Development diary, part 10

Steel beams, yesterday.As a wise man once said, there are as many rules of music engraving as there are music engravers. This is a joke, of course, but perhaps only half of one: although there are certain aspects of how music notation should be presented that more or less everybody can agree on, there are many more that provoke lively debate.

One of these latter areas is beaming; specifically, the placement of beams relative to stave lines, and the amount of slant or slope that should be applied to beams given notes of different pitches, the amount of horizontal space occupied, and so on.

Advance warning: this post goes into a lot of detail about subtle aspects of engraving practice concerning beam placement. Engraving nerds only need apply.

Evolution of beams

Beams first appeared in music notation around the turn of the eighteenth century, and trace their ancestry through the flags used to denote the rhythmic duration of shorter notes, and before that to the ligatures that denoted multiple notes to be sung to the same word or syllable, as shown in the now outmoded practice of beaming notes in vocal music according to the underlay of the words in preference to the rhythmic groupings of the meter. Beams are, simply put, compound flags: instead of drawing individual curly tails at the ends of the stems of successive notes, notes are joined by one or more thick lines. It quickly became standard practice to beam notes together according to the rhythmic groupings of the meter, and the result is music that is both much less visually cluttered (because beam lines are much less fussy than curly tails) and more strongly communicates the rhythms.

To illustrate how the engraving of beams has evolved over the past 300 years, consider the opening four bars of Sonata 1 from Arcangelo Corelli's volume of 12 trio sonatas, Opus 2. Here is how British music publisher John Johnson rendered this music around 1720:

(You can click on all of the images in this post to view them at a larger size.)

All of the stems of the notes in the beamed group are drawn at (more or less) the standard length of 3.5 spaces, or an octave, and in their standard directions (e.g. the final A of bar 3, and the first D of bar 4 in the continuo part). The beam is simply drawn to join them, so that the beam has a concave or convex curve to it, or has an elbow if the stem direction of the notes changes within the beamed group. The beam line is also only moderately thicker than the width of the stem, and there is no evident consideration for where the stem tip ends up relative to the stave lines.

Here, by contrast, is how the German publisher Schott rendered the same music nearly 250 years later, in its Urtext edition of 1966:

Beam lines are now significantly thicker than the stems of notes and the stave lines; in fact, they are now half a space thick (and although this particular example does not contain notes shorter than an eighth (quaver), additional beam lines are typically separated by a quarter of a space). The beams no longer curve inward or outward by following closely the contour of the notes in the beamed group; instead, the beam either slopes upwards, downwards, or is horizontal, and the stems of the notes within the group are lengthened or shortened to meet the beam. The notes that caused beams with elbows in Johnson's edition have had their stem directions reversed, so that the stems of all of the notes within the group have a consistent direction, allowing the beam to be positioned below the stave for those two groups. The beams that lie within the stave (the second group in the first bar, and the second group in the last bar) are carefully positioned relative to the stave lines to avoid small triangular gaps at the start and end of the beam, known as wedges. The beams are overall slanted less steeply, and there is a pleasing balance to the appearance of the music.

Three steps to (beaming) heaven

There are at a high level three distinct processes that go into determining beam placement: firstly, working out the stem directions of the notes in the beamed group; secondly, determining whether the beam should slope upwards or downwards, or should be horizontal; and thirdly, determining the final stem lengths required to produce a beam that is correctly positioned relative to the stave lines, with the desired slant.

Two of the beamed groups in the Corelli example nicely demonstrate the decisions to be made regarding the stem direction of notes in beamed groups. Look, for example, at the second beam group in bar 3 in the continuo part: G3, E3, and A3 should all take down stems; A2 should take an up stem. In this case the majority of notes take down stems, so the group overall takes a down stem. The first group of bar 4 is similar: the first D3 may take either an up or down stem because it is on the middle line of the stave; however, D4, C#3 and A3 all take down stems. Again, the majority of notes take down stems, so the group overall takes a down stem. In the case that an equal number of notes take each stem direction, the stems may all point up or down; to make this decision, you might consider whether the beam groups to either side of the indeterminate group have stems pointing up or down and choose to follow them; or you could simply choose to make the stems point down, which is the default choice in the absence of anything to the contrary.1

Whether the beam should slope upwards or downwards, or be horizontal, is determined by the stave positions of the first and last notes in the beamed group: if the first note is lower than the last note, the beam slopes upwards; if the first note is higher than the last note, the beam slopes downwards; if the first and last notes are at identical stave positions, the beam is horizontal. The beam is also horizontal if the notes in the group are in a repeated pattern of pitches; or if all but one of the notes in the group are the same pitch, and the note of differing pitch is furthest from the beam; or if the notes in the middle of the group are closer to the beam than the notes at both ends of the group (called a concave beamed group).

While there is little debate over either of these first two processes, when it comes to choosing the final stem lengths that determine both the position of the beam line or lines relative to the stave lines and the amount of slant in the desired direction, there is considerably less agreement.

The book that provides the most detail on this subject is The Art of Music Engraving and Processing by Ted Ross2; both Elaine Gould's Behind Bars and Kurt Stone's Music Notation in the Twentieth Century provide a summary of Ross's rules. Ross is considered so authoritative in this regard that engravers refer to "Ross beams", referring to beam positions that broadly follow his recommendations. Frustratingly, despite devoting nearly 50 pages of his roughly 300 page book to beaming, there is still plenty of room for debate.3

A lot of the complication with beam placement comes from what Ross calls the "cardinal rule":

The placement of a beam follows the cardinal rule that when it falls within the staff, its ends must either sit [on], hang [from], or straddle a staff line, whether the beam is single or multiple.

Normally, an unbeamed note's stem is 3.5 spaces long, with the stem tip an octave away from the notehead, though notes on more than one ledger line above or below the stave have their stems extended such that their tips meet the middle line of the stave. A beamed note's stem may be shortened or lengthened in order to produce both the correct placement relative to a stave line, and the correct amount of slant.

There is no agreement on what constitutes the correct amount of slant (even within the pages of Ross's own book his examples are inconsistent), but the general principle is that the amount of slant, determined by the vertical positions of the stem tips of the first and last notes in the beam, is determined by the difference between the stave positions of the first and last notes in the beam. In well-engraved music, the smallest slant, for beams whose first and last notes are separated by one stave position (an interval of a second), is typically 0.25 spaces (i.e. the stem tip of the last note in the beam is 0.25 spaces higher or lower than the stem tip of the first note in the beam), and the maximum slant, for beams whose first and last notes are separated by seven or more stave positions (an interval of a seventh or more), is typically between 1.25 and 1.75 spaces.

Because the angle of a straight line between two points is greater as the two points move closer together, well-engraved music tends to use smaller differences between the positions of the stem tips to produce shallower angles when notes are closely spaced, up to a maximum difference of 0.5 spaces. Gould further suggests that a beamed group consisting of only two notes should never have a slant exceeding 0.5 spaces, regardless of the space between them.

Generally speaking, then, attractive beam slants are relatively shallow, rarely exceeding two spaces even when the intervals between the first and last notes under the beam are very large.

The actual slant that is produced may differ from the ideal slant, however, in order to ensure that the stem tips are placed correctly relative to the stave lines: as Ross's cardinal rule states, each end of a beam must either sit on, hang from, or straddle a stave line. Furthermore, no beam should be closer to a notehead than two spaces, which means that as more beam lines are added (as the duration of the beamed notes get shorter), the natural tendency is for stems to get longer. The exact stem length of the notes in the middle of a beamed group is dictated by the slant of the beam, but the beam itself must take into account the stave positions of those notes to ensure that the beam line or lines do not end up too close to the notes in the middle of the beam.

So a little tug of war must be resolved for every beam group: the stem tips of the first and last notes should be positioned to either sit on, straddle or hang from the stave line, and the stems of the intervening notes under the beam should be sufficiently long to avoid the beam lines getting too close.

Different scoring applications resolve this tug of war in different ways, which is illustrated in the picture below: open it in a new tab or window to see the whole picture.

As a reminder, Product A and Product B are the leading commercial scoring programs, Product C is an open-source engraving program that uses text input files to compile PDF output, Product D is a venerable commercial engraving program dating from the MS-DOS days but still used today by some publishers, and Product E is a popular open-source scoring program with a graphical user interface.

There are three rows of annotations below each example: the first row describes the placement of each end of each beam, using the abbreviations S for sit, St. for straddle, H for hang (the same convention used by Ross in his book), and in a couple of places where the stem tip does not seem to be following any specific placement, ?; the second row gives the length of the stem at each end of each beam, in stave spaces; the third row describes whether the beam is flat (F), slopes upwards (U), or slopes downwards (D), and if it slopes, by how much, again in stave spaces.

Looking at these examples is somewhat instructive. Broadly speaking, all of the various products produce similar default results, but there are some subtle if significant differences.

Product A, for example, by default shortens the stems of all beamed notes by 0.25 spaces, presumably to allow a wider range of placements, while Product B never allows any stem within a beamed group to be shorter than 3.5 spaces, which produces generally longer stems than necessary in many beamed groups, and also never produces slants of less than 0.5 spaces, which means that beams only ever sit or hang, and never straddle. Product B also produces flat beams in more cases than other programs, due to the default choice of beaming style set in its default document. Experienced users of Product B tend to rely on a third-party plug-in to produce stem lengths and beam slants closer to those recommended by Ross, though this of course has the disadvantage that the plug-in has to be run again if the score is edited further after running the plug-in for the first time.

It's also notable that both Product A and Product B produce very similar slants for beamed groups of varying intervals: Product A produces a slant of 1 space for an interval as small as a fourth and as large as a seventh, and a slant of 0.5 spaces for an interval as small as a third; Product B produces a slant of 0.5 spaces for all sloping beams in this particular example. Ross's recommendations are for the slant to increase somewhat as the interval between the notes at either end of the beamed group widens.

Product C generally produces pleasing beam slants for this example, though it is curious that it aggressively shortens the stems of the notes in the final beam group, to the point that the stem of the final note is at the minimum length allowed by Ross (2.5 spaces), which makes that group's beams look particularly stubby.

Product D also produces pleasing beam slants, though it is curious that the beam group describing an interval of a fifth produces a larger slant than the group describing an interval of a seventh, in which the first stem is unnecessarily lengthened and indeed is the only stem tip not to be placed in a sit, straddle or hang position relative to an imaginary stave line below the bottom of the stave (other programs tend to continue placing beams as if there are additional stave lines above or below the stave, as this helps produce more consistent slants regardless of the stave positions of the notes under the beam). Again, experienced users of Product D typically rely on a third-party application that post-processes the default results using enormous numbers of look-up tables for predefined slants based (presumably) on different combinations of numbers of notes in the beamed group, number of beam lines, and position on the stave.

Product E encounters a similar problem with the sixth beam group to Product D, again producing a placement for the first stem tip that neither sits on, straddles nor hangs from the first putative stave line below the stave. It also chooses not to reduce the length of the last stem in the beamed group, so this beamed group also protrudes unnecessarily far from the stave; only Product B, which never allows stems to be shortened from their default length, produces a similarly large protrusion (though it also produces no slope for this group, which is arguably the more problematic aspect of its result).

Our application produces a slightly wider range of slants than the other products in the comparison: the smallest slant it will produce is 0.25 spaces, for an interval of a second, and the largest is 1.5 spaces, for an interval of a seventh or more. Of course, the values for the optimal slant for beamed groups describing each interval from a second up to an octave will be adjustable by the user to suit individual taste or a publisher's specific conventions.

This comparison illustrates that there is a great deal of variation in how beam slants are calculated, even with very simple beam groups consisting of only a single beam line. Only an engraver with a keen eye would perhaps take serious issue with any of the beams produced by these applications, with the exception of those produced by Product B, which are by some distance the furthest away from the recommendations in Ross's book.

Beams between staves

When beams are positioned between the two staves of a keyboard instrument, one of the special considerations is how to handle changes in the duration of the notes under the beam. Additional beam lines can be placed on either side of the primary beam line, but if they are placed on the wrong side, they create unsightly beam corners, as illustrated below.

The comparison below shows how different scoring programs handle cases that are liable to produce beam corners.

Product A produces beam corners for all four examples; Product B appears to have a policy that subdivisions are always placed above the primary beam, which means that it is correct some of the time and incorrect the rest of the time (in this example, the first two beams happen to be incorrect, and the second two happen to be correct); Product E, by comparison, appears always to place subdivisions below the primary beam, so it, too, is correct some of the time and incorrect the rest of the time.

Only our application and Product C correctly handle beams that would otherwise produce beam corners. They can normally be avoided by placing additional beams on the stem side when all the stems of the same subdivision are in the same direction, and when the outer stems of the same subdivision are in the same direction.4

Beaming grace notes

As I mentioned earlier, despite spending more than 50 pages of his book talking about beams, Ted Ross doesn't manage to settle any arguments about beam slants. One aspect that goes completely undiscussed, for example, is how to handle beamed groups of grace notes, and indeed neither Gould, nor Stone, nor Gerou and Lusk weigh in on this issue, either.

Grace notes are particularly challenging because they are smaller than normal notes – typically around two thirds or three fifths the size of normal notes – and as such, noteheads, stems, and beams are all scaled down together. Normally, a beam line is 0.5 spaces thick, and is separated from its neighbour by a gap of 0.25 spaces, but when a beam line is scaled down for a grace note, it becomes around 0.3 spaces, and the gap betwixt it and its neighbour becomes 0.15 spaces.

This makes placement of stem tips relative to sit, straddle or hang positions on stave lines challenging, particularly since the distance between stave lines is relatively much larger than for normal notes, which means that if both ends of the grace note beam are going to be placed correctly, the beam slant must either be shallow (e.g. hang to straddle, straddle to sit, or hang to sit, all relative to the same stave line), or quite steep (e.g. hang from one stave line to hanging from the stave line above or below).

Most existing applications make no attempt to produce good placement for groups of beamed grace notes, as can be seen below.

Product A, Product B, and Product E all take an apparently simplistic approach to positioning beams on grace notes: everything is scaled down by the grace note scale factor, and the stem tips are allowed to fall where they may. This sidesteps the issue of requiring the slants to be either shallow or steep, but does mean that wedges are much more likely to occur; wedges are arguably more problematic for grace note beams than normal-sized beams, since the gap between the beam lines is proportionally smaller and thus easier to fill in with a stave line. However, Product A and Product B do appear to set limits on the amount of slant that can be produced for grace notes, which would in itself be something of a protection against wedges, if only the stem tips were correctly positioned relative to the stave lines, which they are not. In addition, neither Product A nor Product E appear to correctly enforce the rule that no beam line should be closer to a notehead than two spaces (or around 1.2 spaces at the grace note's reduced size), e.g. the penultimate beam group for Product A, and the second bar for Product E, which also features beam slants that are probably too steep for anybody's taste.

Product C takes a different approach: it does produce good placements for the stem tips of the first and last notes in a beamed group of grace notes, but it also increases the size of the gap between the beam lines. In fact, the separation of beam lines is only fractionally narrower for grace note beams than it is for normal beams. This has the advantage of making the problem of wedges no more or less serious than it is for beamed groups of normal-sized notes, but has the disadvantage of making beamed groups of grace notes have practically the same colour (or overall blackness) as normal-sized notes.5

Our application does produce good placements for beamed groups of grace notes, and also limits the amount of slant allowed for a group of only two notes, which helps to avoid very steep slants for beamed groups that are naturally narrowly-spaced. For beamed groups with more than two notes, the stem tip of the note at each end of the group snapped to a valid placement relative to the appropriate stave line. This approach gives results that are very close to traditionally-engraved scores from before the age of computer engraving.

Shearing vs. rotating beams

Most applications draw a flat beam by drawing a rectangle of 0.5 spaces in height. When a sloping beam is required, most applications simply offset the two points at either or both ends of the rectangle by the required vertical distance, making a parallelogram, a kind of transformation known as shearing. This makes the beam line appear progressively thinner as the slant increases, which you can see in this example:

Most applications handle beams this way because it is unusual for beams to have a slant exceeding two spaces, and at small to moderate slants, the effect of the thinning of the beam line is not particularly noticeable.

However, this is not how beams were handled in the days before computer engraving: in traditional engraving, the engraver would use a tool of the appropriate width, and rotate it to match the slant of the beam before making his mark on the plate. As such, beam lines in traditionally-engraved music are always truly half a space thick, regardless of their slant.

We always strive to match even the smallest details of pre-computer engraving, and as such our application rotates beams in the same way that a human engraver would rotate the beam tool, so that the thickness of the beam line is always 0.5 spaces along its whole length, giving a greater visual consistency to the colour of beamed groups with different slants.

You can see the difference between shearing a beam line and rotating it properly in this image.

To our knowledge, no other application does this by default6, and it's another example of the attention to fine detail that runs through all of our work.

Widening beam gaps for very short notes

Another subtle but important detail concerns the treatment of beamed groups of notes requiring three or more beam lines, i.e. 32nd notes (demisemiquavers) or shorter durations. Because three beam lines add up to a total height of two stave spaces (3 x 0.5 space lines, 2 x 0.25 space gaps), this limits the range of slants that will produce valid beam placement. As Gould says:

With the addition of a third beam, the beams must slant a whole stave-space. Any other angle will result in one of the beams beginning or ending in the middle of a stave-space, which is less than satisfactory.7

She goes on to recommend that the distance between beams should be widened slightly in order to allow for a slant of less than one stave space.

Few existing scoring programs widen the gap between beam lines for beamed groups of 32nd notes or shorter by default, as can be seen from the comparison image below of groups of 64th notes (hemidemisemiquavers).

Notice that even for a flat beam, Product A, Product B, and Product D produce poor beam placement, with the stem tips of the notes at either end of the beamed group floating in the middle of a stave space. This is a consequence of not increasing the gap between each beam line: with four beam lines separated by gaps of 0.25 spaces, no correct placement within the stave is possible. Our application, together with Product C and Product E, increase the gap between the beam lines, and thus produce correct placement.

Although Product A follows the recommendation that beamed groups of 32nd notes within the stave should produce a slant of 1 stave space, for 64th notes it allows smaller slants, with the second beamed group in particular (with a slant of 0.25 spaces) producing especially prominent wedges.

Product E consistently follows the recommendation for both 32nd and 64th notes, with the result that it produces a slant of 1 space for both the second and third beamed group; however, it does not ensure the correct amount of minimum space between the innermost beam line and the notehead at the right-hand end of the beam; the correct placement for this beam would be for both stem tips to extend by one additional stave space.

Product C and Product D both produce flat beams for all three of these beamed groups. For 32nd notes, Product C produces slants that do not ensure correct beam placement, despite widening the beams correctly. Product D does not widen beams for either 32nd or 64th notes, and produces no slants for 64th notes, but does produce a slant for large intervals for 32nd notes; however, it does not produce a slant of one stave space, so the placement of the beam relative to the stave lines is incorrect.

In various situations involving slanted beam groups with 32nd notes and shorter durations, therefore, Product A, Product B, Product C and Product D all produce beam placements that result in the stem tip of the note at one or other end of the beamed group landing in the middle of a stave space, which should be avoided wherever possible.

Only our application actually widens the gap between beam lines in order to produce both a smaller slant, the correct placement of the beam relative to the stave lines, and an appropriate minimum distance between the innermost beam line and the notehead closest to the beam. (It should be noted that it may not be to everyone's taste to widen the gap between beam lines in this fashion, so this behaviour is of course optional, and if widening is disabled, our application still produces only valid slants of 1 space for groups of beamed 32nd notes and shorter durations where the beam is placed within the stave.)

In summary

This rather lengthy post has hopefully opened your eyes to some of the more subtle aspects of truly fine music engraving, and demonstrated that we are dedicated to producing output of the highest quality, upholding the traditions developed over the past three centuries, and ensuring that they are carried forwards for the musicians of the future.

There is, as always, much that could be said about many of the other things we're working on, but that will have to wait for another instalment. Until then, beam me up.

There are of course further subtleties here. For example, if one note in the beamed group is very high or very low in comparison with the other notes in the group, the stem direction of that outlying note should determine the stem direction of the beamed group, to avoid having a majority of notes with excessively long stems. There are many more similarly subtle rules, which you can read about in Behind Bars or The Art of Music Engraving and Processing. ↩

Ross's book is not available in print any longer, but a CD-ROM version can be purchased from NPC Imaging. ↩

The Alfred Essential Dictionary of Music Notation by Tom Gerou and Linda Lusk deserves an honourable mention here, as the authors of that book take an admirably practical approach to this thorny issue. A lot of the complexity in Ross's rules arise from attempting to avoid wedges, and they argue that with improvements in the consistency and fidelity of computer output, wedges are much less of an issue today than when using more traditional methods of engraving, and therefore do not recommend going to heroic efforts to avoid them. However, they also state that "adjusting many beams individually on the computer is very time-consuming," which I interpret as the more significant factor in their recommendation. Certainly if you wanted to avoid wedges without it becoming very time-consuming, you would be well advised to choose your scoring application carefully! ↩

See page 316 in Behind Bars. ↩

Furthermore, I've not personally encountered any published editions that use such a large gap between beam lines for groups of beamed grace notes. Perhaps one of the developers of Product C dreamed this up; as a solution to the geometric problem it certainly has merit, but it is unusual. ↩

Users of Product B can achieve this effect with the use of a third-party plug-in, though this carries with it all of the disadvantages of using plug-ins in general, such as needing to repeatedly re-run the plug-in after making further edits to the music. ↩

See page 21 in Behind Bars. ↩