composition de 2ième année pour clarinette solo utilisant la topologie – sous la direction du Dr. michael young – a obtenu un First – (en anglais).

The usual process of composition is to go from a musical idea to a finished score. Here, I choose the reverse process, by going from a finished score to a musical idea. For this purpose, I used Topology[1], and considered the paper used in the conventional 5 lines stave graphical coding of music as a topological space of multiple dimensions, being Pitch, Rhythm, and Dynamics.

I then filled this space with objects, in this case 8th-notes, all of equal properties with respect to pitch (Bb the fundamental tone of my performer’s clarinet), duration, and dynamics, over a space of 168 bars, giving a duration of approximately 5’10’’ at 130Bpm. I therefore obtained a space being isometric in all directions, or, in ordinary language, the equivalent of a flat line.

Next, I set out to sculpt that line, by applying different and asynchronous transformations to the different dimensions of the space, in the following order:

*Pitch*: transformations of several of the pitches of a bar, in the range of -7/+7 diatonic steps, followed by a transformation in the range of -1/+1 chromatic step of one pitch per bar. This chromatic transformation is reflected in the notation I have used in the final score, such Cb or E#, instead of B or F.

*Rhythm*: deletion/punching out of several pitches per bar.

*Dynamics*: transformations of the dynamics of each bar in the discrete range [p, mf, f, ff], each transformation being valid for that bar only.

This is also equivalent to considering musical notes as signifiers linked by logical relationships, the transformations being applied to both signifiers and their intrinsic relationships

To determine on which bar, and where applicable, on which of the pitch(es) of the bar to apply the transformations, I used trigonometric functions, to which random noise was added, in order to attenuate their predictability and therefore “disturb” their periodicity[2]. All the calculations were performed using Wolfram Research’s Mathematica for Students Version 5.2[3]. Where appropriate, the input and/or output variables were renormalized to the corresponding musical range(s), since trigonometric functions require and produce values expressed in radian.

A final revision, based on subjective, and therefore non significant choices, was done on the resulting work.

The choice of the note parameters to be transformed was dictated by a wish for simplification, for both the composition and the performance stages. This avoided the use of unyielding equations as well as obtaining a complex final score, the parameters of which could end up being beyond my own grasp, as I am still finding my way around the use of this method. Also, by establishing an initial smooth and undemanding relationship with my performer, I am hoping to be gradually able to produce more complex works, and therefore a more demanding performance. I consider the relational aspect of this collaborative assignment as important as its creative side.

As a result, I consider Topos 1 for Bb Clarinet to offer a mixture of simplicity and exoticism. Simplicity as it remains mainly in the diatonic domain, presents a large degree of tonality, without being goal orientated. Exoticism, because of the discreet presence of non diatonic notes (unrelated, for the most part, to the C scale in which the piece was written), as well as a fracture of the rhythm which is not based on the normal accentuations of the 4/4 meter, thus superimposing several irregular pulses. In turn, this creates a tantalising hint of a recurring pattern, but which remains evanescent.

Upon the completion of Topos 1, further interesting possibilities have emerged:

– Each time the work is to be performed, a new parsing (with the same functions) could be applied to the initial isometric score, thus giving a slightly different result, each with its own flavour, due to the presence of the random noise parameter in the mathematical functions used.

-Each performance could be recorded, and then combined into a single file with its predecessors, with each performance positioned at different points of the stereo space.

-An initial isometric score could be divided into equal or unequal regions, each region having a different density of occurrence of the transformations. This would be particularly effective using the Lorentz transformation equations[4], as each of the resulting score’s regions could be said to travel at different fractions of the speed of light. (with respect to the both the composer and audience’s frame of reference). This I consider to be different from the notion of polyrhythm. To which extent, however, remains to be determined.

-The use of more complex transformations, such as hose determining the Klein bottle, the Moebius strip[5] or the Borromean knot.

-The application of the transformations on an initial polyphonic isometric score, whereby each part could be considered as individual dimension and made to fold upon one another.

Appendix: functions used for the transformations, and their graphical representations.

[1] Topology is the mathematical study of the properties that are preserved through deformation of objects. For my purposes as a composer, it is the deformation process which holds my interest, not whether invariance is preserved or not.

[2] Cf. Figures.

[3] Cf. Wolfram site <accessed February 2006>.

[4] A set of equations to calculate the mass, length and time of objects inside a frame of reference travelling at a certain fraction of the speed of light, with respect to the observer’s frame of reference.

[5] The Moebius strip in particular, is to my mind a more effective representation of the “circle of fifths” in the equal temperament scale. The demonstration is, however, beyond the scope of this assignment.

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