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\chapter{Code Implementation}  \section{Smith-Waterman} This chapter discusses in brief the manner in which I implemented my code. The full project is over 6,000 lines of C, Python, HTML, and Javascript code. This chapter will only overview notable features of the code.  \subsection{Using \section{Using  the chord alphabet} \subsection{Integer representation of chords}  The Smith-Waterman algorithm is typically used in bioinformatic applications in which the alphabet is 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*} 
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\subsection{Bitwise representation of harmony}  Chord quality can be represented as a 12-bit integer in which each bit corresponds to whether a certain interval is included. This compact representation form  provides a means for quick computation and allows chord distance computations set operations  to be represented expressed with bitwise operators.  , Let $i$ be  a 12-bit integer can represent harmony representing chord quality  in which each bit corresponds to whether a certain interval is included in the chord or not. Since leading 0's are excluded, the representation can start with 1 in the first binary position (right-to-left) to represent the root of the chord. Thus, a concise 12-bit integer representation of harmony consists of a 1 in the first chord and each subsequent  binary position with the following $i$th From right-to-left, the $i$th bit represents $i, 1 \leq i \leq 12$ can represent  whether the interval $i - 1$ $i$ is included  (where 0 is the root note)is included in the chord or not  (see figure~\ref{fig:qualitytable}). All harmonies used in the program and associated intervals can be seen in figure~\ref{fig:qualitybitstable}. Let $h_b$ be the function that extracts the binary mask from a given harmony.  \begin{figure}[h!]  \centering 
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\label{fig:qualitybitstable}  \end{figure}  \item \subsection{Bitwise chord operations}  Basic chord operations can then be constructed. For instance, to get $P_c(c)$ as a 12-bit integer in which each bit $i$ from right-to-left corresponds to whether $p(i) \in P_c(c)$ one need only construct a bit cycling algorithm  \begin{verbatim}  function CycleBits(value, shift)  return ((value << shift) | (value >> (12 - shift))) & b111111111111  \end{verbatim}  where \texttt{b111111111111} refers to the binary bitmask of all 1's for 12 places, \texttt{<<} and \text{>>} are the bit shift left and right operators, and \texttt{|} and \texttt{&} are the bit operators $and$ and $or$, respectively. $P_c(c)$ can then be calculated: \[ P_c(c) = CycleBits(h_b(quality(c)), p(root(c))) | p(bass(c)) \]  Cleverness with representing chords and harmonies as integers and allowing allows effective methods to be constructed with  simple bitwise operations and bitmasks bitmasks.  \subsection{MIPS \section{Smith-Waterman MIPS  Implementation} \item Used BU implementation, extreme speed results, but only for DNA and Protein sequences, the usual use case of Smith-Waterman