Timbre, Shepard Scales, and Ligeti Etudes

Aaron GreiciusLoyola University Chicago

c© 2015 Aaron GreiciusAll Rights Reserved

Timbre

Official definitions of the timbre of a tone are usually equivalent tothe following: timbre is that quality of a tone that cannot beattributed to pitch or loudness.

For example, according to the Acoustical Society of America,timbre is the property of a sound “that allows a listener to judgethat two nonidentical sounds, similarly presented, and having thesame loudness and pitch, are dissimilar”.

In other words, timbre is the property of a sound that allows us todistinguish between a note played on a flute and a piano.

Timbre

Official definitions of the timbre of a tone are usually equivalent tothe following: timbre is that quality of a tone that cannot beattributed to pitch or loudness.

For example, according to the Acoustical Society of America,timbre is the property of a sound “that allows a listener to judgethat two nonidentical sounds, similarly presented, and having thesame loudness and pitch, are dissimilar”.

In other words, timbre is the property of a sound that allows us todistinguish between a note played on a flute and a piano.

Timbre

Official definitions of the timbre of a tone are usually equivalent tothe following: timbre is that quality of a tone that cannot beattributed to pitch or loudness.

For example, according to the Acoustical Society of America,timbre is the property of a sound “that allows a listener to judgethat two nonidentical sounds, similarly presented, and having thesame loudness and pitch, are dissimilar”.

In other words, timbre is the property of a sound that allows us todistinguish between a note played on a flute and a piano.

Physical nature of soundA sound moving past a fixed point is observed as a continuous change in airpressure over time. The function P = f (t) giving the air pressure at time t iscalled the waveform of the sound.

Below you find graphs of waveforms for four different instruments playing anA4 (≈ 440 Hz).

Physical nature of soundA sound moving past a fixed point is observed as a continuous change in airpressure over time. The function P = f (t) giving the air pressure at time t iscalled the waveform of the sound.

Below you find graphs of waveforms for four different instruments playing anA4 (≈ 440 Hz).

Some simple observations about these four waveforms:

1. The four instruments produce four different sounding versions of A4. Wecan see this difference in the shape of their waveforms. Thus thewaveform gives us information about the timbre of the sound.

2. The waveforms are periodic; each is made of a single block (or cycle)that gets repeated indefinitely. The length of this block is called theperiod of the waveform. On the graph this is marked as T . In terms offunction notation, being periodic with period T just means

f (t + T ) = f (t)

for all t. In the example, each waveform has a period of about T = 2.27ms= .00227 s.

3. The frequency f of a periodic waveform is defined as the number ofcycles per second. If a single cycle lasts T seconds, then the number ofcycles per second is f = 1/T . Since T = .0027 s for each of theexamples above, we have f = 1/T ≈ 440.5 Hz.

Some simple observations about these four waveforms:

1. The four instruments produce four different sounding versions of A4. Wecan see this difference in the shape of their waveforms. Thus thewaveform gives us information about the timbre of the sound.

2. The waveforms are periodic; each is made of a single block (or cycle)that gets repeated indefinitely. The length of this block is called theperiod of the waveform. On the graph this is marked as T . In terms offunction notation, being periodic with period T just means

f (t + T ) = f (t)

for all t. In the example, each waveform has a period of about T = 2.27ms= .00227 s.

3. The frequency f of a periodic waveform is defined as the number ofcycles per second. If a single cycle lasts T seconds, then the number ofcycles per second is f = 1/T . Since T = .0027 s for each of theexamples above, we have f = 1/T ≈ 440.5 Hz.

Some simple observations about these four waveforms:

1. The four instruments produce four different sounding versions of A4. Wecan see this difference in the shape of their waveforms. Thus thewaveform gives us information about the timbre of the sound.

2. The waveforms are periodic; each is made of a single block (or cycle)that gets repeated indefinitely. The length of this block is called theperiod of the waveform. On the graph this is marked as T . In terms offunction notation, being periodic with period T just means

f (t + T ) = f (t)

for all t. In the example, each waveform has a period of about T = 2.27ms= .00227 s.

3. The frequency f of a periodic waveform is defined as the number ofcycles per second. If a single cycle lasts T seconds, then the number ofcycles per second is f = 1/T . Since T = .0027 s for each of theexamples above, we have f = 1/T ≈ 440.5 Hz.

Some simple observations about these four waveforms:

1. The four instruments produce four different sounding versions of A4. Wecan see this difference in the shape of their waveforms. Thus thewaveform gives us information about the timbre of the sound.

2. The waveforms are periodic; each is made of a single block (or cycle)that gets repeated indefinitely. The length of this block is called theperiod of the waveform. On the graph this is marked as T . In terms offunction notation, being periodic with period T just means

f (t + T ) = f (t)

for all t. In the example, each waveform has a period of about T = 2.27ms= .00227 s.

3. The frequency f of a periodic waveform is defined as the number ofcycles per second. If a single cycle lasts T seconds, then the number ofcycles per second is f = 1/T . Since T = .0027 s for each of theexamples above, we have f = 1/T ≈ 440.5 Hz.

Sinusoidal pure tonesMost instruments produce sounds with waveforms that are periodic, or at leastvery nearly periodic over small time intervals.

In mathematics we have a whole family of simple periodic functions of the formg(t) = sin(2πft − φ):

T =1

f

Shift due to ϕ

f (t ) = sin(2πft -ϕ)

1/f 2/f 3

f

-1.0

-0.5

0.5

1.0

As indicated in the graph, sin(2πft − φ) has period T = 1/f , and thusfrequency f . The parameter φ is called a phase shift; it moves the graphhorizontally to the left or right.

The corresponding sound with waveform f (t) = sin(2πft − φ) is called a pure

(sinusoidal) tone of frequency f . A sound whose periodic waveform is not one

of these sinusoidal functions is called a complex tone.

Sinusoidal pure tonesMost instruments produce sounds with waveforms that are periodic, or at leastvery nearly periodic over small time intervals.

In mathematics we have a whole family of simple periodic functions of the formg(t) = sin(2πft − φ):

T =1

f

Shift due to ϕ

f (t ) = sin(2πft -ϕ)

1/f 2/f 3

f

-1.0

-0.5

0.5

1.0

As indicated in the graph, sin(2πft − φ) has period T = 1/f , and thusfrequency f . The parameter φ is called a phase shift; it moves the graphhorizontally to the left or right.

The corresponding sound with waveform f (t) = sin(2πft − φ) is called a pure

(sinusoidal) tone of frequency f . A sound whose periodic waveform is not one

of these sinusoidal functions is called a complex tone.

Sinusoidal pure tonesMost instruments produce sounds with waveforms that are periodic, or at leastvery nearly periodic over small time intervals.

In mathematics we have a whole family of simple periodic functions of the formg(t) = sin(2πft − φ):

T =1

f

Shift due to ϕ

f (t ) = sin(2πft -ϕ)

1/f 2/f 3

f

-1.0

-0.5

0.5

1.0

As indicated in the graph, sin(2πft − φ) has period T = 1/f , and thusfrequency f . The parameter φ is called a phase shift; it moves the graphhorizontally to the left or right.

The corresponding sound with waveform f (t) = sin(2πft − φ) is called a pure

(sinusoidal) tone of frequency f . A sound whose periodic waveform is not one

of these sinusoidal functions is called a complex tone.

Sinusoidal pure tonesMost instruments produce sounds with waveforms that are periodic, or at leastvery nearly periodic over small time intervals.

In mathematics we have a whole family of simple periodic functions of the formg(t) = sin(2πft − φ):

T =1

f

Shift due to ϕ

f (t ) = sin(2πft -ϕ)

1/f 2/f 3

f

-1.0

-0.5

0.5

1.0

As indicated in the graph, sin(2πft − φ) has period T = 1/f , and thusfrequency f . The parameter φ is called a phase shift; it moves the graphhorizontally to the left or right.

The corresponding sound with waveform f (t) = sin(2πft − φ) is called a pure

(sinusoidal) tone of frequency f . A sound whose periodic waveform is not one

of these sinusoidal functions is called a complex tone.

Fourier analysisA function

g(t) = A0 + A1 sin(2πf1t − φ1) + A2 sin(2πf2t − φ2) + · · ·Ar sin(2πfr t − φr )

obtained by adding up various multiples (the Ai ) of sinusoidal waveforms ofvarious frequencies (the fi ) and phase shifts (the φi ) will be still be periodic.

More surprisingly, we can go the other way: any periodic function can beapproximated as a sum of sinusoidal functions in this way.

More precisely, given any (nice enough) periodic function f (t), I can write

g(t) ≈ A0 + A1 sin(2πft − φ1) + A2 sin(2π(2f )t − φ2) + · · ·Ar sin(2π(rf )t − φr )

for some constants Ai , φi and f .

In fact, we can replace the ≈ with = if we allow infinite sums; that is we canwrite

g(t) = A0 +∞∑i=1

Ai sin(2π(if )t − φi )

for some constants Ai , φi and f . This fact is by no means obvious. Indeed, it is

a deep theorem of a branch of mathematics called Fourier analysis. The

expression on the right is called a Fourier series representation of f (t).

Fourier analysisA function

g(t) = A0 + A1 sin(2πf1t − φ1) + A2 sin(2πf2t − φ2) + · · ·Ar sin(2πfr t − φr )

obtained by adding up various multiples (the Ai ) of sinusoidal waveforms ofvarious frequencies (the fi ) and phase shifts (the φi ) will be still be periodic.

More surprisingly, we can go the other way: any periodic function can beapproximated as a sum of sinusoidal functions in this way.

More precisely, given any (nice enough) periodic function f (t), I can write

g(t) ≈ A0 + A1 sin(2πft − φ1) + A2 sin(2π(2f )t − φ2) + · · ·Ar sin(2π(rf )t − φr )

for some constants Ai , φi and f .

In fact, we can replace the ≈ with = if we allow infinite sums; that is we canwrite

g(t) = A0 +∞∑i=1

Ai sin(2π(if )t − φi )

for some constants Ai , φi and f . This fact is by no means obvious. Indeed, it is

a deep theorem of a branch of mathematics called Fourier analysis. The

expression on the right is called a Fourier series representation of f (t).

Fourier analysisA function

g(t) = A0 + A1 sin(2πf1t − φ1) + A2 sin(2πf2t − φ2) + · · ·Ar sin(2πfr t − φr )

obtained by adding up various multiples (the Ai ) of sinusoidal waveforms ofvarious frequencies (the fi ) and phase shifts (the φi ) will be still be periodic.

More surprisingly, we can go the other way: any periodic function can beapproximated as a sum of sinusoidal functions in this way.

More precisely, given any (nice enough) periodic function f (t), I can write

g(t) ≈ A0 + A1 sin(2πft − φ1) + A2 sin(2π(2f )t − φ2) + · · ·Ar sin(2π(rf )t − φr )

for some constants Ai , φi and f .

In fact, we can replace the ≈ with = if we allow infinite sums; that is we canwrite

g(t) = A0 +∞∑i=1

Ai sin(2π(if )t − φi )

for some constants Ai , φi and f . This fact is by no means obvious. Indeed, it is

a deep theorem of a branch of mathematics called Fourier analysis. The

expression on the right is called a Fourier series representation of f (t).

Fourier analysisA function

g(t) = A0 + A1 sin(2πf1t − φ1) + A2 sin(2πf2t − φ2) + · · ·Ar sin(2πfr t − φr )

obtained by adding up various multiples (the Ai ) of sinusoidal waveforms ofvarious frequencies (the fi ) and phase shifts (the φi ) will be still be periodic.

More surprisingly, we can go the other way: any periodic function can beapproximated as a sum of sinusoidal functions in this way.

More precisely, given any (nice enough) periodic function f (t), I can write

g(t) ≈ A0 + A1 sin(2πft − φ1) + A2 sin(2π(2f )t − φ2) + · · ·Ar sin(2π(rf )t − φr )

for some constants Ai , φi and f .

In fact, we can replace the ≈ with = if we allow infinite sums; that is we canwrite

g(t) = A0 +∞∑i=1

Ai sin(2π(if )t − φi )

for some constants Ai , φi and f . This fact is by no means obvious. Indeed, it is

a deep theorem of a branch of mathematics called Fourier analysis. The

expression on the right is called a Fourier series representation of f (t).

Let g(t) be a periodic waveform and consider its Fourier series representation

g(t) = A0 +∞∑i=1

Ai sin(2π(if )t − φi ).

1. Note that the frequencies of the various sinusoidal functions involved areall integer multiples of f : i.e., f , 2f , 3f , . . . . The frequency f is called thefundamental frequency of the tone, and is typically identified with thepitch of the sound. The i-th multiple i · f of the fundamental is called thei-th partial.

2. We recognize the sequence f , 2f , 3f . . . as the harmonic series on thefrequency f . Fourier analysis thus tells us that a complex tone offundamental frequency f is “built up” from pure sinusoidal tones withfrequencies taken from the harmonic series on f .

3. The constants Ai function as weights that tell us roughly how much ofthe i-th partial frequency is “mixed into” our complex tone to produce itsparticular color. Thus we can think of the list of pairs(f ,A1), (2f ,A2), (3f ,A3) . . . as a sort of recipe for building our complextone g(t): “Put in A1 teaspoons of frequency f , A2 teaspoons offrequency 2f , A3 teaspoons of frequency 3f , etc.”

Let g(t) be a periodic waveform and consider its Fourier series representation

g(t) = A0 +∞∑i=1

Ai sin(2π(if )t − φi ).

1. Note that the frequencies of the various sinusoidal functions involved areall integer multiples of f : i.e., f , 2f , 3f , . . . . The frequency f is called thefundamental frequency of the tone, and is typically identified with thepitch of the sound. The i-th multiple i · f of the fundamental is called thei-th partial.

2. We recognize the sequence f , 2f , 3f . . . as the harmonic series on thefrequency f . Fourier analysis thus tells us that a complex tone offundamental frequency f is “built up” from pure sinusoidal tones withfrequencies taken from the harmonic series on f .

3. The constants Ai function as weights that tell us roughly how much ofthe i-th partial frequency is “mixed into” our complex tone to produce itsparticular color. Thus we can think of the list of pairs(f ,A1), (2f ,A2), (3f ,A3) . . . as a sort of recipe for building our complextone g(t): “Put in A1 teaspoons of frequency f , A2 teaspoons offrequency 2f , A3 teaspoons of frequency 3f , etc.”

Let g(t) be a periodic waveform and consider its Fourier series representation

g(t) = A0 +∞∑i=1

Ai sin(2π(if )t − φi ).

1. Note that the frequencies of the various sinusoidal functions involved areall integer multiples of f : i.e., f , 2f , 3f , . . . . The frequency f is called thefundamental frequency of the tone, and is typically identified with thepitch of the sound. The i-th multiple i · f of the fundamental is called thei-th partial.

2. We recognize the sequence f , 2f , 3f . . . as the harmonic series on thefrequency f . Fourier analysis thus tells us that a complex tone offundamental frequency f is “built up” from pure sinusoidal tones withfrequencies taken from the harmonic series on f .

3. The constants Ai function as weights that tell us roughly how much ofthe i-th partial frequency is “mixed into” our complex tone to produce itsparticular color. Thus we can think of the list of pairs(f ,A1), (2f ,A2), (3f ,A3) . . . as a sort of recipe for building our complextone g(t): “Put in A1 teaspoons of frequency f , A2 teaspoons offrequency 2f , A3 teaspoons of frequency 3f , etc.”

Let g(t) be a periodic waveform and consider its Fourier series representation

g(t) = A0 +∞∑i=1

Ai sin(2π(if )t − φi ).

1. Note that the frequencies of the various sinusoidal functions involved areall integer multiples of f : i.e., f , 2f , 3f , . . . . The frequency f is called thefundamental frequency of the tone, and is typically identified with thepitch of the sound. The i-th multiple i · f of the fundamental is called thei-th partial.

2. We recognize the sequence f , 2f , 3f . . . as the harmonic series on thefrequency f . Fourier analysis thus tells us that a complex tone offundamental frequency f is “built up” from pure sinusoidal tones withfrequencies taken from the harmonic series on f .

3. The constants Ai function as weights that tell us roughly how much ofthe i-th partial frequency is “mixed into” our complex tone to produce itsparticular color. Thus we can think of the list of pairs(f ,A1), (2f ,A2), (3f ,A3) . . . as a sort of recipe for building our complextone g(t): “Put in A1 teaspoons of frequency f , A2 teaspoons offrequency 2f , A3 teaspoons of frequency 3f , etc.”

Classic 7-Etudes pour piano, 1985-2002, G. LigetiBetween 1985 and 2002 Ligeti composed 18 etudes for piano, published inthree different volumes. These are highly virtuosic pieces of music, representingthe culmination of Ligeti’s long career.

According to Ligeti his thinking was guided by a wide variety of mathematicaland musical influences: the notion of self-similarity from fractal geometry,deformations of spaces, hemiola, sub-Saharan and Central African rhythmicpatterns, the player piano compositions of Conlon Nancarrow, jazzcompositions of Thelonius Monk and Bill Evans, the piano works of Scarlatti,Chopin, Schumann and Debussy.

And yet, as Ligeti himself wrote: “my etudes are neither jazz norChopinesque-Debussian music, neither African nor Nancarrow, and certainlynot mathematical constructs...neither avant-garde nor traditional, not tonaland not atonal–and in no way postmodern...They are virtuosic piano pieces,studies in the pianistic as well as the compositional sense. They start from avery simple core idea and lead from simplicity to a high complexity: theybehave like growing organisms.”

In a similar vein:“ In my music one finds neither that which one might call

‘scientific’ nor the ‘mathematical’, but rather a unification of construction with

poetic, emotional imagination.”

Classic 7-Etudes pour piano, 1985-2002, G. LigetiBetween 1985 and 2002 Ligeti composed 18 etudes for piano, published inthree different volumes. These are highly virtuosic pieces of music, representingthe culmination of Ligeti’s long career.

According to Ligeti his thinking was guided by a wide variety of mathematicaland musical influences: the notion of self-similarity from fractal geometry,deformations of spaces, hemiola, sub-Saharan and Central African rhythmicpatterns, the player piano compositions of Conlon Nancarrow, jazzcompositions of Thelonius Monk and Bill Evans, the piano works of Scarlatti,Chopin, Schumann and Debussy.

And yet, as Ligeti himself wrote: “my etudes are neither jazz norChopinesque-Debussian music, neither African nor Nancarrow, and certainlynot mathematical constructs...neither avant-garde nor traditional, not tonaland not atonal–and in no way postmodern...They are virtuosic piano pieces,studies in the pianistic as well as the compositional sense. They start from avery simple core idea and lead from simplicity to a high complexity: theybehave like growing organisms.”

In a similar vein:“ In my music one finds neither that which one might call

‘scientific’ nor the ‘mathematical’, but rather a unification of construction with

poetic, emotional imagination.”

Classic 7-Etudes pour piano, 1985-2002, G. LigetiBetween 1985 and 2002 Ligeti composed 18 etudes for piano, published inthree different volumes. These are highly virtuosic pieces of music, representingthe culmination of Ligeti’s long career.

According to Ligeti his thinking was guided by a wide variety of mathematicaland musical influences: the notion of self-similarity from fractal geometry,deformations of spaces, hemiola, sub-Saharan and Central African rhythmicpatterns, the player piano compositions of Conlon Nancarrow, jazzcompositions of Thelonius Monk and Bill Evans, the piano works of Scarlatti,Chopin, Schumann and Debussy.

And yet, as Ligeti himself wrote: “my etudes are neither jazz norChopinesque-Debussian music, neither African nor Nancarrow, and certainlynot mathematical constructs...neither avant-garde nor traditional, not tonaland not atonal–and in no way postmodern...They are virtuosic piano pieces,studies in the pianistic as well as the compositional sense. They start from avery simple core idea and lead from simplicity to a high complexity: theybehave like growing organisms.”

In a similar vein:“ In my music one finds neither that which one might call

‘scientific’ nor the ‘mathematical’, but rather a unification of construction with

poetic, emotional imagination.”

Classic 7-Etudes pour piano, 1985-2002, G. LigetiBetween 1985 and 2002 Ligeti composed 18 etudes for piano, published inthree different volumes. These are highly virtuosic pieces of music, representingthe culmination of Ligeti’s long career.

According to Ligeti his thinking was guided by a wide variety of mathematicaland musical influences: the notion of self-similarity from fractal geometry,deformations of spaces, hemiola, sub-Saharan and Central African rhythmicpatterns, the player piano compositions of Conlon Nancarrow, jazzcompositions of Thelonius Monk and Bill Evans, the piano works of Scarlatti,Chopin, Schumann and Debussy.

And yet, as Ligeti himself wrote: “my etudes are neither jazz norChopinesque-Debussian music, neither African nor Nancarrow, and certainlynot mathematical constructs...neither avant-garde nor traditional, not tonaland not atonal–and in no way postmodern...They are virtuosic piano pieces,studies in the pianistic as well as the compositional sense. They start from avery simple core idea and lead from simplicity to a high complexity: theybehave like growing organisms.”

In a similar vein:“ In my music one finds neither that which one might call

‘scientific’ nor the ‘mathematical’, but rather a unification of construction with

poetic, emotional imagination.”

Three of these etudes, namely No. 9 (Vertige), No. 13 (L’escalier du diable)and No. 14 (Columna infinita), are directly concerned with creating an auditoryillusion similar to that of the Shepard scale.

Albrecht Schneider takes a close look at this connection in his article “Washaben Ligetis Etudes pour piano mit Shepard-Skalen zu tun?”. Specifically, heinvestigates to what extent each piece creates the illusion of a continuousascent or descent, while the maintaining a relatively constant brightness(Helligkeit, in German).

As a measure of brightness (also called height) at time t, Schneider includesgraphs of the spectral centroid versus time, as measured from recordings ofperformances of these etudes. Recall that the spectral centroid of a soundwaveform g(t) = A0 +

∑∞i=1 Ai sin(2π(if )t − φi ) is the average frequency∑

||Ai | · (i · f )∑|Ai |

.

Three of these etudes, namely No. 9 (Vertige), No. 13 (L’escalier du diable)and No. 14 (Columna infinita), are directly concerned with creating an auditoryillusion similar to that of the Shepard scale.

Albrecht Schneider takes a close look at this connection in his article “Washaben Ligetis Etudes pour piano mit Shepard-Skalen zu tun?”. Specifically, heinvestigates to what extent each piece creates the illusion of a continuousascent or descent, while the maintaining a relatively constant brightness(Helligkeit, in German).

As a measure of brightness (also called height) at time t, Schneider includesgraphs of the spectral centroid versus time, as measured from recordings ofperformances of these etudes. Recall that the spectral centroid of a soundwaveform g(t) = A0 +

∑∞i=1 Ai sin(2π(if )t − φi ) is the average frequency∑

||Ai | · (i · f )∑|Ai |

.

Three of these etudes, namely No. 9 (Vertige), No. 13 (L’escalier du diable)and No. 14 (Columna infinita), are directly concerned with creating an auditoryillusion similar to that of the Shepard scale.

Albrecht Schneider takes a close look at this connection in his article “Washaben Ligetis Etudes pour piano mit Shepard-Skalen zu tun?”. Specifically, heinvestigates to what extent each piece creates the illusion of a continuousascent or descent, while the maintaining a relatively constant brightness(Helligkeit, in German).

As a measure of brightness (also called height) at time t, Schneider includesgraphs of the spectral centroid versus time, as measured from recordings ofperformances of these etudes. Recall that the spectral centroid of a soundwaveform g(t) = A0 +

∑∞i=1 Ai sin(2π(if )t − φi ) is the average frequency∑

||Ai | · (i · f )∑|Ai |

.

Etude 13, L’escalier du diable, 1993

Etude 9, Vertige, 1990

Ex.l: from 'Vertige'. ? Schott & Co. Ltd and used by kind permission. r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..--- - - - - - .7.-- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..- -

/-/

v- ser r S rh vkaro,,

J

S osr . . . . .. . . . . ..e :E../,. ..,! , , ,

be laminar, then through minuscule oscillations bifurcate (branch out) into more complex forms until, via further oscillations and bifurcations, it becomes chaotic. Multiple 'phase portraits' of such behaviour, in which the initial values are fractionally altered, reveal that different sets of starting condi- tions may settle into similar periodic oscillations or converge on a single point, an 'attractor'; whilst oth- ers trace exotic patterns called 'strange attractors', images which are both strikingly complex yet grace- fully symmetrical.

Fractals are an exceptionally beautiful and intriguing visualisation of dynamical processes dis- tinguished by the property of 'self-similarity'. This means that, at any level of magnification, any small portion can be viewed as a replica of the whole. The Koch snowflake is a famous example. The new com- puter science of irregularity, or fractal geometry, which had such tremendous impact in the 1980s, was first articulated by Benoit Mandelbrot, using a formula of childlike simplicity yet infinitely variable outcome. Well, perhaps not quite childlike. The problem for ordinary mortals is that there are several different concepts of what 'number' means. To embrace fractions, decimals, negative values and more than one dimension, numbers can be natural, real, rational, irrational, integral, even transcenden- tal - all fairly bewildering for non-mathematicians!

Mandelbrot's ingenuity was to look at complex rather than real numbers, ie. referring to coordinates on a two-dimensional plane instead of points on a line. The magic of the Mandelbrot set lies in its con- cept of iteration: take a number, multiply it by itself, add the original number, then repeat the process ad infinitum. This can be summarised as 'the iteration in the complex plane of the mapping z - z2 + c'.10 Although repetitive, this simple arithmetic gives rise to a universe of situations: e.g. steady state, conver- gence towards periodic repetition, or an out-of-con- trol race to infinity

Self-similarity, indeed, is widely evident in nature - for instance in the florets of a cauliflower and the structure of leaves - and so illuminates our under- standing of the universe, pointing to connections between abstract mathematics and everyday experi- ence. The essence of the revolution of the last twenty years lies in the overthrow of an old belief - that simple systems behave simply and complex systems are impenetrable - by its opposite: i.e. simple sys- tems give rise to complex behaviour, complex sys- tems to simple, and the laws of complexity are uni- versal. Mandelbrot based his intellectual life around the proposition that simplicity breeds complexity. This might also be a motto for late Ligeti.

The degeneration of simple premises into chaotic outcomes has a direct parallel in the music of Ligeti,

16 THE MUSICAL TIMES / MARCH 1996

This content downloaded from 147.126.10.145 on Tue, 2 Sep 2014 14:52:48 PMAll use subject to JSTOR Terms and Conditions

Etude 14, Coloana infinita, 1993