ï~~Proceedings of the International Computer Music Conference 2011, University of Huddersfield, UK, 31 July - 5 August 2011
SCHEDULING AND COMPOSING WITH RISSET ETERNAL
ACCELERANDO RHYTHMS
Dan Stowell
Centre for Digital Music, Queen Mary University of London, UK
{ firstname.lastname } @eecs.qmul.ac.uk
ABSTRACT
Jean-Claude Risset described an "eternal accelerando" illusion, related to Shepard tones, in which a rhythm can
be constructed to give the perception of continuous acceleration. The effect can in principle be derived from
any rhythmic template, producing patterns with aspects of
fractal self-similarity. Any attempt to use it compositionally must address the difficult issue of scheduling events
(notes, beats) and structural changes in a way that integrates with the multi-layered self-similar rhythmic structure. In this paper we develop an approach to scheduling
rhythms, melodies and structural changes over a Risset
accelerando (or decelerando) framework. We derive the
mappings which allow note sequences and sample playback to be incorporated. We discuss some compositional
choices available within this framework, and demonstrate
them via audio examples.
1. INTRODUCTION
The well-known "Shepard tones" audio illusion creates a
set of musical notes in which the pitch relation between
the notes is perceived as cyclical rather than linear, disturbing our usual sense of high vs. low pitch [4]. It achieves
this by arranging the partial amplitudes within the notes
so that, as we go up the scale, higher partials fade out and
lower partials fade in, in such a way that after ascending
one octave we have arrived at a set of partials the same as
we had when playing the octave below. Jean-Claude Risset created a similar effect with a continuous upward glissando rather than individual notes [2]. He also described
the rhythmic equivalent of this, in which a beat seems
to accelerate continuously (although he credited Kenneth
Knowlton with being the first to synthesise the effect, in
1974) [3].
The eternal accelerando is created in the same way as
the eternal glissando: versions of a sound at different octaves are amplitude-weighted and combined, though using
tempo octaves rather than pitch octaves. Pitch circularity
has been used in a variety of compositions, both with and
without the formal mapping explored by Shepard and Risset [1]. To our knowledge, Risset rhythms (as we will call
them, encompassing both accelerando and decelerando)
have been very little used in musical works or perceptual
Supported by EPSRC grant EP/I001832/1
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o
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> ~ 0 -- - - - --
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-2 --3
Time Power
Figure 1. Diagrammatic representation of a Risset glissando ascending one octave. The diagonal lines represent
individual partials, each of whose power is frequencydependent as indicated by the bell curve on the right.
experiments. The exploitation of Risset rhythms may be
held back by the complexities of managing accelerating
fractal rhythms within compositional and sound-design
environments, and combining them with non-fractal sequences. This paper aims to facilitate the use of Risset
rhythms in composition by deriving mathematical expressions used to map ordinary rhythms into repeatable Risset rhythms, and exploring some of the consequences of
working in a framework underpinned by Risset rhythmic
structure.
We first describe the construction of a Risset rhythm,
and develop mathematically our approach which leads to
convenient scheduling of such patterns, giving examples
of the rhythms thus created. We then address compositional factors at both the micro and macro level, such
as the use of pitch within Risset rhythms and scheduling
transitions.
2. CONSTRUCTING RISSET RHYTHMS
Figure 1 outlines the elements used to synthesise a Risset
glissando sound. The glissando is composed of sinusoidal
partials, with their time-varying power directly mapped
from their frequency. In the diagram the frequencies are
shown relative to some centre frequency f, and the power
is mapped to a cosine envelope centred on fc and having
a range of 6 octaves. The octave range is a free parame
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