Shakespeare, Beethoven, Wordsworth, Bach:

Chaos, Self-Reference, Order and Chaos

200880236

Contents

Prelude 1

Isomorphism I: Beethoven’s Hammerklavier and Shakespeare’s Richard II 3

Chaos 4

Emergence 8

Order: The Music Plays 12

Isomorphism II: Wordsworth’s The Prelude and Bach’s The Art of Fugue 17

The First Push 19

The Second Push 25

The Third Push: Jumping out of the System 28

Coda 34

Bibliography 37

Prelude

Richard II by Shakespeare, the Hammerklavier Sonata (Sonata 29), by Beethoven, The

Prelude by Wordsworth and The Art of Fugue by Bach might resemble a selection of works chosen

for their greatness but not their relatedness. Nonetheless, these four works, when arranged in a

certain formation, appear to tell us something about each other, about art itself and the world

beyond art.

I am using the term ‘isomorphism’, which Douglas Hofstadter uses a great deal in his book,

Gödel, Escher, Bach. As Hofstadter explains:

The word “isomorphism” applies when two complex structures can be mapped onto

each other, in such a way that to each part of one structure there is a corresponding

part in the other structure, where “corresponding” means that the two parts play

similar roles in their respective structures. This usage of the word "isomorphism" is

derived from a more precise notion in mathematics.1

In Gödel, Escher, Bach, Hofstadter uses isomorphisms between works by his chosen artists

to explore and describe the origin of consciousness from non-conscious systems composed of non-

conscious parts.

I find an isomorphism between Richard II and Hammerklavier, and between The Prelude

and The Art of Fugue. It seems that many such ‘parts’, such as the use of a certain musical phrase in

a musical work, have corresponding parts, such as the use of a poetic image in its literary partner.

The number and placement of these parts is so extensive that it seems that both works in each pair

have the same function, at the level on which I’m examining them. Each isomorphism seems be a

template, furthermore, for a narrative with broader implications for art and nature.

I have paired Richard II and Hammerklavier with a special focus on the Fugue (the last

movement) of Hammerklavier and Richard’s final soliloquy: these sections in the two works appear

to chart the emergence of order out of chaos, shape out of formlessness, understanding and

knowledge from diminution. Conversely, The Prelude and The Art of Fugue appear to present the

transition from a system that gets more and more complex and self-referential to something beyond

it which, somehow, simultaneously combines disorder and order. Both Hammerklavier and Richard

II resemble the emergence of complex systems, as the tree of life developed from its origins in the

1 Hofstadter, Douglas R., Gödel, Escher, Bach: an Eternal Golden Braid (Brighton: Harvester Press, 1979; repr. London: Penguin, 2000), p. 49. All further quotations will refer to this edition.

1

primordial soup. Similarly, The Art of Fugue and The Prelude resemble the way in which certain

systems naturally throw up contradictions, like the wave/particle duality in quantum mechanics,

finding ways to move beyond them and beyond their original genres. Each pair of works seems to

be an artistic model for an important truth about the world. Furthermore, Richard II,

Hammerklavier, The Prelude and The Art of Fugue, appear to exist in an isomorphism with another

structure, which plots the emergence of order from chaos and the emergence of a new chaos within

that order. I believe that I have observed, in this structure, something like that which Hofstadter

sought to describe in his work.

2

Isomorphism I: Beethoven’s Hammerklavier and Shakespeare’s Richard II

The Fugue from the Hammerklavier Sonata (Sonata 29) (composed by Beethoven between

1817 and 1818) and the last soliloquy of Richard in Richard II (written around 1595 by

Shakespeare) seem to present an isomorphism. Both, within their respective frames (music and

verse drama) depict a period of chaos, followed by a period of exploration and quasi-randomness,

followed by the generation of order and beauty from elements that existed within the chaos.

Table I: Schedule of Analogies between Richard II and Hammerklavier

Moment in Richard II (lines within

Act 5, Scene 5)

Moment in Hammerklavier (tempi

within 5th movement)

Status within both

works

Chaos: Reflexive pronoun (1-5) Chaos: Stripped down descending

thirds (Largo)

Self-referential

Emergence: ‘My brain I’ll prove

the female to my soul’ (6-10)

Emergence: Samples Self-affecting

thoughts of: ‘...divine’ (11-17) Harmony (Largo)

‘...ambition’ (18-22) Melody (un poco piú vivace)

‘...content’ (23-30) Counterpoint (Allegro)

‘With nothing shall be pleased...’

(38-41)

Rhythmic crescendo theme (a

tempo)

The breakthrough

Order: Richard calls himself ‘sire’

(41-63)

Order: Fractal symmetry (Allegro

risoluto)

Self-knowing

3

Chaos

I will appeal to ‘chaos’ in both its mythological and its mathematical contexts: chaos as the

formless void that preceded creation in Greek mythology2 and which was a stage of creation in

Christian mythology; chaos, in mathematics, describing a system of which the outcome is so

unpredictable that it appears to be random.3 It has been shown, and is traditional, that order can

arise out of chaos: the Greek primordial gods followed from the chaos;4 the tree of life (probably)

developed from random chemistry. In physics, apparently empty space is characterised by the

constant and random emergence of particles.5 From an artistic perspective, mythological and

mathematical chaos are equivalent: in art, constant change (akin to mathematical chaos) and no

change (akin to mythological chaos) are similarly meaningless. I am not suggesting that Beethoven

and Shakespeare had a premonition of Evolution or vacuum energy; rather, that their works are,

inadvertently, an expression of such concepts.

Beethoven establishes his chaos via music which is (for a time) without form or a frame of

reference. Although Haydn presents the same concept in The Creation of 1798, Beethoven does so

in a very distinct way. At the beginning of the last movement of Hammerklavier, the listener is

confronted with all the Fs on the keyboards of the time, after the previous movement finished with a

chord of F# major. This creates a jarring effect: presenting and remaining on the leading note (E#/F)

of the previous key key (F# minor, finishing in F# major), after it had resolved. Meyer, in his book

Emotion and Meaning in Music, emphasizes the importance of balance between regularity and

change in music: ‘both total segregation and total uniformity will produce sensation, but neither will

be apprehended as pattern or shape.’6 Beethoven combines an extreme form of segregation (via the

curious note) and uniformity (a number of versions of the same note, spread among the octaves).

2 ‘Greek creation’, The Oxford Companion to World Mythology, <www.oxfordreference.com> [accessed 9 December2014].

3 ‘chaos’, The Concise Oxford Dictionary of Mathematics, <www.oxfordreference.com> [accessed 23 December 2014].

4 Hesiod, The Homeric Hymns and Homerica, trans. by Hugh Evelyn-White (London: William Heinmann, 1977), p. 87.

5 ‘vacuum state’, A Dictionary of Physics, <www.oxfordreference.com> [accessed 23 December 2014].6 Leonard B. Meyer, Emotion and Meaning in Music (Chicago: University of Chicago Press, 1956), p. 159.

4

(5th movement; b. 1)7

The voicing of the chords that follow in the right hand is such that it emphasises stasis,

while the bass in the left hand is simply a descending motion, as such, it could be said that there is

no melody. Those descending thirds in the bass recapitulate, however, the same motif that is used

throughout the Sonata (such as in bars 135-65, repeated in the melody). Because the chords in the

right hand consist of a tied demi-semiquaver and semiquaver of which the semiquaver falls on the

strong beat, it is difficult to know what the rhythm actually is (when listening) because there is no

regular (and therefore discernible) frame of reference to which to compare the rhythm in the right

hand. This would be different if the right hand chords consisted of just the semiquaver. The whole

effect is intensified because the barring, which would otherwise have suggested a discernible beat,

cannot be heard by the listener.

Beethoven breaks Meyer’s maxim once more: his ‘segregation’ lies in the indeterminate

rhythm, being so distinct from a regular pulse that one cannot know what it is, it is, however,

‘uniform’ among the four chords. Thus, Beethoven is able to present the reader with just harmony,

without shape or direction. Being without melody, this moment excludes one of the prime means

through which musical ideas affect people. However this moment of segregation and uniformity is,

itself, part of an overall variety which is augmented via its presence.

There can be more formless music, such as a random series of notes or silence; however,

Beethoven presents a moment which – before tone-rows or musical freak-outs, in the formally

wrought context of Classical music and of his own music – represents formlessness while still being

music. It resembles, thus, mythological and mathematical chaos. Within that chaos is potential, the

harmony represents the descending thirds that unite whole sonata, which, to most listeners, would

7 Ludwig Van Beethoven, Complete Piano Sonatas, ed. by Heinrich Schenker (Vienna: Universal Edition, 1920; repr.New York: Dover Publications, 1975) pp.511-556.

5

not be apparent. A Biblical analogy: ‘the earth was without form, and void’.8

In Act 5, Scene 5 of Richard II, there is a personal parallel: Richard is ‘unkinged’ in a

similar fashion to how Beethoven diminishes the germ of Hammerklavier. Kantorowicz observes

the extent to which the play involves medieval theories of monarchy, specifically the ‘two bodies’

of the King: personal and political. Richard loses the metaphysical body of his kingly nature, with

its attendant association with God, and becomes a mere human.9 Historically, Richard was king

before he was a man (since he was ten years old); I argue, because the personal Richard is adapted

to his power and influence, he is now less than a man in the regular sense.

The prison resembles Greek chaos, being a space, almost certainly without furnishing or

comfort which is, with the exception of Richard, empty: it is like a private chaotic world, being all

he can access at this time:—

Richard: I have been studying how I may compare

This prison where I live unto the world;

And for because the world is populous,

And here is not a creature but myself,

I cannot do it. Yet I’ll hammer it out. (5.5.1-5)10

The fact that Richard has been ‘studying’ for an indeterminate time disestablishes a regular

and dependable chronology, because it is not possible for the audience to know how long Richard

has been doing so. Population, people and community seem to be an operative index, in that

Richard, as king, would have been able to rule over all the people England. Here, the opposite is the

case: it is just him, he is unable to affect people or to experience culture.

Metatheatrically, Richard’s loneliness is the antithesis of the earlier performative power that

he had over the other characters, especially when he was being deposed and afterwards. To his

question, ‘Am I both priest and clerk?’ (4. 1. 163 (p. 360)), no one can answer, embarrassing all

present. This rhetorical ability is useless now that he is alone and cannot perform to anyone. He is a

cultural actor in potential, but cannot affect others because he is alone: all he knows is a void. In this

state, Richard is similar to Beethoven’s isolated germ: harmony, alone, without the other facets of

music and thus without the power to affect.

Richard declares that he is comparing the incomparable (his prison and the world): this is

8 The Holy Bible (King James Version) (Cambridge: Cambridge University Press, [n.d.]), Genesis 1. 2.9 Ernst H. Kantorowicz, The King’s Two Bodies: A Study in Mediaeval Political Theology

(Princeton: Princeton University Press, 1957; repr. 1997), p. 40.10 William Shakespeare, Richard II, The Oxford Shakespeare Complete Works, ed. by Stanley Wells,

Gary Taylor, John Jowett & William Montgomery (Oxford: Oxford University Press, 2005), p. 365.

6

not to say that the comparison is inherently invalid, but that his thought process is invalid. Noting a

difference in population, Richard focusses on the material distinctions, missing, for example, the

similarity of the world and the prison in denying the individual certain freedoms. A list generated by

an absurd comparison like this would resemble a random set of qualities, people and things,

resembling chaos, this time in the mathematical sense. Richard’s consciousness might be said to

break Meyer’s maxim, as Beethoven does, in presenting too much change.

His being ‘unkinged’ and his isolation appear to be causal: the former acts as a shock and

diminishes him, while the latter forces him to turn inwards for meaning. There is not much to be

found inwards, given his diminished personality; in that this state is new to him, the initial

consciousness is aimless. His rump personality, social isolation and chaotic thoughts might

represent an overall chaos: disorder and the void. Nonetheless, Richard still maintains, apparently,

his sense of self: he uses personal pronouns five times in the quotation above, the most important of

which being the reflexive pronoun, ‘myself’. Reflexive pronouns signify the interaction with

oneself: this is to say that Richard, despite the duress, is self-aware.

This self-reference in Richard II seems analogous to the descending thirds idea in

Hammerklavier, referring back to the prior parts of the sonata where this motif occurs. Furthermore,

the quoted phrase from Beethoven represents that element with only one form of expression

(harmony), just as Richard uses one of the most basic forms of self-reference (the reflexive

pronoun). Beethoven severs the germ of the sonata from the previous key and presents it,

disorientated and bare, with a diminished power to affect; just as Richard is severed from his kingly

body and is without performative or cultural agency, and with a disordered mindset. Richard throws

off the sense of time within the play, and so does Beethoven’s rhythm create uncertainty with

respect to this index. This analogy is fundamental, moreover, given the importance of time and

duration for music; time, furthermore, will become an important subject for Richard. Richard has

only himself for a frame of reference, like the series of chords in Beethoven, whose rhythm is

impossible to apprehend.

Nonetheless, both creators seem to depict the potential that exists within chaos: in

Beethoven, harmony and the germ of the Sonata still endure, though nearly dormant, while, in

Shakespeare, self-awareness remains. Self-reference is fundamental in that self-similarity (among

the results of self-reference) is one of the most basic tenets of form: as such, within chaos exists the

potential for form.

7

Emergence

In Hammerklavier, what follows the chaos might be called a ‘sampler’ of three different

aspects of music, separated by the same harmonized descending thirds (b. 1-7). The first sample has

melody and pulse, and is definitely not formless. Being, however, a similar harmonic sequence in

the form of broken chords, it seems to represent elaboration on what went before, rather than

composition. The second sample, consisting of free-form scales in both hands, has a melodic

quality, while being broadly Classical in that there is no sequence. The third sample, as Charles

Rosen observes, is an example of Baroque counterpoint.11 I don’t think it would be too much to say

that these three samples represent, in some sense, ‘harmony’, ‘melody’ and ‘counterpoint’,

respectively. The sampler has the character, also, of experimentation, or quasi-randomness, as

though Beethoven were allowing the themes to emerge from the core of his work.

The presentation of these sections – often running into the intervening descending thirds

(the last note, rhythmically, of the first sample is the first of the following chords) or following them

directly – might imply that these brief interludes are generated from the formless germ.

Mathematical chaos produces apparently random outcomes – it is as though Beethoven represents

the harnessing of that chaos in order to produce new material, though with a little randomness still

inherent in it. The sampler seems to represent Beethoven grasping chaos and forcing it to conform

to some sort of structure, as it writhes and rebels.

Meanwhile, Richard describes mental introspection, explaining how, though he is alone, he

can still access population:—

My brain I’ll prove the female to my soul,

My soul the father, and these two beget

A generation of still-breeding thoughts;

And the same thoughts people this little world

In humours like the people of this world. (5.5.1-5 (p. 365))

It seems that Shakespeare uses ‘brain’ to refer to the thinking capacities, while ‘soul’ might

indicate the centre of the self; this interaction would represent consciousness, therefore, given that

consciousness can be described as cognitive potential plus and individuality (the will to think).

Centrally, ‘still-breeding’ suggests that Richard’s thoughts interact, themselves, producing new

11 Charles Rosen, The Classical Style: Haydn, Mozart, Beethoven (New York: W. W. Norton and Company Inc., 1971,repr. 1998), p. 428.

8

thoughts as progeny. Thus, Richard can produce multitudes via the self-reference that is the

interaction of ‘brain’ and ‘soul’. Calling his thoughts ‘people’, Richard implies that he has solved

the loneliness of previously: now he has population and culture within himself; he regains the

metatheatrical power which, earlier, required other people. This realisation seems to result from an

advance on his previous state (self-reference), in that now he is self-affecting: having the ability to

change his mindset and produce new mindsets.

What follows is a series of abstract observations, which are very similar to Beethoven’s

samples, for example:—

[...] The better sort,

As thoughts of things divine, are intermixed

With scruples, and do set the faith itself

Against the faith, as thus: ‘Come, little ones’,

And then again,

‘It is as hard to come as for a camel

To thread the postern of a small needle’s eye.’

Thoughts tending to ambition, they do plot

Unlikely wonders: how these vain weak nails

May tear a passage through the the flinty ribs

Of this hard world, my ragged prison walls; (5.5.12-21 (p. 365))

Richard seems to suggest that it is possible for his consciousness to work against him, with

the twisting of faith as a metaphor. His command ‘Come, little ones’ seems to be directed at his

thoughts, who then rebel, using a perverted form of Jesus’ wisdom.12 Richard seems to depict the

dangers that are associated with thinking and, centrally, the extent to which thoughts have a life of

their own; once his thoughts produced new thoughts he unleashed a chain reaction that is not

entirely controllable. Quite importantly, what seemed to him an absurd suggestion earlier (to

compare his prison with the world) is now totally apt.

While insightful, it is easy to forget that these observations are abstract, with little relevance

to the play as a whole or Richard himself. The samples within Hammerklavier and these thoughts

seem to have a very similar roles: they both represent an almost unrestrained exploration of

possibility, as Beethoven plays with the rules of music, Richard plays with the rules of thought.

Beethoven is taming the chaos, but there is still a bit of wildness in it – Richard has unleashed a

12 The Holy Bible (King James Version) (Cambridge: Cambridge University Press, n.d.), Luke 20. 25.

9

population of thoughts from chaos, but he is not yet master of them. His self-referential and self-

affecting nature seems implicated, too: as each of Beethoven’s samples are bathed in (and in some

cases, derived from) the germ, Richard’s speech is him talking about the content of his mind which

is, itself, the result of him talking to himself.

Meanwhile, the representation of a moment of self-reference, followed by such explosions

of creativity might suggest that merely channelling the chaos is insufficient to create something

worthwhile. Rather, by observing oneself and one’s output, one can select those random ideas which

are worthwhile and develop them. As such, Beethoven and Shakespeare represent the evolutionary

process in art: a system which can affect itself, selecting those parts which work best and discarding

the rest – without this primordial self reference, all that can endure is chaos.13

After his experimentation, Beethoven could be said to conclude with the descending thirds

once more; although, this time, with a strong, pulsating rhythm:—

(5th movement, b. 10)

Of course, this chord-sequence occurred with a regular pulse in the first sample; this time,

however, the pulse is far stronger, with definite chords in both hands and syncopation. This seems to

represent the completion of the period of experimentation: now the theme has a definite rhythm, the

chord sequence appears to be, rather than uncertain or experimental, definitive; it has a motion, it is

going somewhere. Where?

The analogue of this in Richard II appears to be Richard’s maxim:—

13 Both works have three such ‘experimental’ episodes; this similarity, however, seems to be incidental.

10

Nor I, nor any man that but man is,

With nothing shall be pleased till he is eased

With being nothing. (5.5.39-41 (p.365))

It seems that, after a period of experimentation, like Beethoven, Richard has finally reached

a definitive expression of, in his case, knowledge. Donovan Sherman claims that at this stage:

‘Richard’s being persists in affirming a kind of life: he is neither mere matter, nor a rejected scrap.

He becomes a nothing that lives, a story that is performed; in short, he becomes an actor’.14 This is

convincing, though I would take it a step further: Richard was ‘an actor’ during the scene in which

Harry took the crown. Richard was both priest and clerk, now he is both actor and audience, just as

he is able to create a kingdom within his own mind.

Fundamentally, it seems that imprisonment and being de-throned has actually been

personally productive for Richard: he is a greater man and thinker as a result. This experience,

which must have felt like being brought down to nothing, has forced him to accept that he is

‘nothing’, as are all people: limited, insignificant in the grand scheme of things. Grandiosity and

egocentricity cause suffering; realising one’s insignificance and the necessity, therefore, for

creativity within one’s own mind, as Richard has done in the previous imaginings, is the road to

happiness. Richard’s diminution is his spiritual growth. The same seems to be the case for

Beethoven: diminishing the germ of his composition to its most basic form opened it to abstract

creativity, leading to the varied fluctuations and hence a strident and unified iteration of the germ

(as seen just before bar 10), making it a theme in itself. Of course, the Shakespeare quotation above

ends on a half-line, while the Beethoven calls for a crescendo and dramatic diminuendo – in both

works, something is coming.

14 Donovan Sherman, ‘“What more remains?”: Messianic Performance in Richard II’, Shakespeare Quarterly, 65 (1) (2014) 22-48, p. 48.

11

Order: The Music Plays

Beethoven follows his crescendo with a complex fugue theme which, as Rosen explains, is

also based upon the string of descending thirds, (Rosen, p. 430) to be joined by two more voices,

thus forming one of the greatest fugues. It might be said that fugues are the embodiment of order

within music, demanding strict adherence to rules and hoping that, though this, the composer can

achieve beauty. Thus, Beethoven has charted the journey all the way from chaos to order.

Moreover, in his Fugue, Beethoven synthesises the individual musical components from the

samples. There is counterpoint, by definition. Melody is fundamental, moreover, setting this Fugue

apart (especially from the example of counterpoint in the sampler) through Beethoven’s seamless

combination of counterpoint and Classical melody, such as in bar 105:—

(5th movement, bb. 101-109)

One could say that this is one of the few fugues whose dynamics, rich voicing, registration

and changing harmony give a sense of drama. The octave bass-line combines power with a melodic

intensity that approaches that of the vengeful statue in Mozart’s Don Giovanni of 1787.

Beethoven’s tonal structure might be the triumph of the sonata: changing key down a third

for each section (such as an exposition or episode). Thus, imitating the germ, the Fugue is fractally

symmetrical, resembling on the large scale itself on the smallest. Beethoven synthesized

counterpoint, Classical melody and Classical tonality, introducing into an ancient form the deeply

symbolic use of keys that he pioneered. It could be said, therefore, that Beethoven took his music

12

back to the very basics, then allowed it to produce something new; out of disorder: order on a

greater level than before.

In Richard II, immediately after Richard utters his maxim, ‘The music plays’:—

[...] Music do I hear.

Ha, ha; keep time! How sour sweet music is

When time is broke and no proportion kept.

So is it in the music of men’s lives. (5.5.41-44 (p. 365))

The ‘music’ here, starting directly after Richard’s maxim, must be symbolic; having lost its

regular beat, it seems to symbolise, audibly, Richard’s realisation of his wrongs, and hence truth.

What follows seems to be the only example of contrition on the part of Richard in the whole

play:—

And here have I the daintiness of ear

To check time broke in a disordered string

But for the concord of my state and time

Had not an ear to hear my true time broke.

I wasted time, and now doth time waste me,

For now hath time made me his numb’ring clock.

[...]

Now, sire, the sounds that tell what hour it is

Are clamorous groans that strike upon my heart,

Which is the bell. So sighs, and tears, and groans

Show minutes, hours, and times. (5.5.45-50; 55-58 (p. 366))

Indeed, Richard, for the first time, admits that he showed poor judgement while he was

king. Making Richard the subject then the object, ‘I wasted time, and now doth time waste me,’,

might imply that his behaviour was what transformed him from a leader into a serf. Time, the

medium of causality, might be synonymous with natural law: he went against natural law (being

King, but not upholding the law), hence, natural law strikes back. Time means an order to events,

time implies causality: there can be no music without time. Though Richard expresses a tragic

sentiment, he has found order and realises that his actions have consequences and, as such, realises

his sins: he has, as Beethoven expects of the pianist, hammered it out.

13

This is particularly analogous to Hammerklavier, also, in that the curious rhythm in the first

section can be regarded as making the passage of notes with respect to time uncertain – time is hard

to discern. The Fugue, with a proper sequence and definite durations, therefore, closely resembles

Richard’s apprehension of time and its power. The fact that this understanding only came after, and

seems to have resulted from, chaos and emergence, might suggest that power is antithetical to a true

understanding of oneself and, moreover, that seclusion can correct this distortion. Hammerklavier

might say something similar, though with reference to form and genre: it is necessary to discard

them, temporarily, in order to epitomise the former and transcend the latter.

Richard’s ‘sire’ is confusing: to whom is he talking? Perhaps himself: his nature is so clear

to him that he is like another person (he is objective, or symbolic). The subjective experience

usually results in misapprehensions about oneself that are not made with respect to other people;

objectivity has helped him to admit his failings. Richard’s experiences, thus, might make him have a

perspective outside himself, meaning that he has progressed from referring to himself, to affecting

himself, to knowing himself.

This seems to be analogous to the fractal symmetry of Beethoven: the grander scale refers to

the small scale, and vice-versa. This self-reference is removed, in that tonality and tones are such

different aspects of music. Similarly, Richard finds a way to be sufficiently separated from himself

such that a second-person pronoun is appropriate – just as the descending thirds within the tonality

refer to the descending thirds of the theme, though they are not the same. This analogy is intensified

by the fact that the tonal order could be said to observe the thematic order, while Richard

objectively observes himself. On some level, the Fugue might be said to know the germ of

Hammerklavier and, hence, itself.

14

It could be said that, therefore, at the beginning of the speech, Richard is self-aware, he then

becomes self-affecting, then self-knowing. Hammerklavier has a similar relationship with itself: it

reciprocates the unifying symbol of the work, and is thus self-referential; it experiments with it and

the rules of music, so is self-affecting; the whole Fugue maps out the germ: it is, hence, self-

knowing.

15

The reader might be concerned, however, that Richard dies – worse, he rejects his new

wisdom: ‘This music mads me. Let it sound no more,’ (5.5.61). Beethoven’s Fugue, meanwhile,

ends in a glorious cadence, while the synthesis which he produced lived in his final works. Indeed, I

would still class Richard’s experience as ‘order’, despite the fact that it is fleeting. The music does

not ‘mad’ him, quite the opposite, the disturbing realisations are so true, compared to his previous

(and perverse) mindset, that he feels mad. Shakespeare used the tragic history as a container for

Richard’s revelation, it was entirely necessary for him to attain his higher state of thought.

Meanwhile, Richard’s wisdom, unlike the historical Richard, lives on; he fell, that his thoughts

could rise.

It could be said that both works derive order from chaos, power from diminution. They

demonstrate that chaos is order in potential, that self-reference is the key to filtering out meaning

from the its randomness. Beethoven’s demonstration that the prostration of the germ of his sonata

leads to its augmentation resembles Richard’s idea that people have to realise that they are nothing:

both imply that people are too obsessed with building, with more; there is deep wisdom and order in

less and even nothing. It might be said, therefore, that order is merely a self-similar selection of

chaos, and self-knowledge is necessary to derive it.

16

Isomorphism II: Wordsworth’s The Prelude and Bach’s The Art of Fugue

The Prelude by Wordsworth (written in many different iterations from 1798 onwards and

finally published in 1850) and The Art of Fugue (written by Bach between 1738 and 1749) present

another isomorphism, suggesting a broader significance in which both works are united. The

principle of meta-levels is fundamental to my thinking: when I refer to the basic level, I mean the

simplest frame in which a system is meaningful, such as notes and harmony in music, or words and

sentences in literature; when I refer to numbered levels above this, level-I discusses the arrangement

of elements on the basic-level, and level-II the arrangement of elements on level-I, and so on. An

element of art or language can work on multiple levels: the exposition of a musical theme works on

the basic level, while Variation 22 from the Diabelli Variations by Beethoven (for example) works

on level-I as well as on the basic level, because the music discusses itself in relation to Mozart and

the Variations as a set.

There is a great deal of discussion over the ordering of the sections of The Art of Fugue. The

edition which I shall use hereafter calls what is usually ‘Fugue 14’, ‘Contrapunctus 19’. To avoid

confusion, given German translations and alternative numberings, I will call this section (the

unfinished Fugue, in four voices with three themes) ‘Fugue 14’.

Once more, I draw a great deal from Douglas Hofstadter’s Gödel, Escher, Bach, wherein

analysis of meta-levels and self-reference is central. Hofstadter uses the terms ‘push’ (meaning to

ascend to a higher meta-level) and ‘pop’ (meaning to descend to a lower level or to the basic level).

To ‘jump’ out of a particular system is another Hofstadter term, which describes attaining an

overview of the workings of a system in order to explore a problem (or abandoning the system

altogether), rather than continually obeying the system’s rules (Hofstadter, p.128). I will use these

terms hereafter, for brevity. Hofstadter mentions The Art of Fugue and Fugue 14 in his book,

commenting on how Bach’s self-reference (via the BACH theme; German notation, i.e. Bb, A, C,

B) and the abrupt termination of the piece shortly after might represent the problematic nature of

self-reference in the formal systems which he studied.

Both The Prelude and The Art of Fugue appear to undergo two cumulative pushes, and a

third push which is qualitatively different from the previous two, allowing both works to jump out

of their respective systems and become something entirely different, having a significance beyond

their form and genre.

17

Table II: Schedule of Analogies between The Art of Fugue and The Prelude

First push: The iterative and inter-related theme

development during the whole work, referring

back to the original theme

First push: Self-reference through the ‘monitory

voice’ scene, describing the emergence of

consciousness

Second push: During Fugue 14, the combination

of the first theme of The Art of Fugue

(implicitly) with the main theme of Fugue14 and

the BACH theme (bb. 233-299), embodying a

composer

Second push: During the ‘lark’ scene,

Wordsworth describing Wordsworth (the lark)

fly over Wordsworth (the world): a meta-

consciousness

Quandary: Forces a choice between pseudo-

Fibonacci numbers and a complete work,

chooses pseudo-Fibonacci

Quandary: Forces a choice between fluidity and

publication, chooses fluidity

Third push: The incompleteness forms a

framework through which the piece, as a

compositional puzzle, can compose, while

maintaining the pseudo-Fibonacci numbers

Third push: The non-publication enabled

Wordsworth to continue with modifications,

maintaining the active meta-consciousness,

which died when he did

18

The First Push

Consciousness is central to The Prelude, though I will use a slightly uncommon definition

for this concept, from John Locke: ‘the perception of what passes in a man’s own mind’. This

definition necessitates a mind-state that occupies level-I, rather than basic awareness. In the

following passage from the 13-book Prelude, Wordsworth seems to describe a moment in the past

when he was not conscious:—

[...] to bed we went,

With weary joints and a beating mind.

Ah, is there one who ever has been young,

And needs a monitory voice to tame

The pride of virtue, and of intellect?

[...]

A tranquillizing spirit presses now

On my corporeal frame, so wide appears

The vacancy between me and those days,

Which yet have such self-presence in my mind

That, sometimes, when I think of them, I seem

Two consciousnesses—conscious of myself

And of some other being. (II: 17-21; 27-33)15

The ‘monitory voice’ – because of the connotations of the word ‘monitor’ (attention and

responsibility), and because ‘voice’ necessitates speech and communication – suggests the

measured, self-critical part of one’s internal dialogue. In that this is lacking in the younger

Wordsworth, consciousness is absent. The element ‘beating mind’, furthermore, resembling the

unceasing and involuntary action of the heart, might suggest that his thoughts, at this stage, are also

involuntary. Quite importantly, Wordsworth gains this knowledge from his future perspective.

The fact that this is expressed as a question means that it is interpretable as an indictment of

the non-tame behaviour of a young person or as stating that such a phase is natural. Fundamentally,

the act of looking back seems to generate the ‘other being’ for which the contemporary Wordsworth

can be a monitory voice. There is, of course, no young Wordsworth in this scene, only the poet

15 Wordsworth, William, The Prelude 1799, 1805, 1850, ed. by Jonathan Wordsworth, M. H. Abrams and Stephen Gill(New York: W. W. Norton & Company, 1979), p. 66. All further quotations will refer to this edition.

19

communing with his own memories. This is the first push, with Wordsworth ascending to and

operating within the level-I, that is, commenting on and criticising his own thoughts (monitoring),

while the young Wordsworth operated within the basic level of action and thought.

It could be said that consciousness is, therefore, an emergent property of the act of personal

retrospection, which Wordsworth emphasizes in Book VI of the 13-book Prelude:—

Imagination!—lifting up itself

Before the eye and progress of my song

Like an unfathered vapour, here that power,

In all the might of its endowments, came

Athwart me. I was lost as in a cloud,

Halted without a struggle to break through,

And now, recovering, to my soul I say

‘I recognise thy glory’. (VI: 525-532, p. 216)

The fact that imagination lifts itself necessitates that it is not the direct result of some other

object or process, not even of his own free will, as it came ‘Athwart’ Wordsworth. The fact that the

vapour is ‘unfathered’ – without a source – might imply the concept of an emergent property: a

property of a system which is not characteristic of the parts of that system, as culture emerges from

human societies and does not develop in individuals. Wordsworth’s song (presumably The Prelude)

is not declared to be the cause of his imagination; rather, it and the ‘eye’ (perhaps describing the

compass of the poem) are juxtaposed with the vapour. Thus, the process of introspection is

associated with this level of thought in a similar way to how culture is associated with the

interaction of individual people.

Imagination is not synonymous with consciousness, though it is definitely a concept on the

same plane, existing above simple ‘thought’, requiring a combination of directed thought and

inspiration. It seems that Wordsworth describes something greater than the imagination possessed

by young children, given that it can act against him. Together with the previous passage, this seems

to imply that consciousness emerges from the self-reference of introspection and retrospection.

Apparently, Wordsworth has reached the awareness of his thoughts that Richard reached, though

near the beginning of The Prelude, where Richard only reached this point near the very end of the

play.

In The Art of Fugue, Bach undertakes a similar process. Unlike most of his other complex,

large and unified works (such as The Well Tempered Clavier, or The Musical Offering), The Art of

20

Fugue remains in a single key, and the theme of each fugue is a development of the original theme

as it is expressed in Fugue 1. Just as Wordsworth refers back to his former self, highlighting the

subsequent developments, Bach’s iterations make references among themselves.

Here is the original theme:—

(Contrapunctus 1; b. 1-4)16

(Contrapunctus 4; b. 1-4)

Above, in Fugue 4, Bach submits the theme to a simple inversion, making it comply with

the same key by beginning on A.

(Contrapunctus 6; b. 1-4)

Between the first and sixth fugues, the non-inverted theme undergoes minor changes, which

are seen in aggregate in Fugue 6, that is, the descending scale rather than arpeggio in bars 1 and 2,

and the dotted crotchet/quaver rhythms instead of plain crotchets and quavers.

Thereafter, Bach builds on the previous changes, referring back to the archetypes:—

16 Johann Sebastian Bach, Die Kunst der Fuge, ed. by Christof K. Biebricher (Siegburg: Werner, 1999).

21

(Canon Alla Duodecima in Contrapunto Alla Quinta; b. 1-8)

The embellishment, here, falls upon the original theme: bar one (D), bar two (A), bar five

(C#), bar six (D, E) and so on. Meanwhile, the following canon treats the inverted form of the

theme of Fugue 4 with a similar sort of embellishment. This is analogous to, in The Prelude, the

‘monitory voice’ scene, in which Wordsworth refers back to his former self: Bach refers back not

just to the theme, but to an earlier modification of it.

(Canon in Hypodiapason; b. 1-4)

As such, Bach creates a sort of thematic progression, though not one with constant

development or where one variation is followed by a variation on it, but a succession of themes

which are all clearly derivable from the first and which refer back, skipping over many variations,

to earlier variations. It could be said that Bach mirrors in the structure of the whole work that of a

fugue, with the different themes entering at different points, in the original from or inverted, with

the embellishments, perhaps, representing free material. This would demand a finale in which the

original theme emerges with appropriate grandeur – the theme of Fugue 14 is not the original,

though I will argue later that it represents it in a relevant way. As such, it could be said that The Art

of Fugue operates not just on the thematic level, but on the meta-thematic level, just as Wordsworth

presents in The Prelude the meta-level above normal thought. However, Bach’s level-I is music

referring to music, while Wordsworth’s level-I refers to the subject of his narrative. Nonetheless, I

still consider them to be isomorphic, because this is a common difference between music and

literature: music is often about itself; The Art of Fugue even declares itself to be so.

There seems to be a similar relationship between Hammerklavier and The Art of Fugue as I

identified when comparing The Prelude to Richard II. Bach’s level-I operation begins early in his

22

work and compasses nearly all of it; it takes Beethoven until his fifth movement reach level-I. This

difference seems to reflect that each pairing depicts a different aspect of creativity and life.

There is another push in The Art of Fugue, also. In his essay, ‘The mathematical architecture

of Bach's The Art of Fugue’, Loïc Sylvestre describes how Bach very comprehensively wove the

Fibonacci Sequence into his creation. The Fibonacci Sequence refers to the series 1, 1, 2, 3, 5, 8

etcetera, wherein each number after the second is the sum of the two preceding numbers in the

sequence. The sequence has much relevance, especially the fact that the ratio of consecutive terms

tends towards the golden mean. The golden mean holds great importance in art and culture, being a

ratio found in the human body and common among many shapes that are visually pleasing, while

having a beautifully simple explanation: two numbers satisfy the golden ratio if it is the same as the

ratio of the larger to the total.17

Sylvestre’s analysis involved removing parts of the work that were not proper to it, such as

BWV I080/I0a (a preliminary version of BWV I080/I0) and Wenn wir in hoechsten Noethen Canto

Fermo in Canto (added by the publisher) among others.18 The paper describes how the fugues’

length in bars (some individually and some in pairs) closely approximate certain Fibonacci

numbers; moreover, each of the the relevant groupings (such as canons, simple fugues, counter

fugues etcetera) approximate a Fibonacci number, as well as fugues 1-7 and 8-14, and all the

fugues. Because the differences between the terms of the Fibonacci sequence tend towards the

golden mean,19 so does the ratio of total bar numbers between the aforementioned parts of the work.

As Sylvestre notes, Bach was a member of the Corresponding Society of the Musical

Sciences (Sozietät der musicalischen), whose members sought to unify the disciplines of

mathematics, art and science, especially Pythagorean theory. Ruth Tatlow has demonstrated that the

Fibonacci sequence and the golden ratio were known at the time.20 She found, however, no treatise

on their use in music (Tatlow, p. 83), suggesting that it would be plausible for Bach to deliberately

insert the sequence into the work, though it would have been a unique choice.

Sylvestre seeks to prove that the Fibonacci pattern was a deliberate insertion on the part of

the composer. He does not, however, discuss why Bach would not use perfect Fibonacci numbers.

As Wolff notes, handwriting analysis suggests that Bach wrote the last fugue years before he died; it

could not be said that the Fibonacci numbers are imperfect because Bach’s death prematurely ended

17 Emmanuel P. Prokopakis, Ioannis M. Vlastos, Valerie Picavet, Gilbert Nolst Trenité, Regan Thomas, Cemal Cingi, Peter W. Hellings, ‘The golden ratio in facial symmetry’, Rhinology, 51 (1) (2013) 18-21, (p 20).

18 Sylvestre, Loïc, ‘The mathematical architecture of Bach's The Art of Fugue’, Il saggiatore musicale: Rivista semestrale di musicologia, 17 (2) (2010) 175-195, (pp. 176-6). All further quotations will refer to this edition.

19 Megha Garg, ‘Ratio by Using Coefficients of Fibonacci Sequence’, International Journal of Mathematical Combinatorics, 3 (2013), 96-103, (p. 102).

20 Ruth Tatlow, ‘The Use and Abuse of Fibonacci Numbers and the Golden Section in Musicology Today’, Understanding Bach, 1, (2006) 69-85, (p. 81).

23

the composition.21 My discussion of this question, however, will have to wait until a later section,

where it is relevant.

To examine whether these numbers occurred by chance, I performed a similar analysis on

The Well Tempered Clavier,22 wherein I found a distribution of numbers with no tendency for

Fibonacci numbers to predominate. I can say with some confidence that Bach deliberately inserted

pseudo-Fibonacci numbers: what could he mean by doing so? A term of the Fibonacci sequence can

only be defined as the sum of the two previous terms; thus, it could be said that the pseudo-

Fibonacci numbers emphasize the extent to which The Art of Fugue is a self-referential set, derived

from and always referring back to the original theme.

The golden mean is, of course, famous for combining art, nature and mathematics; similarly,

the fugue (and canon) form was treasured for its combination of beauty and formality. The title The

Art of Fugue suggests a definitive exposition of this form. The golden mean is an irrational number:

one which cannot be expressed, only approximated; the ratios of Fibonacci numbers get closer and

closer to it but never reach it. Could Bach be intimating, firstly, that describing beauty is his goal

and, secondly, that it is to be found via an iterative, self-referential process which cannot be finished

in a conventional sense?

This first analogy between Bach and Wordsworth emphasises the fundamental importance,

for consciousness and art, of self-reference: it is the framework through which Wordsworth explains

consciousness and through which Bach constructs the majesty of The Art of Fugue. The poet’s mind

in The Prelude needs to examine its memories and thoughts, per Locke’s definition, so self-

reference is obligatory. The case seems more complicated for The Art of Fugue: that is, the

obsessive, self-contained nature of The Art of Fugue is so complete that, rather than producing

something limited or merely academic, it produces a work which compasses nearly everything:

fugues that delight the uninitiated, dances, tearful melodies and so on. Could the restrictions of

Bach’s manifesto have led him to a form of development, the self-reference of which causes him to

depict something with the scale and nature of a world? The works might intimate, therefore, that

musical beauty and consciousness are not created entities, but emerge, given the right

circumstances.

21 Christoph Wolff, Johann Sebastian Bach, the Learned Musician (New York: Oxford University Press, 2001 (2002 paperback)), p. 447.

22 J. S. Bach, Das Wohltempereierete Klavier, ed. by Otto Von Irmer, 2 books (München-Duisberg: G. Henle Verlag, 1970), books I & II.

24

The Second Push

In Wordsworth, the second push occurs in book XIII of the 13-book Prelude (book XIV of

the 14-book version):—

[...] Anon I rose

As if on wings, and saw beneath me stretched

Vast prospect of the world which I had been,

And was; and hence this song, which like a lark

I have protracted, in the unwearied heavens

Singing, (XIII 378-383, p. 478)

Where, in the previous passages, Wordsworth had been looking back and describing his

former self, here, he looks back on a self that is looking back; there are three Wordsworths, the one

in the form of the world, one in the form of the bird, and the narrator. In that the ‘world’ is a

representation of what Wordsworth was, it could be said that space is a metaphor for the changes in

the personality of the prior Wordsworth throughout time. The imaginative perspective here is

supreme, being fully able to traverse and see any state from his past, while the narrator is stationary

and merely thinking. Given that, during book II, he operated on level-I via referencing himself,

here, observing himself while he observes himself, he operates on level-II. Earlier, Wordsworth

notes that, ‘the termination of my course/is nearer now’ (372-3), suggesting, given that his

manifesto is to set out the ‘Growth of a Poet’s Mind’, that he nears this goal, and that, therefore, this

state of consciousness, involving many layers of discourse, embodies something of the mind that he

hopes to describe.

In the first instance, the Wordsworth of the narrative operated on the basic level, and the

narrative operated in level-I while, in the second instance, the Wordsworth that is described operates

on level-I, the narrative, therefore, operates on level-II. The Prelude – by adding a meta-level with

respect to the Wordsworth of the narrative and predicting his conscious state in book XIII – could

be said to be one step ahead, referring forward, as the narrative looks back: the poem affects its

future states. It could be said that Wordsworth, in assembling a poem that predicts and affects its

future states, while referring backwards to memories, has constructed a mind. The question that

arises, therefore, is, what is to come, given that the bird scene predicts level-III operation?

Bach presents a similar occurrence to the bird scene during the very last bars of Fugue 14,

the lower four lines of music are Bach’s:—

25

(Contrapunctus 19 (Fugue 14); b. 233-9)

Gustav Nottebohm describes how the original theme from The Art of Fugue can be placed

alongside the other themes at this point, in harmony, as I have set above.23 The opening theme of

Fugue 14, seen here in the bass, is sometimes described as being new, not a version of the original

theme of The Art of Fugue. It could be seen, however, as a drastically simplified version of the

original, incorporating important features of some of the successive iterations. It features the rising

fifth from the original then the scale movement downward, as featured in Fugue 6. Meanwhile, the

three note scale up and down from A resembles a simplified version of the scale passage at the end

of the inverse form of the theme. It might be said that the theme from -Ffugue 14 satisfies the

differences between all versions of the theme via its simplicity.

The BACH theme (German notation, i.e. Bb, A, C, B) enters many bars before, marking the

final section of the Fugue, though this moment is the only time during which all three of its themes

are present. It and the ‘lark’ scene in Wordsworth are analogous: the BACH theme of bar 225 aligns

23 Gustav Nottebohm, ‘J. S. Bach's letzte Fuge’, IMSLP <http://imslp.org/imglnks/usimg/f/fc/IMSLP348435-PMLP562864-Nottebohm.pdf> [accessed 22 December 2014] (p. 2).

26

with Wordsworth the narrator (the artist), the opening theme of Fugue 14, entering in bar 224, aligns

with Wordsworth the bird (a past but not original iteration of Wordsworth), while the implied

original theme aligns with Wordsworth as ‘the world of what I was’. Bach operates on level-II, by

juxtaposing himself with two versions of The Art of Fugue theme, discussing the thematic process

that allowed one theme to transition to another. The subject of these bars becomes, therefore,

composing itself and Bach himself as composer. Working on level-II (the level of the composer

when they observe their their work within a composition), the Fugue acts like a composer. Where,

for Wordsworth, the bird scene augments the model mind, this part of Fugue 14 describes

something like a model composer. One might expect Bach to explore the interaction of the themes,

but it ends here: this is the third push.

27

The Third Push: Jumping out of the System

The third push for each work is enabled by the fact that their creator didn’t publish them in a

complete and finalised form. For Wordsworth, this is transformative because it allowed him, while

he lived, to modify his work continuously. Given that he created in The Prelude a model of his

consciousness, a published form would be lacking inasmuch as it would be fixed and dead while, in

Wordsworth’s possession, he holds the ability to change it as he changed. Wordsworth’s quandary,

therefore, is that he could not express his mind through his work to anybody outside his close

friends without publication; but, via publication, The Prelude would lose its fidelity and dynamism.

It could be said that the various Preludes are skeletons of a living work that endured when their

author lived.

For example, the reflective passage after Wordsworth describes the death of his father

changes throughout the different versions of The Prelude, in this case, the 5-book, 13-book and 14-

book:—

And I do not doubt

That in this later time, when storm and rain

Beat on my roof at midnight, or by day

When I am in the woods, unknown to me

The workings of my spirit thence are brought. (V: 385-389)24

[This version is identical to the 5-book version until line 388, after which it

continues, the 5-book version ends here.]

Thou wilt not languish here, O friend, for whom

I travel in these dim uncertain ways—

Thou wilt assist me, as a pilgrim gone

In quest of highest truth. (Wordsworth, 13-book Prelude, XI: 389-392, p. 436)

[...] when storm and rain

Beat on my roof, or, haply, at noon-day,

While in a grove I walk, whose lofty trees,

Laden with summer’s thickest foliage, rock

In a strong wind, some working of the spirit,

24 Wordsworth, William, The Five-Book Prelude, ed. By Duncan Wu (Oxford: Blackwell, 1997), p. 150.

28

Some inward agitations thence are brought,

Whate’er their office, whether to beguile

Thoughts over busy in the course they took,

Or animate an hour of vacant ease. (Wordsworth, 14-book Prelude, XII: 327-335, p.

437)

The addendum, present only in the 13-book version, seems to mark a re-assessment, relative

to the 5-book version, that Wordsworth’s mind-state during this moment of his life actually

represents something of the divine, ‘highest truth’. The 14-book version allows another re-

assessment, making the sentiment during this moment far more ambiguous, perhaps even painful.

The noun ‘agitations’ implies that what Wordsworth experiences is unpleasant, while in the 13-book

version, though he declares that the experience is divine, he does not comment on how it feels. In

the years that passed between the drafting of these versions, Wordsworth re-assessed this memory,

changing it from a simple interaction with nature in the 5-book version to a spiritual quest in the 13-

book version, to an ambiguous and painful experience in the 14-book version. As such, what could

be called the active-Prelude can change its mind.

One could ask, therefore: ‘Why did Wordsworth manoeuvre himself into a position wherein

he would be faced with such a painful choice: to publish a dead text, or to keep it living and have no

one but his close friends read it?’ However, the active-Prelude more closely represents the mind of

Wordsworth inasmuch as, when Wordsworth lived, it lived by his amendments and, when he died,

so did it. Wordsworth embodies death as the cessation of poetic animation: the poem actually dies.

By engineering his work in this fashion, Wordsworth is able to create one of the gravest and most

important facts of life: death. In doing so, Wordsworth jumped out of the system of conventional

literature.

The case is similar for Bach: Indra Hughes’ analysis suggests that the proper completion of

the Fugue 14 would include a further 47 bars.25 Bach had to choose between leaving the Fugue

unfinished and maintaining the numbers, or finishing the Fugue and losing the numbers. This choice

necessitates a similar question: ‘Bach could have engineered The Art of Fugue such that it its total

number of bars, when finished, not unfinished, would fit Fibonacci numbers, why did he go down a

path that would end in such a quandary?’

25 Indra Hughes, ‘Accident or Design? New Theories on the Unfinished Contrapunctus 14 in J. S. Bach’s The Art of Fugue BWV 1080’ (thesis, University of Auckland, 2006), p. 88.

29

This quandary, as for Wordsworth, seems to be meaningful in itself: it might represent the

development of contradictory outcomes in certain systems. In economics, savers benefit when

interest rates are high, and are harmed when they are low; the opposite is the case for borrowers: the

system throws up contradictions. Similarly, in quantum mechanics, some fundamental ‘particles’

can be described as both waves and particles: each alternative is described by very different

30

mathematics. It might be said that Bach’s quandary depicts these contradictions as they exist in

reality and society.

Moreover, by leaving it incomplete, Bach left the unfinished Fugue as an invitation for

others to finish it, as many have. Bach’s model composer is more complete, therefore, in that he

constructed a piece with the power to co-opt the compositional skills of others – in some sense, the

Art of Fugue can finish itself. The question, where can he go, now that he has united maths, himself

and the themes from the whole work can be answered: by jumping out of the system. The beauty

and interrelatedness of The Art of Fugue increased as it progressed, the best option for improving on

what preceded it being to leave the system of conventional musical composition. In doing so, Bach

created a piece of music which exists partly outside music, which incorporates the Fibonacci

sequence, while leaving the completed fugue in potential for others to achieve. This allowed Bach

to solve a quandary in a way that accesses both outcomes, as a photon can be both a particle and a

wave. Not every incomplete composition has these qualities, however: Bach’s composition was so

ordered that within it were the schematics for its completion, such as the presence of the original

theme in the quoted passage.

The same seems to be the case for the discrepancy between Bach’s numbers and Fibonacci

numbers. It might imply that beauty cannot be reduced to a number; by analogy, producing a work

that matched exactly the Fibonacci sequence would resemble a portrait without blemish or

asymmetry. One might claim that Bach’s music hints, therefore, at something more numinous, even,

than the golden mean. Bach’s genius was expressed in choosing numbers that were sufficiently

close to Fibonacci’s such that they are recognisable, while engineering a divergence that makes his

point. Bach can have both outcomes: he can utilise mathematical aesthetics to refer to self-

reference, while questioning the extent to which the golden mean is it. If the golden mean and

Fibonacci can be said to represent ‘order’, Bach may intimate the limits of order in creating great

art, just as he did in leaving his music incomplete and hence (partly) disordered.

Wordsworth and Bach, via their third push, move beyond the traditional confines of the

genres that they used. Wordsworth was able to create something analogous to artificial intelligence,

while Bach, in piece that is about composition, was able to create a composer. All of this, moreover,

relies on the previous layers of meaning and operation that the two had constructed, without which

their pieces would be simply unfinished.

31

Form and craft are fundamental to any sort of creativity, the emphasis being (usually) on

working within the rules and being as creative as is possible within them, even bending them,

without breaking them. For many composers and authors, the process of establishing order is

sufficiently satisfying; fugues are hard enough to write. Wordsworth and Bach both seem to reach a

32

point wherein their mastery of the rules is so great that to move forward they must not ‘break’ the

rules so much as transcend them.

To re-introduce a term with which I began; ‘chaos’ appears to be the ingredient that is most

fundamental to Bach and Wordsworth’s breakthrough: the gaping void of manuscript that followed

Bach’s last notes longs to be finished, while Wordsworth shunned the order of formalised, printed

work in favour of a work which, from the perspective of the general public, never existed. Both

works discard the order of finalised publication and completeness, utilising chaos to move forward

when order can do nothing more for them.

‘Jumping out of the system’ is one of the ways, as Hofstadter observes, in which humans are

often superior to machines (p. 37-38). Hofstadter names a machine with promise, however: a chess

program that lost every game, but, ascending a meta-level and asking whether the match was

winnable, would judge that it wasn’t and yield, rather than pointlessly fighting on to the inevitable

mate. It appears that The Prelude and The Art of Fugue are analogous to this program, within art:

they lose, in the sense that they are not complete and published works, but their nature means that

they fail in a way that is actually more significant than if they had succeeded. In a very broad sense,

therefore, these two works are a step towards humanity, out of the inanimate traditions from which

they stem.

33

Coda

These works seem to be the purest expression that I have found of the concepts that I have

explored thus far. When I first read Richard II, my initial reaction was to compare it with

Beethoven’s Sonata 30 – the composer presents the listener with near desolation, out of which

emerges a wonderful fugue. However, the representation of order out of disorder, as in Richard II, is

far more complete in Hammerklavier, while I could not discern elements such as the fractal

symmetry in Sonata 30. Of course, The Prelude and The Art of Fugue have a more innate

connection through their shunning of the traditional process of publication. Mozart’s Requiem might

have been an appropriate partner, though not with the same autobiographical or summary nature as

The Art of Fugue. The broad genres of each pairing (literature and music) seem fundamental in that

they allow the presentation of time: a piece of music or verse forces the audience to follow a

sequence of events in time in a way that a portrait, for example, cannot. As such, it is fitting that the

concepts that I have identified should be present in these genres, in that they are so heavily reliant

on a sequence of events, such as growth and development.

It seems fair to say that Isomorphism I demonstrates the creation of a system, while

Isomorphism II moves beyond the system: Beethoven demonstrates composition, Shakespeare

demonstrates consciousness; Bach creates a composer, Wordsworth creates a consciousness.

34

The graph above plots the level of order in each isomorphism sequentially; note that Isomorphism II begins

beneath level-I but quickly reaches it; it registers, at its close, as very ordered and disordered.

Isomorphism I suggests that the route from chaos to order necessitates, in the final stages,

the understanding that order is the natural child of chaos and self-reference, as Beethoven and

Richard’s creative wisdom testify. Whence comes this knowledge? In this respect, the moment of

chaos in Hammerklavier and Richard II might resemble the search for meaning in a disorientated

individual: nothing is certain and, to begin, everyone needs a basal axiom on which to build their

identity. It might be said that Shakespeare (via Richard) and Beethoven settle upon an axiom: I am

myself/the germ is Hammerklavier; then demonstrate the fruits of this thinking.

Isomorphism II suggests that multi-layered self-reference in combination with

incompleteness allows escape from a system in a way that is artistically productive. The works’

interconnected and self-referential nature mean that, once they opened the door to chaos, they were

not rendered meaningless, but were augmented by it. As such, chaos is a potent tool, one which can

generate truly new forms, while facilitating development when a system is, apparently, as ordered

as it can get. The conclusion of Isomorphism II is, thus, I am not myself.

The works seem to demonstrate that chaos and order are closely related; order is a self-

similar sample of chaos, while The Prelude and The Art of Fugue can be simultaneously ordered

and chaotic: chaos is a necessary ingredient for augmenting a system of self-similarity. This

35

suggests a cyclic relationship: order can emerge from chaos and, when order is mature, chaos can

emerge within it and the whole process continues.

This is to say that once a system emerges that can express A is A (I am myself), it can, hence,

develop to such an extent that it can express A is not-A26 (I am not myself). To borrow Meyer’s

terminology once more, given that order lies between uniformity and segregation, the uniform

obedience to a system is a disordered state, given that there is no form. This is to say that the curve

above demonstrates the apprehension of a new dimension: Isomorphism II jumps out of its own

system, creating form and hence order with respect to systems. It is, therefore, chaotic on the basic

level while ordered on level-I (with respect to systems). Self-reference is both the catalyst and the

stabilizer: it is the element which helps to draw order from chaos, which allows the re-introduction

of chaos, and which maintains order thereafter.

On a grand scale, the whole scheme resembles the emergence of living organisms, as the first

self-replicating molecules assembled in the primordial soup and generated the tree of life, which,

thereafter, evolved consciousness, thus forming a new chaos as the consciousness was new and

formless. This process may continue if artificial intelligence is ever achieved, given that it was

conceived by the human mind. This is to say that once life evolved it was then possible for self-

knowledge to come about; once there is self-knowledge it is possible for self-transcendence to come

about. Thus, the structure to which I referred in the introduction might be called a schematic for

how matter can transcend itself without the need for the supernatural.

Of course, these writers and composers weren’t aware of this relationship; rather, I think of

their work as analogous to theoretical physics in art. A theoretical physicist like Roger Penrose will

spend weeks formulating a mathematical model that might be useful in describing the Universe:

most of them do not describe anything real, some do. Comparatively, most artistic works do not

have much significance beyond their subject: Shakespeare, Beethoven, Wordsworth and Bach,

because of the breadth and depth of their experimentation, assembled pieces of a model which

depicts how matter becomes conscious, and how consciousness transcends itself; how matter

becomes human and how humans may, one day, transcend themselves.

Da Capo

26 This can be true in quantum mechanics.

36

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