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%!TEX root = intonation_paper.tex
\section{Pitch drift}
\label{sec:drift}


For singers sing with different vocal parts, only one singer can hear another, we observed a significant difference between sing without or with a partner in Comparison Set. The ANOVA results indicates that singers were influenced by their partner when they are singing in terms of MAPE and MAIE, but not significant for MAIES (Table \ref{tab:signi-sim-du-plex} and Table \ref{tab:maies-sim-du-plex}). The acoustic feedback is helpful to reduce the pitch error and interval error for each singer. But for MAIES, it also depends on the test condition. When sing unison (Unison Set), MAIES is less significant than sing duet (Duet Set).

\begin{table}[ht]
 \begin{center}
 %\begin{tabular}{p{2.5cm}|p{4.5cm}}
%\tiny
 \begin{tabular}{|l||c|c|}
  \hline
 & MAPE & MAIE  \\
  \hline
Comparison  & F(1,11926)=70.96*** & F(1,11527)=20.85*** \\
    \hline
Unison & F(1,7684)=25.46***&  F(1,7684)=9.02** \\
    \hline
Duet &F (1,7270)=124.43*** & F(1,6586)=30.36***   \\
    \hline
 \end{tabular}
\end{center}
 \caption{\small The F-test of the factor of simplex and duplex for three data sets and pitch accuracy parameters(***$p<.001$; **$p<.01$; *$p<.05$)}
 \label{tab:signi-sim-du-plex}
\end{table}


\begin{table}[ht]
 \begin{center}
 %\begin{tabular}{p{2.5cm}|p{4.5cm}}
 \begin{tabular}{|l||c|}
  \hline
 &  MAIES\\
  \hline
Comparison  &  F(1,5962)=3.48\\
    \hline
Unison &  F(1,3841)=22.45***\\
    \hline
Duet &F(1,3634)=36.52*** \\
    \hline
 \end{tabular}
\end{center}
 \caption{\small The F-test of the factor of simplex and duplex for three data sets and pitch accuracy parameters(***$p<.001$; **$p<.01$; *$p<.05$)}
 \label{tab:maies-sim-du-plex}
\end{table}


\begin{figure}[ht]
\subfloat[Unison test]{\includegraphics[width=.25\textwidth]{../figures/anova-maies-unison-comparedata.pdf}\label{fig:hy2-soprano3}}
\subfloat[Duet test]{\includegraphics[width=.25\textwidth]{../figures/anova-maies-duet-comparedata.pdf}\label{fig:hy2-alto3}}
\caption{ANOVA results of simplex\solo and simplex duet when sing with different vocal part}
\label{fig:anova-maies-comparedata}
\end{figure}


The Figure \ref{fig:anova-maies-comparedata}
shows the box plot of two test types and listening conditions. Although both factors are significant effect the MAIES, the results are different between unison test ($F(1,3841) = 22.45$, $p<.001$) and duet test ($F(1,2119) = 30.53$, $p<.001$). In the unison test, singing in \duplex condition decrease the MAIES while MAIES increase in duet condition. 

Besides the pitch accuracy, the note stability does not show significant difference among simplex and duplex listening condition in Comparison Set ($F(1,11527)=0.05$, $p=.83$).

\subsection{Accuracy of \solo and duplex}
We suppose that sing solo is more accurate than sing with a partner in different vocal part. In the first duet experiment, we use one-way ANOVA to analysis the influence of listening condition. The results show both soprano and alto have less pitch error when they sing solo than they sing duet. We process the raw data with smoothing window according to the previous study \cite{dai2015analysis}. The data was from 12 groups of duet singers. All the results are base on fitting data which show significant difference among different listening condition for the soprano ($F(1,2420)=10.68$, $p=0.0011$) and the alto ($F(1,2422)=199.42$, $p<.001$). The Figure \ref{fig:hy1} shows the box-plot of MAPE of soprano and alto singers in different listening condition. 

\begin{figure}[ht]
\subfloat[Soprano]{\includegraphics[width=.25\textwidth]{../figures/hy1-soprano.pdf}\label{fig:hy1-soprano4}}
\subfloat[Alto]{\includegraphics[width=.25\textwidth]{../figures/hy1-alto.pdf}\label{fig:hy1-alto4}}
\caption{ANOVA results in different trial conditions when singers sing different vocal part}
\label{fig:hy1}
\end{figure}

As the Table \ref{tab:hy1} shows, both soprano and alto singers make more error when they sing duet. The mean, median and standard deviation value of MAPE of solo singers are less then duet singers. Because we cut the vertical axis from 2 semitones, some extreme points belong to alto duet condition cannot been seen from the Figure \ref{fig:hy1}. It seems that for alto singers in  duet condition are likely to make larger error than the singer sing \solo.  

\begin{table}[ht]
\small
 \begin{center}
 %\begin{tabular}{p{2.5cm}|p{4.5cm}}
 \begin{tabular}{|l||c|c|c|}
  \hline
  MAPE & mean & median & std.dev.\\
  \hline
  Soprano solo & 0.3990 & 0.2274 & 0.4663 \\
    \hline
  Soprano duet & 0.4531 & 0.3072 & 0.4640 \\
    \hline
  Alto solo & 0.2639 & 0.1819 & 0.3884 \\
    \hline
  Alto duet & 0.6818 & 0.3333 & 0.8893 \\
    \hline
 \end{tabular}
\end{center}
 \caption{Solo and duet listening conditions when singers sing different vocal part}
 \label{tab:hy1}
\end{table}
For singers sing the same vocal part show the same significance in different listening condition. They sing more accurate when they sing solo even their partner is singing the same vocal part with them. Soprano ($F(1,2560)=4.28$, $p=0.04$) and alto ($F(1,2560=38.75$, $p<.001$) perform better without their partner in the same vocal part. The Figure \ref{fig:hy12} shows the box-plot of soprano and alto when they sing solo and duet in the same vocal part. 

\begin{figure}[ht]
\subfloat[Soprano]{\includegraphics[width=.25\textwidth]{../figures/hy1-soprano2.pdf}\label{fig:hy1-soprano2}}
\subfloat[Alto]{\includegraphics[width=.25\textwidth]{../figures/hy1-alto2.pdf}\label{fig:hy1-alto2}}
\caption{ANOVA results in different trial conditions when singers sing same vocal part}
\label{fig:hy12}
\end{figure}

Whether sing with different vocal part or sing with same vocal part, sing with a partner is likely to make more pitch error than sing solo. If in the same listen condition, sing with same vocal part makes less error than sing with different vocal part, the comparison about sing with same or different vocal part is in following section.