Composing with Computers,Algorithmic composition, Iterative systems:

Electronic Music Studio TU Berlin

Institute of Communications Research

http://www.kgw.tu-berlin.de/

Prof Eduardo R MirandaVarèse-Gastprofessor

eduardo.miranda@btinternet.com

Music: Sounds organised in space and time.

• Space – vertical, simultaneous relationships between sounds• Time – horizontal, sequential relatiponships

Of course… “space” also means “geographical” distribution of sounds in aPerformance area (sound diffusion in electro-acoustic music, etc.)

Our ability to create and appreciate the organisation of sounds in space and time leads us to the notion that musical composition carry abstract structures.

Computer music – at the fringe of 2 apparently distinct domains of musicalIntelligence:

• Abstract subjectivity (artistic or aesthetic imagination)• Abstract objectivity (logical operations; mathematical ideas)

Abstraction Boundaries

Determines the building blocks of components that form the musical structures ofcomposition.

• Microscopic level

• Note level

• Building-block level

Microscopic Level:

The composers works with “microscopic” sound features such as frequenciesand amplitudes of the individual partials of the sounds.

Composers have access to fine timing controls; in terms of milliseconds.

Musical scores tend to contain lists of numerical values for a digital sound synthesiser (rather than “notes” for performance on musical instruments)

The “performer” is a computer, which interprets the score and synthesisesthe music.

Example: Kenny’s Csound piece

Building-Block Level:

The composers works with larger musical units, lasting several seconds.

Example: rhythmic patters, melodic themes, sampled sound sequences.

Trend encouraged by electroacoustic music, with pieces composed by cutting, pasting and mixing sounds recorded on tape.

Widely practiced today by DJs, pop musicians using samplers, etc.

“Composition” with pre-fabricated sections that can be combined or sequenced.

Note Level:

The atomic element is a single sound event described by a number of attributes:the musical note.

Music traditionally has a well-defined boundary of abstraction for characterising musical note in terms of 4 main attributes:

• Pitch• Duration• Dynamics• Timbre (that is the timbre of the instrument that plays the note)

Composers normally think of a musical piece as structures of notes with carefulcontrol of their attributes.

Approaching composition

• Algorithmic composition: programs that generate music with a certain autonomy.

• Computer-aided composition: programs are tools to help the composer capture and organise ideas. (e.g. MIDI sequencers, software synthesisers, samplers, computer-based studio/production gear, etc.)

We focus here on ALGORITHMIC COMPOSITON: systems and paradigms of a generative nature.

Cognitive Archetypes

• Metaphorical Associations

• Elementary Schemes

Metaphorical Associations

Our cognitive capacities do not work in isolation from one another. Eg.: ability to infer the distance of a sound source is tied to our notion of time and space.

A musical piece is the result of the dynamical unfolding of various musical attributesat different levels (timbre, frequency, pulse, form).

Our ears naturally take in this dynamical unfolding as sequences of sonic eventsthat move from one lapse of time to another.

Sophisticated listening employ a number of cognitive strategies that are similar tothose we employ to understand the events that take place during the course of our lives.

Metaphorical Associations – Music & Literature

Both are multi-layered systems in which sequences of signs convey emotions, ideas, etc. according to specific conventions (harmony & counterpoint in music;grammars & lexicon in language).

A musical style defines a system of relations in musical materials that helpslisteners to infer their musical structures.

Grouping of smaller musical units is the first stage of the process that leads thelistener to mentally build components of large-scale musical forms.

Narrative texts & musical compositions are processes that are presented in smalldiscernible units, strung together in a way that leads the listener to mentally buildlarger sub-processes.

We could think of a composition as musical structures that convey some sort ofnarrative whose plot basically involves transitions from one state of equilibrium toanother.

Equilibrium here is meant in the Cybernetic sense: it describes a stable but notstatic relation between musical elements.

Two states of equilibrium are often separated by a stage of imbalance.

5-stage elementary scheme for narrative in literature, which is a useful startingpoint to study some standard musical forms; e.g., sonata.

• Initial equilibrium => initial section at the tonal key• Disturbance => arrival of elements that conflict with the tonic of the key• Reaction => modulation to the dominant key• Consequence => arrival at the dominant key• Final equilibrium => settlement at this key, etc.

Composers extrapolate …

What is important is to give the listener a sense of direction.

Sense of direction is prescribed by schemes of auditory expectation. (culturaland innate)

Examples:

• Responsorial expectation• Convex curve• Principle of intensification

Responsorial expectation: events seems to call and answer each other.

This phenomenon often manifests itself in a convex arch-like shape, whichpossible mirrors the rising and falling patterns of breath.

Arch-like convex curve: provide an intuitive guide by which one can predict theevolution of given parameters in a process.

These is normally one stage where the music ascends towards a climax,followed by a descending stage.

Both stages should normally evolves at gradual intervals.

Can apply to all domains of music (timbre, forms, etc.)

Convexity: means to achieve well-balanced musical passages; can be used toforge moments of equilibrium.

The opposite of convexity produce moments of tension; e.g. linearity (or nochanges), sudden changes, abrupt zigzags, etc.

Unfulfilled predictions often cause musical tension.

Principle of intensification: create tension and to give a sense of direction forthe musical unfolding.

May apply to various time spans and may be expressed in various ways indifferent parameters through contrasts in a continuous scale:

from few to manyfrom low to highfrom non-rhythmic to rhythmicfrom slow to fastfrom small density to large densityetc.

Algorithmic Composition: the legacy of Xenakis’ formalised music

Use of set theory to represent musical ideas. Example inspired by Xenakis work:

Let us define 3 ordered sets of natural numbers: P (pitch intervals), D (duration)and I (intensity).

A general law of composition for each of these sets may be established asfollows. Let vm be a vector of 3 components, pn, dn and in, such that , arranged in this order: vn ={pn, dn, in}.

A vector is a point in a 3D space:

IiDdPp nnn ∈∈∈ and ,

Considering the algebraic properties of sets introduced earlier, let us define twocompositional laws for vector vn: the additive law and the multiplication law.

)}()(){()(

}),,{(}),,{(

zoynxmji

zyxjonmi

iiddppvv

idpvidpv

+++++=+

⇒=+=

},,{

}),,{(

onmi

onmi

icdcpcv

cidpvc

×××=×

⇒=×

Example:

Let’s define the origins of the co-ordinates as follows: v0 = {60, 4, 64} where:

Elements of set P are defined in terms of MIDI notes.Elements of set D are defined in terms of metric beats (4 = 1 beat in a 4/4 metric)Elements of set I are defined in proportional terms varying from pianissimo (= 0)to fortissimo (= 127).

Let us define the following generative rules, or vector generators (np = n unitsabove or below the origin p.

},,)4{(

},,)12{(

},4,14{

3

2

1

ZiYdpXv

ZiYdpXv

idpv

−=

+=

=

The Fn statements define the formation of sequences of vectors:

],,[

]2,,[

][

33133

2222

11

vvvvF

vvvF

vF

+=

×=

=

Suppose that each Fn statement have been implemented as programs thatgenerate the vectors. In this case, F2 and F3 receive a list of parameters for v (forX, Y and Z), which could be notated as Fn(x, y, z).

A generative score could then be given to the system to generate a musicalpassage.

A generative score could then be given to the system to generate a musical passage:

)]}32,2,0(),48,2,7(),32,2,12[(,)],64,4,10(),48,4,0(),0,4,12[({

:stave Bottom

)]}64,4,24),48,4,12(),0,4,24[()],32,2,0(),48,2,5(),32,2,0[(,{

:stave Top

213

321

−−FFF

FFF

Note that Fn does not need any input parameter because v1 does not need any variableto function; it always produce the same note {14p, 4d, i}

Iterative Process

An iterative process is the repeated application of a mathematicalprocedure where each step is applied to the output of the precedingstep.

A rule that describes the action that is to be repeatedly applied to aninitial value

The outcome is a set of values technically called the orbit; the valuesare called points (of the orbit).

0x

)( 0xOF

{ },...8,6,4,2,0)0(2: 1 =⇒+=+ Fnn OxxF

{ },...125.0,25.0,5.0,1)1()2(: 1 =⇒=+

Fnn O

xxF

Orbit O that arises from the iterated application of arule F to an initial value x.

{ },...770.0,540.0,775.0,5.0)5.0()1(1.3: 1 =⇒−=+ Fnnn OxxxF

An iterative process may produce 3 classes of orbits:

a) whose points tend towards a stable fixed value

b) whose points tend to oscillate between specific elements

c) whose points falls into “chaos” (because it is not possible todistinguish an explicitly recurrent pattern)

It is difficult to identify “chaotic” behaviour (e.g., very largeoscillatory period or chaos?)

Look for 3 principles:

1) high sensitivity to initial conditions

2) period doubling

3) sporadic settlements

{ },...766.0,553.0,768.0,549.0,770.0,540.0,775.0,5.0)5.0()1(1.3: 1

=

⇒−=+F

nnn

O

xxxF

Principle of high sensitivity to initial conditions

Tiny variations in the value of cause significant differentoutcome. “Butterfly effect”, nicknamed by meteorologist EdwardLorenz (model to study global weather).

0x

301.0

3.0

4

)1(:

0

0

1

=

=

=Δ

−Δ=+

x

x

xxxF nnn

Principles of period doubling and sporadic settlement

Successive doubling of the number of different points in the orbit.

As orbits evolve towards chaotic behaviour, quasi-stable trajectoriessporadically emerge, but soon fall back into period doubling.

43

3.0

)1(:

0

1

≤Δ≤

=

−Δ=+x

xxxF nnn

Model to describe the trajectory of a floating object around agravitational body.

Another example Hénon attractor

3.0

4.11

21

=

=

=

−=

+

+

β

α

β

α

nn

nnn

xy

xyx

And another example ...

Mandelbrot set. Depending upon the value of c, the points of theorbit may either grow indefinitely (e.g., c=1) or remain bounded by acertain range of values (e.g., c=-1.38)

9.1

38.1

21

−=

−=

+=+

c

c

cxx nn

Music

People tend to enjoy music that presents a good balance betweenrepetition and novelty.

Orbits that quickly converge towards stable values = not the bestchoice for algorithmic music.

Oscillatory orbits = offers good scope for interesting musical results,particularly if the period is large.

Chaotic orbits = most promising type for music.

Useful properties of chaotic orbits

Tendency to wander through a fixed range of elements, visitingsimilar, though not identical points.

Different initial values produce different orbits; e.g., by varying theinitial values of the same iterative musical system, the composer canproduce variations in a passage, or piece, which begins in a similarmanner and then diverges.

PROBLEM:

Define the abstraction boundary to work with

Find an effective method for mapping the orbits onto the respectiveparameters of the abstraction boundary’s system.

Compositional problems:

Define the abstraction boundary to work with

Find an effective method for mapping the orbits onto the

respective parameters of the abstraction boundary’s system

Further reading: