# A Hidden Order: Revealing the Bonds Between Music and Geometric Art – Part One

• Sama Mara
• Lee Westwood
Living reference work entry

## Abstract

The following chapter describes a method of translating music into geometric art and vice versa. This translation is achieved through an exploration of the mutual foundations – in mathematics and its role in harmony – of both music and geometric art. More specifically, the process involves the implementation of principles derived from traditional Islamic geometric art and contemporary mathematics, including fractal geometry and aperiodic tilings.

The method was discovered by Mara in 2011 and was subsequently developed during his collaboration with composer Lee Westwood on the project A Hidden Order. Examples from this project are used to illustrate parts of this chapter.

Also discussed are the implications of establishing such a connection between music and geometric art. These include the possibility of unique creative processes that combine practices from both visual arts and musical composition, as well as facilitating the application of developments, practices, and creative processes from one discipline to the other.

## Keywords

Music Geometry Visualization Sonification Islamic Art Fractal Aperiodic Tilings Harmony

## Introduction

The exploration of the relationship between music and visual art recurs repeatedly in the arts and sciences over the centuries and includes a variety of different approaches such as the use of theoretical, intuitive, and experiential methods, alongside the exploration of physical phenomena.

The theoretical studies considered include Isaac Newton’s relation of the seven colors from the visible spectrum to the seven notes of the musical scale, detailed in his book Opticks (1704). Also of interest are the various systems of proportion explored in architecture, particularly in the Renaissance period, by scholars such as Alberti, who relates the musical intervals (1:2, 2:3, etc.) to rectangles of the same ratios (Padovan 1999). According to the art historian Wittkower, Leonardo Da Vinci theorizes on the shared harmonic principles at play within music and the practices of perspective in art, stating “The same harmonies reign in music and perspective space” (Wittkower 1953). Recent theoretical approaches include “A Geometry of Music” by Dmitri Tymoczko (2011), which views Western musical tonality through geometric space.

Physical explorations of this relationship between music and art may be seen in experiments conducted in Cymatics, whereby liquids or particles on metal plates are exposed to the vibrations of sound waves, creating symmetric forms dependent on the sonic frequencies. In a similar manner, the harmonograph may be used to create specific geometric shapes via the use of pendulums swinging in relative frequencies analogous to the musical ratios.

Experiential and intuitive approaches in this field include the phenomenon of Synaesthesia, in which the stimulus of one sense may cause an impression in another, most pertinently where a visual sensation is experienced in response to a sound stimulus. There are many examples of composers relating pitch to color, the French composer Olivier Messiaen being perhaps the most well documented, whose experiences directly informed his approach to harmony (Messiaen 2002). Similarly, Scriabin’s later works devised elaborate relationships between color and key centers (Galev and Valechkina 2001). In practice, the intuitive and experiential approaches of relating music to art may be seen in the works of the abstract painters of the 1900s such as Mondrian, Kandinsky, and Klee, as well as animated pieces by the likes of Oskar Fischinger and John Whitney.

Wassily Kandinsky wrote “The sound of colors is so definite that it would be hard to find anyone who would express bright yellow with bass notes, or dark lake with the treble” (Kandinsky 1914). This exemplifies the shared intuitive understanding of a relationship that seems to be common to the human experience of colors and sound. If we omit the references to hues from this statement by Kandinsky, it may be interpreted as relating the pitch of the sound with the brightness of color, in that the higher the pitch, the brighter the color. Within a branch of research called cross-modal correspondence that “refers to consistent associations between features in two different sensory domains” (Griscom 2015), there are many studies exploring the relationship between color and sound which support Kandinsky’s observation. Ward et al. (2006) found that “both synaesthetes and non-synaesthetes associate higher pitches with lighter colors” (Griscom 2015).

The sonification of art is perhaps less explored, although the use of the Golden ratio through the implementation of the Fibonacci series is one area in which we find numerous examples of the influence of geometry and the visual arts on music, as seen in compositions by Joseph Schillinger (Livio 2002), Béla Bartók (Lendvai 2000), and Debussy (Howat 1983).

The approach documented in this chapter is primarily a theoretical one which explores the common root of mathematics at the foundation of music and geometric art. Throughout this method, geometric art is considered in relation to color and pattern and music in terms of sound and rhythm. From these two aspects of music and of visual-art, we relate notes to color, and rhythm to pattern.

The reason for this is due to the relative frequencies of sound, color, rhythm, and pattern and how they are perceived. In the realms of color and musical notes, when considered as light and sound waves, the human sensory systems are not able to distinguish individual cycles of the waves, but rather a general sensation of color or sound. Rhythm and pattern are also of a cyclical nature. However, each unit of the cycle may be experienced individually.

In essence, the translation from sound to color is achieved through understanding both as waveforms and drawing upon the physical properties of waves, such as amplitude and frequency. The relationships derived are supported through research in cross-modal correspondence.

In establishing a relationship between rhythm and pattern, the approach demanded study of the mathematical roots of both. Rhythm is here considered as a division of time and pattern as a division of space. Hence, if both rhythm and pattern are rooted in mathematics, the question of their relationship develops into finding a meaningful and logical association between their respective mathematical roots. By defining rhythm as divisions of time, the theory assumes a pulse in the music. As such, arrhythmic music (e.g., Boulez’s “smooth time” (Boulez 1971)) does not carry a meaningful analogy in the method described here.

## Harmony

At the root of this approach are the mathematical foundations of “harmony of time” (as exemplified by the musical rhythms of many cultures around the world) and “harmony of space” (as understood and applied in Islamic geometric art and many other traditions). We refer to “harmony of time” and “harmony of space,” relating to aural and visual harmony, respectively.

### Harmony of Time

Systems of harmony governing rhythm and pitch in Western music are often based upon simple divisions of time. Concerning rhythm, the bar is divided into a given number of beats, each beat being subsequently divided further into equal parts, thus creating a structure in which the rhythmical aspects of the music may be conceived.

In regard to pitch, and more specifically the intervals between two pitches, it is understood that “when the frequency ratio of the two notes is a ratio of low integers: the simpler the ratio, the more consonant are the two notes” (Bibby 2003). The octave is a ratio of 2:1, meaning that when a note is played at twice the frequency of another note, the interval created is an octave. Likewise, in just intonation, relationships of 3:2 (a fifth), 4:3 (a fourth), 5:4 (a major third), and 6:5 (a minor third) all form intervals found within common musical scales and produce consonant sounds. The tempered scale, now standard in music in the Western world, is also rooted in these harmonies and although it does not employ them precisely apart from the octave, it achieves a harmonious sound by closely approximating these ratios. This is possible because there is a certain tolerance in deviating from these ratios that still creates a harmonious result (Bibby 2003). It is this understanding of harmony of both rhythm and pitch that is used in this method.

### Harmony of Space

In regard to harmony of space, we draw upon the principles applied in Islamic geometric art, which is rooted in Euclidean geometry. These principles are not unique to Islamic art and are true for other systems of proportioning that originate from the regular polygons.

We start with the circle and the equal divisions of its circumference which produce the regular polygons (the equilateral triangle, square, regular pentagon, and so on). The forms and ratios that are revealed through the natural subdivisions of the regular polygons form the myriad of different patterns of Islamic geometric art. The numerical ratios that are at play are different to those in the harmony of time and are far more complex.

Within systems of proportion implemented in art and architecture, the artist Jay Hambidge defined two general approaches as static versus dynamic symmetry. Within static symmetry is included Aberti’s system of proportion as well as the “musical ratios” applied in renaissance art and architecture, consisting of rectangles with edge lengths in whole number ratios. Dynamic symmetry, on the other hand, involves the use of ratios with irrational numbers such as √2:1 and the golden ratio. Although these two systems are not entirely exclusive of one another, it is the approach regarding dynamic symmetry that is explored here.

To serve as a brief introduction to this system of harmony, we shall look at the ratios and forms that appear from the subdivisions of the regular pentagon, hexagon, and octagon. The intent of this is to familiarize the reader with the forms and numbers at play and to illustrate the harmonious interactions between these forms. In each of the examples to follow, we shall derive a dynamic rectangle from the regular polygons – illustrating the inherent harmonic properties they contain – and show an example of them applied in Islamic geometric art.

#### The Pentagon

The regular pentagon contains, within its subdivisions, the well-known golden ratio, as seen in Fig. 1.

The fascinating properties of the golden ratio are well documented, with numerous publications devoted to it observing its occurrence in the visual-arts, architecture, nature, and music, reaching back from Ancient Egypt through the Middle Ages, the Renaissance, and Modernism, up to the present day.

A few of the numerical properties of the golden number may be seen in Fig. 2.
The geometric properties of the golden ratio include the golden rectangle and accompanying spiral, and the golden triangle formed by two diagonals and an edge of a regular pentagon. The golden triangle also has a related spiral, seen in Fig. 3.
From the edge of the pentagon and diagonal of the decagon is a dynamic rectangle whose ratio is related to the golden ratio. Subdivisions within this rectangle display a harmonious arrangement of decagons and pentagons whose interplay suggests the extra levels of subdivisions that may continue indefinitely – see Fig. 4.

Within Islamic geometric art, the regular pentagon and related golden ratio stand as one of a series of harmonious forms among the other regular polygons, each of which possessing their own unique properties of number and form.

#### The Hexagon

The regular hexagon with edge length of 1 has diagonal of length √3. A parallel diagonal reveals a dynamic rectangle referred to as the root-3 rectangle – see Fig. 5.
As the root-3 rectangle originates from the hexagon, it may be subdivided indefinitely with combinations of hexagons and triangles. Figure 6 illustrates some of these harmonic subdivisions.
The root-3 rectangle is often used as a repeat unit in Islamic arts, as shown in Fig. 7.
Interactions between the square and the root-3 rectangle create a series of rectangles with edge lengths in the ratios of 1:√3, 1:√3–1, and 1:√3 + 1 that may be used together at various scales and may be used to create endless possible arrangements and subdivisions within the root-3 rectangle – see Fig. 8.

#### The Octagon

The regular Octagon with edge length 1 has diagonal of length √2 + 1 (Fig. 9). The ratio of 1:√2 + 1 is known as the silver ratio.
The silver rectangle is another dynamic rectangle and may be subdivided by octagons, squares, and related forms, as shown in Fig. 10.
As with the golden rectangle and root-3 rectangle, a repeating sequence of rectangles may be created by the use of squares inside of the silver rectangle. This time, at each generation two squares are placed inside the silver rectangle, leaving a smaller silver rectangle at the next generation, as shown in Fig. 11.
The silver number has interesting numerical properties (see Fig. 12) and is used within Islamic arts to subdivide the square, resulting in interesting interplays of form and harmony – see Fig. 13.

We have seen here a variety of patterns and geometric forms derived from the regular pentagon, hexagon, and octagon, with the intention of appreciating their unique – and at times delightful – properties. We have also seen examples of how these ratios and forms are at the foundation of harmony for geometric art, particularly in the Islamic tradition. These principles also extend to other traditions that are based upon the regular polygons and Euclidean geometry.

Comparing the systems of harmony of time and space, as described here, we find both similarities and differences. In much the same way as the musical bar is divided into beats, which in turn are further subdivided (with each subdivision having its own relevant and defined place in relation to other parts and to the whole rhythmical structure), so we see an equivalent with the elements of a pattern, where each unit is further subdivisible (again, with each subdivision having its place relative to the whole, with no areas left unresolved).

The common root of mathematics is clear in both systems. However, when we observe the numbers at play, two different systems appear. The harmony of time is based upon whole number ratios both in pitch and rhythm: 2:1 resulting in an octave (C to C) and 3:2 in a fifth (C to G), while integer divisions of a bar create a certain number of beats. Visual harmony is also based upon simple integer divisions, but this time of the circumference of the circle. The numbers that unfold over the two dimensional interaction of the intersecting lines are relatively complex and involve irrational numbers. These two number systems are an outcome of the respective dimensions at play within the disciplines of music and art, rhythm being one dimensional (that of time), while pattern is based in two dimensions (those of space). The act of bringing together music and pattern becomes essentially about bridging the gap between one and two dimensions.

## A Mapping between Music and Geometric Art

### Color to Pitch Relationship

The relationship between pitch and color described here is based upon the wave properties of sound and light. The mapping is not new, but is described for the sake of completion, in order to present the whole method together.

Sound and light may both be understood as waves, though they are of differing natures. Light is part of the electromagnetic spectrum and is a transverse wave with the ability to travel through a vacuum, whereas sound waves are longitudinal, requiring a medium such as air or water in which to travel. Studying the wave nature of these two allows for simple correlations to become apparent between the properties of color and sound, these correlations being supported by experiments in cross-modal correspondence.

#### Loudness and Brightness

The amplitude of a wave is defined as the distance between a peak or valley from the equilibrium point – see Fig. 14.

The amplitude of a sound wave corresponds to the loudness of the sound, where increased amplitude results in increased loudness. In regard to light waves, the amplitude relates to the brightness of light. From this we deduce that the amplitude of the wave form relates the loudness of the sound to the brightness of the color.

Studies in cross-modal correspondence support this correlation between loudness of sound and brightness of color. For example, experiments conducted by Stevens and Marks (1965) found consistent correlations between loudness of sound and brightness of color.

The inverse of the above-stated relationship is also of relevance in this context. If the color presented is based upon a white ground rather than black, one might expect a louder sound to have a more intense color, so creating a darker result.

An experiment was conducted that asked subjects to match the loudness of the sound to samples of neutral grey paper of different values, presented on either a white background or dark background. On the dark background, 10 of the 12 participants matched the increasing loudness to increasing brightness of the cards, while 2 matched this with increasing darkness. With the light background, of the 10 subjects involved, 4 matched increasing loudness to increasing brightness, 5 to increasing darkness, with 1 subject matching identical sound pressures to all the greys (Marks 1974). In this experiment, there is a consistent relationship displayed between the increase or decrease in both loudness and brightness: on a darker background the tendency was towards the sequence increasing in brightness, while on a white background it tended towards increased darkness. It is the relationship of increased darkness with increased loudness that is applied in this chapter and was predominantly applied in A Hidden Order. Starting from the white background of the paper, louder notes would leave darker marks, with quieter notes leaving less dark marks. Ultimately silence would be the white of the background.

#### Hue and Pitch

The frequency of a wave is defined by the number of waves passing any given point per second and is measured in hertz (Hz). The frequency of light determines the hue of the color, and in sound it governs the pitch of the note. It is based upon this common root of frequency that we relate the pitch of the sound to the hue of the color.

The human eye is sensitive to wavelengths between 390 nm and 760 nm (Jacobson et al. 2000), a nanometer being 1/1000,000,000 of a meter, the frequency range being approximately 400–770 THz. The wave lengths of sound are much larger, at around 1.3 m for middle C, the frequency range of audible sound being roughly 30–18,000 Hz (Taylor 2003). From these frequency ranges, we can see that humans are able to hear up to around ten octaves (doubling of frequencies), whereas the range of visible frequencies is under one doubling of frequency. This sets up a mismatch in terms of the ranges of the two sets to be mapped to one another.

The auditory experience of the interval of an octave in music is such that two notes an octave apart (with a frequency relationship of 1:2) posses a similar quality, referred to as octave equivalency. This property leads to the formation of cyclical scale systems whereby each note that is double the frequency of another has the same name attributed to it, with a key being defined by the collection of notes within one octave.

The perception of hue is also of a cyclical nature, with violet and red at opposite ends of the visible spectrum, bridged by magenta (not in itself a spectral color, but a combination of the opposite ends of the spectrum) that completes the cyclic gradation of hues.

The means of mapping hue to pitch makes use of the property of octave equivalency and the cyclic nature of both pitch and hue. Specifically, the visible spectrum of colors (plus magenta) is mapped onto the frequencies within one octave of sound, meaning that any pitch an octave apart from another is mapped to the same hue. This creates a continuous mapping between hue and the frequencies of pitch within an octave, independent of any formal system of frequencies, notes, or scales. As each subsequent octave maps to the same range of hues, it is consequently a many-to-one mapping from pitch to hue. Let us now consider how our starting pitch is mapped to a particular hue, from which point all the other frequencies may be mapped relative to this.

##### C Is Green

To map a particular frequency of hue to a particular frequency of pitch, we start from a wavelength of light. For our purposes, we shall choose a mid-green light at a wavelength of 520 nm, though it should be noted that each hue is related to a band of wavelengths on the visible spectrum and is not assigned a particular wavelength as such.

When converted to frequency, a light wave of 520 nm comes to 576 THz. This frequency considered as a sound wave is way beyond the human hearing range. However, recognizing the property of octave-equivalency, the frequency may be halved and still retain the same quality and so the same pitch-to-color mapping. Halving the frequency repeatedly (41 times, in fact), stepping down one octave each time, eventually brings the frequency within audible range to 262.17 Hz, very close to the note of middle C (261.63 Hz) in equal tempered tuning. The authors do not maintain that this relationship between the note of C and the color of green holds any weight, but it is a seemingly logical way of establishing a mapping between pitch and hue.

Once this relationship between the frequency of light and sound is established, the other frequencies are derived in relation to these, resulting in a continuous mapping. Applied to the notes of the chromatic scale, an approximate relationship is derived, as shown in Fig. 15.

#### Brightness, Loudness, and Pitch

A remaining concern is that all the scales are now matched to the same hue, leaving no differentiation between high-pitched notes and low-pitched ones. This brings us back to Kandinsky’s quote and the correlation between an increase in brightness and higher pitch which, as was stated earlier, is supported by studies in cross-modal mappings. As described above, brightness has already been attributed to loudness, meaning that now both the loudness of the note and height of the pitch contribute to the brightness of the color. This relationship of both loudness and pitch to brightness has also been noted in cross-modal studies: Marks states that “visual brightness has at least two structural and functional correlates in the auditory realm – pitch and loudness” (Marks 1989).

The resultant affect of both loudness and pitch relating to brightness is that the same hue and brightness of a color may be achieved by two notes an octave apart, but with a counterbalancing change in loudness.

#### Timbre and Saturation

The “timbre” of a sound is a quality that allows us to determine the difference between two different instruments playing the same note at the same loudness. It may be defined as “the way in which musical sounds differ once they have been equated for pitch, loudness, and duration” (Krumhansl 1989). Timbre is a complex subject and various aspects contribute towards it, one of these aspects being the relationships of the overtones of a sound.

In musical sounds, the overtones typically consist of the harmonic series, whereby each overtone is a multiple of the fundamental frequency. It is the relative amplitudes of these overtones that have an affect upon the timbre of the sound.

In general terms, a tone consisting of a single sine wave and little or no overtones will have a very “pure” sound, where the pitch of the sound is clearly discernible. A sound wave with complex interactions in its harmonic series and less order among them will have a less pure sound, leading ultimately to white noise, where no particular pitch is defined.

Likewise with color, a light source containing just one wavelength will output a color with high saturation, such as the light emitted by a laser. More complex interactions of different frequencies of light will reduce the saturation until ultimately the color is a shade of grey, white, or black.

By the complexity of the interaction of frequencies in either sound or light, this mapping relates aspects of the timbre of sound to the saturation of color. This correlation has also been noted in studies in cross-modal mappings (Caivano 1994).

## A Relationship Between Rhythm and Pattern

### A Unit of Time and a Unit of Space

To explore the relationship between rhythm and pattern, we shall start from a basic premise where one unit of time relates to one unit of space.

For our unit of time, we shall choose a “beat” in music. This assumes that the music in question does in fact have a pulse whereby a beat may be defined, thereby excluding arrhythmic music.

As a unit of space, we shall choose the square. We shall see later that we may equally choose other regular polygons. The decision to start from a polygon is based upon the approach to harmony of space described above.

We now have a beat as our unit of time, represented by a square as our unit of space. The next question follows, what would two beats look like? One option is to place another square next to our original square, creating a double square. This is a valid step and is explored later in this chapter. We shall first look at the approach that represents two “beats” also as a square but twice the area of the original square – see Fig. 16.

The process of doubling the area may be repeated indefinitely, each step representing double the number of beats, and thus creating the sequence 2, 4, 8, 16, 32, 64, and so on (see Fig. 16). We shall refer to each square as a “generation,” so that the first square may be known as the “first generation,” the square of area two as the “second generation,” area 4 as “third generation” and so on.

Beats 1 and 2 are already located within the diagram, the first beat being the original square and the second beat being the second generation minus the first generation square. Beats 3 and 4 are located somewhere in the area defined by the third generation square minus the second generation square. Beats 5–8 are located somewhere within the fourth generation square minus the third generation square.

We may now ask, where are beats 3–8 located? A solution presents itself when we reveal a natural subdivision of these nested squares into a grid of equally sized and shaped cells. Each cell is now the visual representation of a beat in music – see Fig. 17. The convention here shall be that the cell number is correlated to the beat number directly (e.g., cell-3 is the visual representation of beat-3). The grid displays fourfold symmetry about the origin, so from here only the top-left eighth of the grid needs to be considered, as this area is reflected and repeated around the origin.

Within a given generation, there is a choice as to the location of a particular cell number. A logical choice within the third generation square is to place cell-3 neighboring cell-2, and cell-4 neighboring cell-3. This indexing sequence may be continued indefinitely, whereby any two consecutive beats are visually represented by contiguous cells – see Fig. 17.

This sequence describes a version of the Sierpinski space-filling curve from fractal geometry, which may be created by joining the centers of each of the consecutive cells in the grid by a continuous curve, as shown in Fig. 17. A curve is inherently 1- dimensional, the term “space-filling” referring to the fact that this curve will eventually fill a two dimensional space at its limit. As Peitgen describes, “given some patch of the plane, there is a curve which meets every point in that patch” (Peitgen et al. 2004). Successively subdividing the grid and creating the curve creates more and more dense versions (Fig. 18), tending towards its limit of covering every part of the grid. The principles behind space-filling-curves are a perfect concept to meet our aim of crossing the dimensional gap between 1 and 2 dimensions, from line to plane, and from rhythm to pattern.
By implementing this indexing sequence with a selection of rhythms, we see how each rhythm is represented by a unique pattern that may also be read back from pattern to rhythm. In Fig. 19, the shaded areas represent an accented beat, and the white areas represent silence in the music.

The visualizations reveal an inherent problem with the method and indexing sequence so far described, in that simple rhythms do not necessarily relate to simple patterns. For example, one of the most simple rhythms in music – where every other beat is sounded, leaving the intermediate beats silent – creates a relatively complex pattern (see the first pattern in Fig. 19). Within this pattern shapes are formed on the horizontal and vertical axes that differ to those on the diagonal axes, which in turn are different from those not lying on the axes at all, while at the origin only is a square formed. Consequently there are four different forms to represent a two beat repeated rhythm.

It turns out that palindromic rhythms create simple patterns (Fig. 19). Whilst palindromic rhythms are not a standard approach to rhythm in music, this translation between rhythm and pattern is also not consistent in terms of relating the perceived complexity of the rhythm and pattern.

When considering the qualitative aspects of rhythm and pattern, there are basic relationships which ideally should hold true, for a successful visualization of music and vice versa. Two of these aspects are that:
1. 1.

A sparse rhythm should create a sparse pattern, and they should increase in density together.

2. 2.

The complexity of a rhythm should be reflected in its visual counterpart, so a simple rhythm creates a simple pattern.

Of these two requirements the first is already met, but not the second. A solution to meeting both of these requirements would be to re-order the cell indexing, as show in Fig. 20. In a sense, we have embedded the palindromic aspect within the cell order itself, so relieving the rhythmic counterpart of that restriction.
This cell order no longer has the property of the original space-filling curve whereby consecutive beats are represented in contiguous cells. However, on visualizing various rhythms, the results satisfy the requisite where simple rhythms create simple patterns and the complexity increases together – see Fig. 21.
The new indexing sequence is created through a series of reflections, whereby each reflection line runs along the edge of a generation square (see Fig. 22). These reflections govern the ordering of the cells.

Through this process, cell 2 is located via a reflection of cell 1 through reflection-line-1 (rl-1) that runs along on the edge of the original square. Cells 3 and 4 are located by reflecting cells 1 and 2 using rl-2 to map them onto the new cells 3 and 4, respectively. Rl-3 maps each cell from cell 1 to cell 4 to a new cell, as follows: 1 → 5, 2 → 6, 3 → 7, and 4 → 8. The reflection-lines determine the indexing of the cells as each generation is a reflection of all the previous generations, reflecting the origin of the grid out to the vertex of the new generation square and preserving the relative order of cells through the reflection.

#### Binary Counting Grid

An interesting property becomes apparent if the cells are numbered with binary numbers, with the first cell as 0 – see Fig. 23.
Each cell may be located within the grid through a unique combination of reflections. For example, to locate the cell with binary number 1000 (cell number 8) only one reflection is necessary: rl-4 and rl-1, rl-2 and rl-3 are all omitted (Fig. 24). Figure 25 shows another example, locating the cell with binary number 1011, applying rl-1, rl-2 and rl-4, while leaving out rl-3. As a final example, cell number 110 (number 6) is located by applying rl-2 and rl-3, leaving out the first reflection-line. Rl-4 is not applicable as it relates to cell numbers beyond this generation – see Fig. 26.
Table 1 displays the transforms that are applied to locate the cells in the examples. On observing the binary numbers and the reflection lines applied, it becomes apparent that in the above examples the binary expression of the cell numbers encode the instructions as to which reflections to apply in order to locate that given cell within the grid.
Table 1

This table shows the reflection-lines applied to locate three cells within the grid. Note how the reflection-lines applied relate to the binary numbers themselves

Binary Number

Reflection-line-1

Reflection-line-2

Reflection-line-3

Reflection-line-4

1000

x

x

x

1011

x

110

x

x

Reading the digits of the binary numbers from right to left, each digit corresponds to a particular reflection-line in order, whereby the first digit relates to the first reflection, the second digit to the second reflection and so on. If the digit is “1” then we apply the associated reflection-line; if it is “0” then this reflection-line is omitted. It turns out that this may be extended indefinitely: for example, to locate the cell with binary number 111000110110011 within the grid, we apply reflection-lines 1, 2, 5, 6, 8, 9, 13,14, and 15, and omit the others. In a sense this grid and indexing sequence may be considered a visual form of the binary counting system.

#### An Alternative Square Tiling

In the derivation of the previous grid and mapping, we explored the route that represented two beats as a larger square with twice the area of the original square. Now we shall look at two options pursued from an alternative approach, whereby two beats is represented by two squares placed next to each other.

#### Hilbert Curve Tiling

We start from one square representing a beat and the double square representing two beats. A generation may be completed by adding two further squares to the first two, forming a larger square comprised of four squares. To continue the process, we may refer to another space-filling curve known as the Hilbert Curve – see Fig. 27.
As this grid resolves to the next generation at four times the original area, it relates to a time signature based upon the powers of 4 (groups of 4 beats or 16, 64 and so on). On visualizing rhythms using this grid with the standard indexing for the Hilbert curve, we find a similar issue to the Sierpinski space-filling curve, in that simple rhythms result in complex patterns (see Fig. 28). As before, this is overcome by indexing the cells in a different sequence, based upon reflections, as in Fig. 29. In contrast to the first grid explored, we begin to see here how each grid has its own distinctive quality and enables a different language of form and pattern.

#### The Dragon Curve

When starting from a square and a double square, we may alternatively draw upon the space-filling curve called “The Heighway Dragon.” As opposed to the previous grids that are created by the use of reflections, this grid is based upon rotations, creating a distinctively visual quality.

Each generation doubles the area of the grid. Unlike the other grids, each generation does not resolve to the same congruent form, but tends towards a particular form known as the Heighway Dragon (Fig. 30). The grid is indexed using the same principles as before, where each rotation maps the existing cells onto the new cells in the same order – see Fig. 31.

The use of rotations rather than reflections lends the grid a distinctive look and feel that is no longer so reminiscent of traditional forms of geometry (such as those found in Islamic art), but is distinctive of fractal geometry. This grid may lend itself to certain styles of music, as opposed to the hard crystalline quality of the reflection-based grids.

### Hexagons

The above examples all explored possibilities on the basis of a square representing a beat. What happens when we choose a hexagon to represent a beat? What implications does this have upon the system for translating rhythm to pattern?

Following a similar process as implemented in the first of our grids, we start by assigning a regular hexagon to represent a beat. Two beats is represented by a hexagram, so doubling the area. At the next step the area of the original form is tripled and the sequence returns to a regular hexagon, so completing one generation (see Fig. 32). Continuing this process creates a sequence of nested hexagons that triples the area with each generation (1, 3, 9, 27, etc.).
This series of nested hexagons will naturally subdivide into a grid of triangles, as shown in Fig. 33. As the grid resolves to each new generation at powers of 3, it relates to musical time signatures based upon the same numbers of beats per bar such as 3, 9 or 27 and so on.
The grid may be created, and cells located, by a series of reflections in the same manner as the first grid, though now there are three possible transformations (no reflection, 1 reflection, or 2 reflections) – see Fig. 34.

As with the square grid and its relation to the binary counting system, this grid and indexing system relates to the ternary counting system. Here, each cell number written as a ternary number contains the information as to which reflections to apply and which to omit to locate any given cell within the grid.

#### More Hexagons

Returning to the sequence from hexagon to hexagram, then to a larger hexagon, we may continue another step to arrive once more at a larger hexagon which is four times the original area, again continuing this sequence to quadruple the area in each generation – see Fig. 35.

Because the hexagon resolves with an increase of three times or four times the original area, hexagonal grids may be created that resolve after groups of 3 cells, 4 cells, or any combinations of these (e.g., 9, 12, 16, 18, 36). This enables the possibility of exploring a variety of different note groupings, time signatures, and accordingly, musical styles.

#### Rhythmic Motifs

Within music a rhythmic motif may define an entire style of music. A Flamenco compás, the clave from Central American music, the waltz and so on, all have unique qualities defined by the arrangement of beats, accents, and rests. The instrumentation and performance of these motifs is of vital importance to the music, but the DNA of the style, as it were, is encoded within the rhythm itself.

Figure 36 shows examples of a variety of traditional rhythms translated into their corresponding patterns. These examples are all 12 beat motifs based upon a hexagon divided into a 12 cell sequence.

As with the rhythmic motifs, each visual pattern has its own unique character and expression, created purely by the different arrangements of the shaded cells within the grid.

#### Rotations

Figure 37 shows an example of a hexagonal grid created through rotations. The aesthetic quality is in line with the Heighway dragon, but this grid has sixfold symmetry rather than fourfold, and relates to a time signature with 6 or 12 beats per bar. Further grids created through rotations are possible, exploring different symmetries and related time signatures.

#### Grid Symmetry, Time Signature, and Structure of the Composition

The symmetry of the grid relates to the time signature of the music. The growth of the areas between subsequent generations of the grid determines the number of beats in a bar. For example, in the first square grid described, a square resolves to a larger square with a doubling of the area, relating to either a 2-beat bar, a 4-beat bar, 8-beat bar, or any number in that doubling sequence. The hexagon resolves to a larger hexagon with a tripling of the area, relating to a 3-beat bar, but may also resolve to the next generation at 4-times the original area, so relating to a 4-beat bar, or a combination of these.

The structure of the grid also determines the structure of the musical piece beyond a bar length. It determines the arrangement of the bars into sections and ultimately of the sections into the whole composition. For example, with the square gird, we may choose a bar length of 4 beats. These bars themselves would also be structured into sections relating to the grid sequence, such as 16 bars. These sections would then also be structured according to the grid and could have four sections, creating a macro level pattern over the whole piece, where the juxtaposition of one section against another will create an overall pattern. Just as when exploring rhythms within a bar and the related visual motifs, the macro level of the pattern and overall structure of the piece is open to creative exploration.

### Pentagonal Symmetry

The grids explored so far involve the square and regular hexagon. These two regular polygons, as well as the equilateral triangle, create the three regular tilings that use just one type of regular polygon to tile the plane, leaving no gaps or overlaps. These tilings translate to time-signatures commonly found within music, such as 4/4, 3/4, and 12/8.

In recent decades, there has been progress in tiling theory − particularly regarding aperiodic tilings – the Penrose Tiling being the most well known of these, which tiles the plane indefinitely with fivefold symmetry and has two cell types. Building a grid based upon the Penrose tiling and other aperiodic tilings reveals interesting implications for the rhythmical and structural counterpart in music, as we will see.

To derive a Penrose tiling, we shall use a similar approach to the other grids and start from a regular decagon. This time, rather than expanding outwards, the areas shall be subdivided to reveal the tiling. This process is known as substitution tiling, whereby each cell of a given shape is replaced by a specific grouping of cells to reveal the next generation of the tiling. The two substitutions applied here are as shown in Fig. 38. The cell which forms one tenth of the decagon (hereby called a type “a” tile) is substituted with one smaller version of itself and a new, wider tile called type “b.” The type-b tile, in turn, is substituted with two smaller type-b tiles and one type-a tile. These are known as Robinson Triangles. Figure 39 shows this substitution applied to the decagon four times to create a Penrose tiling. There are particular orientations in which this substitution must take place. For further information, see Grünbaum and Shephard (1987).

The amount of cells in one segment of the decagon in each generation increases in the sequence 1, 2, 5, 13, 34. These are alternate numbers from the Fibonacci series, the remaining numbers of the Fibonacci series being revealed where the grid resolves as a pentagonal shape comprised of type-b tiles – see Fig. 40.

#### Fibonacci, Bar Length, and Structure of Composition

The implications of the generations of this grid being based on the Fibonacci series are that, when explored as music, the Fibonacci series will govern the structure at every scale from beat to bar, to section, to the whole piece – see Fig. 41.

The two cell types, although different in size, are both considered to be the same beat length in music and so represent equal lengths of time. This seems acceptable given that both cells may be considered as projections from the same higher dimensional cubic structure (Senechal 1995). This does raise a concern, however, in that at larger scales of the grid -- for example, bar lengths -- the two areas which are congruent to these two forms contain different amounts of cells, and so correspond to different lengths of time.

The shapes congruent to the type-a and type-b cells on any given scale contain consecutive numbers of cells from the Fibonacci series (Fig. 41), meaning that in musical form there will be two bar lengths within a piece using consecutive Fibonacci numbers. For example, we may choose a bar length of 5 beats and a bar of 8 beats. These bars shall then also be grouped into sequences based upon the Fibonacci series, and so on, up until the level of the whole piece (Fig. 41). To make meaningful use of the grid and the relationship of the forms that are naturally occurring within them, the composer and designer must consider the order of “a” type cells and “b” type cells within the bar. What this all adds up to is a rhythmical structure tightly governed by the Fibonacci series at every scale of the piece.

#### Indexing the Penrose Tiling

One form of indexing the Penrose tiling would be to follow the sequence of type-a and type-b tiles created from the substitution itself. Figure 42 shows this applied to the first 13 cells of the Penrose tiling. This sequence presents an issue, in that congruent shapes within the tiling sometimes have a different route through them, as shown in Fig. 42.

Other indexing sequences have been explored, though an entirely satisfactory grouping of cells which works at every level of the substitution with the Penrose tiling remains illusive.

However, the Penrose cartwheel tiling (Fig. 43) has been indexed by Fred Lunnon (Grünbaum and Shephard 1987) in a manner that works at every level of the sequence figure and may be applied in this translation method.

### Octagonal Symmetry

Another aperiodic tiling which may be applied in this method is the Ammann-Beenker tiling, with eightfold symmetry – see Fig. 44.

Another eightfold aperiodic tiling discovered by Mara, inspired by the aesthetics of Islamic arts, is shown in Fig. 45.

## Summary

In summary, we have seen a process for translating rhythm to pattern and vice versa, derived from a simple premise and following logical steps in geometry. This results in a geometric grid standing as the visual counterpart of the temporal structure of music, whereby each cell in a grid represents a particular beat, allowing for creative explorations in either rhythm or pattern to be represented in the other. We subsequently explored a selection of tilings, looking at their different symmetries and related time signatures. We have observed how each tiling has its own visual qualities and vocabulary of forms and have examined the contrasting quality of tilings created through reflections or rotations, as well as the regular tilings versus aperiodic tilings.

## References

1. Bibby N (2003) Tuning and temperament: closing the spiral. In: John F, Flodd R, Wilson R (eds) Music and mathematics from pythagoras to fractals, 1st edn. Oxford University Press, New York, pp 13–14Google Scholar
2. Boulez P (1971) Notes of an apprenticeship. Faber & Faber, LondonGoogle Scholar
3. Caivano JL (1994) Color and sound: physical and psychophysical relations. Color Res Appl 19(2):126–133Google Scholar
4. Galeev BM, Valechkina IL (2001) Was Scriabin a Synesthete? Leonardo 34(4):357–361
5. Griscom W (2015) Visualizing sound: cross-modal mapping between music and color. PhD, University of California, Berkeley, P1Google Scholar
6. Grünbaum B, Shephard GC (1986) Tilings and patterns. In: Klee V (ed) . W.H. Freeman and company, New York, pp 540–570Google Scholar
7. Howat R (1983) Debussy in proportion. Cambridge University Press, CambridgeGoogle Scholar
8. Jacobson R, Ray SF, Attridge GG, Axford NR (2000) The manual of photography, digital and photographic imaging, 9th edn. Focal Press, OxfordGoogle Scholar
9. Kandinsky W (1914) Concerning the spiritual in art. Dover Publications, New York, p 25. (1977)Google Scholar
10. Krumhansl C (1989) Why is musical timbre so hard to understand? In: Olsson O, Nielzén S (eds) Structure and perception of electroacoustic sound and music, 1. Excerpta Medica, Amsterdam, pp 43–53Google Scholar
11. Lendvai E (2000) Bela bartok: an analysis of his music. Kahn & Averill, LondonGoogle Scholar
12. Livio M (2002) The golden ratio: the story of phi, the extraordinary number of nature, art and beauty. Review, London, p 193Google Scholar
13. Marks LE (1974) On associations of light and sound: the mediation of brightness, pitch, and loudness. Am J Psychol 87:173–188
14. Marks LE (1989) On cross-modal similarity: the perceptual structure of pitch, loudness, and brightness. J Exp Psychol Hum Percept Perform 15(3):598
15. Messiaen O (2002) Traite De Rythme, De Couleur, Et D’Ornithologie, tome VII. Alphonse Leduc, Paris, pp 95–191Google Scholar
16. Padovan R (1999) Proportion. Routledge, New York, pp 221–227, 2008Google Scholar
17. Peitgen HO, Jürgens H, Saupe D (2004) Chaos and fractals: new frontiers of science, 2nd edn. Springer Science+Business Media Inc, New York, p 92
18. Senechal M (1995) Quasicrystals and geometry. Cambridge University Press, Cambridge, p 195
19. Steinitz R (2013) György ligeti: music of the imagination. Faber & Faber, London, pp 267–269Google Scholar
20. Stevens JC, Marks LE (1965) Cross-modality matching of brightness and loudness. Proceedings of the National Academy of Sciences of the United States of America 54(2):407–411
21. Taylor C (2003) The science of musical sound. In: John F, Flodd R, Wilson R (eds) Music and mathematics from pythagoras to fractals, 1st edn. Oxford University Press, New York, p 51Google Scholar
22. Tymoczko D (2011) A geometry of music: harmony and counterpoint in the extended common practice. Oxford University Press, New YorkGoogle Scholar
23. Wittkower R (1953) Brunelleschi and “proportion and perspective”. J Warburg Courtauld Inst XVI. Idea and Image, Thames & Hudson, London, p 131, 1978Google Scholar