deletions | additions       diff --git a/music2.tex b/music2.tex  index 4cc5170..34556ae 100644  --- a/music2.tex  +++ b/music2.tex 
...
Western music is based on a division of 12 distinct frequencies per \textit{octave}. An octave is an \textit{interval}, or distance between two frequencies, that corresponds to a power of 2 multiplication. Musical pitch is perceived in a logarithmic scale---one octave above a given perceived frequency is double that frequency; one octave below is half that frequency. The progression of notes containing all 12 pitches in succession in an octave is called a \textit{chromatic scale}. A \textit{semitone}, or \textit{half-step}, is the smallest interval, equal to $1/12$ of an octave. $n$ semitones above a given frequency $f_0$ or $-n$ below can be calculated as $f_0 \cdot 2^{n/12}$.  \textit{Note names} are used to classify the pitches in the chromatic scale. Note names consist of a base name and 0 or more \textit{accidentals}. The base names of a note correspond to the white keys on a piano---in any one given octave there are the following names: $C$, $D$, $E$, $F$, $G$, $A$, and $B$. A base note name can optionally be decorated with an indefinite number of sharps ($\#$) or flats ($b$), but not both, in the note name. These are referred to as accidentals. Each additional $\#$ increases the pitch to which the note name refers by 1 semitone; likewise, each $b$ decreases the pitch by 1 semitone. The black keys on the piano represent pitches 1 semitone in between This can be illustrated with  the surrounding white keys. Each white key is either 1 semitone or 2 semitones apart, depending on if a black key is context-free grammar  inthe middle. For instance, $C$ and $D$ are 2 semitones apart, whereas $E$ and $F$ are 1 semitone apart. See  figure ~\ref{fig:piano}. ~\ref{fig:cfgnote}.  \begin{figure}[h!]  \begin{center}  \begin{align}  Note NoteName  &\to BaseNote \mid BaseNote\``\#''\  SharpAccidentals \mid BaseNote\``b''\  FlatAccidentals \\ BaseNote &\to C \mid D \mid E \mid F \mid G \mid A \mid B \\  SharpAccidentals &\to "\#"\ (\#)\  SharpAccidentals \mid \epsilon (\#)  \\ FlatAccidentals &\to "b"\ (b)\  FlatAccidentals \mid \epsilon (b)  \end{align} \caption{Context-free grammar of a note name}  \label{fig:cfgnote}  \end{center}  \end{figure}  These are referred to as accidentals. Each additional $\#$ increases the pitch to which the note name refers by 1 semitone; likewise, each $b$ decreases the pitch by 1 semitone. The black keys on the piano represent pitches 1 semitone in between the surrounding white keys. Each white key is either 1 semitone or 2 semitones apart, depending on if a black key is in the middle. For instance, $C$ and $D$ are 2 semitones apart, whereas $E$ and $F$ are 1 semitone apart. See figure ~\ref{fig:piano}.