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A Hidden Order: Revealing the Bonds BetweenMusic and Geometric Art – Part Two

Lee Westwood and Sama Mara

Contents

Structure of Final Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2Indefinite Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2Linear Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Compound Grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Creative Implications of a Translation Between Music and Art . . . . . . . . . . . . . . . . . . . . . . . . . 14Creative Approaches Explored in A Hidden Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Applying Musical Composition Techniques to Geometric Artwork . . . . . . . . . . . . . . . . . . . 14Aperiodic Rhythms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17Some Final Thoughts on the Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

Abstract

This chapter follows on from Part 1 in describing a method of translating musicinto geometric art and vice versa. Here we look at the macro level structure ofthe whole piece.

Also discussed are the implications of establishing such a connection betweenmusic and geometric art. These include the possibility of unique creativeprocesses that combine practices from both visual arts and musical composition,

S. Mara (�)Musical Forms, London, UKe-mail: [email protected]

L. WestwoodUniversity of Sussex, Brighton, UKe-mail: [email protected]

© Springer International Publishing AG, part of Springer Nature 2018B. Sriraman (ed.), Handbook of the Mathematics of the Arts and Sciences,https://doi.org/10.1007/978-3-319-70658-0_19-1

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mailto:[email protected]

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https://doi.org/10.1007/978-3-319-70658-0_19-1

2 S. Mara and L. Westwood

as well as facilitating the application of developments, practices and creativeprocesses from one discipline to the other.

KeywordsMusic · Geometry · Visualisation · Sonification · Islamic · Art · Fractal ·Aperiodic · Tilings · Harmony

Structure of Final Design

In the following text, works from the A Hidden Order project shall be used toillustrate the points being made. Created by Mara and Westwood, this projectexplored the relationship between music and art, implementing the process fortranslating music into geometric art discussed here. The project is comprised ofa series of geometric artworks and a corresponding suite of compositions formixed ensemble, short films, an album, an interactive platform, and a live audio-visual performance. Upon completion of the design and compositional process,the works were performed by a small ensemble comprising of flute, cor anglais,cello, and marimba/percussion, which were recorded professionally in the studio.These recordings were entered into a bespoke computer program implementing thetranslation described here, resulting in a series of high quality digital prints and theiraccompanying animations.

Indefinite Growth

The mapping between pattern and rhythm discussed here describes an ever-expanding form that grows relative to the length of time of the music. The nature ofhow the grids are constructed means that two notes or sections of music separatedby long periods of time may neighbor one another in the work’s visual form (Fig. 1).This can lead to visual results that do not correspond to the auditory experience. Forexample, two patterns neighboring each other may appear visually dissonant, butgiven the distance in time separating these sections in its musical translation, thereis no corresponding dissonance aurally.

Creating visually harmonious forms that bridge these areas, maintaining logicalstructures, is of course possible. There are many musical forms that follow similarlysimple and rigid structures, and this was the approach taken towards a number ofpieces with the A Hidden Order project, including Octagon I – Flute & Marimba(Fig. 2), whereby the geometric structure of the grid dictated the musical structureprecisely. However, always sticking to a predefined and rigid structure ultimatelyleads to creative restrictions on the musical composition.

To free up the compositional requirements when working with the translationfrom rhythm to pattern, different structural systems may be applied at differentscales of the piece so that, for example, one system of symmetry may be used upto the level of a bar, with another that arranges the bars and sections themselves.

A Hidden Order: Revealing the Bonds Between Music and Geometric Art – Part Two 3

Fig. 1 An example of two cells neighboring each other which are temporally distant. The grid(right) shows four sections of the structure of a piece. The grid represents 768 beats. At 120 bpmeach cell represents half a second, so the grid would represent 6 min and 24 s. The magnified area(left) shows two cells neighboring each other shaded orange and blue. These cells exist in differentsections and represent beats that are 96 s apart in the musical structure

Fig. 2 Octagon I – Flute &Marimba

This becomes relevant when considering the structure of a piece of music, wherebydifferent scales of the piece, from beat, to bar, to section, to the whole piece mayfollow different structural arrangements, and where each scale may be consideredas an independent object within the whole, one not necessarily entwined with, ordependent on, another distant part.


Linear Layout

One example of a change of structure applied at different scales is to introduce thelinear element of music into the overall structure of the visual pattern. To illustratethis, we shall look at the piece Octagon Square–Marimba (Fig. 3), where each baris visually represented as a square. These squares are arranged in a simple linearsequence. This arrangement allows more freedom for the musical composition, asthe linear reading of the piece frees up the consideration of how disparate parts ofthe piece might juxtapose visually.

Change of Time SignatureIt has not yet been possible to have a logical and continuous visual representationof a change of time-signature, as the symmetry of the grid would need to changein accordance with this. The linear layout provides an opportunity to allow forthis to happen in a seamless manner (see Fig. 4). Any time-signature change maybe accommodated by the use of a triangular or rectangular section of a grid thatcontains the required number of cells.

Compound Grids

Drawing inspiration from Islamic geometric artworks once again, we see beautifulworks that display local areas of radial symmetry at different scales of a piece(see Fig. 5). This artwork displays a symmetric cell unit, with an immediatesurrounding area also displaying radial symmetry and ultimately a symmetriccomposition of the whole piece. Visually this is very satisfying and something thatwas felt appropriate to integrate within the system itself.

This is not only an aesthetic consideration but also has an impact on theeffectiveness of the translation method. By applying an inherent symmetry atvarious levels, the bars, motifs, and sections of the musical composition are in asense visually self-contained within an area of the piece, sitting alongside othersections rather than being entwined together. This liberates the dependence of onearea of the piece from another, more akin to the way music is heard and experienced.

It is possible to achieve radial symmetry at various levels of the work followingthe original approach, and this was explored most notably in the pieces PentagonIII – ‘Roundels’ Ensemble and Octagon III – Solo Conga (Fig. 6). However,as discussed above, this is achieved only through the use of a restrictive set ofcompositional requirements.

Following is presented a solution to achieving local areas of symmetry at variouslevels of scale within the visual design while maintaining more freedom in thecompositional process. The approach again draws upon techniques from fractalgeometry and is to construct radially symmetric groupings, which themselves arearranged and repeated so as to create a larger symmetric group, repeating thisprocess up until the scale of the whole piece. Examples of this approach from A


Fig. 3 Octagon Square–Marimba. Each bar of music is represented by a square. These square arearranged in a linear sequence starting from top-left and ending bottom right

Hidden Order are Hexagon I – Ensemble (Fig. 7), Square – Ensemble (Fig. 8), andTriangle – Ensemble (Fig. 9).

To illustrate the process, we shall examine the underlying structure of HexagonI – Ensemble. Each bar of music is comprised of 12-beats and is represented by ahexagon which itself is radially symmetric. The hexagons are grouped into foursand arranged together, with the centers of each hexagon lying on the vertices ofa rhombus comprised of two equilateral triangles (Fig. 10). This group of four


Fig. 4 A visualization of achange in time-signature.Each row represents one barof music. For example, thesquares (bars 1–4) couldrepresent 4-beat bars, thetriangles 9-beat bars, and therectangles above could bebased on another symmetryand represent bar lengths of5-beats

hexagons is then copied and rotated three times to create a section of radialsymmetry (Fig. 11).

These four bar sections themselves are arranged together with other four barsections on the vertices of a triangle (Fig. 12), which are then copied and rotated tocreate another level of radial symmetry (Fig. 13).

This process may be repeated up until the final arrangement of the piece. InHexagon I – Ensemble, the final arrangement was left without radial symmetry foraesthetic reasons.

At various levels of the piece, we see local symmetry displayed. First on the levelof a bar, then within four bar sections, then 12 bar sections and so on. This approachleads to visually satisfying pieces and frees up the creative constraints from havingto strictly consider the relationship of disparate parts of the composition. Noweach area contains its own symmetry and is therefore visually more self-contained,though the juxtaposition of these areas still has a visual impact, and there is stillthe ability to create pieces with consideration for the larger scales of the musicalstructure and visual pattern.


Fig. 5 An example of Zellijtiling from Morocco, from theMoulay Idriss II. Within thisartwork are areas of localsymmetry that are repeatedand arranged to create asymmetric overallcomposition (Image ©Ghulam Hyder Daudpota)

Fig. 6 Octagon III – SoloConga, from the A HiddenOrder collection. The areahighlighted in purple is asymmetric motif, and this isrepeated eight times aroundthe blue highlighted area tocreate a larger symmetricmotif in orange. The orangearea itself is repeated eighttimes around the center of thewhole piece

Performance Dynamics and AccentsIn exploring the relationship between rhythm and pattern, we have built upwardsfrom a “beat” and a “cell,” towards bars, sections, and a whole piece. It is clear tosee that the beat and cell may also be subdivided in the same manner as has already


Fig. 7 Hexagon I – Ensemble

Fig. 8 Square – Ensemble

been described, by working down to smaller generations rather than up to largerones.

This subdivision may be performed indefinitely, leading to shorter and shorterbeats and ultimately enabling the analysis of the contours of a sound within


Fig. 9 Triangle – Ensemble – Variation I

Fig. 10 Each hexagonrepresents one bar of music.The diagram shows four barsarranged together based on arhombus

a particular beat. The process is best performed by the use of a computer toachieve more detailed and accurate results, given the short lengths of time that arebeing analyzed. This fine-detailed analysis of sounds within a beat allows for theperformance dynamics and even aspects of the timbre of the music to be visuallyrepresented in the rendering of the pattern, creating artworks with subtle nuancesand variations which lend a natural feel to an otherwise digital process.


Fig. 11 The four bar sectionis repeated in thevisualization three times tocreate a radially symmetricarea. This whole arearepresents four bars, as therepetition is applied to thevisual representation only

Fig. 12 The four bar sectionis combined with two other4-bar sections in a triangularformation, visuallyrepresenting a 12-bar section

In Fig. 14, we see the rendering of a section of a percussive piece performed onthe conga. The piece was performed by a musician, recorded, and processed througha computer program which implements the method described here. The two colorsare created by the two different-pitched congas, one bass one treble. Within each cellof the grid, there is a gradation of intensity from a rich, saturated dark color at themore acute vertex, fading gradually to the white of the background (Fig. 15). This


Fig. 13 The 12-bar section isrepeated visually three times,arranged on an equilateraltriangle to create a radiallysymmetric 12-bar section,comprised of three sections offour bars, where each four barsection is also radiallysymmetric

Fig. 14 Detail of Octagon –Solo Conga


Fig. 15 A single beat(outlined in orange) from thepiece Octagon III – SoloConga. The attack of thesound starts at the sharpestvertex of the triangle,represented by the darker andmore saturated area, and fadestowards the opposite side

gradation is created by the attack of the note as the conga is struck, and subsequentdecay of the sound as it returns to silence.

Across the area of the artwork are subtle fluctuations of the colors, created bythe variations within the dynamics of the performance, when the drum is struckslightly harder or softer, or in a slightly different manner that resulted in a changeof instrumental tone (Fig. 16).

Timbre and TextureAt an even finer level of detail, the timbre of the sound affects the texture of thepattern. There are many aspects that contribute towards the timbre of a sound.According to Griscom, “timbre is particularly interesting as a musical featurebecause it is inherently multidimensional, consisting of aspects that are temporal,spectral, and spectro-temporal in nature” (Griscom 2015). The spectral aspects oftimbre were discussed earlier and relate to the saturation of the color. Here it is thetemporal and spectro-temporal aspects which result in a textural variation within thepattern.

Figure 17 shows visualizations of the note E, performed at similar amplitudeson two different instruments. The resolution of the analyses is high, each cellrepresenting 0.06 ms, and the visualization in total representing 1 s. On a generallevel, we see the variations in attack and decay of the two notes. Also evident here isa variation of the texture itself, created by the difference in the harmonic spectrumof the two instruments. The subtle variations in hue, brightness, and saturation overeach piece are due to the interactions of the harmonics of the sounds varying overtime.


Fig. 16 The octagram shownin blue is comprised of thesame drum being played 16different times, resulting inslight variations in thesections of the octagram

Fig. 17 The note E played on marimba (left) and flute (right). The darker center of the marimbashows the sharp attack of the note, with the sound quickly fading to silence as the color fades backto the white of the ground. Inversely, the flute has a slow attack, with a paler center, the curve ofthe sound reaching its loudest in the area with the most saturated purple. The note then starts tofade away as the colors return towards white. At a finer level of detail, the specific texture withinthe forms is created through the analyses of the harmonics of the sound and how they vary over thecourse of the note length. The texture of the flute’s visualization is more consistent and smoother,whereas the marimba is more erratic and has more contrast, resulting in a rougher texture


Creative Implications of a Translation Between Music and Art

In the above sections, we have described a translation between music and geometricart, working from the mathematical foundations of both. The process strives to placeminimal restrictions upon musical composition, allowing for a wide spectrum ofcreative expression. Such translation enables music to have a visual form and forgeometric artwork to be heard.

This direct dialogue between music and art allows for a unique creative processwhereby, at any stage during the development of a piece, it may be consideredin either its aural or visual form. As such, it is possible that a piece may bedeveloped by applying techniques and practices from either the discipline of musicalcomposition or geometric design. A further outcome of this interaction betweenmusic and art is the ability to bring in developments of one practice to inform andadvance the other.

Creative Approaches Explored in A Hidden Order

Within the development of the works of A Hidden Order, Mara and Westwoodexplored a range of creative possibilities:

1. The piece was composed purely from aural considerations, using a time signa-ture, and structure related to the chosen grid. The piece was then visualized uponcompletion of the composition process (Figs. 7 and 8).

2. Starting from a selection of musical motifs, these motifs were then visualizedand selected based upon their aesthetic qualities. The selected motifs weresubsequently developed into a final musical composition (Fig. 9).

3. Starting from a selection of visual motifs, these were translated into theirrhythmical counterparts and developed into a final musical composition (Fig. 3).

4. A visual template was created that governed the rhythmic and overall structureof the piece. This was then developed musically, with the addition of pitchesand harmonic accompaniment, and once again visualized. A to-and-fro processbetween the visual and sonic states took place before arriving at a final composi-tion based upon the composers’ aural and aesthetic considerations (Fig. 2).

5. A complete geometric artwork was designed, based purely upon aestheticconsiderations. This was transcribed to its musical counterpart (Fig. 6).

Applying Musical Composition Techniques to Geometric Artwork

The translation between music and geometric art allows for practices and creativedecisions of one medium to be directly applied to the other, creating the possibilitythat developments from one field are able to benefit the other.


Introduction/Contrasting SectionsWithin the structure of a musical piece, it is common to have a variety of contrastingsections. The introduction, for example, may be more sparse, where themes areslowly introduced and an atmosphere is set. Developing over time, there may bea chorus that acts in contrast to a verse, and so on. In geometric art, the use ofcontrasting sections and the development of themes occurs, but is not standardpractice. There is usually one theme within the work, repeated across the imagefield, and there will usually be only one color scheme, with a consistent dynamicacross the artwork (Fig. 18). Exceptions to this exist: for example, within the Zillijtiling work of Morocco (Fig. 5), we see works that are not purely one design, butinstead develop over the area of the artwork. However, even in this example, thereis only one color scheme and a consistent dynamic across the piece.

In Triangle – Ensemble – Variation I, a piece primarily created through musicalcomposition, the central area (Fig. 9) is the introduction to the piece. Here, theareas of white are created by the silences within the music, with the accented notescreating the strong green. Gradually, longer notes are introduced, creating largerareas of color. Following this, more defined rhythms come into play, creating areaswith more cohesive patterns. Here we see a more lyrical approach to geometricart, created from the narrative and development that is so natural to musicalcomposition.

The sparse introduction to the piece stands in contrast to the two sections depictedeither side of the introduction (Fig. 9). In these two sections, the full ensemble offlute, cor anglais, cello, and marimba is performing together, creating a rich tapestryof colors and a greater depth of pattern, resulting from the complex interactionsof the harmonies and dynamics of the musical piece. Again this use of layeredmotifs and rich overlaying textures is not standard practice within geometric art,but translates very successfully and opens many possibilities that are hitherto lessexplored.

Fig. 18 A classic Islamicpattern from the Alhambra inGranada, Spain. The use of arepeated, unaltered motifacross the image plane isstandard practice in Islamicarts. (Image © Ghulam HyderDaudpota)


Fig. 19 Detail from Triangle – Ensemble, showing a series of motifs that are variations on a theme

Another example of compositional approaches applied to geometric artworkincludes is the development and manipulation of motifs (Fig. 19). Within the foursections, there is a common theme which may be seen in the overall structure ofthe section, though the details vary and the level of complexity within each of themgrows as the theme is developed.

Aperiodic Rhythms

Through this translation, we have also been able to bring in developments fromgeometric art into musical composition. An example of this in A Hidden Orderis of the application of aperiodic sequences as a musical structure. As mentionedearlier, the Fibonacci series has been implemented within musical composition. Thisis usually applied to the overall structure, or at specific scales of the piece. Withinthe method described here, however, the Fibonacci series permeates every level,from beat, to bar, to motif, to section, to the structure of the work as a whole, allinformed and guided by its visual counterpart. Pentagon III – ‘Roundels’ Ensemblewas created by examining the Penrose tiling and locating the areas which displayedlocal five-fold symmetry at various sizes of cell groupings (Fig. 20). It was ofinterest to see if the visual and geometric structure that governs the occurrence andplacement of these roundels (and the grid as a whole) would have a harmonic valuewhen translated into a time based structure.

In this compositional game, each instrument in the quartet was assigned adifferent scale of cell groupings. The lengths chosen were of 2, 5, 13, 34, and 89beats. To create these areas of radial symmetry, the given instrument would need torepeat the same motif at specific parts of the composition, governed by the structureof the grid. The subsequent interplay of these motifs would be an outcome of thenature of the tiling.


Fig. 20 The Penrose tiling,with areas of local symmetryat various cell groupingshighlighted. Each size of cellgrouping was assigned to adifferent instrument andgoverned the structure of theirrespective parts

Though this is a very particular setup, it gives us the opportunity to exploresequences standard in mathematics and geometry but not common in music, to hearhow these sequences would sound and feel, and to interact with them in a creativemanner musically. There is still much to be explored, but the first inroads seempromising, as the aperiodic sequences, although unfamiliar, do not feel unnatural,and they do allow for creative expression within their structures.

Conclusion

Some Final Thoughts on the Research

Beyond outlining the theory and methodology involved in this creative process, it isworth considering some of the practical, aesthetic, and philosophical implicationsthat were brought to light throughout the course of the research.

Pain and Gain Through RestrictionsWhile there is undoubtedly an enormous amount of creative freedom to be foundwithin the system underlined throughout this chapter, these acts of compositionalso come with a number of practical considerations. These include: the necessityof a fixed bpm; a limitation on the kinds of time signature that might be usedwithin one composition (an area in which progress has been made subsequent to


the A Hidden Order project); and the musically unfamiliar structural demands ofaperiodic tiling. Such considerations might be viewed as restrictive, but in practicethey can also prove to be the most liberating or innovative side to the process. Inthe author’s experience, working within the very specific requirements of a briefcan facilitate one’s creativity, freeing the composer from paralysis in the face ofthe dauntingly infinite possibilities of a blank page. Using rules adopted from otherworlds – scientific, mathematical, philosophical – can also help to shape the workin unexpected ways, giving a fresh slant to old methods, and guiding us awayfrom habit. Most fundamentally though, in this case, the conceptually rigorousfoundations for Mara’s translation between sound and pattern circumvented anyefforts in musical visualization which, having a pleasing surface quality to them,would nonetheless be founded on arbitrary relationships (consider your averagecomputer screensaver, for example).

While it is common practice across many cultures to write music that followsthe 4/4, 12/8, 3/4, or 6/8 time signatures characteristic of square and triangularsymmetry, and even to structure such music in ways sympathetic with the unfoldingof higher generations within these grids (doubling, tripling, etc.), forms based onpentagonal or octagonal symmetry have quite plausibly not been explored in thesame strict manner undertaken in the A Hidden Order suite. The music, guided bythe geometry and related numerical sequences, must follow a very specific sequenceof mixed time signature changes, with stresses and relationships between musicallyunusual beats of the bar. Enlarging the musical structure from one generation of thegrid to the next also requires the composition of a highly specific and exponentiallylarge amount of material entirely unfamiliar to the doubling or ternary sequencesamiable to, say, Classical music (consider the implications of the Fibonacci series:1,1,2,3,5,8,13,21...).

Reflecting on the experience of working within these various symmetries, it isinteresting to consider how each one promoted a slightly different approach tothe writing practice, resulting in the creation of a very diverse and colorful suiteof compositions. From a musical perspective, while composing within square andtriangle grids felt as natural as putting on one’s own shoes, writing in the unfamiliarenvironments of pentagonal and octagonal symmetry was most fruitful when weapproached them with geometric sketches as a starting point, working music intothese designs (or rather, working the music out of them) as we went.

A particularly exciting systemic method of composition was adopted for thework Pentagon III – ‘Roundels’ Ensemble, as described above in Section AperiodicRhythms. Here, following Fibonacci proportions faithfully through what becamea gradual build-up, break-down, and final concentration of melodic material trans-posed into a very natural musical structure. At other times, desirable changes inthe geometry led to the need for frustrating edits to musical details (and vice versa)which, in their original medium, had worked very well, as was the case with OctagonI – Flute & Marimba. Having carefully crafted a melodic line, requests such as“could you change the second and seventh beat of every fourth bar so that it’sblue” (to paraphrase a certain geometer) can be painful! This, of course, goes withthe territory of any collaborative work, and learning how to navigate the parallel-


composition process resulted in what are perhaps A Hidden Order’s most excitingand successful works.

These differences should not suggest that the two worlds are unconnected orthat this research forces a nonexistent relationship between music and geometry.Rather, one might speculate that they are symptomatic of modes of expressionwhich hold a common root – a shared mathematical genome, if you will – butwhich have diverged over many centuries of cultural evolution, adopting theirown idiosyncrasies along the way. Whatever the case, there were times whenit came as a surprise to find that a very natural musical gesture would lookawkward, or where a pleasing geometric formation exhibited less musical interestthan anticipated. Consequently, when working in one medium exclusively until asatisfactory outcome was reached, it could not necessarily be expected that theresults would translate with equal success. This was more typical of grids obeyingthe golden and silver ratio than those based on triangular, hexagonal, or squaresymmetry, although it might simply be put down to a lack of experience workingwith these kinds of structure in a musical context. Broadly speaking, however, whenthe compositional process was founded on continual feedback from one side to theother, the results yielded were hugely rewarding. Hexagon II – Cello & Percussion,for example, made recourse to some of our earliest research into simple rhythmsand rudimentary, repeated geometric patterns, providing a very reliable foundationon which to build the work. Furthermore, its visual construction benefitted fromthe linear layout discussed above in Part I Section Linear Layout, freeing the musicfrom the necessity of strict interactions over large time-spans.

A Multidimensional Artistic ObjectAs mentioned in Part I Section Performance Dynamics and Accents, the use ofFourier analysis, and the consequent application of the rhythm-to-pattern translationat higher resolutions, allowed the rendering of a fine level of detail from the recordedaudio, one far exceeding the information available in the original score. This meantsubtle nuances in performance, dynamics, timbre, and timing became visible in thefinal image, making the geometry a blueprint not just for the music, but for therecording of that particular performance.

This brings to light a number of interesting philosophical questions surroundingthe nature and function of the score: if this printed language, originally a seriesof symbolic instructions for recreating sounds, could now be used as a map ofanother image, in turn, could this geometric image be considered a score for themusic, and vice versa? At a certain point in the creative process, it became apparentthat what was being created was not specifically an image or a musical work, butrather a multidimensional artistic object that existed independently of either of thesemediums, one which could, however, be “viewed” through a number of differentgoggles: you might put on your “music goggles” and experience the object as sound;or you might wear your “geometry goggles” and experience it from a differentdimension, one of pattern... In either of these cases, the object itself was essentiallysomething else, something intangible yet all-pervading, universal.


One very exciting outcome of this mode of creation is that each pair of gogglesoffers a unique vantage point, from which the viewer can best experience differingqualities of the object. Using our “music goggles,” we are better positioned to distin-guish the various layers of a dense “polyphonic” texture. Our ears may effortlesslyisolate the activities of different overlaid voices using timbral information, a claritywe struggled to replicate in our 2-dimensional grids. For example, it is far harderfor us to draw out the cello line with our eyes from Triangle – Ensemble than itis with our ears. Similarly, through our “geometry goggles,” where sound in timeis mapped onto 2-dimensional space, we are able to visualize the entire work onone canvas, allowing us to gather an impression of the entire structure at once, incontrast to the necessarily moment-to-moment experiential nature of music. Here, inthis visual realm, free from the restrictions of memory, disparate points in time mayneighbor one another and form more explicit relationships which otherwise may gooverlooked or forgotten.

Time as SpaceDealing with temporality in 2-dimensional space led to the consideration of anumber of alternatives in the way that time unfolded geometrically. While themusical works can be visualized as a static image – one containing all the musicalinformation at a glance – it is also possible to render them in real-time, allowing thecreation of animations for each composition, and the performance of the suite livein a multimedia concert environment.

Mara’s original method of visualization involved working from the centeroutwards, eventually culminating in an entire image which contained the fullduration of the musical work, much like a printed score. This provided the idealmeans of displaying the architecture of the piece as a whole, but resulted inoccasional difficulties where musical sections otherwise conceived of as unrelatedwould neighbor one another visually and thus have to be molded to suit (see PartI Section Indefinite Growth). Striving for something that emulated the momentaryexperience of music, Mara then developed a process whereby each visualized barof music was tessellated across the visual field which, on completion, began tofade and write over itself. This offered an excellent means of exploring musicalworks with more unusual structures, as well as works that were through-composed,or improvisatory in nature. In summary, the two methods employed allowed thework to proceed on an equal footing with each medium, or alternatively with a biastowards either musical or geometric freedom.

A number of other philosophical considerations arose from these issues intranslation. For one, the geometric work was founded on the principle of rotationalsymmetry. As a consequence, each moment in time appears at numerous points inspace simultaneously. A second perceptual dilemma (discussed in Part I SectionA Hidden Order: Revealing the Bonds Between Music and Geometric Art – PartI) stems from the fact that cells of different sizes within the aperiodic tiling ofpentagonal and octagonal grids (type-A and type-B shape cells) correspond to thesame beat length. In tow with the foundation of the theory which relates length oftime directly to area of space, it may prove fascinating to experiment with music


that embraces these two distinct beat lengths (in the case of the Penrose tiling theresulting two beat lengths are in the ratio 1:ϕ). A future consideration might be toexplore such relationships through the use of electronic music, bringing to mind the“impossible” rhythms found in the works of Conlon Nancarrow, who used pianorolls to achieve the otherwise unplayable (Steinitz 2013).

Looking AheadAt the time of creating A Hidden Order, it was not yet clear how a grid might beconstructed which would be capable of dealing with a sequence of time signaturesor tempi that relate to changing symmetries (see Part I Section Change of TimeSignature). Having now overcome the issue of time signatures through linear layoutsopens up vast avenues for future research. Reflecting on such restrictions, one mightalso ask how much the still-developing software that aids the visualization has beenallowed to guide the creative process. While this is something that should continueto be reviewed, it is without doubt that the implementation of these technologiesin A Hidden Order has taken the resulting art to levels that had not originallybeen foreseen, not only making very time-consuming processes a reality, but alsorevealing some of the most exciting aspects of the work, ones which would haveremained hidden had the authors proceeded by hand.

Perhaps the most poignant dilemma when working between the arts and sciencesconcerns how we define the quality of “beauty” and how one judges what is aesthet-ically “true.” For the most part – and as the authors of this paper, one can only speakfrom the point of view of one’s own ears and eyes – the works of A Hidden Orderdisplay a staggering fluency between the musical and geometric world that hintsat a universal harmony transcending artistic mediums, a harmony deeply rooted inthe world of mathematics. This sense is strengthened by a perceived success in thecompositions derived from musically nontraditional structures proportions, such asPentagon III – ‘Roundels’ Ensemble and Octagon I – Flute & Marimba.

However, this dependence upon the senses becomes especially contentious whenevaluating those endeavors throughout the research that were deemed by us to beless successful. When a composition follows a certain musical logic, but to our eyedoes not immediately look right (or vice versa), can this be considered evidence ofthe work not exhibiting a “true” or “common” harmony across both mediums, orare we simply responding to what we are culturally conditioned to see as “right”?Is it conceivable that we are viewing a geometry we are not yet accustomed to? It isimportant, as practitioners, to keep an open mind and to continually ask the questionof what guides our aesthetic judgments – whether we should look back to tradition,rely on instinct, or embrace new musical and geometric harmonies which push theboundaries of those established within our culture.


Cross-References

�A Hidden Order: Revealing the Bonds Between Music and Geometric Art – PartI

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