4 \documentclass{article}
6 \title{Not the Font-En-Tja font}
8 \def\preMudelaExample{}
9 \def\postMudelaExample{}
14 \section{Introduction}
16 This document are some design notes of the Feta font, and other
17 symbols related to LilyPond. Feta (not an abbreviation of
18 Font-En-Tja) is a font of music symbols. All MetaFont sources are
19 original. The symbols are modelled after various editions of music,
20 notably \begin{itemize} \item B\"arenreiter \item Hofmeister \item
21 Breitkopf \item Durand \& C'ie \end{itemize}
23 The best references on Music engraving are Wanske\cite{wanske} and
24 Ross\cite{ross} some of their insights were used. Although it is a
25 matter of taste, I'd say that B\"arenreiter has the finest typography
29 \section{Bezier curves for slurs}
31 Objective: slurs in music are curved objects designating that notes
32 should fluently bound. They are drawn as smooth curves, with their
33 center thicker and the endings tapered.
35 There are some variants: the simplest slur shape only has the width as
36 parameter. Then we give some suggestions for tuning the shapes. The
37 simple slur algorithm is used for drawing ties as well.
41 \subsection{Simple slurs}
43 Long slurs are flat, whereas short slurs look like small circle arcs.
44 Details are given in Wanske\cite{ross} and Ross\cite{wanske}. The
45 shape of a slur can be given as a Bezier curve with four control
49 B(t) &=& (1-t)^3c_1 +3(1-t)^2tc_2 + 3(1-t)t^2c_3 + t^3c_4.
52 We will assume that the slur connects two notes of the same
53 pitch. Different slurs can be created by rotating the derived shape.
54 We will also assume that the slur has a vertical axis of symmetry
55 through its center. The left point will be the origin. So we have
56 the following equations for the control points $c_1\kdots c_4$.
65 The quantity $b$ is given, it is the width of the slur. The
66 conditions on the shape of the slur for small and large $b$ transform
69 h \to h_{\infty} , &&\quad b \to \infty\\
70 h \approx r_{0} b, &&\quad b \to 0.
72 To tackle this, we will assume that $h = F(b)$, for some kind of
73 $F(\cdot)$. One function that satisfies the above conditions is
75 F(b) = h_{\infty} \frac{2}{\pi} \arctan \left( \frac{\pi r_0}{2
76 h_{\infty}} b \right).
79 For satisfying results we choose $h_{\infty} = 2\cdot \texttt{interline}$
82 \subsection{Height correction}
84 Aside from being a smooth curve, slurs should avoid crossing
85 enclosed notes and their stems.
87 An easy way to achieve this is to extend the slur's height,
88 so that the slur will curve just above any disturbing notes.
90 The parameter $i$ determines the flatness of the curve. Satisfying
91 results have been obtained with $i = h$.
93 The formula can be generalised to allow for corrections in the shape,
102 i' = h(b) (1 + i_{corr}), \quad h' = h(b) (1 + h_{corr}).
105 The default values for these corrections are $0$. A $h_{corr}$ that is
106 negative, makes the curve flatter in the center. A $h_{corr}$ that is
107 positive make the curve higher.
109 At every encompassed note's x position the difference $\delta _y$
110 between the slur's height and the note is calculated. The greatest
111 $\delta _y$ is used to calculate $h_{corr}$ is by lineair extrapolation.
113 However, this simple method produces satisfactory results only for
114 small and symmetric disturbances.
117 \subsection{Tangent method correction}
119 A somewhat more elaborate\footnote{While staying in the realm
120 of empiric computer science} way of having a slur avoid
121 disturbing notes is by first defining the slur's ideal shape
122 and then using the height correction. The ideal shape of a
123 slur can be guessed by calculating the tangents of the disturbing
125 % a picture wouldn't hurt...
127 y_{disturb,l} &=& \rm{rc}_l x\\
128 y_{disturb,r} &=& \rm{rc}_r + c_{3,x},
132 \rm{rc}_l &=& \frac{y_{disturb,l} - y_{encompass,1}}
133 {x_{disturb,l} - x_{encompass,1}}\dot x\\
134 \rm{rc}_r &=& \frac{y_{encompass,n} - y_{disturb,r}}
135 {x_{encompass,n} - x_{disturb,r}} \dot x + c_{3,x}.
138 We assume that having the control points $c_2$ and $c_3$ located
139 on tangent$_1$ and tangent$_2$ resp.
142 y_{tangent,l} &=& \alpha \rm{rc}_l x\\
143 y_{tangent,r} &=& \alpha \rm{rc}_r + c_{3,x}.
146 Beautiful slurs have rather strong curvature at the extreme
147 control points. That's why we'll have $\alpha > 1$.
148 Satisfactory resulsts have been obtained with
153 The positions of control points $c_2$ and $c_3$ are obtained
154 by solving with the height-line
156 y_h &=& \rm{rc}_h + c_h.
159 The top-line runs through the points disturb$_{left}$ and
160 disturb$_{right}$. In the case that
162 z_{disturb,l} = z_{disturb,r},
166 \angle(y_{tangent,l},y_h) = \angle(y_{tangent,r},y_h).
173 Traditional engraving uses a set of 9 standardised sizes for Staffs
174 (running from 0 to 8).
176 We have tried to measure these (helped by a magnifying glass), and
177 found the staffsizes in table~\ref{fonts:staff-size}. One should note that
178 these are estimates, so I think there could be a measuring error of ~
179 .5 pt. Moreover [Ross] states that not all engravers use exactly
185 Staffsize &Numbers &Name\\
188 22.6pt &No. 1 &Giant/English\\
189 21.3pt &No. 2 &Giant/English\\
190 19.9pt &No. 3 &Regular, Ordinary, Common\\
191 19.1pt &No. 4 &Peter\\
192 17.1pt &No. 5 &Large middle\\
193 15.9pt &No. 6 &Small middle\\
194 13.7pt &No. 7 &Cadenza\\
195 11.1pt &No. 8 &Pearl\\
198 \label{fonts:staff-size}
209 Traditionally, beam slopes are computed by following a large and hairy
210 set of rules. Some of these are talked-about in Wanske, a more
211 recipy-like description can be found in Ross.
213 There are some problems when trying to follow these rules:
216 \item the set is not complete
218 \item they are not formulated as a general rule with exceptions, but
219 rather as a huge case of individual rules\cite{ross}
221 \item in some cases, the result is wrong or ugly (or both)
223 \item they try to solve a couple of problems at a time (e.g. Ross
224 handles ideal slope and slope-quantisation as a paired problem)
226 Reading Ross it is clear that the rules presented there are certainly
227 not the ultimate idea of what beam(slope)s should look like, but
228 rather a (very much) simplified hands-on recipy for a human engraver.
230 There are good reasons not to follow those rules:
233 \item One cannot expect a human engraver to solve least-squares
234 problems for every beam
236 \item A human engravers will allways trust themselves in judging the
237 outcome of the applied recipy. If, in a complicated case, the result
238 "doesn't look good", they will ignore the rules and draw their own
239 beams, based on experience.
241 \item The exact rules probably don't "really exist" but in the minds
242 of good engravers, in the form of experience
245 We'll propose to do a least-squares solve. This seems to be the best
246 way to calculate the slope for a computerised engraver such as Lily.
248 It would be nice to have some rules to catch and handle "ugly" cases,
249 though. In general, the slope of the beam should mirror the pitches
250 of the notes. If this can't be done because there simply is no
251 uniform trend, it would probably be best to set the slope to zero.
254 \subsection{Quantising}
256 The beams should be prevented to conflict with the stafflines,
257 especially at small slopes. Traditionally, poor printing techniques
258 imposed rather strict rules for quantisation. In modern (post 1955)
259 music printing we see that quality has improved substantially and
260 obsoleted the technical justification for following some of these
261 strict rules, notably the avoiding of so-called wedges.
264 \subsection{Thickness and spacing}
266 The spacing of double and triple beams (sixteenth and thirtysecond beams)
267 is the same, quadruple and quintuple (thirtyfourth and hundredtwentyeighth
269 All beams are equally thick. Using the layout of triple beams and the
270 beam-thickness $bt$ we can calculate the inter-beam spacing $ib$.
272 Three beams span two interlines, including stafflines:
275 ib &=& (2 il - bt) / 2
283 \subsubsection{Quadruple beams}
285 If we have more than three beams they must open-up
286 in order to not collide with staff lines. The only valid
287 position that remains is for the upper beam to hang.
290 3 ib_{4+} + bt &=& 3 il\\
291 ib_{4+} &=& (3 il - bt) / 3
295 \section{Layout of the source files}
297 The main font (with the fixed size music glyphs) uses a the \TeX\
298 logfile as a communication device. Use the specialised macros to
299 create and export glyphs.
301 \bibliographystyle{plain}
302 \bibliography{engraving}
309 Paul Terry <paul@musonix.demon.co.uk> writes:
311 Ross states that the dies (the stamps to make the symbols) come in
314 >Can you tell me how big rastrals are?
316 According to the Score manual:
318 Rastral Size Height in millimetres
328 I must say, despite having been a music setter for many years, I had to
329 look these up - none of the publishers I work for deal in Rastral sizes
330 these days (they all use millimetres).