THE HORIZONTAL SPACING OF GRAPHICAL NOTATION

Mike Solomon

The University of FloridaSchool of Music, Gainesville, FL, USA

mike@apollinemike.com

ABSTRACT

While several articles have explored the automated spac-ing of a line of music, little attention has been paid to theautomated spacing of graphical notation in the horizon-tal domain. This article proposes an algorithm to stretchand compress images so that they are able to be integratedinto a traditional horizontal spacing model while still at-taining certain “target points” in a score. This is done viaa linear program that distributes stretching error over thegraphical object. The linear program is described througha presentation of its constituent matrices along with sev-eral musical examples that demonstrate its functionality.The article concludes with ideas regarding the program’sbroader applicability to the domain of graphical notation.

1. INTRODUCTION

Since the ascendence of graphical notation in the 1960s,a major concern for composers and engravers alike hasbeen the most efficient and musical manner to mix graph-ics with traditional notation. Thus far, three major ap-proaches have emerged. The first, used by Mark Apple-baum in hisOn the Metaphysics of Notation, is to createscores that make use of both traditional and unconven-tional glyphs without specifying ordinal relationships be-tween these objects. A second approach, used throughoutCornelius Cardew’sTreatise, is to use locally-traditionalsnippets of notation that are merged into a document withan unconventional layout. A third approach, used by Krzys-ztof Penderecki in his operaDiably Z Loudun, is the inte-gration of non-traditional glyphs and images into the lay-out of a traditional score (time on the X-axis, instrumentand pitch on the Y-axis).1 Because of the highly individ-ualized nature of these artworks, it is difficult to proposeany unifying digital tool that can help composers auto-mate certain aspects of their layout decisions. Headwayhas been made by libraries such asBelle, Bonne, Sage[3],which provides an easily-callable collection of rescalablemusical glyphs. However, context-agnostic tools such asthis do not draw upon the many musical engraving con-ventions to which performers have grown accustomed, de-

1In this taxonomy, I am excluding works such as Stockhausen’sZy-klus, which uses a system of organized glyphs that is intended to com-municate precise musical information. In the present study, “graphicalnotation” refers to ambiguous notational schemes that are intended toevoke an interpretive response above and beyond that which “traditional”notation could presumably engender.

manding a significant coding overhead from the user wish-ing to mix received knowledge with experimental nota-tion. This paper offers a partial solution to this problem,using a smoothing algorithm that adapts vector graphicsto the dicta of conventional horizontal spacing as devel-oped in [16], [18], [4], [8], [2], [9], [7], [17], and [1]. Saidalgorithm uses linear programming to constrain a graphicso that it attains certain horizontal “target points” alonga line of music, smoothly distributing displacement errorover the graphic so that it is minimally distorted from itsoriginal version and so that any distortion is not localizedat the target points to which the graphic is anchored.

The paper will proceed as follows. First, it will surveycanonical literature on horizontal spacing in engraving,according special attention to the algorithmic automationof this task. It will then propose a model for the horizontalspacing of graphical notation that uses data from this gen-eral spacing model. Finally, it will present examples ofthis algorithm’s results and provide suggestions for futureresearch.

2. HORIZONTAL SPACING

This section explores horizontal spacing from a traditionalengraving approach. After surveying common horizontalspacing practices in music engraving, it then summarizesa popular horizontal spacing algorithm [8] that has beenimplemented by several digital engravers such as Lily-pond [15] and GUIDO [10].

2.1. Traditional Horizontal Engraving Considerations

Horizontal spacing is traditionally calculated based on twoprincipal constraints: fixed distances between glyphs (here-after called ‘rods’) and variable distances between adja-cent glyphs (hereafter called ‘springs’). Ross [18] dis-cusses the use of several of these rods in traditional mu-sic engraving, such as the standard distance between clefsand noteheads and the standard distance between clefs andtime signatures (see Figure 1). Read [16] establishes aprecursor to the present-day notion of springs through hischarts of least-common-multiple relationships between du-rations (see Figure 2). Few canonical engraving texts,however, confront the issue of the area of music over whichthese calculations are relevant. Instead, they give exam-ples that function within fixed widths that are often a mea-sure long. Algorithmic horizontal engraving has sought to

mailto:mike@apollinemike.com

Figure 1. “Rod” spacing suggestions by Ross [18, p.145].

Figure 2. Reproduction of a horizontal spacing configu-ration showing a 5:8 metric division [16, p. 201]. Thismethod of partitioning the horizontal space has been gen-eralized via the [8] algorithm.

remedy this problem with a Spring-Rod model that can beapplied to various lengths of staff space.

2.2. Algorithmic Horizontal Engraving

The issue of horizontal spacing in computer typesettingwas first given comprehensive and publicly-available schol-arly attention by Gourlay [8], who proposed a Box-Gluemodel2 in which horizontal spacing of objects on lines ofmusic is initiated by applying force to a line of notes withsprings and rods between them and evaluating where theseobjects come to rest when the springs reach an equilibriumpoint after the force has dissipated. Haken and Blostein[9] introduce the terminology ‘Spring’ and ‘Rod’ to de-scribe the Gourlay model, refining it to perform better inits handling of rods, and Renz [17] offers an improved al-gorithm that treats objects on the line as “neighborhoods”with flexible borders that can be adjusted over multiplepasses of the Gourlay algorithm.

Currently, one of the more comprehensive versions ofthe algorithm, which takes into account all of the majorresearch on the subject, is found in GNU Lilypond. Nien-huys [14] summarizes the algorithm as such:

• Each time-point gets a paper column that includessimultaneous events. For example, notes in a chord,

2This model is itself is based on the one developed for text layout inTEX [11].

articulations, dots, and accidentals all may fall intoone column. This column takes into account poly-phonic voices and multiple systems.

• The spacing engine computes an ideal for each col-umn. This position takes the form of a spring ofideal length with a stretch factor to account for dif-ference in fixed space (symbols) and stretchable space(white space between symbols) upon stretching orcompressing.

• Springs are only implemented between adjacent columns.Each column pair has just one spring.

• Arbitrary (including non-adjacent) columns may spec-ify minimum distances (rods) that represent the min-imum widths necessary for preventing collisions.

• In monophonic situations, spacing is counted fromthe left edge of the note head to the next left-edgeof the head (assuming that there are no chords withseconds or unmergeable unisons in them – other-wise, different spacing constants are applied thatrepresent the expanded horizontal extent of the col-umn).

• The common shortest noteS in a piece is spacedwith 2W, whereW is the width of a black notehead.

• Spaces are logarithmically assigned, so ifS is thecommon shortest duration in the piece, thenD(2S)=3W, D(4S) = 4W etc., whereD is a function calcu-lating the duration space.

• Calculate the shortest duration for each measure.The common shortest duration (CSD) is the one whichappears in most measures. This is to prevent a sin-gle short note from stretching the entire piece.

• For durations R shorter thanCSD, the space isW+W×R

S .

• For polyphonic situations, the space isDTSP×D(SP),whereSP is the shortest note playing at that timepoint, andDT is the time difference between thecolumns.

• Spacing from the first clef to the first note retainsa constant space (or rod) irrespective of the resultsabove.

• The final result is achieved by applying force to fi-nal column in a line (for example, a double bar-line)and letting the force move through the springs un-til the upper and lower bound on the measures fora given system become equal. Optimal line breaksare calculated based onCSDas defined above.

Because this algorithm calculates line breaks based on theshortest duration in an entire work, it achieves a sense ofglobal visual constancy that would not be possible if linebreak decisions were exogenous to the Spring-Rod model.

The choice to do this is, of course, a subtle aesthetic one,and the unity it confers onto a piece’s layout is not im-mediately perceptible to the musician reading through awork. However, a crucial argument in favor of this ap-proach centers around spanners, a group of musical ob-jects that is only cursorily treated in the above-cited arti-cles on horizontal spacing. Spanners, such ascrescendoanddiminuendomarkings, beams, slurs, andottava indi-cations, have the ability to reach over systems — and incertain extreme cases over long spans of pieces — to unitedisparate regions of a score. While theGestaltmecha-nism that tells a reader that notes “belong” to a slur maynot lead her to desire uniform spacing across line breaks,other groups, like beams, have a strong grouping effecton the notes they contain because of the way they shapehow performers cognitively assimilate musical informa-tion. Cole [5] writes:

When we read our own language, we recog-nize familiar word-profiles, pick up clues asto function and structure of sentences fromcharacteristic successions of letters or words.We recognize grammatical situations as werecognize faces – not by this or that detail,but on an over-all basis. The same processapplies in music reading.. . . One of the greatadvantages of the graphical features on thenotational map is the advance warning thatwe get of the approximate density, direction,and type of movement from a single glance atthe bars ahead (pp. 23–26).

It is thus natural that we would want different regions ofthe same beam, which may cut across several line breaks,to be as uniform as possible in their representation of the“densities” about which Cole speaks.

In this context, the relationship between graphical no-tation and a piece-wide Spring-Rod algorithm takes on vi-tal importance. If a composer creates a graphic that issupposed to hit certain target points — or span — over apiece that spans multiple lines, one would assume that thecomposer would want consistent spacing rules that con-trolled the horizontal layout at all of these target points.Otherwise, the composer would have difficulty estimatinghow much staff space the graphic would occupy over itsduration, and furthermore, slight changes to the contentof a line would potentially result in appreciable changesin the layout of a graphic that spanned several lines. Byusing the horizontal spacing algorithm presented above tosmoothly stretch and contract graphical notation, one canengrave images that are spanned across large regions of ascore without sacrificing the traditional horizontal spacingfrom which performers glean crucial musical information.

3. THE GRAPHICAL NOTATION ALGORITHM

This section will present how a graphical notation algo-rithm, by horizontal spacing data, can smoothly engravegraphics in scores. It will first develop a linear-program

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Figure 3. Score fragment to which graphical notation willbe appended.

Figure 4. Graphic to be appended.

that effectuates this typesetting, then showing several ex-amples of this algorithm as applied to musical examples.

3.1. The Graphical Notation Linear Program

In order to frame the algorithm proposed in this subsec-tion, consider the following problem. A composer, work-ing with the score fragment in Figure 3, wants to super-pose the graphic in Figure 4 such that it hits the targetpoints specified in Figure 5.3 That is, the vector graphicshould have exhausted 28% of its width when it reachesthe left end of the C♮ on beat two of the second measure,and it should have exhausted 72% of its width when itreaches the left end of the B♮ on beat one of the thirdmeasure. Absent of any spacing algorithm, the composercould simply cut the graphic at these points, stretch thepieces to hit their targets, and obtain a result such as thatin Figure 6. One will notice that the different regions ofthis graphic are scaled using visibly different stretch fac-tors, and even more problematically, awkward kinks areintroduced into the figure at the borders established by thetarget points.

The solution to the dilemma articulated above is for-mulated as the following linear program. The programtakes the standard form

maximizecTx

subject toAx≤ b(1)

3All graphics in this paper are assumed to be vector graphics com-prised of lines and B́ezier curves.

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Figure 5. Target points in the original score. Note that therequested distances between the targets do not match upwith the actual staff space consumed, thus necessitating astretching function for the graphic.

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Figure 6. Naive stretching mechanism that changesstretching and contracting factors at target points, givingan uneven look across the graphic and awkward kinks atthe target points.

0

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Figure 7. Acceptable smoothing polynomial for the mu-sical example presented in Figure 3.

where the upper rows of matrixA represent equations de-scribing a piecewise linear polynomial that is henceforthcalled a “smoothing polynomial.” The domain of the poly-nomial represents the length of the vector graphic (nor-malized from 0 to 1), and the integral of the polynomialrepresents staff space consumed. This means that, by def-inition, the Y value of the function at any point is the rateof change of the consumed staff space (∆ staff space∆ graphic length). Thegoal, then is to have the pieces of the polynomial con-strained to touch each other at target points, thus ensur-ing that there is no disruption in the rate at which thegraphic is changing. For example, an acceptable solutionto achieve the spacing values requested in Figure 3 wouldbe the one shown in Figure 7. This solution has the func-tion:

f (x)=

−2.29378561x+1.4505339 x≤ 0.28−0.39902167x+0.92 0.28< x≤ 0.724.8365928x−2.84964242 0.72< x

(2)One can further verify that the integral of the functionover the [0,1] interval yields the appropriate staff spaces

reported in Figure 5.4

∫ 0.28

0.0(−2.29378561x+1.4505339)dx= 0.3162 (3)

∫ 0.72

0.28(−0.39902167x+0.92)dx= 0.3170 (4)

∫ 1.0

0.72(4.8365928x−2.84964242)dx= 0.3668 (5)

This solution to the spacing problem is not the only pos-sible one. Multiple potential solutions stem from the factthat there are less constraining equations (five (5) in total– three (3) for the area under the pieces of the polyno-mial, and two (2) to set the polynomials equal at the targetpoints) than there are variables in the equations that de-scribe the polynomial (six (6) – two (2) for each of the(3) linear equationsy = mx+ b). This mismatch meansthat certain solutions will force the polynomial below 0,as seen in Figure 8 and verified in Equations 6–9 below:

f (x)=

−17.63092847x+3.5977339 x≤ 0.289.36097833x−3.96 0.28< x≤ 0.72−10.32107333x+9.24287473 0.72< x

(6)∫ 0.28

0.0(−17.63092847x+3.5977339)dx= 0.3162 (7)

∫ 0.72

0.28(9.36097833x−3.96)dx= 0.3170 (8)

∫ 1.0

0.72(−10.32107333x+9.24287473)dx= 0.3668 (9)

Anywhere the polynomial accumulates negative area, thegraphic consumes “negative” staff space, or regresses fromright to left. Preventing the polynomial from falling be-low 0 is the reason that one needs to use a linear pro-gram (whose constraint takes the formAx≤ b) as opposedto a LaGrangian optimization (whose constraint takes theform Ax= b). In the present linear program, the lowerrows of matrixA represent the inequalities necessary toassure that each polynomial in the piecewise polynomialstays above 0 at the target points, which guarantees viathe intermediate value theorem that the polynomial willbe positive at all points in the bounded region.

What follows is a simplified version ofAx≤ b in theform of two equations and one inequality for target pointsX = [0,x1, . . . ,xi−1,1], staff spacesY= [0,y1, . . . ,yi−1,1],5

and slopes and y-intercepts of a linear piecewise polyno-mial MRB= [(m0,b0), . . . ,(mi ,bi)]whose boundaries alongthe X-axis are the points in setX. Equation 11 representsthe constraint that adjacent pieces of the polynomial be

4Note that the staff space points from Figure 5 are normalized tofall between 0 and 1. For example, the second integral’s result repre-sents the staff space consumed between the 66.32% and 31.62% points( 63.32−31.62100 = 0.3170).

50 and 1 bookend thex andy series of values to represent the initialand terminal value for target points and staff spaces respectively.

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Figure 8. Unacceptable smoothing polynomial for themusical example presented in Figure 3. It attains the samearea under the curve as the acceptable solution presentedin Figure 7, but accumulates negative area.

connected. Equation 12 assures that the area under eachpiece of the polynomial is equal to the appropriate amountof staff space to be consumed between two target points.Note that

g(a,b) =a2−b2

2(10)

Equation 13 guarantees that the piecewise polynomial ispositive at each target point. In order for these relationsto be turned into a linear program, the equality constraintswould have to be rewritten as inequalities using slack vari-ables. Numerous articles and textbooks explain how thesevariables may be introduced [13][19].

x1 1 −x1 −1 . . . 0 0 0 0...

...0 0 0 0 . . . xi−1 1 −xi−1 −1

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0...0

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n1−0...

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Figure 9. The smoothing linear program applied to thegraphic in Figure 6. Note that the graphic hits its targetpoints in a much smoother fashion.

≤

0...0

(13)

The optimization portion of the linear program — maxi-mizec

Tx — can be designed independent of the constraint

matrix described above, and several approaches are pos-sible. For example, one could minimize the average dis-tance that any given point in the graphic is from whereit would have originally been if it were not stretched tohit intermediary target points. However, the problem withthis approach is that it does not guarantee the avoidanceof local anomalies around the target points, which is whatthis algorithm is supposed to eliminate. Thus, in my im-plementation of the linear program, I have chosen insteadto minimize thesum of the absolute value of the differencein slope of adjacent line segments. Using the slopes (m)from Equations 11–13, this can be expressed as

min(i−1

∑n=0

|mn+1−mn|) (14)

Or, in other words, minimize the extent to which the slopechanges at the target points. By pushing error away fromthe target points and over the entire graphic, the distortionis less noticeable on a local level without suffering appre-ciably on the whole, as seen in Figure 9.

The smoothing polynomial does not need to confineitself to interpolation at target points. By further split-ting up the piecewise polynomial into smaller lines, it canbetter approximate curves of higher degrees, thus guar-anteeing smoother transitions between rates of change inthe stretching of the graphic. In some cases this addeddetail is the only way to find an acceptable solution. Toachieve the results in Figure 10, the first example must usethree times as many line segments in its smoothing poly-nomial to arrive at a successful approximation, whereasthe second must use four times as many. Not surprisingly,in these more computationally demanding situations, thesuccess of the algorithm is more apparent when comparedto an untreated graphic, as seen in Figure 11.

3.2. Alternative Approaches and Criticism

In order to achieve smooth derivatives along the piece-wise polynomial (and thus smoother interpolation at thetarget points), it would be necessary to use polynomials

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Figure 10. By adding more segments to the piecewisesmoother polynomial, more severe stretching scenarioscan be handled smoothly.

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Figure 11. The top graphic is smoothed by a linear ap-proximation of a quartic polynomial to avoid regressing.Below is the same graphic without the algorithm applied.One can see that the unevenness and kinks are more pro-nounced that those of Figure 6, whose targets require lessstretching.

of higher degrees. As stated above, one manner of do-ing this is by approximating non-linear curves with linesegments. Another possibility is the use of semidefiniteprogramming to create positive polynomials of arbitrarilyhigh degrees. If a polynomial is verified to be the sumof squares over a region, then all of its values in that re-gion will be positive. Because sum-of-squares polynomi-als can be expressed in the formx

TAx, which will yield a

real symmetric matrix with nonnegative eigenvalues, theircoefficients can be solved for by traversing matrices onthe semidefinite cone [20]. While this approach can leadto smoother results, it suffers from a number of compu-tational and practical problems. First, matrices of simi-lar density take longer to solve in semidefinite than linearprogramming. Second, approximating non-linear poly-nomials leads to a greater chance of rounding error thanworking with a group of line segments. Lastly, and per-haps most importantly, any greater accuracy that semidef-inite programming affords is not visible at the orders ofresolution at which typeset music is usually printed.

Irrespective of the algorithm used, a criticism of thisapproach is that it can distort the graphic so severely asto denature the original aesthetic statement that the com-poser intended to make with the image. It is true that,in using this algorithm, severe stretching between targetpoints may cause the composer to reconsider a given graphic,and only below a certain threshold of visual tolerance (i.e.Figure 9) will one fail to immediately notice the stretch-ing being applied to the image. However, because of thealgorithm’s speed and ability to be invoked directly froma markup language,6 the composer’s decision to modify asource graphic can be made quickly with respect to theevolving visual output. Additionally, insofar as graph-ics provide an opportunity for variation and transforma-tion that is akin to that which one would find in musicalmotivic manipulation, this algorithm permits an evolutionof graphical objects over scores that would be painstak-ing to calculate, make, and remake by merging music andgraphics in a vector graphics editor. Lastly, for objectswhose graphical integrity depends on their relative spac-ing with respect to noteheads (i.e. waveforms and spec-trograms), this linear program provides a manner to pre-serve the alignment of arrival points without sacrificingthe score’s horizontal layout.

3.3. Musical Examples

Having established the manner in which horizontal spac-ing data can be harvested by a linear program to createsmoothing polynomials, I will now show some of the addedadvantages that come from this approach in more devel-oped musical contexts.

As mentioned in the first section, a major concern ofthe horizontal spacing algorithm in [15] is preserving con-tinuity of staff spacing over line breaks. Using data from

6The musical figures in this paper, for example, were made usingGNU Lilypond [15]. The linear programming solver is the simplexmethod [6] as implemented by the GNU Linear Programming Kit [12].

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this algorithm to create a smoothing polynomial allowsgraphical notation to span over line breaks without al-tering its rate of spacing on either side of the break, asseen in Figure 12. Furthermore, because horizontal spac-ing calculations factor in data from all staves in a system,graphics in different staves that have similar horizontaltarget points will line align vertically as seen in Figure13. Lastly, the same linear program implemented in thehorizontal domain is implemented in the vertical domainin Figure 14, achieving a constancy of error distributionalong the Y-axis that further contributes to the visual unityof the score.

4. CONCLUSION

This paper has proposed a model for the spanning of graph-ical notation that uses data from canonical horizontal spac-ing algorithms. By feeding this data into a linear pro-gram that distributes stretching error over a graphic whilepreventing right-to-left regressions, one can take a joltygraphic (see Figure 6) and make it acceptably smooth (seeFigure 9). Because this approach is programmatically in-tegrated into score markup languages, it can easily con-trol the distribution of graphics over multiple regions of ascore. It is perhaps this aspect of the graphical notationlinear program that is most promising for future research.As stated in the introduction, a disadvantage of using tra-ditional horizontal spacing mechanisms is that they limitthe scope of experimentation one can do with graphics.

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However, by generating fragments of traditional score ma-terial that are governed by and even fed into the graphicalnotation algorithm, one can eventually create the bend-ing staves and curving beams that are characteristic ofmore ambitious experiments such as Burnson’sBike Rideor those found in Crumb’sMakrocosmos. Doing so canachieve a sense of traditional horizontal proportionalityinan otherwise advanced graphical aesthetic.

5. REFERENCES

[1] P. Bellini and P. Nesi, “Automatic Justification andLine-Breaking of Music Sheets,”International Jour-nal of Human-Computer Studies, vol. 61, pp. 104–137, July 2004.

[2] D. Blostein and L. Haken, “Justification of PrintedMusic,” Communications of the ACM, vol. 33, pp.88–99, March 1991.

[3] W. A. Burnson, “Belle, Bonne, Sage: The Beau-tiful, Good, Wise C++ Vector-Graphics Libraryfor Music Notation,” 2010. [Online]. Available:http://bellebonnesage.sourceforge.net

[4] D. Byrd, “Music Notation by Computer,” Ph.D. dis-sertation, Indiana University, 1984.

[5] H. Cole,Sounds and Signs. London: Oxford Uni-versity Press, 1974.

http://bellebonnesage.sourceforge.net

[6] G. Dantzig, Linear Programming and Extensions.Princeton: Princeton University Press, 1963.

[7] M. Gieseking, “Code-basierte Generierung interak-tiver Notengraphik,” Ph.D. dissertation, UniversitätOsnabr̈uck, 2001.

[8] J. S. Gourlay, “Spacing a Line of Music,” Depart-ment of Computer and Information Science, TheOhio State University, Tech. Rep. OSU-CISRC-10/87-TR35, 1987.

[9] L. Haken and D. Blostein, “A new algorithm for hor-izontal spacing of printed music,”Proceedings of theInternational Computer Music Conference, pp. 118–119, 1995.

[10] H. H. Hoos, “The GUIDO Notation Format:A Novel Approach for Adequately RepresentingScore-Level Music,”Proceedings of the Interna-tional Computer Music Association, pp. 451–454,1998.

[11] D. E. Kunth and M. F. Plass, “Breaking Paragraphsinto Lines,” Software – Practice and Experience,vol. 11, pp. 1119–1184, November 1981.

[12] A. Makhorin, “GNU Linear ProgrammingKit, version 4.45,” 2010. [Online]. Available:http://www.gnu.org/software/glpk

[13] H. Nabli, “An Overview on the Simplex Algo-rithm,” Applied Mathematics & Computation, vol.210, no. 2, pp. 479–489, 2009.

[14] H.-W. Nienhuys, “Re: ICMC Paper,” ElectronicMail, January 2011.

[15] H.-W. Nienhuys, J. Nieuwenhuizenet al., “Lily-pond, the GNU Music Typesetter, version 2.13.47,”2011. [Online]. Available: http://www.lilypond.org

[16] G. Read,Music Notation, A Manual of ModernPractice. Boston: Allyn and Bacon, 1964.

[17] K. Renz, “An improved algorithm for spacing a lineof music,” Proceedings of the International Com-puter Music Conference, pp. 475–481, 2002.

[18] T. Ross,The Art of Music Engraving and Process-ing: A Complete Manual, Reference and Text Bookon Preparing Music for Reproduction and Print.Miami: Hansen Books, 1970.

[19] R. I. Rothenberg,Linear Programming. New York:North Holland, 1979.

[20] C. W. Scherer and C. W. J. Hol, “Matrix Sum-of-Squares Relaxations for Robust Semi-Definite Pro-grams,” Mathematical Programming, vol. 107, no.1/2, pp. 189 – 211, 2006.

http://www.gnu.org/software/glpkhttp://www.lilypond.org

Introduction Horizontal Spacing Traditional Horizontal Engraving Considerations Algorithmic Horizontal Engraving

The Graphical Notation Algorithm The Graphical Notation Linear Program Alternative Approaches and Criticism Musical Examples

Conclusion References