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\subsection{Using the chord alphabet}  The Smith-Waterman algorithm is typically used in bioinformatic applications in which the alphabet istypically  restricted to DNA or protein characters. To use an alphabet that contains all the chord symbols, a bijective function can be established between every type of chord and a unique 16 bit integer. Recall a chord can be described with the following grammar: \begin{align*}  Chord &\to Root\ Harmony \ Bass \mid \textbf{NoChord} \\ 
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Notice that $|Root| = |Bass| = |PitchClass| = 12$ and $|Harmony| = 12$. A bijective function $p$ between $PitchClass$ and an integer from 0 through 11 can be established, along with a bijective function $h$ between $Harmony$.  \begin{tabular}{llll} \[\begin{tabular}{llcll}  \toprule  $PitchClass$ & $p(PitchClass)$ & \hspace*{2cm} &  $Harmony$ & $h(Harmony)$ \\ \cmidrule(r){1-2} \cmidrule(r){3-4} \\  \textbf{A} & 0 & &  \textbf{maj} & 0 \\ \textbf{A#/Bb} & 1 & &  \textbf{6} & 1 \\ \textbf{B} & 2 & &  \textbf{maj7} & 2 \\ \textbf{C} & 3 & &  \textbf{m} & 3 \\ \textbf{C#/Db} & 4 & &  \textbf{m6} & 4 \\ \textbf{D} & 5 & &  \textbf{m7} & 5 \\ \textbf{D#/Eb} & 6 & &  \textbf{7} & 6 \\ \textbf{E} & 7 & &  \textbf{aug} & 7 \\ \textbf{F} & 8 & &  \textbf{dim} & 8 \\ \textbf{F#/Gb} & 9 & &  \textbf{dim7} & 9 \\ \textbf{G} & 10 & &  \textbf{m7b5} & 10 \\ \textbf{G#/Ab} & 11 & &  \textbf{UnknownHarmony} & 11  \\ \end{tabular} \end{tabular}\]  In base 12, a chord can be represented as integer.