\section{Introduction}
-Feta (not an abbreviation of Font-En-Tja) is a font of music symbols.
-All MetaFont %ugh
-sources are original. The symbols are modelled after
-various editions of music, notably
+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
+%ugh sources are original. The symbols are modelled after various
+editions of music, notably
\begin{itemize}
\item B\"arenreiter
\item Hofmeister
typography of all.
-\section{Bezier curves for simple slurs}
+\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. We will assume that the slur connects two notes of the same
+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
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 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.
+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.
-\section{Sizes}
+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).
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