3 \documentclass{article}
5 \title{Not the Font-En-Tja font}
7 \def\preMudelaExample{}
8 \def\postMudelaExample{}
13 \section{Introduction}
15 This document are some design notes of the Feta font, and other
16 symbols related to LilyPond. Feta (not an abbreviation of
17 Font-En-Tja) is a font of music symbols. All MetaFont sources are
18 original. The symbols are modelled after various editions of music,
19 notably \begin{itemize} \item B\"arenreiter \item Hofmeister \item
20 Breitkopf \item Durand \& C'ie \end{itemize}
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
28 % stupid test to check convert-mudela.py.
31 % well, mudela-book:fragment is a bit broken
32 \begin{mudela}[fragment,center]
33 \notes {\clef bass; c1 \clef alto; c'2 \clef treble; c''4}
37 \section{Bezier curves for slurs}
39 Objective: slurs in music are curved objects designating that notes
40 should fluently bound. They are drawn as smooth curves, with their
41 center thicker and the endings tapered.
43 There are some variants: the simplest slur shape only has the width as
44 parameter. Then we give some suggestions for tuning the shapes. The
45 simple slur algorithm is used for drawing ties as well.
49 \subsection{Simple slurs}
51 Long slurs are flat, whereas short slurs look like small circle arcs.
52 Details are given in Wanske\cite{ross} and Ross\cite{wanske}. The
53 shape of a slur can be given as a Bezier curve with four control
57 B(t) &=& (1-t)^3c_1 +3(1-t)^2tc_2 + 3(1-t)t^2c_3 + t^3c_4.
60 We will assume that the slur connects two notes of the same
61 pitch. Different slurs can be created by rotating the derived shape.
62 We will also assume that the slur has a vertical axis of symmetry
63 through its center. The left point will be the origin. So we have
64 the following equations for the control points $c_1\kdots c_4$.
73 The quantity $b$ is given, it is the width of the slur. The
74 conditions on the shape of the slur for small and large $b$ transform
77 h \to h_{\infty} , &&\quad b \to \infty\\
78 h \approx r_{0} b, &&\quad b \to 0.
80 To tackle this, we will assume that $h = F(b)$, for some kind of
81 $F(\cdot)$. One function that satisfies the above conditions is
83 F(b) = h_{\infty} \frac{2}{\pi} \arctan \left( \frac{\pi r_0}{2
84 h_{\infty}} b \right).
87 For satisfying results we choose $h_{\infty} = 2\cdot \texttt{interline}$
90 \subsection{Height correction}
92 Aside from being a smooth curve, slurs should avoid crossing
93 enclosed notes and their stems.
95 An easy way to achieve this is to extend the slur's height,
96 so that the slur will curve just above any disturbing notes.
98 The parameter $i$ determines the flatness of the curve. Satisfying
99 results have been obtained with $i = h$.
101 The formula can be generalised to allow for corrections in the shape,
110 i' = h(b) (1 + i_{corr}), \quad h' = h(b) (1 + h_{corr}).
113 The default values for these corrections are $0$. A $h_{corr}$ that is
114 negative, makes the curve flatter in the center. A $h_{corr}$ that is
115 positive make the curve higher.
117 At every encompassed note's x position the difference $\delta _y$
118 between the slur's height and the note is calculated. The greatest
119 $\delta _y$ is used to calculate $h_{corr}$ is by lineair extrapolation.
121 However, this simple method produces satisfactory results only for
122 small and symmetric disturbances.
125 \subsection{Tangent method correction}
127 A somewhat more elaborate\footnote{While staying in the realm
128 of emperic computer science} way of having a slur avoid
129 disturbing notes is by first defining the slur's ideal shape
130 and then using the height correction. The ideal shape of a
131 slur can be guessed by calculating the tangents of the disturbing
133 % a picture wouldn't hurt...
135 y_{disturb,l} &=& \rm{rc}_l x\\
136 y_{disturb,r} &=& \rm{rc}_r + c_{3,x},
140 \rm{rc}_l &=& \frac{y_{disturb,l} - y_{encompass,1}}
141 {x_{disturb,l} - x_{encompass,1}}\dot x\\
142 \rm{rc}_r &=& \frac{y_{encompass,n} - y_{disturb,r}}
143 {x_{encompass,n} - x_{disturb,r}} \dot x + c_{3,x}.
146 We assume that having the control points $c_2$ and $c_3$ located
147 on tangent$_1$ and tangent$_2$ resp.
150 y_{tangent,l} &=& \alpha \rm{rc}_l x\\
151 y_{tangent,r} &=& \alpha \rm{rc}_r + c_{3,x}.
154 Beautiful slurs have rather strong curvature at the extreme
155 control points. That's why we'll have $\alpha > 1$.
156 Satisfactory resulsts have been obtained with
161 The positions of control points $c_2$ and $c_3$ are obtained
162 by solving with the height-line
164 y_h &=& \rm{rc}_h + c_h.
167 The top-line runs through the points disturb$_{left}$ and
168 disturb$_{right}$. In the case that
170 z_{disturb,l} = z_{disturb,r},
174 \angle(y_{tangent,l},y_h) = \angle(y_{tangent,r},y_h).
181 Traditional engraving uses a set of 9 standardised sizes for Staffs
182 (running from 0 to 8).
184 We have tried to measure these (helped by a magnifying glass), and
185 found the staffsizes in table~\ref{fonts:staff-size}. One should note that
186 these are estimates, so I think there could be a measuring error of ~
187 .5 pt. Moreover [Ross] states that not all engravers use exactly
193 Staffsize &Numbers &Name\\
196 22.6pt &No. 1 &Giant/English\\
197 21.3pt &No. 2 &Giant/English\\
198 19.9pt &No. 3 &Regular, Ordinary, Common\\
199 19.1pt &No. 4 &Peter\\
200 17.1pt &No. 5 &Large middle\\
201 15.9pt &No. 6 &Small middle\\
202 13.7pt &No. 7 &Cadenza\\
203 11.1pt &No. 8 &Pearl\\
207 \label{fonts:staff-size}
211 Ross states that the dies (the stamps to make the symbols) come in
218 Traditionally, beam slopes are computed by following a large and hairy
219 set of rules. Some of these are talked-about in Wanske, a more
220 recipy-like description can be found in Ross.
222 There are some problems when trying to follow these rules:
225 \item the set is not complete
227 \item they are not formulated as a general rule with exceptions, but
228 rather as a huge case of individual rules\cite{ross}
230 \item in some cases, the result is wrong or ugly (or both)
232 \item they try to solve a couple of problems at a time (e.g. Ross
233 handles ideal slope and slope-quantisation as a paired problem)
235 Reading Ross it is clear that the rules presented there are certainly
236 not the ultimate idea of what beam(slope)s should look like, but
237 rather a (very much) simplified hands-on recipy for a human engraver.
239 There are good reasons not to follow those rules:
242 \item One cannot expect a human engraver to solve least-squares
243 problems for every beam
245 \item A human engravers will allways trust themselves in judging the
246 outcome of the applied recipy. If, in a complicated case, the result
247 "doesn't look good", they will ignore the rules and draw their own
248 beams, based on experience.
250 \item The exact rules probably don't "really exist" but in the minds
251 of good engravers, in the form of experience
254 We'll propose to do a least-squares solve. This seems to be the best
255 way to calculate the slope for a computerised engraver such as Lily.
257 It would be nice to have some rules to catch and handle "ugly" cases,
258 though. In general, the slope of the beam should mirror the pitches
259 of the notes. If this can't be done because there simply is no
260 uniform trend, it would probably be best to set the slope to zero.
263 \subsection{Quantising}
265 The beams should be prevented to conflict with the stafflines,
266 especially at small slopes. Traditionally, poor printing techniques
267 imposed rather strict rules for quantisation. In modern (post 1955)
268 music printing we see that quality has improved substantially and
269 obsoleted the technical justification for following some of these
270 strict rules, notably the avoiding of so-called wedges.
273 \subsection{Thickness and spacing}
275 The spacing of double and triple beams (sixteenth and thirtysecond beams)
276 is the same, quadruple and quintuple (thirtyfourth and hundredtwentyeighth
278 All beams are equally thick. Using the layout of triple beams and the
279 beam-thickness $bt$ we can calculate the inter-beam spacing $ib$.
281 Three beams span two interlines, including stafflines:
284 ib &=& (2 il - bt) / 2
292 \subsubsection{Quadruple beams}
294 If we have more than three beams they must open-up
295 in order to not collide with staff lines. The only valid
296 position that remains is for the upper beam to hang.
299 3 ib_{4+} + bt &=& 3 il\\
300 ib_{4+} &=& (3 il - bt) / 3
304 \section{Layout of the source files}
306 The main font (with the fixed size music glyphs) uses a the \TeX\
307 logfile as a communication device. Use the specialised macros to
308 create and export glyphs.
310 \bibliographystyle{plain}
311 \bibliography{engraving}