Music
Notation Font Design:
Technical Specifications for Note Head Font Design
Originally
published © 1999 Sunhawk Corporation (now Sunhawk Digital
Music LLC)
Seattle, WA
authored by Gary Martin, Engraving/Production Manager
As
a result of this paper and engraving samples that accompanied
it, Sunhawk's music notation font "Sinfonia" was
accepted as meeting the MPA standard according to the minutes
of the Music Publisher's Association Board Meeting, February
3, 1999. The Sinfonia font, used in Sunhawk's Solero® Music
Editing software, was designed and created by Gary Martin
in the early part of 1995. The paper has been slightly modified
for presentation on this web page.
[To
go directly to the Table of Values and engraving samples,
click here.]
1.1
QUARTER NOTE (CROTCHET, VIERTELNOTE)
1.1.1
Elliptical Shape
1.1.1.1
Ellipse
and Modifications to an Ellipse
The
regular ellipse is the design shape of choice for regular
quarter note heads. Font and graphics programs make it possible
to modify the regular ellipse in numerous ways. A few examples
are shown below. The only departure from the regular ellipse
that might come under consideration for note head design would
be Example 2 of Series A, with a slight elongation of the
horizontal nodes.
1.1.1.2.
Angle of Slant: NonStandard Examples
In
some older music one occasionally finds note heads slanted
with the long axis at a negative angle with respect to the
staff lines (below left), or with the long axis horizontal
or parallel with the staff lines (below right), but these
are generally felt to be of inferior aesthetic quality.
1.1.1.3.
Angle of Slant:
Standard Practice
1.1.2.
Design Parameters
There
are four mathematical values that play a role in note head
design: (1) height,(2)
width, (3) angle, (4) eccentricity,
or the relationship of long axis to short axis. Below are
examples of varying the values in different combinations:
1.1.2.1.
Note heads
with identical angles and eccentricities, but with varying
heights and widths:
1.1.2.2.
Note
heads with identical heights and eccentricities, but with
varying angles and widths:
1.1.2.3.
Note heads
with identical heights and angles, but with varying eccentricities
and widths:
1.1.2.4.
Depending on the constraints we put on the note head design,
these four values will depend upon each other in different
ways. We begin by addressing the first primary constraint
— height.
1.1.3.
HEIGHT
1.1.3.1.
Any height that is less than the distance between two staff
lines will result in a spatial separation of notes in a chord
and a size that is simply too small, except for use as cues
or graces (below left). Any height that is greater than the
distance between two staff lines will result in excessive
overlapping of notes in a chord, yielding an unclear or blurred
effect (below right).
The
height of a note head, regardless of its angle or eccentricity,
should thus be constrained to the distance between two staff
lines.
1.1.3.2.
Effect of
Staff Line Thickness in Height Determination.
As
insignificant as it may seem, the staff line thickness itself
has to be taken into consideration in the precise determination
of the height. One may consider three options for setting
the height of a note head between two staff lines:
1.1.3.2.1.
Top of note head touching the bottom of a staff line above
it,
Bottom of note head touching the top of a staff line below
it—(1) below.
1.1.3.2.2
Top of note head reaching to center of a staff line above
it,
Bottom of note head reaching to center of a staff line below
it—(2) below.
1.1.3.2.3.
Top of note head reaching to top of a staff line above it,
Bottom of note head reaching to bottom of a staff line below
it—(3) below.
If
we allow proportional increase of width as we increase height,
the increase in overall size (surface area) increases significantly.
In fact, in the particular case illustrated, the surface area
of ellipse (2) is ca. 13% greater than that of ellipse (1),
and the surface area of ellipse (3) is over 28% greater than
that of ellipse (1). If we constrain width and increase height
only, the difference between (3) and (1) is still significant
(slightly over 8% greater surface area).
1.1.3.3.
Staff Line
Thickness Relative to Staff Line Spacing.
In
the illustration above, the staff line thickness and staff
line spacing are shown in the ratio used in the Solero® Editor
(1:16). Different ratios will lessen or exaggerate the surface
area differences shown.
1.1.3.4.
Overlap of
Note Heads in a Stacked Chord.
If
we include staff line thickness in the height parameter of
note head design, there may occur some small overlap of note
heads in stacked chords, depending on angle and eccentricity
parameters. However, the overlap zone is contained entirely
within the staff line thickness and does not cause any significant
blurring effect.
1.1.3.5.
HEIGHT RULE
In
the interest of keeping note head sizes as large as possible,
while avoiding extension beyond staff line thickness zones,
we arrive at the following rule for note head height:
The
height of a note head shall be the distance between
two staff lines including the staff line thickness. 
1.1.4.
ANGLE AND ECCENTRICITY: DESIGN OPTION 1
1.1.4.1.
A possible design option for setting the angle of note heads
takes as its starting point the graphical relationship of
note heads a second apart in pitch on the same stem. The rule,
which really consists of two parts, is expressed as follows:
The
long axes of seconds shall share the same line. The seconds
shall touch each other but not overlap. 
1.1.4.2.
It is of interest to note that if these two conditions are
accepted, angle and eccentricity will be mathematically dependent
on each other—stipulate the one, and the other is automatically
determined. If you decide you want to see the note heads slanted
at a particular angle, you will thereby have defined the major
and minor axes of the note head. If on the other hand you
decide you want to see a particular relationship of the axes,
you will thereby have defined the angle. Furthermore, the
width is also determined. It remains then only to find a suitable
pair of angle/eccentricity values, and the note head design
is finished.
1.1.4.3.
To understand the effect of the two constraints stated above
on note head design, let’s begin by drawing a line at some
arbitrary angle on a staff and see how choice of angle determines
the shape of the note head, given the fact that we have also
already established a height rule. The shaded lines represent
the staff lines.
1.1.4.4.
The line shown above will be the line common to the major
axes of our note heads to be placed a second apart. By reason
of vertical symmetry, the centers of the note heads must be
located either centered in the thickness a staff line, or
exactly half way between the centers of two staff lines. This
gives us two points we can now put on the line, which will
be the centers of our note heads.
1.1.4.5.
Imagine a point on the line exactly half way between the two
points shown. That halfway point will be where the two note
heads meet. Therefore the distance between either point and
the halfway point equals the length of the semimajor axis
of the note heads. Let’s draw in the note heads to see this
more clearly, and add a dotted horizontal line passing through
the center of the lower note head as well as a vertical line
passing through the center of the second note head.
1.1.4.6.
Since the vertical distance between the centers of the two
note heads is precisely known (half the distance between the
centers of the staff lines), the length of the semimajor
axis can be calculated: sin^{ }q
= 0.5s/2a where s is the vertical distance between the centers
of two staff lines. The minor axis is now also constrained
to a unique value that results in the topmost and bottommost
points of the ellipse touching the preset vertical boundaries
established by the height rule. Thus, angle
determines eccentricity.
1.1.4.7.
The
general behavior of note head shapes as the angle changes
is illustrated in the following examples.
1.1.4.7.1.
Consider
first the case in which the major and minor axes of an ellipse
equal each other (a = b) resulting in a circle (eccentricity
= 0).
Let
s = the distance between the centers
of two staff lines, and t = the staff line thickness. The
diameter of the circle = s+t and the radius = ½(s+t). In the
triangle shown, the hypotenuse equals 2r, and the vertical
side equals half of the distance between two staff lines (½s).
Solving for the angle [sin^{
}j
= (½s)/2r = s/2(s+t)] and using Solero® Editor units of s
= 512 and t = 32, angle j
= 28°. [In a notation system where the note heads reach only
to the centers of the staff lines, q
=
30°, since the thickness factor (t) falls out and the equation
is then simply: sin q
= s/2s = 0.5.]
1.1.4.7.2.
Any angle above 28° will create note heads with negative angles.
Below shows what happens when the angle is increased to 40°.
1.1.4.7.3.
When
the angle is kept below 28° the note heads will slant with
a positive angle. Below is the case when the angle equals
20°.
1.1.4.7.4.
It
remains only to find the angle/eccentricity combination that
seems most aesthetically pleasing. All the values can be determined
precisely using basic mathematical principles of geometry,
which are shown next, first for the condition not including
staff line thickness, then modifying for it.
1.1.4.8.
First we rotate the ellipse so that the major axis is horizontal
and becomes the xaxis. The known quantities are h
(a preset value which equals half the distance between the
centers of two staff lines), and angle j,
the preset angle. We have a line tangent to the ellipse at
point (x_{1},y_{1}) intersecting the xaxis
at point P(2a,0). We can solve for values of a
(semimajor axis) and b
(semiminor axis).
1.1.4.9.
SemiMajor
Axis, a.
The length 2a equals the length of the hypotenuse of the triangle
with short side h.
Thus:
1.1.4.10.
SemiMajor Axis, b. We start by constructing two equations
for the tangent line.
(1) The general equation for the line tangent to the point
(x_{1},y_{1}) on the ellipse is:
At
point P(2a,0) we have:
(2)
Another
equation for the tangent line is constructed by finding the
slope and the yintersect.
Slope:
y intersect (y_{o}):
So
the equation of the line is:
At
point (x_{1},y_{1}), substituting for x_{1}
from equation (1):
or
in terms of h using
Now
substitute the values for x_{1} from equation (1)
and y_{1} from equation (2) into the equation for
the ellipse. First solve for b in the general equation:
Then
substitute and simplify:
1.1.4.11.
Summary.
The equations for the semimajor and semiminor axes are:
where
the values h and
j
are given.
1.1.4.12.
Conclusion of this Section.
In a given notation system where the distance between the
centers of two staff lines has been set, the choice of angle
of the shared major axes for the note heads of seconds automatically
determines the values for major and minor axes (and therefore
also the width as we will see).
1.1.4.13.
Modifying to Include Staff Line Thickness in Height of Note
Head.
We want to include the staff line thickness in the overall
height of the note head, but not adjust the length of the
major axis. What effect does increasing h
have on b? In the illustration shown below, the staff line is represented
by the shaded zone, which has been enlarged to show the detail.
We
want to increase h
by the length ½t.
The new values for the point (x_{2},y_{2})
in terms of (x_{1},y_{1}) are:
Using
earlier expressions for x_{1} and y_{1}, we
have:
Using
known values for h,
t, and j,
calculate x_{2} and y_{2,} then solve for
b using the standard
ellipse equation. The final equation for b
in terms of known quantities is:
which
reduces to the earlier equation for b
when t = 0 (see
1.1.5.11).
1.1.4.14.
Width.
To find the width, consider first the diagram shown below:
One
equation for the line passing through the Tangent Point is
[see 1.1.5.10 (1)]:
which
results in the yintercept value of:
Now,
so,
Another
equation for the same line is found by using the slope/intersect
method [see 1.1.5.10 (2)]:
Slope:
so,
since
the slopes from the two equations are equal.
Thus,
To
find the yintercept, let l
be the length from y_{1} to y_{o.}
So
the yintercept is at:
(y_{1}
is negative)
Since
the yintercepts from the two equations are also equal, we
find that:
Substituting
this expression for x_{1} into the equation for y_{1}
above, we obtain:
Now
we can solve for w
substituting this expression for y_{1} in
which
results in:
Note
that if j =
0°, w = a.
(Empirical tests show also that as j
approaches 90°, w
approaches b.)
1.1.4.15.
Table of Values.
Based on experimental tests, the practical range of angles
for note heads will lie between 18° and 24°. Using the equations
derived in the preceding sections, the following table of
values has been generated. (Separation is the horizontal distance
between centers of note heads). Units are Solero® Editor units.
Click
on a row to load a pdf sample of an engraved music page. Each
sample will load in a different window to facilitate comparison.
Click
here for the original sample provided by the MPA.
1.1.4.16.
Graphical
Samples (created from Table of Values).
The samples below show the alignment of seconds and the
effect of stacked notes in a chord for angles from 18° to
25°. Note the tiny overlap of the note heads in the stacked
chords, as mentioned in 1.1.4.4 (confined to thickness zone).
1.1.4.17.
Construction
of Note Heads in Font or Graphic Software.
Using the Table of Values, it is an easy matter to construct
note heads that will precisely meet all the criteria addressed
in “Option 1” of note head design. Here is the procedure for
creating the note head with angle of 20°.
1.1.4.17.1.
Enter a circle of diameter equal to the minor axis (=512).
1.1.4.17.2. Scale the circle horizontally only by a factor
of 748/512 (=146.09%).
1.1.4.17.3. Rotate the scaled circle, now an ellipse, by +20°.
1.1.4.17.4. Set the position in the emsquare according to
font design specifications.
1.1.4.18.
Engraving Samples. [Not included here.]
1.1.4.19.
Evaluation.
Based on the graphical samples above and the attached engraving
samples, one can eliminate at least the 18° sample as too
wide and the 25° as too rounded for good note head design.
The 21° and 22° samples seem match the older plate engraving
sample provided by MPA the best.
1.2.
HALF
NOTE (MINIM, HALBENOTE)
1.2.1.
Design Relationship To Quarter Note
1.2.1.1.
For
consistency in appearance, the general design criteria adopted
for the quarter note head should be applied to the half note
head. For example, if quarter note heads are to be aligned
with their major axes along the same line, half note heads
should follow the same pattern at the same angle.
1.2.1.2.
The only new design aspect of half note heads is the shape
and size of the counter (the enclosed white space inside the
half note head).
1.2.2.
Half Note Head Counter Design
1.2.2.1.
The
half note head is not merely a “hollow” version of the quarter
note head. The counter is not just a smaller version of the
half note head, as shown below.
Not this:
1.2.2.2.
The
counter should be proportionally narrow at the minor axes
than at the major axes, as shown below.
1.2.2.3.
Counter
Values.
1.2.2.3.1.
Major Axis of counter is ca. 85% that of the major axis of
the note head.
1.2.2.3.2. Minor Axis of counter is ca. 45% that of the minor
axis of the note head.
1.2.2.3.3. The vertical node separation of the counter is
increased to 154% of the ellipse vertical node separation.
The half note head counter is simply a vertical node stretch
on an ellipse with a higher eccentricity than the ellipse
used for the illustration there.
1.2.2.4.
The values listed above should not be viewed as absolute.
Certainly a range about these values will yield acceptable
results.
1.3
WHOLE NOTE (SEMIBREVE,
GANZENOTE)
1.3.1.
General
Design Aspects
1.3.1.1.
The
whole note head design is complex, which has given rise to
perhaps a wider variety of designs among music font designers
than is the case with the quarter and half note heads.
1.3.1.2.
The general design parameters include:
1.3.1.2.1. Ratio of width to height.
1.3.1.2.2. Shape of outer perimeter.
1.3.1.2.3. Shape and size of counter.
1.3.2.
Whole Note Head Width/Height Ratio
1.3.2.1.
The width/height ratio for the whole note head used in the
Solero® Editor was decided upon after evaluating whole notes
from a variety of music publishers. Some publishers use wider
whole notes, others use narrower ones. A compromise position
was struck for the Sinfonia font.
1.3.2.2.
The ratio for the whole note head in Sinfonia is: 1.957 (1002
units wide by 512 units high). Unlike the quarter and half
note heads, the height of the whole note head was kept at
the distance between the centers of two staff lines to avoid an extensive vertical overlap
zone along the horizontal that would be created due to the
flatter curves at the top and bottom of the whole note head.
1.3.3.
Shape of Outer Perimeter
1.3.3.1.
The outer perimeter is a slightly skewed ellipse. The leftmost
point is shifted up from the vertical center slightly (10
units, or ca. 2% of the height), and the rightmost point
is shifted down from the vertical center by the same amount.
The effect on the horizontal axis is to give it a slight negative
value.
1.3.3.2.
The topmost point is shifted to the left of the horizontal
center slightly (29 units, or ca. 3% of the width), and the
bottommost point is shifted to the right of the horizontal
center by the same amount.
1.3.4.
Size and Shape of Counter
1.3.4.1.
The counter is most complex of all and will only be dealt
with here in general terms.
1.3.4.2. Width/height ratio of counter is 1.19 (514 units
wide, 432 units high). The ratio of note width to counter
width is thus 1.95 (1002:514) and the ratio of note height
to counter height is 1.19 (512:432).
1.3.4.3. The approximate angle of the counter is 35° counterclockwise
from the vertical.
