
csong@49: \section{Background}
csong@49: 
csong@49: \subsection{Rhythm representation}
csong@49: In this section, we introduce some key concepts to assist readers in understanding the mechanisms of 
csong@49: each syncopation model. Please refer to \cite{Song14thesis} for the detailed explanation all relevant rhythmic concepts in music theory and their mathematical notations.
csong@49: 
csong@49: \begin{figure}
csong@49: \centerline{\epsfig{figure=images/general3.pdf, width=\columnwidth}}
csong@49: \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.}
csong@49: \label{fig:general}
csong@49: \end{figure}
csong@49: 
csong@49: \subsubsection{Time-span}
csong@49: 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}.
csong@49: 
csong@49: 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.
csong@49: 
csong@49: \begin{figure}
csong@49: \centerline{\epsfig{figure=images/clave_tpq.pdf, width=\columnwidth}}
csong@49: \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.}
csong@49: \label{fig:clave}
csong@49: \end{figure}
csong@49: 
csong@49: \subsubsection{Note and rhythm}
csong@49: 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 $\velocity > 0$.
csong@49: 
csong@49: This allows us to represent an arbitrary rhythm as a note sequence $\sequence$, ordered in time
csong@49: \begin{equation}
csong@49: \label{eq:def_sequence}
csong@49: \sequence = \langle\note_0, \note_1, \cdots,  \note_{\sequencelength-1}\rangle
csong@49: \end{equation}
csong@49: 
csong@49: Suppose TQP is set as 4, an example note sequence for the clave rhythm in Figure~\ref{fig:clave} can be: 
csong@49: \begin{equation}
csong@49: \label{eq:note_sequence}
csong@49: \sequence = \langle (0,3,2),(3,1,1),(6,2,2),(10,2,1),(12,4,1) \rangle
csong@49: \end{equation}
csong@49: 
csong@49: The higher $velocity$ values of the first and third notes reflect that these notes are accented.
csong@49: 
csong@49: An alternative representation of a rhythm is the \emph{velocity sequence}.  This is a sequence of values representing equally spaced points in time; the values corresponding to the normalised velocity of a note onset if one is present at that time or zero otherwise. 
csong@49: 
csong@49: The velocity sequence for the above clave rhythm can be derived as
csong@49: \begin{equation}
csong@49: \label{eq:velocity_sequence}
csong@49: \spanvector = \langle 1,0,0,0.5,0,0,1,0,0,0,0.5,0,0.5,0,0,0 \rangle
csong@49: \end{equation}
csong@49: 
csong@49: It should be noted that the conversion between note sequence and velocity sequence is not commutative, because the note duration information is lost when converting from note sequence to velocity sequence. For example, the resulting note sequence converted from Equation~\ref{eq:velocity_sequence} would be 
csong@49: \begin{equation}
csong@49: \label{eq:note_sequence}
csong@49: \sequence' = \langle (0,3,2),(3,3,1),(6,4,2),(10,2,1),(12,4,1) \rangle
csong@49: \end{equation}
csong@49: , which is different from the original note sequence in Equation~\ref{eq:note_sequence}. 
csong@49: 
csong@49: \subsubsection{Metrical levels and Time-signature}
csong@49: \begin{figure}
csong@49: \centerline{\epsfig{figure=figs/ch_model/meter_hierarchy7.pdf, width=0.85\columnwidth}}
csong@49: \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 simple-triple hierarchy dividing a bar into three beats, each of which is subdivided by two (e.g. 3/4 time-signature). (c) 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 $\metricvector_\metriclevel$ (see Equations~\ref{eq:def_metricvector2} and \ref{eq:def_metricvector}).} %showing strong and weak beat positions along with the metrical `weights' associated with each node in the tree: 
csong@49: \label{fig:meter-hierarchy}
csong@49: \end{figure}
csong@49: 
csong@49: 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}. 
csong@49: 
csong@49: 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 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}.
csong@49: 
csong@49: \subsection{Syncopation models}
csong@49: 
csong@49: \subsection{Longuet-Higgins and Lee 1984 (\lhl)}
csong@49: 
csong@49: 
csong@49: \subsection{Pressing 1997 (\pressing)}
csong@49: 
csong@49: \subsection{Toussaint 2002 `Metric Complexity'  (\metrical)}
csong@49: 
csong@49: \subsection{Sioros and Guedes 2011 (\sioros)}
csong@49: 
csong@49: \subsection{Keith 1991 (\keith)}
csong@49: 
csong@49: \subsection{Toussaint 2005 `Off-Beatness' (\offbeat)}
csong@49: 
csong@49: \subsection{G\'omez 2005 `Weighted Note-to-Beat Distance' (WNBD)}
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