Documentation/fonts.tex

   1 \documentclass{article}
   2 \def\kdots{,\ldots,}
   3 \title{Not the Font-En-Tja font}
   4 \author{HWN \& JCN}
   5 \begin{document}
   6 \maketitle
   7
   8
   9 \section{Introduction}
  10
  11 This document are some design notes of the Feta font.  Feta (not an
  12 abbreviation of Font-En-Tja) is a font of music symbols.  All MetaFont
  13 sources are original.  The symbols are modelled after various editions
  14 of music, notably
  15 \begin{itemize}
  16 \item B\"arenreiter
  17 \item Hofmeister
  18 \item Breitkopf
  19 \item Durand \& C'ie
  20 \end{itemize}
  21
  22 The best references on Music engraving are Wanske\cite{wanske} and
  23 Ross\cite{ross} quite some of their insights were used.  Although it
  24 is a matter of taste, I'd say that B\"arenreiter has the finest
  25 typography of all.
  26
  27
  28 \section{Bezier curves for slurs}
  29
  30 Objective:  slurs in music are curved objects designating that notes
  31 should fluently bound.  They are drawn as smooth curves, with their
  32 center thicker and the endings tapered.
  33
  34 There are some variants: the simplest slur shape only has the width as
  35 parameter.  Then we give some suggestions for tuning the shapes.  The
  36 simple slur algorithm is used for drawing ties as well.
  37
  38
  39
  40 \subsection{Simple slurs}
  41
  42 Long slurs are flat, whereas short slurs look like small circle arcs.
  43 Details are given in Wanske\cite{ross} and Ross\cite{wanske}.  The
  44 shape of a slur can be given as a Bezier curve with four control
  45 points:
  46
  47 \begin{eqnarray*}
  48   B(t) &=& (1-t)^3c_1 +3(1-t)^2tc_2 + 3(1-t)t^2c_3 + t^3c_4.
  49 \end{eqnarray*}
  50
  51 We will assume that the slur connects two notes of the same
  52 pitch.  Different slurs can be created by rotating the derived shape.
  53 We will also assume that the slur has a vertical axis of symmetry
  54 through its center.  The left point will be the origin.     So we have
  55 the following equations for the control points $c_1\kdots c_4$.
  56
  57 \begin{eqnarray*}
  58 c_1&=& (0,0)\\
  59 c_2&=& (i, h)\\
  60 c_3&=& (b-i, h)\\
  61 c_4&=& (b, 0)
  62 \end{eqnarray*}
  63
  64 The quantity $b$ is given, it is the width of the slur.  The
  65 conditions on the shape of the slur for small and large $b$ transform
  66 to
  67 \begin{eqnarray*}
  68  h \to h_{\infty} , &&\quad b \to \infty\\
  69  h \approx r_{0} b, &&\quad b \to 0.
  70 \end{eqnarray*}
  71 To tackle  this, we  will  assume that $h   = F(b)$, for  some kind of
  72 $F(\cdot)$.  One function that satisfies the above conditions is
  73 $$
  74 F(b) = h_{\infty} \frac{2}{\pi} \arctan \left( \frac{\pi r_0}{2
  75 h_{\infty}} b \right).
  76 $$
  77
  78 For satisfying results we choose $h_{\infty} = 2\cdot \texttt{interline}$
  79 and $r_0 = \frac 13$.
  80
  81 \subsection{Height correction}
  82
  83 Aside from being a smooth curve, slurs should avoid crossing
  84 enclosed notes and their stems.
  85
  86 An easy way to achieve this is to extend the slur's height,
  87 so that the slur will curve just above any disturbing notes.
  88
  89 The parameter $i$ determines the flatness of the curve.  Satisfying
  90 results have been obtained with $i = h$.
  91
  92 The formula can be generalised to allow for corrections in the shape,
  93 \begin{eqnarray*}
  94 c_1&=& (0,0)\\
  95 c_2&=& (i', h')\\
  96 c_3&=& (b-i', h')\\
  97 c_4&=& (b, 0)
  98 \end{eqnarray*}
  99 Where
 100 $$
 101 i' = h(b) (1 + i_{corr}), \quad h' = h(b) (1 + h_{corr}).
 102 $$
 103
 104 The default values for these corrections are $0$.  A $h_{corr}$ that is
 105 negative, makes the curve flatter in the center.  A $h_{corr}$ that is
 106 positive make the curve higher.
 107
 108 At every encompassed note's x position the difference $\delta _y$
 109 between the slur's height and the note is calculated.  The greatest
 110 $\delta _y$ is used to calculate $h_{corr}$ is by lineair extrapolation.
 111
 112 However, this simple method produces satisfactory results only for
 113 small and symmetric disturbances.
 114
 115
 116 \subsection{Tangent method correction}
 117
 118 A somewhat more elaborate\footnote{While staying in the realm
 119 of emperic computer science} way of having a slur avoid
 120 disturbing notes is by first defining the slur's ideal shape
 121 and then using the height correction.  The ideal shape of a
 122 slur can be guessed by calculating the tangents of the disturbing
 123 notes:
 124 % a picture wouldn't hurt...
 125 \begin{eqnarray*}
 126   y_{disturb,l} &=& \rm{rc}_l x\\
 127   y_{disturb,r} &=& \rm{rc}_r + c_{3,x},
 128 \end{eqnarray*}
 129 where
 130 \begin{eqnarray*}
 131   \rm{rc}_l &=& \frac{y_{disturb,l} - y_{encompass,1}}
 132     {x_{disturb,l} - x_{encompass,1}}\dot x\\
 133   \rm{rc}_r &=& \frac{y_{encompass,n} - y_{disturb,r}}
 134     {x_{encompass,n} - x_{disturb,r}} \dot x + c_{3,x}.
 135 \end{eqnarray*}
 136
 137 We assume that having the control points $c_2$ and $c_3$ located
 138 on tangent$_1$ and tangent$_2$ resp.
 139 % t: tangent
 140 \begin{eqnarray*}
 141   y_{tangent,l} &=& \alpha \rm{rc}_l x\\
 142   y_{tangent,r} &=& \alpha \rm{rc}_r + c_{3,x}.
 143 \end{eqnarray*}
 144
 145 Beautiful slurs have rather strong curvature at the extreme
 146 control points.  That's why we'll have $\alpha > 1$.
 147 Satisfactory resulsts have been obtained with
 148 $$
 149   \alpha \approx 2.4.
 150 $$
 151
 152 The positions of control points $c_2$ and $c_3$ are obtained
 153 by solving with the height-line
 154 \begin{eqnarray*}
 155   y_h &=& \rm{rc}_h + c_h.
 156 \end{eqnarray*}
 157
 158 The top-line runs through the points disturb$_{left}$ and
 159 disturb$_{right}$.  In the case that
 160 $$
 161 z_{disturb,l} = z_{disturb,r},
 162 $$
 163 we'll have
 164 $$
 165   \angle(y_{tangent,l},y_h) = \angle(y_{tangent,r},y_h).
 166 $$
 167
 168
 169
 170 \section{Sizes}
 171
 172 Traditional engraving uses a set of 9 standardised sizes for Staffs
 173 (running from 0 to 8).
 174
 175 We have tried to measure these (helped by a magnifying glass), and
 176 found the staffsizes in  table~\ref{fonts:staff-size}.  One should note that
 177 these are estimates, so I think there could be a measuring error of ~
 178 .5 pt.  Moreover [Ross] states that not all engravers use exactly
 179 those sizes.
 180
 181 \begin{table}[h]
 182   \begin{center}
 183     \begin{tabular}{lll}
 184 Staffsize       &Numbers                &Name\\
 185 \hline\\
 186 26.2pt  &No. 0\\
 187 22.6pt  &No. 1          &Giant/English\\
 188 21.3pt  &No. 2          &Giant/English\\
 189 19.9pt  &No. 3          &Regular, Ordinary, Common\\
 190 19.1pt  &No. 4          &Peter\\
 191 17.1pt  &No. 5          &Large middle\\
 192 15.9pt  &No. 6          &Small middle\\
 193 13.7pt  &No. 7          &Cadenza\\
 194 11.1pt  &No. 8          &Pearl\\
 195
 196     \end{tabular}
 197     \caption{Foo}
 198     \label{fonts:staff-size}
 199   \end{center}
 200 \end{table}
 201
 202 Ross states that the dies (the stamps to make the symbols) come in
 203 12 different sizes.
 204
 205 \bibliographystyle{plain}
 206 \bibliography{engraving}
 207
 208
 209
 210 \end{document}