release: 1.0.17

[lilypond.git] / Documentation / tex / fonts.doc

% -*-LaTeX-*-

\documentclass{article}
\def\kdots{,\ldots,}
\title{Not the Font-En-Tja font}
\author{HWN \& JCN} 
\def\preMudelaExample{}
\def\postMudelaExample{}
\begin{document}
\maketitle


\section{Introduction}

This document are some design notes of the Feta font, and other
symbols related to LilyPond.  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.

%
% stupid test to check convert-mudela.py.
%
% (so sue me)
\begin{mudela}[fragment,center]
  \clef bass; c1 \clef alto; c'2  \clef treble; c''4
\end{mudela}


\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.

\section{Beams}

\subsection{Slope}

Traditionally, beam slopes are computed by following a large and hairy
set of rules.  Some of these are talked-about in Wanske, a more
recipy-like description can be found in Ross.

There are some problems when trying to follow these rules:
\begin{itemize}

\item the set is not complete

\item they are not formulated as a general rule with exceptions, but
rather as a huge case of individual rules\cite{ross}

\item in some cases, the result is wrong or ugly (or both)

\item they try to solve a couple of problems at a time (e.g. Ross
handles ideal slope and slope-quantisation as a paired problem)
\end{itemize}
Reading Ross it is clear that the rules presented there are certainly
not the ultimate idea of what beam(slope)s should look like, but
rather a (very much) simplified hands-on recipy for a human engraver.

There are good reasons not to follow those rules:

\begin{itemize}
\item One cannot expect a human engraver to solve least-squares
problems for every beam
  
\item A human engravers will allways trust themselves in judging the
outcome of the applied recipy.  If, in a complicated case, the result
"doesn't look good", they will ignore the rules and draw their own
beams, based on experience.

\item The exact rules probably don't "really exist" but in the minds
  of good engravers, in the form of experience
\end{itemize}

We'll propose to do a least-squares solve.  This seems to be the best
way to calculate the slope for a computerised engraver such as Lily.

It would be nice to have some rules to catch and handle "ugly" cases,
though.  In general, the slope of the beam should mirror the pitches
of the notes.  If this can't be done because there simply is no
uniform trend, it would probably be best to set the slope to zero.


\subsection{Quantising}

The beams should be prevented to conflict with the stafflines,
especially at small slopes.  Traditionally, poor printing techniques
imposed rather strict rules for quantisation.  In modern (post 1955)
music printing we see that quality has improved substantially and
obsoleted the technical justification for following some of these
strict rules, notably the avoiding of so-called wedges.


\subsection{Thickness and spacing}

The spacing of double and triple beams (sixteenth and thirtysecond beams)
is the same, quadruple and quintuple (thirtyfourth and hundredtwentyeighth
beams) is wider.
All beams are equally thick.  Using the layout of triple beams and the 
beam-thickness $bt$ we can calculate the inter-beam spacing $ib$.

Three beams span two interlines, including stafflines:
\begin{eqnarray*}
 2 ib + bt &=& 2 il\\
 ib &=& (2 il - bt) / 2
\end{eqnarray*}

We choose
\begin{eqnarray*}
  bt &=& 0.48(il - st)
\end{eqnarray*}

\subsubsection{Quadruple beams}

If we have more than three beams they must open-up
in order to not collide with staff lines.  The only valid
position that remains is for the upper beam to hang.

\begin{eqnarray*}
 3 ib_{4+} + bt &=& 3 il\\
 ib_{4+} &=& (3 il - bt) / 3
\end{eqnarray*}


\section{Layout of the source files}

The main font (with the fixed size music glyphs) uses a the \TeX\
logfile as a communication device.  Use the specialised macros to
create and export glyphs.

\bibliographystyle{plain}
\bibliography{engraving}



\end{document}