diff SMC2015latex/section/background.tex @ 62:e2b9ccb92973

updating things

--- a/SMC2015latex/section/background.tex	Mon Apr 27 13:10:10 2015 +0100
+++ b/SMC2015latex/section/background.tex	Mon Apr 27 16:30:13 2015 +0100
@@ -4,26 +4,32 @@
 \subsection{Rhythm representation}
 \label{sec:background:rhythm}
 In this section, we introduce some key concepts to assist readers in understanding the mechanisms of 
-each syncopation model. Please refer to \cite{Song14thesis} for the detailed explanation all relevant rhythmic concepts in music theory and their mathematical notations.
+each syncopation model. Please refer to \cite{Song15thesis} for the detailed explanation all relevant rhythmic concepts in music theory and their mathematical notations.
 
-\begin{figure}
-\centerline{\epsfig{figure=images/general3.pdf, width=\columnwidth}}
-\caption{\textbf{An example note sequence}. Two note events $\note_0$ and $\note_1$ occur in the time-span between time origin $\timeorigin$ and end time $\timeend$. The time-span duration $\timespan$ is three quarter-note periods. The rests at the start and end of the bar are not explicitly represented as objects in their own right here but as periods where no notes sound.}
+\begin{figure}[t]
+\centering
+\includegraphics[width=\columnwidth]{images/general3.pdf}
+\caption{An example note sequence. Two note events $\note_0$ and $\note_1$ occur in the time-span between time origin $\timeorigin$ and end time $\timeend$. The time-span duration $\timespan$ is three quarter-note periods. The rests at the start and end of the bar are not explicitly represented as objects in their own right here but as periods where no notes sound.}
 \label{fig:general}
 \end{figure}
 
+
+
 \subsubsection{Time-span}
 \label{sec:background:rhythm:timespan}
 The term \emph{time-span} has been defined as the period between two points in time, including all time points in between \cite{Lerdahl_Jackendoff83GTTM}. To represent a given rhythm, we must specify the time-span within which it occurs by defining a reference time origin $\timeorigin$ and end time $\timeend$, the total duration $\timespan$ of which is $\timespan = \timeend-\timeorigin$ (Figure~\ref{fig:general}.
 
 The basic time unit is in \emph{ticks} as opposed to seconds, therefore we set the parameter Ticks Per Quarter-note (TPQ) to describe the time-span of a length of rhythm. The minimum TPQ is determined by the rhythm-pattern so that all the events can be represented. As demonstrated in Figure~\ref{fig:clave}, the \emph{Son} clave rhythm pattern could be represented both at 8 and 4 ticks per quarter-note but the minimum representable resolution would be 4.
 
-\begin{figure}
-\centerline{\epsfig{figure=images/clave_tpq.pdf, width=\columnwidth}}
-\caption{\textbf{The representation of \emph{Son} clave rhythm in different settings of Ticks Per Quarter-note (TPQ)}. Each quarter-note is represented by 8 and 4 ticks in (a) and (b) respectively, thus all the sounded notes are captured (highlighted by the blue circles); however in (c) where TQP is 2, the second note cannot be represented by this resolution.}
+\begin{figure}[t]
+\centering
+\includegraphics[width=0.85\columnwidth]{images/clave_tpq.pdf}
+\caption{The representation of \emph{Son} clave rhythm in different settings of Ticks Per Quarter-note (TPQ). Each quarter-note is represented by 8 and 4 ticks in (a) and (b) respectively, thus all the sounded notes are captured (highlighted by the blue circles); however in (c) where TQP is 2, the second note cannot be represented by this resolution.}
 \label{fig:clave}
 \end{figure}
 
+
+
 \subsubsection{Note and rhythm}
 \label{sec:background:rhythm:note}
 A single, \emph{note} event $\note$ occurring in this time-span may be described by the tuple $(\starttime, \durationtime, \velocity)$ as shown in Figure~\ref{fig:general}, where $\starttime$ represents start or \emph{onset} time relative to $\timeorigin$, $\durationtime$ represents note duration in the same units and $\velocity$ represents the note \emph{velocity} (i.e. the dynamic; how loud or accented the event is relative to others), where $\velocity > 0$.
@@ -59,29 +65,41 @@
 
 \subsubsection{Metrical structure and time-signature}
 \label{sec:background:rhythm:meter}
-\begin{figure}
-\centerline{\epsfig{figure=figs/ch_model/meter_hierarchy7.pdf, width=0.85\columnwidth}}
-\shortCap{Metrical hierarchies for different time-signatures.}{(a) A simple-duple hierarchy dividing the bar into two groups of two (as with a 4/4 time-signature). (b) A compound-duple hierarchy dividing a bar into two beats, each of which is subdivided by three (e.g. 6/8 time-signature). Reading the weights from left to right in any level $\metriclevel$ gives the elements in sequence $\metricvector_\metriclevel$}
+
+\begin{figure}[t]
+\centering
+\includegraphics[width=\columnwidth]{images/meter_hierarchy7.pdf}
+\caption{Metrical hierarchies for different time-signatures.(a) A simple-duple hierarchy dividing the bar into two groups of two (as with a 4/4 time-signature). (b) A compound-duple hierarchy dividing a bar into two beats, each of which is subdivided by three (e.g. 6/8 time-signature). Reading the weights from left to right in any level $\metriclevel$ gives the elements in sequence $\metricvector_\metriclevel$}
 \label{fig:meter-hierarchy}
 \end{figure}
 
+
+
 Isochronous-meter is formed with a multi-level hierarchical metrical structure~\cite{Lerdahl_Jackendoff83GTTM, London04Meter}. As shown in Figure~\ref{fig:meter-hierarchy}, under a certain metrical hierarchy, a bar is divided by a subdivision factor $\subdivision$ at each metrical level with index $\metriclevel$ where $\metriclevel \in [0, \levelmax]$. The list of subdivision factors is referred as a \emph{subdivision sequence}. 
 
 Events at different metrical positions vary in perceptual salience or \emph{metrical weight}~\cite{Palmer_Krumhansl90}. These weights may be represented as a \emph{weight sequence} $\metricweightset = \langle \metricweight_0, \metricweight_1, ... \metricweight_{\levelmax}\rangle$. The prevailing hypothesis for the assignment of weights in the metrical hierarchy is that a time point that exists in both the current metrical level and the level above is said to have a \emph{strong} weight compared gto time points that are not also present in the level above~\cite{Lerdahl_Jackendoff83GTTM}. The choice of values for the weights in $\metricweightset$ can vary between different models but the assignment of weights to nodes is common to all as in ~\cite{Lerdahl_Jackendoff83GTTM}.
 
 \subsection{Syncopation models}
 \label{sec:background:models}
-In this section we give a brief review of each implemented syncopation model, including their general hypothesis, mechanism and scope of capabilities. 
+In this section we give a brief review of each implemented syncopation model, including their general hypothesis and mechanism.   To compare the capabilities of each model, we give an overview of the musical features each captures in Table~\ref{ta:capabilites}. For a detailed review of these models see \cite{Song15thesis}.
 
 \subsubsection{Longuet-Higgins and Lee 1984 (\lhl)}
 \label{sec:background:models:lhl}
 
-Longuet-Higgins and Lee's \cite{LHL84} decomposes rhythm pattern into a tree structure from Section~\ref{sec:background:rhythm:meter} with metrical weights as $\metricweight_\metriclevel = -\metriclevel$ for all $\metricweight_\metriclevel \in \metricweightset$ i.e. $\metricweightset = \langle 0,-1,-2, ... \rangle$.
-The hypothesis of this model is that a syncopation occurs when a rest ($\RestNode$) in one metrical position follows a note ($\NoteNode$) in a weaker position.  Where such a note-rest pair occurs, the difference in their metrical weights is taken as a local syncopation score. Summing  the local scores produces the syncopation prediction for the whole rhythm sequence. 
+Longuet-Higgins and Lee's model \cite{LHL84} decomposes rhythm patterns into a tree structure as described in Section~\ref{sec:background:rhythm:meter} with metrical weights $\metricweight_\metriclevel = -\metriclevel$ for all $\metricweight_\metriclevel \in \metricweightset$ i.e. $\metricweightset = \langle 0,-1,-2, ... \rangle$.
+The hypothesis of this model is that a syncopation occurs when a rest ($\RestNode$) in one metrical position follows a note ($\NoteNode$) in a weaker position.  Where such a note-rest pair occurs, the difference in their metrical weights is taken as a local syncopation score. Summing the local scores produces the syncopation prediction for the whole rhythm sequence. 
 
 \subsubsection{Pressing 1997 (\pressing)}
 \label{sec:background:models:prs}
-Pressing's cognitive complexity model~\cite{Pressing97,Pressing93} specifies six prototype binary sequences and ranks them in terms of \emph{cognitive cost}. For example, the \emph{filled} prototype that has a note in ever position of the sequence (e.g. $\langle 0,1,1,1,0,1,1,1 \rangle$) cost less than the \emph{syncopated} prototype that has a 0 in the first, strongest metrical position (e.g. $\langle 0,1,1,1,0,1,1,1 \rangle$)(refer~\cite{Song14thesis} for details). The model analyses the cost for the whole rhythm-pattern and its sub-sequences at each metrical level determined by $\subdivision_\metriclevel$. The final output will be a weighted sum of the costs by the number of sub-sequences in each level.
+Pressing's cognitive complexity model~\cite{Pressing97,Pressing93} specifies six prototype binary sequences and ranks them in terms of \emph{cognitive cost}. For example, the lowest cost is the \emph{null} prototype that contains either a rest or a single note whereas the \emph{filled} prototype that has a note in every position of the sequence e.g. 
+$
+\langle 1,1,1,1 \rangle \nonumber
+$  
+which, in turn, has a lower cost than the \emph{syncopated} prototype that has a 0 in the first (i.e.\ strongest) metrical position e.g.  
+$
+\langle 0,1,1,1 \rangle \nonumber
+$.
+The model analyses the cost for the whole rhythm-pattern and its sub-sequences at each metrical level determined by $\subdivision_\metriclevel$. The final output is a weighted sum of the costs by the number of sub-sequences in each level.
 
 \subsubsection{Toussaint 2002 `Metric Complexity' (\metrical)}
 \label{sec:background:models:tmc}
@@ -103,16 +121,14 @@
 \label{sec:background:models:wnbd}
 
 
-\subsubsection{Capabilities of models}
-\label{sec:background:models:capabilities}
+\begin{table}
+\renewcommand{\arraystretch}{1.2}
+\centering
 
-\begin{table}
-\centering
-\caption{Comparisons of the properties of syncopation models.}
-\label{ta:capabilites}
-\begin{tabular}{c | c c c c c c c}
-\hline
-Property & \lhl & \pressing & \metrical & \sioros & \keith & \offbeat & \wnbd \\
+{\footnotesize
+\begin{tabular}{lccccccc}
+%\hline
+Property & \lhl  & \pressing  & \metrical  & \sioros & \keith  & \offbeat  & \wnbd \\
 \hline
 Onset & \checkmark & \checkmark & \checkmark & \checkmark & \checkmark & \checkmark & \checkmark \\
 Duration & & & & & \checkmark & & \checkmark \\
@@ -122,27 +138,20 @@
 Duple & \checkmark & \checkmark & \checkmark & \checkmark & \checkmark & \checkmark & \checkmark \\
 Triple & \checkmark & \checkmark & \checkmark & \checkmark & & \checkmark & \checkmark \\
 \hline
-%\hline
-%Model & Basis & Onset & Dynamics & Melody & Mono & Poly & Duple & Triple \\
-%\hline
-%\lhl & H & \checkmark & & & \checkmark & & \checkmark & \checkmark\\
-%\keith & C & \checkmark & & & \checkmark & \checkmark & \checkmark & \\
-%\pressing & H,C & \checkmark & & & \checkmark & & \checkmark & \checkmark\\
-%\metrical & H & \checkmark & & & \checkmark & & \checkmark & \checkmark\\
-%\offbeat & O & \checkmark & & & \checkmark & \checkmark & \checkmark & \checkmark\\
-%\wnbd & O & \checkmark & & & \checkmark & \checkmark & \checkmark & \checkmark\\
-%\sioros & H & \checkmark & \checkmark & & \checkmark & & \checkmark & \checkmark\\
-%\ksa & A & \checkmark & \checkmark & \checkmark & \checkmark & & \checkmark & \\
-%\hline
 \end{tabular}
+}
+\caption{Musical properties captured by the different syncopation models. All models use note onsets, but only two use note duration rather than inter-onset intervals. Only SG takes dynamics (i.e. variation in note velocity) into account. All models handle monorhythms but the four models based on hierarchical decomposition of rhythm patterns are unable to handle polyrhythmic patterns. All models can process both duple and triple meters with the exception of KTH that can only process duple.}
+\label{ta:capabilites}
 \end{table}
 
-To summarise the seven syncopation models, we compare their capabilities in terms of musical features they can capture in Table~\ref{ta:capabilites}. All the models use temporal features (i.e. onset time point and/or note duration) in the modelling. The SG model also process dynamic information of musical events (i.e. note velocity). We use the term \emph{monorhythm} to refer to any rhythm-pattern that is not polyrhythmic. All the models can measure syncopation of monorhythms, but only KTH, TOB and WNBD models can deal with polyrhythms. Finally, all the models can deal with rhythms (notated) in duple meter, but all models except KTH can cope with rhythms in a triple meter.
 
 
+%All the models use temporal features (i.e. onset time point and/or note duration) in the modelling. The SG model also process dynamic information of musical events (i.e. note velocity). We use the term \emph{monorhythm} to refer to any rhythm-pattern that is not polyrhythmic. All the models can measure syncopation of monorhythms, but only KTH, TOB and WNBD models can deal with polyrhythms. Finally, all the models can deal with rhythms (notated) in duple meter, but all models except KTH can cope with rhythms in a triple meter.
 
 
 
 
 
 
+
+