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07-02-ACA-Tonal-Music.tex
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07-02-ACA-Tonal-Music.tex
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% move all configuration stuff into includes file so we can focus on the content
\input{include}
\subtitle{module 7.2: representation of pitch in music}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
% generate title page
\input{include/titlepage}
\section[overview]{lecture overview}
\begin{frame}{introduction}{overview}
\begin{block}{corresponding textbook section}
%\href{http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=6331122}{Chapter 5~---~Tonal Analysis}: pp.~82--91
section~7.2
\end{block}
\begin{itemize}
\item \textbf{lecture content}
\begin{itemize}
\item pitch-related music terminology: interval, mode, tonic, chord
\end{itemize}
\bigskip
\item<2-> \textbf{learning objectives}
\begin{itemize}
\item name musical intervals and notate them in score notation
\item explain pitch distance
\item discuss whether a chord is a harmony
\end{itemize}
\end{itemize}
\inserticon{directions}
\end{frame}
\section[musical pitch]{pitch in music}
\begin{frame}{musical pitch}{notation and names}
each octave (freq factor $2$) is split into $12$ pitch classes
\begin{columns}
\column{0.35\linewidth}
\begin{figure}
\scalebox{.8}
{
\input{pict/pitch_pitchclass}
}
\end{figure}
\column{0.5\linewidth}
\begin{figure}
\includegraphics{graph/pitch_pitches}
\end{figure}
\end{columns}
\begin{footnotesize}
\begin{table}
\centering
\begin{tabular}{cccccccccccc} %{\textwidth}{@{\extracolsep{\fill}}ccccccccccccc}
\\ \hline
\bf{\emph{$0$}} & \bf{\emph{$1$}} & \bf{\emph{$2$}} & \bf{\emph{$3$}} & \bf{\emph{$4$}} & \bf{\emph{$5$}} & \bf{\emph{$6$}} & \bf{\emph{$7$}} & \bf{\emph{$8$}} & \bf{\emph{$9$}} & \bf{\emph{$10$}} & \bf{\emph{$11$}}\\
\hline
\bf{$C$} & $C\sharp / D\flat$ & $D$ & $D\sharp / E\flat$ & E & $F$ & $F\sharp / G\flat$ & $G$ & $G\sharp / A\flat$ & $A$ & $A\sharp / B\flat$ & $B$\\
\end{tabular}
\end{table}
\end{footnotesize}
\end{frame}
\section{intervals}
\begin{frame}{musical pitch}{intervals}
\vspace{-8mm}
\begin{columns}
\column{0.47\linewidth}
\begin{scriptsize}
\begin{table}
\centering
\begin{tabular}{lccccccccccc} %{\textwidth}{@{\extracolsep{\fill}}ccccccccccccc}
\\ \hline
\bf{\emph{Interval}} & \bf{\emph{Enharmonic Equivalent}} & \bf{\emph{$\Delta\mathrm{ST}$}}\\
\hline
\bf{Unison} & Diminished Second & 0\\
\bf{Minor Second} & Augmented Unison & 1\\
\bf{(Major) Second} & Diminished Third & 2\\
\bf{Minor Third} & Augmented Second & 3\\
\bf{Major Third} & Diminished Fourth & 4\\
\bf{(Perfect) Fourth} & Augmented Third & 5\\
\bf{Augmented Fourth} & Diminished Fifth/Tritone & 6\\
\bf{(Perfect) Fifth} & Diminished Sixth & 7\\
\bf{Minor Sixth} & Augmented Fifth & 8\\
\bf{Major Sixth} & Diminished Seventh & 9\\
\bf{Minor Seventh} & Augmented Sixth & 10\\
\bf{Major Seventh} & Diminished Octave & 11\\
\bf{(Perfect) Octave} & Augmented Seventh & 12\\
\end{tabular}
\end{table}
\end{scriptsize}
\column{0.5\linewidth}
\begin{figure}
\includegraphics[scale=.8]{graph/pitch_intervals}
\end{figure}
\end{columns}
\end{frame}
\section{MIDI pitch}
\begin{frame}{musical pitch}{ MIDI pitch}
\begin{eqnarray*}\label{eq:midi_pitch}
\mathfrak{p}(f) &=& 69 + 12\cdot\log_2\left(\frac{f}{f_\mathrm{A4}}\right) \\
f(\mathfrak{p}) &=& f_{A4}\cdot2^{\frac{\mathfrak{p}-69}{12}}
\end{eqnarray*}
\pause
MIDI pitch mapping to \textit{pitch class}
\begin{equation*}\label{eq:pcidx}
\mathrm{PC}(\mathfrak{p}) = \mod(\mathfrak{p}, 12)
\end{equation*}
\end{frame}
\section{Cent}
\begin{frame}{musical pitch}{ (MIDI) pitch distance}
\textbf{cent}: pitch distance between two frequencies
\begin{footnotesize}
\begin{eqnarray*}\label{eq:cent}
\Delta C(f_1,f_2) &=& 100\cdot\big(\mathfrak{p}(f_1) - \mathfrak{p}(f_2)\big)\\
\pause
&=& 100\cdot\left(\left(69 + 12\cdot\log_2\left(\frac{f_1}{f_{A4}}\right)\right) - \left(69 + 12\cdot\log_2\left(\frac{f_2}{f_{A4}}\right)\right)\right)\\
\pause
&=& 1200\cdot\log_2\left( \frac{f_1}{f_2} \right)
\end{eqnarray*}
\end{footnotesize}
\bigskip
$\Rightarrow$ $\unit[100]{cents}$ span one semitone
\end{frame}
\section{temperament}
\begin{frame}{musical pitch}{ temperament}
\begin{itemize}
\item equally tempered scale:
\begin{itemize}
\item octave split into 12 equidistant notes (on log scale)
\item not key dependent,any modulation possible
\item enharmonic equivalence: $C\sharp = D\flat$
\item typical scale for keyboard instruments
\end{itemize}
%\item<3-> deviations of other scales
%\begin{tiny}\begin{table}
%\centering
%\begin{tabular}{lcccccc} %{c|p{12mm}p{12mm}p{12mm}p{12mm}p{12mm}p{12mm}p{12mm}}
%\\ \hline
%\bf{\emph{Pitch Class}} & \bf{\emph{Equally}} & \bf{\emph{Pythagorean}} & \bf{\emph{Meantone}} & \bf{\emph{Diatonic Major}} & \bf{\emph{Diatonic Minor}}\\
%\hline
%\bf{$C$} & $0$ & $0$ & $0$ & $0$ & $0$\\
%\bf{$C^\#$} & $0$ & $-$ & $-$ & $-$ & $-$\\
%\bf{$D$} & $0$ & $+3.9$ & $-6.9$ & $+3.9$ & $+3.9$\\
%\bf{$E^b$} & $0$ & $-$ & $-$ & $-$ & $+15.6$\\
%\bf{$E$} & $0$ & $+7.8$ & $-13.7$ & $-13.7$ & $-$\\
%\bf{$F$} & $0$ & $-2.0$ & $+3.4$ & $-2.0$ & $-2.0$\\
%\bf{$F^\#$} & $0$ & $-$ & $-$ & $-$ & $-$\\
%\bf{$G$} & $0$ & $+2.0$ & $-3.5$ & $+2.0$ & $+2.0$\\
%\bf{$A^b$} & $0$ & $-$ & $-$ & $-$ & $+13.7$\\
%\bf{$A$} & $0$ & $+5.9$ & $-10.2$ & $-15.6$ & $-$\\
%\bf{$B^b$} & $0$ & $-$ & $-$ & $-$ & $+17.6$\\
%\bf{$B$} & $0$ & $+9.8$ & $-17.1$ & $-11.7$ & $-$\\
%\end{tabular}
%\end{table}\end{tiny}
\end{itemize}
\bigskip
\begin{equation*}
\frac{f_1}{f_2} = 2^{\nicefrac{N}{12}}
\end{equation*}
\begin{itemize}
\item other scales can sound purer for specific keys but are less commonly used
\end{itemize}
\end{frame}
\section{intonation}
\begin{frame}{musical pitch}{ intonation \& vibrato}
\begin{itemize}
\item \textbf{expressive intonation}: deviation of pitch frequency from temperament depending on musical context
\begin{itemize}
\item leading tones
\item ``pure'' intervals
\end{itemize}
\item<2-> \textbf{vibrato}
\begin{itemize}
\item periodic modulation around mean pitch
\pause
\item frequency: app. 4--10 \unit{Hz}, range: app. 20--300 \unit{cents}
\end{itemize}
\bigskip
\item<3-> applies only to instruments with
\begin{itemize}
\item<4-> continuous frequency scales: vocals, string instruments, trombone, \ldots
\item<5-> other possibilities to adjust frequency: guitar, wind instruments, \ldots
\end{itemize}
\end{itemize}
\end{frame}
\section{summary}
\begin{frame}{summary}{lecture content}
\begin{itemize}
\item \textbf{pitch}
\begin{itemize}
\item each octave split into 12 pitches
\item \textit{pitch class} is an octave-independent representation of pitch
\end{itemize}
\smallskip
\item \textbf{intervals}
\begin{itemize}
\item distance between two pitches
\end{itemize}
\smallskip
\item \textbf{cent}
\begin{itemize}
\item metric for pitch distance
\end{itemize}
\end{itemize}
\inserticon{summary}
\end{frame}
\end{document}