--- /dev/null
+\documentclass{article}
+\def\kdots{,\ldots,}
+\title{Not the Font-En-Tja font}
+\author{HWN \& JCN}
+\begin{document}
+\maketitle
+
+
+\section{Introduction}
+
+This document are some design notes of the Feta font. Feta (not an
+abbreviation of Font-En-Tja) is a font of music symbols. All MetaFont
+sources are original. The symbols are modelled after various editions
+of music, notably
+\begin{itemize}
+\item B\"arenreiter
+\item Hofmeister
+\item Breitkopf
+\item Durand \& C'ie
+\end{itemize}
+
+The best references on Music engraving are Wanske\cite{wanske} and
+Ross\cite{ross} quite some of their insights were used. Although it
+is a matter of taste, I'd say that B\"arenreiter has the finest
+typography of all.
+
+
+\section{Bezier curves for slurs}
+
+Objective: slurs in music are curved objects designating that notes
+should fluently bound. They are drawn as smooth curves, with their
+center thicker and the endings tapered.
+
+There are some variants: the simplest slur shape only has the width as
+parameter. Then we give some suggestions for tuning the shapes. The
+simple slur algorithm is used for drawing ties as well.
+
+
+
+\subsection{Simple slurs}
+
+Long slurs are flat, whereas short slurs look like small circle arcs.
+Details are given in Wanske\cite{ross} and Ross\cite{wanske}. The
+shape of a slur can be given as a Bezier curve with four control
+points:
+
+\begin{eqnarray*}
+ B(t) &=& (1-t)^3c_1 +3(1-t)^2tc_2 + 3(1-t)t^2c_3 + t^3c_4.
+\end{eqnarray*}
+
+We will assume that the slur connects two notes of the same
+pitch. Different slurs can be created by rotating the derived shape.
+We will also assume that the slur has a vertical axis of symmetry
+through its center. The left point will be the origin. So we have
+the following equations for the control points $c_1\kdots c_4$.
+
+\begin{eqnarray*}
+c_1&=& (0,0)\\
+c_2&=& (i, h)\\
+c_3&=& (b-i, h)\\
+c_4&=& (b, 0)
+\end{eqnarray*}
+
+The quantity $b$ is given, it is the width of the slur. The
+conditions on the shape of the slur for small and large $b$ transform
+to
+\begin{eqnarray*}
+ h \to h_{\infty} , &&\quad b \to \infty\\
+ h \approx r_{0} b, &&\quad b \to 0.
+\end{eqnarray*}
+To tackle this, we will assume that $h = F(b)$, for some kind of
+$F(\cdot)$. One function that satisfies the above conditions is
+$$
+F(b) = h_{\infty} \frac{2}{\pi} \arctan \left( \frac{\pi r_0}{2
+h_{\infty}} b \right).
+$$
+
+For satisfying results we choose $h_{\infty} = 2\cdot \texttt{interline}$
+and $r_0 = \frac 13$.
+
+\subsection{Height correction}
+
+Aside from being a smooth curve, slurs should avoid crossing
+enclosed notes and their stems.
+
+An easy way to achieve this is to extend the slur's height,
+so that the slur will curve just above any disturbing notes.
+
+The parameter $i$ determines the flatness of the curve. Satisfying
+results have been obtained with $i = h$.
+
+The formula can be generalised to allow for corrections in the shape,
+\begin{eqnarray*}
+c_1&=& (0,0)\\
+c_2&=& (i', h')\\
+c_3&=& (b-i', h')\\
+c_4&=& (b, 0)
+\end{eqnarray*}
+Where
+$$
+i' = h(b) (1 + i_{corr}), \quad h' = h(b) (1 + h_{corr}).
+$$
+
+The default values for these corrections are $0$. A $h_{corr}$ that is
+negative, makes the curve flatter in the center. A $h_{corr}$ that is
+positive make the curve higher.
+
+At every encompassed note's x position the difference $\delta _y$
+between the slur's height and the note is calculated. The greatest
+$\delta _y$ is used to calculate $h_{corr}$ is by lineair extrapolation.
+
+However, this simple method produces satisfactory results only for
+small and symmetric disturbances.
+
+
+\subsection{Tangent method correction}
+
+A somewhat more elaborate\footnote{While staying in the realm
+of emperic computer science} way of having a slur avoid
+disturbing notes is by first defining the slur's ideal shape
+and then using the height correction. The ideal shape of a
+slur can be guessed by calculating the tangents of the disturbing
+notes:
+% a picture wouldn't hurt...
+\begin{eqnarray*}
+ y_{disturb,l} &=& \rm{rc}_l x\\
+ y_{disturb,r} &=& \rm{rc}_r + c_{3,x},
+\end{eqnarray*}
+where
+\begin{eqnarray*}
+ \rm{rc}_l &=& \frac{y_{disturb,l} - y_{encompass,1}}
+ {x_{disturb,l} - x_{encompass,1}}\dot x\\
+ \rm{rc}_r &=& \frac{y_{encompass,n} - y_{disturb,r}}
+ {x_{encompass,n} - x_{disturb,r}} \dot x + c_{3,x}.
+\end{eqnarray*}
+
+We assume that having the control points $c_2$ and $c_3$ located
+on tangent$_1$ and tangent$_2$ resp.
+% t: tangent
+\begin{eqnarray*}
+ y_{tangent,l} &=& \alpha \rm{rc}_l x\\
+ y_{tangent,r} &=& \alpha \rm{rc}_r + c_{3,x}.
+\end{eqnarray*}
+
+Beautiful slurs have rather strong curvature at the extreme
+control points. That's why we'll have $\alpha > 1$.
+Satisfactory resulsts have been obtained with
+$$
+ \alpha \approx 2.4.
+$$
+
+The positions of control points $c_2$ and $c_3$ are obtained
+by solving with the height-line
+\begin{eqnarray*}
+ y_h &=& \rm{rc}_h + c_h.
+\end{eqnarray*}
+
+The top-line runs through the points disturb$_{left}$ and
+disturb$_{right}$. In the case that
+$$
+z_{disturb,l} = z_{disturb,r},
+$$
+we'll have
+$$
+ \angle(y_{tangent,l},y_h) = \angle(y_{tangent,r},y_h).
+$$
+
+
+
+\section{Sizes}
+
+Traditional engraving uses a set of 9 standardised sizes for Staffs
+(running from 0 to 8).
+
+We have tried to measure these (helped by a magnifying glass), and
+found the staffsizes in table~\ref{fonts:staff-size}. One should note that
+these are estimates, so I think there could be a measuring error of ~
+.5 pt. Moreover [Ross] states that not all engravers use exactly
+those sizes.
+
+\begin{table}[h]
+ \begin{center}
+ \begin{tabular}{lll}
+Staffsize &Numbers &Name\\
+\hline\\
+26.2pt &No. 0\\
+22.6pt &No. 1 &Giant/English\\
+21.3pt &No. 2 &Giant/English\\
+19.9pt &No. 3 &Regular, Ordinary, Common\\
+19.1pt &No. 4 &Peter\\
+17.1pt &No. 5 &Large middle\\
+15.9pt &No. 6 &Small middle\\
+13.7pt &No. 7 &Cadenza\\
+11.1pt &No. 8 &Pearl\\
+
+ \end{tabular}
+ \caption{Foo}
+ \label{fonts:staff-size}
+ \end{center}
+\end{table}
+
+Ross states that the dies (the stamps to make the symbols) come in
+12 different sizes.
+
+\bibliographystyle{plain}
+\bibliography{engraving}
+
+
+
+\end{document}
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