On my first trip to India more than twenty years ago, I discovered ragas, and with them, the pleasure of improvising according to other musical rules.

At the time I was a professional jazz musician who had been taught to play music in a tonally focused manner. As such, my discovery of India and its traditions proved a turning point in my personal explorations.

Over the years, I improvised both in jazz and Indian music, and this led me to search for a system to organise musical notes so that I could combine elements from these two kinds of music. This is how the idea of the rajazz was born. It is a hybrid reality, because, like ragas, a rajazz consists of a single melodic material, but it nevertheless allows for modulations of this individual material in all its transpositions—which I call reflections—on the basis of 12 tonics.

I have been exploring rajazz for almost 20 years, and the research grant that I was awarded by Fonds de Recherche en Art, managed by the Belgian National Fund for Scientific Research (FNRS), gave me the opportunity to take this work a step further, specifically by completing my development of this musical theory in all its details, and by exploring the astonishing ramifications that this system has for mathematics, geometry in particular.

The first challenge for my research was to connect this musical theory to mathematical tools that could facilitate its use and its comprehension, and especially optimise the visualisation of its possibilities.

Rajazz in fact exhibit interesting possibilities in terms of modulation, but they are hard to measure, calculate, and conceptualise in some way. So, I began by developing geometric graphics, such as musical mandalas and modulation wheels, to depict the dynamics of this musical theory with far greater clarity and synthesis than traditional musical notation would allow. I then used combinatorics to compile all the correlations between rajazz and the smallest sub-groups of notes (triads, tetratonic scales), and, conversely, between rajazz and the broader modes (heptatonic scales and modes of limited transposition). I ended up proposing the dodecahedron as the geometric form symbolising rajazz and the dynamics of its multiple connections.

The second challenge lay in putting this theory into practice by issuing a call for compositions. The idea was to generate a dynamic of collective creative intelligence and thereby to bring together a diversity of creative approaches that all had the same theoretical foundation. Various composers responded, and their works were recorded. All of this documentation was posted to the Rajazz Collective website to highlight the possibilities that rajazz offer.

Rajazz theory

The rajazz theory proposes a new way of structuring musical material. It gives us a new way to manage modes and it can be applied in many different ways that provide composers with new tools.

The rajazz theory is based on three essential rules.

Rule 1

The basic material of rajazz is always composed of five notes.

In this respect, rajazz borrows from the Indian raga the tendency to limit itself to an invariable melodic material.

The name “rajazz” is a contraction of “raga” and “jazz.” The reference to the raga pays tribute to India’s modal culture, whose desire to stick with a single colour and explore it in depth is echoed here. The reference to jazz highlights both the propensity to modulate that is typical of Western tonal music and the fact that this musical theory lends itself as much to composition as to improvisation. However, beyond the nominative reference to ragas and to jazz, rajazz does not correspond to a particular style in itself; it is simply a system for organising notes according to a predetermined type of material and a few well-defined rules. As a result, composers are free to create rajazz according to an aesthetic of their own choosing.

The idea behind rajazz stems from the wish for a musical technique situated exactly between modality and tonality, rooted both in the desire to explore a single musical material and the desire to be able to “modulate” it.

Rule 2

In a rajazz, no interval between two successive notes should exceed a major third.

Mathematically speaking, the twelve chromatic notes yield 330 possible combinations of pentatonics that include the root. That said, we quickly realise that the overwhelming majority of these possibilities seem more like fragments of heptatonic modes than actual, properly balanced modes. Which is why we have decided that, in a rajazz, no interval between two successive notes shall exceed a major third. By itself, this rule guarantees “the most equitable distribution possible” of the five notes within the octave. When we apply this rule as a filter, it yields a total of 155 pentatonic modes, which we use to define the framework of a rajazz in terms of its base material.

Two ways to look at a mode

The degree-based notation reflects the focus on the five notes selected, namely [1 ♭3 4 5 6].

The interval-based notation corresponds to a view of these five intervals as line segments. To quantify these intervals, they will be measured as semitones. Given that we have limited the intervals between successive notes to a maximum of a major third, we will never have more than four semitones in the interval-based notation of a rajazz, which will be expressed in semitones from one to four:

The minor second = 1
The major second = 2
The minor third = 3
The major third = 4

Given that the mode is constructed within an octave, the total of the intervals will always equal twelve. The interval structure of this particular rajazz, which includes the following series of intervals [m3+M2+M2+M2+m3], will translate into the following notation: [3 2 2 2 3]

Cousin modes

Cousin modes share the same material, except that their centre of gravity moves from one note of to the other within the same mode. ach rajazz may have five cousins modes. The rajazz 1 therefore has five modes, as shown in illustration 1 in the centre spread.

Each one of these five modes has its own interval structure, but they all have the same interval sequence, which is why we will call them “cousin” modes.

The families

A family contains all the combinations of modes constructed on the basis of the same intervals, regardless of their order within the structure or their interval sequence.

For example, we may group together within the same family all the possible permutations of these five intervals: 4 / 3 / 3 / 1 / 1.

This yields the following six interval sequences: 43311, 43131, 43113, 41331, 41313, and 41133, which correspond to six rajazz, each one of which will have five sibling modes, each with its own interval structure.

Interval sequence and structure

Our rajazz are thus organised as follows:

• Six families, totalling 31 sequences.
• 31 interval sequences in total, corresponding to the 31 rajazz
• 5 cousin modes per rajazz (= 5 interval structures for each sequence)

This yields a total of 5 × 31 = 155 pentatonic modes.

Rule 3

A rajazz may be transposed from 12 tonics, so as to obtain a series of “reflections” of itself. To move from one reflection to another, a “pivot” interval is used that is made up of two common notes and which serves as a hinge.

The first sentence in Rule 3 consists of rendering our colour on the basis of the 12 tonics in the chromatic circle, which gives us 12 transpositions, or reflections, of the same mode.

A reflection is the result of a transposition of the musical material. This term helps us see the difference between the act of transposing and the result obtained. It also avoids confusion with the more common meaning of transposition applied to a melody or a harmony, as here only the material is transposed. This notion of a reflection also acquires significance in the context of the visual representation of a rajazz.

The second sentence defines the procedure for shifting from one transposition to another through the use of a common pivot interval that functions as a melodic-harmonic hinge.

For all rajazz, the intervals that may serve as pivots are: the minor second, the major second, the minor third, the major third, and the tritone. The perfect fourths and fifths are ruled out at first, but not necessarily prohibited, and the sixths and sevenths can be understood as inversions of the thirds and seconds.

There are two types of pivot intervals: those between the successive notes in a mode, depicted in black, and those between non-successive notes, depicted in mauve. (illustration 2.1, p.13)

A sensation of modulation

The inherent dynamic of a rajazz consists of moving from one reflection to another, each shift eliciting a sensation of modulation. A rajazz is explored through all the possibilities that its pivot intervals offer. They embody all the potential relationships between the different reflections within a rajazz.

The movement from one reflection to another is made by means of a pivot interval. If we look at the first example we made of rajazz 1: [1 ♭3 4 5 6], it has a minor third between 1 and ♭3 and another between 6 and 1. In C, my third [1-♭3] becomes C-E♭. When transposed to E♭, this same rajazz 1 has a third [6-1] that also becomes [C-E♭]. This minor third [C-E♭] can thus serve as the pivot interval between these two reflections.

In visual terms, the pentagon representing the reflection in rajazz 1 in E♭ must first of all be oriented in a way that its side [6-1] corresponds to the orientation of the [1-♭3] side in C. Then, the two sides serving as a pivot will be overlaid to form an axis. The pivot interval shared by both reflections visually expresses its role as a hinge.  (illustration 2.2, p.14)

Let us look at all the pivot intervals in rajazz 1, between both successive notes and non-successive notes. As we can see, the geometric representation of the rajazz visualises this dynamic. Each pivot interval points to one or more reflections in the mode from other tonics. It was this manner of proceeding that revealed the first link between rajazz and mathematics to me. (illustration 3, p.15-16)

Research between music and mathematics

When I was conducting this research into rajazz theory, I used several mathematical tools that I selected with the aim of serving a practical purpose for the composer: various tables based on combinatorics and set theory; 2D geometry and graphs, which help visualise the internal relationships within each rajazz; and finally, 3D geometry and the dodecahedron, which makes it possible to develop the symbolic axis of the system.

Geometry

A musical mandala

For each rajazz, we therefore have a well-defined set of pivot intervals and a set of correlated modulations. There are musical mandalas, with their respective reference sheets, for each of the 31 rajazz.

Considered in its entirety, a musical mandala visualises all of a rajazz’s potential modulations, starting with a reflection on a given tonic placed at the centre of the diagram.

If we want a diagram that is as universal as possible and which is valid for any reflection based on the twelve existing tonics, we may reformulate it beginning with a tonic X, which lets us visualise our mandala in the following manner:

There are two kinds of mandalas: open circuit and closed circuit. We talk about a “closed” circuit rajazz when we remain within one sole rajazz, limiting ourselves to moving between the reflections of a single colour through a limited number of pivot intervals. We talk about an “open” circuit when we mix one or more rajazz, thus multiplying the number of pivot intervals and potential modulations. In some cases, this form is identical for both circuits. (illustration 4, p.17-18)

The modulation wheels

Modulation wheels allow us to depict, in a single image, all the potential links between the 12 reflections within the same rajazz. The 12 points of the circle here represent the tonics of the 12 transpositions, or reflections, of the same pentatonic mode, of the same rajazz. Within a wheel, any line segment connecting two tonics indicates the potential for direct modulation between them via at least one common pivot interval. (illustration 5, p.19-20)

It is important to note, however, that the size of the pivot intervals is not represented in the modulation wheels. Furthermore, when the same modulation is accessible via several pivot intervals, this is not readable in the modulation wheel. It is therefore advisable to continue consulting the musical mandala musical in parallel.

There are 13 of these modulation wheels, and they correspond to all the possible scenarios encountered in the 31 rajazz. Some of these modulation wheels apply to more than one rajazz.

The dodecahedron–a 3D model of rajazz

The third step involves trying to represent a rajazz geometrically in 3D space. The search for a shape that could represent a pentatonic mode simultaneously in its twelve reflections naturally brought to my mind the dodecahedron, which is a polyhedron with twelve faces, each of which is equivalent to a pentagon. (illustration 6, p.19-20)

• Each of the twelve faces represents one of the twelve reflections from a different tonic.
• Each face, or reflection always consists of a pentagon, the vertices of which represent the five notes of a pentatonic mode.

The structure of the dodecahedron is therefore particularly illustrative of the rajazz system and its balance. There is, however, one difference between the geometric representation of the dodecahedron and that of the mandalas: in this 3D vision, the pentagon is always regular. In the musical mandalas, the sides of the pentagon are always proportional to the intervals between the successive notes of the mode, and each of the 31 rajazz is represented by an irregular pentagon of a different shape. In our dodecahedron, however, the sides of each pentagon are always identical for all 31 rajazz, which means that they represent the intervals between the successive notes of the mode symbolically, rather than proportionally.

A symbolic representation

Because the dodecahedron is identical for the 31 rajazz, its shape serves purely as an analogy; in other words, this geometric representation of the rajazz is merely symbolic in nature. It nevertheless remains valuable because envisioning the musical form symbolically changes our perspective on the role of the composer or improviser.

As stated by Ernst Cassirer, a philosopher who was passionate about the notion of the symbolic, the symbols themselves embody the very activity of the human mind.¹ And in our perception of the symbolic dimension, when it is expressed artistically, it touches a part of our soul that nourishes our overall perception of life and the world.

Combinatorics

In mathematics, combinatorial analysis is the set of techniques used to count, enumerate, and categorise certain finite structures.

This is why I have compiled tables answering very practical questions raised by the use of rajazz in various forms:

• Which pivot intervals, triads and tetratonics are found in which rajazz?
• How can rajazz material be combined with more classical harmonic contexts?
• How to combine several rajazz?

The research covers all the responses, essentially in the form of tables.

Why and how to use rajazz?

The perception of rajazz

Why use rajazz? For what purpose? With what musical objective and with the aim of eliciting what kind of sensation and reaction in the listener? This is a very concrete question. After all, these theoretical aspects are meaningful only if the experience of rajazz is distinctive enough to justify its use as an alternative, or at least as a complementary system for organising musical material.

Let us see what this experience is based on:

A system that maintains an intermediate equilibrium

The mechanics of rajazz strike a balance between the exploration of a single, specific colour, as in ragas, and the capacity for modulation, as in tonal music. We navigate somewhere in between.

Refreshing material

Some of the 155 pentatonic modes in the 31 rajazz sound familiar, others less so. Choosing a material different from what we are used to refreshes the ear and stimulates creativity.

Internal structure of the material

The transposition of a pentatonic mode into reflections from 12 tonics generates an invariant intervallic structure within each rajazz. This order, inherent in the musical material itself, is perceived by the listener, regardless of the melodic line, and generates a particular balance for each rajazz, as well as for each combination of rajazz.

The spirit of rajazz

Three factors inherited from the East and Middle East give rajazz its characteristic spirit: the exploration of the full breadth of the interiority-exteriority axis, the tendency to link a quality of intention to each melodic-harmonic colour, and the awareness of the unfolding of time.

A balance between freedom and constraints

This system generates a lovely balance between, on the one hand, great melodic freedom from very limited material and, on the other hand, a strong constraint on how to modulate, but supported by a vast choice of potential modulation paths. Finally, each individual musician can strike their own balance between composition and improvisation.

All these axes together contribute to the distinctive character of rajazz.

Flexibility

The right balance between constraints and freedoms has always seemed essential to me, both as an improviser and as a composer. Four parameters thus contribute to generating great flexibility and freedom in the use of rajazz.

• Flexibility in terms of work structure
• Stylistic flexibility
• A great deal of melodic freedom
• Flexible constraints in terms of the material

Forms of using rajazz

At a practical level, whether through composition or improvisation, rajazz are therefore meant to be highly flexible and can be explored in the following ways:

Rajazz a la raga

Each of the 155 pentatonic modes can be treated on its own, just as you would play an Indian raga whose scale corresponds to the pentatonic material of one of the rajazz. We choose a single tonic, a single pentatonic mode, and explore it without ever going beyond this framework, which amounts to developing a modal style of playing on a pentatonic. The mode’s central gravitational force is ideally expressed by a drone. The construction of the musical narrative can also draw inspiration from certain procedures in Indian classical music: mastering the gradual expansion of the texture, mastering the evolution of the density of the number of notes in the melodic development, and developing a melodic-rhythmic playing style that unfolds progressively.

Closed or open circuit

The rajazz algorithm lends itself mainly to these two modes of playing.

Closed circuit

To play in a closed circuit, the three rules developed at the beginning of this text must be combined: choose one of the 31 rajazz, restrict yourself to this single mode, and always switch from one reflection to another using potential modulations via a pivot interval.

Open circuit with two or more rajazz

To play in an open circuit, it is simply a matter of combining several rajazz and adapting the same rules. The number of existing pivot intervals and the number of potential modulations will increase in proportion to the number of rajazz combined.

It is possible to combine 2 rajazz, which already yields 465 combinations. It is also possible to combine more than two rajazz, in which case the possibilities increase exponentially:

3 rajazz = 4,495 combinations, 4 rajazz = 31,465 combinations, 5 rajazz = 169,911 combinations, and so on.

Because the number of possibilities is enormous, a simple online program developed by Gonzalo Romero-Garcia allows for the calculation of all pivot intervals and their correlated modulations for every imaginable combination of rajazz. Here is the link: www.rajazz.manuelhermia.com

Parallel cousin modes

This solution consists of using the 5 cousin modes cousins of the same rajazz, but transposing them from the same tonic.

The sub-colors

The 155 pentatonic modes that form the basis of rajazz already constitute, in themselves, a considerable basis of modal pentatonic language. If we were to disregard the third rule of rajazz, we would still have all this pentatonic material, usable in a modal context, in a tonal context, or in a hybrid context.

These 155 pentatonic modes can be used as “sub-colours” of heptatonic scales. By sub-colour I mean a mode with a set of notes in common, but fewer in number. [1 2 4 5 7] is therefore a sub-colour of [1 2 3 4 5 6 7]. For the sake of completeness, I have extended the field of the heptatonic modes of the melakarta (72 modes) to the extended melakarta (117 modes). I now propose a mapping of 189 heptatonic modes, providing a complete and precise reference for investigations on rajazz as sub-colours. The tables reproducing the 155 rajazz modes as sub-colours are then related to these 189 heptatonic modes.

These different ways of using rajazz constitute a toolbox from which the musician can draw at will. Each of these tools can be self-sufficient, combined, or simply added to one’s usual writing and/or improvisation techniques.

Research process

The first year consisted of conducting all the theory-related research and developing mathematical representations using the various tools mentioned above. This yielded a pdf document of some 250 pages.

At the end of the first year, two public presentations on the theory of rajazz were given in Brussels, Belgium, aimed primarily at composers from all generations and styles: the first was held at the Maison des Musiques, organised by the Forum de la Création Musicale, and the second, at the Royal Conservatory of Brussels, which was organised in-house. These two presentations were accompanied by a “call for scores” that invited any composer who wanted to compose a work based on rajazz in any of its possible forms. To date, a dozen composers have answered the call.

The second year of research consisted of composing myself a series of works based on rajazz for different configurations and in different styles. At the same time, I followed up with the composers who answered the call, and those who were interested were given a half-day in a studio to record their piece to document the research. I would like to give my heartfelt thanks to those who answered the call with this first wave of compositions: Andrei Beuthe, Damien Brassart, Cécile Broché, Clément Cerovecki, Maureen Kellendonk, Yann Lecollaire, Alain Pierre, Lydie Thonnard, Juliette Recasens, Mathieu Robert, Catherine Smet, and Gaspard de Trazegnies.