using both the immediate context of the piece as well as using previous musical experience.  As the music unfolds, listeners continually revise their model; in other words, they revise their own, subjective probabilistic belief state. These changes in probabilistic beliefs can be associated with
quantities of information; these are the focus of information dynamics.



\section{The Melody Triangle}\label{sec:The_Melody_triangle}
%%%How we created the transition matrixes and created the triangle.
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 before \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.

The Melody Triangle enables the discovery of melodic content matching a set of information theoretic criteria.   Positions within the triangle correspond with pairs of values of entropy rate and redundancy. %The relationship with the predictive information rate is not explicitly controlled as this would require a three-dimensional interface, but an implicit relationship emerges, which is described in section \ref{makingthetriangle}.  
The physical interface to the Triangle has so far been realised in two forms: as an interactive installation and as a screen based interface.  

Given the information coordinates corresponding to a point in the triangle, we select from a pre-built
library of random processes, choosing one whose entropy rate and redundancy match the desired
values.  The implementations discussed in this paper use first order Markov chains as the content generator,
since it is easy to compute the theoretically exact values of entropy rate, redundancy and predictive
information rate given the transition matrix of the Markov chain. However, in principle, any generative system could be used to create the library of sequences, given an appropriate probabilistic listener model supporting
the estimation of entropy rate and redundancy.

The Markov chain based implementation generates streams of symbols in the abstract; the alphabet of symbols is then mapped to a set of distinct sounds, such as pitched notes in a scale or a set of percussive
sounds.  Further by layering these streams intricate musical textures can be created. The selection of
notes or sounds is arbitrary, as long as they are all distinguishable.
%)le is not a part of the Melody Triangle's core functionality, i
Indeed, the symbols could be mapped to even non sonic outputs such as visible shapes, colours, or movements.

Any sequence of symbols can be analysed and information theoretic measures estimated from it.  
The novelty of the Melody Triangle lies in that we reverse this mapping: given desired values for these measures, as determined from the user interface, we return a stream of symbols with the desired properties.
In the next section we describe the three information theoretic measures that we use.  


\section{Sequential Information Measures}\label{sec:Sequential_Information_Measures}
The \emph{entropy rate} of a random process is a basic measure of its randomness or
unpredictablity. Consider the viewpoint of an observer at a certain time, and split the
sequence into an infinite \emph{past}, as single symbol in the \emph{present}, and the 
infinite \emph{future}. The entropy rate is a conditional entropy; informally:
\begin{equation}
	\mathrm{EntropyRate} = H( \mathrm{Present} | \mathrm{Past}),
\end{equation}
that is, it represents our average uncertainty about the present symbol \emph{given}
that we have observed everything before it. Processes with zero entropy rate can
be predicted perfectly given enough of the preceeding context.

The \emph{redundancy} of the a process, in the sense we are using the term here, is
a measure of how much the predictability of the process depends on knowing the
preceeding context. It is the difference between the entropy of a single element of the
sequence in isolation (imagine chosing a note from a musical score at random with your 
eyes closed and then trying to guess the note) and its entropy after taking into account
the preceeding context:
\begin{equation}
	\mathrm{Redundancy} = H( \mathrm{Present} ) - H(\mathrm{Present} | \mathrm{Past}).
\end{equation}
If the previous symbols reduce our uncertainty about present symbol a great deal, then 
the redundancy is high. For example, if we know that a sequence consists of a repeating
cycle such as $ \ldots b, c, d, a, b, c, d, a \ldots$, but we don't know which was the first
symbol, then the redundancy is high, as $H(\mathrm{Present})$ is high (because we
have no idea about the present symbol in isolation, but $H(\mathrm{Present}|\mathrm{Past})$
is zero, because knowing the previous symbol immediately tells us what the present symbol is.

The \emph{predictive information rate} (PIR) brings in our uncertainty about the future. It is a
measure of how much each symbol reduces our uncertainty about the future as it is
observed, \emph{given} that we have observed the past:
\begin{equation}
	\mathrm{PIR} = H(\mathrm{Future} | \mathrm{Past}) - H(\mathrm{Future} | \mathrm{Present}, \mathrm{Past}).
\end{equation}
It is a measure of the \emph{new} information in each symbol.
Notice that if the past completely determines both the present and the future (as in the cyclic
pattern above) the PIR is zero, since the present symbol brings no new information. However,
if the symbols in a sequence are generated completely independently, e.g. by rolling a die for each
one, then again, the present symbol provides no information about the future and the PIR
is zero. 

%However,  there do exist processes that have high predictive information rates as compared
%with their entropy rates: within the class of Markov chains, these are neither the periodic nor the sequentially uncorrellated ones. Rather they tend to yield sequences that have certain recognisable patterns or motifs,
%but which occur at irregular times. A certain symbol might tell us about which one of the characteristic patterns will appear next.  Each symbol tell a us little bit about the future; in order to make good predictions,
%the listener must continually pay attention, building up expectations on the basis of each new observation.
%% but only a limited amount about the infinite future, we only learn about that as time goes on; there is continual building of prediction.
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 \cite{Wundt1897} that stimuli are most
pleasing at intermediate levels of novelty or disorder, where there is a balance between
`order' and `chaos'. 

  \begin{fig}{wundt}
    \raisebox{-4em}{\colfig[0.43]{wundt}}
 %  {\ \shortstack{{\Large$\longrightarrow$}\\ {\scriptsize\emph{exposure}}}\ }
    {\ {\large$\longrightarrow$}\ }
    \raisebox{-4em}{\colfig[0.43]{wundt2}}
    \caption{
      The Wundt curve relating randomness/complexity with
      perceived value. Repeated exposure sometimes results
      in a move to the left along the curve \cite{Berlyne71}.
    }
  \end{fig}


\begin{figure}
\centering
\includegraphics[width=0.2\textwidth]{figures/PeriodicMatrix.png}
\includegraphics[width=0.2\textwidth]{figures/NonDeterministicMatrix_bw.png}
\caption{Two transition matrixes.  The shade of white represents the probabilities of transition from one symbol to the next (black=0, white=1). The current symbol is along the bottom, and in this case there are twelve possibilities (mapped to a chromatic scale).  The left hand matrix has no uncertainty; it represents a periodic pattern. The right hand matrix contains unpredictability but nonetheless is not completely without perceivable structure, it is of a higher entropy rate. \label{TransitionMatrixes}}
\end{figure}



 \begin{fig}{mtriscat}
	\colfig[0.9]{mtriscat}
	\caption{The population of transition matrices in the 3D space of 
	entropy rate, redundancy and PIR, 
	all in bits.
	The concentrations of points along the redundancy axis correspond
	to Markov chains which are roughly periodic with periods of 2 (redundancy 1 bit),
	3, 4, \etc all the way to period 7 (redundancy 2.8 bits). The colour of each point
	represents its PIR---note that the highest values are found at intermediate entropy
	and redundancy, and that the distribution as a whole makes a curved triangle. Although
	not visible in this plot, it is largely hollow in the middle.  \label{InfoDynEngine}}
\end{fig}



%\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}
 \begin{figure}[h]
\centering
\includegraphics[width=0.5\textwidth]{figures/TheTriangle.pdf}
\caption{The Melody Triangle\label{TheTriangle}}
\end{figure}

\subsection{Populating the triangle}\label{makingthetriangle}

%
%The properties of the predictive information rate (PIR) 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.
%
%
%
%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, redundancy  and
%predictive information rate. The points along the `redundancy' axis correspond
%to periodic Markov chains. Those along the `entropy' axis 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', a graphical interface
%for exploring the melodic patterns generated by each of the Markov chains represented
%as points in \figrf{mtriscat}.


Before the Melody Triangle can used, it has to be `populated' 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 \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.  



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 `Melody Triangle' and forms the interface by which the system is controlled.
 
 Though the interface is 2D, the third dimension (PIR) is implicitly present, as 
transition matrices retrieved from
along the centre line of the triangle will tend to have higher PIR.
We hypothesise that, under
the appropriate conditions, these will be perceived as more `interesting' or 
`melodic.'

   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 \ref{TheTriangle} shows how the triangle maps to different measures of redundancy, entropy rate and predictive information rate.  
 
%%%paragraph explaining what the different parts of the triangle are like.
Each corner corresponds to three different extremes of predictability and unpredictability, which could be loosely characterised as `periodicity', `noise' and `repetition'. %Melodies from the `noise' 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 `periodicity' to `repetition' edge are all deterministic loops that get shorter as we approach the `repetition' corner, until it becomes just one repeating note.  It is the areas in between the extremes that provide the more `interesting' melodies. That is, those that have some level of unpredictability, but are not completely random.  Or, conversely, that are predictable, but not entirely so.  This triangular space allows for an intuitive exploration of expectation and surprise in temporal sequences based on a simple model of how one might guess the next event given the previous one.     
In our experiments with visualising and sonifying sequences sampled from
	first order Markov chains \cite{AbdallahPlumbley2009}, we found that
	the measures of redundancy rate, entropy rate and predictive information rate correspond to perceptible
	characteristics, and that the transition matrices maximising or minimising
	each of these quantities are quite distinct. High entropy rates are associated
	with completely uncorrelated sequences with no recognisable temporal structure
	High values of redundancy rate are associated with long periodic cycles (and low PIR
	and entropy rate). High values of predictive information rate are associated with intermediate values
	of redundancy rate and entropy rate, and recognisable, but not completely predictable,
	temporal structures. 


\section{User Interfaces}
Any number of interfaces could be developed for the Melody Triangle\footnote{The Melody Triangle was developed in Prolog and MatLab. It can be controlled with OpenSoundControl messages, and thus is independent of any specific interface implementation.}. We have developed two; a standard screen based interface where a user moves tokens with a mouse in and around a triangle on screen, and a multi-user interactive installation where a Kinect\footnote{http://www.xbox.com/en-GB/Kinect} camera tracks individuals in a space and maps their positions in the space to the triangle.  

\subsection{The Multi-User Installation}

\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{figures/kinnect.pdf}
\caption{The depth map as seen by the Kinect, and the bounding box outlines the blobs detected by OpenNI.\label{Kinect}}
\end{figure}

As a Kinect camera overlooks a space, its range naturally forms a triangle.  As visitors/users comes into the range of the camera, they start generating a melody, the statistical properties of this melody determined by the mapping of physical space to statistical space as discussed above. Thus by exploring the physical space the participant changes the predictability of the generated melodic content.  When multiple people are in the space they can cooperate to create interweaving melodies, forming intricate polyphonic textures.

The streams of symbols are mapped to MIDI and then played with software instruments in Logic.  The tracking system was capable of detecting gestures, and these were mapped to different musical effects such as tempo changes, periodicity changes (going to the off-beat), instrument/register changes and volume (see Figure \ref{gestures}).     
  
\subsubsection{Tracking and Control}

Tracking and control was done using the OpenNI libraries' API\footnote{http://OpenNi.org/} and high level middle-ware for tracking with Kinect.  This provided reliable blob tracking of humanoid forms in 2d space.  By triangulating this to the Kinect's depth map it became possible to get reliable coordinate of visitors' positions in the space.

By detecting the bounding box of the 2d blobs of individuals in the space, and then normalising these based on the distance of the depth map it became possible to work out if an individual had an arm stretched out or if they were crouching.  With this it was possible to define a series of gestures for controlling the system without the use of any controllers(see table \ref{gestures}).  Thus for instance by sticking out one's left arm quickly, the melody doubles in tempo.  By pulling one's left arm in at the same time as sticking the right arm out the melody would shift onto the offbeat.   Sending out both arms would change the instrument being `played'.    

\begin{table}
\centering
%\includegraphics[width=0.5\textwidth]{InstructionsText.pdf}
\caption{Gestures and their resulting effect\label{gestures}}
\begin{tabular}{ l c l }
left arm & right arm & meaning\\
\hline
  out & static & double tempo \\
  in & static & halve tempo \\
  static & out & triple tempo \\
  static & in & one-third tempo\\
  out & in & shift to off-beat \\
  out & out & change instrument\\
  in & in & reset tempo\\
\end{tabular}
\end{table}

\subsubsection{Observations}
Although visitors would need an initial bit of training they would then quickly be able to collaboratively design musical textures.  For example, one person could lay down a predictable repeating bass line by keeping themselves to the periodicity/repetition side of the room, while a companion can generate a freer melodic line by being nearer the 'noise' part of the space. 


The collaborative nature of this installation is an area that merits attention.  By not having one user be able to control the whole narrative, the participants would communicate verbally and direct each other in the goals of learning to use the system and finding interesting musical textures.  This collaboration added an element of playfulness and enjoyment that was clearly apparent. 

As an artefact this installation is an exploratory prototype and occupies an ambiguous role in terms of purpose; it is in a nebulous middle ground between instrument, art installation and technical demonstration.  It is clear however, that as a vehicle for communicating ideas related to the expectation, pattern and predictability in music to the public, it is very effective.   

\subsection{The Screen Based Interface}

\begin{figure}
\centering
\includegraphics[width=0.3\textwidth]{figures/UIscreenshot.png}
\caption{Screen shot of the screen based interface for the Melody Triangle\label{UIScreenShot}}
\end{figure}

%The Melody Triangle can also be explored with a standard screen, keyboard and mouse interface.  A triangle is drawn on the screen, screen space thus mapped to the statistical space of the Melody Triangle.   A number of round tokens, each representing a melody can be dragged in and around the triangle.  When a token is dragged into the triangle, the system will start generating the sequence of notes with statistical properties that correspond to its position in the triangle.  
%
%Additionally there are a number of keyboard controls.  These include controls for changing the overall tempo, for enabling and disabling individual voices, changing registers, going to off-beats and changing the speed of individual voices.  The system gives visual feedback to indicate when a token has locked on to a new melody, and contains a buffer zone for allowing tokens to be pushed right to the edges of the triangle without falling out.  
%
%In this mode, the Melody Triangle can be used as a kind of composition assistant for the generation of interesting musical textures and melodies. However unlike other computer aided composition tools or programming environments, here the composer engages with music on the high and abstract level of expectation, randomness and predictability.

The screen based interface can serve as a compositional tool.
%%A triangle is drawn on the screen, screen space thus mapped to the statistical
%space of the Melody Triangle.  
A number of tokens, each representing a
sonification stream or `voice', can be dragged in and around the triangle.
For each token, a sequence of symbols is sampled using the corresponding
transition matrix, which
%statistical properties that correspond to the token's position is generated.  These
%symbols 
are then mapped to notes of a scale or percussive sounds%
\footnote{The sampled sequence could easily be mapped to other musical processes, possibly over
different time scales, such as chords, dynamics and timbres. It would also be possible
to map the symbols to visual or other outputs.}%
.  Keyboard commands give control over other musical parameters such
as pitch register and inter-onset interval.  
%The possibilities afforded by the Melody Triangle in these other domains remains to be investigated.}.
%
The system is capable of generating quite intricate musical textures when multiple tokens
are in the triangle, but unlike other computer aided composition tools or programming
environments, the composer excercises control at the abstract level of information-dynamic
properties.


\section{User Trials}

We are currently in the process of using the screen-based
Melody Triangle user interface to investigate the relationship between the information-dynamic
characteristics of sonified Markov chains and subjective musical preference.
We carried out a pilot study with six participants, who were asked
to use a simplified form of the user interface (a single controllable token,
and no rhythmic, registral or timbral controls) under two conditions:
one where a single sequence was sonified under user control, and another
where an additional sequence was sonified in a different register, as if generated
by a fixed invisible token in one of four regions of the triangle. In addition, subjects
were asked to press a key if they `liked' what they were hearing.

After the study the participants were surveyed with the Goldsmiths Musical Sophistication Index\cite{Mullensiefen:2011ts} to elicit their prior musical experience. 

We recorded subjects' behaviour as well as points which they marked
with a key press.
Some results for two of the subjects are shown in \figrf{mtri-results}. Though
we have not been able to detect any systematic across-subjects preference for any particular
region of the triangle, subjects do seem to exhibit distinct kinds of exploratory behaviour.
Our initial hypothesis, that subjects would linger longer in regions of the triangle
that produced aesthetically preferable sequences, and that this would tend to be towards the
centre line of the triangle for all subjects, was not confirmed. However, it is possible
that the design of the experiment encouraged an initial exploration of the space (sometimes
very systematic, as for subject c) aimed at \emph{understanding} %the parameter space and
how the system works, rather than finding musical patterns. It is also possible that the
system encourages users to create musically interesting output by \emph{moving the token},
rather than finding a particular spot in the triangle which produces a musically interesting
sequence by itself.

\begin{fig}{mtri-results}
	\def\scat#1{\colfig[0.42]{mtri/#1}}
	\def\subj#1{\scat{scat_dwells_subj_#1} & \scat{scat_marks_subj_#1}}
	\begin{tabular}{cc}
%		\subj{a} \\
		\subj{b} \\
		\subj{c} \\
		\subj{d}
	\end{tabular}
	\caption{Dwell times and mark positions from user trials with the
	on-screen Melody Triangle interface, for three subjects. The left-hand column shows
	the positions in a 2D information space (entropy rate vs multi-information rate
	in bits) where each spent their time; the area of each circle is proportional
	to the time spent there. The right-hand column shows point which subjects
	`liked'; the area of the circles here is proportional to the duration spent at
	that point before the point was marked.}
\end{fig}

Comments collected from the subjects
%during and after the experiment 
suggest that
the information-dynamic characteristics of the patterns were readily apparent
to most: several noticed the main organisation of the triangle,
with repetitive notes at the top, cyclic patterns along one edge, and unpredictable
notes towards the opposite corner. Some described their systematic exploration of the space.
Two felt that the right side was `more controllable' than the left (a consequence
of their ability to return to a particular distinctive pattern and recognise it
as one heard previously). Two reported that they became bored towards the end,
but another felt there wasn't enough time to `hear out' the patterns properly.
One subject did not `enjoy' the patterns in the lower region, but another said the lower
central regions were more `melodic' and `interesting'.

We plan to continue the trials with a slightly less restricted user interface in order
make the experience more enjoyable and thereby give subjects longer to use the interface;
this may allow them to get beyond the initial exploratory phase and give a clearer
picture of their aesthetic preferences. In addition, we plan to conduct a
study under more restrictive conditions, where subjects will have no control over the patterns
other than to signal (a) which of two alternatives they prefer in a forced
choice paradigm, and (b) when they are bored of listening to a given sequence.





\section{Further Work}
The Melody Triangle has so far only been used with first-order Markov chains for generating content.  This mean that the melodies generated don't have any long term structure or form and hence don't seem to `go anywhere'.  As such the system in its current form is better suited to creating textures and short phrases as oppose to composing over-arching musical structures.  

We are currently investigating how higher-order Markov models can be mapped to information theoretic measures and adapting the Melody Triangle to those models.  This would generate higher level patterns and provide more long-term structures.  Further more sophisticated listener models\cite{Pearce:2005wr}\cite{Potter:2007tt} could be used for computing information measures for more conventional or ecologically valid music.
 
 As it stands, the streams of symbols generated are only mapped to note values.  However they could just as well be applied to any other musical property, such as intervals, chords, dynamics, timbres, structures and key changes.  The possibilities for the Melody Triangle to be compositional guide in these other domains remains to be investigated. 
 
%The Melody Triangle in its current form however forms an ideal tool for investigations into musical preference and their relationship to the information dynamics models, and as such more detailed studies under wider experimental conditions and with more participants will be carried out.  

Although our initial data on aesthetic preference are inconclusive, there is still
plenty of work to be done in this area: where-ever there are probabilistic models,
information dynamics can shed light on their behaviour. 

\section{acknowledgments}
This work is supported by EPSRC Doctoral Training Centre EP/G03723X/1 (HE), GR/S82213/01 and EP/E045235/1(SA), an EPSRC Leadership Fellowship, EP/G007144/1 (MDP) and EPSRC IDyOM2 EP/H013059/1.  Thanks to Louie McCallum and Davie Smith from QMUL EECS for Kinect programming support.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%bibliography here
\bibliography{smc2012template,nime,all,c4dm}


%%%%%TO INCLUDE
%In our experiments with visualising and sonifying sequences sampled from
%	first order Markov chains \cite{AbdallahPlumbley2009}, we found that
%	the measures $h_\mu$, $\rho_\mu$ and $b_\mu$ correspond to perceptible
%	characteristics, and that the transition matrices maximising or minimising
%	each of these quantities are quite distinct. High entropy rates are associated
%	with completely uncorrelated sequences with no recognisable temporal structure
%	(and low $\rho_\mu$ and $b_\mu$). 
%	High values of $\rho_\mu$ are associated with long periodic cycles (and low $h_\mu$
%	and $b_\mu$). High values of $b_\mu$ are associated with intermediate values
%	of $\rho_\mu$ and $h_\mu$, and recognisable, but not completely predictable,
%	temporal structures. 



%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.

%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' axis 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', a graphical interface
%for exploring the melodic patterns generated by each of the Markov chains represented
%as points in \figrf{mtriscat}.



\end{document}
