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\begin{document}

\centerline{{\Large {\bf {\em From Music to Mathematics: Exploring the Connections}}}}

\ls

\centerline{{\large {\bf Gareth E. Roberts}}}

\vs

\centerline{\Large {\bf Sir Peter Maxwell Davies and Magic Squares}}

\ls

\centerline{\large {\bf Worksheet for Section 8.1}}

\ls


\noindent   {\bf Definition:}  An $n \times n$ square array containing the numbers 
$$
1, \,  2, \,  3, \,  4, \,  \cdots , \,  n^2 - 1, \,  n^2 ,
$$ 
each occurring precisely once, and having each row, column, and main diagonal sum to the same amount,
is called a {\em magic square}.  


\renewcommand{\arraystretch}{1.8}
\begin{table}[h!]
\begin{center}
\begin{tabular}{|c|c|c|}
\hline 
\makebox[0.3in]{8}   & \makebox[0.3in]{1}   & \makebox[0.3in]{6} \\
\hline
3  & 5 & 7  \\
\hline
4 & 9  & 2 \\
\hline
\end{tabular}
\caption{The {\em Lo Shu} magic square, known to the ancient Chinese over 3,000 years ago.  Each row, column, and main diagonal sum to 15.}
\label{Table:LoShu}
\end{center}
\end{table}
\renewcommand{\arraystretch}{1.0}

\vspace{-0.1in}

\renewcommand{\arraystretch}{1.8}
\begin{table}[h!]
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline 
\makebox[0.3in]{16}   & \makebox[0.3in]{3}   & \makebox[0.3in]{2}  & \makebox[0.3in]{13} \\
\hline
5   &  10 &  11 & 8 \\
\hline
9   &  6   &   7   & 12 \\
\hline
4   &  15 &  14  & 1 \\
\hline
\end{tabular}
\caption{The $4 \times 4$ magic square used by D\"{u}rer in his famous engraving {\em Melancholia I}.
Each row, column, and main diagonal sum to 34.
The date of the carving, 1514, can be found in the middle of the last row of the magic square.}
\end{center}
\end{table}
\renewcommand{\arraystretch}{1.0}



\noindent   {\bf Magic Squares Fun Fact:}   The rows, columns, and main diagonals of any 
$n \times n$ magic square always sum up to the same value, the {\em magic constant}.  
This value is given by the formula 
$$
M_n \;  = \;  \displaystyle{ \frac{n (n^2 + 1)}{2} }.
$$


\renewcommand{\arraystretch}{1.7}
\begin{table}[h!]
\begin{center}
\begin{tabular}{c|ccccccccc}
\makebox[0.4in]{$n$}   &
\makebox[0.2in]{2}  &  \makebox[0.2in]{3}     & \makebox[0.2in]{4}     &  \makebox[0.2in]{5}    & \makebox[0.2in]{6}     & 
\makebox[0.25in]{7}    & \makebox[0.25in]{8}     &  \makebox[0.25in]{9}     &  \makebox[0.25in]{10}    \\
\hline
\makebox[0.6in]{$M_n$}   &
5 &  15   &  34 &  65 &  111 &  175 &  260 &  369 & 505 
\end{tabular}
\end{center}
\vspace{-0.15in}
\caption{The magic constant $M_n$ for different values of $n$.}
\label{Table:MagicConstant}
\end{table}
\renewcommand{\arraystretch}{1.0}



%%%%%%%%%%%%
\pagebreak
%%%%%%%%%%%%


\centerline{ {\bf A Mirror of Whitening Light}, Peter Maxwell Davies (Opus \#75; 1977) }

\vspace{0.1in}

\renewcommand{\arraystretch}{1.8}
\begin{table}[h!]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline 
\makebox[0.3in]{C}  & \makebox[0.3in]{A}  & \makebox[0.3in]{\Ash}  & \makebox[0.3in]{\Fsh}  &  \makebox[0.3in]{D}  
&  \makebox[0.3in]{D}  &  \makebox[0.3in]{\Csh}  &  \makebox[0.3in]{G}  \\
\hline
\Gsh  & F  &  \Fsh  &  \Dsh &   G  &  D  &  \Csh  &  \Csh   \\
\hline
A  &  \Gfl  &  G  &  E   &   \Afl \footnotesize{(D)}  &   \Efl   &   D   &   D \footnotesize{(\Afl)} \\
\hline
G   &  \Dsh  &   E   &   \Csh  &   A \footnotesize{(D)} &   \Gsh  &  G   &   D \footnotesize{(A)}   \\
\hline
B   &  B  &   C  &  F   &   \Csh   &   E   &   \Efl   &   \Fsh  \\
\hline
F  &   \Bfl  &  B    &   C  &   E   &  G   &   \Fsh  &   \Bfl  \\
\hline
E   &   A   &  \Ash   &  B   &   \Dsh   &  \Fsh  &   F   &   A  \\
\hline
F   &   E   &    F   &  B   &   G   &  A   &  \Gsh  &   C \\
\hline
\end{tabular}
\caption{Davies' special matrix of notes derived from a plainchant melody, transposition of that melody, and the
$8 \times 8$ magic square of Mercury.  The notes in parentheses are adjustments to the 
square that were used by the composer.  The notes and durations for the piece come from following different patterns through this
special musical magic square.}
\label{Table:MagicSquareNotes}
\end{center}
\end{table}
\renewcommand{\arraystretch}{1.0}



%------------------------------------------------------------------------------------------------------------------------------------------
\begin{figure}[h!]
\begin{center}
\includegraphics[width = 6.6in]{DaviesFlute.pdf}
\end{center}
\vspace{-0.3in}
\caption{The flute part starting at rehearsal letter Z in {\em A Mirror of Whitening Light}.  \copyright 1978 by Boosey \& Hawkes Music Publishers Ltd.}
\label{Fig:FluteDavies}
\end{figure}
%------------------------------------------------------------------------------------------------------------------------------------------




\pagebreak



\renewcommand{\arraystretch}{1.8}
\begin{table}[h!]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline 
\makebox[0.3in]{8}   & \makebox[0.3in]{2}   & \makebox[0.3in]{3}  & \makebox[0.3in]{5} & \makebox[0.3in]{4} & \makebox[0.3in]{6} & \makebox[0.3in]{7} & \makebox[0.3in]{1}\\
\hline
1 & 7  &  6 & 4 & 5  & 3 & 2 & 8   \\
\hline
1 &  7  &  6  &  4  & 5 (8) & 3 & 2 & 8 (5) \\
\hline
8   &  2 & 3  & 5  & 4 (1)  & 6 &  7 & 1 (4)  \\
\hline
8 & 2   &  3 & 5 & 4  & 6 & 7 & 1  \\
\hline
1  & 7 &  6  &  4  &  5 &  3 & 2 &  8  \\
\hline
1  & 7 &  6  &  4 & 5 &  3 & 2 & 8  \\
\hline
8  & 2 &  3  &  5 & 4 &  6 & 7 & 1  \\
\hline
\end{tabular}
\caption{The $8 \times 8$ magic square of Mercury reduced modulo 8, except that any number equivalent to 0,
that is, any multiple of 8, has been written as an 8.}
\label{Table:MagicMercuryReduced}
\end{center}
\end{table}
\renewcommand{\arraystretch}{1.0}


\noindent   {\bf Exercises:}
\vspace{-0.1in}
\begin{enumerate}


\item  By writing the specific letter names of the notes and their durations (count by eighth notes and include rests),
confirm that the flute part shown in Figure~\ref{Fig:FluteDavies} corresponds to a counterclockwise
spiral through Davies' magic square for the notes (Table~\ref{Table:MagicSquareNotes}) and 
their durations (Table~\ref{Table:MagicMercuryReduced}).  Some pairs of consecutive notes should be treated
as the same note, with their durations combined.


\ls

\item  Figure~\ref{Fig:DaviesViolin} shows the first violin part for {\em A Mirror of Whitening Light}, beginning
at rehearsal letter F1.    \index{Davies, Peter Maxwell!Mirror@{\em A Mirror of Whitening Light}} 
\index{Mirror@{\em A Mirror of Whitening Light}}

\begin{itemize}
\item[{\bf a.}]   List the letter names of the notes consecutively and find the pattern in Table~\ref{Table:MagicSquareNotes}
used by Davies to construct this sequence of notes.

\item[{\bf b.}]   Consider the durations of each note, including rests.  Find the pattern in the reduced magic square
(Table~\ref{Table:MagicMercuryReduced}) corresponding to the lengths of each note.  
For the entire excerpt, count using the quarter note as one beat rather than an eighth note.
\end{itemize}

%------------------------------------------------------------------------------------------------------------------------------------------
\begin{figure}[h!]
\begin{center}
\includegraphics[width = 6.8in]{DaviesViolin.pdf}
\end{center}
\vspace{-0.3in}
\caption{The first violin part in Davies' {\em A Mirror of Whitening Light}, starting at rehearsal letter~F1.  \copyright 1978 by Boosey \& Hawkes Music Publishers Ltd.}
\label{Fig:DaviesViolin}
\end{figure}
%------------------------------------------------------------------------------------------------------------------------------------------



\pagebreak

\item  Complete the following $4 \times 4$ magic square.  There is one, unique solution.

\renewcommand{\arraystretch}{2.0}
\begin{table}[h!]
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline 
\makebox[0.3in]{}   & \makebox[0.3in]{12}   & \makebox[0.3in]{3}  & \makebox[0.3in]{} \\
\hline
2  &  &  & 9 \\
\hline
16 &  &  & 7 \\
\hline
     & 6 & 13 &  \\
\hline
\end{tabular}
\end{center}
\end{table}
\renewcommand{\arraystretch}{1.0}


\vs


\item  Complete the following $6 \times 6$ magic square. 

\renewcommand{\arraystretch}{2.0}
\begin{table}[h!]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
\hline 
\makebox[0.3in]{}   & \makebox[0.3in]{1}   & \makebox[0.3in]{6}  & \makebox[0.3in]{} & \makebox[0.3in]{19} & \makebox[0.3in]{24} \\
\hline
3  & 32  & 7 & 21 &    & 25 \\
\hline
31 &      & 2 &    & 27  & 20  \\
\hline
8   & 28 & 33 &   & 10  &     \\
\hline
30 & 5   &  34 & 12 &    & 16  \\
\hline
4   & 36 &       &  13 & 18 &   \\
\hline
\end{tabular}
\end{center}
\end{table}
\renewcommand{\arraystretch}{1.0}

\vs


\item  Prove that the sum of the corners in any $4 \times 4$ magic square is also equal to 34, the
magic constant.



\end{enumerate}


\end{document}
