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Strange Attractors: A Commentary on Applications

of Indeterminacy in my Recent Music

Scott Graeme Mc Laughlin

A portfolio of compositions and commentary submitted to

the University of Huddersfield in partial fulfilment of the

requirements for the degree of Doctor of Philosophy

December 2008

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Table of Contents

Acknowledgements.................................................................................................5

Copyright Credits...................................................................................................5

Abstract................................................................................................................6

List of Works Submitted..........................................................................................7

1: Chapter 1 - Context...................................................................................................9

1.1: Introduction....................................................................................................9

1.2: Compositional Language.................................................................................12

1.2.1: Sound, Ambiguity and Gestalt.................................................................12

1.2.2: Chaos and Strange Attractors.................................................................16

1.2.3: Sonification, and not Sonification.............................................................18

1.2.4: Feedback..............................................................................................21

1.2.5: Evolution..............................................................................................23

1.2.6: Indeterminacy......................................................................................27

1.2.7: Vertical Indeterminacies.........................................................................28

1.2.7.1: Inharmonic Sound and Strange Attractors..........................................29

1.2.7.2: Frequency Modulation......................................................................33

1.2.7.3: Spectral modelling...........................................................................34

1.2.7.4: Detuning and Notational Imprecision.................................................36

1.2.7.5: Vertical Text Processes.....................................................................38

1.2.8: Horizontal Indeterminacies.....................................................................39

1.2.8.1: Notational Imprecision.....................................................................40

1.2.8.2: Bounded Improvisation....................................................................40

1.2.8.3: Horizontal Text Processes.................................................................41

1.2.9: Teleology..............................................................................................42

1.2.10: Modes of Delivery................................................................................43

1.3: Musical Contextualisation................................................................................45

1.3.1: Experimentalism...................................................................................45

1.3.1.1: Beats and clusters...........................................................................49

1.3.1.2: Feldman and Memory.......................................................................55

1.3.2: Spectralism..........................................................................................57

2: Chapter 2 - Works...................................................................................................61

2.1: Commentaries...............................................................................................61

2.1.1: Tracheids (2006)...................................................................................61

2.1.2: The Lady with the Hammer (2006)..........................................................62

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2.1.3: Primes (2006).......................................................................................66

2.1.4: Filament (2006)....................................................................................69

2.1.5: Inner Shadow (2006).............................................................................71

2.1.6: Distemper (2006)..................................................................................73

2.1.7: Five Bells for Elliott Carter (2006)............................................................74

Five Bells for Elliott Carter – Orch. (2008)..........................................................75

2.1.8: Whitewater (2006)................................................................................79

2.1.8.1: Cellular Automata............................................................................79

2.1.8.2: Bounded Improvisation and Performance ...........................................84

2.1.9: For Garrett Sholdice (2007)....................................................................86

2.1.10: Bifurcations (2007) .............................................................................89

2.1.11: Torsion (2007)....................................................................................92

2.1.12: Nano (2007) ......................................................................................93

2.1.13: Intra- (2007)......................................................................................94

2.1.14: Whitewater II (2008)...........................................................................98

2.1.15: The Well-Tempered Prism (2008).........................................................101

2.1.16: Lorenz (2008)...................................................................................103

2.1.17: Marx (2008)......................................................................................106

2.2: Conclusions.................................................................................................109

2.2.1: The Supporting Diagonal .....................................................................109

2.2.2: The Inbetween-ness Paradox................................................................110

2.2.3: The future..........................................................................................112

Appendix 1: Javascript code for spectralConway.....................................................113

Appendix 2: Scores written during PhD study but not included in portfolio.................118

Appendix 3: Scores of Pieces written before PhD....................................................119

Sources Consulted: Bibliography...........................................................................120

Webography.......................................................................................................125

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Illustration Index

Illustration 1: The Necker Cube......................................................................................12

Illustration 2: Five Bells for Elliott Carter, gestalt principles................................................15

Illustration 3: The Koch Snowflake..................................................................................18

Illustration 4: Koch Snowflake Bounded...........................................................................18

Illustration 5: Symmetrically coupled non-linear system used by Rolf Wallin........................20

Illustration 6: The Mandelbrot Set..................................................................................22

Illustration 7: Recursive detail in the Mandelbrot Set – 'Seahorse Valley'.............................22

Illustration 8: Sound objects and their methods of evolution..............................................26

Illustration 9: spectral analysis of saxophone multiphonic..................................................30

Illustration 10: G. Douglas Barrett - Surfaces I-IV............................................................35

Illustration 11: notation of microtonal accidentals.............................................................38

Illustration 12: Lutoslawski indeterminate fingerings in violin solo part................................54

Illustration 13: The Lady with the Hammer score page 3...................................................64

Illustration 14: The Lady with the Hammer close harmony.................................................66

Illustration 15: Prime-numbered partials from fundamental G? = 1.62hz............................68

Illustration 16: Filament harmony...................................................................................70

Illustration 17: Distemper, section D. Upper stave is live part and lower shows computer part

(playback of recorded live sections, hairpins indicate reversed sounds)...............................74

Illustration 18: transformations of Carter's chord in Five Bells for Elliott Carter.....................76

Illustration 19: three successive generations (l-r) of Conway's Game of Life.........................80

Illustration 20: For Garrett Sholdice, ambiguity created through subtle variation..................88

Illustration 21: Bifurcations (mm.14-20), detail of linear descents and ascent......................90

Illustration 22: Bifurcations (mm.85-end), multiphonic clustering.......................................90

Illustration 23: Bifurcations section-I harmonic motion......................................................91

Illustration 24: Bifurcations section-II harmonic motion.....................................................91

Illustration 25: a generation from Dawkins' Biomorph application.......................................95

Illustration 26: Intra- (mm.59-61) showing guitar and violin pitch interaction......................97

Illustration 27: Whitewater II pitch-set partitioned by instrument range............................100

Illustration 28: The Well-Tempered Prism, opening bars...................................................103

Illustration 29: Lorenz attractor as a 2D shape being outlined by 1D points........................105

Illustration 30: Clusters and sidebands in frequency modulation.......................................107

Illustration 31: AfterImages, spectral Schubert melody....................................................107

Illustration 32: Marx: synthesiser notation (staves 3 & 4)................................................109

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Acknowledgements

I would like to thank Dr. Pierre Alexandre Tremblay and Dr. Bryn Harrison for their

supervision of my work and their exceptional support in bringing it to fruition. I would also

like to thank especially Dr. James Saunders and Prof. Christopher Fox for their supervision

and inspiration early on in my studies, and the following people for their advice and

support which has all been invaluable over the past three years: Simon Anderson, Aaron

Cassidy, Michael Clarke, Lisa Colton, The Contemporary Music Centre (Dublin), Liz Dobson,

Samuel Freeman, Richard Glover, Iain Harrison, the HCMF, Scott Hewitt, Ben Isaacs, Joe

Kudirka, Alvin Lucier, John McLachlan, Eleri Pound, Heather Roche, Rebecca Saunders,

Benedict Schlepper Connelly, Garrett Sholdice, Mic Spencer, Philip Thomas, Nick Williams,

Stewart Worthy, the YCC.

Special thanks go to Jessica and our families for their love and support.

Copyright Credits

MathWorld--A Wolfram Web Resource <http://mathworld.wolfram.com>. Images of

strange attractors, Mandelbrot set, Koch curve. Used by permission.

Wikicommons copyright free images. <http://en.wikipedia.org/>. Image of necker cube.

G. Douglas Barrett, Surfaces I-IV score extract used with permission of the composer.

Witold Lutoslawki, Chain 2 score extract used with permission of the publishers.

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Abstract

This commentary reflects on how indeterminacy has been used in the music I have written

over the period of my doctoral studies, 2005-2008. Non-musical ideas play a major role in

my compositional language and this is reflected in the the use of 'strange attractors' as a

metaphor for the philosophical and aesthetic stance behind composing with indeterminacy.

Chapter 1 introduces aspects of my compositional language and the context for my music.

Short sections introduce my readings of important concepts such as inharmonic sound or

'strange attractors' and explain how they inform the music. Specific compositional

techniques are then discussed both in terms of how they are applied in my compositional

language and their original contexts.

The discussion of indeterminacy is divided into the vertical and the horizontal. Vertical

indeterminacy focuses on applications of indeterminacy to pitch organisation techniques

such as spectral modelling and frequency modulation are examined as part of a

frequency-based harmonic continuum. Different methods of generating ambiguous pitch

percepts that sit at the boundaries of the harmony/timbre duality are considered in pieces

with text processes. Horizontal indeterminacy includes the effects of using time-space

notation to remove some control at the note-level. This leads to an examination of

techniques for indeterminacy from larger temporal units through bounded-improvisation

and text processes.

Musical contextualisation is provided by looking in particular at the American experimental

school of composers such as Alvin Lucier.

Chapter 2 is the commentaries on individual pieces. These commentaries will discuss the

composition of the work, with particular reference to the technical and aesthetic concepts

that have been discussed in chapter 1. As not all of the works discussed can be considered

as successful, these commentaries will also address issues of language and problems that

have arisen through performance of the works.

The conclusion of the commentary briefly examines the points of crossover between the

horizontal and the vertical aspects and the importance of developing an approach in which

they co-exist.

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List of Works Submitted

Tracheids (2006)

- flute, clarinet, bass guitar, percussion: 3 mins

The Lady with the Hammer (2006)

- 2 tuba, 2 trombone, 2 euphonium, 2 piano: 12 mins

Primes (2006)

- for high string instrument and Max/MSP: 10+ mins

Filament (2006)

- guitar quartet: 1 min

Inner Shadow (2006)

- violin, cello, bass clarinet, accordion, acoustic metronome:

Distemper (2006)

- any equal-tempered plucked/struck string instrument and Max/MSP: 10+ mins

Five Bells for Elliott Carter (2006)

- string quartet: 8mins

Whitewater (2006)

- wind instrument(s) and Max/MSP: 10+ mins

For Garrett Sholdice (2007)

- violin, oboe, piano: 6 mins

Bifurcations (2007)

- clarinet quartet: 8 mins

Nano (2007)

- clarinet: 15+ mins

Torsion (2007)

- eighth-tone trumpet: 4 mins

Intra- (2007)

- violin, prepared guitar: 13 mins

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Whitewater II (2008)

- 8 or more saxophones: 8-10 mins

The Well-Tempered Prism (2008)

- clarinet, piano: 6 mins

Lorenz (2008)

- saxophone, contrabass, cello, viola, perc: 10 mins

Marx (2008)

- mezzo soprano, viola d'amore and synthesiser: 13 mins

Five Bells for Elliott Carter – Orch. (2008)

- string orchestra: 10mins

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1: Chapter 1 - Context

1.1: Introduction

This commentary relates to the works written between 2005 and 2008 (submitted in the

accompanying portfolio) and focuses specifically on the varied techniques of indeterminacy

used in their composition. The strange attractor is used as a metaphor for my musical

application of indeterminacy. Aspects of chaos theory are explored in their role as

influences on my way of thinking about both harmonic and temporal relationships, and

ultimately the development of my compositional language. The commentary is divided into

two main chapters. Chapter 1 examines musical language and contextualisation, while

Chapter 2 consists of the commentaries on individual works and an overall conclusion.

In Chapter 1, several facets of my compositional language are examined with respect to

technical details, and to musical and extra-musical context. Section 1.2 takes specific

terms and ideas and attempt to define generally how my music is informed by them: in

some cases this includes examining examples from the portfolio and musical

contextualisation. The basic principles of my music as organised sound are covered in

1.2.1: Sound, Ambiguity and Gestalt. In 1.2.2: Chaos and Strange Attractors, I will briefly

cover my reading of chaos theory and relate it to musical indeterminacy: this will draw

mainly on two works of popular science, Chaos: Making a New Science by James Gleick,

and Complexity: The Emerging Science at the Edge of Order and Chaos by M. Mitchell

Waldrop. 1.2.3: Sonification, and not Sonification briefly discusses the idea of sonification

and examines the work of various composers who have taken inspiration directly from

scientific ideas. 1.2.4: Feedback discusses the concept of feedback in a dynamic system

and applies it to musical indeterminacy, while 1.2.5: Evolution examines the formal and

temporal aspects of my work and argues that in much of my musical language it is more

accurate to view the works as an idea's evolution rather than its variation.

Sections 1.2.6-1.2.8 discuss the more technical aspects of indeterminacy in my language.

I have split this into two specific areas which I term vertical indeterminacy and horizontal

indeterminacy. The vertical deals with the application of indeterminacy to aspects of

harmony and pitch organisation, and the horizontal to temporal aspects such as duration,

pitch-motion and form. The reason for this separation is twofold: the techniques used by

the two different aspects are sufficiently different to demand separate examination, and

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both are based in different musical traditions. The vertical aspects are developed from

techniques of pitch organisation and generation derived largely from studying the Spectral

school of composition focusing on the works of Gérard Grisey and Tristan Murail.

Horizontal indeterminacy owes more to the American experimental school as exemplified

by Alvin Lucier, James Tenney, Morton Feldman and others. This distinction is of course

slightly artificial and areas of crossover will quickly become clear to the reader, a point

which will be summed up in 2.2: Conclusions by looking at 'inbetween-ness'.

Section 1.3 is a more involved look at specific musical traditions and their relationship to

my music and development as a composer. 1.3.1: Experimentalism discusses my relation

to the American experimental school, and focuses on comparing Cagean indeterminacy

with that of Alvin Lucier with respect to soundworlds and intention. Aspects of the

European approach to indeterminacy are discussed with particular focus on the work of

György Ligeti and Witold Lutoslawski. 1.3.2: Spectralism looks at the influence of the

Karlheinz Stockhausen, Gérard Grisey and Tristan Murail in particular. This section expands

on the technical aspects of spectral music that is covered in 1.2.7: Vertical

Indeterminacies by briefly exploring my relationship with the spectral approach to time

and form.

Chapter 2 is the commentaries on all the works that I have composed during my PhD

studies. This portfolio of works is the body of the PhD, the works themselves are my

research and these commentaries are in place to discuss aspects of their composition and

context. Five of these pieces are text scores and thus have a strong element of

indeterminacy, but also some works which are presented here as fixed scores. The

indeterminacy in the fixed pieces is not always immediately clear to the score reader, but

equally, I would argue that in a blind test it would not be immediately clear to a listener

how many of the pieces were text scores. The indeterminacy is not always something

which arises from performance, it is an inherent quality built into the basic concepts of my

compositional language. Elements such as inharmonic material and mistunings lead to

acoustical beating patterns perceivable either as discrete pitches or as fused sound

complexes: the musical equivalent of the multistability of the Necker Cube (see illustration

1 below; p.12).

1

For instance, sustained musical textures can attempt to freeze time, or

given enough room would allow a listener to take in a harmony before it is replaced by the

flow of the piece's evolution. A fixed score may specify what players do on each step of

the metric grid, but that is only a scaffolding for the music. The qualities of the piece come

1

The Necker Cube is a well-known optical illusion. The lack of depth perception in the image allows the brain to see it in

two possible aspects, but not simultaneously; a phenomenon referred to as multistable perception. See Kruse, P &

Stalder, M (eds.), Ambiguity in Mind and Nature: Multistable Cognitive Phenomena, (New York, 1995).

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from an indeterminacy in the nature of sound itself which can be harnessed to music by

different types of notation: for example, it has long been accepted in digital sound

synthesis that an element of randomness in the form of LFOs and noise makes a sound

more 'real'. Music, for me, is an organisation of sound that has in past models focused on

'beautiful sound' rather than simply 'sound'. The civilising aspect of music often aims to

smooth out the sonic bumps which nature insists on and takes us a little further from

nature, indeterminacy allows some of nature's rough unpredictability back in.

Correspondingly, Alvin Lucier's oeuvre sets itself against the normalising tendency in

music that dismisses so many interesting sound phenomena as undesirable:

It's the idea of control and sameness, like balance in a chord; you want it to be the

same as the other chords. Balance is only one idea. There's imbalance too.

2

To extend Lucier's point, control is only one idea, it can be more interesting to move the

control back a few steps and let some chaos in. The results may look messy and

incoherent to one who is concerned with controlling every last detail, but messiness does

not have to be 'error' in the same way that detail is not an absolute necessity in creating

coherence; sometimes more sense can be made of the global when the local detail is in

flux. Making a similar point in the context of science, James Gleick describes Benoit

Mandelbrot's seminal work on fractals thus:

Mandelbrot's work made a claim about the world, and the claim was that the odd

shapes carry meaning. The pits and tangles are more than blemishes distorting the

classic shapes of [classical] Euclidean geometry. They are often the keys to the

essence of the thing.

3

I use indeterminacy as a compositional technique, but also as a political and philosophical

stance. Lack of determinism does not have to mean absence of control, it is more about

what one chooses to be free and what one needs to control in order to make the whole

work. The accompanying portfolio of work aims to express this ideal.

2

Lucier, Alvin, (transcribed by Anne Guthrie.), Ostrava Days 2005 Report, (Ostrava, 2006), 120.

3

Gleick, James, Chaos: Making a New Science, (London, 1987), 94.

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1.2: Compositional Language

1.2.1: Sound, Ambiguity and Gestalt

My music is composed around a dialectic between clarity and ambiguity in sound. The

processes that I use in both the vertical and horizontal aim to either create ambiguity

through indeterminate or determinate means, or to use clarity to contain, negate or

refocus that ambiguity. Generally, the sound or sound object is the material and the

music's form is created or arises from the material's proliferation and the ambiguity

between its repetition, transformation and negation.

The material that I work with is often a sound that is static

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or linear but has an

independent inner motion, such as a sustained multiphonic or split-tone, hovering

ambiguously between chord and timbre. The archetypal sound for me is a bell, a sound

that is made of a multitude of oscillating pitches—some interacting and some independent

—all decaying at their own rate but still perceived as a single sound—in gestalt perceptual

theory this is the principle of common fate, sounds that start together and continue along

the same vector are perceived as one (see below for the Gestalt principles of grouping).

The ambiguity in the vertical component of my music involves harmony that may be

perceived as a chord or as a timbre, a multistable harmony. Illustration 1 above shows the

Necker cube, a 2D drawing that demonstrates the phenomenon of multistable perception,

where an ambiguous figure can be interpreted in more than one way by the mind but not

simultaneously. In my music, inharmonic sounds—such as those of bells, where the

partials do not conform to the harmonic series—can be considered as analogous to the

4

'Static' meaning 'motionless', not meaning 'static electricity' or its sound.

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Illustration 1: The Necker Cube.

Necker cube because the pitch of the sound as perceived by a listener can be changed if

they refocus their listening from that pitch and settle on another of the bell's pitches:

most inharmonic sounds have several stable pitch centres with varying degrees of relation

to each other.

This ambiguity between chord and timbre can be composed in acoustic music to a certain

extent but also relies on environmental variables such as acoustics and the resonant

characteristics of the individual instruments so it cannot always be relied on for specific

effect. But I do not aim to create music where ambiguity is explored as a phenomenon for

its own sake in the manner of Alvin Lucier (this will be explained in more detail in 1.3.1:

Experimentalism, p.45), I prefer to think of this ambiguity as being always present as an

attribute of the musical language.

Horizontal indeterminacy creates ambiguity as to whether it is the music or the listener

that is changing. The music hovers between repetition and variation. The technique of

evolving sound objects in my music relies on this perceptual ambiguity. In Nano for

example, the listener will rapidly lose the ability to hear the piece as a whole form

unfolding and will only be able to make sense of formal implications from the most recent

set of events: the greater the level of difference between events the more likely that the

event can be linked to a previous similar event. But by corollary, the more subtle the

differences, the greater the perceptual ambiguity and so the more the form becomes

about moment to moment difference. Time is expanded as information is reduced: the

more information (contrast between moments) that there is, then the easier it is to make

long range temporal connections at the macro-formal level. Intra-, in its first section,

makes use of a perceptual ambiguity in a similar way. By limiting the material to three

archetypes and crossbreeding them, it becomes unclear whether elements are being

altered or literally repeated. This generates formal ambiguity that allows a play of tension

and release.

For ambiguity to play such a part in my formal thinking I need a framework for

understanding the perception of sound. Apart from a general understanding of the

properties of sound and the auditory system, I take great inspiration as a composer from

the gestalt principles of grouping, especially with regards to the blending of sounds and

timbral fusion—many individual sounds fusing to form a single timbre or percept—as this

occurs in practically all of my works.

The gestalt principles of grouping were developed by psychologist Max Wertheimer in the

early twentieth century. They are outlined here by psychoacoustician Roger Shepherd:

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•

Proximity: Things that are located close together are likely to be grouped together.

•

Similarity: When objects are equally spaced, the ones that appear similar tend to

be grouped as being related. If objects are similar in shape they are most probably

related.

•

Symmetry: Because random unrelated objects in the world are not expected to

exhibit symmetry, it would be most improbable for unrelated objects to exhibit

symmetric relationships.

•

Good Continuation: If objects are colinear, or arranged in such a way that it

appears likely that they continue each other, they tend to be grouped perceptually.

•

Common Fate: Objects that move together are likely to be connected. This is the

strongest principle in Gestalt theory.

5

In my music these principles may be applied to different perceptual attributes from piece

to piece, depending on how the elements need to be grouped. For example, the similarity

principle is applied in Five Bells for Elliott Carter where different spectra within the same

chord are played with different timbres (sul ponticello or sul tasto). The principle of

common fate is used in the same context by applying the same dynamic swell to notes

from the same spectra; in illustration 2 below there are three different spectra within the

chord and each has its own dynamic swell position.

5

Shepard, Roger, 'Cognitive Psychology and Music', Music, Cognition and Computerised Sound: An Introduction to

Psychoacoustics ed. Cook, Perry R. (London, 1999), 32-33.

14

15

Illustration 2: Five Bells for Elliott Carter, gestalt principles.

1.2.2: Chaos and Strange Attractors

The basic principle of chaos theory is that certain dynamic systems exhibit 'sensitivity to

initial conditions'; that imperceptibly small changes in the starting condition of a process

may disproportionally affect its future states. My application of chaos theory to music is

largely metaphorical, by applying the principles of chaotic dynamics to music rather than

engaging in the literal sonification of a chaotic system: Bifurcations comes close to being a

sonification but only in the very general sense that its structure is based on a bifurcation

diagram.

The concept of strange attractors comes from dynamical systems theory; a branch of

chaos theory which looks at the mechanisms of how complex systems evolve and change.

This aspect of science focuses on process rather than state. The ideas are applied to fields

as diverse as condensed matter physics, fluid mechanics, economics and sociology.

Strange attractors are a particular class in the general field of attractors, defined by the

American Heritage Science Dictionary as:

A set of states of a dynamic physical system toward which that system tends to

evolve, regardless of the starting conditions of the system.

6

There are three main types of attractor: point attractors, periodic attractors and strange

attractors. The simplest attractor is a point attractor, where the system tends towards a

single state. An example of a point attractor would be a marble rolling in a bowl as it will

always end up in the centre of the bowl (at the bowl's flattest point). Periodic attractors

settle into a state where any point tends towards one of multiple possible solutions, or it

may oscillate forever between them. A strange attractor has an infinite amount of possible

stable states which all occupy a finite space; meaning that a point may never settle into a

predictable state or intersect its own path. The difference between periodic attractors and

strange attractors can be summed up by comparing the solar orbits of planet. A periodic

attractor relates to Newtonian physics, where the planets orbit the Sun like a complex

clock and ultimately return to a predictable starting point, while the strange attractor is

closer to the real orbit of the planets as the complexity of the competing gravities

constantly perturbs their paths and ensures they are never in the same place twice.

7

Music, under certain conditions, can be seen as a dynamic system, especially where there

6

No author, 'attractor', The American Heritage® Science Dictionary, Houghton Mifflin Company, (n.p., n.d.),

<http://dictionary.reference.com/browse/attractor>, accessed 04/08/08.

7

Ibid.

16

is a feedback system that allows the real-time state of the performance to affect its future

states. For this to happen the music needs to have an element of freedom and a method

of interaction between parts, it is this concept of the musical work as being a collection of

dynamically interacting parts that I find interesting. In this way the music can be seen

holistically as form that arises out of the interacting parts and is greater than their sum.

Strange attractors are a stimulating metaphor for the idea behind my use of

indeterminacy for several reasons. They exhibit infinite variety within a fixed space, as will

be demonstrated further on in this section by looking at the Koch curve. This is extended

by analogy with music to the infinite possibilities for different pitches within an interval in

frequency-based pitch space. Strange attractors move between levels of chaos, order and

periodicity which is essential for music as a structure whose form and morphology is

based on difference across time. This unpredictable mixture of pattern and spontaneous

change imbues attractors with a lifelike aspect which is the basis of much of their

fascination for me. These co-principles of evolution and constant change within bounded

parameters are the basis for the unpredictable conjunctions arising from indeterminate

durations in Lorenz, Whitewater II, and the pieces which use what I call 'bounded

improvisation', Primes and Whitewater.

Lastly, strange attractors are often powered by feedback loops which contribute their non-

linearity. From my point of view, most music can not be considered as a dynamic system,

it is a programme which begins and ends when the instructions—usually, the score—end.

However, the inclusion of a feedback loop whereby, in performance, some element of the

music can effect another element leads to structures that evolve and change. In pieces

such as Whitewater and Primes, feedback loops are used in conjunction with bounded

improvisation in order to create pieces that are designed to self-organise from the bottom-

up—meaning that I compose small components and the form emerges from their

interactions rather than being imposed by a pre-composed (top-down) structure.

The Koch curve is a good example of feedback and of infinity within a boundary. The Koch

curve is constructed by the repeated application of a simple transformation as follows:

take an equilateral triangle, divide the sides into thirds, in the middle third attach a new

triangle which is the same shape but a third the size: the feedback in this system is that

each transformation acts upon the result of the previous transformation, thus the system

exhibits hysteresis; dependence upon the history of its inputs. This first step in the

transformation process transforms the triangle into a six-sided shaped, the Star of David.

Repeating this process on the new triangles and then on their new triangles and so on,

17

and the boundary of the shape very quickly becomes complex as shown in illustration 3.

Also known as the 'Koch Snowflake', this is an early example of the fractal shapes which

can be found throughout nature.

8

The interesting part of this from a musical point of view—as will be shown below—is that

the total area of the shape remains finite, because the infinities in the curve fold in upon

themselves; this can be shown simply by drawing a circle around the shape as it changes,

as shown in illustration 4. This idea of a bounded set having infinite possibilities is

extended in to the idea of bounded improvisation: see 1.2.8.2: Bounded Improvisation

(p.40).

1.2.3: Sonification, and not Sonification

Start all over again with listening and understand what happens without any

knowledge of what you have read or heard before. Of course, if you come with

some well-defined rules and you compare them with what you hear, you will be lost

because the rules don’t exist a priori. They should not be a priori, they should be

born out of what you hear, otherwise you’re repeating, you’re making an imitation

of something that you have as a memory.

- Iannis Xenakis, interviewed by Morton Feldman

9

8

Gleick, op cit., 99.

9

Xenakis, Iannis, Feldman, Morton, 'Interview', <http://grace.evergreen.edu/arunc/texts/music/xenakisFeldman.pdf>,

accessed 05/08/08.

18

Illustration 4: Koch Snowflake Bounded

Although ideas taken from science inform much of my music, these pieces are in no way

an attempt to sonify these ideas, any more so than they are a narrative or mimetic

representation of the ideas. The idea is interesting because of its structural properties and

in how those structural properties could be applied in a musical context.

Michael Winter, a composer and student of James Tenney's, defines sonification as follows:

The parametric profiles of the changing characteristics of sound can be expressed

as mathematical functions of time. By using physical scenarios and mathematical

functions to determine these parameters, one can distance oneself from aesthetic

opinions and attempt to relinquish intuition. This provides another way to create a

new music.

10

Outside the domain of electroacoustic music, Iannis Xenakis is the composer most

associated with the mapping of scientific and mathematical concepts onto musical

parameters. His biographer Nouritza Matossian has this to say about Xenakis' sonification

of such ideas:

Xenakis never claimed that a rigorous mathematical basis is sufficient to produce a

well-formed piece of music. Those who are partially informed about the

mathematical theory expect the music to be a mirror of mathematical processes

and equations. Pithoprakta is no more a translation of probability theory than an

artichoke or a celery is a translation [Matossian's italics] of the Fibonacci series, or

a flowing river is a translation of random functions. [...] Even though underlying

structures are shared, particularity of musical resources ensures uniqueness in each

one.

11

Many composers have used fractals in composition and have focused on the aspect of self-

similarity because it maps easily to the compositional atom of the cell or motif. Ligeti

claimed influence from fractals in the Piano Concerto (1988)

12

some of his Piano Etudes

from the 1980s, such as 'Vertige' from Book Two

13

and 'Autumn in Warsaw' from Book

One,

14

in relation to the development of rhythm and pitch cells across the piece.

15

He takes

primarily poetic interest in fractals by applying the 'idea' to his own compositional

language. Ligeti's inspiration is a version of an ancient theme in Western Art, that of

'unity', a deeper order behind a surface seemingly chaotic.

16

This feeling is echoed by the Danish composer Per Nørgård: 'Chaos is fascinating because

you have an order behind chaos and chaos behind order. That fits very well with my

10

Winter, Michael, On Sonification <http://www.mat.ucsb.edu/mwinter/writings.html> (n.p, 2005), (accessed 04/01/07).

11

Matossian, Nouritza, Xenakis (London, 1990), 106.

12

Ligeti, György, Concerto for Piano and Orchestra, (Mainz, 1988).

13

Ligeti, György, Piano Etudes: Book II, (London, 1994).

14

Ligeti, György, Piano Etudes, (London, 1986).

15

Steinitz, Richard, 'Music, Maths & Chaos', The Musical Times, 137/1837 (Mar., 1996), 17.

16

Steinitz, Richard, 'The Dynamics of Disorder', The Musical Times, 137/1839 (May, 1996), 8.

19

feeling of life too'.

17

Nørgård devised what he termed an 'infinity series' based on fractals

for melodic variation, which he describes thus: 'it's always developing, it's never repeated.

It's just one row working out on many levels.' By this of course he doesn't mean a serial

row, although there's surely some inspiration, rather a row of numbers which is put

through a fractal-based function to produce a constantly changing series.

18

Norwegian composer Rolf Wallin takes a more literal-scientific approach by using the

fractal's equations to create stochastic maps in pre-composition. In a 1989 lecture given

at the Nordic Symposium for Computer Assisted Composition he outlined the contribution

of fractals to Onda di Ghiaccio, the piece he was writing at the time and his first to use

fractals. The piece progresses as a series of 'fields' which use two equations, one for

macro-form and one for micro. The stochastic maps generated by the fractals control for

each field parameters such as pitch center, pitch range, field duration, 'rhythmic influence'

and others. The equations he used are for a 3D 'symmetrically coupled nonlinear system'

in which three parameters mutually feed back into each other to create the structure

shown in illustration 5 below.

19

Wallin does not go into details about how this particular

structure was applied to the piece, but it is visually clear that the alternating chaotic and

ordered areas could have interesting musical applications

Wallin follows on from Xenakis in using stochastic procedures to generate and proliferate

material. Like Xenakis, Wallin is using the cutting edge of scientific ideas with which to

create his music; where Xenakis had the IBM 7090 computer running stochastic

calculation and basic cellular automata, Wallin has chaos theory and non-linear systems

modelling. But like Xenakis, Wallin uses indeterminacy only as part of stochastic

17

Anderson, Martin, 'The Many Patterns of Per Nørgård', Tempo, New Series, No. 202 (Oct., 1997), 6.

18

Ibid., 5.

19

Wallin, Rolf, 'Fractal Music: Red Herring or Promised Land', (n.p., 1989), <http://www.rolfwallin.org/Fractalarticle.html>,

accessed 04/08/08.

20

Illustration 5: Symmetrically coupled non-linear system used by Rolf Wallin.

calculations and doesn't allow it into the live music. Xenakis considered improvisation—

presumably only in scored music—and aleatoricism to be 'an abuse of language and an

abrogation of the composer's function', although he had more respect for chance when it

applied only to pre-formed musical sections as in Boulez' 3rd Piano Sonata.

20

He was also

more forgiving of chance procedures when they were part of an overall calculation such as

the game strategies used in his own pieces Duel

21

(1959) and Stratégie

22

(1962). Equally,

the stochastic and probability-based composition that was his major preoccupation

contained many elements which were by definition random, but acceptable in his language

because the randomness was itself 'controlled' by the composer.

In my music, the mix of order and disorder which so fascinates composers is extended to

the live domain by introducing it into open works and text processes. I cannot argue that

my approach is any more a 'real' application of chaos theory to music than such

approaches as Ligeti and Wallin's, it is simply the approach which resonates most strongly

for me. Further connections with sonification will be examined in the upcoming section on

spectral modelling (1.2.7.4 - p.34).

I have taken great inspiration from the approach of Ligeti and Xenakis to chaos theory and

their attempts to bridge C.P. Snow's Art-Science gap

23

(whether either composer knew

explicitly about Snow is not known). Chaos theory and music connect for these composers

at the level where they perceive a connection between the mechanisms of their musical

language—or their understanding of musical function—and the mechanisms of chaos

theory. The connection for me is the same but within a musical language which owes more

to the American experimental tradition than Western Art Music. My language is based on

the processes involved in the scientific idea, and this informs a process that occurs in real-

time, rather than a compositional process that can be manipulated until the desired sonic

configurations appear. To fully explore the ideas present in chaos theory it is necessary to

include feedback. To do this one needs to move past the fixed score and explore the

possibilities of text processes which allow the music to evolve in real-time.

1.2.4: Feedback

Strange attractors are iterative functions which derive their strangeness in part from

feedback. The 'sensitivity to initial conditions', that characterises chaotic systems, works

because at each iteration of the function, the value from the previous iteration is fed back

20

Xenakis, Iannis, interviewed by Mario Bois in Iannis Xenakis: The Man and his Music, (London, 1967), 12.

21

Xenakis, Iannis, Duel, (London, 1959).

22

Xenakis, Iannis, Stratégie, (London, 1962).

23

Steinitz, Richard, 'Music, Maths & Chaos', The Musical Times, vol. 137, No. 1837 (Mar., 1996), 14.

21

in. As seen previously in the Koch curve, feedback and recursion are fundamental

properties of fractals such as the well known Mandelbrot set, as shown in illustration 6:

with the recursive detail shown in illustration 7, the small red rectangle in the left image is

shown greatly magnified in the right image.

24

The text processes in my music use feedback to varying degrees. Feedback is important in

Nano as the piece is dependent on each event being different from previous events within

the space of a limited set of parameters. Each event is an iteration of the text process.

The player's memory of what values of the variables they have already played informs

them on how to proceed. In Nano the feedback is not linearly carried from iterations of a

transformation, instead, the memories of previous events build up and inform each other

and each new iteration.

Whitewater and Primes are completely dependent on the feedback loop between player

24

Weisstein, Eric W. 'Mandelbrot Set', MathWorld--A Wolfram Web Resource.

<http://mathworld.wolfram.com/MandelbrotSet.html>, accessed 04/08/08.

22

and computer to evolve both material and form. The computer and player are reacting to

each other through the text process and contribute to the other's part. In a continuous

dynamical system such as music, feedback also increases the overall complexity by

effectively smearing interactions across time. The consequences of one action are usually

still happening while a feedback induced reaction begins to happen. In my music the

generous durations and slow timescale can lead to several overlapping cause and effect

loops, each of which is feeding into the others.

The relevance of this to the music owes much again to the fact that music is a complex

dynamical system. Indeterminacy is part of a feedback process which allows the music to

evolve in response to its environment in terms of players and spatio-temporal elements.

This music is not 'better' in any way than music which allows no indeterminacy, it is simply

at another position on the plane of musical possibilities. As James Gleick says of chaotic

processes:

The Euclidean and Cartesian methods of turning equations into curves are familiar

to anyone who has studied high school geometry or has found a point on a map

using two coordinates. Standard geometry takes an equation and asks for a set of

numbers that satisfy it. [...] But when a geometer iterates an equation instead of

solving it the equation becomes a process instead of a solution, dynamic instead of

static.

25

Xenakis' biographer Noutitza Matossian quotes Ligeti as saying that it was 'worthwhile to

try and achieve a compositional design of the process of change'. I see the use of

feedback and indeterminacy as being similar to that, the compositions aim to model the

process of change by making allowing each change to have a direct effect on the future

state of the piece.

1.2.5: Evolution

The world of things living, like the world of things inanimate, grows of itself, and

pursues its ceaseless course of creative evolution.

- D'Arcy Wentworth Thompson (author of On Growth and Form)

26

For reasons both musical and otherwise conceptual, the term 'evolution' will be used

throughout this document in reference to the transformation of material across time.

Semantically speaking, 'variation' may be the more correct term, but the musical baggage

25

Gleick, James, Chaos: Making a New Science, (London, 1987), 226-227.

26

Thompson, D'Arcy Wentworth, On Growth and Form, abridged version, abridged ed. John Tyler Bonner, (London,

1961), 3.

23

that it has accumulated make it unsuitable. 'Variation' presupposes the existence of a

'theme', a notion which is tied to ideas of absolute and objective truth. This also involves

rescuing 'evolution' from the anthropocentric notion of an evolutionary teleology: the idea

that things can evolve 'towards' something. The pieces under discussion here have no goal

to evolve towards, only the idea that difference is interesting for itself because difference

highlights essence.

It is more interesting to consider the idea of evolution, in which there is no beginning or

end, only change. Evolution can also mean different possible realisations of a single idea

and this relates strongly to the form of my music, where a single idea is repeated and

each successive instance of the idea is a different way to realise this idea. Different

performances of a piece will of course be different due to the level of indeterminacy. But

within each piece there is also a difference between each event because each event is

generated from the same text, thus each is an instance of the idea. The score is the

instructions on how to realise an event. The piece consists of successive events,

successive instances of that idea.

In the biological meaning of evolution, organisms have two forms, genotype and

phenotype. The genotype is the basic idea of the organism as encoded as a genome. This

genetic information is actualised as the phenotype, which is a specific instance of the

organism itself. To be more exact, the phenotype is the 'observable characteristics of an

individual determined by its genetic make-up and the environment'.

27

In musical terms

this is analogous to the text score being the piece's genotype, and each instance is a

phenotype wherein the specific instructions in the score are subject to modification by

environmental conditions—such as performers and performance space, and the history of

previous instances. The indeterminacies in these works will create differences between

instances in the same way that minute indeterminacies in cell division and environmental

conditions will create different children from the same parents.

28

This form of repeated instances of an idea is best demonstrated in Nano. The instructions

in the score describe the steps necessary to form a single event: a descending scale in

sub-semitonal intervals. The final instruction is to repeat the event an indeterminate

number of times, with each instance being different.

29

Indeterminacy is generated from a

clear and simple mechanism in Nano as each event is a different application of the basic

formula for event construction. The piece exhibits modular characteristics in that the four

27

'Phenotype', Compact Oxford English Dictionary, Eds, Soames, Catherine and Hawker, Sara, 3

rd

ed. (Oxford, 2005),

763.

28

Obviously the mechanisms are vastly different, but I believe as a simplified model of environmental forces and

indeterminate variables acting on a system, the analogy holds up.

29

The score defines a minimum of 23 times but there is no maximum set.

24

variable parameters (starting pitch, tempo, number of notes in event, interval size

between notes

30

) are altered for each new instance: new numbers are plugged into the

variables. Whitewater is another example of this form of repeated instances, but much

more involved because of the two-agent feedback process. The instance being repeated

here is two-part, the multiphonic played live and its altered re-iteration by the computer.

The feedback process means that the two agents co-evolve the soundworld and the

material accumulates between them, thus the instance is being smeared across time and

blended with other instances.

It should be clear from this example that there is no theme to vary because the

compositions are built on an idea which cannot be reduced to a single perfect realisation in

sound. The piece is a series of instances, each being a possible rendering of the idea but

none more perfect than the last. Illustration 8 below is a summary of the musical objects

in each piece and the method by which they are transformed. Note that Lorenz and

Distemper do not have specific musical objects to transform, these two pieces do not have

material in the same manner as the other pieces, they work on relationships instead: see

their respective commentaries below (2.1.17 Lorenz and 2.1.6 Distemper) for more

thorough discussions of their construction.

30

Interval size between each note can be different. Not the same interval size for each note of the event.

25

26

Illustration 8: Sound objects and their methods of evolution.

1.2.6: Indeterminacy

Chance, understood as connections between independent worldlines, has the

liberating power to release us from the limitations of our expectations. Given a

headstart by subversive intentionality, chance has a chance to outrun intention and

thereby open us to the world as it is, not merely as we think it will or should be.

31

The above quotation comes from the chaos theorist Katherine Hayles, writing about John

Cage. Such a quotation is apposite and pertinent to open a commentary aiming to bring

together aspects of physics and music. While very few of my pieces directly use models

from science, my whole aesthetic outlook is informed by that way of looking at the world.

32

Heisenberg's uncertainty principle and quantum theory closed the door on Newtonian

physics by postulating that uncertainty was in the nature of the Universe.

33

The

Uncertainty Principle states that:

One cannot assign exact simultaneous values to the position and momentum of a

physical system. Rather, these quantities can only be determined with some

characteristic ‘uncertainties’ that cannot become arbitrarily small simultaneously.

34

Allied to the rise of ambiguity in physical reality—as described by physicists—are post-

modern and post-structural ideas of subjectivity and contextualisation. The notion that the

World is made meaningful by interactions and connections rather than fixed identities is

both musically and philosophically attractive. Seyla Benhabib, Professor of Government at

Harvard University, explains the re-positioning of the subject thus:

The subject is replaced by a system of structures, oppositions and differances

35

which, to be intelligible, need not be viewed as products of a living subjectivity at

all. You and I are the mere 'sites' of such conflicting languages of power, and the

self is merely another position in language.

36

This statement can be translated into a musical context where the subject—as a sound

event which can be considered complete in gestalt perceptual terms—is viewed as

something in flux, with no fixed or basic identity, only those generated from the

31

Hayles, Katherine, 'Chance Operations: Cagean Paradox and Contemporary Science', in John Cage: Composed in

America, eds. Perloff, Marjorie and Junkermann, Charles, (London, 1994), 231.

32

As I am not skilled mathematically, nor trained to any great degree in scientific disciplines—I completed six months of a

degree in Materials Science at the University of Limerick before dropping out in 1992. I must include as a disclaimer

that much of the scientific thought drawn upon to produce this work are understood by me on a level sufficient for

metaphorical use, but that is devoid of true mathematical understanding and is researched only in the popular literature

on the subject: I relate to the concepts but not the technicalities. However, this does not in any way impede my use of

these ideas in composition, a subject which is all about the free play of connections, metaphorical or other.

33

Although this interpretation of quantum theory is currently under attack from a growing body of theory supporting

Bohmian mechanics. See Mark Buchanan's New Scientist article 'Quantum randomness may not be random' in the

issue of 22/03/08.

34

Hilgevoord, Jan and Jos Uffink, "The Uncertainty Principle", The Stanford Encyclopaedia of Philosophy (Fall 2006

Edition), Edward N. Zalta (ed.), <http://plato.stanford.edu/archives/fall2006/entries/qt-uncertainty/>, accessed 02/08/08

35

Not a mis-spelling, 'différance' is a term coined by Jacques Derrida with many subtle meanings, the most prominent

being a deferral of meaning by referring to other words and concepts.

36

Benhabib, Seyla, cited in Butler, Christopher, Postmodernism: A Very Short Introduction, (Oxford, 2002), 51.

27

differences with others and with its environment. This corresponds with a fundamental

formal approach in my music in which a single musical event evolves through interaction

with other agents. In Primes and Whitewater this agent is a computer program (Max/MSP)

that has a level of agency in creating locally unpredictable patterns while in Lorenz and

Whitewater II the agents are other players and the rule systems that control them, and in

Nano it is the player's memory.

Over the course of my PhD studies I have developed techniques involving varying levels

and functions of indeterminacy. The following sections will look at vertical (1.2.7 Vertical

Indeterminacies , p.28) and horizontal indeterminacies (1.2.8 Horizontal Indeterminacies,

p.39), their techniques and applications in my music, as well as some discussion of the

context of these techniques in relation to other composers' work.

1.2.7: Vertical Indeterminacies

Works of Art do not reproduce the visible, rather they make visible.

37

- Paul Klee

This section will outline different ways in which vertical material is generated and

organised, with focus on the compositional techniques used. 'Vertical' can be taken here to

be concerned only with pitch generation and organisation. Issues of instrumental timbre

and orchestration are largely ignored except where they have particular relevance, while

issues relating to time, pitch prolongation and proliferation will be examined in 1.2.8

Horizontal Indeterminacies, p.39.

This section begins with an overview of the subject matter, presenting some of its aims

and relating it to the ‘strange attractor’ metaphor. With regard to strict pitch generation,

the context and details of the techniques of spectral modelling and frequency modulation

are examined with particular reference to the works of the (then) IRCAM-based Spectral

school, most notably Gérard Grisey and Tristan Murail: Grisey died in 1998, and Murail

now lives and teaches in the USA. Then the more indeterminate approaches to pitch are

discussed, such as detuning, notational imprecision and generation from text processes.

The techniques used here arose from a concern for beauty in sound and, to echo Paul

Klee's dictum above, a fascination with the sounds that are not there—such as

combination tones and acoustic phenomena. I can speak retrospectively about why I

37

Klee, Paul, cited in Schneider, Peter, 'How Not to Teach Architectural History', Journal of Architectural Education, 34/1,

(Autumn, 1980), 23.

28

choose to compose this way, but must include the disclaimer that reasoning in this way is

done after the fact: these reasons were not on my mind while composing, rather they

have become clear to me through writing these commentaries. At the time, I was

fascinated by the sound of the non-equal-tempered harmony in these pieces, the strange

sense of ambiguity between note, chord, timbre and noise. I tried to compose with

quartertones but could not understand how to create a harmony from them, so I searched

for a system and found spectralism, the indeterminacy followed later. What I am trying to

achieve in using vertical indeterminacy is the generation and proliferation of complex and

dynamic evolving harmonic structures. Indeterminacy is used to create ambiguity between

groups of single pitch percepts—such as a chord—and complex pitch percepts formed from

fusing single pitches—such as the synthesised multiphonics in Whitewater. This ambiguity

based on the duality between harmony and timbre drives evolution of the material and in

the process generates form out of difference.

1.2.7.1:

Inharmonic Sound and Strange Attractors

Many of the works in this portfolio use inharmonic sound as their basic material. By

inharmonic sound, I mean that the material is either inharmonic in itself—such as

multiphonics or prepared guitar—or it is an inharmonic complex created by combining

harmonic sounds—such as the modelling of a multiphonic in Whitewater II using sustained

saxophone notes where simultaneous harmonics fuse into a single pitch percept.

Perceptual fusion is a concept central to most of my work; I use slow sustained

soundworlds partly because of a predilection for the sound and partly because that is the

speed at which sound most likely to fuse. My preference is for the sounds of bells and

other metalophones. These sounds have complex spectra which do not resolve simply into

one pitch percept but instead split into multiple simultaneous percepts.

As was discussed previously (1.2.2 Chaos and Strange Attractors, p.16), the strange

attractor metaphor connects with inharmonic pitch structures by relating the infinite

amount of points in a fractal space to the infinite amount of possible pitches in a bounded

frequency-based pitch continuum. For example, the indeterminate text process in

Whitewater II instructs the players to choose at random from a pitch set which defines a

vertical sonority (the pitch bank for the sonority is derived from analysis of a saxophone

multiphonic). If this text process were repeated over and over, the accumulation of

different pitches would sketch the original multiphonic from which the vertical was

derived; the clusters around nodal pitch areas becoming ever denser as the repeats

continue. This is analogous to the shape of a strange attractor becoming clearer as the

iterations of points accumulate on a graph.

29

Illustration 9 shows a spectral analysis of a saxophone line where a single pitch becomes a

multiphonic and then collapses into a different single pitch. The opening pitch is B?4 +14¢

showing 1

st

and 2

nd

partials,

38

while the closing pitch is the 1

st

to 4

th

partials of E5 +12¢.

Because this event is performed as a single gesture across one fingering, the single

pitches are slightly thin and lack some of their higher partials. However, the analysis

clearly shows the difference between the wide and evenly-spaced partials of the harmonic

single pitches and the dense spectral complexity of the multiphonic. This complexity

results from the perceptual fusion of the strong partials, and the creation of many

combination tones due to the intermodulation of the strong pitches.

39

The resultant

clusters of pitches which form around the strong partials are clear in this image.

All of my techniques for pitch generation result in microtonal clustering—including octave

displaced clusters—as substructures of larger pitch structures; this can be seen in

38

Or possibly the 2

nd

and 4

th

partials of a lower fundamental. The analysis is unclear here because while there is a lower

fundamental—and what appears to be a 3

rd

partial between the two strong partials—this fundamental is not linearly

related to the pitches above it. The fundamental is 213hz, compared to the strong pitches which are 470hz and 942hz

respectively. However, in my experience, the lowest partials in inharmonic sounds have a tendency to be flat so this

may not be abnormal.

39

Scavone, Gary Paul, 'An Acoustic Analysis of Single-Reed Woodwind Instruments with an Emphasis on Design and

Performance Issues and Digital Waveguide Modelling Techniques', (PhD Dissertation, Stanford University, 1997), 90-9.

30

Illustration 9: spectral analysis of saxophone multiphonic

illustration 9 where many two-note clusters propagate up through the registers.

Whitewater, for example, takes such multiphonics and mixes them with intonationally

different clones bred by a computer patch to create even more dense structures, while

Lorenz uses a text process to make string instrument clusters aggregate around a

multiphonic, again making a constantly shifting set of clusters. These clusters are on the

boundary between being perceived as either a single complex timbre percept or sets of

individual pitch percepts, and this ambiguity drives the vertical indeterminacy. This

multistable perception can be applied in a musical context where there is ambiguity

between aural interpretation of a sound event as single or multiple percepts. Vertical

indeterminacy is often used to vary a vertical pitch group between timbral and chordal

states. To extend the metaphor, many of my works can be visualised as a series of Necker

Cubes endlessly evolving and inter-penetrating.

When the basic material is inharmonic, there is no clear boundary between 'in-tune' and

'out of tune'. While most of my work uses standard western classical instruments with

harmonic spectra, they are usually combined in ways which create an inharmonic whole.

Again, the exceptional piece here is Primes, which reverses the paradigm by using only

harmonic partials, but in a complex way; the computer plays mathematically perfect

pitches but the live player can never play as perfectly as that so there is always friction in

the form of beats between the acoustic instrument and the computer: see 2.1.3 Primes,

p.69.

Pitch generation techniques will be examined in the context of when and how they are

used to generate indeterminacy. The generative processes are sometimes applied strictly,

such as in Marx where the pitches generated through ring modulating the voice and viola

parts are fixed in the score as the synthesiser part. In other pieces, the generative

processes have a strict compositional method but are loosely applied; for example, the

spectral modelling in Whitewater II is strictly generated but is subject to an indeterminate

text process which means that in performance only some of the generated pitches will be

used. At the extremes of indeterminacy, some pieces have almost no pitch generation

outside the performance context. Game pieces and text scores such as Nano rely only on

rules of interaction to describe what pitches may be played. In these pieces, the act of

composition in the vertical realm is a matter of defining these interactions, and by

corollary, the boundaries of the piece.

In my pieces with strict pitch generation there are two main archetypal processes:

spectral modelling and frequency modulation. Both tend to generate pitch structures rich

in inharmonic pitches and clustering, and both of these techniques are derived from

studying the works of the European Spectralists; Grisey and Murail's own writings, and

31

writings about them, give considerable insight into the specifics of their techniques for

pitch generation and proliferation.

My interest in the idea of frequency-based harmony originates with Tristan Murail, who

used the term 'frequencial harmony' to describe pitch organisations that he says are

'conceived outside the domain of equal temperament [...] and form an unlimited harmonic

realm, which happens to be contiguous with timbral space'.

40

He is referring specifically

here to a kind of spectral modelling by creating harmonies which are, as he says; '[...]

completely invented, through analogies to the spectra found in nature'.

41

This method of

thinking about harmony in terms of frequency is present in all of the pieces in this

portfolio. A method for dealing with the continuum nature of frequency is an important

step towards working intelligently with inharmonic sounds such as the woodwind

multiphonics used in Whitewater, Whitewater II and Lorenz, or the metalophone sounds

used in Intra- and Five Bells for Elliott Carter.

The analogy that I am teasing out draws on the idea of the strange attractor as a model

for inharmonic sounds because they are often not reducible to a single stable spectrum.

The strange attractor is a shape made of fuzzy lines, a shape of indeterminate edges

whose indeterminacy sometimes only becomes clear as time goes on. The inharmonic

sounds which I compose with such as multiphonics and metalophones are similarly fuzzy

in pitch terms because they do not follow the ideal, mathematically perfect acoustic

spectrum. It is fair to say that like real planetary orbits, real instruments never have

mathematically perfect spectra; physical realities in the form of variable string masses and

imperfect conical bores intrude on the maths. But as far as our ears are concerned these

slight imperfections generally go unnoticed. However, in the case of inharmonic sounds,

the spectra are grossly misformed— relative to that of pitched instruments—and the ear

can interpret a single sound as anything from a timbral complex to several distinct

pitches.

The spectra of inharmonic sounds are complex because the partials do not line up

mathematically and so are not resolvable by the ear into a single pitch. For me as a

composer, this lack of alignment allows room for an indeterminacy of pitch that only subtly

alters the sound and may allow for imperceptible morphology based on sound event

identity. I experimented during the composing of Whitewater by making a spectral model

of a multiphonic using the strongest sixteen partials and found that the sound's identity

did not change significantly if a few individual partials were moved by up to a semitone.

42

40

Murail, Tristan, 'After-Thoughts', Contemporary Music Review, 19/3 (2000), 8.

41

Murail, op cit., 8.

42

For this to work there had to be no pattern to the displacement of partials; it had to be random amounts in different

directions. Moving a significant amount of partials by the same amount in the same direction simply amounts to 'dirty'

transposition and is perceived as such.

32

Of course this is a subjective claim based on my own perceptions, but it allowed me to

proceed with composition based on fuzzy pitches.

1.2.7.2:

Frequency Modulation

The technique referred to here as frequency modulation is a simplified version of the

mathematics behind frequency modulation. In simplifying the process, calculation of

amplitudes was ignored entirely; although still represented intuitively by lowering the

dynamic weighting of sideband pitches as they become further from the fundamental

frequencies of the carrier and the modulator.

The basic process is to derive pitch sets of any size from two input frequencies f1 and f2,

achieved by calculating summation tones and difference tones between two pitches,

according the following formulas:

sum tones {x = f1 + f2, 2f1 + f2, 2f1 + 2f2, 3f1 + 2f2...nf1 + nf2}

difference tones {x = f1 - f2, 2f1 - f2, 2f1 - 2f2, 3f1 - 2f2...nf1 - nf2}

Examples of other works using these formulas are numerous, Tristan Murail's Gondwana

43

(1980) and Gerard Grisey's Modulations

44

(1978) are both analysed briefly by Julian

Anderson in his Grove article on Spectral Music; in which he gives a basic demonstration

of their application of this formula.

45

The mathematics first caught my attention at a time

when I was searching for a formalised method for composition with microtones, but after

some research into Spectral Music I found that it was not the technique but rather the

sounds and the ideas in Spectral Music which fascinated me. Learning about the frequency

modulation techniques in particular introduced me to psychoacoustics and sound

perception which led to a fascination with sounds that are technically present but masked

by other sounds. I see the frequency modulation technique as a way of bringing these

sounds into the foreground.

Frequency modulation as a technique in my work has not been applied strictly in recent

works, but the concept of modulation and the generation of material by the interaction of

pitch at the atomic level is present in most of my techniques. Multiphonics and inharmonic

sound in general contain elements of modulation between their multiple pitches. The

products of this are then applied through spectral modelling.

43

Murail, Tristan, Gondwana, (Paris, 1980).

44

Grisey, Gérard, Modulations, (Paris, 1978).

45

Anderson, Julian, 'Spectral Music', Grove Music Online, ed. Laura Macy, accessed 07/08/08.

33

1.2.7.3:

Spectral modelling

Compared with FM, spectral modelling is in some ways the simpler and more freely

applied technique of those I have used. Sound material is identified and its spectrum

analysed, then the subsequent data is intuitively filtered and reduced to those pitches

which present the most interesting harmonies (subject to the composer's taste) and those

which provide a close match to the original material. All of my uses of spectral modelling

have tended towards the same purposes, to create beat-rich and complex pitch structures

by allowing a measure of indeterminacy to blur the spectral model. This is never intended

to be a literal representation of the original material, analysis allows the material to be

abstracted and freely manipulated; this approach is shared by the Spectralists.

A brief example from the Spectral canon will serve to demonstrate this. In Grisey's

Partiels (1975), the spectrum of a trombone's low E (82.41hz) is orchestrated across

eighteen players.

46

This chord is repeated slowly, to allow for simulated decaying of

partials, and with each subsequent repetition the pure spectrum is altered by introducing

non-harmonic pitches until after several minutes an inharmonic spectrum is reached and

the piece moves on to another process. A much later example of spectral music is Murail's

Le Partage des Eaux

47

(1996) for large orchestra. Murail takes advantage of the greater

analytical power of modern computers—he has worked with IRCAM since the early 1980s

—to compose with spectral models of water breaking on rocks.

48

The first piece in which I successfully applied this technique was M.Grisey, his Galliard

49

(2005), for 11 strings (6,2,2,1). In this piece, recordings of bells and other metalophones

were analysed and chords derived from the resultant data, the piece's main structural

argument involved slow metamorphoses between chords. The group of strings was

sufficiently large to make the harmonies fuse into complex timbres during the more

intense chordal sections.

All of my uses of spectral modelling have tended towards the same purposes, to create

beat-rich and complex pitch structures by allowing a measure of indeterminacy to blur the

spectral model. Other examples of composers using spectral modelling in acoustic music

include James Tenney and Clarence Barlow. Tenney's Three Indigenous Songs (1979)

50

and

Barlow's Im Januar am Nil (1984)

51

both use instrumental resynthesis to render speech as

instrumental pieces by orchestrating the frequencies of the speech phonemes' partials.

46

Grisey, Gérard, Partiels, (Paris, 1975).

47

Murail, Tristan, Le Partage des Eaux, (Paris, 1996).

48

Murail, Tristan, interviewed by Joshua Cody at Cody, Joshua, 'The Ensemble Sospeso - Tristan Murail',

<http://www.sospeso.com/contents/articles/murail_p4.html>, (no date), accessed 07/08/08.

49

See appendix 3 for score.

50

Wannamaker, Robert A., 'The spectral music of James Tenney', Contemporary Music Review, 27/1, (February 2008),

107.

51

Barlow, Clarence, Im Januar am nil, (Cologne, 1984).

34

Barlow focuses on vowel sounds and their formants

52

while Tenney also includes noise

elements from the percussion to approximate phoneme transients such as plosives and

fricatives.

53

G. Douglas Barrett, a young American composer and former student of Tenney's,

continues this line with more advanced technology by using automatic notation generators

to create, as he says, 'experimental transcriptions of recorded material—field recordings

made on Hollywood street corners, video documentation of performance, recordings of

other pieces of music…'.

54

Barrett's Surfaces I-IV

55

(2007) is the transcription of four

tibetan bowls into a glass-like sequence of overlapping microtonal lines for string quartet;

illustration 10 is a section from the score.

These pieces all share an experimental attitude in that they seem fascinated with seeking

greater exactitude in finding compositional analogues to various phenomena —such as

Tenney using different percussion for the plosive and fricative phonemes— while

acknowledging that this work is not an attempt at literal representation of the object being

modelled. At the same time, the fascination seems to require pitch accuracy as a function

of the analysis rather than compositional necessity. Spectral analysis is a powerful tool

which can analyse pitch to a resolution far beyond the capabilities of human perception.

Of the pieces in my portfolio, Intra-, Whitewater, Whitewater II and Five Bells for Elliott

52

Ox, Jack, 'Im Januar am Nil stillshots and animations', <http://www.jackox.net/pages/introImJan.html>, (No date),

accessed 07/08/08.

53

Wannamaker, op cit., 109.

54

Barrett, G. Douglas, 'Bio', <http://synthia.caset.buffalo.edu/gbarrett/bio.html>, accessed 06/08/08.

55

Barrett, G. Douglas, Surfaces I-IV, self published (Buffalo, 2007).

35

Illustration 10: G. Douglas Barrett - Surfaces I-IV

Carter all use spectral modelling as their basic pitch generation technique. Intra- and

Whitewater II generate pitches pre-compositionally, which are written into the score, while

in Whitewater the modelling is done in real-time performance by the computer. Five Bells

for Elliott Carter takes the most intuitive approach, repeatedly altering the intonation and

dynamics within a single chord to make it more like the structure of a bell spectrum.

1.2.7.4:

Detuning and Notational Imprecision

I made a slow transition from notational accuracy to indeterminacy. Early works in this

portfolio such as Tracheids and The Lady with the Hammer use accidentals that are

annotated in the score with specific cent deviations from equal temperament. I was trying

to bring out specific difference tones or create timbral fusion by calculating the exact

frequencies that these pitches would have, but I was slowly realising that it was the

beating sounds and the ambiguous harmonies that I was really searching for. It took

several years of spectral experiments for me to understand that rather than specifying

difficult intonations which a performer must play with precision, I could simply ask them

to detune their instrument relative to others in the ensemble before playing. This would

mean that everything would be 'out of tune' and it would be easier to achieve beats and

complex clusters. In these cases it was also more practical to have them use a special

fingering to reach the occasional 'in tune' note, than to be tuned normally and have many

special fingerings for the 'out of tune' notes. When players would ask how sharp or flat I

wanted a particular note to be I never had an exact answer. It took time for me to trust

the instinct which said that the note simply had to be 'out', and to realise that precision

tuning was not the issue. This technique is used in Bifurcations and Whitewater II

specifically.

Notational imprecision is similar to detuning in that the indeterminacy is introduced

against the grain of the player's perceptual training and intuition, demanding that they act

independently while still working together as a group. The notation in Whitewater II, Inner

Shadow, Bifurcations and Five Bells for Elliott Carter uses accidentals which are

deliberately undefined and explicitly requests that players do not attempt to tune to each

other or even to be 'in tune' (as shown in the previous section on detuning), players are

expected to play what they see regardless of what is happening around them: there are

some subtleties to this which will be examined later.

This may make players speculate as to how far they should trust the fidelity of the

notation (or the piece...) if this most basic performative assumption is removed, but

36

removing this basic tradition does not necessarily topple the edifice of harmony entirely,

the players are simply being asked to play with intonation which is defined as a field

rather than a point.

To illustrate this, an example from Alvin Lucier is required. In a reply to a question about

using intonation systems he says that the critical band is too wide for him, because most

of his music is concerned only with the phenomena which occur within intervals smaller

than 16hz.

56

He goes on to say:

I don't want my players to be anxious about tuning; they have enough problems in

evenly glissading a semitone over such long time spans.

57

Lucier doesn't mean that accuracy is not important, simply that whatever happens with

the field boundaries is acceptable. In this case the boundaries are defined relatively by

micro-pitch; as long as the players are within that boundary then the piece will be correct

because anything within that proximity will generate the beats which he wishes to make

explicit. Further to this, he emphasises the point about imprecision here:

I wrote a little orchestra piece a few years ago in which the string players sweep

slowly up and down. They would often lag behind and speed up quickly to get to

the pitch they were aiming for. I should have told them that they could just as well

have missed the pitch as long as the sweep was even, but I didn't think of it in

time.

58

Again, what Lucier and I (and others) are looking for is not the strict point-intonation

required by hundreds of years of pitch-centric western musical tradition, rather a field

condition of being in the right area for interesting things to happen. Often in chaos theory,

one finds that the most interesting things happen within a few narrow bands of possibility

on the edge of plateaus of non-information—that which is too complex to find meaningful

patterns, or that of such utter regularity to be meaningless. Mathematician David Ruelle

describes below the response to his early papers on—the then unknown—fractals and their

ability to fold an infinite line within a finite space:

The reaction by the scientific public to our proposal was quite cold, In particular, the

notion that continuous spectrum would be associated with a few degrees of

freedom was viewed as heretical by many physicists.

59

While I am not suggesting that musicians find this field harmony concept to be heretical, it

does require a different way of thinking.

56

Alvin Lucier (transcribed by Anne Guthrie.), Ostrava Days 2005 Report, (Ostrava, 2006), 117.

57

Ibid.

58

Ibid.

59

David Ruelle, cited in Gleick, James, Chaos: Making a New Science, (London, 1997) 139.

37

The notational imprecision in my works uses the set of accidentals indicated in illustration

11, which includes standard accidentals as well as those with arrows to indicate direction

of deviation. Pitches in twelve-tone equal temperament and the fairly standard

quartertone symbols will have standard intonations—a solid body of pedagogical work

exists

60

to provide fingerings and techniques for quartertones on most instruments—but

the accidentals inbetween have no fixed interpretation other than in specific performing

traditions. In most music these ambiguous accidentals would simply be a tuning question

to be 'fixed' before the performance as an issue that is open to the players’ interpretation

and therefore requires consensus before the group can move on: even the most

knowledgeable performer of microtonal music would make sure they understood which

system the composer intended by this notation. In my music there is no Rosetta stone for

intonation. The performers are expected and encouraged to interpret the accidentals

independently and accept the group harmony which arises from this: in some scores,

performers are advised simply to aim for beats with their pitch neighbours.

1.2.7.5:

Vertical Text Processes

Text processes in Whitewater, Primes, Nano, and Lorenz have a direct relationship with

the pitch organisation and the horizontal indeterminacy to the point where in several of

these pieces the two are intertwined: the pitch generation is not an object that the

indeterminacy simply acts upon, pitch is generated by the indeterminate process. The

intertwining extends also to mixing the vertical and the horizontal aspects of the music,

where pieces have no pitch organisation other than organising how to get from one pitch

set to the next.

Nano is the most involved example of this in the portfolio. As it is a piece for solo

instrument, technically there is no vertical harmony; the text process defines a falling

scale whose constituent intervals must all be smaller than a semitone, this process is then

repeated indefinitely. It is included here because the pitches generated fall into the same

60

Such as Bartolozzi, Bruno, Brindle, Reginald Smith, New Sounds for Woodwind, 2nd edn. (London, 1982), and

Rehfeldt, Phillip, New Directions for Clarinet, (London, 1977).

38

Illustration 11: notation of microtonal accidentals.

categories as the other pieces in the portfolio. The process in Nano generates what is in

effect a microtonal cluster in arpeggiated-chord form. It could be interesting to write a

second version piece in which the clarinet was offstage and a computer patch played the

descending scales as clusters of sine wave chords—an inversion of the process, but in

essence it is the same piece even with the radical change of soundworld.

In this hypothetical chordal version the vertical indeterminacy would be clearer as the

piece would be a series of cluster chords all of the same family; the familial relationship

comes from the generative rules in the text process. This leads to a kind of topology of

harmony where all the objects generated by the process are stretched, twisted and

displaced versions of a generic object which is an ideal solution to the process.

Mathematician Eric Weisstein defines topology as:

[...] the mathematical study of the properties that are preserved through

deformations, twistings, and stretchings of objects. Tearing, however, is not

allowed. A circle is topologically equivalent to an ellipse (into which it can be

deformed by stretching) and a sphere is equivalent to an ellipsoid.

61

This topological relationship between the process and the pitch structures which it

produces can be seen in all of the pieces using text processes. For example, the harmony

of Whitewater and Primes depend on the feedback that is built into their processes, thus

the harmony of each new event is a consequence of a previous event; hysteresis in these

pieces combines with the restrictions of the process to produce events which are always

different but of the same form.

1.2.8: Horizontal Indeterminacies

This section is concerned with time in my compositions, its processes and various levels of

indeterminacy. Horizontal indeterminacy involves processes which determine the motion

or displacement of vertical material in time. Three main areas of horizontal decision

making will be examined: notational imprecision, bounded improvisation and text

processes.

Firstly, a brief note on terminology. In his seminal work on time and gestalt theory in

music, Meta + Hodos, James Tenney devised an analytical method based around what he

later termed 'temporal gestalt units' which split a work into different functional and

61

Weisstein, Eric W. 'Topology', MathWorld--A Wolfram Web Resource. <http://mathworld.wolfram.com/Topology.html>,

accessed 07/08/08. Topology can also be summarised by the following joke: a topologist is a person who can't tell the

difference between a doughnut and a coffee mug.

39

perceptual levels.

62

While a full application of his method to my music would not be

appropriate here, I will apply his principles to the forthcoming discussion by defining four

levels of function in my music:

Note-level: single sounds: notes, chords, complexes.

Phrase-level: sequences of sounds which are complete in themselves: events. The

terms 'phrase', 'event' and 'object' are used more or less interchangeably

throughout this thesis to mean phrase-level units.

Section-level: only applicable in pieces with well defined macro-formal sections

such as Bifurcations and Intra-.

Macro-form: the whole piece.

1.2.8.1:

Notational Imprecision

This is the simplest form of horizontal indeterminacy, removing rhythmic information from

the notation; where there are no strict timings, metric grid or rhythmic notation, temporal

aspects of the music are made indeterminate. This notation cedes control of the precise

rhythmic placement at note-level to the performer, but still specifies the majority of the

event's characteristics.

Notational imprecision in the time domain is a common device used in much experimental

music—and a lot of music which is not so experimental—where it is usually called time-

space notation; examples include Lutoslawski 's ad libitum pieces such as Jeux Vénitiens

63

,

Berio's Sequenza no.1

64

and Cage's Atlas Eclipticalis

65

. I have only avoided that term here

so that the connection with the notational imprecision in the vertical domain which was

examined in the previous chapter is more clear.

1.2.8.2:

Bounded Improvisation

Bounded Improvisation is a term which refers to an extended form of free notation

operating at the phrase level and involving improvisation within strict boundaries. It

occupies a grey area between improvisation and free notation: essentially it is an

extended text notation but sufficiently free that it requires a performer familiar with

improvisation to play it with confidence. Bounded improvisation is emphatically not free

improvisation, because the performers may only play freely within specific boundaries; the

act of composition in these pieces is mainly concerned with defining these boundaries.

62

Tenney, James, Meta + Hodos and META Meta + Hodos, 2

nd

ed. (no place, 1988), 101.

63

Lutoslawski, Witold, Jeux Vénitiens (Paris, 1962).

64

Berio, Luciano, Sequenza no.1 (Milan, 1958).

65

Cage, John, Atlas Eclipticalis, (New York, 1963).

40

Bounded improvisation means that the performer is only given descriptive boundaries of

what and how to play, along with an example score. This could be seen as improvisation in

the same way as a violinist may improvise a cadenza in a Mozart concerto by adhering to

the classical performance tradition: its rules of style, material, gesture etc. Stockhausen's

approach to total serialism applied serial processes to parameters of music that Jonathan

Harvey describes as 'things normally left to good taste and fine feelings' and left them, as

Harvey says 'no longer emotionally determined, but formally determined',

66

bounded

improvisation takes the same parameters and allows the performer to sculpt them but

under the direction of my score. In my works I am creating a performance tradition for

each piece. The performer is free to improvise within the 'tradition' that I set out in the

score. Other works using this kind of improvisation include Christopher Fox's series of

Generic Compositions

67

and Michael Parsons' Independent Pulses

68

.

1.2.8.3:

Horizontal Text Processes

Text processes are another technique for generating indeterminacy. Nano, Lorenz and

Whitewater II all use text processes which describe strict actions to be carried out by the

performers. The main difference with bounded improvisation is that in these pieces, the

score is approached like a recipe: no personal interpretation is required, the performers

simply do as the score says. There may of course be instructions with multiple possible

interpretations, but always with a certain sound-outcome in mind which guides the

performer.

For instance, the percussion and string players in Lorenz follow text processes where the

way that they choose notes is strict, but is also dependent on perceptual elements. The

strings are always chasing a moving pitch target which in theory they should get closer to

the longer it stays in the same area. The saxophone has control over its pitch sequence

(the strings' target) and the rate at which it passes. These two strategies form a limited

feedback loop similar to that of Primes: the saxophone 'leads' the strings around but will

not let them get too close, unless he runs out of notes. Notwithstanding differences in

aural skills, the strings will tend to get better at tracking the saxophone pitch as the piece

goes on. The general form of the piece should be a series of wedges, defined by the

strings becoming closer to the saxophone and the saxophone moving away but restricted

by its need to repeat pitches.

66

Harvey, Jonathan, The Music of Stockhausen (London, 1975), 16.

67

Fox, Christopher, Generic Compositions, (Fox Edition, 1989-93).

68

Parsons, Michael, Independent Pulses, (London, 1992).

41

1.2.9: Teleology

My music is largely concerned with teleology on the local level and non-teleology at the

level of entire pieces. The standard formal archetype here is repetition and

transformation of a single object or event. Local teleology is where there is goal

orientation within the event itself, but where there is no macro-formal teleology: the

succession of events is largely arbitrary. Because the objects are simply being repeated

and transformed, they follow no prescribed path and could, theoretically, continue with the

process of repetition and transformation into infinity.

Pieces using bounded improvisation (Primes, Whitewater) have a certain local teleology

thrust upon them in the form of phrase-level improvisation, but again their macro-form is

non-teleological. In Primes the player is working from a model of melody + drone. This

has a clear teleological shape to it, and constantly evolves as the interaction with the

computer is unpredictable. Phrases in Whitewater are built from interacting with the

computer on a note-by-note basis; the feedback loop involved is self-reinforcing so a local

teleology will inevitability arise from similarities and conjunctions between both parts. But

in both cases this local goal-orientation does not extend to the macro-form. Phrase-level

ordering of events is arbitrary and theoretically endless: the piece may stop after three

events or three thousand.

Because the performer is improvising, they will always bring their own sense of phrase

structure to the piece, and will probably try to influence the macro-form with long term

strategies: this is perfectly acceptable. However, the nature of the computer part and the

feedback loop, especially in Whitewater, means that any form imposed by the player will

be constantly modified by the computer's reaction. In Primes, the player has a

considerable amount of control over the macro-form; the computer's agency is felt in the

evolution of the note-level events. The parity of agency in Whitewater between player and

computer means that both agents are evolving their own material with reference to each

other, so neither has the upper hand in macro-formal decisions.

However, saying that the event succession is completely arbitrary would be an

exaggeration. Ideally, each event is an instance of the genotype described in the score

with no direct reference to each other. In practice, the events are being composed in real-

time by the agents involved, and future events will be informed by their memories,

emotions and personalities. There are always causes for each event to start and begin,

and these will be quite clear from examining a recording of any given performance. My

point here is that there is no long-term pattern or causality for events built in to the piece

42

other than the general principle that they be varied instances of the piece's central idea.

The macro-form is an emergent aspect of the patterns arising from constant

transformation inscribed in the lowest level instructions of the piece.

1.2.10: Modes of Delivery

In my music I use evolution as a means of transformation. Evolution implies constant, but

gradual change from one iteration to the next but without any predefined final goal or

teleology, implying a continuum of change across infinite time. This extends to the idea

that what the listener hears in a concert/installation is only a temporal slice from an

infinite piece of music. According to free improviser Mark Wastell, Derek Bailey claimed

that '[Bailey's] improvisation was continuous, only broken by moments when he sets down

his guitar'; this echoes a feeling shared by many improvisers about their own playing.

69

Swiss composer Manfred Werder actualises this idea in his piece 2 ausführende (1999)

which is performed in chunks. For each performance the composer sends off the next few

pages of the score to be played; the entire score would take several days to perform

continuously. James Saunders' modular work #[untitled] (2000 - ) occupies similar

territory in that each performance is a new piece built from an evolving and growing set of

musical modules. Each performance is a phenotype of Untitled#, optimised for that

particular performance context. My own Primes and Whitewater depend heavily on

improvising players to transform the material continuously. The performers create the

material in real-time—according to the score's rules—and the piece is the evolution of that

material as part of a feedback loop with the computer. There is no solution to the piece,

and the points of beginning and end are effectively arbitrary.

At this point the biological analogy may break down a little in relation to my own works

because the pieces themselves are not truly evolving in time. The score-genotype does

not evolve, only the instance-phenotype; because every new performance requires the

piece to start again. It could be said that over multiple performances by the same player,

their approach will evolve across time. But this doesn't change the fact that the piece of

paper remains the same.

One of the beautiful aspects of the strange attractor is how they develop asymptotically

over time: always becoming clearer but never reaching perfection. This aspect translates

linearly to my music in that the longer an observer spends immersed in the music the

69

Wastell, Mark, in Blocks of Consciousness..., (eds) Mark Wastell, Brian Marley (London, 2005), 7.

43

more interesting it hopefully becomes. Assuming that the material and its transformations

are fundamentally interesting to the observer, then the piece can go on for as long as the

observer remains fascinated by the process and the inner life of the sound. With that in

mind it seems logical that many of these works would be more suited to the installation

context than the concert hall. I think that both versions are viable as long as it is clear to

the audience how they are expected to listen in order to engage as much as possible with

the music.

Many of these works exist in both installation and concert versions. The fixed score works

in my portfolio are more suited to the concert hall as they have definite boundaries in their

beginnings and endings, while the open works simply begin and then simply stop; there is

a certain amount of licence in the more improvised works for the player to shape the

form, but there are no formalised ending functions. The pieces themselves do not change

much (if at all), but the context suggests that the listener put on 'different ears'. Most of

my work involves a single object which transforms again and again, requiring the listener

to keep track of many parameters. In practice, as illustrated by information theory, it is

more likely that the listener will drift between parameters as the amount of change they

exhibit differs: the more change there is, the more that parameter will draw the listener

in.

For example, Whitewater has been submitted in three recorded versions: a 10 minute solo

concert version, a 40 minute solo studio version, and a 26 minute installation version with

three players. The only real difference is that the installation version has many players

simultaneously moving around a space, as opposed to the single, unmoving, performer in

the concert and studio versions: the installation version has four microphones positioned

around the space, so the computer only receives input when a player is close to a

microphone. This means that in the installation version there is more time for the

computer to evolve sounds without being interrupted by new input; new input will

eventually re-seed the cellular automaton, thus interrupting evolution and changing the

sound.

The situation is similar for the live players. They interact with the computer sound, thus

the more it changes, the more they change. But they also interact amongst themselves to

find common pitches and create beating patterns. Much of this interaction will not be

picked up by the computer as the players spend most of their time moving between

microphones (although they are instructed to spend enough time at the microphone

interacting with the computer that they can hear the results). Ultimately, all of this should

allow for more expansive evolution as the feedback loop between players' improvisation

and computer's transformation is wider; the computer and the players both have more

44

time to evolve their parts independently, before the parts interact via occasional

microphonic contact.

Installations solve the formal issue of beginnings and endings that many of my pieces

have. Even to call it a 'problem' is a relativism because the fact that the pieces have no

formal boundaries is only a problem in the concert context and then only to listeners who

expect such elements. As an installation the works can be appreciated in their own

environment where the listener's beginning is where they pick up the thread. The form of

these pieces relies on local teleologies leading to local difference, thus the piece is

different for everyone in that for every listener starting at a new place there is a new

series of differences: one person's section A

?

B

?

C could be another listener's G

?

F

?

D.

Apart from the intractable problem that some listeners simply do not like to listen to music

non-teleologically, this way of listening may render meaningless the trajectories and

relationships that different performers may bring to the works, especially over long time

frames; there is not much can be done about this.

Manfred Werder's solution solves the problem by allowing the listener to appeal to a sense

that this work is part of something larger which is being revealed one section at a time.

This offers some level of comfort to those who require a sense of closure, at least partially,

to their listening. James Saunders' works invert the situation because they all stem from

one meta-work and the individual pieces are complete in themselves. Saunders also has

the advantage of always composing to order, thus he can fit the material and the form to

the performing context.

1.3: Musical Contextualisation

1.3.1: Experimentalism

I want to find the music, not to compose it.

- Tom Johnson

70

The work of the American Experimental school is greatly varied in terms of both sound

70

Johnson, Tom, 'I want to find the music', Editions75, Date Unknown, <http://www.editions75.com/Articles/I%20WANT

%20TO%20FIND%20THE%20MUSIC.pdf>, accessed 01/06/08.

45

and technique. While I admit that their use of indeterminacy has few direct connections

with Chaos Theory, what it does have is a commitment to process and the critical function

of asking 'what can be music?'. This attitude is ideologically centred around John Cage's

classic dictum that composers should 'let the sounds be themselves'.

71

What is most

important for me here is that the freedom of sound in Cage's compositions gave so many

other composers the permission to take bold steps in applying new ideas to musical form.

From the surrealist processes of Fluxus

72

to the literal mathematical processes of Tom

Johnson, process in experimental music takes ideas and gives them sonic clothing,

sometimes to shock and sometimes to elucidate. The idea of indeterminacy in 20

th

Century

Art music owes most to the American experimentalists, who were always more willing to

simply let a sound idea simply 'be' rather than attempting to assimilate it into the metrical

grid as many Europeans such as Lutoslawski and Xenakis did.

73

The combination of rule-

based processes and indeterminate performance follows the experimental thinking in

science where hypothesis leads to practice to test a theory. The act of composition is

largely about finding the right rules. As Tom Johnson says above, 'finding the music'

rather than composing it.

While Cage is the composer most readily associated with indeterminacy, his methods have

always been at odds with how I compose. His music often focuses on the philosophical

concept of freeing the sounds at the expense of an intended sonic result. For example,

works such as Music of Changes (1951) use the Chinese I-Ching or other quasi-

randomising processes to decide on the details of material and its placement. The

indeterminacy here is limited to the set of sound types chosen by the composer—such as

instrument type or ranges of pitch and dynamics—but the ordering and placement in time

obeys only the logic of chance with no over-riding sonic goal. At the extreme of this line of

thought are pieces like Variations II (1961). The score of which consists of six transparent

sheets, some with lines and some with dots. The sheets are overlaid by the performer and

the resulting conjunctions of dots and lines interpreted as sound parameters such as pitch

and duration. In his article on Cage in the Grove Dictionary of Music and Musicians, James

Pritchett describes the 'flexibility' of the piece as 'being such that it could theoretically

describe any imaginable combination of sounds.'

74

To me this flexibility is a step too far as

71

Cage conversation with Bill Womack (1979), in Kostelanetz, Richard, ed.,Conversing with Cage (New York, 1988), 42.

72

A worldwide movement which had many American members and very close ties with the American Experimental

school, for that reason I include it here.

73

Xenakis conformed to the metrical grid notion to a lesser degree than most composers of the time; in that the surface

of his work often obscured the grid underneath. However I still consider his work to be conceptually tied to the grid

because his sound ideas are mostly translated into a metrical idea or at least notated in this way: his notation pushed

the grid around but this is still a relationship to the grid, many experimental composers simply ignored the grid and

concentrated on time as perceived by the performer.

74

Pritchett, James, 'Cage, John', Grove Music Online. Oxford Music Online.

<http://www.oxfordmusiconline.com/subscriber/article/grove/music/49908>, accessed 05/08/08.

46

the piece has no sound objective, no intention. Cage, I am sure, following his own

doctrine of non-intention, would happily agree.

My music and Cage's differ greatly due to our different applications of indeterminacy and

the differing level of importance we place on specificity of sound intention, but they also

connect in one way because his compositional indeterminacy makes structure a perceptual

issue unique to each listener. By allowing chance to determine the sequence and nature of

sound events Cage creates a music where structure is not forced on the listener but arises

out of their own perception: each listener brings their own unique perceptual apparatus

and history to bear on the quasi-random musical events and interpret it as patterns

accordingly. In my music, a similar but more conditioned experience is created in the

works involving emergent structures—such as Whitewater, Whitewater II and Nano—

where the structure is not imposed on the listener but rather emerges as a memory

process in the listener, an esthesic process where patterns equating to structure arise in

the accumulation of similarities.

Alvin Lucier, like most of the post-Cage experimentalists, had his ideas about composition

turned upside by an encounter with Cage's music but of course he took Cage's influence

and made his own way with the new ideas.

75

Where Cage concentrated on the freedom of

sound's identity, Lucier's composition is all about freeing sound's physicality. Lucier uses

processes built on diverse scientific sources to explore sound phenomena such as standing

waves or acoustic beating patterns as well as occasionally rendering other types of

phenomena as sound. Lucier's pieces almost always take the form of a single linear

process which he describes thus: 'an action or process, set into motion and sustained

throughout the course of the work, produces unexpected and complex results.'

76

Lucier's aesthetic has greatly influenced how I think about both musical sound and form,

and much of my music's procedures are indebted to his work. Apart from the beauty and

innovation of his soundworlds, two aspects of his composition have particular interest for

me: (1) the way in which he tightly controls the boundaries of indeterminacy in his music;

(2) how he designs emergent processes and people processes to control the horizontal

aspects of the music. Both of these can be demonstrated by examining his 1968 piece

Vespers.

77

In Vespers, performers use sondols—'sonic dolphins': devices which produce a loud click,

75

Lucier, Alvin, 'Origins of a Form: Acoustical Exploration, Science and Incessancy' Leonardo Music Journal, vol. 8,

Ghosts and Monsters: Technology and Personality in Contemporary Music (1998), 5.

76

Lucier, Alvin, 'Origins of a Form: Acoustical Exploration, Science and Incessancy' Leonardo Music Journal, vol. 8,

Ghosts and Monsters: Technology and Personality in Contemporary Music (1998), 11.

77

Lucier, Alvin, Vespers, in Reflections (Cologne, 1995).

47

used for echolocation—to make their way through a space. The music in this piece is the

sound of the space itself, in the form of echoes produced by the sondols. In the score for

Vespers, it's clear how concerned Lucier is that the performers understand their role, here

is an extract from the instructions to performers:

Decisions as to speed and direction of outgoing clicks must be made only on the

usefulness in the process of echolocation. Any situations that arise from personal

preferences based on ideas of texture, density, improvisation, or composition that

do not directly serve to articulate the sound personality of the environment should

be considered deviations from the task of echolocation.

The music is emergent from the people process which is set in motion by the text process

of the score: as we can see, the music is not explicitly inscribed in the score but emerges

from its processes. People processes is a term that I use to distinguish processes which

involve actions carried out by people to achieve a sonic result from specifically musical

processes such as augmentation of a rhythm by a set duration in a specified time. People

processes include actions like Lucier's echo-following performers in Vespers or the string

players in my own Lorenz who must attempt to match the pitch of the saxophone part

according to strict rules. People processes rely on an explicitly scored set of instructions to

underscore the hidden sound or artistic purpose. It is imperative that the people in

question are open to the music itself and committed to performing it with integrity. Pianist

and experimental music specialist Philip Thomas takes a similar viewpoint, describing the

role of the performer in Lucier’s music as 'both catalyst and medium for a particular sonic

peculiarity [which Lucier's music presents] clearly and honestly with only the minimum of

performer intervention.'

78

Indeterminacy in Lucier's music is tightly bound to the intended sound of each piece. He

creates a process whereby a limited space is explored for variation on a specific sonic

outcome. As shown above in Vespers, the instructions to the performers expressly prohibit

actions which do anything other than realise the piece's central sound-idea. Thomas DeLio

sums up Lucier's approach below in relation to Music for Pure Waves, Bass Drums and

Acoustic Pendulums (1980):

79

Central to any understanding of [Lucier's Music for Pure Waves, Bass Drums and

Acoustic Pendulums] is the recognition that its form is identified exclusively with the

isolation and magnification of one particular acoustic phenomenon.

[...]

[regarding the installation version] At no time however are these parameters [sine

78

Thomas, Philip, 'A prescription for action: a common approach to performing simple, complex, graphic and verbal

scores '[lowercasing by author], Ashgate Research Companion to Experimental Music, ed . James Saunders,

(unpublished draft, 2008), 29.

79

Lucier, Alvin, Music for Pure Waves, Bass Drums and Acoustic Pendulums in Reflections (Cologne, 1995).

48

freq/amp, drum tension etc.] subject to manipulation by the composer. Instead he

accepts the random and spontaneous sonic/visual display which their fortuitous

actions produce.

80

A lot is made in writings on Lucier about how he freely accepts the results of 'fortuitous

actions', but this disregards the fact that the piece is set up in a very controlled manner to

produce a specific soundworld. Lucier creates the conditions necessary for a certain

phenomenon to happen and his concern is with the sound of the phenomenon. Once the

boundaries are in place then the performer is free to act in any way as long as it is within

the boundaries. Lucier creates a box whose walls are defined acoustically and gesturally,

and as with any wall the purpose is both to keep things out as well as to keep things in.

The boundaries ensure that if the performance is done with goodwill and commitment that

the required phenomenon and its sound result will be all that happens, nothing will

happen to break the spell of the performance by introducing extraneous elements.

There are many possible versions and outcomes of an idea. Like a well designed

experiment, exploration of these possibilities using indeterminacy removes certain

preformed expectations and allows us to experience the unexpected while not removing

the intended.

1.3.1.1:

Beats and clusters

In the previous sections on techniques, beats are often referred to as a prominent

intended sound outcome in my music. This is an aspect of my fascination with physical

sound and making sounds visible. Lucier is a powerful example of a composer who makes

sounds visible. He takes phenomena such as acoustic beating and composes to show this

alone, as he puts it himself, 'having to pare away any musical gestures in a work in order

to uncover the true idea in the piece'.

81

But Lucier has no interest in abstract systems used

to generate pitches for composition other than the use of number systems to order events

and pitches. As was shown above, his interest lies in exposing the physical phenomena

itself.

His work gave me permission to focus on a single evolving sound and reveal the

complexity that can be found in simple processes. In a certain light, Lucier epitomises the

experimental composer; as he says himself:

Some of my works look like scientific experiments. I have pieces where I simply run

a pure wave from the low to high without altering it. If I did alter it, you would miss

something. In a scientific experiment you scan all the possibilities. If you start

choosing one thing over another, you're relating to personal experience.

82

80

Delio, op cit., 100.

81

Lucier, Alvin, being interviewed by William Duckworth, Reflections (Cologne, 1995), 40.

82

Lucier, Alvin, (transcribed by Anne Guthrie.), Ostrava Days 2005 Report, (Ostrava, 2006), 116.

49

Alvin Lucier's work centers on pieces which expose certain acoustic phenomena, his chief

compositional strategy is to reduce the music to only the elements which take part in this

phenomena and remove or reduce all other parameters. In his 1983 article on Lucier,

Thomas DeLio compares this strategy to that of visual artist Robert Irwin, citing Irwin's

remark that 'the reduction was a reduction of imagery to get a physicality, a reduction of

metaphor to get at presence'

83

[Irwin's italics]. My music follows on from Lucier's in

reducing the music to it's materials and processes for themselves rather than putting

them in the service of an emotion or narrative purpose.

Lucier's piece In Memoriam Jon Higgins (1984) is one of many which focus on the

phenomena of acoustic beating patterns formed by the simultaneous sounding of pitches

whose fundamental—or occasionally higher partials—frequencies are less than

approximately 20hz apart. In Memoriam Jon Higgins, written for clarinet and sine wave

generator, is a single process wherein a single sine tone slowly rises through a frequency

space while the clarinet sustains single pitches. The clarinet's pitches are placed so that

they lie just above the sine tone's frequency at that moment, as the clarinet sustains, the

sine tone passes through it and the proximity generates beats which accelerate and

decelerate as it approaches and passes the clarinet.

Every part of In Memoriam Jon Higgins is directed towards demonstrating the phenomena

of beating, all other parameters are suppressed to highlight this and reduce the possibility

that there is any other way to listen to the piece other than as Lucier intends. The overall

trajectory of the piece is a simple linear ascension from X-hz to Y-hz, no distracting

changes of direction. The notes of the clarinet enter and leave as silently as possible, if

the player is good enough, the clarinet is only noticeable when the listener perceives the

beats. The following anecdote from a lecture he gave at the Ostrava Days festival in 2005

illustrates this perfectly.

Frederic Rzewski asked me why I didn't vary the speed of the rising wave [the sine

wave in In Memoriam Jon Higgins]. If I did that I would be invoking an idea which

comes from other music, that is, to change your idea to interest the audience. I

would never do that, it just confuses the issue. You simply set something in motion,

don't interfere with it. If you interfere with it, you'll never discover the

unexpected.

84

Where my music differs most from Lucier's is in the focus on phenomena for their own

sakes. I take from him the technique and aesthetic of removing compositional elements

that are extraneous to the realisation of an idea, but I am not interested in the

83

DeLio, Thomas, Circumscribing the Open Universe, (London, 1984), 92.

84

Ibid.

50

phenomena themselves except as components of a larger language. Lucier generates

harmonic and temporal ambiguity as an artifact of his focusing on the sound itself, I focus

on the sound itself in order to bring out these ambiguities.

The best example of another composer working in the field of sustained clusters in motion

is Phill Niblock. Using multi-track recording and close-micing he creates a very rich and

somewhat unrelenting soundworld of sustained drone clusters. His cluster harmony is built

up intuitively without reference to systems or models: as he puts it himself:

I'd rather not be too specific. I'm not into just intonation or overtone systems. What

I'm interested in is just big clouds of sound.

85

Specificity can also work to achieve goals such as Niblock's. Ligeti is an example of a

composer using different tunings to widen the intonational space in his soundworld. In his

Ramifications

86

(1969) for string orchestra he detunes half the orchestra by a quartertone.

This is a cheap but effective way to use quartertones without creating too many difficulties

for the players (as long as they don't listen too hard to the other half of the orchestra).

However, he may have been too exacting here in his use of quartertones. In the light of

Ligeti's other intonational experiments it may be that he simply wanted to make the sound

a little more 'dirty', rather than extending equal temperament to 24 pitches per octave.

This is exemplified in the detuned organ required for Harmonies from the Two Studies for

Organ

87

(1962) and Volumina

88

(1966). In Harmonies, Ligeti asks for reduced wind

pressure in the organ, producing a sound described by Janet Owen Thomas as 'wheezy'

and 'indefinite' with 'fluctuations of dynamics and intonation (micro-intervals, glissandos

etc) [that] cause a denaturing of the tone colours to produce the “pale, strange, vitiated

ones” Ligeti was after.'

89

After Ligeti had largely abandoned cluster-based harmony in the 1980s, he continued to

be interested in denaturing his harmony, which was now much more tonal. An example of

an early demonstration of interest in non-12-tone equal temperament harmonies is the

use of natural harmonics on the horn in the Trio

90

(1982); an idea which was extended to

a horn soloist and a quartet of natural horns in the Hamburg Concerto

91

(1999). A more

systematic approach to mixing intonations was developed in the concertos for violin

92

and

85

Niblock, Phill, (transcribed by Anne Guthrie.), Ostrava Days 2005 Report, (Ostrava, 2006), 139.

86

Ligeti, György, Ramifications, (London, 1969).

87

Ligeti, György, Two Studies for Organ, (London, 1962).

88

Ligeti, György, Volumina, (London, 1966).

89

Owen Thomas, Janet, 'Ligeti's Organ Music', The Musical Times, Vol. 124, No. 1683 (May, 1983), pp. 319.

90

Ligeti, György, Trio: for Violin Horn and Piano, (Mainz, 1982).

91

Ligeti, György, Hamburg Concerto, (Mainz, 1999).

92

Ligeti, György, Concerto for Violin and Orchestra, (Mainz, 1992).

51

piano

93

(1988 and 1992 respectively). The Konzert for Violin and Orchestra involved

tuning entire instruments to natural string harmonics. The 7

th

partial of the double bass's

G string is used as the reference pitch for one violin from the orchestra, which tunes its E

string up to the resulting F5 -34¢: the violin then retunes its remaining strings as fifths

below this F5. By the same process, a single viola from the orchestra retunes by

referencing its D string relative to the 5

th

partial of the bass's A string. This means that

one viola and one violin are in standard tuning relative to themselves but are respectively

31¢ flat and 14¢ flat relative to the orchestra. His performance instructions are very clear

about maintaining the integrity of the different tunings, as seen in this quote from the

performance instructions prefacing the Konzert for Violin and Orchestra:

To reduce deviations in intonation, both scordatura soloists play non vibrato

throughout and are careful not to adjust stopped notes to the rest of the

orchestra.

94

Ligeti had no practical interest in tuning systems other than as a method for adding

'dirtiness' to the soundworld of the piece.

95

In a 1978 interview with Péter Várnai he

claimed to ‘abhor all fixed systems’,

96

and later in the same interview talked about wanting

music ‘not based on quartertones, but mistuned music’.

97

Further examples of this pitch

indeterminacy in the concertos for violin and piano involve instruments of uncertain pitch

such as recorders

98

, slide whistles and ocarinas. The 2

nd

movement of the Konzert for

Violin and Orchestra features a chorale for four ocarinas which is beautifully mistuned.

Ligeti uses both indeterminate processes such as the reduced wind pressure in

Harmonies, and specific actions such as the simultaneous conflicting tunings in the

concertos, all in order to enrich the soundworld by creating intonational conflict. Detuning

instruments out of 12-tone equal temperament is often used by composers for quite the

opposite purpose as Ligeti and myself, they wish to make it more precise. Horatio

Radulescu and James Tenney are two composers who have used scordatura tuned to

natural harmonics so that players can easier perform justly tuned intervals. In Tenney's

string quartet Arbor Vitae

99

(2007), all the open strings are octave equivalents of partials

1, 3, 5, 7 and 11.

100

This scordatura allows him to create a complex tree structure of just

intervals in branching harmonic relationships, described by Tenney's student Michael

Winter as 'a series of related tonalities modulating through a richly populated extended

93

Ligeti, György, Concerto for Piano and Orchestra, (Mainz, 1988).

94

Ligeti, György, Konzert for Violin and Orchestra, (Cologne, 1992), performance instructions (no page number).

95

Steinitz, Richard, György Ligeti: Music of the Imagination, (London, 2003), 171.

96

Ligeti, György, interviewed by Péter Várnai in Ligeti in Conversation, (London, 1983), 54.

97

Ibid., 55.

98

Ligeti seemed to consider these as instruments of indeterminate pitch. They are played here by percussionists.

99

Tenney, James, Arbor Vitae, (Lebanon NH, 2006).

100

Winter, Michael, 'On James Tenney's Arbor Vitae for string quartet', Contemporary Music Review, 27/1, (February

2008), 146.

52

just intonation pitch space.'

101

In this piece, Tenney follows the compositional school of Just

Intonation (JI)—which grew mainly out of Harry Partch's work—by concentrating pitch

accuracy to ensure the purity of the just interval. Tenney was highly eclectic and many of

his other pieces—such as Beast

102

(1971)—uses vertical indeterminacy, this example

merely highlights a different purpose for intonation-based scordatura. Here, the vertical

aspect, as with Lutoslawski and Xenakis, is tightly controlled.

Another example of this kind of thinking can be seen in the work of Witold Lutoslawski.

The violin solo parts in the 7

th

prelude from the Preludes and Fugue

103

(1970-72) and in

Chain 2

104

(1988) include the technique of 'placing the fingers close together to produce

approximate microtones';

105

see illustration 12. Lutoslawski believed that specific

microtonal intervals on the violin would be too closely spaced for violinists to play easily

and so accepted the approximation provided by closely placed fingers.

106

This technique is

an isolated example of free microtones in Lutoslawski's output. When he uses

quartertones in works with larger string instruments such the Concerto for Cello and

Orchestra

107

—and in other works in general—he uses tempered pitches with specific

values.

108

This vertical indeterminacy was a singular occurrence Lutoslawski was only interested in

horizontal freedom for the performer, and then only in a limited fashion. He advocated a

'clear-cut division between the role of the composer and that of the performer' and did not

'wish to even partially relinquish the authorship of the music [he] had written.'

109

101

Ibid., 131.

102

Tenney, James, Beast, (Baltimore MD, 1971).

103

Lutoslawski, Witold, Preludes and Fugue: for 13 Solo Strings, (London, 1973).

104

Lutoslawski, Witold, Chain 2 for Violin and Orchestra, (London, 1988).

105

Petersen, Peter, 'Microtones in Lutoslawski's Music' Lutoslawski Studies, ed. Skowron, Zbigniew, (Oxford, 2001), 355.

106

Petersen, op cit., 355.

107

Lutoslawski, Witold, Concerto for Cello and Orchestra, (London, 1971).

108

Petersen, op cit., 344.

109

Stucky, Steven, Lutoslawski and his Music, (Cambridge, 1981), 110.

53

Other examples of Lutoslawski's horizontal indeterminacy include his use of part-scores in

the String Quartet (1964), where each player has an independent line. He developed this

idea further in the ad libitum sections of large scale works such as Jeux Vénitiens (1960-

61) where players freely follow their own lines within sections; a clever system of repeat

bars and conducted cues keeps the players synchronised by section. The indeterminacy in

this context is a way to give inner motion to harmonic constructs which are actually static

moments in a larger form. Lutoslawski's form is definitely teleological, he described his

own music as 'finipetal'—tending towards the finale.

110

Lutoslawski's reasons for using

limited aleatoricism are described by Steven Stucky as 'a means, not an end'; he

continues:

'The goal is twofold: to accommodate the performer by restoring his interpretative

role, and to realize specific textures of enormous microrhythmic complexity and

variety.'

111

Lutoslawski's reasons for using indeterminacy as a way to free the performer resonate

with the use of bounded improvisation in my own work. Although he claims not to count

'even to the least extent [...] on the possible creative abilities of the performers' our

positions on performer choice is similar as he allows them to interpret the rhythm in their

own way, allowing for rubato and expression.

112

I would eschew such expressive devices,

110

Rae, Charles Bodman, The Music of Lutoslawski, (London, 1994), 24.

111

Stucky, Steven, Lutoslawski and his Music, (Cambridge, 1981), 110.

112

Ibid., 110.

54

Illustration 12: Lutoslawski indeterminate fingerings in violin solo part.

but go a step further in general indeterminacy by allowing the performers to form phrases

and develop material from pitch material which is defined by me. Lutoslawski's performers

come from the orchestral tradition where interpretation is expected but not improvisation,

thus he is right to limit them to what they know. I expect performers from the

contemporary music tradition who have experience in improvisation and my music reflects

this. I restrict their ability to improvise in a similar way to that of Lutoslawski in his own

time, but in the context of the performance tradition of experimental music.

1.3.1.2:

Feldman and Memory

Towards the end of my PhD studies, I became more aware of the role of memory in the

form of my music. The forms that I had intuitively been using since around 2004-05 I was

now applying more consciously, allowing me to think more critically about their function.

Specifically this relates to the form wherein a single sound-object evolves through

repetition and (mostly) minimal variation. The key to this is summed up by Morton

Feldman's description of the text provided for him by Samuel Beckett for the opera

Neither:

Finally I see that every line is really the same thought said in another way. And yet

the continuity acts as if something else is happening. Nothing else is happening.

What you're doing in an almost proustian way is getting deeper and deeper

saturated into the thought.

113

As I outlined in 1.2.5 Evolution, this form takes firstly a compositional idea, a shell of a

thought that exists only in the mind, and renders it as a sound-object. Then there follows

another rendition that will be different in some way, then another that is different again

and onwards through a potentially endless series of renderings. Each sound-object is a

valid rendition of the idea, but none is 'the idea' itself, because there is no perfect

rendition. The form is what is generated from this iteration. From one object to the next

there may be a sharp difference or there may be no difference (pure repetition), the

patterns in time created by these differences make the form. Memory plays an important

role here because each new iteration may depend in some way on the previous: as with

the feedback mechanisms of strange attractors, this system relies on hysteresis.

The work of Morton Feldman exerts a strong influence on me in the development of this

form. The above quotation in reference to Beckett could easily have been about Feldman's

own work, as John Cage said in Silence, 'Feldman's music seems more to continue than to

113

Feldman, Morton, cited in Rich, Alan, San Francisco Contemporary Music Players, Mosko, Stephen (Musical Dir.), For

Samuel Beckett (sleevenotes), (1991) Newport Classics NPD 85506.

55

change'.

114

This sense of the work as an evolving world rather than a closed piece became

more obvious in his late works with their famously extreme durations. As an example, the

90 minute long Piano and String Quartet

115

is essentially a series of transformations on a

single sound-object consisting of an arpeggiated chord on the piano and a sustained chord

in the strings. Apart from a few direct repetitions, the chords never appear the same way

twice, but the differences between them are on the edge of what is perceivable. This is

achieved in the score in two ways. Firstly, the material itself is often quite neutral, clusters

with octave-displaced pitches, simple rhythmic patterns in irrational rhythms that do not

establish any 'clear-cut rhythmic shape'.

116

Secondly, the changes that he makes to the

material are so slight that they can go almost unnoticed By moving a pitch here or

changing a rhythmic value there Feldman is operating on the border of meaningful

change, and in this creates a language of ambiguity. However, by working with such small

levels of difference Feldman can now create play with this language and use levels of

difference as a method to shape the form. The most extreme examples of this are when

he chooses to place a single fff event into a piece that is primarily ppp: this occurs in

many of his works such as For Bunita Marcus (1985) and Trio (1980).

My own music too aims for this ambiguity between change and non-change.

Music can achieve aspects of immobility, or the illusion of it. [...] The degrees of

stasis, found in a Rothko or a Guston, were perhaps the most significant elements

that I brought to my music from painting.

117

By concentrating on the fine details of transforming his materials in this way, Feldman

creates the illusion of motion and simultaneously the illusion of stasis, a formal

multistability. In his writings, Feldman speaks of creating and breaking patterns, asking

'when does a pattern become a pattern?'.

118

This concern with the perception of structure

on a micro-level is part of Feldman's need for the music to exist in the now, to avoid grand

teleological structures. For example, many works from the 1970s such as Violin and

Orchestra

119

(1979) proceed as a series of blocks of instrumental colour—each negates the

last but maybe a repetition from earlier in the score—and the form emerges out of the

conjunctions and connections made by memory from these juxtaposed blocks: Boulez said

that Messiaen did not compose, he juxtaposed,

120

perhaps Messiaen was searching for the

same sense of stasis sometimes in his music. Feldman takes influence from Cage's

114

Cage, John, cited in Rich, Alan, op. cit.

115

Feldman, Morton, Piano and String Quartet (London, 1985).

116

Feldman, Morton, 'Crippled Symmetry', Give my Regards to Eighth Street, ed. B.H. Friedman (Cambridge MA, 2000),

142.

117

Feldman, Morton, cited in Rockwell, John, 'Morton Feldman (and Crippled Symmetry)', Morton Feldman Page, <http://

www.cnvill.net/mfrockwl.htm> accessed 11/12/08

118

Feldman, 'Crippled Symmetry', op. cit., 139.

119

Feldman, Morton, Violin and Orchestra (London, 1979).

120

Boulez, Pierre, cited in Cross, Jonathan, The Stravinsky Legacy (Cambridge, 1998), 55.

56

method of disassociating the parameters of sound from their structural roles by (in effect)

randomising their ordering, and says that it is 'not involved with the grammar of design'.

In my own music, I follow Feldman's techniques for keeping the music hovering on the

edge of change. The main difference between myself and Feldman is that my music has

moved away from notating this subtlety of change and instead creates performance

situations, such as Whitewater and Nano, where the principle of subtle change is

embedded in the rules. This means that the form of the music is not just realised in the

moment of perception but also at the moment of utterance. To apply Feldman's words to

myself; 'Before, my pieces were like objects; now, they're like evolving things'.

121

1.3.2: Spectralism

As is mentioned above, the music of the spectral composers Tristan Murail and Gérard

Grisey was very important for the development of my musical language both in terms of

techniques and aesthetics. When I discovered spectral music, it was the techniques that I

was most attracted to because they allowed me to create the microtonal soundworlds that

I was interested in. It took time for me to realise that it was the sound itself that was

interesting to me, and the ways in which these sounds could be used structurally. My PhD

studies show a fairly linear progression from using spectral techniques abstractly through

to a greater understanding of the nature of sound and perception. I have discussed above

(1.2.7 Vertical Indeterminacies) what I have learned from the spectralists in terms of

techniques, here I will briefly discuss what I learned from them in relation to time.

The first composer to mention must be the proto-spectralist Karlheinz Stockhausen. I have

only been dimly aware of his contribution to spectral music until recently, many of his

discoveries are simply presumed in the writings of the spectral school, or considered

common knowledge to anyone who had ever slowed down a digital sample (or tape reel,

or vinyl record) to the point that the pitch became pulse. When I wrote my early spectral

pieces I was completely unaware of his formulation of the relationship between pitch and

tempo that Tracheids, The Lady with the Hammer and Primes take as their starting point.

It is almost impossible to generalise about Stockhausen's soundworlds and compositional

techniques, at best I can say that while I was never attracted to much of his music in

more than a passing manner, there are plenty of instances where I have found something

fascinating in a technique that he has used or a sound that he has created.

121

Rockwell, op. cit.

57

Stockhausen's work in spectralism may have been, for me, overshadowed by that of

Grisey at al, but in researching his work I have some across a surprising number of

connections in relation to his use of indeterminacy: Stockhausen would have heard Cage's

lectures at Darmstadt in the 1950s,

122

but he also had an interest in probabilities and

information theory that would have contributed to this, he studied Communication theory

and phonetics with Werner Meyer-Eppler in Bonn from 1954-1956.

123

In Stockhausen's

Klavierst

ü

ck XI

124

(1956) there are examples of structural indeterminacy where the pianist

may choose from any of 19 groups of notes 'playing the first that catches his eye'. This

technique was hardly novel—see many open-form pieces or those influenced by the

mobiles of Alexander Calder, such as Boulez' 3

rd

Piano Sonata

125

(1963) and many works

by Earle Brown—but Jonathan Harvey notes as the most interesting aspect of this piece

Stockhausen's suggestion that the piece be played twice or more in a single concert.

Stockhausen considered the power of the open form to be the possibility that the more

one hears the piece the more clear the structure will become, based as it is on

probabilities. My own work with bounded improvisation leads to the same conclusions, in

that works such as Whitewater and Nano have more power to communicate their

relationships and structure over a longer time scale of listening. The more bounded the

area, the clearer the details are under repeated or extended listening, and the greater the

contrasts appear: performer Heather Roche explained to me that while performing Nano

she is amazed at how wide a semitone begins to sound.

126

Stockhausen uses indeterminacy to varying levels in many of his other works, especially

after his development of 'moment-form' in the late 1950s where he seeks a 'concentration

on the Now' to make 'vertical sections which penetrate across a horizontal portrayal of

time to a state of timelessness'.

127

This is another example of where Stockhausen and I

agree in principle but not always in execution. My work also attempts to sometimes freeze

the horizontal—in Whitewater and Primes the harmony is smeared across time by the

computer part, and in Five Bells for Elliott Carter a chord from Carter's 3

rd

String Quartet

is suspended in a set of spectral transformations—but with the exception of Five Bells for

Elliott Carter the moment is always a product of the past. Harvey describes Stockhausen's

'impatience with the teleology of musical moments which are always a result of past

musical events'.

128

Many of my forms rely on the feedback system where the 'now' can

only be a product of a particular past, and each performance of the piece will have a

122

Harvey, Jonathan, The music of Stockhausen (London, 1975), 13.

123

Ibid., 124-125.

124

Stockhausen, Karlheinz, Klavierst

ü

ck XI (London,1956).

125

Boulez, Pierre, 3

rd

Piano Sonata (Paris, 1963).

126

Personal communication, 21/06/08.

127

Harvey, op cit., 85.

128

Ibid.

58

series of different 'now' that depend on the previous 'now'.

Although written without the influence of Stockhausen, some of the text scores in my

portfolio—Primes, Whitewater and Lorenz especially—are similar in several ways to text

scores of his such as Ylem

129

(1972) and Stimmung

130

(1968). Both of our scores have

complex texts that must be internalised by the performer before the piece can be

performed. This is different to a standard musical score because while there is always the

possibility that even the most difficult score can be sight-read (assuming that the notation

in this case is a direct representation of the intended sound), these text scores exhibit a

kind of 'irreducible complexity'

131

in that all aspects of the score must be learned and then

put together. These scores are also dependent on performer input outside the normal

range of musicality, the performer is often required to create their own phrase-level

structures or supply the musical detail where only a structure and a principle or intended

sound-outcome are given.

Stockhausen's music also extends out to extremes of conceptual art and musical theatre,

much of which has open-ness and indeterminacy built in but bound by a context or quasi-

narrative. These works I consider to be largely outside what I can discuss in relation to my

own music.

The composers of the Paris-based spectral school, Grisey and Murail, have a much more

direct relationship with my work than Stockhausen does because, as seen above, their

techniques of pitch generation and transformation were an important influence on the

development of my own compositional language. Apart from the techniques of spectral

modeling and frequency modulation, I take from the spectralists their concepts and

aesthetics that relate to time, as Grisey states:

Strengthened by an ecology of sounds, spectral music no longer integrates time as

an external element imposed upon a sonic material considered as being 'outside-

time,' but instead treats it as a constituent element of sound itself.

132

He goes on to talk of the 'hypnotic power of slowness' and the spectral 'obsession with

'continuity, thresholds, transience and dynamic forms.'

133

All of this is clearly set out in the

seminal spectral works of the 1970s such as Grisey's Partiels (1975)—see 1.2.7.4 Spectral

129

Stockhausen, Karlheinz, Ylem (Kürten, 1972).

130

Stockhausen, Karlheinz, Stimmung (London, 1968).

131

An argument used by the supporters of creationism and I.D. (intelligent design) to suggest that some biological

systems and objects must have been designed (by a 'creator') because they could not have evolved from less complex

predecessors as they could not function at a lower level of complexity. The eye is frequently, and wrongly, used as an

example of this.

132

Grisey, Gérard, 'Did you say Spectral?', trans. Fineberg, Joshua, Contemporary Music Review, 19/3, (January 2000),

2.

133

Ibid.

59

Modeling (p.34). Partiels begins with a long section where a pure spectrum on E is

repeated while each new iteration is further distorted until it reaches a threshold of

inharmonicity and breaks apart. The second section is another linear transformation from

alternation of different harmonies and timbres (the line between these is blurred

smoewhat in this case) to a homogeneous consonance, and this again ruptures into a new

section, flurrying but harmonically static. The fourth section is another contrast of timbre

and harmony and again is reached via a linear transformation that culminates in an abrupt

change and a negation of the preceding soundworld, and this then slowly loses energy and

collapses into noise and silence. Tristan Murail commented on these early pieces—which

were for me, the most interesting and influential—that 'there is continuity, but there are

also ruptures and many other types of transition'.

134

While I definitely absorbed all of these formal concepts in my own music, I can see some

distinct differences between my own work and the spectral composers. Some of my earlier

works, such as Poetics of Knots (2003), After-Images (2004) and Prisma (2005), are

similar to these early spectral works in that successive sections use the same spectral (or

proto-spectral) material but create noticeably different soundworlds; a sectional approach

to form but with an underlying unity. As my language developed I moved more towards

the same type of formal models as before and that are exemplified by Partiels—those that

rely on linear continuities within a section and a rupture to signal sectional change—but

increasingly brought the material to the fore by reducing the level of difference from

section to section; M.Grisey, his Galliard (2005) alternates dense and sparse material that

is essentially the same, with occasional ruptures (mm.46-51, m.76 and m.96). By the

beginning of my portfolio my work is settling into the mature archetype where the

material has become a single idea and each section is another way to render that idea as

sound, later as exemplified in later pieces such as Nano and Whitewater. Here, the

spectral goal, described by Murail as 'the capacity to control the finest degree of

change',

135

is put into service as the facilitator of a language of ambiguity.

134

Murail, Tristan, 'After-Thoughts', Contemporary Music Review, 19/3, (January 2000), 7.

135

Ibid.

60

2: Chapter 2 - Works

2.1: Commentaries

2.1.1: Tracheids (2006)

Flute, clarinet, bass guitar, percussion: 4 minutes

Tracheids was written partly as an experiment to see if a strict relationship between

frequency and tempo would be perceivable to a listener: though the piece is not written as

a study, the tempo/harmony relationship was the primary material and the piece was

composed in a manner that would highlight this aspect. Tempo and harmony are related

through fundamental tones that are sufficiently low to be perceived as pulses rather than

pitches. These pulses form the tempo for a given section, as marked by constant crotchets

on a tenor drum, and the same pulse is also the fundamental tone for the harmony in the

section, for example, 1.71667Hz = 103bpm = 'A' (concert 'A' at 440Hz is the 256

th

partial

of this fundamental). The tenor drum keeps a constant pulse while the other three

instruments sustain a harmony derived from the given related spectrum. Successive

sections are marked by abrupt changes of tempo and harmony which, because they can

be very subtle—such as the tempo change from 109bpm to 115bpm at m.20—happen

mainly on a point between two crotchets to emphasise the change.

At the time of composition, I was unaware of Stockhausen's work in relating tempo to

frequency; from the mid 1950s, 'the “prime mover” of his his music'.

136

In his article

'...how Time Passes...',

137

he explains how 'duration and pitch are only different areas in

one time scale reaching from an impulse every eight seconds to 6000 impulses every

second',

138

and how this was incorporated into his use of the serial method to create a

tempo scale or tempo series where the tempi were related to the durations and pitches.

Although the basic concept of relating frequency to tempo is the same, many instances of

Stockhausen's application of the concept are diametrically opposed to mine in terms of

sound. In works such as Gruppen (1955-57), Stockhausen continues equates rhythms

and tempi with harmonies or tonalities and, as Harvey describes it, Stockhausen...

[...] fluctuates his tempi within his twelve-step scale': Harvey elaborates on this by

saying that if the music stayed in one tempo 'it would be the equivalent of tonal

136

Stockhausen, Karlheinz, cited in Harvey, Jonathan, The Music of Stockhausen (London, 1975), 30.

137

Stockhausen, Karlheinz, '...how Time Passes...', Die Reihe 3, (Pennsylvania, 1959).

138

Harvey, Jonathan, The Music of Stockhausen (London, 1975), 30.

61

music (having only one tonal centre instead of many).'

139

In my music I believe that the listener needs some time and repetition in order to perceive

these relationships and so my music tends to have a very slow rate of change. Jonathan

Harvey, being a spectral composer himself, appears to be in sympathy with this when he

later says that 'this is not to say that Stockhausen has written the ideal music for the

purpose; it is by no means educational music'.

140

While writing Tracheids, I was still trying to understand the techniques of spectral music

and as a result there are several aspects which are unsuccessful. Because I was interested

at the time in very dissonant harmonies, I made no effort to select pitches that would

emphasise the fundamental pitch of each section, so there is no sense of that section

having a specific pitch weighting. The opening five bar section is based on a fundamental

tone of A, but the pitches used are B2 +52¢, A?4, D5 +57¢, these pitches are obscure

harmonics of A which will not in any way promote a sense of A being the fundamental

pitch. In this case, my concern for 'interesting' harmony undermined the piece's idea. But

with that in mind, if I had made more effort to emphasise the fundamental pitch then the

sense that the tempo could be explicitly linked to harmony would also have been lost as

the harmony would have been overstated and there would be no reason to even consider

linking the tempo to the harmony. The correct approach I feel would have been to find a

common ground where the harmony only changes very subtly, so allowing the tempo

changes to come to the fore.

Ultimately, I consider this piece as an unsuccessful experiment for two reasons. Firstly, I

do not believe that the human perceptual apparatus can connect tempo to harmony in this

way, the tempo differences in some cases are almost imperceptible—for example,

A=103bpm, A?=109bpm—and secondly because I did not choose my harmony with care to

emphasise its relation to the tempo. I am satisfied with some compositional elements and

it is clear that the piece is not inconsistent with my compositional language. The piece

uses a block-sectional structure where the sectional boundaries are sharply emphasised

and the sectional material is a sound object which evolves as the piece progresses: this

can also be seen in works such as Intra-, Bifurcations, Whitewater II, and Inner Shadow.

2.1.2: The Lady with the Hammer (2006)

2 tuba, 2 bass trombone, 2 euphonium, 2 piano: 12 minutes

139

Ibid., 32.

140

Ibid., 34.

62

The title refers to Galina Ustvolskya, the Russian composer whose unusual orchestrations

—such as Composition no.2: Dies Irae for eight double basses, piano and wooden box

141

—

and use of obstinately repeated clusters had a strong influence on this piece. The

influence of the spectral composers is also unavoidable in the piece's harmonic structure

as all the harmonies are derived from prime-numbered partials of very low E (1.29Hz),

and in the middle section (mm.146-175) from G? (1.62Hz): see Table 1 below for a list of

prime-numbered partials and their sub-audio fundamentals. As with Tracheids, these sub-

audio fundamentals also act as the tempo for each section.

The basic technique in this piece is to have the brass acting in pairs to sustain dyads that

all activate the same combination tone. The two pianos act as resonators for this

combination tone by using silently depressed keys to let the string at the combination

tone pitch vibrate in sympathy with the brass. In illustration 13 the pianos are holding G5

and G4 open (as well as the low E1, which is left open for most of the score for general

resonance), these are the 304

th

and 152

nd

partials of E (1.29Hz) and both are octave

multiples of the 19

th

partial (G = 24.47Hz, a tone below the low A on a grand piano). The

brass pairs play dyads that sum to either 304 or 152, thus in theory they should all

produce combination tones that will cause sympathetic vibrations in the open piano strings

G4 and G5:

Trombones:

partials 251 and 53

Euphoniums:

partials 233 and 71

Tubas:

partials 109 and 43

(mm.14-16) tuba and euphonium: partials 263 and 41

141

Ustvolskaya, Galina, Composition No.2: Dies Irae (Hamburg, 1973).

63

The concept is taken to its logical conclusion in the final bars of the work where there are

pitchless tongue sounds at tuplet speeds which represent the prime-numbered partials

below audibility: in this scheme triplets are the 3

rd

partial, quintuplets are the 5

th

partial

64

Illustration 13: The Lady with the Hammer score page 3.

and so on.

This piece was written closely after Tracheids (around Christmas 2006) and can be

considered as an unsuccessful experiment for the same reasons as Tracheids, that is, too

rigorous an application of theory without fully understanding the practical implications; but

I did learn a lot about my own harmonic language in the process of writing it. The main

problem is that the combination tones are not audible, and do not cause the open piano

strings to vibrate. I had assumed that because acoustically generated difference tones

were audible (first order difference tones at least) then combination tones would also be

audible, but as Rossing, Moore and Wheeler point out in The Science of Sound, 'no one

has presented convincing evidence that even simple sum tones (f1 + f2) can be heard'.

142

Even if the combination tones had worked, there was also the problem that the

performance space would need to have a very dry acoustic in order to hear the piano

resonance above the reverberation of the brass fading away. The brass should

theoretically have enough acoustic power to cause the piano strings to vibrate at any pitch

(assuming the sympathetic vibration did not work), but the open piano strings used in the

piece were often too high to sustain for any length of time or with any volume, thus the

sound of the brass chords dying away would mask any piano resonance.

It would be possible to rewrite this piece with another more effective way to represent the

combination tones—any instrument with the correct range would work really—but I was

dissatisfied with the overall sound of the piece and so would rather rethink it as a new

piece than try to fix the existing version. I found that I was most satisfied with the close

harmonies, where notes less than a semitone apart grind against each other (see

illustration 14 below) this led to me considering harmony more in terms of closely-spaced

clusters and formed the model for a lot of my subsequent harmonic ideas.

142

Rossing, Thomas D., Moore, F. Richard, Wheeler, Paul A., The Science of Sound, 3

rd

edn. (San Francisco, 2002), 160.

65

2.1.3: Primes (2006)

Any high string instrument and Max/MSP: 10 minutes or more

Primes is an early incarnation of the interaction model that developed into Whitewater, but

with a very different harmonic idea. Primes is a text piece based on the pure acoustic

spectrum, with focus on the dialectic between the computer's perfect spectrum and the

near impossibility of the human performer's task to play intervals in perfectly tuned just

intonation.

In Primes, the performer has a single basic gesture which they repeat throughout the

piece, modifying it in response to their own improvisatory skills and to the response of the

computer. The gesture is in two-parts: first, an improvised melodic fragment (the score

defines a simple melodic model) in long tones which uses only pitches from the harmonic

spectrum, and secondly, this then becomes a drone which generates acoustic beats

through near unison playing with the computer part: a main source of tension in the piece

is this shifting duality of foreground and background as the player moves between melody

and drone. The computer has two simultaneous processes, (1) quasi-randomly generating

66

Illustration 14: The Lady with the Hammer close harmony.

a cloud of sine waves which are prime-numbered partials of a fundamental (signalled by

the performer), and (2) generating whichever prime partial is closest to the player's

current pitch. The computer part is a quasi-random slave to the performer, it must follow

the performer's choice of fundamental (which can be altered in real-time) and has no

agency to steer or end the piece.

The performer's part uses ‘bounded improvisation’: where a boundary is put on the

accepted types of material and how it may be used, but within this boundary are infinite

possibilities. The score defines only the moment-to-moment interactions, thus the form is

emergent, a function of a limited feedback loop between performer and computer. The

computer reacts to the player in a very limited fashion, but is constantly undermining and

limiting the player's response by its unpredictable actions. Form emerges from a bottom-

up process whereby the same macro-gesture is repeated many times, and each repetition

is part of an evolving form. Of the two pieces in this portfolio that use bounded

improvisation, Primes is simpler than Whitewater because the level of interaction between

computer and performer is largely one way: the performer seeds the computer part and

then reacts to the computer's material, reseeding it in the process: this forms a loop of

?

?

seeding

reaction/reseeding

reaction.

Use of bounded improvisation in this piece is extensive and functions to create horizontal

indeterminacy at all levels. The player is supplied with harmonic material and a phrase

archetype, then expected to improvise around this model. The indeterminacy is double in

that the player's own improvisation is indeterminate and then this is subject to

modification by the player in response to the computer part, which is itself changing in

response to the player. Together this forms a contextual indeterminacy, where the agents

act inside a feedback loop of change in response to change. Added to this is the

unpredictable nature of the computer patch, errors in analysing the input pitch mean that

the computer's output is not always predictable, adding another cause of indeterminacy. It

is possible that a poor signal between violin and computer could lead to so many errors

that the piece would be undermined—for example, if the trackers never read the pitch

correctly and consistently played pitches that the violin could not generate beats from—

this can only be mitigated against by careful rehearsal and management of the

equipment.

As was mentioned in the chapter introduction, Primes stands out from the later pieces in

the portfolio because its material is the natural harmonic series rather than the inharmonic

soundworlds of most of my works: although like earlier pieces such as Tracheids and The

Lady with the Hammer it does make use of prime-numbered partials to create and

inharmonic sounding world from the harmonic series. Primes however, also attempts to

67

introduce indeterminacy and impurity to the harmonic series by contrasting the

mathematical perfection of the series with the imperfect ‘live-ness’ of the performer: a

similar contrast is used in Marx (2008) between the synthesiser and the acoustic voice

and viola d'amore.

Like its successor Whitewater, Primes is written for solo performer and computer. The

computer's material is sine waves whose frequencies are prime-numbered partials of a

fundamental that is defined in real-time by the player: the patch listens for three

percussive taps in a row, and the pitch-class that follows that is taken as the new

fundamental. Primes uses prime-numbered partials up to the 503

rd

and these are, unlike

the low end of the harmonic series, very far from tonal sounding. They often have

harmonic characteristics that are very close to the inharmonic sounds I usually employ,

such as broken octaves or fifths, and local clustering: prime numbers can clump together

in small groups, such as {41, 43} or {277, 281, 283} and these create very close clusters

as high partials. Assuming a very low fundamental of G? = 1.62hz [equivalent to a tempo

of 97.2bpm] partials 277, 281 and 283 are a cluster of A4 + 31¢, A4 + 60¢ and A4 + 71¢.

Because the part of the patch controlling these sine waves is quasi-random, it is only

through chance conjunction that a cluster such as this will arise. But this chance is

increased because prime numbers tend to happen in clumps; illustration 15 shows a

section of prime-numbered partials on G? = 1.62hz to demonstrate this.

Illustration 15: Prime-numbered partials from fundamental G? = 1.62hz

The fundamental pitch was also mapped to the tempo of the randomly generated prime

partial sine waves; the fundamental was the tempo. Using the example above, a

fundamental of G? = 1.62hz is equivalent to a tempo of 97.2bpm. The frequency of the

fundamental supplied by the violin at the start of each section is simply divided by 64 to

give the tempo for the section. The patch assumes that the violin note sounded is the 64

th

partial, the player is instructed only to use the octave G3-G4 to sound fundamentals: this

reflects that the first performance was by a violinist.

In the harmony of Primes there is a conflict between the mathematically perfect harmonic

pitches played by the computer and the live player's attempt to match this or to create

68

beats by mistuning to it. Tension in the piece is created by the live player moving between

two roles: a foreground role where they play melodies against the web of prime-numbered

partials from the computer, and a role where they blend into the computer part by playing

long drones around its pitches. This tension is increased by the computer using two

'tracker' drones to detect the pitch of the violin and play the nearest prime partial to it.

Beats are generated firstly as a by-product of the violin attempting to play perfectly in just

intonation while in close proximity to the perfectly intoned computer pitches, and then

deliberately by the live player's drones mixing with the computer's.

The computer is essentially a processor rather than a truly interactive component. The

patch follows the performer's pitch, has no agency to change sections or end the piece,

and has limited agency to create pitches of its own: only the quasi-random generation of

background pitches. The performer has their own material that seeds the computer

(providing the patch with something to react to), then the performer responds to the

computer's performances. The most suitable metaphor for interaction in Primes is that of

the ventriloquist act, where the computer is a life-like dummy, under the performer's

control.

Primes creates a beautiful soundworld and works well in itself, but falls short of some of its

original aspirations. The piece was written early in my research into spectral music and as

such it incorporates some ideas which are impractical or not implemented correctly. The

main idea was that there would be a clear sense of 'key change' when the violin alters the

fundamental during the piece, but this exaggerates the importance of spectral pitches

alone in determining key identity: this problem is in the same category as the problem of

'key' identity in Tracheids. The point of using prime partials is that they are the least tonal

sounding but at the time I failed to make the simple link that tonality requires tonal

relationships to be perceivable.

On the recording of Primes submitted for this thesis, the performer required a fixed score

as she could not fully integrate the improvised element in time for the performance: see

Whitewater commentary below (2.1.8 Whitewater – p.79) for more discussion of this

issue.

2.1.4: Filament (2006)

Guitar quartet: 1 minute

69

Filament consolidated my work on creating sound objects which are harmonically static on

the macro-scale but exhibit complex motion on the scale of the micro-interval. The macro-

level structure of the piece is a single gesture that becomes a sustained tremolando chord

across the quartet. At the micro-level the chord is based on an E-major/minor triad but

uses constant sub-semitonal glissando motion across all the instruments to avoid the

chord settling on a single harmonic identity. As illustration 16 shows, the chord begins

with three non-tempered pitches, G5 and two G?5 with different intonations, bounded by

F5 and B5. This sound object is both ambiguous at the level of pitch and (under the right

acoustic circumstances) will fuse into a single sound-mass as there are no strong attacks

that would allow one instrument to dominate the texture and all the instruments are

constantly in motion (both through glissando and tremolando). This chord is sustained for

about two thirds of the piece before a recapitulation of the opening gesture leads to a

more stable E triad with the upper and lower pitches constant but still with the

major/minor ambiguity as G and A? glissade

143

across each other.

143

The word 'glissade' is taken from the writings of Alvin Lucier and James Tenney as a more elegant alternative to using

'glissando' as a verb: compare 'glissading' to 'glissandoing'. 'Glissade' has a similar root to 'glissando' in that it also

means in French 'to slide'. See the Lucier quotation on p.37 (footnote 57).

70

Illustration 16: Filament harmony.

Creating this pitch ambiguity derived from my interest in inharmonic spectra and is based

on my attempts to model my harmony on the harmonic attributes that I had observed in

bell-like sounds. The Filament chord is thus based on a triad but with multiple

simultaneous contradicting pitches such as the multiple major and minor thirds that are all

unequally tempered to some degree, as well as the flattened dominant (B5 -50¢) and the

sharpened tonic (E5 +50¢, or sometimes a semitone sharp at F5): there is always one

tempered triadic pitch present in the chord in any given bar, but this is always

compromised by other untempered triadic pitches. This harmonic development was a step

towards vertical indeterminacy, the pitches within the chord are unstable but it is a

carefully defined instability and localised to a specified harmony and tonic.

2.1.5: Inner Shadow (2006)

Violin, cello, bass clarinet, accordion, acoustic metronome: 6 minutes

Inner Shadow was commissioned as part of the Samuel Beckett centenary celebrations in

Dublin. Samuel Beckett has always been a powerful source of inspiration for me,

especially in his construction of ambiguity, both in language and in form. The influence of

Beckett in this piece may also explain the use of theatrical gestures in this piece as they

appear in almost no other work of mine: the last minute of the piece the violin and bass

clarinet slowly walk off-stage while still playing, and there is an acoustic metronome which

beats at 60bpm for the duration of the piece, thus the piece is bound by the gesture of the

metronome being switched on and off.

This piece continues my development of close harmonies with microtonal clusters and

perceptually fused sound objects. Formally the piece is a series of sections based on the

same material and marked by abrupt transitions: sometimes to silence and sometimes to

a static bridging passage. The material in each section is based around a microtonal

cluster and as in Filament the harmonic identity of the chord is in flux because at any

given moment at least one pitch in the chord is glissading slowly across a sub-semitonal

interval. This internal motion imbues the chord, and thus the sound-object as a whole,

with an inner life made up of beating patterns and timbral fusion between the

instruments, all of which leads to a level of ambiguity between chord and timbre/sound-

object.

Inner Shadow is an early score of mine to include indeterminacy: one of my first

experiments in that area. Despite the metric notation—although the first version was a

quasi-graphic score—some of the main sound attributes are not fully notated and thus

71

become indeterminate to a certain degree; indeterminate within certain boundaries. The

score is metric in order to keep the players together but the pitches are often microtonally

indeterminate (a little sharp or a little flat) and the texture is dependent on the pulsing of

the instruments, this is only included as a text instruction in the score so the exact speed

of pulsing is indeterminate.

The graphic score (which, unfortunately, has since been lost) was not abandoned because

it did not represent the music properly; I moved to a metric notation purely because the

ensemble found the graphic score too awkward to learn quickly, they wanted something in

a language that they could assimilate quickly. I believe that the graphic score was a more

truthful representation in that musical elements that were loops were portrayed once and

the player looped them, this is to my mind a more truthful representation of the nature of

the loop than the metrical grid representation. The graphic score also showed more clearly

when instruments played together—a gesture for example—and when they were

independent, while the metric notation at least implies rhythmic concurrence if not

outright defining it, and this undermines the attempt to portray certain phrases as being

rhythmically independent. The graphic notation may not be perfect, but it is closer to my

intentions. The metric notation is not ideal, but at the time it was a fair compromise given

the limited rehearsal time.

The use of the metronome was not originally intended to be theatrical, it ended up being

so because it seemed (to the ensemble) to make more visual sense if I walked to the

stage and started the metronome, rather than a member of the ensemble doing it: I

appear to 'switch on' the piece. The ticking of the metronome is a constant element in

Inner Shadow that acts to gauge how static the ensemble music is. When the ensemble

music is active, the metronome fades into the background as one more mechanism in a

cluster of mechanisms. When the ensemble has more static music, the metronome comes

insistently to the foreground. There is no counterpoint between the two because the

rhythm of the instruments is mostly an indeterminate but constant pulsing that the

metronome simply blends into. When the instruments are static and sustaining the

metronome stands out as the only moving sound: in gestalt perceptual theory, this is the

principle of figure and ground, a combination of principles that (mainly) describes the

grouping of moving objects against a relatively static background.

As an aside, after the first performance of Inner Shadow, a member of the audience

mentioned to me that Beckett often used a metronome in his rehearsals to train the actors

to move with a certain timing, I am unable to find reference to this in sources on Beckett

but would like to believe that this is accurate, not to validate my use of the metronome

but merely as an interesting coincidence.

72

2.1.6: Distemper (2006)

Solo equal-tempered plucked/struck string instrument and Max/MSP: 10+ mins

Distemper is the first in a projected series of works called The Book of Functional

Harmony, all of which use Max/MSP to deal with the problem of writing in my microtonal

language for equal-tempered instruments. Each piece solves the problem in a different

way and with a different aesthetic, some are game pieces and some fully composed.

The approach taken in Distemper is to record long sections (typically 1 minute long) of

audio in real-time and then play them backwards—by reversing the file—and slightly

faster or slower to alter the pitch. One objective of this technique is to create beating

patterns between the real and recorded parts. For the other objective, the recorded part is

played back in reverse to allow the player to create sound events without attacks. The

player must time their attack so that it coincides with the reversed attack in the recorded

layer. This should create a sound object with a long fade in (the reversed tail of the note)

that crescendos to the doubled attack of live and recorded note and then fades out on the

long tail of the live note. These recorded layers then build up in a manner that owes a

large debt to Alvin Lucier's I am Sitting in a Room, partly because of the increasing

density of the piece and partly because the piece echoes Lucier's wish (as he says in his

piece) to 'smooth out any irregularities [his] speech might have'

144

by smoothing out the

instrument's attacks. Table 1 shows the layering of sections as they alternate between

normal and reversed playback.

Section (time?)

Live

A

B

C

D

E

F

G

H

I

J

(tacet)

Computer

A rev

B rev

C rev

D rev

E rev

F rev

G rev

H rev

I rev

A

B

C

D

E

F

G

H

A rev

B rev

C rev

D rev

E rev

F rev

G rev

A

B

C

D

E

F

A rev

B rev

C rev

D rev

E rev

A

B

C

D

A rev

B rev

C rev

A rev

B rev

A

Table 1: sections in Distemper showing layering of recorded parts.

144

Lucier, Alvin, I am Sitting in a Room (New York, 1970).

73

The speed alterations are applied more intuitively. To avoid a sensation of sudden pitch

change at the beginning of each section, the speed alterations are applied gradually; ten

seconds at each end of the file playback are devoted to accelerating or decelerating the

recorded part—to and from normal speed—as required. As the piece builds up layers the

various speed changes cause layers of beating in different registers. In Illustration 17 the

computer part (lower stave) shows reversed sounds with hairpins indicating approximate

length of tail and the letters 'd' or 'u' indicating whether the pitch will be down or up

relative to tempered pitch. This version of Distemper uses a single pitch class (E) almost

exclusively in order to highlight the beating patterns: this also draws on Lucier's approach

in that the composition is limited to the phenomena that the composer wants the listener

to hear, eschewing other surface musical details that may distract without contributing to

the phenomena (in this case, the beating effect between closely tuned pitches, and the

smoothing of the attacks by introducing the reversed sound-tail).

The version of Distemper submitted with this portfolio is fully notated. I was interested in

doing a version that allowed bounded improvisation but never found the right set of

boundaries, this remains a possibility for a future version.

2.1.7: Five Bells for Elliott Carter (2006)

String quartet: 8mins

74

Illustration 17: Distemper, section D. Upper stave is live part and lower shows computer part

(playback of recorded live sections, hairpins indicate reversed sounds).

Five Bells for Elliott Carter – Orch. (2008)

String Orchestra - 12 violin, 6 viola, 6 cello, 2 double bass: 10mins

This work—in both versions—begins with the premise that the twelve note chord which

both opens and closes Elliott Carter's 3

rd

String Quartet

145

can be frozen in time. The chord

is frozen in a process which alters the intonation and dynamic weighting of the pitches in

order to make them more like a generic bell spectrum. Many possible bells can be made

from Carter's chord, the piece's moves through several of these before returning to the

Carter original. To stay close to the Carter—and allow my piece to act metaphorically as an

extension of it—this piece was written for string quartet, but that was found to be

unsatisfactory for various reasons and so I rewrote the piece for string orchestra—scores

for both have been submitted in the portfolio but only the string quartet version has been

recorded so far. The following discussion will begin with the quartet version and then move

on to the rewritten version and the reasons for the rewrite.

Carter's twelve-note chord requires all the performers to play triple-stops, a technique

that requires very heavy bow pressure and cannot be sustained for long. The intensity of

the triple-stopped playing then gives way to the static tableaux of the chord frozen in slow

legato arpeggios. The players maintain the same pitch-classes for eight minutes, with

subtly altered intonation and gentle dynamic shifts as the piece progresses through a

series of bell-like configurations. The bell spectra are created by emphasising two

opposing harmonic centres within the chord and so creating a multistable harmonic object.

This is achieved by altering the intonation of pitches in such a way as to create ambiguity

around which harmonic centre each pitch belongs.

Illustration 18 is an excerpt from my score showing the original Carter chord and the first

few transformations—note that transformations are randomly ordered rather than

sequential. Dynamics have been removed from this example but the reader can assume

that single pitches are focal, thus have higher dynamics, and small noteheads in chords

are pitches to played very quietly. The second chord in illustration 18 is made bell-like by

choosing the viola B?3 as the strong partial. To support this the violin's top G?6 is flattened

slightly to make it a 7

th

partial and the second violin's F5 is the 5

th

partial. Common

attributes of bell-like spectra include the prominent minor 9

th

, B5 in this chord, and

placement of the major third D (D3 on the cello) below the perceived fundamental while

the minor third C?4 is above it; the minor triad sound is especially characteristic of church

bells, a well known example being those of St Paul's Cathedral in London. Beats are

created between the viola's F5 and flattened G?4. A secondary strong tone G3 should also

145

Carter, Elliott, String Quartet no.3, (New York, 1973).

75

be apparent due to the strong support of its 5

th

partial B5 in the violin and the cello's D3.

In Five Bells for Elliott Carter, the degree of mistuning due to notational imprecision is not

significant. As each separate chord in the piece is a different alteration of Carter's original

total chromatic chord in such a way as to imply a bell spectrum, the degree of mistuning is

not as important as the fact that successive chords are different, thus contributing to the

form of the piece. As was shown above, the bell spectra pitches involve either altering a

pitch in twelve-tone equal temperament so that it matches a harmonic partial to support a

lower pitch in the chord, or altering a pitch to create unison or octave beats within the

chord. Because the bell spectra are inherently inexact in terms of strict mathematical

relationships between partials there is no need for notational precision here.

The score controls the macro-form of the chord sequence by specifying the timing,

dynamic and intonation for each section. The moment to moment details of the note

movement are left indeterminate. A simple example of this is in defining events by bow

and breath lengths—seen also in Lorenz, Nano and Whitewater II. This is partially used to

avoid notes and events being broken up by a bow re-taken or a new breath, which is

never smooth enough not to break a still sound. The other reason is to generate a time

unit which is variable in duration but within limits. The duration of a violin bowstroke for

example is dependent on parameters such as register, dynamics and tone colour but will

generally last between six and ten seconds. This may seem like a wide variation—and in a

faster and more rhythmically dynamic piece it would be unacceptably wide—but in slow

76

Illustration 18: transformations of Carter's chord in Five Bells for Elliott Carter.

music such as this the differences at note-level are not in the perceptual foreground: they

are all simply 'long notes'. In Five Bells for Elliott Carter this is used to blur the chord

changes. The players are bowing independently, with full bow draws each time, but must

complete each bow before changing pitch. So when the time indication for a chord change

arrives, they must finish their bow before changing, thus staggering the chord.

The piece has been performed on two occasions; the premiere was by a student group

who were not a regular quartet, and the second performance was at the Ostrava Days

Festival 2007 by Sonar Streichquartett. While both performances were well received,

neither performance of the work has been entirely convincing to me because the expected

harmonic fusion was not perceivable; the changes to the intonation and pitch weighting

did not create bell-like sounds. There is no perceptual fusion because the players are

bowing slowly alternating double stops rather than sustained chords and no matter how

smoothly they bow it is still possible to perceive each player separately and so the pitches

do not fuse into a unified—although multistable—timbre as expected. I assumed after the

first performance that these problems were due to inexperienced players, but the Sonar

Streichquartett performance showed that the problems were built into the piece.

On another level the piece still 'works' because there is a sense that something is always

changing and the piece keeps moving. But the piece does not work in the way it was

originally supposed to. However, I still consider the quartet version to be a viable piece

because I like how it sounds and accept that the thought processes used to compose a

piece are not necessarily related to how a piece should be heard. The quartet version

sounds like a piece of music, not like a failed experiment. The failure of the idea behind

the piece to work opens up the possibility of a new version that addresses its problems, I

see the problems as being that:

The string quartet as a medium is too thin for the proliferation method I used. If all

players attacked the chord simultaneously each time then perceptual fusion may

have been more successful, but the overlapping entries allowed individual

instruments to stand out too much, this undermines the perception of all four

instruments as a single timbre.

The 1

st

violin part is very high, even good players cannot sustain these double-

stops without some squeaking: the Sonar Quartet's 1

st

violinist used a mute in their

performance, this alleviated the problem slightly. Perhaps replacing the upper note

with harmonics would work, if this were feasible. Lowering the A-string to G would

be an elegant solution but would require that the top note be fixed, with no

77

intonational changes possible. This would be acceptable as the note is so high that

it does not blend well with lower pitches but a harmonic may be blend better.

The solution to these problems was to rewrite the piece for a larger string group. In 2008,

Five Bells for Elliott Carter – Orch. was written for 12 violins, 6 violas, 6 cellos and one

double bass: with the exception of the double bass, there are two players per part in the

score, this can be scaled up for larger ensembles. The first alteration is that the strings no

longer play arpeggios so that there is less motion within the chords. Also, this means that

all the pitches are audible continuously, thus increasing the possibility of spectral fusion.

Because each player now only has a single pitch—or occasional double-stop—to sustain,

there is more room for controlling the morphology of the chord through dynamic swells.

These link particular pitches together due to the gestalt law of common fate: if there is no

other dynamic motion within the sound mass then a group of pitches swelling together

should be treated by the ear as a single object. If there are multiple groups of

independently swelling pitch groups within the chord then this should foster ambiguity as

to which is the pitch centre, or which relationships the ear should follow.

Many decisions about register and orchestration have been taken with respect to spectral

fusion. For example, at time index 5'35” the strongest note is A4 in violin-6, but this is

played on the G-string where it is so far up the neck of the instrument that the short

string length creates a very muted sound, compounded by the instruction to play sul

tasto. This means that although it is the loudest pitch by dynamic it should not stand out

from the chord as it is weak in high frequency overtones, the overtones of the other

pitches will mask these and allow the A to blend. The muted sound for one group of

pitches in this chord is contrasted with a mistuned G? minor/major triad in violins 1, 2 and

5 played sul ponticello that due to its low dynamic may or may not be perceivable as a

separate sound object, I would hope that the effect is subtle enough to cause a

disturbance in the overall perception of the chord and not enough for it to become a

separate percept.

In some other cases the strong pitches in the chords are allowed to stand out, not all the

chords are orchestrated to create such ambiguity. I elected to avoid an overly dogmatic

approach to orchestrating the chords, the variation between orchestrational principles

allows each chord to have its own character, rather like bells in the real world.

Other alterations include the maintaining of the string quartet for the loud beginning and

endings of the work, this keeps in place the metaphorical association with Carter's quartet

and also means that when the full string ensemble comes in there is a change of space

from the chamber to the orchestral, to reflect the change in how time will be perceived

78

relative to the soundworld of Carter. The piece has also been extended slightly by making

all the sections 20-30% longer, to allow more time for each new chord to settle and for

the ear to be pulled into the new harmony more—thus more time for psychoacoustic

effects to come into play and for the mind's ear to create relationships in the harmony.

2.1.8: Whitewater (2006)

Any wind instrument(s) and Max/MSP: 15 minutes or more

The aim of Whitewater is to use multiphonics as harmonic material and focus on their

inharmonicity. The piece also aims to build on the interaction model used in Primes and

make a piece that is truly interactive on the harmonic level. To achieve this, Whitewater

uses bounded improvisation with a choice of material that is wider (any stable multiphonic

and its constituent single pitches) but with a more robust harmonic identity than Primes.

Whitewater also improves on the interaction by using a computer patch that has equal

agency to the live player in terms of leading the evolution of the piece.

To contribute meaningfully to the harmonic evolution of the piece and on an equal footing

with the live player, the computer uses a cellular automaton as a life-like spectral filter.

The computer analyses the input multiphonics in real-time and responds with its own

version in the form of sine waves, the computer then breeds new versions of the

multiphonic by treating each partial of the spectral data as a cell in a cellular automaton—

see below for a detailed discussion of the cellular automaton. The player's subsequent

interaction with these 'bred' multiphonics creates a loop whereby the cellular automaton

keeps altering the data and at the same time is constantly reseeded by new data from the

performer, which is itself is a response to the what the computer does. Unlike Primes, the

computer here has complete agency, it has the same level of responsibility as the

performer to alter or stop the piece. The computer's agency is quasi-random and not a

true intelligence like the live player, but the material and the interactions allowed by the

score are simple enough for both agents to contribute equally: in effect, the live player is

brought down to the level of the computer in terms of decision making. Left to its own

devices, the computer may become locked in a loop or may simply stop, depending on the

particular material in circulation at that moment, the player's job is to lead the computer –

as a charmer leads a snake.

2.1.8.1:

Cellular Automata

For me, if one chooses to accept a definition of music as organised sound, the organisation

79

needs to be perceivable on some level. For music to be understandable it must have a

level of perceivable organisation which falls between two possible states—the extremes

and boundaries of which differ for each listener. At one extreme is the state of order

146

,

with minimal variety and maximal predictability, while the other extreme is that of

maximal variety and correspondingly minimal predictability known as chaos. Music is a

lifelike entity which exists in the area between these two states.

147

Strange attractors also exist in this 'inbetween' state which is characterised by a variable

mixing of order and chaos. The attractor never comes to rest in either state but also

doesn't necessarily fall into a periodic oscillation; it may be infinitely varied. Strange

attractors sit at this boundary between chaos and order in the same way that

indeterminacies in music may oscillate in a seemingly random way inside a fixed space. In

my portfolio are examples that represent this process in sound, such as the ever shifting

harmonic relationships between live multiphonics and the evolving computer-generated

multiphonics in Whitewater, or the overlapping independent lines of Lorenz.

Cellular automata are a class of logical automaton processes in which the repeated

application of simple interactive rules to a set of two state (on-off) cells on a grid can

produce an astonishing variety of complex systems.

The best known example of these is John Conway's 'Game of Life' from 1970, which owes

its name to the surprisingly life-like dynamic structures which can arise from many

starting conditions. Illustration 19 shows three successive generations taken from a Game

of Life run. The continuous evolution of the shapes is clear even in a small sample such as

this. Cellular automata are used in my piece Whitewater as the engine by which the

computer evolves material from the multiphonics received as live input.

From my musical perspective, systems that lie between chaos and order—represented by

146

As an aside. In the Discworld novels of Terry Pratchett there are minor characters called 'the Auditors' whose purpose

it is to reduce the Universe to extreme order by removing all variety and individuality. They occasionally fall foul of a

fatal attack of vanity by referring to themselves in the first person. See Pratchett, Terry, Reaper Man, (London, 1991) or

Pratchett, Terry, Hogfather, (London, 1996).

147

Eco, Umberto, The Open Work, (Cambridge MA, 1989), 51.

80

Illustration 19: three successive generations (l-r) of Conway's Game of Life

cellular automata such as Game of Life—are the most interesting because they maintain

the balance between known and unknown that is essential for musical interest. This is put

to work in my portfolio most clearly in the works which use text processes and bounded

improvisation rather than fixed scores. The rules defined in the text score allow the piece

to self organise in a manner analogous to the self-organisation of cellular automata and

strange attractors. By limiting the mechanism of transformation, the identity of the piece

stays within acceptable limits to ensure it always sounds like the same piece. By working

within a continuum—such as frequency values instead of discrete pitches—there is an

infinite range of possibilities for evolution so that the space is never exhausted.

The Max/MSP patch for Whitewater uses a cellular automaton to filter partials. Under the

cellular automaton rule system, cells may tend to the extreme states of either chaotic

proliferation, or total order if all the cells die. What is more typical is the middle ground

solution, that the grid settles into one of an infinite number of possible states of periodic

oscillation: where some or all of the cells are locked into a predictable pattern. For

Whitewater, this is the most musically interesting phenomenon as it allows the player

greater options for interaction.

However, the preceding description of generic cellular automata refers only to the process

itself, the system becomes more complex once we examine its application in Whitewater

where the cellular automaton is a process embedded in a two-agent text process. In this

situation, rather than being left alone to evolve after the initial seeding of the cells, the

interaction between player and computer is constantly reseeding the automaton with new

cells, which leads to a considerable degree of indeterminacy and iterative feedback.

The cellular automaton in Whitewater is in effect a complex filter acting discretely on the

partials of the input sound. This approach appears to differ from previous implementations

of cellular automata in music, which mainly fall into two large paradigms: (a) note-to-cell

mapping which focuses on localised pattern variation in midi/note-based music; (b)

Granular synthesis implementations, focusing on stochastic pattern variation and swarm

effects in timbre-based music. For a more detailed history of cellular automata

implementations in music, please refer to the article by Burraston and Edmonds titled

'Cellular Automata in Generative Electronic Music and Sonic Art: A Historical and Technical

Review'.

148

For Whitewater, neither of these earlier approaches were suitable. Because Whitewater

uses spectral definition and discrete frequency content rather than quantised pitch-values,

148

Burraston, D., Edmonds, E., 'Cellular Automata in Generative Electronic Music and Sonic Art: A Historical and

Technical Review', Digital Creativity 16/3, (2005), 165-185.

81

I wanted to avoid implementations which were based on the mapping of graphical cells to

fixed pitch values. Similarly, the swarm-effect paradigm doesn't allow sufficient control at

the frequency level. Rather than controlling stochastic distributions of frequencies, I

needed to be able to manipulate individual partials (frequency/amplitude pairs) which

change over time.

My initial idea was to take the partials of a sound and treat them as though they were

cells on a cellular automaton grid. This idea was refined by Pierre Alexandre Tremblay who

suggested replacing the unworkable grid concept with a frequency continuum—an

approach which fits much better with my compositional language in general. In this

approach, the cell's neighbourhood is defined as a musical interval. Thus, applying the

same rules as above, if there are either three or four partials alive in the neighbourhood

then the partial being tested remains alive, otherwise it dies: for example, if the

neighbourhood chosen is a major 2

nd

, then if there are three or four partials within a major

2

nd

of the partial being tested then that partial is not filtered out. This mechanism was

coded by Dr Tremblay as the spectralConway object; in the Whitewater patch

spectralConway is used in its javascript version and further references in this document to

spectralConway will always mean the javascript version—see Appendix 1 for the complete

javascript code.

The Max/MSP patch contains a javascript object that is emulating the cellular automata,

while Max/MSP itself is only doing two relatively simple tasks: pitch analysis is carried out

by the fiddle object, and the resynthesis is handled by two ioscbank objects that

process alternating pitch sets and overlap each other in time to ensure there is no break

in the sound. The pitch analysis happens in two sections:

1. The raw spectral data is constantly being filtered through sprintf into a coll.

2. Fiddle registers an attack then (after a 10ms delay to bypass the attack

transient) sends three 'dump' messages to the coll, with 1000ms delays between

each after the first. Fiddle is configured to read the 15 strongest partials in the

signal but often only finds six or seven. The three dumps method ensures that

enough partials are sent from the coll into the spectralConway javascript engine

Once the data is sent into spectralConway then the javascript takes over.

1. A temp array is created to handle the incoming frequency/amplitude pairs and

these are sorted by frequency.

2. The index of dead and alive cells is checked and if there is at least one dead cell

then that is replaced with the next new frequency/amplitude pair from the temp

82

array. This process continues until all the dead cells have been replaced or the temp

array is empty.

3. The new array of live cells is output into Max/MSP as frequency/amplitude pairs to

be rendered as audio by the ioscbank objects.

The cellular automata in spectralConway will continue to run as long as the 'evolve'

message keeps being sent from the Max/MSP patch. The 'evolve' function in

spectralConway runs steps 1-3 above and processes a new generation of cells in the

automata. The purpose of the automata is to vary the spectra from the live sound and

provide a foil for the live improviser, and if there was only a single sound played live then

the automata process would tend towards its states of silence, chaos or periodicity. Each

time there is new input from the performer then the automata is being reseeded with new

data, thus the computer part is constantly being interrupted by the player. Equally, the

player's attempts to form phrase-level objects are interrupted by the computers evolving

harmony. This relates to the form of Lorenz in that the different agents are impeding each

other, and from this struggle arises the form.

The basic mapping of the multiphonic onto the cellular automaton is the beginning of the

evolution process. The cellular automaton is itself an indeterminate process (dependent on

hysteresis and sensitive to tiny changes in input) inside the web of indeterminate

processes and feedback systems that make up Whitewater. The computer part begins as a

sonic near-match to the live multiphonic and then follows an indeterminate path to several

possible states; including the extreme states of silence and chaos. However, the most

likely end state is a periodic oscillation between a small set of pitches or timbre

complexes.

The computer's output then becomes an element in the live player's text process. Because

the live player is trying to create beats with the computer part, there is an internal

feedback loop that pulls the player back to the pitch of the computer. But the interaction

with the computer is a component of the computer's own feedback loop which modulates

its partial set with the most recent live pitch. This could be a static process but for the

indeterminate nature of both agents' responses. The live player will deviate slightly from

the computer's pitch in order to create beats, and this deviation will in turn become

multiplied through a frequency modulation process. Whatever is generated from this

frequency modulation process will then be filtered by the cellular automaton—which is

indeterminate in itself—before being returned to the live player who starts the process

again. To add a level of complexity, the computer patch inserts an extra frequency

modulation process as an internal feedback process between the live input and the

83

filtering cellular automaton. The filtering process is continuous, while the frequency

modulation happens only when there is an input, thus the computer part tends to vary in

cycles related to the activity of the live player. This is one of the main sites of vertical

indeterminacy in the piece. The player has a level of control over the multiphonics and

pitches but their evolution is dependent on a multivalent feedback process with several

rule-sets operating simultaneously.

In Whitewater, spectral modelling is performed by the computer by analysing the real-

time input of the solo performer. The computer takes the multiphonics of the player and

'breeds' eccentric variations before resynthesising them from sine waves and playing them

back. The breeding process is performed by a cellular automaton acting as a filter that

removes partials according to a proximity-based rule system; the rules favour small

clusters separated by large intervals. Because the cellular automaton only removes

partials, the computer patch includes an internal feedback system which adds extra

partials derived by modulating the incoming partials against the most recent pitch played

by the live player. As was seen earlier, a high degree of pitch material in multiphonics is

derived from modulation and this extra process adds to the material in an indeterminate

way which is dependent on the immediate pitch context as it evolves—which happens as

some partials are filtered out and others added in. The evolution of this material is entirely

dependent on both the text process in the score and the computer patch.

2.1.8.2:

Bounded Improvisation and Performance

Whitewater takes the principles of bounded improvisation and interaction in Primes one

step further by giving the computer part complete agency. This is achieved by allowing the

computer to vary its own material via a cellular automaton. The feedback model described

for Primes is essentially the same, but increased in complexity as the computer part is

now undergoing spontaneous transformation: the ventriloquist's dummy takes on a life of

its own, making the ventriloquist speak in tongues.

The bounded improviser playing Whitewater may use any stable multiphonic and its

constituent single pitches (and intonational variants on these) as material. The phrase

level ordering is not prescribed other than that long notes must be used: as is largely the

case in Primes, this is both aesthetically pleasing and allows the computer time to react.

As in Primes, the improviser is using the computer part as a duet partner, but here the

partnership is more evenly balanced as the machine has its own agency. In Primes, the

player could stop providing input and the patch would continue to play, giving the

84

impression of agency. But in reality this would only be the randomly generated partials

sounding, the trackers would simply maintain the last pitch they detected. Whitewater on

the other hand can continue indefinitely without input (apart from the initial seeding). The

piece would then be subject to the whim of the automaton's life-like processes, but it

would still be evolving and 'performing'.

This raises the question of why bounded improvisation is necessary at all: why not simply

write a fixed score for Primes and Whitewater? For it to be necessary, the improvisation

must be doing something extra and unique. On a practical level, the improvisation is a

way to deal with the inequities of the patch's response to the live input. There are both

deliberate and accidental elements of unpredictability inherent in these patches, the only

way to respond to these is to have an equally unpredictable live part. More importantly,

the feedback loop which is central to these pieces needs this constant evolution to keep

going, and to maintain the tension of repetition against variation. Without the feedback

loop, the live player may as well be soloing karaoke-style over a tape part.

There is also the social element, that a piece like these requires the player's total

commitment; these pieces cannot be sight-read, they require immersive learning before

they can be performed. This is about making more of the experience for the player, to

take the time to have true ownership of a piece, if it is what interests them. As the

American painter Georgia O'Keeffe said:

Nobody sees a flower, really — it is so small — we haven't time, and to see takes

time, like to have a friend takes time.

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Bounded improvisation and free notation leave temporal decisions in the hands of the

performer. The indeterminacy in these pieces is tempered by what the performer brings to

the piece in terms of their own voice and performance reflexes both positively and

negatively. On the positive side, each performer's version explores a different area of the

piece's phase space (the set of possibilities within the boundaries defined by the score),

ultimately enriching the piece. On the negative side, the performer may bring too much

from other music and end up applying forms inappropriate to the pieces. There is no

stricture against this in the score because the specifics are unforseeable; it is a risk that I

accept, and can be mitigated against in practice through rehearsal and careful choice of

performer. As an example of this, I took the piece to bassoonist and free improviser Mick

Beck, who specialises in the manipulation of multiphonics. While the initial few runs went

well, it soon became clear that the player found the piece's limitations to be too stifling.

The piece can only support long sustained pitches, and this became aesthetically tiring for

the player. We tried some variations where short notes were used and this was fine as

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long there was a lot of space between them, but any attempt to play melodic or gestural

music into the patch was unsatisfactory as the computer could not respond in kind. We

considered the option of using a footswitch to occasionally cut off the input to the patch so

that the player could improvise in whatever way they chose without affecting the patch's

output. This would allow an improviser to take more control but it would also destroy the

feedback loop between player and computer: it would turn the computer into an

accompanist.

As with Primes, the score for Whitewater defines the interaction from the bottom up

without defining an overall structure. The piece's form emerges from the interactions

being always different but the same. Whitewater has a robust identity due again to the

limited soundworld. The flexibility of the form means that it can be performed as a concert

piece or as an installation. In the past Whitewater has almost exclusively been played as a

solo concert work, but recently there has been a very successful version with three

simultaneous players who moved through a space with multiple microphones and speaker

arrays. This is something that has a lot of potential for the future of this and other pieces.

The first few performances required that a loose score be prepared for the player. This was

mainly so that the performer could be confident that they were playing the piece

'correctly'. Using a fixed score creates a performance problem, because the computer's

response cannot be predicted from version to version (the patch is sensitive to initial

conditions), the performer would become disoriented when they are following a score

which the computer is not referring to: this leads to the computer making decisions that

the performer must react to but which bear no relation to the score. Hence the piece only

truly works with improvising players, a score is fundamentally incompatible with the piece

but is acceptable as a learning tool. However, the piece is also problematic for improvisers

due to its limitations. As was mentioned above, the player has infinite possibilities within a

small area, but the simplicity of this in some ways reduces them to the machine level of

decision making. Players who come from certain free improvising traditions can find this

stifling and become disinterested. This piece works best with players who can strike a

balance between reacting to an ever changing context and not straying outside its

boundaries.

2.1.9: For Garrett Sholdice (2007)

Oboe, violin, piano: 6 minutes

This piece is one that I enjoy but do not consider a successful piece simply because the

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soundworld and construction betray the influence of Morton Feldman too strongly:

Feldman is omnipresent in my musical thinking, his methods and aesthetics are echoed in

much of my music, but this piece actually sounds like Feldman and this is a step too far

for me. Once I had realised how much Feldman was in the piece I decided to title it in the

manner of Feldman. Garrett Sholdice is a young Irish composer whose work I greatly

admire, the decision to title the piece for him came after hearing an excellent work of his

which reminded me a little of Feldman, and activated my feelings of being too close to

another composer's sound.

The most Feldmanesque element is the piano writing with its softly arpeggiated chromatic

clusters across several octaves. Feldman made great use of such distended clusters in his

music, especially the late piano works such as Triadic Memories where many of the

harmonic configurations come from displacing the individual notes of a cluster by one or

more octaves. Feldman often created ambiguity between subsequent events by slightly

altering their rhythmic profiles—especially where the pitch content stays largely the same

from event to event—so that the level of change was on the edge of what can be

perceived.

In For Garrett Sholdice, the same type of ambiguity is explored through repeating the

same ascending arpeggio gesture for long periods and subtly altering each iteration. With

each iteration the pitch set changes slightly, the pitches do not appear in the same

register in subsequent bars, and each arpeggio is always in a crotchet rhythm but the

length of the bars—and by corollary, the number of pitches in each arpeggio—changes

from bar to bar: see illustration 20. The oboe and violin contribute to this same ambiguity

through very slight alterations to their respective intonations.

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Small amounts of indeterminacy are present in this piece. The piano part in some sections

is left rhythmically free and uses time-space notation: this looks forward to the

synthesiser part in Marx which uses the same technique. More importantly, the oboe and

violin use notational imprecision throughout the piece to again create ambiguity as to the

harmonic identity of each event. This is sometimes used locally to unsettle a series of

harmonies, and sometimes is used across a whole section as a trajectory: for example,

the oboe between mm.19-29 moves slowly from A5 to A?5(but slightly sharp), sometimes

repeating pitches and sometimes making imperceptibly small alterations.

The piece's block structure and simple repetition also stem from Feldman (among other

influences) and look forward to the repetitions of Intra- and to a certain extent also to

Nano. Formally, the piece is a dialectic between two sets of material, the repeated and the

sustained. The repeated material is the previously discussed arpeggio figures based in the

piano, and the violin. The tendency of the oboe towards sustained notes is amplified in the

opening gesture where the oboe slowly overblows a minor ninth interval (on the same

fingering): the slow overblowing means that for a brief but glorious moment the oboe is

suspended between D4 and a very flat E?5. During this long note the piano is playing a

free arpeggio and violin assumes its role as arbiter between arpeggio and sustained notes.

The two types of material move between states of stable identity ambiguity, often the

state is only clear when one type of material interrupts the other.

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Illustration 20: For Garrett Sholdice, ambiguity created through subtle variation.

2.1.10: Bifurcations (2007)

Clarinet quartet: 8 minutes

This piece takes a structural idea from complexity theory

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, where a bifurcation is a

sudden phase state change occurring at a point in a steadily changing parameter.

Bifurcations begins as a single chord, which slowly pulls itself apart until the chord

bifurcates into two chords, upon which the process begins again with these two chords

which subsequently bifurcate into four. The piece stops after this third section as it wasn't

practical for four clarinets to maintain eight chords simultaneously without departing

significantly from the sustained texture.

Clarinets are ideal for this piece due to their almost unique ability to begin and end a note

in silence, this lends the surface a gently undulating smoothness as the instruments

overlap each other: see illustration 21 below (in both illustrations 21 and 22, clarinets 2

and 3 are both tuned ¼-tone flat, the examples are at written pitch). The harmony

changes within each section are at the level of 25¢ to 50¢ per alteration because I am

aiming for imperceptible change, while the sectional breaks are the opposite, violent

events that reset the material.

Indeterminacy is only a marginal concern in Bifurcations; before playing, two of the four

clarinets detune by approximately a quartertone in order to increase the probability of

beats between the close parts. The four players each follow a linear trajectory in

descending or ascending eighth-tones that covers about 130¢ to converge on two clusters

over three minutes. The detuning is largely a practical concern in this piece as it ensures

that the student players—whom the piece was written for—can play very tight clusters

without having to use difficult or awkward fingerings. The third section also benefits

because the strongly perceived pitch of the multiphonics are mostly in twelve-tone equal

temperament, thus the detuning increases the friction between them. This is even more

acute when the same multiphonic is played simultaneously by more than one player, as

shown in illustration 22; this final chord aims for maximum cluster density. While the first

and second sections could have worked just as well with more complex fingerings, the

third section would be much less dense and interesting without the detuning as it is

almost impossible to use the embouchure to bend the multiphonic out of tune without the

sound becoming unstable.

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Wolfram, Stephen, 'Complex Systems Theory', (1984), <http://www.stephenwolfram.com/publications/articles/ca/84-

complex/2/text.html>, accessed 07/08/08.

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Illustration 21: Bifurcations (mm.14-20), detail of linear descents and ascent.

Illustration 22: Bifurcations (mm.85-end), multiphonic clustering.

Formally speaking, Bifurcations is modelled on the bifurcation phenomena, where regular

change in a single parameter results in period-doubling. The piece uses this idea as a

macroform, creating three sections which divide from one chord to two chords to four

chords.

The sections themselves each have strict teleological processes within them. For example,

section-I begins as a single chord of four pitches which gradually separates out into two

distinct clusters. The evenly spaced chord (D, F, G, A) narrows to two microtonal clusters

around D?4 and G?4 (see illustration 23), with perceptual focus shifting from a single chord

to two beating sound complexes.

Section-II has a similar process but without any memory of the first. This new process has

different pitches and register—separated as it is into two distinct chords—and a slightly

different morphology in that the lower chord inverts itself through an 'X' pattern rather

than simply narrowing to a cluster: see illustration 24.

The global teleology of the piece is to bifurcate again and again. For practical reasons the

piece is truncated after two bifurcations, but this teleology would doubtless be clearer if it

were possible to continue the process for longer; I have considered a longer version of the

piece that would explore more sections of the bifurcation model and make clearer the

global teleology, but this would require larger instrumental forces. This global teleology is

separate from the local teleology at work within each section. Each bifurcation point starts

the work again with twice the amount of lines, so each section is treated as a tabula rasa.

As with the pieces discussed already, each section in Bifurcations is an instance of the

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Illustration 23: Bifurcations section-I harmonic motion.

Illustration 24: Bifurcations section-II harmonic motion.

piece's main idea. In this case though there are two ideas: the teleological processes

within each section, and the macro-formal process of bifurcations.

In the piece's originating metaphor these two processes inform each other; the

macro-formal bifurcations being an emergent property of the system under the

continued morphology of one parameter. This aspect has been metaphorically

transformed here so that the parameter undergoing change across the piece is a

metaphorical topological constant controlling the changing identity of each sectional

instance with reference to the ur-object (the central sound idea). In this way the

strange attractor is invoked inside the bifurcating system in slow motion: where each

bifurcation brings up a new point on the attractor.

2.1.11: Torsion (2007)

Eighth-tone trumpet: 4 minutes

A short piece for Steven Altoft's eighth-tone trumpet: a standard trumpet with a fourth

valve added to allow quartertone divisions, the eighth-tones are achieved by using the

tuning slides. I had great difficulty writing a piece for solo instrument, because there is no

possibility of harmony.

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There was always the option to include tape or some sort of

automated drone but I wished to see the challenge of writing a monophonic piece through.

In keeping with my work on ambiguities, I devised a simple process to take advantage of

Steve's ability to play eighth-tone divisions. There is a four-note cell (F4, A4, G?4, F?4)—

conforming to one of my standard models, the ambiguous minor-major cell—that over the

length of the piece is slowly compressed into a single G4 pitch. This cell has a

counterbalance in the form of a single sustained G5 that sounds after each of the four-

note cells, but isolated in registral and temporal space. Across the piece this G5 slowly

becomes as G?5 so that by the end of the piece we are left with the minor ninth G4-G?5

between the upper and lower voices. The counterpoint is also between the motion of the

four-note cell with its quasi-lyrical contour—a simple rhythmic permutational process

ensures that each iteration of the cell has a different temporal shape—and the single high

G5, static in space.

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Torsion was begun before Nano and it is possible that the difficulties in writing Torsion contributed to the conception of

Nano. Perhaps I needed to try and write a notated monophonic piece before I could consider other approaches.

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