deletions | additions       diff --git a/chorddistfuncs.tex b/chorddistfuncs.tex  index 3c44237..9ecbd68 100644  --- a/chorddistfuncs.tex  +++ b/chorddistfuncs.tex 
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This metric has the advantage that it is very versatile and fast, but potentially loses some information about a chord in disregarding distinctions of elements within the chord, instead focusing only on the pitch classes within.  \subsection{Tonal Pitch Step} Tonal Pitch Step is a distance measure that is grounded in cognitive psychology, algebra, and tonal music theory, proposed by Fred Lerdahl in \cite{lerdahl1988tonal}. The measure is unique from the others in that it takes into account the key of a song, which can be recalled as:  \begin{align*}  Key &\to Root\ Mode \\  Root &\to NoteName \\  Mode &\to \text{Major Minor} \\  \end{align*}  Given all possible combinations of a pitch class describing $root$ and $mode$, there are 24 distinct key signatures. It is important to note that the acceptable pitch classes within \textit{relative} keys are identical, such as $C$ major and $A$ minor.  The tonal pitch step algorithm is calculated, with revisions courtesy of \cite{de2008tonal} to only include one key, as  \[ C_d(c_1, c_2) = i(root(c_1), root(c_2)) + j(c_1, c_2) \]  where $i$ calculates the \textit{circle-of-fifths} distance between two pitch classes. The circle-of-fifths (see figure~\ref{circlefifths}) distance describes how many $\pm7$ semitone traversals are needed to reach one note from another.  \subsubsection{Key Finding Using Tonal Pitch Step}