Relating the Formal Characteristics of the Sonnet: A ... · Relating the Formal Characteristics of the ... Sonnet Writing 137 3.1.6 The Problem of Aggregation ... following formal

Relating the Formal Characteristics of the Sonnet:

A Theory of Centred Form

by

Kevin J. M. Keane

October, 2015

© 2011, 2015 (revised) Kevin J M Keane All rights reserved.

Kevin J M Keane Abstract

Relating the Formal Characteristics of the Sonnet: A Theory of Centred Form

The sonnet is characterized formally by the separated categories of number of lines,

stanzaic form, volta, isometry and rhyme scheme. This inquiry sets out instead to

uncover and weigh evidence for the claim that a sonnet unfolds from its centre to

form a pattern in which its formal characteristics inhere. This idea is referred to as

the theory of centred form. Theoretical evidence is provided by the construction of a

working model from first principles and the subsequent modelling of the formal

characteristics of five classic sonnet traditions. From simplified rhyme schemes,

centre arrays and two-array centre matrices of four and five elements are deduced

and tested by developing them into array models. In each of the models presented,

equivalents of the sonnet's formal characteristics unfold from the model's centre: the

equivalent of isometry results from the development of a fixed array of elements; the

equivalent of the volta is deemed to occur at the point of greatest contrast between

directionality flows in the models; the equivalent of stanzaic form results from

changes in directionality; rhyme scheme equivalents result from cyclicity in array

development; and the equivalent of fourteen line sonnet length in the models is

effected by the limit between array innovation and redundancy. To mitigate the risk

of error and bias in the array models, a second type of model is developed

independently of them to act as a cross-check on their results. Finally, practical

evidence for the claim is furnished in the centred form sonnet cycle, Memorial Day:

the Unmaking of a Sonnet. The balance of evidence strongly supports the claim: A

simple binary pattern unfolding from the equivalent of the sonnet's centre relates

equivalents of the sonnet’s so-called formal characteristics and, in so doing,

suspends the boundary between reflective thought and creative writing.

iii

Contents Page

Page Part 1 The Sonnet as Centred Form

1.0 Introduction 1

1.1 Rationale for the Inquiry 4

1.2 Working Model 9

1.2.1 Concepts, Definitions and Rules 9 1.2.2 Step-by-step Description of Array Model Development 18 1.2.3 Assessment 26 1.2.4 Conclusion 32

Part 2 Modelling Sonnet Traditions

2.0 Introduction 33

2.1 Early Italian Tradition 33

2.1.1 Simplified Rhyme Schemes 33 2.1.2 Centre Array Derivation and Array Model Development 38 2.1.3 Assessment 53 2.1.4 Conclusion and Outlook 53

2.2 Petrarchan Tradition 55

2.2.1 Simplified Rhyme Schemes 55 2.2.2 Centre Matrix 59 2.2.3 Step-by Step Description of Array Model Development 60 2.2.4 Assessment 74 2.2.5 Conclusion 75

2.3 Pleadean Tradition 77

2.3.1 Simplified Rhyme Schemes 77 2.3.2 Pleadean 1: Centre Arrays and Array Models 78 2.3.3 Assessment: Pleadean 1 Array Models 80 2.3.4 Pleadean 1: Centre Sequence and Triangle Models 83 2.3.5 Pleadean 2: Array Models 87 2.3.6 Assessment: Pleadean 2 Array Models 89 2.3.7 Pleadean 1 & 2: Sequence Models 89 2.3.8 Conclusion 101

(cntd.)

iv

Contents Page

Page Part 2 Modelling Sonnet Traditions (cntd.)

2.4 Shakespearean Tradition 102

2.4.0 Introduction 102 2.4.1 Simplified Rhyme Scheme 102 2.4.2 Centre Array 103 2.4.3 Simple Array and Triangle Models 103 2.4.4 Complex Array Model: Step-by-step Description 108 2.4.5 Assessment 114 2.4.6 Complex Triangle Model: Step-by-Step Description 115 2.4.7 Conclusion and Retrospective 121

Part 3 Centred Writing

3.0 Introduction 128

3.1 Memorial Day: The Unmaking of a Sonnet 128

3.1.1 Sonnet Cycle Centre Matrix: Sonnet 8 128 3.1.2 Rules for Array Development 130 3.1.3 Memorial Day Array Model: Step-by-step Description 131 3.1.4 Assessment 136 3.1.5 Link between Sonnet Pattern and Sonnet Writing 137 3.1.6 The Problem of Aggregation 139 3.1.7 Conclusion 140

Part 4

4.0 Summary and Outlook 141

References 142

Appendices

A Early Italian Model: Unsuitability of Two-Type, Three Element Arrays 146 B Early Italian Model: Centre Array Derivation 149 C Petrarchan Model: Centre Matrix Derivation 163 D Pleadean Models: Centre Array Derivation 172 E Shakespearean Model: Centre Array Derivation 175 F Memorial Day: The Unmaking of a Sonnet, Poems 186

v

List of Tables

Table Page

1 Cyclicity 9

2 Array Elements 10

3 Individuation of Array Elements 11

4 Directionality Change 12

5 Array Innovation and Redundancy 13

6 Summary of Array Development 17

7 Initial Leftwards versus Rightwards Development 25

8 Working Array Model 31

9 Working Model (WM) vs. ‘Simplified’ Early Italian (EI) Rhyme Scheme: Comparison of End-Array Element Changes 36

10 Unsuitability of Two-Element Centre Array 39

11 Unsuitability of Two-Element Centre Array: Change of Directionality at Array 6 40

12 Unsuitability of Three-Element Centre Array: Leftwards & Rightwards Development 41

13 Unsuitability of 4:1 Distribution of Centre Array Elements 42

14 Redundancy in a 4:1 Distribution of Centre Array Elements: Leftwards Development with Directionality Change in Array 6 43

15 Early Italian Model: Centre Array Candidates 1 44

16 Early Italian Model: Centre Array Candidates 2 45

17 Development of Centre Array through Arrays 5 & 11 47

18 Development of Arrays 4–1 and 12–15 48

19 Continuous Redundancy in Array Development 49

20 Early Italian 15-Array Models ‘b b a b a’ and ‘b a b b a’ 50

(cntd.)

vi

List of Tables

Table Page

21 Identical 14-Array Sub-Models 51

22 Early Italian Array Models 52

23 Petrarchan Model: Overlapping Paired and Alternating Rhyme Equivalents 1 55

24 Petrarchan Model: Overlapping Embracing and Alternating Rhyme Equivalents 2 56

25 Petrarchan Flow Pattern 1 57

26 Petrarchan Flow Pattern 2: Sub-Patterns (a), (b), (c) & (d) 58

27 Petrarchan Centre Matrix 59

28 Petrarchan Array Models 72

29 Identical Petrarchan Array and Triangle Models 73

30 Pleadean 1 & 2: Comparison of Simplified Rhyme Schemes 77

31 Pleadean 1: Simplified Rhyme Scheme with Symmetry 77

32 Pleadean 2: Simplified Rhyme Scheme without Symmetry 78

33 Pleadean 1: Array Models 79

34 Pleadean 1: Array Series Redundancy 81

35 Distribution of Four Stresses, (x), in a Four-Element Array 82

36 Pleadean 1: Accommodation of Conventional Rhyme Scheme 82

37 Pleadean 1: Triangle Models’ Centre Sequence 83

38 Pleadean 2: Array Models 88

39 Pleadean 1: Sequence Model: Leftwards Directionality 91

40 Pleadean 2: Sequence Model: Leftwards Directionality 92

41 Pleadean 2: Sequence Model: Chirality 95

42 Pleadean 1: Sequence Model: Chirality between Model Halves 97

(cntd.)

vii

List of Tables

Table Page

43 Pleadean 1: Sequence Model: Symmetry within Model Halves 97

44 Pleadean 1: Sequence Model: Leftwards and Rightwards Directionality 99

45 Pleadean 2: Sequence Model: Leftwards and Rightwards Directionality 100

46 Simple Shakespearean Array Model 104

47 Simple Shakespearean Triangle Model: Binary Expansion 105–107

4 8 Identical Simple Shakespearean Array and Triangle Models 108

49 Identical Shakespearean Sub-Models 112

50 Complex Shakespearean Array Models: Leftwards and Rightwards Developments 113

51 Shakespearean Complex Array and Triangle Models 120

52 Related Early Italian and Shakespearean Models 1 122

53 EIM (RHS) and Shakespearean Models: Shared Centre Sequence 123

54 EIM (LHS) and Shakespearean Models: Shared Centre Sequence 124

55 Related Early Italian and Shakespearean Models 2 125

56 Memorial Day: Centre Matrix: Internal Elements 129

57 Memorial Day: Centre Matrix Buildup 129

58 Memorial Day: Array Model 135

59 Memorial Day: Array 8 as Volta Equivalent 136

60 Centre Matrix: Distribution of Key Vowels 1 138

61 Centre Matrix: Distribution of Key Vowels 2 138

viii

Acknowledgements

I should like to thank the librarians and staff of Gray Herbarium, Harvard College Library, Schlesinger Library, Radcliffe Institute for Advanced Study, Harvard University and the Bayerische Staatsbibliothek for their professional and sympathetic support during my research on this project.

My heartfelt thanks go also to the late Henry Grunbaum and his son, Mark Grunbaum, of Cambridge, MA, for opening their home to me over the past four summers and offering me their friendship.

I should also like to express my gratitude to Uta Knolle-Tiesler for her interest in my research, the many lively discussions whether over the magic of literature, the process of discovery, or the imponderables of translation, and for her persistent socratic questioning, and friendship.

Finally, I should like to thank my wife, Hui Hsing, for her encouragement during difficult times, her support during long periods apart, and for keeping her sense of humour and maintaining her confidence throughout.

ix

for my parents

x

Part 1: The Sonnet as Centred Form

1.0 Introduction

This paper introduces the concept of centred form and develops methods

for the analysis of different sonnet traditions. The main poetic result is a sonnet

symmetry pattern that unfolds from about the sonnet’s centre and in which

equivalents of the sonnet’s formal characteristics evolve and inhere. This result 1

flows from the construction and testing of a centred form working hypothesis

model that is used to model five classic European sonnet traditions and inform the

writing of a centred form sonnet cycle. By model, I mean a complete and

consistent description of the assumptions, rules and methods by which a sonnet

pattern develops so that equivalents of the formal characteristics of a particular

sonnet tradition are seen to originate and evolve within it; and by centred form, I

mean an enabling pattern for reflective thought and creative writing, a means by

which to develop a small number of parts, two or three words, say, into a

harmonious whole and a whole, here, a short poem, into a harmony of parts.

Hobsbaum (1996) lists the following formal characteristics of the sonnet:

The form as practised in English has five main characteristics: (1) It has fourteen lines; (2) these fourteen lines are divided into a group of eight (octave) and a group of six (sestet); (3) the sonnet has a volta, or turning-point in thought, usually situated at the end of the octave or the beginning of the sestet; (4) it is written in five-stress lines (though very occasionally six-stress lines have been used); (5) it has a pre-set rhyme scheme, involving an extent of alternation of rhyme. All this is description, based on the practice of poets; not a prescription of what future poets might do (pp. 154–55).

The problem, and the challenge, posed by these characteristics is the many open

questions they raise: Why does a sonnet characteristically have fourteen lines and

The first two sentences of this paper are an homage to the mathematician John Nash.1

!1

not, say, thirteen or fifteen? Why is it split into an octave and a sestet and not, for

instance, into a sestet and an octave, or nonet and quintet? Why is the volta

usually to be found at the end of the octave or at the beginning of the sestet and

not at the end of the first quatrain or at the beginning of the second tercet? Why

does it have a volta in the first place? Why should a sonnet be written in five- or

very occasionally six-stress lines and not, for example, in three- or four-stress

lines? Why does it have a pre-set rhyme scheme involving an extent of alternation

of rhyme, rather than no rhyme scheme at all?

Do, moreover, the formal characteristics relate to each other? And, if so,

how do they relate? How might fourteen lines entail division into an octave and a

sestet? Or the octave division into quatrains and the sestet into tercets? How might

the octave and sestet be connected to the volta? Or the volta linked to five- or six-

stress lines? How might the number of stresses in a line be tied to a pre-set rhyme

scheme?

Whilst much scholarly effort has been invested in analysing and

interpret-ing sonnets by considering the overall contribution made by the sonnet’s

formal characteristics taken separately, much less attention has been paid to

considering the separate contribution made by its formal characteristics taken as a

whole. This is perhaps not surprising as such an approach presupposes that the

sonnet’s formal characteristics are but aspects of an ordering principle that is hard

to discern.

From the positing of such a principle, however, follows the inference of

a cohesive, underlying sonnet pattern in which the sonnet’s formal characteristics

inhere. If such a sonnet pattern could be found, it might reveal not only how the

!2

formal characteristics relate to each other, but also how they arise and thus,

perhaps, contribute to a better understanding of why the sonnet has proved so

popular for so long across so many cultures. Yet, does such a sonnet pattern exist

and, if so, what sort of pattern might it be? How, moreover, is it to be found? The

cost of not finding answers to these questions is considerable as it means

admitting that the form that has produced no small amount of Western

civilization’s best writing and thought over the past eight centuries remains not

only inexplicable, but inexplicably inexplicable.

I shall claim a possible answer to these questions based on the idea that

the formal characteristics of the sonnet are byproducts of a pattern of elements

originating in and unfolding from the sonnet’s centre. In other words, I shall seek

to support the claim that the sonnet’s formal characteristics are better understood

by considering that they all begin at the centre of the sonnet and are developed

from it in a pattern that manifests all the characteristics noted by Hobsbaum.

In sympathy with the view that considers poetic form not as object to be

exploited, but as possibility to be discovered (Lennard, 1996, p. 25), I shall refer

to this idea as the theory of centred form, and sonnet writing based on it as centred

writing. To show that this claim is reasonable by providing good evidence for it,

that is, evidence both sound and sufficient (Turabian, 2007, p. 60), I shall offer:

1. a rule-based development of a sonnet working model from a centre

array of three elements;

!3

2. five models of classic sonnet forms from the scuola siciliana, Petrar-

chan, Pleadean, from which two, and Shakespearean traditions based on

the principle of centred form to show how the claim works in theory;

3. my own centred form model and sonnet cycle to show how centred

sonnet writing works in practice.

For clarity’s sake, I should like to emphasize that my desired aim in

undertaking this inquiry is to change the way in which the sonnet at the formal

level is understood and represented today. My main aim is not to provide a means

to analyse or interpret the work of other poets. If some of the ideas presented here

do help in the better appreciation of their work, that would, naturally, be very

gratifying. However, that is not my chief concern. This inquiry has been

undertaken because I want to know whether and, if so, how the formal elements of

the sonnet relate to each other in order that readers might have a deeper

appreciation of the sonnet’s beauty and so find more pleasure in their reading.

1.1 Rationale for the Inquiry

The idea of centred form and centred writing occurred to me when,

happening to glance at a copy of Shakespeare’s Sonnets lying open on a table one 2

day, a few words from the middle of Sonnet 12 caught my eye:

I see

all girded up

with white

the wastes

The Arden Shakespeare. Ed. Duncan-Jones, K. 2005. 2

!4

It struck me that these words succinctly captured what I considered to be

the sonnet’s generally sombre tone. There was also the development of an imaged

representation of death in the two phrases “all girded up with white” to its more

abstract representation in “I see the wastes”, a binary construct typical of sonnet

writing. As a consequence, I wondered, speculatively enough, to be sure, whether

the writing of the sonnet might have started at this point. The words made sense

when read linearly, of course, that is, conventionally, from top left to bottom right.

However, they could also be made sense of, with some grammatical tolerance,

when read from bottom to top. These observations encouraged me in my

speculation since they suggested that writing from the centre could create

meaning or, at least, start to create meaning. Expecting, though, that I was

probably making too much of a glance, I decided to check further. Words from the

middle of equivalent lines, that is, lines 5, 7, 8, and 10, in Sonnet 13 seemed to

reveal the same pattern:

beauty

after

issue

honour

The diction here appears to develop the ‘persuasion to marry’ theme

regarded as persistent in this group of sonnets (Ellrodt, 1986, p. 38) and to do so

with the same movement from imaged to abstract representation seen in sonnet

12. The words from the middle of equivalent lines in sonnet 14, however,

disabused me:

!5

fortune

princes

I

in them

Even broadly construed, “princes I fortune in them” would hardly bear

interepretation as a continuation of the ‘persuasion to marry’ motif. The step from

concrete to abstract representation was also missing. When sonnet 15 offered:

men

youthful sap

brave state

rich in youth

I started to think that my speculation was probably just that. After a quick glance

through the remaining sonnets, I became quite convinced of it. Here, for example,

is the diction from the centre of sonnet 154:

votary

general

virgin

took heat

There appeared then to be little evidence from the sonnets as a whole to

support the notion that the writing of a sonnet might start with verses 7 and 8.

However, the idea that it might be possible to write a sonnet from its centre

nevertheless took hold for it seemed plausible in at least two instances that some

kind of pattern was being established that might then be developed into a sonnet.

!6

Yet how was I to try out this idea? Where was the evidence for it to be collected?

There appeared to be little in the scholarly literature on the sonnet to suggest that

it was designed to be written from its centre. On the contrary, Ernest Hatch

Wilkins (1915, 1959), cited by Borgstedt (2009, pp. 120-121), at first supported

the hypothesis that the sonnet’s octave was inspired by one of the earliest Italian

verse forms, the eight verse, single stanza Sicilian strambotto, with the sestet pos-

sibly coming from a Sicilian variety of the arabic zajal (1915, p. 494). Wilkins

then later viewed a Sicilian form of the strambotto, the canzuna, as the source of

the octave and the sestet as “a wonderfully appropriate conclusion” (1959, p. 39)

devised by the probable inventor of the form, Giacomo da Lentino. Jost (1989)

favours instead the romano-provençal and sicilian-arabic spheres as the primary

influence on the sonnet's genesis (p. 39). Oppenheimer (1989) radically extends

these opinions by suggesting that “the sonnet’s peculiar fourteen-line structure...is

traceable to Plato’s Timaeus, with its mathematical description of the architecture

of the human soul and of heaven” (p. 3). Kemp (2002) prefers to divide theories

of the sonnet’s origins between the Provençal canzone and the strambotto (p. 46).,

whilst Stephen Burt and David Mickics (2010) emphasize the contribution of the

“scientific advances of Islamic North Africa along with the chivalric habits and

troubadour poetry of southern France” as influencing the sonnet’s origins ( p. 6).

I do not propose to weigh here the relative merits of these opinions as

there is evidence to be found within the sonnet tradition itself for the relevance of

patterning to sonnet writing. Francesco Petrarca, perhaps the most celebrated

sonnet writer of the Italian Renaissance, was undoubtedly familiar with the poetic

potential of patterning as the sestinas in his Canzoniere reveal a close familiarity

!7

with the interlacing retrogradatio cruciata pattern of Arnault Danièl’s emblematic

sestina ‘Lo ferm voler’ (Shapiro, 1980; Spanos, 1978). This might only be

circumstancial evidence for the use of patterning in his sonnet writing, a sestina is

not after all a sonnet, yet the presence along with sonnets in Petrarch’s collection

made the likelihood of an equivalent underlying pattern for the sonnet more

plausible. 3

Furthermore, the presuppositions behind the creation of the sonnet being

unknown, there was no compelling reason to assume that the sonnet was

conceived with any or all of the formal characteristics noted by Hobsbaum in

mind. It was still an open question therefore whether the formal characteristics

themselves were no more than a scholarly fata morgana: various useful,

stimulating, even necessary, categories for sonnet analysis and interpretation, yet

also, possibly, an anachronistic distraction and hindrance to a more cohesive

appreciation of the genre.

A methodological advantage of assuming a single ordering principle to

account for the sonnet’s formal characteristics was its inference that the formal

characteristics were inextricably linked. This meant that starting at any one of

them would lead to all of the others, greatly simplifying the work involved: If

such a connection could be found, it would be a strong indication of a common

sonnet ordering principle; if not, it would make it more likely that a sonnet pattern

I should like to thank the American author, poet, drama and literary critic, Richard Lord,3 for raising the question of a possible connection between sestina and sonnet patterns at the book launch of Memorial Day: the unmaking of a sonnet at BooksActually in Singapore in June, 2010.

!8

did not exist, or that the approach adopted was inadequate, in either case putting a

quick end to the inquiry.

1.2 Working Model

1.2.1 Concepts, Definitions and Rules

I should like to begin the discussion of the working model by showing

how ‘parts’ may together form a ‘whole’ that in turn becomes a part of a greater

whole. To illustrate this idea, I shall introduce the concept of cyclicity, by which 4

I mean the property of recurring at regular intervals. By way of illustrating this

concept consider, as shown in Table 1 below, the cyclic number 142,857 in its

decimal fraction form, 0.142857, along with two of its variations:

Table 1 Cyclicity

0. 142857 ≈ 1:7

0. 285714 ≈ 2:7

+ 0. 571428 ≈ 4:7

≈ 1. 000000 ≈ 7:7

The primary sonnet interest here lies neither in the numeric value of the

fractions, nor in their sum, but in the way the regular development of digit pairs

forms a spiral pattern as they cycle about each other from one fraction to the next.

It is just such a patterning mechanism that I shall apply in the models to develop

the equivalents of the sonnet’s formal characteristics from a simple array.

For background, see: Weisstein, Eric W. "Cyclic Number." From MathWorld–A Wolfram 4 Web Resource. (http://mathworld.wolfram.com/CyclicNumber.html)

!9

This idea of a patterning mechanism is illustrated in table 2 below using

the example from Table 1. I have dropped the zeroes and the decimal points, the

addition sign and the sum total and, in addition, have arranged the digit pairs of

each decimal fraction into three separate columns with line numbers and letters

added for ease of reference; lastly, arrows highlight the axial movement of the

digit pairs from one array to the next.

Table 2 Array Elements a b c

1. 14 28 57 ↙ ↙ 2. 28 57 14 ↙ ↙ 3. 57 14 28

To distinguish patterning as a concept from the process by which it is achieved, I

shall term the building-up of arrays and concomitant changes in the positions of

digit pairs from array to array development, the digit pairs elements and define

patterning as the rule-based development of arrays.

It is apparent from Table 2 that development is a consequence of three

conditions: first, the ordering of elements into an array in line 1; second, the

cyclicity of the elements, that is, the regular recurrence of the elements about each

other and, finally, the direction of cyclicity from one array to the next. Now, the

relative positions of array elements being fixed in line 1 and cyclicity assumed as

a latent property of the array, development depends, therefore, on the direction of

cyclicity from the first, or ‘start’, array. This initial directionality, as I shall call it,

!10

is clearly determined by two simple choices, first, the choice between an

‘upwards’ or ‘downwards’ development from the start array and, second, the

choice between a ‘right’ or ‘left’ shifting of the digit pairs from one array to the

next. Development in the example above may thus be seen upon inspection to be

‘down’ and ‘left’, or ‘downwards left’. To avoid confusion as to whether

directionality is to the left or right, it is helpful to take a cue from the development

of the middle element in each array: If the middle element in column ‘b’ develops

to the left, the other elements in the array do so as well; if to the right, the other

elements follow suit.

One consequence of directionality is the individuation of array elements.

Consider array 4, in Table 3, the result of a continuation of ‘downwards left’

development:

Table 3 Individuation of Array Elements

a b c

1. 14 28 57 ↙ ↙ 2. 28 57 14 ↙ ↙ 3. 57 14 28 ↙ ↙ 4. 14 28 57

Array 4 has the same order of elements as array 1, yet they are not identical as

their respective cyclical properties differ. To see this, assume that each element

has both a symbolic property, here represented by two digits, and cyclical

properties of one or both of the following two types, a flow towards another

element and, additionally, or, alternatively, a flow away from itself, hereinafter

!11

termed towards flow and away flow, respectively. These two cyclical properties

are represented in Table 3 by the same arrows for, evidently, the away flow from

the perspective of one element is the towards flow of the corresponding element

in the subsequent array developed. For example, the element at ‘b1’ has a

downwards left away flow, as does every element in the first array, and each of

these away flows, from the perspective of the elements in the second array, is a

towards flow. Thus, array 4 and array 1, despite having the same elements in the

same order, are not identical as their respective cyclical properties differ, the

former having towards flows, the latter none. Thus, it results that an array element

is only defined when both its symbolic and cyclical properties have been

determined.

If development is continued the away flows from array 4 to array 5 repeat

array 2. To avoid this repetition there is a change in directionality of development

from downwards left to downwards right follows, as shown in Table 4:

Table 4 Directionality Change

4. 14 28 57 ↘ ↘ 5. 57 14 28

Yet, is this not arbitrary? Why should arrays not repeat themselves direc-

tionally as well as symbolically? To see why this is not the case here requires

considering the role that innovation and redundancy are deemed to play in the

model. Let the model aim to develop a maximum of innovation and a minimum of

redundancy in array development. Furthermore, let innovation be understood as

the continuous creation of unique arrays and redundancy as the loss of uniqueness

!12

through array repetition. Array development therefore consists of two phases, an

initial innovative phase developed from a start array, and a second redundancy

phase, which, as redundancy is to be minimized, when it occurs, marks the end of

array development and the completion of the model.

Now, given these assumptions and definitions, if a maximum of

innovation is to be achieved and redundancy minimized, then a repetition in the

symbolic prop-erties of an array, that is, the recurrence of two arrays having the

same digit pairs in the same order must, wherever possible, be counterbalanced by

innovation in its cyclical properties. Thus, in the development of array 5,

innovation takes the form of directional change from left to right, as shown in

Table 5:

Table 5 Array Innovation and Redundancy

1. 14 28 57 ↙ ↙ 2. 28 57 14 ↙ ↙ 3. 57 14 28 ↙ ↙ 4. 14 28 57 ↘ ↘ 5. 57 14 28

Before turning these ideas into operational rules for array development,

in order to establish a link between the model’s elements and the sonnet’s formal

characteristics, it is necessary to draw one final distinction between the symbolic

and cyclical properties of array elements on the one hand and their placeholder

!13

function on the other. Every element in the working model is deemed to have a

dual function, first, a combined symbolic and cyclical function to create difference

vis-à-vis other array elements and thus provide for the generation of innovative

arrays and, second, a placeholder function that creates a distinct pattern of flows

between symbolically similar array elements throughout the model. These

functions are not, incidentally, affected by the type of element used in the model.

As long as difference between elements is established, other element types are

permissible. Thus, just as the elements ‘14’, ‘28’ and ‘57’ occupy the first array, so

might in principle any other group of three elements, such as musical notes,

colours, letters, or indeed any combination of elements from these or any other

symbolic category, for it is the possession of a differentiable property that is the

condition for its inclusion as an element in the model, and decidedly not the type

of element per se.

This placeholder function of array elements is central to the main claim

of the inquiry as it is the final pattern of placeholders and the flows between them

that create several of the equivalents of the formal characteristics of the sonnet. In

other words, these equivalents shall be seen to be different aspects of a particular

distribu-tion of placeholders throughout the model and, crucially, of the

directionality and changes in directionality of the flows between them that result.

With distinctions between innovation and redundancy, and elements and

placeholders, drawn, three rules for array development may now be defined.

These rules, the result of numerous tests of possible array developments, are based

on the principle of centred form, in which the maximizing of array innovation and

minimizing of array redundancy plays the key role in limiting the number of

!14

arrays in the model, just as the pattern of flows between placeholders determines

the emergence of the equivalents of isometry, stanzaic form, volta and rhyme

scheme.

However, before turning to the rules for array development, it seems

appropriate now to start to attempt to formalize terms in order to show how the

working model relates to the sonnet. The complete set of arrays to be developed

shall, therefore, be referred to as the model; the initial array from which the other

arrays are developed as the centre array; a line, in this case, of three elements as

an array, and any one digit pair, as noted earlier, as an array element, or simply

element. In a perforce adumbrated way, given the still early stage of the inquiry,

these terms are deemed to correspond to the following aspects of a written sonnet:

model = sonnet

array = sonnet line, or verse

centre array = eighth sonnet line, or verse

element = word, or word group

The working model has three rules for array development:

Rule 1. From the centre array, arrays develop alternately upwards

and downwards, in each direction either to the left or right,

but not to the left and right, nor to the right and left;

Rule 2. Symbolic repetition of the centre array causes a change in the

direction of development;

Rule 3. Symbolic repetition of any array and cyclical repetition of its

towards flows halts development and completes the model.

!15

Having defined operational rules for development of the working model,

it is also now not inappropriate to describe the ordering principle underlying it. It

was mentioned in the discussion of directionality that development may be

upwards as well as downwards from the start array. This is just another way of

saying that arrays develop in opposite directions from a ‘middle’, or ‘centre’, thus

connecting the sonnet’s formal characteristics, as presupposed. This principle of

centred form may be described as follows: A sonnet unfolds from its centre

towards its beginning and end to form a pattern in which its formal

characteristics inhere.

A corollary of the centred form principle is its implication of a

fundamental distinction between the manner of reading a sonnet and the writing of

it. Whereas the former is customarily linear, the principle of centred form suggests

that the writing of a sonnet may begin with the creation of a centre and then

continue outwards towards the sonnet’s start and finish while maintaining the

linear legibility of the lines so created. This is not to say, of course, that the

writing of any sonnet starts, or must start, from its centre, that would be absurd; it

is to claim, however, that such writing can elicit the sonnet’s formal characteristics

and, in so doing, offers a testable theory for their presence.

To make it easier to follow the detailed description of the working

model’s construction that follows, Table 6 presents a summary of array

development. Beginning with the centre array, array 8, on the left of the table,

arrays develop alternately towards the top and bottom of the model to end with

array 1:

!16

Table 6 Summary of Array Development

step 1 2 3 4 5 6 7 8

array 7 6 5 4 3 2 ↗

array 8 ↓ ↓ ↓ ↓ ↓ ↓ 1 ↗

array 9 ⤴ 10 ⤴ 11 ⤴ 12 ⤴ 13 ⤴ 14

Development consists of eight steps resulting in a model of fourteen

arrays. Step 1 represents the choice of centre array, the equivalent of line 8 in a

fourteen-line sonnet. Then follow six steps in which symbolically identical, but

cyclically dissimilar, pairs of arrays are developed alternately towards the top and

bottom of the model. The eighth and final step develops array 1, the equivalent of

line 1 in a sonnet. As array 1 is symbolically and cyclically a repeat of array 7,

redundancy is introduced into the model and, according to Rule 3, completes it.

To show how this summary description of the model works in detail, a step-by-

step description of the development of the arrays that create the working model

now follows.

!17

1.2.2 Step-by-step Description of Array Model Development

Step 1 centre array

8. 14 28 57

The centre array consists, as remarked above, of three mutually distinguishable

symbolic elements with cyclical properties. It is developed alternately both

upwards and downwards, in both cases initially either to the left or right, and has

only ‘away flows’, but no ‘towards flows’ as the centre array forms the starting

point from which all other arrays are developed and, as such, ‘towards flows’ for

it are undefined.

Step 2 arrays 7 & 9

7. 28 57 14 ↖ ↖ 8. 14 28 57 ↙ ↙ 9. 28 57 14

As Rule 1 permits a choice between leftwards and rightwards development from

the centre array, let development start leftwards to create arrays 7, then 9. These

two arrays are identical symbolically, but differ in their directionality, array 7

having upwards left and array 9 downwards left directionality.

!18

Step 3 arrays 6 & 10

6. 57 14 28 ↖ ↖ 7. 28 57 14 ↖ ↖ 8. 14 28 57 ↙ ↙ 9. 28 57 14 ↙ ↙ 10. 57 14 28

With of course no repetition of the centre array in Step 2, in Step 3 development

continues upwards then downwards to the left to form arrays 6 and 10.


5. 14 28 57 ↖ ↖ 6. 57 14 28 ↖ ↖ 7. 28 57 14 ↖ ↖ 8. 14 28 57 ↙ ↙ 9. 28 57 14 ↙ ↙ 10. 57 14 28 ↙ ↙ 11. 14 28 57

The development of arrays 5 and 11 in Step 4 results in symbolic, but not cyclical

identity with array 8 for, unlike arrays 5 and 11, the centre array elements have no

towards flows. Neither are arrays 5 and 11 identical for, although their towards

flows are both leftwards, the directionality of the former is upwards from the

centre array, that of the latter, downwards. Therefore, due to symbolic, but not

!19

cyclical identity with the centre array, Rule 2 is applied, and directionality in away

flows changes from left to right.


4. 57 14 28 ↗ ↗ 5. 14 28 57 ↖ ↖ 6. 57 14 28 ↖ ↖ 7. 28 57 14 ↖ ↖ 8. 14 28 57 ↙ ↙ 9. 28 57 14 ↙ ↙ 10. 57 14 28 ↙ ↙ 11. 14 28 57 ↘ ↘ 12. 57 14 28

In Step 5, arrays 4 and 12 are symbolically the same as arrays 6 and 10, but, as

their towards flows are cyclically different, development continues.

!20


3. 28 57 14 ↗ ↗ 4. 57 14 28 ↗ ↗ 5. 14 28 57 ↖ ↖ 6. 57 14 28 ↖ ↖ 7. 28 57 14 ↖ ↖ 8. 14 28 57 ↙ ↙ 9. 28 57 14 ↙ ↙ 10. 57 14 28 ↙ ↙ 11. 14 28 57 ↘ ↘ 12. 57 14 28 ↘ ↘ 13. 28 57 14

In Step 6, lack of either repetition of the centre array or both symbolic and

cyclical repetition of any previous arrays means development continues in the

same rightwards direction.

!21


2. 14 28 57 ↗ ↗ 3. 28 57 14 ↗ ↗ 4. 57 14 28 ↗ ↗ 5. 14 28 57 ↖ ↖ 6. 57 14 28 ↖ ↖ 7. 28 57 14 ↖ ↖ 8. 14 28 57 ↙ ↙ 9. 28 57 14 ↙ ↙ 10. 57 14 28 ↙ ↙ 11. 14 28 57 ↘ ↘ 12. 57 14 28 ↘ ↘ 13. 28 57 14 ↘ ↘ 14. 14 28 57

In Step 7, array 8, the centre array, re-emerges symbolically in arrays 2 and 14.

Now, although the centre array is developed here for the fourth and fifth times,

there is as yet no repetition of its cyclical properties: The centre array itself has no

towards flows and, whilst arrays 5 and 11 have leftward towards flows, arrays 2

and 14 have rightward towards flows. As there is thus no repetition of both the

symbolic and cyclical properties of the centre array, development continues with,

however, according to Rule 2, a change in directionality from right to left.

!22

Step 8 array 1

1. 28 57 14 ↖ ↖ 2. 14 28 57 ↗ ↗ 3. 28 57 14 ↗ ↗ 4. 57 14 28 ↗ ↗ 5. 14 28 57 ↖ ↖ 6. 57 14 28 ↖ ↖ 7. 28 57 14 ↖ ↖ 8. 14 28 57 ↙ ↙ 9. 28 57 14 ↙ ↙ 10. 57 14 28 ↙ ↙ 11. 14 28 57 ↘ ↘ 12. 57 14 28 ↘ ↘ 13. 28 57 14 ↘ ↘ 14. 14 28 57

The development of array 1 in Step 8 results in the symbolic and cyclical

repetition of the towards flows of array 7, thus introducing redundancy into the

model. Further development is halted, according to Rule 3, and the model

complete. Full development has thus taken eight steps, resulting in a model of

fourteen arrays.

!23

What model is created, however, if development from the centre array is

initially rightwards instead of leftwards? The alternative developments are juxta-

posed in Table 7 further below. Symbolically, the centre array ‘14 28 57’ recurs in

the same arrays in both models, namely, in arrays 5 and 11, and 2 and 14. The

other two arrays ‘28 57 14’ and ‘57 14 28’ substitute for each other, that is, where

‘28 57 14’ occurs in the leftwards model, it is replaced by ‘57 14 28’ in the

rightwards model, and vice versa. Cyclically, the models’ flows are mirrored.

Table 7 below underscores the symmetry within and between these two

models. The differences between them are not, however, inconsequential. When,

in the second part of the inquiry, it comes to the attempt to model a number of

traditional sonnet forms, it shall be seen that the initial choice between leftwards

and rightwards development has a bearing on the type of rhyme scheme that

results. For the moment, however, the choice between the two models is

indifferent and as its development has been described in detail the model with

leftwards development will be adopted for the purposes of the discussion to

follow.

!24

Table 7 Initial Leftwards versus Rightwards Development

leftwards rightwards

1. 28 57 14 57 14 28 ↖ ↖ ↗ ↗ 2. 14 28 57 14 28 57 ↗ ↗ ↖ ↖ 3. 28 57 14 57 14 28 ↗ ↗ ↖ ↖ 4. 57 14 28 28 57 14 ↗ ↗ ↖ ↖ 5. 14 28 57 14 28 57 ↖ ↖ ↗ ↗ 6. 57 14 28 28 57 14 ↖ ↖ ↗ ↗ 7. 28 57 14 57 14 28 ↖ ↖ ↗ ↗ 8. 14 28 57 14 28 57 ↙ ↙ ↘ ↘ 9. 28 57 14 57 14 28 ↙ ↙ ↘ ↘ 10. 57 14 28 28 57 14 ↙ ↙ ↘ ↘ 11. 14 28 57 14 28 57 ↘ ↘ ↙ ↙ 12. 57 14 28 28 57 14 ↘ ↘ ↙ ↙ 13. 28 57 14 57 14 28 ↘ ↘ ↙ ↙ 14. 14 28 57 14 28 57

With this comparison of the two alternative directions of development

from the centre array, the initial description of the working model ends, the

assumptions and principles, definitions and rules for development of a three-

element centre array up to and including its end-array having been defined and

stepwise illustrated.

!25

1.2.3 Assessment

How well, then, does the model describe the sonnet’s formal

characteristics? Might it not be objected, for example, that the equivalent of the

sonnet’s fourteen lines is accounted for inadequately by the theory of innovation

and redundancy? Surely a maximum of innovation and a minimum of redundancy

implies an absolute amount of the former and an absence of the latter, resulting in

a model of thirteen, and not fourteen, arrays? This objection is based on the

misapprehension that innovation and redun-dancy in the working model are

mutually exclusive, whereas they are, on the contrary, mutually dependent. This is

necessarily so, as the cyclical properties of the elements in the centre array mean

that there cannot be infinite innovation in array development due to the limited

number of possible combinations of the symbolic and cyclical properties of the

elements themselves. Innovation in the working model is, as may be seen upon

inspection of Table 7, limited to six arrays upwards and six downwards away from

the centre.

If innovation is, then, limited, and redundancy preprogrammed, is there

not all the more reason for the number of arrays in the model to be limited to the

thirteen innovative arrays with the redundant fourteenth array omitted? The

difficulty with halting development after the thirteenth array is that, from the

perspective of development within the model, it is not certain whether innovation

continues into the fourteenth array, or not. The only way to find out is to continue

development until symbolic and cyclical repetition occur, as defined by Rule 3.

The difficulty with repetition on the other hand is, of course, that it has to repeat

itself to be repetitive. The development of the fourteenth array, by causing Rule 3

!26

to be invoked and completing the model, addresses these two difficulties by

simultaneously ending both innovation and redundancy, thus maximizing the

former as it minimizes the latter.

Even allowing for this, however, does not array 2 and its away flows

represent symbolic and cyclical repetition of array 8 and, this being so, introduce

redundancy into the model thus restricting the total number of arrays to thirteen?

Rule 3 says that it is not repetition in the away flows, but in the towards flows of

an array, together with symbolic repetition, that creates redundancy and halts

development. As an array’s towards flows are the same as the away flows from

the preceding array, this argument might appear specious. However, an array’s

towards flows mark the incipient development of an array, whilst away flows

show that an array has already been developed or, as in the case of the centre

array, assumed. Hence, development continues from array 2 to develop the

towards flows of array 1 and fulfil the conditions necessary for redundancy.

Is it certain, though, that innovation, seen from the perspective of

development within the model, is at an end after thirteen arrays just because one

redundant array has been developed? Is it not possible that an innovative array

might be developed at the fifteenth array, or later still? Let it be assumed that such

innovation occurs. Now, for innovation to occur it must develop from an array that

is innovative. However, development has resulted in an array that is redundant.

Hence, any further development at the fifteenth array, or later, cannot be

innovative. Alternatively put, if there can be no innovation from redundancy, then

from redundancy, there can only be more redundancy. This implies that

redundancy is not simply repetition of the same arrays, but accretion of

!27

supplementary arrays that share the same symbolic and cyclical properties as

previously developed redundant arrays. In other words, any upward and

downward extension of the working models is continuous away from the centre.

From this it may be further deduced that there is no formal closing of the circle in

the model: The principle of centred form described here assumes that

development in the working model would continue indefinitely in opposite

directions away from its centre were it not restricted by the principle of innovation

and redundancy to fourteen arrays.

How, then, does the model account for the characteristic division of the

sonnet into octave and sestet, with their respective subsequent divisions into

quatrains and tercets? Furthermore, what of the volta and isometry? How are they

related by the principle of centred form? The traditional stanzaic structure of the

sonnet may be thought of hierarchically as a division of fourteen lines into two

major parts, the octave and sestet, followed by two further divisions separating the

octave and sestet into quatrains and tercets, respectively. These three divisions

then correspond to the points in the working model where a contrast in

directionality produces innovation. The directionality of the two away flows of

array 8 contrast with each other, whilst in the upwards and downwards

development of the model, the away flows of arrays 5 and 11 contrast in

directionality with respect to their towards flows. Array 8, having the relatively

starker contrast between its flows due to its lack of towards flows is accordingly

deemed to represent the volta and mark the major stanzaic division between

octave and sestet, leaving arrays 5 and 11 to mark the divisions between quatrains

and tercets. Isometry refers to the constant number of prosodic markers, or

!28

stresses, per line of verse. In the working model, each array has three elements,

each of which is deemed to have the same number of stresses as the other two.

Hence, every array throughout the model has the same number of stresses making

the model isometric.

What, then, of the change in directionality between arrays 2 and 1? To

which stanzaic division does it correspond? How, moreover, is the stanzaic form

of the Shakespearean sonnet with its three quatrains and final couplet to be

satisfied? As noted above, it is only the directionality change introduced by the

symbolic repetition of the centre array when it produces innovation that is

pertinent to the traditional stanzaic form of the sonnet. The change in

directionality between arrays 2 and 1 leads not to innovation, but to redundancy

and is, therefore, as irrelevant to stanzaic form, as it is pertinent to sonnet length.

The stanzaic form of the Shakespearean sonnet, as shall be seen below, results

from a different centre array. In this respect, the working model does not, indeed

cannot, account for it.

Whilst the model so far appears to represent adequately the sonnet equi-

valents of length, stanzaic form, volta and isometric verses, its limitations become

apparent when the question of accommodating the sonnet traditions’ rhyme

schemes arises. This difficulty appears daunting when considered broadly for, as

Lennard (1996, pp. 25–26) has calculated, and Queneau (1961) has demonstrated,

the number of potential rhyme schemes in a sonnet is very large. I shall deal with

this problem of the exhaustion of inexhaustibility by ignoring it for the moment

and limit the inquiry to the five rhyme schemes identified by Kircher (1979) as

representative of the Italian, French and English language sonnet traditions.

!29

According to Bermann (1988) in her study of the sonnets of Petrarch, Shakespeare

and Baudelaire, these poets’ sonnets reflect “the lyric’s enormous potential for

difference” (p. 2), a view that may serve to justify the number and linguistic

variety of sonnet traditions selected for this inquiry. I think any fewer would be

too weak a test for the claim, any more, I hope, superfluous.

Before turning to these traditions, the numerical elements used to

construct the working model with leftwards directionality are replaced in Table 8

below by variables qua lower case roman letters to represent any elements

fulfilling the symbolic and cyclical conditions given in the definition of array

elements above. The 14, 28 and 57 of the centre array, array 8, are thus

represented from now on by the variables a, b and c, respectively. To better

highlight how the contrasting flow patterns of the working model reflect the

equivalent of sonnet stanzaic form, in the presentation of the final working model

in Table 8 below, I have removed the flow lines between the equivalents of octave

and sestet and quatrains and tercets and widened the spaces between them slightly.

!30

Table 8 Working Array Model

Working Array Model

b c a ↖ ↖ a b c ↗ ↗ b c a ↗ ↗ c a b

a b c ↖ ↖ c a b ↖ ↖ b c a ↖ ↖ a b c

b c a ↙ ↙ c a b ↙ ↙ a b c

c a b ↘ ↘ b c a ↘ ↘ a b c

!31

1.2.4 Conclusion

How can a model consisting of numbers and variables have anything

trenchant to say about poetic form? Whilst a sonnet is always, arguably, made of

words, words are necessary in a model only insofar as they help elucidate the

subject matter of the inquiry, in this case, the idea that a sonnet unfolds from its

centre to form a pattern in which its formal characteristics inhere. For such an

undertaking, symbols as numbers to illustrate, and variables as letters to

generalize, suffice. This is inevitably so for the model presupposes that the

relationship between its elements and the formal characteristics of the sonnet is

constitutively the same, namely, patterned.

!32

Part 2: Modelling Sonnet Traditions

2.0 Introduction

In this second part of the inquiry, the principle of centred form as

developed in the working model in Part 1 is applied in the construction of array

models of the Early Italian, Petrarchan, Pleadean and Shakespearean sonnet

traditions. After the problem of the multiplicity of rhyme schemes is addressed in

the discussion of the Early Italian tradition, concern about the risk of error and

bias in model development leads to the search for an alternative, independently

constructed centred form model to cross-check array model results. The

theoretical basis for this second model, to be termed triangle model due to its

origins in, and geometric similarity with, a binary expansion, is consolidated at

the start of the Shakespearean section. The Shakespearean triangle model’s

subsequent construction, a comparison between its results and those of the

Shakespearean array model and an analysis of the relatedness of the

Shakespearean and Early Italian models close the second part of the inquiry.

2.1 Early Italian Tradition

2.1.1 Simplified Rhyme Schemes

Kircher (p. 414) notes two rhyme schemes as characteristic of the Italian

sonnet, namely, either alternating or embracing rhymes in the octave with two

variations in the sestet:

a b a b / a b a b or a b b a / a b b a , c d c / d c d or c d e / c d e.

As the embracing, or arching, rhyme in the octave is characteristic of the

Petrarchan and Pleadean models described below, discussion here is limited to

!33

what shall be termed the Early Italian sonnet, dating from the scuola siciliana of

the early 13th century, with its alternating rhyme in the octave and two variations

in the sestet: a b a b a b a b , c d c d c d or c d e c d e . 1

Several facts about the formal characteristics of the Early Italian sonnet

may of course be gleaned from these conventional rhyme schemes inter alia that

the sonnets are fourteen lines in length, that they have a two quatrain, two tercet

stanzaic form and that the rhyme schemes themselves have two pairs of rhymes in

the quatrains and either two ternary rhymes or three pairs of rhymes in the sestet.

Yet, the rhyme schemes also reveal something else that at first glance appears

quite mundane, but shall prove helpful to the inquiry, namely, that from one verse

to the next the end rhyme always changes. From ‘a’ in the first verse to ‘b’ in the

second, change; from ‘b’ in the second to ‘a’ in the third, change; from ‘a’ in the

third to ‘b’ in the fourth, change, and so on. Generalizing this observation, the two

variations in the sestet merge so that the rhyme scheme shows continuous change

throughout: a b a b a b a b a b a b a b

This array relates to the working model in the following manner: Each ‘a’

and ‘b’ in the array corresponds to its equivalent end-array element in each of the

working model’s arrays. The final, that is, rightmost, element of each array in the

working model has, therefore, one more function than the other elements: a

symbolic and cyclical function, a placeholder function and the function of

representing the rhyme scheme in terms of ‘change’ and ‘no change’. If the spaces

Thirty of the thirty-one sonnets Wilkins (1915, p. 83) recognises as belonging to the 1 group of earliest sonnets have these rhyme schemes.

!34

marking stanzaic division into quatrains and tercets in the array above are now

removed, the following simple alternating array results:

a b a b a b a b a b a b a b

This simplified rhyme scheme shows that it is possible for a simple array

of two alternating letters to hide complex information about the number of lines,

stanzaic form and rhyme scheme of a traditional sonnet form and that the rhyme

scheme of the Early Italian sonnet may be understood not only in terms of ternary

and paired rhymes, but also in terms of ‘change’ and ‘no change’. The question

that now naturally arises is how well the simplified Early Italian rhyme scheme is

described by the working model.

Consider the ‘change / no change’ column in Table 9 below situated

between the working model (WM) and the Early Italian model (EI). In this table,

change or lack of change from one end-array element to the next in the working

model is compared with change or lack of change within the simplified rhyme

scheme of the Early Italian sonnet developed above.

!35

Table 9 Working Model (WM) vs. ‘Simplified’ Early Italian (EI) Rhyme Scheme: Comparison of End-Array Element Changes

array/line WM change / no change EI 1. a change start a

2. c change change b

3. a change change a

4. b change change b

5. c change change a



8. c start change b



11. c change change a



14. c change change b

Although the starting points for the working model and the simplified

Early Italian rhyme scheme are different, array 8 and array 1, respectively, the

result is the same: change from one array to the next throughout. If the working

model were to express change from array 1 instead of from array 8, it would

entirely coincide with the simplified Early Italian rhyme scheme as would the

latter with the former were it developed in ‘centred form’ fashion from array 8.

!36

From these findings, it can be deduced, in terms of ‘change’ or ‘no change’, that

the working and Early Italian models’ simplified rhyme schemes are identical,

with both providing a basis for a traditional alternating rhyme scheme consisting

of paired rhymes in the octave and ternary or paired rhymes in the sestet.

It might be objected here that it is not a conventional rhyme scheme that

is being compared with the working model's end-array elements, but a simplified

Early Italian rhyme scheme, implying that end rhymes are not being compared at

all. This objection stems, though, from a conflation of the placeholder and

symbolic functions of the elements in the working model and Early Italian

simplified rhyme scheme. The Early Italian simplified rhyme scheme highlights

the placeholder function of the conventional Early Italian rhyme schemes, thus

making a comparison with the working model’s end-array placeholders possible

and appropriate. In other words, it is not end rhymes that are being compared in

Table 9, but their placeholders. The equivalents of the octave and sestet

placeholders may still of course be filled with paired rhymes and paired or ternary

rhymes, respectively: Change from one end-array to the next at the placeholder

level not only does not preclude, it corresponds with an alternating rhyme scheme

at the symbolic level. The end rhymes of the Early Italian sonnet are indeed,

therefore, being compared with the end-array elements of the working model, but

as placeholders, not as rhymes.

To summarize the discussion so far, the rhyme schemes noted by Kircher,

and here characterized as Early Italian, are:

a b a b a b a b c d c d c d

& a b a b a b a b c d e c d e

!37

From each of these, in simplified form, the following alternating rhyme scheme

was derived: a b a b a b a b a b a b a b

This same simplified, alternating rhyme scheme equivalent was also seen

to be derivable from the working model, from which it follows that the working

model satisfactorily describes the rhyme scheme of the Early Italian sonnet at the

placeholder level. From this same simplified rhyme scheme, it may also be

inferred that two types of element, ‘a’ and ‘b’, suffice to construct the Early Italian

model. The question now is how to decide on the correct number, mix and order

of these two types of element for the model’s centre array.

2.1.2 Centre Array Derivation and Array Model Development

In the following, the derivation of the Early Italian model’s centre array

is presented in detail, that is, exemplarily for the Petrarchan, Pleadean and Shake-

spearean models.

As an approach to answering questions concerning the makeup of the

Early Italian model’s centre array, let the five characteristics of the sonnet noted

by Hobsbaum henceforth be considered as five conditions needing to be satisfied

simultaneously by any model. Let this stringency furthermore be extended to the

types and number of elements in the centre array by applying the following rule:

as few types and number of elements as possible, as many of either as necessary

to satisfy all five sonnet conditions. Additionally, let types take precedence over

number: A solution with fewer types of element and a greater number of

individual elements is hence preferable to a solution with more types and fewer

elements. Let this principle be called the principle of economy.

!38

The methodological advantage of this approach is that it simplifies the

selection of centre arrays that potentially satisfy all five sonnet conditions. For

example, if two types of element, ‘a’ and ‘b’, are indeed sufficient to construct the

Early Italian Model, the development of a centre array consisting of only one of

each of these is clearly insufficient: Two-element arrays, whilst able to represent

an alternating rhyme scheme, produce, in working model terms, precipitate

redundant arrays and thus fail to satisfy the conditions of sonnet length and

stanzaic form. The unsuitability of such centre arrays may be seen in Table 10, in

which the symbolic and cyclical repetition of array 7 occurring in array 5 leads to

redundancy and halts development before the completion of fourteen arrays. For

the purposes of exposition, only upward development from the centre array

through array 5 is shown. In addition, as ‘a b’ is a transposition of ‘b a’, and may,

therefore, stand in lieu of it, only the development of the array ‘a b’ is included.

Furthermore, as leftwards development in a two-array model is tantamount to

rightwards development, only leftwards development is shown in the examples

below.

Table 10 Unsuitability of Two-Element Centre Array

5. b a = array 7 ↖ 6. a b ↖ 7. b a ↖ 8. a b

!39

The alternative of changing directionality at array 6 to avoid redundancy

only postpones redundancy until array 3, itself a symbolic and cyclical repetition

of array 5, as may be seen in Table 11 below. The possibilities for the satisfactory

development of a two-element array being exhausted, it is concluded that a two-

element array is unsuitable for constructing a model of the Early Italian sonnet.

Table 11 Unsuitability of Two-Element Centre Array: Change of Directionality at Array 6

3. b a = array 5 ↗ 4. a b ↗ 5. b a ↗ 6. a b ↖ 7. b a ↖ 8. a b

Similarly, in a centre array numbering three elements of two element

types, for example, ‘b a b’, formal sonnet conditions are no closer to being

fulfilled for development necessarily results in the impossibility of

accommodating an alternating rhyme scheme. Table 12, below, shows that in a

leftwards development of the array ‘b a b’, for example, consecutive end-array

elements are immediately produced in array 7. In a rightwards development,

without a change in directionality, similar end-array elements are repeated in the

consecutive arrays 6 and 5. Changing directionality at array 6 to avoid redundancy

results in repeating end-array elements in arrays 3 and 4, failing to satisfy the

alternating rhyme condition. The other two possible three-element distributions of

!40

the centre array, ‘a b b’ or ‘b b a’, produce similar results when developed, as may

be seen in Appendix A.

Table 12 Unsuitability of Three-Element Centre Array: Leftwards & Rightwards Development

i) Leftwards Development

7. a b b ↖ ↖ 8. b a b

ii) Rightwards Development

5. b a b ↗ ↗ 6. a b b ↗ ↗ 7. b b a ↗ ↗ 8. b a b

iii) Rightwards Development with Change in Directionality in Array 6

3. a b b ↖ ↖ 4. b a b ↖ ↖ 5. b b a ↖ ↖ 6. a b b ↗ ↗ 7. b b a ↗ ↗ 8. b a b

!41

Neither does a centre array of four elements suffice to satisfy all sonnet

conditions. In this case, the ratio and distribution of two element types must

necessarily be either 2:2, 3:1 or 1:3. In the last two cases an alternating rhyme

scheme is not possible for inevitably, as just remarked, the same two types of

element must follow each other by the end of the third array developed. An even

distribution of elements between element types, ‘a b a b’ or ‘a a b b’, for example,

has the same disadvantage as a two-element ‘a b’ array: It can alternate, but it

cannot be stanzaic.

A centre array of two element types and five elements can, however, as

shall be seen below, satisfy all five sonnet conditions due to the sufficient number

of combinatorial possibilities inherent in its symbolic and cyclical properties. To

see this, let distributions that are clearly unsuitable first be excluded. Consider, for

example, a distribution of 4:1 elements divided between two element types, as in,

for example, the arrays ‘a b b b b’ or ‘a a a a b’. Neither of these alternatives will

do, as, by simple inspection of Table 13, alternation of placeholders beyond the

development of array 6 is impossible:

Table 13 Unsuitability of 4:1 Distribution of Centre Array Elements

leftwards development rightwards development

5. b b a b b ↖ ↖ ↖↖ 6. b b b a b ↖ ↖ ↖↖ 7. b b b b a b a b b b ↖ ↖↖ ↖ ↗↗ ↗↗ 8. a b b b b a b b b b

!42

A change in directionality at array 6 does not help as it leads to a breakdown in

the alternation of end-array elements in array 3, as shown in Table 14, below:

Table 14

Redundancy in a 4:1 Distribution of Centre Array Elements: Leftwards Development with Directionality Change in Array 6

3. b a b b b ↗ ↗ ↗ ↗ 4. a b b b b ↗ ↗ ↗ ↗ 5. b b b b a ↗ ↗ ↗ ↗ 6. b b b a b ↖ ↖ ↖ ↖ 7. b b b b a ↖ ↖ ↖ ↖ 8. a b b b b

It follows that the mix of element types and their number in the centre array, is in

the ratio 3:2, either three ‘a’s and two ‘b’s, or three ‘b’s and two ‘a’s. As ‘a a b b b’

is a transposition of ‘b b a a a’ and does not affect the pattern of placeholders in

the model, nothing is lost by choosing one distribution over the other. Let the

distribution then be three ‘b’s and two ‘a’s. Now, the pattern each element traces

during development depends only on the rules for array development, which are a

priori the same for all elements. That is, as far as the ordering of elements within

the centre array is concerned, it is not necessary to distinguish between elements

within each element type: One a or b is as good as any other. Hence, the number

of ways that elements in the centre array may be ordered is governed by the

!43

mathematical rule for combinations. This rule gives the number of possible 2

centre arrays as ten, which confirms the ten arrays, presumed exhaustive in the

ordering of their elements, listed in Table 15, below:

Table 15 Early Italian Model: Centre Array Candidates 1

i.) a a b b b

ii.) a b a b b

iii.) a b b a b

iv.) a b b b a

v.) b a a b b

vi.) b a b a b

vii.) b a b b a

viii.) b b a a b

ix.) b b a b a

x.) b b b a a

As remarked above, any centre array with a contiguity of three similar

elements is unsuitable because it compromises alternation of end-array elements

in a fourteen array model therefore ruling out the equivalent of an alternating

rhyme scheme. For this reason, arrays i.), iv.) and x.) do not pass muster. Slightly

less obviously, perhaps, but for the same reason, neither do the arrays v.) and

viii.): Due to the elements’ cyclical properties, the three ‘b’ elements in both cases

Assuming C (n, k) for two element types, b and a, and five elements distributed in the 2 ratio 3:2, there are in all, (5!/(5-2)!(2)! = 120/12 = 10 combinations of elements.

!44

are contiguous. There are, therefore, only five centre arrays, listed in Table 16,

that might still satisfy all five sonnet conditions:

Table 16 Early Italian Model: Centre Array Candidates 2

ii.) a b a b b

iii.) a b b a b

vi.) b a b a b

vii.) b a b b a

ix.) b b a b a

Combined with the choice of either leftwards or rightwards directionality,

there are, then, in all twice five, or ten, candidate centre arrays. Now, it was seen

in the discussion of the working model that it was changes in directionality that

represent the equivalents of stanzaic form and lead to redundancy and model

completion. From this finding, two further criteria for excluding candidate centre

arrays follow. First, if to achieve the equivalent of alternation in end-array

elements a directionality change is needed in an array other than the fourth or fifth

and eleventh or twelfth arrays, then that candidate is unsuitable for it cannot

satisfy the Early Italian sonnet’s two quatrain, two tercet stanzaic form condition.

Second, if a candidate’s centre array is not redeveloped in array 2, then it also fails

for there is no mechanism to develop a redundant array in array 1, complete the

model in fourteen arrays and satisfy the sonnet condition for number of lines.

With the application of these additional criteria, eight of the ten

remaining candidates may be excluded: arrays ii.), iii.), vi.), with rightwards and

!45

leftwards, vii.) with rightwards and ix.) with leftwards development: Either they

do not provide alternation in end-arrays, or their centre array is not redeveloped in

array 2. Detailed workings in support of this conclusion may be found in

Appendix B. Hence, there are only two arrays that meet the criteria established so

far: vii.), ‘b a b b a’ with leftwards, and ix.) ‘b b a b a’ with rightwards,

development.

The rules for development of the ‘b a b b a’ array are similar to those for

the working model. The differences are as follows. First, array pairs rather than

single arrays are developed simultaneously and not alternately, resulting initially

in models of fifteen rather than fourteen arrays; second, arrays are developed only

with leftwards directionality from the centre array rather than with either

leftwards or rightwards directionality; finally, in addition to the centre array,

development of the array ‘b a b a b’ also leads to directionality change. Including

a rule for model completion that maximises array innovation and minimises array

redundancy, there are, then, three rules for array development in all:

Rule 1. From the centre array, arrays develop simultaneously

upwards and downwards to the left;

Rule 2. Symbolic repetition of the centre array or the array

‘b a b a b’ causes a change in directionality;

Rule 3. Symbolic repetition of any array and cyclical repetition of

its towards flows halts development and completes the

model.

!46

Leftwards development from the centre array ‘b a b b a’ creates the

following first three pairs of arrays, 7–5 and 9–11, as shown in Table 17 below:

Table 17 Development of Centre Array through Arrays 5 & 11

5. b a b a b ↖ ↖ ↖ ↖ 6. b b a b a ↖ ↖ ↖ ↖ 7. a b b a b ↖ ↖ ↖ ↖ 8. b a b b a ↙ ↙ ↙ ↙ 9. a b b a b ↙ ↙ ↙ ↙ 10. b b a b a ↙ ↙ ↙ ↙ 11. b a b a b

According to Rule 2, a change in directionality is introduced in arrays 5

and 11 to avoid the development of successive identical end-array elements. The

development of arrays 4–1 and 12–15 completes the model, as shown in Table 18

below:

!47

Table 18 Development of Arrays 4–1 and 12–15

1. a b b a b ↖↖↖↖ 2. b a b b a ↗↗ ↗↗ 3. a b b a b ↗↗ ↗↗ 4. b b a b a ↗↗ ↗↗ 5. b a b a b ↖↖↖↖ 6. b b a b a ↖↖ ↖↖ 7. a b b a b ↖↖ ↖ ↖ 8. b a b b a ↙ ↙↙ ↙ 9. a b b a b ↙ ↙ ↙ ↙ 10. b b a b a ↙ ↙ ↙ ↙ 11. b a b a b ↘ ↘ ↘ ↘ 12. b b a b a ↘ ↘ ↘ ↘ 13. a b b a b ↘ ↘ ↘ ↘ 14. b a b b a ↘ ↘ ↘ ↘ 15. a b b a b

The last pair of arrays developed, arrays 1 and 15, repeats arrays 7 and 9 symbol-

ically and their towards flows cyclically. According to Rule 3, development is,

therefore, halted and the model complete. Development stops at this point for the

same reason as in the working model: the maximization of array innovation and

the minimization of array redundancy. As may be seen in Table 19 below, any

!48

further development of arrays results only in the development of additional

redundant arrays. These redundant arrays then repeat indefinitely for, as argued in

the first part of the inquiry, from redundant arrays only more redundant arrays are

developed.

Table 19 Continuous Redundancy in Array Development

-1. b b a b a = 6. ↖ ↖ ↖ ↖ 1. a b b a b = 7.

...

15. a b b a b = 9. ↙ ↙ ↙ ↙ 16. b b a b a = 10.

The second centre array presumed to satisfy all five Early Italian sonnet

conditions is, as noted above, the array ‘b b a b a’ with rightwards development.

In this model's construction, development of the array ‘a b a b b’, as well as the

centre array, leads to a change in directionality. The two complete array models

are juxtaposed in Table 20 below.

!49

Table 20 Early Italian Fifteen Array Models ‘b b a b a’ and ‘b a b b a’

‘b b a b a’ ‘b a b b a’

1. a b b a b a b b a b 1. ↗ ↗↗ ↗ ↖ ↖↖ ↖ 2. b b a b a b a b b a 2. ↖↖ ↖ ↖ ↗ ↗ ↗↗ 3. a b b a b a b b a b 3. ↖ ↖ ↖ ↖ ↗ ↗ ↗↗ 4. b a b b a b b a b a 4. ↖ ↖ ↖ ↖ ↗ ↗ ↗↗ 5. a b a b b b a b a b 5. ↗↗ ↗ ↗ ↖ ↖ ↖ ↖ 6. b a b b a b b a b a 6. ↗↗ ↗ ↗ ↖ ↖ ↖ ↖ 7. a b b a b a b b a b 7. ↗↗ ↗ ↗ ↖ ↖ ↖ ↖ 8. b b a b a b a b b a 8. ↘ ↘ ↘ ↘ ↙ ↙ ↙ ↙ 9. a b b a b a b b a b 9. ↘ ↘ ↘ ↘ ↙ ↙ ↙ ↙ 10. b a b b a b b a b a 10. ↘ ↘ ↘ ↘ ↙ ↙ ↙ ↙ 11. a b a b b b a b a b 11. ↙↙ ↙↙ ↘ ↘↘ ↘ 12. b a b b a b b a b a 12. ↙ ↙↙↙ ↘ ↘ ↘ ↘ 13. a b b a b a b b a b 13. ↙ ↙ ↙↙ ↘ ↘↘ ↘ 14. b b a b a b a b b a 14. ↘ ↘↘ ↘ ↙ ↙ ↙ ↙ 15. a b b a b a b b a b 15.

Table 21 below shows, using the example of the model on the right in the

table above, that each of these fifteen array models comprises two identical

fourteen array sub-models. That is, when the arrays 15–2 are read from bottom to

top, they are identical symbolically and cyclically with arrays 1–14.

!50

Table 21 Identical 14-Array Sub-Models

1. a b b a b a b b a b 15. ↖ ↖ ↖ ↖ ↖↖ ↖ ↖ 2. b a b b a b a b b a 14. ↗↗ ↗ ↗ ↗ ↗ ↗ ↗ 3. a b b a b a b b a b 13. ↗↗↗ ↗ ↗↗ ↗↗ 4. b b a b a b b a b a 12. ↗↗ ↗↗ ↗ ↗↗ ↗ 5. b a b a b b a b a b 11. ↖↖ ↖↖ ↖ ↖ ↖ ↖ 6. b b a b a b b a b a 10. ↖ ↖↖ ↖ ↖ ↖ ↖ ↖ 7. a b b a b a b b a b 9. ↖↖ ↖ ↖ ↖ ↖ ↖ ↖ 8. b a b b a b a b b a 8. ↙ ↙↙ ↙ ↙ ↙ ↙↙ 9. a b b a b a b b a b 7. ↙ ↙ ↙↙ ↙ ↙ ↙ ↙ 10. b b a b a b b a b a 6. ↙ ↙ ↙↙ ↙ ↙ ↙ ↙ 11. b a b a b b a b a b 5. ↘ ↘↘↘ ↘ ↘↘↘ 12. b b a b a b b a b a 4. ↘ ↘↘↘ ↘ ↘ ↘ ↘ 13. a b b a b a b b a b 3. ↘ ↘↘↘ ↘ ↘ ↘ ↘ 14. b a b b a b a b b a 2.

The final fourteen array Early Italian models developed from the centre

arrays ‘b b a b a’ and ‘b a b b a’ are shown in Table 22 below. They are presented

in a form profiling model equivalents of Early Italian sonnet conditions to make it

easier to follow the subsequent discussion. Array numbering has been removed in

the table, as have the arrows between the equivalents of the sonnet's stanzaic

divisions.

!51

Table 22 Early Italian Array Models

Early Italian Array Models

a b b a b a b b a b ↗ ↗ ↗↗ ↖ ↖ ↖↖ b b a b a b a b b a ↖ ↖ ↖↖ ↗ ↗ ↗↗ a b b a b a b b a b ↖ ↖ ↖↖ ↗ ↗ ↗↗ b a b b a b b a b a a b a b b b a b a b ↗ ↗ ↗ ↗ ↖ ↖ ↖↖ b a b b a b b a b a ↗ ↗ ↗ ↗ ↖ ↖ ↖ ↖ a b b a b a b b a b ↗ ↗ ↗↗ ↖ ↖ ↖↖ b b a b a b a b b a a b b a b a b b a b ↘ ↘ ↘ ↘ ↙↙ ↙ ↙ b a b b a b b a b a ↘ ↘ ↘ ↘ ↙↙ ↙ ↙ a b a b b b a b a b b a b b a b b a b a ↙ ↙ ↙ ↙ ↘ ↘ ↘ ↘ a b b a b a b b a b ↙ ↙ ↙ ↙ ↘ ↘ ↘ ↘ b b a b a b a b b a

!52

2.1.3 Assessment

How well, then, do these models describe the equivalents of the Early

Italian sonnet’s formal characteristics? The fourteen arrays of the model are

deemed equivalent to the fourteen lines of the Early Italian sonnet. The contrast in

away flows from array 8 marks the equivalent of the division of the sonnet into

octave and sestet just as the changes in directionality at arrays 5 and 11 represent

the equivalent division of the octave into quatrains and the sestet into tercets,

respectively. By dint of the contrast in flow directionality in the model being

greatest at array 8, this array is deemed to represent the equivalent of the volta.

Each array having five elements fulfils the condition for isometry. These five

elements, as noted in the first part of the inquiry, may represent five prosodic

markers, or stresses, and can thus also accommodate the accentuation rules of the

endecasyllabi sciolti, or free hendecasyllables of the standard Italian verse line in

its verso tronco, piano or sdrucciolo forms of ten, eleven or twelve syllables,

respectively. Finally, alternating end-array elements throughout the models are

deemed equivalent to the sonnet’s alternating rhyme scheme. The models thus

appear to describe satisfactorily equivalents of the Early Italian sonnet tradition.

2.1.4 Conclusion and Outlook

The problem posed by the multiplicity of potential sonnet rhymes schemes

for model construction raised at the end of Part 1 is addressed by considering the

formal characteristics of the Early Italian sonnet tradition as conditions to be

satisfied simultaneously. This has the effect of drastically reducing the number of

simplified rhyme schemes that might satisfy all sonnet conditions. By then

!53

applying a principle of economy to this general condition, from a corpus of ten

candidate arrays, two centre arrays that are presumed to satisfy all conditions are

elicited by first excluding those arrays that clearly do not. The two centre arrays

are then developed into array models. Encouragingly, both models appear to

describe satisfactorily key characteristics of the Early Italian sonnet, thus

providing support for the claim and making it reasonable to want to seek more

corroborative evidence in the modelling of other sonnet traditions. However, the

somewhat Procrustean approach to defining array development rules to satisfy

sonnet conditions raises concerns about error and bias in the final models’ results.

Moreover, why are there two array models that satisfy sonnet conditions, rather

than one? Although their mirrored flows suggest that they could be complete parts

of a broader pattern, what that pattern might be is as yet unclear. Therefore, in

search of not only more evidence in support of the claim by way of array models

that satisfy the sonnet conditions of other traditions, but also a means to mitigate

the risk of error and bias in model results, as well as a broader pattern that might

relate the two Early Italian array models developed above, the inquiry now turns

to a consideration of the Petrarchan sonnet tradition.

!54

2.2 Petrarchan Tradition


The two other rhyme schemes noted by Kircher (p. 414) as characteristic of

the Italian sonnet tradition, those with embracing, or arched, rather than alternating

rhymes in the octave, are famously associated with the Petrarchan sonnet:

a b b a a b b a c d c d c d

& a b b a a b b a c d e c d e

Transforming both by following the same procedure as for the Early Italian

tradition, but retaining stanzaic divisions for the moment, results in the following

simplified rhyme scheme: a b b a a b b a b a b a b a. Its most obvious feature is,

of course, as with the conventional Petrarchan rhyme schemes, the contrast between

the equivalents of embracing rhymes in the octave and alternating rhymes in the

sestet. Consider, however, the simplified rhyme scheme not only as contrast, but also

as balance struck between the equivalents of different types of rhyme. To see this,

suspend the assumption of linearity imposed by a conventional rhyme scheme

presentation and see the simplified rhyme scheme instead as an equilibrium between

the equivalents of embracing, alternating and paired rhymes overlapping in its centre

and developing outwards from it. This is shown for paired and alternating rhyme

equivalents in Table 23:

Table 23 Petrarchan Model: Overlapping Paired and Alternating Rhyme Equivalents 1

12 13 14 1 2 3 4 5 6 7 8 9 10 11 12 13 14 1

a b ∣ a a b b a a b b a b a b a b a ∣ a ∣ paired ⟵⟶ alternating ∣ b a b a b a b a

!55

The result is a balance of four paired and four alternating rhyme equivalents

on either side of a shared central element, element 7. It can be seen that element 14

in the simplified rhyme scheme is also shared by both the paired and alternating

rhyme equivalents. Continued development of these rhyme types is, therefore, out of

the question as no further rhyme pair is possible to the left just as there is no alter-

nation possible to the right. These limits are marked by bars in the table.

The embracing and alternating rhyme equivalents, as shown in Table 24,

overlap in elements 7 and 8:

Table 24 Petrarchan Model: Overlapping Paired and Alternating Rhyme Equivalents 2

12 13 14 1 2 3 4 5 6 7 8 9 10 11 12 13 14 1

a b a ∣ a b b a a b b a b a b a b a embracing ⟵ ⟶ alternating a b b a a b b a b a b a b a ∣ a

Again each rhyme type equivalent is developed as far as possible outwards from the

simplified rhyme scheme’s centre, resulting in a balance of eight elements in each.

These findings suggest that, besides the so-called asymmetry between the

embracing and alternating rhymes of a conventional Petrarchan rhyme scheme, there

is also a simpler, underlying bilateral symmetry that evolves from its centre. To

underscore this symmetry, notwithstanding the risk of momentarily getting ahead of

the discussion, the pattern traced by the flows between the Petrarchan array models’

placeholders is presented in Tables 25 and 26 below.

!56

Table 25 Petrarchan Flow Pattern 1

This flow pattern results from superposing the halves of the Petrarchan

array model and is composed, therefore, not of fourteen single arrays, but of seven

array pairs. The pattern may be disaggregated, as in Table 26 below, into four

similar, mirrored sub-patterns:

!57

Table 26 Petrarchan Flow Pattern 2: Sub-Patterns (a), (b), (c) & (d)

(a) (b) (c) (d)

1. b b a a b b a a b b a a b b a a

2. b a a b b a a b b a a b b a a b

3. a a b b a a b b a a b b a a b b

4. a b b a a b b a a b b a a b b a

5. b b a a b b a a b b a a b b a a

6. b a a b b a a b b a a b b a a b

7. a a b b a a b b a a b b a a b b

8. b a b a b a b a b a b a b a b a

9. a b a b a b a b a b a b a b a b






The lines in black in the figures represent flows between placeholders in the

top half of the array model, those in blue between placeholders in its bottom half,

!58

whilst the italicised and underlined black and blue letters in the tables show how the

development of arrays and flow-lines pace each other.

2.2.2 Centre Matrix

Regarding the construction of the Petrarchan array model, as a single array

cannot at the same time develop the equivalents of different rhyme types, embracing

and alternating rhymes, for instance, the model has two ‘centre arrays’. These two

arrays shall be termed centre matrix to avoid possible confusion between the terms

‘centre array’ and ‘centre arrays’. Not only are towards flows for each array in the

centre matrix undefined, as the centre matrix develops different rhyme types, neither

are flows between its arrays. The number, types and mix of elements in the centre

matrix are derived in the same way as for the Early Italian model. A discussion of

the derivation of the Petrarchan centre matrix may be found in Appendix C. Two

centre matrices comprising arrays (7) and (8) are presumed to satisfy the conditions

of the Petrarchan sonnet, first, ‘(7) a a b b, (8) b a b a’ with rightwards and, second,

‘(7) a b b a, (8) a b a b’ with leftwards directionality. The former is shown in Table

27 below and is developed exemplarily for the latter with the difference in direction-

ality between them being taken into account subsequently.

Table 27 Petrarchan Centre Matrix

Centre Matrix: Arrays 7 & 8

7. a a b b

8. b a b a

!59

As there are no changes in directionality in the Petrarchan model, it has one

less development rule than those for the working and Early Italian models. That is,

there is one rule for the start and one for the finish of development:

Rule 1. From the centre matrix, arrays develop simultaneously up-

wards and downwards to the right;

Rule 2. Symbolic repetition of any two consecutive array pairs and

cyclical repetition of the flows towards them halts array

development and completes the model.

Array model completion matches completion of the pattern and sub-

patterns shown in Tables 25 and 26 above so that just as the array model marks the

limit between array innovation and redundancy, the flow patterns mark the limit

between spatial innovation and redundancy. To show this, there follows a step-by-

step description of the development of both the array model and Sub-Pattern (a),

which serves as proxy for the other sub-patterns.

2.2.3 Step-by-Step Description of Array Model Development

Step 1 Centre Matrix: Arrays 7 & 8

7. a a b b

8. b a b a

The derivation of the centre matrix is discussed in Appendix C, as noted

above. The start elements for the development of the flow-lines in Sub-Pattern (a)

are highlighted in arrays 7 and 8. Arrays 6 and 9 and the initial flow lines of Sub-

!60

Pattern (a) that are developed in Step 2 along with their description are placed

together below to help make their relationship clear at the outset.

(The remainder of this page is deliberately blank to better presents the tables and

figures that follow.)

!61

Step 2 arrays 6 and 9

6. b a a b ↗ ↗ ↗ 7. a a b b

8. b a b a ↘ ↘ ↘ 9. a b a b

Development starts in opposite directions rightwards from the centre

matrix, according to Rule 1. The underlined elements correspond to the origin and

initial development of the sub-pattern.

Petrarchan sonnet pattern: Development of Sub-Pattern (a)

1. * * 8.

2. * * 9. 3. * * * * 10.

4. * * * * 11.

5. * * * * 12.

6. * * 13.

7. * * 14.

With the development of the second array pair in the model, the flow lines

of Sub-Pattern (a) start to emerge between placeholders. The asterisks refer to the

placeholders resulting from the direct translation of one half of the model onto the

other. The numbering on each side of the pattern serves to show that array 1 overlays

!62

array 8, as array 2 does array 9, and so forth. Thus, the blue line in the top half

represents the flow from the highlighted element in array 8 to the highlighted

element in array 9, while the black line in the bottom half represents the flow from

the first element in array 7 to the second element in array 6. In the sonnet pattern

diagrams to follow, these lines continue to accompany the development of arrays

throughout the model.

Step 3 Arrays 5 and 10

5. b b a a ↗ ↗ ↗ 6. b a a b ↗ ↗ ↗ 7. a a b b

8. b a b a ↘ ↘ ↘ 9. a b a b ↘ ↘ ↘ 10. b a b a

With no repetition of array pairs, development continues rightwards,

according to Rule 1, to develop the new array pair, 5 &10.

!63


1. 8.

2. 9. 3. 10.

4. * * * * 11.

5. * 12.

6. * 13.

7. * 14.

The development of flow lines accompanies that of their corresponding elements.


4. a b b a ↗ ↗ ↗ 5. b b a a ↗ ↗ ↗ 6. b a a b ↗ ↗ ↗ 7. a a b b

8. b a b a ↘ ↘ ↘ 9. a b a b ↘ ↘ ↘ 10. b a b a ↘ ↘ ↘ 11. a b a b

In step 4, the next array pair, 4 & 11, is developed, according to Rule 1.

!64


1. 8.

2. 9. 3. 10.

4. 11.

5. 12.

6. * 13.

7. * 14.

The development of flow lines continues concurrently with the

development of the new arrays.

!65


3. a a b b ↗ ↗ ↗ 4. a b b a ↗ ↗ ↗ 5. b b a a ↗ ↗ ↗ 6. b a a b ↗ ↗ ↗ 7. a a b b

8. b a b a ↘ ↘ ↘ 9. a b a b ↘ ↘ ↘ 10. b a b a ↘ ↘ ↘ 11. a b a b ↘ ↘ ↘ 12. b a b a

In Step 5 the array pair, 3 & 12, is developed. Array 3 repeats array 7

symbolically, but not cyclically as array 7 has no towards flows. Development,

therefore, continues, according to Rule 1.

!66


1. 8.

2. 9. 3. 10.

4. 11.

5. 12.

6. * 13.

7. * 14.

The first of the sub-pattern’s three similar triangles is created.

!67


2. b a a b ↗ ↗ ↗ 3. a a b b ↗ ↗ ↗ 4. a b b a ↗ ↗ ↗ 5. b b a a ↗ ↗ ↗ 6. b a a b ↗ ↗ ↗ 7. a a b b

8. b a b a ↘ ↘ ↘ 9. a b a b ↘ ↘ ↘ 10. b a b a ↘ ↘ ↘ 11. a b a b ↘ ↘ ↘ 12. b a b a ↘ ↘ ↘ 13. a b a b

The array pair 2 & 13 repeats arrays 6 & 9 symbolically and their towards

flows cyclically, thus fulfilling the first part of Rule 2’s condition for redundancy.

!68


1. 8.

2. 9. 3. 10.

4. 11.

5. 12.

6. 13.

7. 14.

Within the pattern there is as yet no duplication of the first triangle, no

redundancy and, hence, continuation of development.

!69


1. b b a a ↗ ↗ ↗ 2. b a a b ↗ ↗ ↗ 3. a a b b ↗ ↗ ↗ 4. a b b a ↗ ↗ ↗ 5. b b a a ↗ ↗ ↗ 6. b a a b ↗ ↗ ↗ 7. a a b b

8. b a b a ↘ ↘ ↘ 9. a b a b ↘ ↘ ↘ 10. b a b a ↘ ↘ ↘ 11. a b a b ↘ ↘ ↘ 12. b a b a ↘ ↘ ↘ 13. a b a b ↘ ↘ ↘ 14. b a b a

The array pair, 1 & 14, repeats the array pair 5 & 10 symbolically and its

towards flows cyclically. The two consecutive array pairs, 2 & 13 and 1 & 14, thus

trigger Rule 2, halt development and complete the model.

!70


1. 8.

2. 9. 3. 10.

4. 11.

5. 12.

6. 13.

7. 14.

Completion of the array model is mirrored by the double duplication of the

central triangle in Sub-Pattern (a). In this way, array and spatial developments mark

the limit between array and spatial innovation and redundancy. To confirm this,

consider that any further development of arrays results in the nascent repetition of

array series already developed in both halves of the model, namely, the series 4–1

and 8–9. Now, although the two-array series 8–9 is developed nearly four times by

array 14, the series 4–1 is only developed for the first time with completion of the

model, and any further development beyond fourteen arrays results in the incipient

redevelopment of both series. Hence, a maximum of innovation and a minimum of

array series redundancy in both halves of the model is achieved when fourteen

arrays have been developed.

The second centre matrix presumed to satisfy the conditions of the

Petrarchan sonnet, ‘7. a b b a’ and ‘8. a b a b’ with leftwards directionality, has

almost identical rules for development as the first, the only difference between them

!71

being their initial directionalities. The two completed Petrarchan array models are,

therefore, placed side by side in Table 28 in such a way as to highlight the model

equivalents of the Petrarchan tradition's sonnet characteristics.

Table 28 Petrarchan Array Models

Petrarchan Array Models

(II) (I)

b a a b b b a a ↖ ↖ ↖ ↗ ↗ ↗ b b a a b a a b ↖ ↖ ↖ ↗ ↗ ↗ a b b a a a b b ↖ ↖ ↖ ↗ ↗ ↗ a a b b a b b a b a a b b b a a ↖ ↖ ↖ ↗ ↗ ↗ b b a a b a a b ↖ ↖ ↖ ↗ ↗ ↗ a b b a a a b b

a b a b b a b a

b a b a a b a b ↙ ↙ ↙ ↘ ↘ ↘ a b a b b a b a ↙ ↙ ↙ ↘ ↘ ↘ b a b a a b a b a b a b b a b a ↙ ↙ ↙ ↘ ↘ ↘ b a b a a b a b ↙ ↙ ↙ ↘ ↘ ↘ a b a b b a b a

!72

Results identical to these two array models may be developed alternatively

by binary expansion, as shown in Table 29 below:

Table 29 Identical Petrarchan Array and Triangle Models

b b

b b a b

b b a a a b

b b a a b a a b ↗ ↗ ↗ ↗ ↖ ↖ ↖ ↖

b b a a b b b a a b ↗ ↗ ↗ ↗ ↖ ↖ ↖ ↖

b b a a b b a b b a a b ↗ ↗ ↗ ↗ ↖ ↖ ↖ ↖

b b a a b b a a a b b a a b ↗ ↗ ↗ ↗ ↖ ↖ ↖ ↖

b b a a b b a a b a a b b a a b ↗ ↗ ↗ ↗ ↖ ↖ ↖ ↖ b b a a b b a a b b b a a b b a a b ↗ ↗ ↗ ↗ ↖ ↖ ↖ ↖ 7. ↗ b b a a b b a a b b a b b a a b b a a b ↖7. (I) (II) 8. ↘ b a b a b a b a b a a b a b a b a b a b ↙8. ↘ ↘ ↘ ↘ ↙ ↙ ↙ ↙ b a b a b a b a b b a b a b a b a b ↘ ↘ ↘ ↘ ↙ ↙ ↙ ↙ b a b a b a b a a b a b a b a b ↘ ↘ ↘ ↘ ↙ ↙ ↙ ↙ b a b a b a b b a b a b a b ↘ ↘ ↘ ↘ ↙ ↙ ↙ ↙ b a b a b a a b a b a b ↘ ↘ ↘ ↘ ↙ ↙ ↙ ↙ b a b a b b a b a b ↘ ↘ ↘ ↘ ↙ ↙ ↙ ↙ b a b a a b a b

b a b b a b

b a a b

b b

!73

The rightwards and leftwards expansions are termed ‘binary’ as the indiv-

idual elements in what shall be termed their centre sequences, designated ‘7.’ and

‘8.’ in the table, are developed upwards and downwards simultaneously from them.

To distinguish these results from those of the array models, the models in Table 29

are termed triangle models. The derivation of the triangle models’ centre sequences

is discussed in more detail in the Pleadean and Shakespearean sections below. For

now, let it be seen that the triangle models’ results, underlined in the expansions, are

identical to those of their corresponding Petrarchan array models’, designated (I) and

(II), in Table 28. Given this, it follows that the only difference between the models is

their different methods of development, a difference that is presumed to offer an

independent means to cross-check the results of the array models and, hence, reduce

the risk of error and bias in their design.

2.2.4 Assessment

However, do the Petrarchan array models' results even satisfy the sonnet

conditions of the Petrarchan tradition? The number of lines condition is satisfied by

the simultaneous maximization of innovation and minimization of redundancy in

array series and spatial developments. Isometry is satisfied by the constant number

of elements per array. The equivalent of the volta is deemed to occur, as in the

working and Early Italian models, at the point of greatest contrastive directionality

in flows between arrays, which here falls between arrays 7 and 8. Array 8 is still

deemed, however, to represent the equivalent of the volta for, flows between and to-

wards the arrays of the centre matrix being undefined, array 8 is the first array to

show a change in flow directionality in a conventional, linear reading of the model.

!74

In order to mark the equivalent of the division of the sonnet into octave and sestet,

this change in directionality is not made explicit in Table 28, however. It may, none-

theless, be straightforwardly deduced by taking into consideration the change in the

ordering of elements from array 8 to array 9. Subsequent division into quatrain and

tercet equivalents is quasi-translational for the quatrains, and rotational for the

tercets. The quatrain equivalents may be discerned by the identical arrays that make

up the first three lines of each. That the ordering of the elements in the final arrays

of the quatrain equivalents differs is due to the need for alternating elements in array

8 of the centre matrix. The tercets are defined by 180 degree rotational symmetry

about a centre lying between arrays 11 and 12, which maps each element of one

tercet directly onto its counterpart in the other. Equivalents of both Petrarchan rhyme

schemes are accommodated by the order of end-array elements in the model: The

end-array elements of arrays 9–14 allow for equivalents of either the two ternary

rhymes or three rhyme pairs of the Petrarchan sonnet noted by Kircher, which, in

terms of the simplified rhyme scheme, are both alternating, and the end-array

elements of arrays 1–8 may be seen upon inspection to accommodate equivalents of

embracing rhyme pairs. The Petrarchan array models thus appear, on balance, to

satisfy the formal conditions of the Petrarchan sonnet traditions under discussion.

2.2.5 Conclusion

The Petrarchan array models show how the principle of centred form can

relate equivalents of the formal characteristics of the Petrarchan sonnet, while the

triangle models help mitigate, but not entirely eliminate, the risk of error and bias in

array model results by providing an independent means to cross-check them. To this

!75

extent, the Petrarchan array and triangle models provide further evidence in support

of the claim. However, the value of the evidence is weakened by the absence of

development rules for, and lack of testing of, the triangle models. With the aim of

addressing these weaknesses, the inquiry now turns to the Pleadean sonnet tradition.

!76

2.3 Pleadean Tradition


Kircher (415) notes two rhyme schemes as characteristic of the Pleadean

tradition: a b b a a b b a c c d e e d

& a b b a a b b a c c d e d e . 1

Transforming both into simplified form and referring to them, somewhat

chicly, as the Pleadean 1 and 2 traditions, respectively, brings even more to the fore,

as shown in Table 30, the single difference that separates them, namely, of course,

the reversed order of their final two end rhymes:

Table 30 Pleadean 1 & 2: Comparison of Simplified Rhyme Schemes

Pleadean 1 a b b a a b b a b b a b b a

Pleadean 2 a b b a a b b a b b a b a b

It may also be seen that the Pleadean 1 array comprises two mirrored sub-

arrays, as shown in Table 31. The inserted bars show how the two arrays are related:

Table 31 Pleadean 1: Simplified Rhyme Scheme with Symmetry Line: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Pleadean 1 a b b a a b b a b | b a b b a

Mirrored Sub-Arrays: b a a b b a b | b a b b a a b 3 4 5 6 7 8 9 10 11 12 13 14 1 2

A conventional rhyme scheme representation emphasizing the so-called

asymmetry of the octave–sestet relationship is, therefore, to the extent that it

masks this symmetry, something of a trompe-l’œil.

Just under four of five Pleadean sonnets make use of these rhyme schemes in a ratio 1 heavily in favour of the Pleadean 1 tradition (nearly 5:1). For Ronsard, the ratio is just over 2:1. These findings are drawn from Olmsted's (1897, pp. 59-109) tablulations.

! 77

The simplified Pleadean 2 rhyme scheme shows no such symmetry. If the

symmetrical principle above is extended as far as possible within it, the mirrored

sub-arrays shown in Table 32 result. It can be seen that the pairs of sixth and seventh

elements, developed from the centre and underlined in the table, do not correspond:

Table 32 Pleadean 2: Simplified Rhyme Scheme without Symmetry

Line: 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Pleadean 2 a b b a | a b b a b b a b a b

Mirrored Sub-Arrays b a b a b b a | a b b a b b a

12 13 14 1 2 3 4 5 6 7 8 9 10 11

What might appear as a poetic bagatelle, a difference in the order of two

final end rhymes between two conventional rhyme schemes, distinguishes, as shall

be evidenced below with the development of the Pleadean models, two quite differ-

ent structural principles, namely, symmetry and chirality.

2.3.2 Pleadean 1: Centre Arrays and Array Models

Following the procedure adopted in the development of previous models,

and as discussed in Appendix D, the two centre arrays presumed to satisfy the form-

al conditions of the Pleadean 1 tradition are ‘a a b b’ with leftwards, and ‘a b b a’

with rightwards development. Apart from their initial directionality, their develop-

mental rules are identical:

Rule 1. From the centre array, arrays develop alternately upwards and down-

wards to the left for the ‘a a b b’ and to the right for the ‘a b b a’

centre array;

! 78

Rule 2. Directionality changes in array 11;

Rule 3. Series redevelopment of the first four arrays in upwards, and sym-

metrical series redevelopment of the first three arrays in downwards

development completes the model.

The final Pleadean 1 array models are shown in Table 33:

Table 33 Pleadean 1: Array Models

Pleadean 1 Array Models

b a a b b b a a ↖ ↖ ↖ ↗ ↗ ↗ b b a a b a a b ↖ ↖ ↖ ↗ ↗ ↗ a b b a a a b b ↖ ↖ ↖ ↗ ↗ ↗ a a b b a b b a ↖ ↖ ↖ ↗ ↗ ↗ b a a b b b a a ↖ ↖ ↖ ↗ ↗ ↗ b b a a b a a b ↖ ↖ ↖ ↗ ↗ ↗ a b b a a a b b ↖ ↖ ↖ ↗ ↗ ↗ a a b b a b b a ↙ ↙ ↙ ↘ ↘ ↘ a b b a a a b b ↙ ↙ ↙ ↘ ↘ ↘ b b a a b a a b ↙ ↙ ↙ ↘ ↘ ↘ b a a b b b a a

↘ ↘ ↘ ↙ ↙ ↙ b b a a b a a b ↘ ↘ ↘ ↙ ↙ ↙ a b b a a a b b ↘ ↘ ↘ ↙ ↙ ↙ a a b b a b b a

! 79

2.3.3 Assessment: Pleadean 1 Array Models

Do, then, these models satisfy the formal conditions of the Pleadean 1

sonnet? The models' fourteen arrays are equivalent to the traditional Pleadean sonnet

length of fourteen lines. Development is halted after fourteen arrays not because of

the development of a single redundant array, there is categorically a repetition of

array 6 in array 2 in both, but because, with the development of array 1, the potential

for the innovative symmetrical development of array series is exhausted. To see

this, consider that the array series 8–5 is repeated symbolically and cyclically in the

series 4–1 as is the series 8–10 in the series 12–14, when reflected in array 11.

Extending development beyond the models results in the repetition of these series,

as shown in Table 34 below, using the ‘a b b a’ centre array model as exemplary for

both models. Strictly speaking, array 4 is not a repetition of array 8, which is why it

is bracketed in the column to the right of the table: The former has towards flows

that for the centre array are undefined. This may be regarded as a limitation of the

model. It is also clear from Table 34, however, that the next array developed

upwards is a repetition of the symbolic and cyclical properties of array 4, creating

duplication of the series in arrays 7–4. The model thus represents, with the

aforementioned limitation, a maximum of array series innovation and a minimum of

array series redundancy.

! 80

Table 34 Pleadean 1: Array Series Redundancy

-4. b b a a = 1.

-3. b a a b = 2.

-2. a a b b = 3.

-1. a b b a = 4.

1. b b a a = 5.

2. b a a b = 6.

3. a a b b = 7.

4. a b b a = (8), -1.

5. b b a a

6. b a a b

7. a a b b

8. a b b a

9. a a b b

10. b a a b

11. b b a a

12. b a a b = 10.

13. a a b b = 9.

14. a b b a = 8.

15. b b a a = 11.

16. a b b a = 14.

17. a a b b = 13.

18. b a a b = 12.

19. b b a a = 11.

The equivalent of the volta is deemed to occur at the point of greatest

contrastive flows, that is in array 8. As to the equivalents of stanzaic form, the

! 81

octave is separated into quartets and the sestet into tercets by the identity of arrays in

the first and reflective identity of arrays about array 11 in the second. The isometry

condition is satisfied by the constant four elements per array, which also accom-

modates the equivalent of the usual four accents of the four measured tetrameter of

the preferred twelve syllable Pleadean verse, the alexandrine, as shown in Table 35.

Table 35 Distribution of Four Stresses, (x), in a Four-Element Array

1 2 3 4

– – x – – x – – x – – x

Finally, as may be seen in Table 36, the model’s simplified rhyme scheme

accommodates, of course, as this was after all the starting point for the analysis, the

end rhymes of the Pleadean 1 rhyme scheme noted by Kircher:

Table 36 Pleadean 1: Accommodation of Conventional Rhyme Scheme

Pleadean Sonnet 1 Rhyme Scheme a b b a a b b a c c d e e d

Pleadean 1 Simplified Rhyme Scheme a b b a a b b a b b a b b a

The model, then, is able to relate equivalents of the formal characteristics of

the Pleadean 1 tradition. Yet how reliable are its results? In the conclusion to the

discussion of the Early Italian tradition, it was suggested that the symmetry be-2

tween its array models indicated a broader pattern of which each was a part, a

pattern that might serve to confirm their results independently. Then, in the

discussion of the Petrarchan model, it was shown that a binary expansion, in the

form of a triangle model, could provide such a pattern. It now seems appropriate to

p. 542

! 82

test whether a similar triangle model is capable of relating the two Pleadean 1 array

models shown in Table 33. The purpose of such a model, it may be recalled, is to

show whether and, if so, how, independently of the array models, the formal charac-

teristics of a particular sonnet tradition might be developed and related. If a Pleadean

triangle model could achieve this, it would serve to cross-check the array models’

results. How then is such a model to be constructed for the Pleadean 1 tradition?

2.3.4 Pleadean 1: Centre Sequence and Triangle Models

One approach would be to take the Pleadean 1 simplified rhyme scheme as

a centre sequence for development into a triangle model by, first, substituting it for

the Petrarchan centre sequence that led to the development of the Petrarchan triangle

models and, second, by applying the same development rules to it. As noted earlier,

and as shown in Table 37, the Pleadean 1 centre sequence comprises two mirrored

sub-arrays. Let the bar representing the line of symmetry between them serve as a

point of orientation in the discussion to follow.

Table 37 Pleadean 1: Triangle Models’ Centre Sequence

b a a b b a b | b a b b a a b

Let the same developmental rule as for the Petrarchan triangle model now

be applied to this centre sequence, that is, let development continue without a

change in directionality from the outer elements towards the centre. As development

ends when elements meet in the centre, as shown in the stepwise description of the

! 83

triangle model’s construction to follow, a second rule for ending development is

unnecessary:

Rule 1 From its centre sequence, elements to the left of the line of

symmetry are developed upwards and downwards to the

right, those to its right are developed upwards and

downwards to the left.

A step-by-step description of the development of the triangle models now

follows. As it is a matter of testing whether the models might relate the two

previously developed Pleadean 1 array models, sequences shall be developed only

from the point in the centre sequence where the centre arrays of these models,

underlined in Step 1 below, are located.

Step 1 Centre Sequence with Array Model Centre Arrays

b a a b b a b | b a b b a a b (b)

The bracketed element on the right signifies that this element falls outside

the centre sequence in Table 37. However, here, as in the development of the

Petrarchan triangle model, although it was not made explicit at the time for purposes

of exposition, it is assumed that the centre sequence develops infinitely away from

the centre thus allowing for the inclusion of the next element in the sequence. That

the element ‘b’ is indeed the next element is inferred from the rightmost ‘b’ element

of the second array of the Pleadean 1 array model, counting from the top, in Table

33. As shall be seen below, the triangle model can only be fully developed if this

end-array element is included in the centre sequence.

! 84

Step 2 Development of Arrays 7–5 and 9–11

5. b b a a b a a b

6. b a a b b b a a

7. a a b b a b b a ↗ ↗ ↗ ↖ ↖ ↖ 8. ... a b b a a b b a b | b a b b a a b b a a b b ... ↘ ↘ ↘ ↙ ↙ ↙ 9. a a b b a b b a

10. b a a b b b a a

11. b b a a b a a b

Pairs of new sequences are developed in binary fashion by the successive

division and distribution of individual elements entering from the left and right of

the centre sequence. With the development of each new sequence, the element

closest to the centre of the sequence exits the models. The triangular shape resulting

from development, familiar from the Petrarchan triangle model, is omitted here to

highlight the parts of the sequences that correspond to the Pleadean 1 array models.

! 85

Step 3 Development of Arrays 4–1 and 12–14

1. b b a a b a a b

2. b a a b b b a a

3. a a b b a b b a

4. a b b a a a b b

5. b b a a b a a b

6. b a a b b b a a

7. a a b b a b b a ... ↗↗↗↗↗↗↗ ↖↖↖↖↖↖↖ ... 8. ... b b a a b b a a b b a b | b a b b a a b b a a b b a a b ... ... ↘↘↘↘↘↘↘ ↙↙↙↙↙↙↙ ... 9. a a b b a b b a

10. b a a b b b a a

11. b b a a b a a b

12. (a b b a) (a a b b)

13. (a a b b) (a b b a)

14. (b a a b) (b b a a)

The models complete, the Pleadean triangle models’ equivalents of arrays

12–14 of the array models are bracketed because they do not correspond to the

results of the Pleadean 1 array models developed above. That is, the triangle model

for the Petrarchan tradition is not transferable to the Pleadean. The reason is due of

course to the directionality change in array 11 of the Pleadean 1 array models, a

development unknown in the Petrarchan models. How, then, are these directionality

! 86

changes to be understood and reproduced independently so as to support the

Pleadean 1 array model results? This question is considered now in the discussion of

the second Pleadean tradition.

2.3.5 Pleadean 2: Array Models

The array model for the Pleadean 2 tradition is more complex than that for

the Pleadean 1 in that not one, but three directionality changes are needed in its final

six arrays to satisfy sonnet conditions. As discussed in Appendix D, the centre arrays

for both Pleadean traditions are the same, and their rules for array development

differ only insofar as extra directionality changes need to be taken into account. This

being the case, the array models for the second Pleadean tradition are presented

complete in Table 38 below.

! 87

Table 38 Pleadean 2: Array Models

Pleadean 2 Array Models

b a a b b b a a ↖ ↖ ↖ ↗ ↗ ↗ b b a a b a a b ↖ ↖ ↖ ↗ ↗ ↗ a b b a a a b b ↖ ↖ ↖ ↗ ↗ ↗ a a b b a b b a ↖ ↖ ↖ ↗ ↗ ↗ b a a b b b a a ↖ ↖ ↖ ↗ ↗ ↗ b b a a b a a b ↖ ↖ ↖ ↗ ↗ ↗ a b b a a a b b ↖ ↖ ↖ ↗ ↗ ↗ a a b b a b b a ↙ ↙ ↙ ↘ ↘ ↘ a b b a a a b b ↙ ↙ ↙ ↘ ↘ ↘ b b a a b a a b ↙ ↙ ↙ ↘ ↘ ↘ b a a b b b a a ↘ ↘↘ ↙ ↙ ↙ b b a a b a a b ↙ ↙ ↙ ↘ ↘ ↘ b a a b b b a a ↘ ↘ ↘ ↙ ↙ ↙ b b a a b a a b

! 88

2.3.6 Assessment: Pleadean 2 Array Models

The equivalent of sonnet length is determined by the same considerations of

array series innovation and redundancy as for the Pleadean 1 array model. The same

is also the case for the equivalents of the volta, isometry and rhyme scheme. The

equivalent of stanzaic form is represented by repetition of arrays in the quartet

equivalents and, differently from the Pleadean 1 model, a contrast between the

constant directionality of the flows in the first tercet equivalent and their constant

variability in the second.

2.3.7 Pleadean 1 & 2: Sequence Models

How then are the results of the Pleadean array models to be accounted for

independently? Given that the Petrarchan triangle model proved inadequate due to

the directionality change introduced in the lower half of the Pleadean 1 model, how

much more inadequate would it prove in dealing with the three directionality

changes of the Pleadean 2 model? Is there, then, another way of defining the triangle

model’s developmental rules that might prove more satisfactory?

One possibility would be to think of the Pleadean 1 and 2 triangle models’

centre sequences as developing according to the same cyclicity principle applied in

the working, Early Italian and Petrarchan array models, wherein development in one

half of the model is mirrored by development in the other. Developing the Pleadean

1 and 2 centre sequences according to this principle, however, cannot account for the

changes in directionality of the Pleadean models as identical sequences would be

created in both halves of the model whereas directionality changes occur only in the

Pleadean models’ lower halves. It would seem then that the patterns created by the

! 89

array and triangle models are not broad enough in the sense that they do not provide

enough, what might be thought of as, ‘patterned data’ to allow for the emergence of

an intelligible difference that might account for the directionality changes in the

Pleadean models.

In order to try out the idea of creating a broader pattern, let the Pleadean 1

and 2 centre sequences instead be developed cyclically into what shall be termed

sequence models with a constant fourteen elements per sequence throughout, as

opposed to the four elements per array in the array models and the constantly

diminishing number of elements per sequence in the triangle models. To avoid a

possible confusion of terms, let the centre sequences of the triangle models be called

start sequences when used to develop sequence models. The development rule for

the sequence models is uncomplicated: The start sequences for both the Pleadean 1

and 2 sequence models are developed leftwards without any change in directionality.

This rule results in the creation of models of twenty-seven sequences, as shown in

Tables 39 and 40 below.

! 90

Table 39 Pleadean 1: Sequence Model: Leftwards Directionality

(...)

(b a a b b a b b a b b a a b) = 3. (b b a a b b a b b a b b a a) = 2. (a b b a a b b a b b a b b a) = 1.

a a b b a a b b a b b a b b 14. b a a b b a a b b a b b a b 13. b b a a b b a a b b a b b a 12. a b b a a b b a a b b a b b 11. b a b b a a b b a a b b a b 10. b b a b b a a b b a a b b a 9. a b b a b b a a b b a a b b 8. b a b b a b b a a b b a a b 7. b b a b b a b b a a b b a a 6. a b b a b b a b b a a b b a 5. a a b b a b b a b b a a b b 4. b a a b b a b b a b b a a b 3. b b a a b b a b b a b b a a 2. ↖ a b b a a b b a b b a b b a 1. * b b a a b b a b b a b b a a 2. ↙ b a a b b a b b a b b a a b 3. a a b b a b b a b b a a b b 4. a b b a b b a b b a a b b a 5. b b a b b a b b a a b b a a 6. b a b b a b b a a b b a a b 7. a b b a b b a a b b a a b b 8. b b a b b a a b b a a b b a 9. b a b b a a b b a a b b a b 10. a b b a a b b a a b b a b b 11. b b a a b b a a b b a b b a 12. b a a b b a a b b a b b a b 13. a a b b a a b b a b b a b b 14.

(a b b a a b b a b b a b b a) = 1. (b b a a b b a b b a b b a a) = 2. (b a a b b a b b a b b a a b) = 3.

(...)

! 91

Table 40 Pleadean 2: Sequence Model: Leftwards Directionality

(...) (b a a b b a b b a b a b a b) = 3 (b b a a b b a b b a b a b a) = 2. (a b b a a b b a b b a b a b) = 1.

b a b b a a b b a b b a b a 14. a b a b b a a b b a b b a b 13. b a b a b b a a b b a b b a 12. a b a b a b b a a b b a b b 11. b a b a b a b b a a b b a b 10. b b a b a b a b b a a b b a 9. a b b a b a b a b b a a b b 8. b a b b a b a b a b b a a b 7. b b a b b a b a b a b b a a 6. a b b a b b a b a b a b b a 5. a a b b a b b a b a b a b b 4. b a a b b a b b a b a b a b 3. b b a a b b a b b a b a b a 2. ↖ a b b a a b b a b b a b a b 1. * b b a a b b a b b a b a b a 2. ↙ b a a b b a b b a b a b a b 3. a a b b a b b a b a b a b b 4. a b b a b b a b a b a b b a 5. b b a b b a b a b a b b a a 6. b a b b a b a b a b b a a b 7. a b b a b a b a b b a a b b 8. b b a b a b a b b a a b b a 9. b a b a b a b b a a b b a b 10. a b a b a b b a a b b a b b 11. b a b a b b a a b b a b b a 12. a b a b b a a b b a b b a b 13. b a b b a a b b a b b a b a 14.

(a b b a a b b a b b a b a b) = 1. (b b a a b b a b b a b a b a) = 2. (b a a b b a b b a b a b a b) = 3.

(...)

! 92

These sequence models comprise two continuous leftward developments

upwards and downwards from the model’s centre. That is, differently from the

Pleadean array models, the Pleadean sequence models are constructed without any

change in directionality. This being the case, directionality changes in the Pleadean

array models may be understood grosso modo as resulting from the developmental

compromises required by a centre array of only four elements needing to maintain

cyclical coherency while at the same time developing equivalents of the tradition’s

formal characteristics. If this roughly describes directionality changes within the

array models, what, however, of the specific differences in directionality changes

between them? How is the single change in directionality in the Pleadean 1 model

versus the three in the Pleadean 2 to be made sense of? Is there, for instance, a

common principle that relates them? Moreover, could such a principle help describe

more precisely the directionality changes within the array models as well, thus

providing independent support for their results?

It is with these questions that discussion returns to the two different struc-

tural principles mentioned at the outset of the discussion on the Pleadean tradition:

the principles of symmetry and chirality. Wehrli (2008), drawing on Nakahara

(2003) and Kelvin (1893), describes the difference between these two ideas:

Chirality is an attribute of symmetry. A figure is called symmetrical when there exists a non-identical congruent isomorphism of itself....An object without any non-identical congruent image is chiral....Chirality, as I choose to understand the term, is possible in spaces with any number of dimensions. (p. 61)

! 93

Symmetry, therefore, unlike chirality, requires that one object coincide perfectly

with another. To describe chirality more fully, Wehrli paraphrases Kelvin, to whom

he ascribes the introduction of the term into the natural sciences:

Chirality means handedness. Our right hand is the mirror image of the left. Although both hands are isometric, they cannot be brought to coincidence with each other, i.e., perfectly aligned if one was placed on top of the other. So they are different from each other, although they are the same metrically. We say they are chiral. A third hand, which is likewise isometrical to the right and left hand and which nevertheless cannot be aligned with either in the same way, does not exist. For every hand there is one, and only one, counterpart opposing handedness. An object is chiral, when it has a mirror image which is not identical with it. (p. 60)

Table 40 above shows how chirality originates and evolves within the

sequence model of the Pleadean 2 tradition. The simplified rhyme scheme in Table

32, as noted, serves as the model’s start sequence. It is developed, as seen, upwards

and downwards with leftwards directionality. It also serves as a proxy for rightwards

directionality as each is just the reverse of the other. Arrows on the right-hand side

of the model indicate the constant directionality involved throughout. The model

comprises two groups of fourteen sequences with the start sequence serving as both

first sequence to the fourteen array model developed in the upper half and first

sequence to its mirror image in the lower half. That two models are developed is due

of course to the binary development of the start sequence. The bracketed sequences

and ellipses serve to show that the sequences 1–14 repeat themselves indefinitely

upwards and downwards beyond the model’s limits. Now, the only two simplified

rhyme schemes of the Pleadean 2 tradition developed within the model are in bold in

its leftmost column. The simplified rhyme schemes overlap at the underlined first

! 94

element of the start sequence. That is, the underlined element forms the first element

of the simplified rhyme scheme developed in the lower half, and the first element of

the simplified rhyme scheme developed in the upper half of the model. Each half of

the model mirrors the other, as remarked above, yet, if the simplified rhyme schemes

developed in the model are superposed, they do not coincide, as shown in Table 41

below. The simplified rhyme schemes in Table 40 are, therefore, by the definitions

of symmetry and chirality given above, chiral in one dimension and symmetrical in

three.

Table 41 Pleadean 2: Sequence Model: Chirality

lower half upper half

1. a b 14. 2. b a 13. 3. b b 12. 4. a a 11. 5. a b 10. 6. b b 9. 7. b a 8. 8. a b 7. 9. b b 6. 10. b a 5. 11. a a 4. 12. b b 3. 13. a b 2. 14. b a 1.

Let the Pleadean 2 sequence model now be compared with that of the

Pleadean 1 sequence model in Table 39 above. Differently from the Pleadean 2

sequence model, the Pleadean 1 model develops its simplified rhyme scheme twice.

These are in bold in the first and fifth columns of Table 39, counting from the left-

hand side of the model. Once again, the simplified rhyme schemes overlap in the

! 95

underlined elements in the start sequence. There is, however, a difference in the

order of the elements between the two Pleadean 1 simplified rhyme schemes devel-

oped in this sequence model. The leftmost rhyme scheme is ordered, so to speak,

from within to without, that is, from 1–14 upwards and 1–14 downwards, whilst the

rightmost is ordered from without to within, that is, from the top and bottom of the

model, from 14–1 in both cases. Now, as in the Pleadean 2 sequence model, each

half of the simplified rhyme schemes developed in the Pleadean 1 model mirrors the

other, coinciding if folded, not coinciding if superposed. As in the Pleadean 2

sequence model, therefore, the two halves are chiral in one dimension and

symmetrical in three. Differently from the Pleadean 2 model, however, the rhyme

schemes within each of the top and bottom halves of the Pleadean 1 model are also

symmetrical in two dimensions: Each rhyme scheme may be rotated about the

midpoints lying between them to map onto the other. This chirality between, and

symmetry within, the two halves of the Pleadean 1 sequence model is shown

respectively in Tables 42 and 43 below. In Table 42, the developed simplified rhyme

scheme on the left-hand side in Table 39 serves also to exemplify chirality for the

alternative development to its right, while in Table 43 below, the lower halves of

both developments in Table 39 stand as proxy for those in the upper half. The

rotational sign between rows 7 and 8 is deemed to signal rotation in either direction.

! 96

Table 42 Pleadean 1: Sequence Model: Chirality between Model Halves

lower half upper half

1. a a 14. 2. b b 13. 3. b b 12. 4. a a 11. 5. a b 10. 6. b b 9. 7. b a 8. 8. a b 7. 9. b b 6. 10. b a 5. 11. a a 4. 12. b b 3. 13. b b 2. 14. a a 1.

Table 43 Pleadean 1: Sequence Model: Symmetry within Model Halves

lower half 1. a a 2. b b 3. b b 4. a a 5. a b 6. b b 7. b a ↺ 8. a b 9. b b 10. b a 11. a a 12. b b 13. b b 14. a a

! 97

This analysis of the equivalents of the two Pleadean sonnet traditions under

consideration shows that while the relationship between the bottom and top halves

of each model is chiral and symmetrical, the development within the Pleadean 2

sequence model is uniquely so as its simplified rhyme scheme is developed only

once in each of the model’s halves. Its development twice within each half of the

Pleadean 1 sequence model, on the other hand, allows for symmetry in an additional

dimension, as well as chirality, to be included in the one model. Thus, between them,

these two models, taking the two simplified Pleadean rhyme schemes as their

starting point, permit with maximum economy a minimal representation of chirality

and symmetry, qualities traceable back to the one difference in the conventional

representation of their rhyme schemes highlighted in Table 30 above. 3

It perhaps needs emphasizing that exclusively leftwards or rightwards

binary development from a start sequence or centre array is inherently chiral, and

that it is these specific directionalities that are able to bring out the symmetrical and

chiral aspects of the simplified rhyme schemes. To underscore this point, the two

other possible initial developments from the Pleadean start sequences, upwards left

and downwards right, and downwards left and upwards right, result merely in the

repeated symmetrical development of the Pleadean’s simplified rhymes schemes.

This is shown by the elements in bold in Tables 44 and 45 below which are

developed from the Pleadean 1 and 2 start sequences. In the tables, upwards left and

downwards right development, being simply the reverse of an upwards right and

downwards left development, stands as a proxy for it.

p. 773

! 98

Table 44 Pleadean 1: Sequence Model: Leftwards and Rightwards Directionality

...

(b a a b b a b b a b b a a b) = 3. (b b a a b b a b b a b b a a) = 2. (a b b a a b b a b b a b b a) = 1.

a a b b a a b b a b b a b b 14. b a a b b a a b b a b b a b 13. b b a a b b a a b b a b b a 12. a b b a a b b a a b b a b b 11. b a b b a a b b a a b b a b 10. b b a b b a a b b a a b b a 9. a b b a b b a a b b a a b b 8. b a b b a b b a a b b a a b 7. b b a b b a b b a a b b a a 6. a b b a b b a b b a a b b a 5. a a b b a b b a b b a a b b 4. b a a b b a b b a b b a a b 3. b b a a b b a b b a b b a a 2. ↖ a b b a a b b a b b a b b a 1. * a a b b a a b b a b b a b b 2. ↘ b a a b b a a b b a b b a b 3. b b a a b b a a b b a b b a 4. a b b a a b b a a b b a b b 5. b a b b a a b b a a b b a b 6. b b a b b a a b b a a b b a 7. a b b a b b a a b b a a b b 8. b a b b a b b a a b b a a b 9. b b a b b a b b a a b b a a 10. a b b a b b a b b a a b b a 11. a a b b a b b a b b a a b b 12. b a a b b a b b a b b a a b 13. b b a a b b a b b a b b a a 14.

(a b b a a b b a b b a b b a) = 1. (b a b b a a b b a b b a b b) = 2. (b b a b b a a b b a b b a b) = 3.

...

! 99

Table 45 Pleadean 2: Sequence Model: Leftwards and Rightwards Directionality

... (b a a b b a b b a b a b a b) = 3. (b b a a b b a b b a b a b a) = 2. (a b b a a b b a b b a b a b) = 1.

b a b b a a b b a b b a b a 14. a b a b b a a b b a b b a b 13. b a b a b b a a b b a b b a 12. a b a b a b b a a b b a b b 11. b a b a b a b b a a b b a b 10. b b a b a b a b b a a b b a 9. a b b a b a b a b b a a b b 8. b a b b a b a b a b b a a b 7. b b a b b a b a b a b b a a 6. a b b a b b a b a b a b b a 5. a a b b a b b a b a b a b b 4. b a a b b a b b a b a b a b 3. b b a a b b a b b a b a b a 2. ↖ a b b a a b b a b b a b a b 1. * b a b b a a b b a b b a b a 2. ↘ a b a b b a a b b a b b a b 3. b a b a b b a a b b a b b a 4. a b a b a b b a a b b a b b 5. b a b a b a b b a a b b a b 6. b b a b a b a b b a a b b a 7. a b b a b a b a b b a a b b 8. b a b b a b a b a b b a a b 9. b b a b b a b a b a b b a a 10. a b b a b b a b a b a b b a 11. a a b b a b b a b a b a b b 12. b a a b b a b b a b a b a b 13. b b a a b b a b b a b a b a 14.

(a b b a a b b a b b a b a b) = 1. (b a b b a a b b a b b a b a) = 2. (a b a b b a a b b a b b a b) = 3.

...

! 100

2.3.8 Conclusion

The Pleadean sonnet traditions bring symmetry and chirality within what

Seamus Heaney (1980) in his essay Feeling into Words, cited by Vendler (1998, p.

8), refers to as the jurisdiction of poetic form: The sequence models with leftwards

directionality reveal the latent symmetrical and chiral properties of the conventional

Pleadean rhyme schemes. The two Pleadean traditions considered separately are

therefore only partially appreciated, rather more so when viewed as counterparts.

The only equivalents of the sonnet’s formal characteristics to be developed

in the sequence models are, however, the rhyme scheme and sonnet length. The

Pleadean sequence and array models, therefore, only partly coincide, and it still

remains to be shown how array model results might be fully developed indepen-

dently when directionality changes are involved. To seek an answer to this question,

discussion now turns to the final sonnet tradition to be considered in the inquiry, the

Shakespearean.

! 101

2.4 Shakespearean Tradition

2.4.0 Introduction

In seeking an approach that relates equivalents of the formal

characteristics of the Shakespearean sonnet independently of the array models, I

shall adopt a disarmingly straightforward principle from the field called by Gell-

Mann ‘plectics’, namely, that complexity evolves from simplicity (Gell-Mann,

1996, p. 3). In applying this principle, I shall term the array models discussed so

far complex to the extent that they involve directionality changes. I shall then

infer that these complex models are derived from simpler models without

directionality changes. From this, I shall further infer that these simpler models

are part of a more general binary expansion pattern, the origin and evolution of

which offers an alternative approach, independent of the array models, to the

modelling of the Shakespearean sonnet form.

2.4.1 Simplified Rhyme Scheme

Kircher (p. 415) notes the following rhyme scheme for the

Shakespearean sonnet: a b a b c d c d e f e f g g

In simplified form, but retaining stanzaic markers for the moment, this

gives the equivalent of an alternating rhyme scheme with a final rhymed couplet:

a b a b a b a b a b a b a a

The absence of contrasting embracing and alternating rhyme equivalents suggests

that a centre array rather than a centre matrix suffices to develop the array model.

!102

2.4.2 Centre Array

Analysis and testing, following the same procedure as in previous

models, indicates the array ‘a b b a b’ with leftwards and rightwards directionality

as a centre array presumed to satisfy the formal conditions of the Shakespearean

sonnet. The detailed workings may be referred to in Appendix E. However, rather

than, as with previous models, straightaway defining rules to develop this centre

array, in order to seek an independent approach based on the principle of

complexity evolving from simplicity, let a simple array model first be constructed

by developing the centre array with no changes in directionality.

2.4.3 Simple Array and Triangle Models

The resulting simple array model appears in Table 46 below.

Immediately following it in Table 47, a second, alternative model to be termed

simple triangle model is developed without directionality changes. Its centre

sequence is composed of the simple array model's leftmost array elements and

centre sequence. It can be seen from the juxtaposing of the two models in Table 48

below that their results are, not unexpectedly, identical. Now, as it is in principle

possible to reverse this procedure, that is, to derive the simple array model from

the simple triangle model, it is deduced that both are identical aspects of the same

binary expansion, differing only in their alternative methods of development.

!103

Table 46 Simple Shakespearean Array Model

a b a b b 1. ↗ ↗ ↗ ↗ b a b b a 2. ↗ ↗ ↗ ↗ a b b a b 3. ↗ ↗ ↗ ↗ b b a b a 4. ↗ ↗ ↗ ↗ b a b a b 5. ↗ ↗ ↗ ↗ a b a b b 6. ↗ ↗ ↗ ↗ b a b b a 7. ↗ ↗ ↗ ↗ ↗ ✶ a b b a b 8. ↘ ↘ ↘ ↘↘ b a b b a 9. ↘ ↘ ↘↘ a b a b b 10. ↘ ↘ ↘↘ b a b a b 11. ↘ ↘ ↘↘ b b a b a 12. ↘ ↘ ↘↘ a b b a b 13. ↘ ↘ ↘↘ b a b b a 14. ↘ ↘ ↘↘ a b a b b 15.

!104

Table 47 Simple Shakespearean Triangle Model: Binary Expansion

Steps 1–9. (Steps 10–11 are overleaf).

a a a b a a b a b a 1. a 2. a b 3. a b a 4. a b a b a a b a b a a a b a

a a a b a a b a b a a b a b a a b a b a b a a b a b a b a b b a b a b a b a b b a b a b b a 5. a b a b b 6. a b a b b a 7. a b a b b a b a b a b a b a b b a b a b b a a b a a b a b a b b b b a b a b a a b a b a a b a b a a a b a a a a b a b a b a a b a a b a b a b a b a b a b b a b a b b a b a b b a a b a b b a a b a b b a b a b a b b a b a b a b b a b a 8. a b a b b a b a 9. a b a b b a b a b a b a b b a b a b a b b a b a a b a b b a a b a b b a b a b a b b a b a b b a a b a b a b a b b a b a a b a b a b a b a a a b a

!105

Table 47 (cntd.) Simple Shakespearean Triangle Model: Binary Expansion.

Steps 10–11. (Step 12 is overleaf).

a a a b a b a b a a b a a b a b a b a b a b a b b a b a b b a b a b b a a b a b b a a b a b b a b a b a b b a b a b a b b a b a a b a b b a b a a b a b b a b a b a b a b b a b a b a b a b b a b a b b 10. a b a b b a b a b b 11. a b a b b a b a b b a a b a b b a b a b a b a b b a b a b b a b a b b a b a a b a b b a b a b a b a b b a b a b a b b a b a a b a b b a a b a b b a b a b a b b a b a b b a a b a b a b a b b a b a a b a b a b a b a a a b a

!106

Table 47 (cntd.) Simple Shakespearean Triangle Model: Binary Expansion.

Step 12.

a a b a b a Simple Triangle Model a b a b

a b a b b a b a b b 1. a b a b b a b a b b a 2. a b a b b a b a b b a b 3. a b a b b a b a b b a b a 4. a b a b b a b a b b a b a b 5. a b a b b a b a b b a b a b b 6. a b a b b a b a b b a b a b b a 7. a b a b b a b a b b a b a b b a b 8. a b a b b a b a b b a b a b b a 9. a b a b b a b a b b a b a b b 10. a b a b b a b a b b a b a b 11. a b a b b a b a b b a b a 12. a b a b b a b a b b a b 13. a b a b b a b a b b a 14. a b a b b a b a b b 15.

a b a b a b a a b a

!107

Table 48 Identical Simple Shakespearean Triangle and Array Models

Array Model Triangle Model (Table 46) (Table 47)

a b a b b a b a b b 1. b a b b a b a b b a 2. a b b a b a b b a b 3. b b a b a b b a b a 4. b a b a b b a b a b 5. a b a b b a b a b b 6. ↗ b a b b a b a b b a 7. ✶ a b b a b a b b a b 8. ↘ b a b b a b a b b a 9. a b a b b a b a b b 10. b a b a b b a b a b 11. b b a b a b b a b a 12. a b b a b a b b a b 13. b a b b a b a b b a 14. a b a b b a b a b b 15. Let these simple Shakespearean array and triangle models without direc-

tionality changes now be distinguished from the more complex Shakespearean

array and triangle models with directionality changes.

2.4.4 Complex Array Model: Step-by-Step Description

Having shown that the simple Shakespearean array and triangle models

are identical, to demonstrate two independent approaches to the construction of a

complex Shakespearean model two conditions need to be satisfied. First, a

complex array model, developed from a centre array, must be able to relate and

describe equivalents of the formal characteristics of the Shakespearean sonnet.

Second, a complex triangle model that is identical to the complex array model

must be able to be constructed from the simple triangle model described above.

!108

Assuming these conditions fulfilled, it follows, first, that the complex triangle

model also relates and describes equivalents of the formal characteristics of the

Shakespearean sonnet and, second, that both the complex array and triangle

models may be considered identical aspects of the same binary expansion, only

developed independently. In this way, two independent approaches to the

construction of a Shakespearean model with directionality changes may be

demonstrated.

Turning to the demonstration itself, to construct a complex array model

that relates the equivalents of the formal characteristics of the Shakespearean

sonnet, once again understood as five conditions to be satisfied simultaneously,

there are three rules for development of the centre array, ‘a b b a b’:

Rule 1. From the centre array, arrays develop simultaneously

upwards and downwards to the right;

Rule 2. Development of the array ‘a b a b b’ causes a change in

directionality from right to left;

Rule 3. Symbolic repetition of successive end-array elements halts

development and completes the model.

There now follows a step-by-step description of the application of these

rules. For clarity of presentation, the development with leftwards directionality is

omitted. Its final model is shown alongside the model with rightwards

development in Table 50 further below.

!109

Steps 1 & 2 Development of arrays 7 & 8 and 6 & 10

6. a b a b b ↗↗↗↗ 7. b a b b a ↗↗↗↗ 8. a b b a b ↘↘↘↘ 9. b a b b a ↘↘↘↘ 10. a b a b b

Development begins, according to Rule 1, with the simultaneous creation

of array pairs upwards and downwards to the right from array 8, the centre array.

Step 3 Development of arrays 5 & 11

5. b a b b a ↖↖↖↖ 6. a b a b b ↗↗↗↗ 7. b a b b a ↗↗↗↗ 8. a b b a b ↘↘↘↘ 9. b a b b a ↘↘↘↘ 10. a b a b b ↙↙↙↙ 11. b a b b a

With the development of the array ‘a b a b b’ in arrays 6 & 10, direction-

ality changes from right to left, according to Rule 2.

!110

Steps 4, 5, 6 & 7: Development of arrays 4–1 & 12–15

1. a b a b b ↖↖↖↖ 2. b a b a b ↖↖↖↖ 3. b b a b a ↖↖↖↖ 4. a b b a b ↖↖↖↖ 5. b a b b a ↖↖↖↖ 6. a b a b b ↗↗↗↗ 7. b a b b a ↗↗↗↗ 8. a b b a b ↘↘↘↘ 9. b a b b a ↘↘↘↘ 10. a b a b b ↙↙↙↙ 11. b a b b a ↙↙↙↙ 12. a b b a b ↙↙↙↙ 13. b b a b a ↙↙↙↙ 14. b a b a b ↙↙↙↙ 15. a b a b b

Development continues leftwards. However, with the repetition of suc-

cessive end-array elements in Arrays 2 & 1 and 14 & 15, array development is

halted and the model complete, according to Rule 3. As in the previous models,

this model has fifteen arrays as it comprises not one, but two identical fourteen

!111

array sub-models, arrays 2–15 and 14–1. Juxtaposing these sub-models, as shown

in Table 49, makes their identity manifest:

Table 49 Identical Shakespearean Sub-Models

2. b a b a b b a b a b 14. ↖↖↖↖ ↖↖↖↖ 3. b b a b a b b a b a 13. ↖↖↖↖ ↖↖↖↖ 4. a b b a b a b b a b 12. ↖↖↖↖ ↖↖↖↖ 5. b a b b a b a b b a 11. ↖↖↖↖ ↖↖↖↖ 6. a b a b b a b a b b 10. ↗↗↗↗ ↗↗↗↗ 7. b a b b a b a b b a 9. ↗↗↗↗ ↗↗↗↗ 8. a b b a b a b b a b 8. ↘↘↘↘ ↘↘↘↘ 9. b a b b a b a b b a 7. ↘↘↘↘ ↘↘↘↘ 10. a b a b b a b a b b 6. ↙↙↙↙ ↙↙↙↙ 11. b a b b a b a b b a 5. ↙↙↙↙ ↙↙↙↙ 12. a b b a b a b b a b 4. ↙↙↙↙ ↙↙↙↙ 13. b b a b a b b a b a 3. ↙↙↙↙ ↙↙↙↙ 14. b a b a b b a b a b 2. ↙↙↙↙ ↙↙↙↙ 15. a b a b b a b a b b 1.

Combining the two sub-models gives the complex Shakespearean array

model of fourteen arrays, as shown on the right-hand side of Table 50 below. On

the left-hand side, in its completed form, is its counterpart with leftwards

development.

!112

Table 50 Complex Shakespearean Array Models: Leftwards and Rightwards Developments

a b a b b b a b a b ↗ ↗ ↗ ↗ ↖ ↖ ↖ ↖ b a b b a b b a b a ↗ ↗ ↗ ↗ ↖ ↖ ↖ ↖ a b b a b a b b a b ↗ ↗ ↗ ↗ ↖ ↖ ↖ ↖ b b a b a b a b b a

b a b a b a b a b b ↖ ↖ ↖ ↖ ↗ ↗ ↗↗ b b a b a b a b b a ↖ ↖ ↖ ↖ ↗↗ ↗↗ a b b a b a b b a b ↙ ↙ ↙ ↙ ↘ ↘ ↘↘ b b a b a b a b b a

b a b a b a b a b b ↘ ↘ ↘ ↘ ↙ ↙ ↙ ↙ b b a b a b a b b a ↘ ↘ ↘ ↘ ↙ ↙ ↙ ↙ a b b a b a b b a b ↘ ↘ ↘ ↘ ↙ ↙ ↙ ↙ b a b b a b b a b a

a b a b b b a b a b ↘ ↘ ↘ ↘ ↙ ↙ ↙↙ b a b a b a b a b b

!113

2.4.5 Assessment

Are, then, the formal characteristics of the Shakespearean sonnet

described by this array model? The fourteen arrays are deemed equivalent to the

sonnet length condition of fourteen lines. Redundancy is triggered by successive

end-array elements with the same symbolic and cyclical properties. Changes in

directionality at arrays 5 and 9 divide the first twelve arrays into three groups of

four arrays equivalent to the three quatrains of a Shakespearean sonnet, whilst the

final two arrays with similar end-array elements represent the equivalent of the

couplet, thus fulfilling the condition for stanzaic form. With all other end-array

elements alternating, the condition for the Shakespearean rhyme scheme is also

satisfied. As to the volta equivalent, the point of greatest contrastive development

before the disaggregation and combination of the sub-models to create the final

complex array models is at the centre array, array 8, as may be seen in Steps 4, 5,

6 & 7 above. However, in the final complex array model array 8 becomes array 7,

so that by the criterion of greatest contrastive flows, the volta equivalent does not

occur in line 8, but in line 7. This may appear to be a limitation of the model,

though, when read linearly, that is, conventionally from top to bottom, array 8 is

the first array in the model developed after the point of greatest contrastive flows.

As such, it is deemed to represent the equivalent of the volta. With the five

elements in each array able to accommodate the equivalent of the five stresses of

the dominant Shakespearean sonnet metre, the iambic pentameter, the isometry

condition is also satisfied. It seems, on balance, therefore, reasonable to conclude

that the models satisfactorily describe and relate equivalents of the formal

!114

characteristics of the Shakespearean sonnet, fulfilling the first condition for the

construction of independent complex models.

2.4.6 Complex Triangle Model: Step-by-step Description

As remarked above, fulfilment of the second condition requires, first, that

a complex triangle model's results be the same as those of its complex array model

counterpart and, second, that the complex triangle model be developed from its

simple triangle model. Only then will it have been shown that the same complex

models can be developed independently both from within a binary expansion and

from a centre array.

To show how a simple triangle model may be developed into a complex

triangle model with results identical to those of the complex array model, let

arrays 1 & 15 of the simple triangle model shown in Table 47 above be the

starting point for its development. It is, of course, possible to construct the

complex triangle model by beginning development from the centre sequence of

the simple triangle model. However, to make it easier to see the changes in

directionality in the complex triangle model and to emphasize its independent

approach to producing the same results as the complex array model, I have chosen

to start from arrays 1 and 15 of the simple triangle model. That these arrays

represent the start of the centre sequence of the simple triangle model, may be

seen in Step 12 of the triangle model development in Table 47. There are three

rules for development of arrays 1 &15:

!115

Rule 1. Arrays develop simultaneously downwards and upwards

to the right from arrays 1 & 15, respectively, of the simple

triangle model;

Rule 2. Symbolic repetition of arrays 1 & 15 leads to a change in

directionality;

Rule 3. When the centre array, array 8, is developed, development

is halted and the model complete.

A stepwise construction of the complex triangle model now follows.

Step 1 Arrays 1 & 15

1. a b a b b

a b a b b

15. a b a b b

As just remarked, the underlined array on the left represents the first five

elements of the simple triangle model’s centre sequence, whilst arrays 1 & 15

represent these elements’ development within the simple triangle model, as shown

in the models on the left- and right-hand sides of Table 47, respectively. As there

can have been no changes in directionality at this point in the model’s

development, arrays 1 & 15 of both the simple and complex triangle models are of

course identical.

!116


1. a b a b b ↘ ↘ ↘ ↘ 2. b a b a b …

14. b a b a b ↗ ↗ ↗ ↗ 15. a b a b b

In Step 2, the final elements in arrays 1 and 15, reading from the left,

become the first in the next arrays to be developed, the first becomes the second,

the second the third, and so forth, according to the principle of cyclicity.

Step 3 Arrays 3–6 & 13–10

1. a b a b b ↘ ↘ ↘ ↘ 2. b a b a b ↘ ↘ ↘ ↘ 3. b b a b a ↘ ↘ ↘ ↘ 4. a b b a b ↘ ↘ ↘ ↘ 5. b a b b a ↘ ↘ ↘ ↘ 6. a b a b b

...

10. a b a b b ↗ ↗ ↗ ↗ 11. b a b b a ↗ ↗ ↗ ↗ 12. a b b a b ↗ ↗ ↗ ↗ 13. b b a b a ↗ ↗ ↗ ↗ 14. b a b a b ↗ ↗ ↗ ↗ 15. a b a b b

!117

With the symbolic repetition of array ‘a b a b b’ in arrays 6 and 10,

directionality changes, according to Rule 2, from right to left.

Step 4 Arrays 7 & 9

1. a b a b b ↘ ↘ ↘ ↘ 2. b a b a b ↘ ↘ ↘ ↘ 3. b b a b a ↘ ↘ ↘ ↘ 4. a b b a b ↘ ↘ ↘ ↘ 5. b a b b a ↘ ↘ ↘ ↘ 6. a b a b b ↙ ↙ ↙ ↙ 7. b a b b a

8.

9. b a b b a ↖ ↖ ↖ ↖ 10. a b a b b ↗ ↗ ↗ ↗ 11. b a b b a ↗ ↗ ↗ ↗ 12. a b b a b ↗ ↗ ↗ ↗ 13. b b a b a ↗ ↗ ↗ ↗ 14. b a b a b ↗ ↗ ↗ ↗ 15. a b a b b

The array ‘a b a b b’ not, of course, being reproduced in arrays 7 and 9,

development continues leftwards, according to Rule 2.

!118

Step 5 Array 8

1. a b a b b ↘ ↘ ↘ ↘ 2. b a b a b ↘ ↘ ↘ ↘ 3. b b a b a ↘ ↘ ↘ ↘ 4. a b b a b ↘ ↘ ↘ ↘ 5. b a b b a ↘ ↘ ↘ ↘ 6. a b a b b ↙ ↙ ↙ ↙ 7. b a b b a ↙ ↙ ↙ ↙ 8. a b b a b ↖ ↖ ↖ ↖ 9. b a b b a ↖ ↖ ↖ ↖ 10. a b a b b ↗ ↗ ↗ ↗ 11. b a b b a ↗ ↗ ↗ ↗ 12. a b b a b ↗ ↗ ↗ ↗ 13. b b a b a ↗ ↗ ↗ ↗ 14. b a b a b ↗ ↗ ↗ ↗ 15. a b a b b

With the development of the centre array, the model is complete,

according to Rule 3. The final model comprises two sub-models, arrays 2–15 and

14–1, each of which represents the final complex triangle model. In Table 51

below, the complex array model from the right-hand side of Table 50 is compared

with the complex triangle model and seen to be identical with it, the only

!119

immaterial difference between them being that their flows move in opposite

directions due to their developments’ different points of departure.

Table 51 Shakespearean Complex Array and Triangle Models

Complex Array Model Complex Triangle Model

2. b a b a b b a b a b ↖ ↖ ↖↖ ↘ ↘ ↘ ↘ 3. b b a b a b b a b a ↖ ↖ ↖↖ ↘ ↘ ↘ ↘ 4. a b b a b a b b a b ↖ ↖ ↖↖ ↘ ↘ ↘ ↘ 5. b a b b a b a b b a ↖ ↖ ↖↖ ↘ ↘ ↘ ↘ 6. a b a b b a b a b b ↗ ↗ ↗ ↗ ↙ ↙ ↙ ↙ 7. b a b b a b a b b a ↗ ↗ ↗ ↗ ↙ ↙ ↙ ↙ 8. a b b a b a b b a b ↘ ↘ ↘ ↘ ↖ ↖ ↖ ↖ 9. b a b b a b a b b a ↘ ↘ ↘ ↘ ↖ ↖ ↖ ↖ 10. a b a b b a b a b b ↙ ↙ ↙ ↙ ↗ ↗ ↗ ↗ 11. b a b b a b a b b a ↙ ↙ ↙ ↙ ↗ ↗ ↗ ↗ 12. a b b a b a b b a b ↙ ↙ ↙ ↙ ↗ ↗ ↗ ↗ 13. b b a b a b b a b a ↙ ↙ ↙ ↙ ↗ ↗ ↗ ↗ 14. b a b a b b a b a b ↙ ↙ ↙ ↙ ↗ ↗ ↗ ↗ 15. a b a b b a b a b b

!120

2.4.7 Conclusion and Retrospective

It has been shown to be theoretically possible within a binary expansion

to transform a simple triangle model into a complex triangle model identical to the

complex array model developed from a centre array. As the complex array model

is able to describe satisfactorily how equivalents of the formal characteristics of

the Shakespearean sonnet are related by the principle of centred form, it follows

that the complex triangle model also does so. Furthermore, as each model is

developed independently from different starting points, they provide a means to

cross-check each other’s results and mitigate the risk of error and bias in their

results. By this means, the complex Shakespearean models’ results are presumed

more robust.

The two ideas, first, of a sonnet pattern developing as part of a binary

expansion and, second, complexity evolving from simplicity also help to clarify

the relationship between the two Early Italian array models (EIM). As discussed

above, their mirrored flows, as seen in Table 20, suggested that the models might

be part of a broader pattern. As their elements were by inspection clearly not

symbolically mirrored, however, the question arose as to just how they were

related, how their results might be corroborated and whether one was a better

representation of its tradition than the other. To seek answers to these questions,

let arrays 5 through 11 of the Early Italian model (RHS) of Table 20 now be

reversed so that, leaving aside arrays 1 and 15 for the moment, the model has no

directionality changes. What, as a result, amounts to a Simple Early Italian (RHS)

array model is seen to be identical with the Simple Shakespearean array and

triangle models, as shown below in Table 52.

!121

Table 52 Related Early Italian and Shakespearean Models 1

EIM (RHS) EIM (RHS) Simple Shakespearean arrays 5–11 Array & Triangle reversed Models

1. a b b a b a b b a b a b a b b ↖ ↖ ↖↖ ↖ ↖ ↖↖ ↗ ↗ ↗↗ 2. b a b b a b a b b a b a b b a ↗ ↗ ↗↗ ↗ ↗ ↗↗ ↗ ↗ ↗↗ 3. a b b a b a b b a b a b b a b ↗ ↗ ↗↗ ↗ ↗ ↗↗ ↗ ↗ ↗↗ 4. b b a b a b b a b a b b a b a 5. b a b a b b a b a b b a b a b ↖ ↖ ↖↖ ↗ ↗ ↗↗ ↗ ↗ ↗↗ 6. b b a b a a b a b b a b a b b ↖ ↖ ↖ ↖ ↗ ↗ ↗↗ ↗ ↗ ↗↗ 7. a b b a b b a b b a b a b b a ↖ ↖ ↖↖ ↗ ↗ ↗↗ ↗ ↗ ↗↗ 8. b a b b a a b b a b a b b a b

9. a b b a b b a b b a b a b b a ↙↙ ↙ ↙ ↘↘ ↘ ↘ ↘↘ ↘ ↘ 10. b b a b a a b a b b a b a b b ↙↙ ↙ ↙ ↘↘ ↘ ↘ ↘↘ ↘ ↘ 11. b a b a b b a b a b b a b a b

12. b b a b a b b a b a b b a b a ↘↘ ↘↘ ↘↘ ↘↘ ↘↘ ↘ ↘ 13. a b b a b a b b a b a b b a b ↘ ↘ ↘↘ ↘ ↘ ↘↘ ↘↘ ↘ ↘ 14. b a b b a b a b b a b a b b a ↙↙ ↙ ↙ ↙↙ ↙ ↙ ↘↘ ↘ ↘ 15. a b b a b a b b a b a b a b b

It follows, ignoring arrays 1 and 15 for a moment longer, that the simple

RHS Early Italian and Shakespearean array and triangle models share the same

centre array in the same centre sequence. The elements of the Early Italian

!122

model’s centre sequence may be deduced from its simple array model, that is, the

model in the middle of Table 52 above, by letting the initial elements of its arrays

7–2 or 9–14, ‘b a b b a b’, be arranged in order to the left of its centre array, ‘a b b

a b’. This permits, as shown in Table 53, a comparison of the Italian (RHS) and

Shakespearean centre sequences, a comparison which shows them to be almost

identical.

Table 53 EIM (RHS) and Shakespearean Models: Shared Centre Sequence

( ) b a b b a b a b b a b EIM (RHS)

a b a b b a b a b b a b Shakespearean

The lack of identity is due to the bracketed, missing element on the left-

hand side of the EIM (RHS) sequence. That the element is missing, is due to the

difference between the first and fifteenth arrays of the Early Italian and Shake-

spearean models, which in turn is due to the differing assumptions underlying the

models’ redundancy rules: The Early Italian models’ rules are based on the

working model’s rules that were constructed within the logic of a fourteen-array

model, whereas the Shakespearean model’s rules are developed within the logic of

fifteen-array array and triangle models. Now, due to its two independent

approaches to model construction, the results of the Shakespearean model are

presumed to be more error-resistant than those of the Early Italian models. Its

results shall therefore be preferred. The desirable quality in a working model of

being productively incomplete is reflected accordingly in the rest of the discussion

by foregoing the directionality change in array 2 of the Early Italian array models.

!123

This leads to the development of the array ‘a b a b b’ in its arrays 1 and 15 and

identity with the simple Shakespearean array and triangle models.

Turning now to the other Early Italian array model on the left-hand side

(LHS) of Table 20, as it shares mirrored flows with the RHS model and as the

RHS model can be related to the simple Shakespearean models, it follows that all

three models share the same centre sequence. The different centre array of the

EIM (LHS) model suggests merely that it starts development at a different point

along the centre sequence than the other models. This is shown in Table 54.

Table 54

EIM (LHS) and Shakespearean Models: Shared Centre Sequence

a b a b b a b a b b a b a EIM (LHS)

a b a b b a b a b b a b Shakespearean

The only point within the original centre sequence of the EIM (LHS)

from which the centre array ‘b b a b a’ might be developed is that beginning at the

fourth element from the left-hand side. The underlined version of the EIM (LHS)

centre array is preferred for comparative purposes, however, as it is in principle

the same, and more clearly shows the relationship between the Early Italian LHS

model and the simple, Shakespearean and Early Italian RHS models. The detailed

relationship is shown in Table 55 below. As the LHS model is displaced by one

element with respect to the Shakespearean and the Early Italian RHS models, in

the table’s second column the final element in each of its arrays is moved to the

head of the array to compensate. These elements are underlined in array 1, by way

of example. When the away flows of the model’s arrays 5–2 and 11–14 are then

!124

reversed, to undo its complexity, so to speak, the Early Italian LHS model is seen

to be identical with the simple Shakespearean and Early Italian RHS models.

Table 55 Related Early Italian and Shakespearean Models 2

EIM (LHS) EIM (LHS) EIM (LHS) Simple final element mirrored Shakespearean & heads array EIM (RHS)

1. b a b a b b b a b a a b a b b a b a b b ↖ ↖ ↖↖ ↖ ↖ ↖↖ ↗ ↗ ↗ ↗ ↗ ↗ ↗↗ 2. b b a b a a b b a b b a b b a b a b b a ↖ ↖ ↖↖ ↖ ↖ ↖↖ ↗ ↗ ↗↗ ↗ ↗ ↗↗ 3. a b b a b b a b b a a b b a b a b b a b ↖ ↖ ↖↖ ↖ ↖ ↖↖ ↗ ↗ ↗↗ ↗ ↗ ↗↗ 4. b a b b a a b b a b b b a b a b b a b a 5. a b a b b b a b a b b a b a b b a b a b ↗ ↗ ↗↗ ↗ ↗ ↗↗ ↗ ↗ ↗↗ ↗ ↗ ↗↗ 6. b a b b a a b a b b a b a b b a b a b b ↗ ↗ ↗↗ ↗ ↗ ↗↗ ↗ ↗ ↗↗ ↗ ↗ ↗↗ 7. a b b a b b a b b a b a b b a b a b b a ↗ ↗ ↗↗ ↗ ↗ ↗↗ ↗ ↗ ↗↗ ↗ ↗ ↗↗ 8. b b a b a a b b a b a b b a b a b b a b 9. a b b a b b a b b a b a b b a b a b b a ↘↘ ↘ ↘ ↘↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘↘ ↘ ↘ 10. b a b b a a b a b b a b a b b a b a b b ↘↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘↘ ↘ ↘ 11. a b a b b b a b a b b a b a b b a b a b 12. b a b b a a b a b b b b a b a b b a b a ↙↙↙ ↙ ↙ ↙ ↙ ↙ ↘↘ ↘↘ ↘ ↘ ↘ ↘ 13. a b b a b b a b b a a b b a b a b b a b ↙↙↙ ↙ ↙↙ ↙ ↙ ↘ ↘↘↘ ↘↘ ↘ ↘ 14. b b a b a a b b a b b a b b a b a b b a ↙↙↙ ↙ ↙ ↙↙ ↙ ↘↘ ↘ ↘ ↘↘ ↘ ↘ 15. b a b a b b b a b a a b a b b a b a b b

!125

It may be concluded, therefore, first, that the Early Italian and Shake-

spearean array models are related by the same centre sequence and only differ in

their final construction due to differences in starting points, the initial

directionality of their developments and the arrays in which changes in direc-

tionality are introduced. Second, the results of both Early Italian models are, after

adjustments in arrays 1 and 15, corroborated by the simple Shakespearean array

and triangle models. Third, as may be seen perhaps most clearly in Step 5 of the

development of the complex Shakespearean triangle model, arrays 7–1 and 9–15

are non-superposable yet foldable mirror images of each other about array 8. The

model's halves are therefore chiral in one dimension and symmetrical in three, a

finding which also holds for the Early Italian models, as perhaps most easily

observable in Table 18. Differently from the Pleadean models, however, these

chiral and symmetrical properties are immanent within the sonnet patterns

themselves, and are masked only to the extent of their transformation from

fifteen- into fourteen-array models. It seems reasonable to conclude, therefore,

that both symmetry and chirality inhere in the Early Italian and Shakespearean

sonnet patterns. Finally, neither Early Italian array model is a better representation

of the Early Italian tradition than the other as each simply represents different yet

related developments from a common centre sequence.

If the principle of centred form is thus able to describe and relate equival-

ents of the formal characteristics of the sonnet within and across sonnet traditions

in theory, it still remains to be shown whether it can be applied in practice. In the

third part of the inquiry that now follows, practical evidence for the claim is

!126

provided by a brief introduction to centred writing by way of the author’s sonnet

cycle, Memorial Day: the Unmaking of a Sonnet.

!127

Part 3: Centred Writing

3.0 Introduction

That this part of the inquiry does not start out from another traditional

sonnet form might seem presumptuous. It does serve to show, however, that creative

use may be made of traditional sonnet theory. The sonnet pattern underpinning

Memorial Day: the Unmaking of a Sonnet was not derived from the formal charac-

teristics of a particular sonnet tradition, but rather developed to try out whether

centred form was an idea capable of relating the formal characteristics of the sonnet

in practice. Chronologically, the Memorial Day sonnet pattern was developed in

parallel with Part 1 and is based on the same first principles. Initially, a centre matrix

was developed by applying the principle of cyclicity to three types of elements.

These elements then materialized as a distribution of key vowels and accentuations

to create the final sonnet pattern. In presenting the practical evidence for the claim,

discussion begins, therefore, with a description of the rationale for the centre matrix

of Sonnet 8 of Memorial Day, the central sonnet of the cycle, and the first written. 1

3.1 Memorial Day: The Unmaking of a Sonnet

3.1.1 Sonnet Cycle Centre Matrix: Sonnet 8

The internal elements of the Memorial Day centre matrix are arranged into

two diagonally opposed element pairs of three different elements, one pair with

identical, the other with non-identical elements, as shown in Table 56 below:

The poems may be found in Appendix F.1

!128

Table 56 Memorial Day: Centre Matrix: Internal Elements

7. c a ↖ ↗ ↙ ↘ 8. b c

The arrows indicate that the elements in the upper and lower halves of the

model are to unfold in diagonally opposite directions away from its centre. The idea

behind this arrangement is to establish, from the outset, a simple, continuous cycle

of tension and resolution between elements. This tension, represented, prosaically

enough, by inequality in the number of different element types, here, two ‘c’s, one

‘a’ and one ‘b’, is resolved initially as the elements reach numerical equality in the

centre matrix, as shown in Table 57 below. Once set, these initial conditions ensure a

recurrence of tension and resolution between elements throughout the sonnet cycle.

Resolution is ultimately achieved with the development of the cycle's final sonnet

pattern, as described in the discussion of aggregation in section 3.1.6 further below.

Table 57 Memorial Day: Centre Matrix Buildup

Step 1: Internal Elements Frequency and Distribution

7. c a a b c

8. b c 1 1 2

!129

Step 2: Intermediary Elements Frequency and Distribution

7. b c a c a b c

8. a b c b 2 3 3

Step 3: External Elements Frequency and Distribution

7. a b c a c b a b c

8. c a b c b a 4 4 4

The rules for development of the centre matrix shown in Step 3 and a step-

by-step description of the array model’s construction now follow.

3.1.2 Rules for Array Development

There are three rules for array development in the Memorial Day model:

Rule 1. The halves of the centre matrix’s array pairs develop

simultaneously upwards and downwards in four diagonally opposite directions away

from the centre matrix;

Rule 2. Arrays develop without any change in directionality;

Rule 3. When a series of three consecutive arrays, symbolically and

cyclically with regard to its flows towards, is repeated,

redundancy enters the model, development is halted and the model complete.

A step-by-step description of the model’s construction now follows.

!130

3.1.3 Memorial Day Array Model: Step-by-step Description

Step 1 Centre Matrix

7. a b c a c b ↖ ↗ ↙ ↘ 8. c a b c b a

The rationale for the centre matrix is discussed above.

Step 2 Arrays 6 & 9

6. b c a b a c ↖↖ ↗ ↗ 7. a b c a c b

8. c a b c b a ↙↙ ↘↘ 9. a b c a c b

Development begins, according to Rule 1, with each half of the centre

matrix’s array pairs being simultaneously developed upwards and downwards

diagonally away from the centre matrix to create arrays 6 and 9.

!131


5. c a b c b a ↖ ↖ ↗ ↗ 6. b c a b a c ↖ ↖ ↗ ↗ 7. a b c a c b

8. c a b c b a ↙ ↙ ↘ ↘ 9. a b c a c b ↙ ↙ ↘ ↘ 10. b c a b a c

As there is, as yet, of course, no series repetition, development continues

according to Rule 2.

!132


4. a b c a c b ↖ ↖ ↗ ↗ 5. c a b c b a ↖ ↖ ↗ ↗ 6. b c a b a c ↖ ↖ ↗ ↗ 7. a b c a c b

8. c a b c b a ↙ ↙ ↘ ↘ 9. a b c a c b ↙ ↙ ↘ ↘ 10. b c a b a c ↙ ↙ ↘ ↘ 11. c a b c b a

The first series of three symbolically and, with respect to flows towards,

cyclically identical arrays, 6–4 and 9–11, is now complete. Development continues

according to Rule 2.

!133

Step 5 Arrays 3–1 & 12–14

1. a b c a c b ↖ ↖ ↗ ↗ 2. c a b c b a ↖ ↖ ↗ ↗ 3. b c a b a c ↖ ↖ ↗ ↗ 4. a b c a c b ↖ ↖ ↗ ↗ 5. c a b c b a ↖ ↖ ↗ ↗ 6. b c a b a c ↖ ↖ ↗ ↗ 7. a b c a c b

8. c a b c b a ↙ ↙ ↘ ↘ 9. a b c a c b ↙ ↙ ↘ ↘ 10. b c a b a c ↙ ↙ ↘ ↘ 11. c a b c b a ↙ ↙ ↘ ↘ 12. a b c a c b ↙ ↙ ↘ ↘ 13. b c a b a c ↙ ↙ ↘ ↘ 14. c a b c b a

The array series 3–1 and 12–14 being symbolic and cyclical repetitions of

the array series 6–4 and 9–11, respectively, redundancy enters the model, develop-

ment is halted and the model complete, according to Rule 3. The final array model is

presented in Table 58 below in a manner emphasizing how centred development

creates the equivalent of a traditional two quatrain, two tercet sonnet form.

!134

Table 58 Memorial Day: Array Model

Memorial Day Array Model

a b c a c b ↖ ↖ ↗ ↗ c a b c b a ↖ ↖ ↗ ↗ b c a b a c ↖ ↖ ↗ ↗ a b c a c b c a b c b a ↖ ↖ ↗ ↗ b c a b a c ↖ ↖ ↗ ↗ a b c a c b c a b c b a ↙ ↙ ↘ ↘ a b c a c b ↙ ↙ ↘ ↘ b c a b a c ↙ ↙ ↘ ↘ c a b c b a a b c a c b ↙ ↙ ↘ ↘ b c a b a c ↙ ↙ ↘ ↘ c a b c b a

!135

3.1.4 Assessment

The constant number of elements per array throughout the model represents

the equivalent of isometry. The volta equivalent is deemed to occur at the point of

greatest contrast between flows. In a linear reading of the model, as opposed to its

centred form construction, in which flows between arrays 7 & 8 are undefined, this

occurs in array 8, as shown in Table 59.

Table 59 Memorial Day: Array 8 as Volta Equivalent

7. a b c a c b ↘ ↘ ↙↙ 8. c a b c b a ↙↙ ↘↘ 9. a b c a c b

It might be argued, however, that array 9 is the better equivalent of the volta

as, in a linear reading, its towards and away flows contrast with the flows of arrays

1–7, whilst towards flows are undefined for array 8. As there is no contradiction

between this argument and Hobsbaum’s observation, noted at the start of the inquiry,

that the volta is “usually situated at the end of the octave or the beginning of the

sestet”, array 9, situated at the beginning of the sestet equivalent, is deemed to

represent the volta.

With array 9 marking the equivalent of the division into octave and sestet,

the model in Table 58 may be seen to represent the traditional Italian and French,

two quatrain, two tercet form: Overlapping, quasi-embracing rhyme pair equivalents

delimit the quatrains, just as identical groups of three arrays do the tercets.

Although, given that there is change from one end-array element to the next, the

equivalent of an alternating rhyme scheme is accommodated by the model, a

!136

principle that distributes key vowels and accentuation throughout the model’s arrays

is preferred, as described in the next section. Finally, the equivalent of fourteen line

sonnet length is determined by the limit between innovation and redundancy in array

series, as described in Rule 3. The model thus represents the sonnet characteristics

noted by Hobsbaum at the outset of the inquiry. As there is no doubt about the

design of the array model, there is no need for its results to be corroborated indepen-

dently by a triangle model, the development of which is therefore omitted.

3.1.5 Link between Sonnet Pattern and Sonnet Writing

The internal elements of the centre matrix serve to establish the relative

positions of the sonnet’s key vowels and accentuation. The cycle begins with the 2

three words: silent, pages and leaves. In Table 60 below, the bottom right ‘internal’ 3

‘c’ element of the Memorial Day centre matrix is replaced by the key vowel sound

of the word ‘silent’. Equivalently, the element ‘b’ assumes the key vowel of ‘leaves’

and the element ‘a’ that of ‘pages’. These three vowels are then distributed respect-

ively among the remaining elements in the matrix, as shown in Table 61 also below.

This idea occurred to me after remembering reading a vocalic analysis of Verlaine’s Il 2 pleure dans mon coeur (Chiss, Filliolet et Maingeneau, 1977, II, pp. 123–124) that presented itself during a later reading of Sylvester’s idea of Phonetic Syzygy (1870, p. 11).

These words were occasioned by a fall of light on the pages of a book I was reading one 3 Sunday in May 2009 in the garden of the Isabella Stewart Gardner Museum in Boston, MA, U.S.A.

!137

Table 60 Centre Matrix: Distribution of Key Vowels 1

Centre Matrix: Internal Elements key vowels

7. c a a b c 8. b c /eɪ/ /iː/ /aɪ./

‘pages’ ‘leaves’ ‘silent’

Table 61 Centre Matrix: Distribution of Key Vowels 2

variables key vowels

7. a b c a c b /eɪ/ /iː/ /aɪ./ pages /aɪ./ /iː/

8. c a b c b a /aɪ./ /eɪ/ leaves silent /iː/ /eɪ/

This schema then helped prompt the opening lines of the cycle, the lines 7 & 8 of

sonnet 8:

7. how fey, how free, the mitered pages mild do sheen

8. and shimmer and sway with leaves of silent beechen gray,

The key vowel sounds and their accompanying accentuation were then

distributed line by line throughout the remainder of the sonnet’s arrays as the writing

of the poem unfolded. The rule I adopted was not to stick rigidly to a particular

vowel sound, but rather to stray either by a very little or quite a lot therefrom

depending on the poetic possibilities that presented themselves. In Sonnet 8, about

two-thirds of the groups of three elements comprising each array conform to the key

vowel distribution described above.

!138

3.1.6 The Problem of Aggregation

One final aspect of Memorial Day: the Unmaking of a Sonnet lends support

to the claim of a centred form to sonnet writing: The method applied to combine

individual sonnets into a sonnet cycle. This problem is referred to by Spiller (1997)

as the problem of aggregation. In discussing the return to vogue of sonnet sequences

in nineteenth century English and American literature, he describes the difficulty as

follows:

There is, as we shall see, plenty of scope for originality. However, one problem always presents itself, in any age or place, because of the nature of the sonnet: the problem of aggregation into a whole of items that are also meaningful separately–a difficulty no other genre, in prose or verse, presents. (p. 20)

In Memorial Day, I address this problem by taking alternating pairs of

arrays from the central sonnet, Sonnet 8, for use as the centre matrices of the

subsequent sonnets in the cycle. Thus, as arrays 7 and 8 form the centre matrix of

sonnet 8, so sonnet 8’s arrays 6 and 7 form the centre matrix of sonnet 7, sonnet 8’s

arrays 8 and 9, the centre matrix of sonnet 9, and so forth, until the final sonnet to be

written, sonnet 1, avails itself for its centre matrix of arrays 14 and 1 from sonnet 8.

Apart from providing cohesiveness between the sonnet patterns in the cycle, this

approach has the formal advantage, with regard to the central sonnet, of exhausting

the principle of unfolding from the centre.

This order of writing the sonnets is reflected in the cycle’s preludium and

postludium. The top line of the preludium is the seventh line of the first sonnet to be

written, sonnet 8, as the top line of the postludium is the eighth line of the same

sonnet; the second line of the preludium is the seventh line of sonnet 7, the second

!139

sonnet to be written, as the second line of the postludium is the eighth line of sonnet

7, and so forth throughout. Accordingly, the last lines of the preludium and post-

ludium are the seventh and eighth lines respectively of the last sonnet written for the

cycle, sonnet 1. In the pre- and postludiums, the lines from the individual sonnets

are reproduced without their punctuation.

3.1.7 Conclusion

This brief introduction to centred writing, the marshalling of elements into

a centre matrix, the construction of an array model, the equivalents of the sonnet’s

formal characteristics within the model, the link between sonnet pattern and sonnet

diction, the description of how individual sonnets are aggregated into a sonnet cycle

and the fact of the written sonnet cycle itself constitutes the practical evidence in

support of the claim. The theoretical and practical evidence for the claim furnished,

the inquiry now closes with a general conclusion.

!140

Part 4

Summary and Outlook

This inquiry sets out to uncover and weigh evidence for the claim that a

sonnet unfolds from its centre to form a pattern in which its formal characteristics

inhere. That is, it seeks to try out the idea that the formal characteristics of the

sonnet might be better understood not as a list of somehow-connected, empirical

categories, but as complex byproducts of a simple pattern originating in, and

developed from, the sonnet’s centre. The balance of evidence presented supports

the claim. The initial evidence provided in Part 1 is naturally tenuous as it consists

of a working hypothesis model constructed from first principles. In Part 2, by

applying and extending these principles, however, evidence is furnished by way of

theoretical models showing how complex equivalents of the formal charac-

teristics of five sonnet traditions, the so-called Early Italian, the Petrarchan, two

Pleadean and the Shakespearean, can be related by the principle of centred form.

Finally, in Part 3, practical evidence for the claim is provided by the centred form

sonnet cycle Memorial Day: the Unmaking of a Sonnet. In each of the models

presented, equivalents of the sonnet’s formal characteristics unfold from the

models’ centre: the equivalent of isometry results from the development of a fixed

array of elements; the equivalent of the volta is deemed to occur at the point of

starkest contrast in directionality flows between arrays; the equivalent of stanzaic

form results from symmetries in the pattern of flows between placeholders in the

models, just as rhyme scheme equivalents, for their part, result from cyclicity in

!141

array development; and, finally, the equivalent of fourteen line sonnet length is

effected by the limit between innovation and redundancy in array development. To

mitigate the risk of error and bias in the evidence presented, in addition to the

array model, a triangle model is developed to independently cross-check array

model results. It is also shown that the array and triangle models were situated

within a broader pattern of binary expansion with symmetrical and, in some

traditions, chiral properties. These expansions, furthermore, hold out the prospect

of an independent basis for the comparison of sonnets across traditions as is seen,

for example, in the relatedness of the Early Italian and Shakespearean models.

Research questions raised as a result of the inquiry might turn on whether the

theory is supported by evidence from the sonnet corpus itself and perhaps the

degree to which the theory helps further our understanding of other sonnet

traditions. The establishment of symmetrical and chiral properties in the models

also suggests potential for interdisciplinary research. Additionally, hard questions

regarding the nature of the relationship between reading and writing are also

raised. These questions resolve themselves broadly into a set of three distinctions:

1. structure / pattern

2. separated category form / centred form

3. linear reading / centred writing

The sonnet is as much chastised as it praised for being hard, old and elitist. The

findings and conclusions of this inquiry reveal it to be instead a simple, enabling

pattern for reflective thought and creative writing for anyone, anywhere.

!142

References

Bermann, Sandra L. (1988). The sonnet over time: A study in the sonnets of Petrarch,

Shakespeare, and Baudelaire. (University of North Carolina studies in comparative

literature: No. 63) Chapel Hill, NC: University of North Carolina Press.

Borgstedt, Thomas. (2009). Topik des Sonetts: Gattungstheorie und Gattungs-

geschichte. [Topology of the sonnet: Theory and history of genre (sic)]. Tübingen: Max

Niemeyer Verlag.

Burt, Stephen and Mikics, David. (2010). The art of the sonnet. Cambridge, MA: Belknap

Press of Harvard University Press.

Chiss, Jean-Louis, Filliolet, Jacques et Maingeneau, Dominique. (2001). Introduction à la

linguistique française t. II: Syntaxe, communication, poétique. [Introduction to French

linguistics, Vol. 2: Syntax, communication, poetics]. Paris: Hachette.

Ellrodt, Robert. (1986). The Cambridge companion to Shakespeare studies.

Cambridge [Cambridgeshire]; New York: Cambridge University Press.

Gell-Mann, Murray. (1996). Let's call it plectics. John Wiley and Sons, Inc.:

Complexity Vol.1, no. 5.

Heaney, Seamus. (1980). Preoccupations: selected prose. 1968-1978. London; Boston:

Faber and Faber.

Hobsbaum, Philip. (1996). Metre, rhythm and verse form. London; New York, NY:

Routledge.

Jost, François. (1989). Le sonnet de Pétrarque à Baudelaire: modes et modulations. [The

sonnet from Petrarch to Baudelaire: modes and modulations]. Berne; Francfort-s. Main;

New York, NY; Paris: Lang.

!143

Kelvin, William Thomson, Baron. (1904). Robert Boyle Lecture, delivered before the Oxford

University Junior Scientific Club on Tuesday, May 16, 1893. London: Clay.

Kemp, Friedhelm. (2002). Das europäische Sonett. Band 1. [The European sonnet. Vol. 1].

Göttingen, BRD: Wallstein Verlag.

Kircher, Hartmut. (1979). Deutsche Sonette. [German sonnets]. Stuttgart, BRD: Reclam.

Lennard, John. (1996). The poetry handbook: a guide to reading poetry for pleasure and

practical criticism. Oxford; New York, NY: Oxford University Press.

Nakahara, Mikio. (2003). Geometry, topology and physics. Bristol; Philadelphia: Institute

of Physics Publishing. 2nd ed.

Olmsted, Everett Ward. (1897). The sonnet in French literature and the development of the

French sonnet form. (Doctoral dissertation, Cornell University). Ithaca, N.Y.

Oppenheimer, Paul. (1989). The birth of the modern mind: self, consciousness, and the

invention of the sonnet. New York, NY: Oxford University Press.

Queneau, Raymond. (1961). Cent mille milliards de poèmes. [One hundred million million

poems]. Paris: Gallimard.

Shapiro, Marianne. (1980). Hieroglyph of time: The Petrarchan sestina. Minneapolis:

University of Minnesota Press.

Spanos, Margaret. (1978). The sestina: An exploration of the dynamics of poetic structure.

Speculum, V. 53, pp. 545-557.

Spiller, Michael R. G. (1997). The sonnet sequence: a study of it's strategies. New York,

NY: Twayne Publishers.

Sylvester, J. J. (1870). The laws of verse or principles of versification exemplified in

metrical translations. London: Longmans, Green, and Co.

!144

Turabian, Kate L. (2007). A manual for writers of research papers, theses, and

dissertations: Chicago style for students and researchers / Kate L. Turabian: revised by

Wayne C. Booth, Gregory G. Colomb, Joseph M. Williams and University of Chicago Press

editorial staff. 7th ed. Chicago, IL and London: The University of Chicago Press.

Vendler, Helen H. (1998). Seamus Heaney. Cambridge, MA: Harvard University Press.

Wehrli, Hans. (2008). Metaphysics: Chirality as the basic principle of physics. Zürich:

Author. Retrieved from http://www.hanswehrli.ch/en/buch_1.htm January, 2011.

Wilkins, Ernest Hatch. (1915). The invention of the sonnet. Modern Philology, V. 13, no. 8,

December 1915, pp. 463-494.

————————— (1959). The invention of the sonnet and other studies in Italian

literature. Rome: Edizioni di Storia e Litteratura.

!145

Appendix A

Early Italian Model: Unsuitability of Two-Type, Three-Element Arrays

A1.0 Two-Type, Three Element Arrays

As may be seen by inspection of Tables A1.1, A1.2 and A1.3, neither of the

possible remaining two type, three element arrays, ‘a b b’ nor ‘b b a’, produces a

pattern of continuous ‘change’ in end-array elements, that is, neither can accom-

modate the equivalent of the alternating rhyme scheme of the Early Italian tradition.

This is the case even when, for the centre array ‘a b b’, there is a change in

directionality in array 6, as shown in Table A1.2. As these results apply equally to

downwards development in the tables, only upwards development from the centre

array, array 8, is shown.

Table A1.1: Centre Array ‘a b b’: Leftwards and Rightwards Development


3. 3.

4. 4.

5. a b b 5.

↖ ↖

6. b a b 6.

↖ ↖

7. b b a 7. b a b

↖ ↖ ↗ ↗

8. a b b 8. a b b

!146

Appendix A (cntd.)

Table A1.2 Centre Array ‘a b b’: Leftwards Development: Directionality Change in Array 6

leftwards

1.

2.

3. b a b

↗ ↗

4. a b b

↗ ↗

5. b b a

↗ ↗

6. b a b

↖ ↖

7. b b a

↖ ↖

8. a b b

Leftwards development for the centre array ‘a b b’, as shown in Table A1.1, produces

successive identical placeholders in arrays 6 and 5. Changing directionality in array 6

to avoid this, shown in Table A1.2, only postpones its redevelopment until arrays 4

and 3. In rightwards development, this occurs immediately in arrays 8 and 7. Thus, in

both cases, it is not possible for end-array placeholders to accommodate alternating

rhyme schemes equivalents.

!147

Table A1.3: Centre Array ‘b b a’: Leftwards and Rightwards, Upwards Development


1. 1.

2. 2.

3. 3.

4. 4.

5. 5.

6. a b b 6. b a b

↖ ↖ ↗ ↗

7. b a b 7. a b b

↖ ↖ ↗ ↗

8. b b a 8. b b a

Both leftwards and rightwards development of the centre array ‘b b a’ lead to

successive similar placeholders in end-array elements, as shown in Table A1.3, thus

preventing alternating rhyme scheme equivalents. None of the three two-type, three-

element centre arrays is, therefore, suitable for satisfying the rhyme scheme

condition of the Early Italian tradition.

!148

Appendix B

Early Italian Model: Centre Array Derivation

B1.0 Centre Array Candidates

In Table B1.1 are listed the Early Italian centre array candidates from Table 16.

Table B1.1 Early Italian Model: Centre Array Candidates

ii.) a b a b b

iii.) a b b a b

vi.) b a b a b

vii.) b a b b a

ix). b b a b a

B2.0 Centre Array: ii.) ‘a b a b b’

In Table B2.1 below, it may be seen that rightwards development from the

centre array ‘a b a b b’ leads immediately to three identical end-array elements,

making the equivalent of the alternating rhyme scheme required by the Early Italian

tradition impossible. This may be seen in the development on the right hand side of

the table. Leftwards development of the centre array leads to the same result in

arrays 4 and 3, and 12 and 13. A change in directionality in arrays 4 and 12 to try to

circumvent this, as shown in Table B2.2, does not produce the necessary redundancy

mechanism, that is, repetition of the centre array in arrays 2 and 14. Now, if a

change in directionality is introduced in arrays 5 and 11, a suitable redundancy

mechanism does develop. However, when in later models this redundancy

!149

mechanism becomes superfluous (p. 120 ff.), the centre array no longer fulfils the

alternating rhyme scheme condition for the Early Italian tradition. The array ‘a b a b

b’ is thus excluded as a possible candidate for the Early Italian model.

Table B2.1 Centre Array: ii.) ‘a b a b b’


1. 1.

2. 2.

3. a b a b b 3. ↖↖ ↖ ↖ 4. b a b a b 4. ↖↖ ↖ ↖ 5. b b a b a 5. ↖↖ ↖ ↖ 6. a b b a b 6. ↖↖ ↖ ↖ 7. b a b b a 7. b a b a b ↖↖ ↖ ↖ ↗ ↗ ↗ ↗ 8. a b a b b 8. a b a b b ↙ ↙ ↙ ↙ ↘ ↘ ↘ ↘ 9. b a b b a 9. b a b a b ↙ ↙ ↙ ↙ 10. a b b a b 10. ↙ ↙ ↙ ↙ 11. b b a b a 11. ↙ ↙ ↙ ↙ 12. b a b a b 12. ↙ ↙ ↙ ↙ 13. a b a b b 13.

14. 14.

15. 15.

!150

Table B2.2 Centre Array: ii.) ‘a b a b b’ No Redundancy Mechanism in Arrays 2 and 14

leftwards

1. 2. a b b a b ↗ ↗ ↗ ↗ 3. b b a b a ↗ ↗ ↗ ↗ 4. b a b a b ↖↖ ↖ ↖ 5. b b a b a ↖↖ ↖ ↖ 6. a b b a b ↖↖ ↖ ↖ 7. b a b b a ↖↖ ↖ ↖ 8. a b a b b ↙ ↙ ↙ ↙ 9. b a b b a ↙ ↙ ↙ ↙ 10. a b b a b ↙ ↙ ↙ ↙ 11. b b a b a ↙ ↙ ↙ ↙ 12. b a b a b ↘ ↘ ↘ ↘ 13. b b a b a ↘ ↘ ↘ ↘ 14. a b b a b

15.

B3.0 Centre Array: iii). ‘a b b a b’

Table B3.1 shows leftwards and rightwards development from the centre array

‘a b b a b’. In order to avoid the rupture of an alternating rhyme scheme equivalent,

as occurs in arrays 6-5 and 10-11 of both leftwards and rightwards development, a

!151

change in directionality has to be introduced in the sixth and tenth arrays. This,

however, denies the possibility of satisfying the two quatrain, two tercet stanzaic

form condition for the Early Italian tradition. Thus, the array ‘a b b a b’ is also

excluded as a centre array candidate for the Early Italian model.

Table B3.1 Centre Array: iii.) ‘a b b a b’


1. 1. 2. 2. 3. 3. 4. 4.

5. a b a b b 5. b a b a b ↖↖ ↖ ↖ ↗ ↗ ↗ ↗ 6. b a b a b 6. a b a b b ↖↖ ↖ ↖ ↗ ↗ ↗ ↗ 7. b b a b a 7. b a b b a ↖↖ ↖ ↖ ↗ ↗ ↗ ↗ 8. a b b a b 8. a b b a b ↙ ↙ ↙ ↙ ↘ ↘ ↘ ↘ 9. b b a b a 9. b a b b a ↙ ↙ ↙ ↙ ↘ ↘ ↘ ↘ 10. b a b a b 10. a b a b b ↙ ↙ ↙ ↙ ↘ ↘ ↘ ↘ 11. a b a b b 11. b a b a b

12. 12. 13. 13. 14. 14. 15. 15.

B4.0 Centre Array: vi). ‘b a b a b’

In Table B4.1 below, it can be seen that leftwards development of the centre

array ‘b a b a b’ immediately fails the alternating rhyme equivalent condition. In

!152

rightwards development, a change in directionality is necessary in arrays 4 and 12

as, by inspection, continued development in the same direction leads to the dis-

ruption of an alternating rhyme scheme equivalent. Such a change, however,

although resulting in alternation, does not provide a redundancy mechanism in array

2 or 14, as may be seen in Table B4.2 further below. As with the centre array

candidate 'a b a b b', if a change in directionality is introduced in arrays 5 and 11, a

suitable redundancy mechanism is developed in arrays 2 and 14. Once again, how-

ever, when this redundancy mechanism becomes superfluous due to the subsequent

development of a a triangle model within a binary expansion, the centre array 'b a b

a b' no longer fulfils the alternating rhyme scheme condition for the Early Italian

tradition. Thus, the ‘b a b a b’ centre array is also excluded as a centre array

candidate for the Early Italian model.

!153

Table B4.1 Centre Array: vi.) ‘b a b a b’


1. 1.

2. 2.

3. 3. b a b a b ↗ ↗ ↗ ↗ 4. 4. a b a b b ↗ ↗ ↗ ↗ 5. 5. b a b b a ↗ ↗ ↗ ↗ 6. 6. a b b a b ↗ ↗ ↗ ↗ 7. a b a b b 7. b b a b a ↖↖ ↖ ↖ ↗ ↗ ↗ ↗ 8. b a b a b 8. b a b a b ↙ ↙ ↙ ↙ ↘ ↘ ↘ ↘ 9. a b a b b 9. b b a b a ↘ ↘ ↘ ↘ 10. 10. a b b a b ↘ ↘ ↘ ↘ 11. 11. b a b b a ↘ ↘ ↘ ↘ 12. 12. a b a b b ↘ ↘ ↘ ↘ 13. 13. b a b a b

14. 14.

15. 15.

!154

Table B4.2 Centre Array: vi). ‘b a b a b’ No Redundancy Mechanism in Arrays 2 and 14

rightwards

1. b b a b a ↖↖ ↖ ↖ 2. a b b a b ↖↖ ↖ ↖ 3. b a b b a ↖↖ ↖ ↖ 4. a b a b b ↗ ↗ ↗ ↗ 5. b a b b a ↗ ↗ ↗ ↗ 6. a b b a b ↗ ↗ ↗ ↗ 7. b b a b a ↗ ↗ ↗ ↗ 8. b a b a b ↘ ↘ ↘ ↘ 9. b b a b a ↘ ↘ ↘ ↘ 10. a b b a b ↘ ↘ ↘ ↘ 11. b a b b a ↘ ↘ ↘ ↘ 12. a b a b b ↙ ↙ ↙ ↙ 13. b a b b a ↙ ↙ ↙ ↙ 14. a b b a b ↙ ↙ ↙ ↙ 15. b b a b a

!155

B5.0 Centre Array: vii.) ‘b a b b a’

As shown in Table B5.1 below, rightwards development of the centre array ‘b

a b b a’ produces successive identical elements in arrays 7 & 6 and 9 & 10, so that

this candidate is unsuitable for developing the alternating rhyme scheme equivalent

of the Early Italian sonnet. A change in direction at array 7 fails to develop the

equivalent of stanzaic form, which requires that directionality changes mark the

transitions between stanzas. In the leftwards development of the centre array,

however, a change of direction at arrays 5 and 11 does develop an appropriate

redundancy mechanism in arrays 2 and 14 with the redevelopment of the centre

array, as shown in Table B5.2 further below. Thus, the array ‘b a b b a’ with

leftwards directionality satisfies the conditions of the Early Italian sonnet.

!156

Table B5.1 Centre Array: vii.) ‘b a b b a’


1. 1.

2. 2.

3. 3.

4. a b a b b 4. ↖↖ ↖ ↖ 5. b a b a b 5. ↖↖ ↖ ↖ 6. b b a b a 6. b a b a b ↖↖ ↖ ↖ ↗ ↗ ↗ ↗ 7. a b b a b 7. a b a b b ↖↖ ↖ ↖ ↗ ↗ ↗ ↗ 8. b a b b a 8. b a b b a ↙ ↙ ↙ ↙ ↘ ↘ ↘ ↘ 9. a b b a b 9. a b a b b ↙ ↙ ↙ ↙ ↘ ↘ ↘ ↘ 10. b b a b a 10. b a b a b ↙ ↙ ↙ ↙ 11. b a b a b 11. ↙ ↙ ↙ ↙ 12. a b a b b 12.

13. 13.

14. 14.

15. 15.

!157

Table B5.2 Centre Array: vii.) ‘b a b b a’

leftwards

1. a b b a b ↖↖ ↖ ↖ 2. b a b b a ↗ ↗ ↗ ↗ 3. a b b a b ↗ ↗ ↗ ↗ 4. b b a b a ↗ ↗ ↗ ↗ 5. b a b a b ↖↖ ↖ ↖ 6. b b a b a ↖↖ ↖ ↖ 7. a b b a b ↖↖ ↖ ↖ 8. b a b b a ↙ ↙ ↙ ↙ 9. a b b a b ↙ ↙ ↙ ↙ 10. b b a b a ↙ ↙ ↙ ↙ 11. b a b a b ↘ ↘ ↘ ↘ 12. b b a b a ↘ ↘ ↘ ↘ 13. a b b a b ↘ ↘ ↘ ↘ 14. b a b b a ↙ ↙ ↙ ↙ 15. a b b a b

!158

B6.0 Centre Array: ix.) ‘b b a b a’

Table B6.1 below shows that leftwards development of the ‘b b a b a’ centre

array almost immediately leads to an unsuitable development in end-array elements.

However, rightwards development, with a change in directionality in arrays 5 and

11, as shown in Table B6.2, redevelops the centre array in arrays 2 and 14 leading to

redundancy and the fulfillment of the number of lines, and all other, conditions for

the Early Italian tradition.

!159

Table B6.1 Centre Array: ix.) ‘b b a b a’


1. 1.

2. 2.

3. 3.

4. 4. b a b a b ↗ ↗ ↗ ↗ 5. 5. a b a b b ↗ ↗ ↗ ↗ 6. a b a b b 6. b a b b a ↖↖ ↖ ↖ ↗ ↗ ↗ ↗ 7. b a b a b 7. a b b a b ↖↖ ↖ ↖ ↗ ↗ ↗ ↗ 8. b b a b a 8. b b a b a ↙ ↙ ↙ ↙ ↘ ↘ ↘ ↘ 9. b a b a b 9. a b b a b ↙ ↙ ↙ ↙ ↘ ↘ ↘ ↘ 10. a b a b b 10. b a b b a ↘ ↘ ↘ ↘ 11. 11. a b a b b ↘ ↘ ↘ ↘ 12. 12. b a b a b

13. 13.

14. 14.

15. 15.

!160

Table B6.2 Centre Array: ix). ‘b b a b a’

rightwards

1. a b b a b ↗ ↗ ↗ ↗ 2. b b a b a ↖↖ ↖ ↖ 3. a b b a b ↖↖ ↖ ↖ 4. b a b b a ↖↖ ↖ ↖ 5. a b a b b ↗ ↗ ↗ ↗ 6. b a b b a ↗ ↗ ↗ ↗ 7. a b b a b ↗ ↗ ↗ ↗ 8. b b a b a ↘ ↘ ↘ ↘ 9. a b b a b ↘ ↘ ↘ ↘ 10. b a b b a ↘ ↘ ↘ ↘ 11. a b a b b ↙ ↙ ↙ ↙ 12. b a b b a ↙ ↙ ↙ ↙ 13. a b b a b ↙ ↙ ↙ ↙ 14. b b a b a ↘ ↘ ↘ ↘ 15. a b b a b

!161

B7.0 Solutions

There are thus two solutions from the corpus of ten centre arrays listed in

Table 16: ‘b a b b a’ with leftwards development and ‘b b a b a’ with rightwards

development. Both models are shown in Table B7.1.

Table B7.1 Early Italian Model Centre Array Solutions

1. a b b a b 1. a b b a b ↗ ↗ ↗ ↗ ↖↖ ↖ ↖ 2. b b a b a 2. b a b b a ↖↖ ↖ ↖ ↗ ↗ ↗ ↗ 3. a b b a b 3. a b b a b ↖↖ ↖ ↖ ↗ ↗ ↗ ↗ 4. b a b b a 4. b b a b a ↖↖ ↖ ↖ ↗ ↗ ↗ ↗ 5. a b a b b 5. b a b a b ↗ ↗ ↗ ↗ ↖↖ ↖ ↖ 6. b a b b a 6. b b a b a ↗ ↗ ↗ ↗ ↖↖ ↖ ↖ 7. a b b a b 7. a b b a b ↗ ↗ ↗ ↗ ↖↖ ↖ ↖ 8. b b a b a 8. b a b b a ↘ ↘ ↘ ↘ ↙ ↙ ↙ ↙ 9. a b b a b 9. a b b a b ↘ ↘ ↘ ↘ ↙ ↙ ↙ ↙ 10. b a b b a 10. b b a b a ↘ ↘ ↘ ↘ ↙ ↙ ↙ ↙ 11. a b a b b 11. b a b a b ↙ ↙ ↙ ↙ ↘ ↘ ↘ ↘ 12. b a b b a 12. b b a b a ↙ ↙ ↙ ↙ ↘ ↘ ↘ ↘ 13. a b b a b 13. a b b a b ↙ ↙ ↙ ↙ ↘ ↘ ↘ ↘ 14. b b a b a 14. b a b b a ↘ ↘ ↘ ↘ ↙ ↙ ↙ ↙ 15. a b b a b 15. a b b a b

!162

Appendix C

Petrarchan Model: Centre Matrix Derivation

C1.0 Petrarchan Centre Array Candidates

As the Petrarchan simplified rhyme scheme shows an of embracing and

paired rhymes in its upper half and alternating end rhymes in its lower, it is assumed

that a centre matrix of two arrays, each composed of four elements of two element

types, ‘a’ and ‘b’, are necessary to satisfy Petrarchan sonnet conditions. There are

thus, as listed in Table A5.3.1.1, C (n,k), that is, 4!/ (4-2)!2! = 24/4 = 6, candidate

arrays in all:

Table C1.1: Petrarchan Centre Matrix Candidates 1

i.) a a b b

ii.) a b a b

iii.) a b b a

iv.) b b a a

v.) b a b a

vi.) b a a b

As only ii.) and v). can represent the continuous change in end-array elements to be

found in the lower half of the Petrarchan sonnet, only they might represent array 8.

As any of the remaining four arrays might represent array 7, there are in all eight

possible centre matrices, as shown in Table C1.2 below:

!163

Table C1.2 Petrarchan Centre Matrix Candidates 2

i.) a a b b

a b a b

ii.) a a b b

b a b a

iii.) a b b a

a b a b

iv.) a b b a

b a b a

v.) b b a a

a b a b

vi.) b b a a

b a b a

vii.) b a a b

a b a b

viii.) b a a b

b a b a

From the simplified rhyme scheme, shown in Table C1.3 below, it can be

seen that from the end-array in array 7 to the end-array in array 8 there is change

from ‘b’ to ‘a’ which by transposition is equivalent to ‘a’ to ‘b’. Thus, the candidate

!164

matrices i.), iv.), vi.) and vii.) may immediately be excluded as their end-array

elements show no change from one array to the next.

Table C1.3 Petrarchan Simplified Rhyme Scheme

1 2 3 4 5 6 7 8 9 10 11 12 13 14

a b b a a b b a b a b a b a

The remaining candidates are shown in Table C1.4.

Table C1.4 Petrarchan Centre Array Candidates 3

ii.) a a b b

b a b a

iii.) a b b a

a b a b

v.) b b a a

a b a b

viii) b a a b

b a b a

C2.0 Directionality: Symmetry and Chirality

In Tables C3.1–4 further below, each of the final candidate matrices is

tested with leftwards and rightwards directionality only. Leftwards and rightwards,

!165

and rightwards and leftwards developments are excluded for the reasons given in the

brief discussion of symmetry and chirality which now follows. A detailed

description of symmetry and chirality is found in the Pleadean section .

In each of the matrices’ lower arrays there is change from one element to

the next. Whether they are developed rightwards or leftwards therefore produces,

symbolically, the same result. For example, the array ‘b a b a’ from the matrix ii.)

above when developed leftwards creates the array ‘a b a b’, when developed

rightwards the symbolically identical array ‘a b a b’. As a result, for every array that

satisfies Petrarchan sonnet conditions in the model’s upper half, there are two

solutions in the lower.

If development of the lower arrays is symbolically the same irrespective of

the directionality of development, the flows between placeholders differ, depending

on whether development is either rightwards or leftwards. The flows in the lower

half of the model will, therefore, either be symmetrical with regard to the flows in

the upper half, or chiral. If they are symmetrical, then the flows in the two halves of

the model coincide and there is no perception of the spatial relationships in the form

of geometrical shapes shown in Tables 25 and 26, no mechanism for introducing

redundancy into the model and, consequently, failure to fulfill the sonnet’s number

of lines condition. Therefore, only developments that allow for chirality between the

flows in the model’s halves is are considered. As such, only models where the

directionality of development is either rightwards or leftwards, but not rightwards

and leftwards or leftwards and rightwards, are tested for only they result in chiral

relationships between the flows in each of the models’ halves.

!166

C3.0 Testing of Centre Matrices

In Table C3.1 below, leftwards development of the centre matrix candidate

ii.) does not create the equivalent of an embracing rhyme. Rightwards development,

however, is satisfactory, as may be seen by inspection.

Table C3.1 Development: Centre Matrix ii.)


1. 1. b b a a ↗ ↗ ↗ 2. 2. b a a b ↗ ↗ ↗ 3. 3. a a b b ↗ ↗ ↗ 4. 4. a b b a ↗ ↗ ↗ 5. 5. b b a a ↗ ↗ ↗ 6. a b b a 6. b a a b ↖ ↖ ↖ ↗ ↗ ↗ 7. a a b b 7. a a b b

8. b a b a 8. b a b a ↙ ↙ ↙ ↘ ↘ ↘ 9. a b a b 9. a b a b ↘ ↘ ↘ 10. 10. b a b a ↘ ↘ ↘ 11. 11. a b a b ↘ ↘ ↘ 12. 12. b a b a ↘ ↘ ↘ 13. 13. a b a b ↘ ↘ ↘ 14. 14. b a b a ↘ ↘ ↘ 15. 15. a b a b

!167

The centre matrix candidate iii.) is tested for suitability in Table C3.2. Leftwards

directionality is satisfactory, rightwards is not, as no equivalent to embracing rhymes

can be developed.

Table C3.2 Development: Centre Matrix iii.)


1. b a a b 1. ↖ ↖ ↖ 2. b b a a 2. ↖ ↖ ↖ 3. a b b a 3. ↖ ↖ ↖ 4. a a b b 4. ↖ ↖ ↖ 5. b a a b 5. ↖ ↖ ↖ 6. b b a a 6. a a b b ↖ ↖ ↖ ↗ ↗ ↗ 7. a b b a 7. a b b a

8. a b a b 8. a b a b ↙ ↙ ↙ 9. b a b a 9. ↙ ↙ ↙ 10. a b a b 10. ↙ ↙ ↙ 11. b a b a 11. ↙ ↙ ↙ 12. a b a b 12. ↙ ↙ ↙ 13. b a b a 13. ↙ ↙ ↙ 14. a b a b 14. ↙ ↙ ↙ 15. b a b a 15.

!168

Leftwards development of the centre matrix v.) in Table C3.3 creates no equivalent to an embracing rhyme. Rightwards development is satisfactory.

Table C3.3 Development: Centre Matrix v.) leftwards rightwards

1. 1. a a b b ↗ ↗ ↗

2. 2. a b b a ↗ ↗ ↗

3. 3. b b a a ↗ ↗ ↗

4. 4. b a a b ↗ ↗ ↗

5. 5. a a b b ↗ ↗ ↗

6. b a a b 6. a b b a ↖ ↖ ↖ ↗ ↗ ↗

7. b b a a 7. b b a a

8. a b a b 8. a b a b ↘ ↘ ↘ 9. 9. b a b a ↘ ↘ ↘ 10. 10. a b a b ↘ ↘ ↘ 11. 11. b a b a ↘ ↘ ↘ 12. 12. a b a b ↘ ↘ ↘ 13. 13. b a b a ↘ ↘ ↘ 14. 14. a b a b ↘ ↘ ↘ 15 15. b a b a

!169

Leftwards development of the centre matrix viii.) shown in Table C3.4 satisfies

Petrarchan sonnet conditions, rightwards does not as an embracing rhyme cannot be

developed.

Table C3.4 Development: Centre Matrix viii.)


1. a b b a 1. ↖ ↖ ↖ 2. a a b b 2. ↖ ↖ ↖ 3. b a a b 3. ↖ ↖ ↖ 4. b b a a 4. ↖ ↖ ↖ 5. a b b a 5. ↖ ↖ ↖ 6. a a b b 6. b b a a ↖ ↖ ↖ ↗ ↗ ↗ 7. b a a b 7. b a a b

8. b a b a 8. b a b a ↙ ↙ ↙ 9. a b a b 9. ↙ ↙ ↙ 10. b a b a 10. ↙ ↙ ↙ 11. a b a b 11. ↙ ↙ ↙ 12. b a b a 12. ↙ ↙ ↙ 13. a b a b 13. ↙ ↙ ↙ 14. b a b a 14. ↙ ↙ ↙ 15 a b a b 15.

!170

C4.0 Solutions

There are, therefore, four solutions:

array ii.) rightwards

array iii.) leftwards

array v.) rightwards

and array viii.) leftwards

As arrays ii.) and v.), and iii.) and viii.) are transpositions, either of each

may satisfy. Array ii.) with rightwards and array iii.) with leftwards directionality are

chosen for the models themselves.

!171

Appendix D

Pleadean Models: Centre Array Derivation

D1.0 Pleadean Centre Array Derivation

As the Pleadean 1 and 2 simplified rhyme schemes in Table 30 show

predominantly embracing rhymes, as in the upper half of the Petrarchan model, it is

assumed that a four element, two element type centre array offers the best means of

developing equivalents of the Pleadean formal characteristics for both traditions. For

this mix there are thus for each tradition C (n,k), that is, 4!/ (4-2)!2! = 24/4 = 6

centre array candidates, as listed in Table C1.1. They are incidentally the same

candidate arrays as for the Petrarchan centre matrix (Appendix C).

Table D1.1: Pleadean Centre Array Candidates 1

i.) a a b b

ii.) a b b a

iii.) a b a b

iv.) b b a a

v.) b a a b

vi.) b a b a

Arrays i.), ii.) and iii.) being transpositions of iv.), v.) and vi.) either of the two

groups of three may represent the other. Let the first group be chosen, as in Table

D1.2 below.

!172

Table D1.2 Pleadean Centre Array Candidates 2

i.) a a b b

ii.) a b b a

iii.) a b a b

The array ‘a b a b’ can only develop alternating arrays and may therefore be

eliminated. As shown then in Table D1.3, there are thus just two candidates times

two directionalities, that is four final centre array candidates in all when the choice

between leftwards and rightwards directionality from the centre array is taken into

account,

Table D1.3 Pleadean Centre Array Candidates 3

i.) a a b b leftwards

ii.) a a b b rightwards

iii.) a b b a leftwards

iv.) a b b a rightwards

Array ii.), a a b b with rightwards development from array 8, immediately

develops three similar end-array elements, as shown in Table D1.4 below, thus fails

the Pleadean 1 rhyme scheme test and is eliminated as a candidate.

!173

Table D1.4 Unsuitability of Array ii.)

7. b a a b

↗

✶ 8. a a b b

↘

9. b a a b

The same fate befalls array iii), a b b a with leftwards development, as

shown in Table D1.5. It is, therefore, also eliminated.

Table D1.5 Unsuitability of Array iii.)

7. b b a a

↖

✶ 8. a b b a

↙

9. b b a a

D2.0 Solutions

This leaves just two candidates, array i.), ‘a a b b’ with leftwards and array

iv.), ‘a b b a’ with rightwards development.

!174

Appendix E

Shakespearean Model: Centre Array Derivation

E1.0 Shakespearean Centre Array Candidates

From the discussions of the previous models’ centre arrays and matrices

(Appendices B, C and D) and the analysis of the Shakespearean simplified rhyme

scheme, it is assumed that the development of a centre array of five elements with

two element types suffices to render equivalents to the formal characteristics of the

Shakespearean sonnet. As this is the same assumption made for the Early Italian

model, the same logic applies to the selection of centre array candidates for the

Shakespearean model. The list of candidate arrays is therefore the same as that in

Table 16 for the Early Italian model. The list is shown in Table E1.1.

Table E1.1 Shakespearean Centre Array Candidates

ii.) a b a b b

iii.) a b b a b

vi.) b a b a b

vii.) b a b b a

ix.) b b a b a

With either leftwards or rightwards development there are thus five times two

candidates. From the discussion in the appendix to the Early Italian centre arrays,

two of them may be eliminated as their development immediately results in lack of

alternation of end-array elements: ‘a b a b b’ with rightwards and ‘b a b a b’ with

!175

leftwards development. There now follows the test results for the remaining

candidate arrays.

E2.0 Centre Array: ii.) ‘a b a b b’

As may be seen in the model on the left of Table E2.1 below, leftwards

development requires a change in directionality in the fourth and twelfth arrays if

alternation of end-array elements is to be maintained. This change, as may be seen in

the right hand model, does not create the equivalent of a rhyming couplet. It is

therefore eliminated as a candidate.

!176

Table E2.1 Centre Array: ii.) ‘a b a b b’

leftwards leftwards 1. 1. b a b b a ↗ ↗ ↗ ↗ 2. 2. a b b a b ↗ ↗ ↗ ↗ 3. a b a b b 3. b b a b a ↖↖ ↖ ↖ ↗ ↗ ↗ ↗ 4. b a b a b 4. b a b a b ↖↖ ↖ ↖ ↖↖ ↖ ↖ 5. b b a b a 5. b b a b a ↖↖ ↖ ↖ ↖↖ ↖ ↖ 6. a b b a b 6. a b b a b ↖↖ ↖ ↖ ↖↖ ↖ ↖ 7. b a b b a 7. b a b b a ↖↖ ↖ ↖ ↖↖ ↖ ↖ 8. a b a b b 8. a b a b b ↙ ↙ ↙ ↙ ↙ ↙ ↙ ↙ 9. b a b b a 9. b a b b a ↙ ↙ ↙ ↙ ↙ ↙ ↙ ↙ 10. a b b a b 10. a b b a b ↙ ↙ ↙ ↙ ↙ ↙ ↙ ↙ 11. b b a b a 11. b b a b a ↙ ↙ ↙ ↙ ↙ ↙ ↙ ↙ 12. b a b a b 12. b a b a b . ↙ ↙ ↙ ↙ ↘ ↘ ↘ ↘ 13. a b a b b 13. b b a b a ↘ ↘ ↘ ↘ 14. 14. a b b a b ↘ ↘ ↘ ↘ 15. 15. b a b b a

!177

E3.0 Centre Array: iii.) ‘a b b a b’

Table E3.1 shows leftwards and rightwards development from the centre

array ‘a b b a b’. In order to avoid the rupture of an alternating rhyme scheme

equivalent, a change in directionality has to be introduced in the sixth and tenth

arrays. These changes are shown in Table E3.2 below.

Table E3.1 Centre Array: iii.) ‘a b b a b’


1. 1.

2. 2.

3. 3.

4. 4.

5. a b a b b 5. b a b a b ↖↖ ↖ ↖ ↗ ↗ ↗ ↗ 6. b a b a b 6. a b a b b ↖↖ ↖ ↖ ↗ ↗ ↗ ↗ 7. b b a b a 7. b a b b a ↖↖ ↖ ↖ ↗ ↗ ↗ ↗ 8. a b b a b 8. a b b a b ↙ ↙ ↙ ↙ ↘ ↘ ↘ ↘ 9. b b a b a 9. b a b b a ↙ ↙ ↙ ↙ ↘ ↘ ↘ ↘ 10. b a b a b 10. a b a b b ↙ ↙ ↙ ↙ ↘ ↘ ↘ ↘ 11. a b a b b 11. b a b a b

12. 12.

13. 13.

14. 14.

15. 15.

!178

The change in directionality results in two models that satisfy the conditions

for the Shakespearean sonnet.

Table E3.2 Centre Array: iii.) ‘a b b a b’


1. b a b a b 1. a b a b b ↗ ↗ ↗ ↗ ↖ ↖ ↖ ↖ 2. a b a b b 2. b a b a b ↗ ↗ ↗ ↗ ↖ ↖ ↖ ↖ 3. b a b b a 3. b b a b a ↗ ↗ ↗ ↗ ↖ ↖ ↖ ↖ 4. a b b a b 4. a b b a b ↗ ↗ ↗ ↗ ↖ ↖ ↖ ↖ 5. b b a b a 5. b a b b a ↗ ↗ ↗ ↗ ↖ ↖ ↖ ↖ 6. b a b a b 6. a b a b b ↖↖ ↖ ↖ ↗ ↗ ↗ ↗ 7. b b a b a 7. b a b b a ↖↖ ↖ ↖ ↗ ↗ ↗ ↗ 8. a b b a b 8. a b b a b ↙ ↙ ↙ ↙ ↘ ↘ ↘ ↘ 9. b b a b a 9. b a b b a ↙ ↙ ↙ ↙ ↘ ↘ ↘ ↘ 10. b a b a b 10. a b a b b ↘ ↘ ↘ ↘ ↙↙ ↙ ↙ 11. b b a b a 11. b a b b a ↘ ↘ ↘ ↘ ↙↙ ↙ ↙ 12. a b b a b 12. a b b a b ↘ ↘ ↘ ↘ ↙↙↙ ↙ 13. b a b b a 13. b b a b a ↘ ↘ ↘ ↘ ↙↙ ↙ ↙ 14. a b a b b 14. b a b a b ↘ ↘ ↘ ↘ ↙↙ ↙ ↙ 15. b a b a b 15. a b a b b

!179

E4.0 Centre Array: vi.) ‘b a b a b’

The array ‘b a b a b’ with leftwards directionality is excluded as noted

earlier. In the left hand model in Table E4.1, a change in directionality in arrays 4

and 12 is needed to avoid the same problem. This change may be seen in the column

on the right. As there are no placeholders to accommodate the equivalent of the final

Shakespearean couplet, the array ‘b a b a b’ may therefore be excluded.

Table E4.1

Centre Array: vi.) ‘b a b a b’

rightwards rightwards

1. 1. b b a b a ↖ ↖ ↖ ↖ 2. 2. a b b a b ↖ ↖ ↖ ↖ 3. b a b a b 3. b a b b a ↗ ↗ ↗ ↗ ↖ ↖ ↖ ↖ 4. a b a b b 4. a b a b b ↗ ↗ ↗ ↗ ↗ ↗ ↗ ↗ 5. b a b b a 5. b a b b a ↗ ↗ ↗ ↗ ↗ ↗ ↗ ↗ 6. a b b a b 6. a b b a b ↗ ↗ ↗ ↗ ↗ ↗ ↗ ↗ 7. b b a b a 7. b b a b a ↗ ↗ ↗ ↗ ↗ ↗ ↗ ↗ 8. b a b a b 8. b a b a b ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ 9. b b a b a 9. b b a b a ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ 10. a b b a b 10. a b b a b ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ 11. b a b b a 11. b a b b a ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ 12. a b a b b 12. a b a b b ↘ ↘ ↘ ↘ ↙ ↙ ↙ ↙ 13. b a b a b 13. b a b b a ↙ ↙ ↙ ↙ 14. 14. a b b a b ↙ ↙ ↙ ↙ 15. 15. b b a b a

!180

E5.0 Centre Array: vii.) ‘b a b b a’

In the model on the right in Table E5.1 it can be seen that similar end-array

elements are created in arrays 7 and 6. As a change in directionality in array 7 comes

too early to result in the equivalent of Shakespearean stanzaic form, this model is

eliminated. Leftwards development requires a change in directionality in array 5.

This is shown in Table E5.2 below.

Table E5.1 Centre Array: vii.) ‘b a b b a’


1. 1.

2. 2.

3. 3.

4. a b a b b 4. ↖↖ ↖ ↖ 5. b a b a b 5. ↖↖ ↖ ↖ 6. b b a b a 6. b a b a b ↖↖ ↖ ↖ ↗ ↗ ↗ ↗ 7. a b b a b 7. a b a b b ↖↖ ↖ ↖ ↗ ↗ ↗ ↗ 8. b a b b a 8. b a b b a ↙ ↙ ↙ ↙ ↘ ↘ ↘ ↘ 9. a b b a b 9. a b a b b ↙ ↙ ↙ ↙ ↘ ↘ ↘ ↘ 10. b b a b a 10. b a b a b ↙ ↙ ↙ ↙ 11. b a b a b 11. ↙ ↙ ↙ ↙ 12. a b a b b 12.

13. 13.

14. 14.

15. 15.

!181

The result in the change in directionality is the development of a model

which does not develop placeholders that accommodate the requisite equivalent of

the final couplet. This centre array is therefore also eliminated from the list of

possible centre array candidates.

Table E5.2 Centre Array: vii.) ‘b a b b a’

leftwards

1. a b a b b ↗ ↗ ↗ ↗ 2. b a b b a ↗ ↗ ↗ ↗ 3. a b b a b ↗ ↗ ↗ ↗ 4. b b a b a ↗ ↗ ↗ ↗ 5. b a b a b ↖↖ ↖ ↖ 6. b b a b a ↖↖ ↖ ↖ 7. a b b a b ↖↖ ↖ ↖ 8. b a b b a ↙ ↙ ↙ ↙ 9. a b b a b ↙ ↙ ↙ ↙ 10. b b a b a ↙ ↙ ↙ ↙ 11. b a b a b ↘ ↘ ↘ ↘ 12. b b a b a ↘ ↘ ↘ ↘ 13. a b b a b ↘ ↘ ↘ ↘ 14. b a b b a ↘ ↘ ↘ ↘ 15. a b a b b

!182

E6.0 Centre Array: ix.) ‘b b a b a’

In table E6.1 below, the model with leftwards directionality develops

similar end-array elements in arrays 7 and 6, and 9 and 10, marring the development

of the equivalent of Shakespearean stanzaic form. This centre array is therefore

excluded. Rightwards development requires change in directionality in arrays 5 and

11 which is shown in Table E6.2 below.

Table E6.1 Centre Array: ix.) ‘b b a b a’


1. 1.

2. 2.

3. 3.

4. 4. b a b a b ↗ ↗ ↗ ↗ 5. 5. a b a b b ↗ ↗ ↗ ↗ 6. a b a b b 6. b a b b a ↖↖ ↖ ↖ ↗ ↗ ↗ ↗ 7. b a b a b 7. a b b a b ↖↖ ↖ ↖ ↗ ↗ ↗ ↗ 8. b b a b a 8. b b a b a ↙ ↙ ↙ ↙ ↘ ↘ ↘ ↘ 9. b a b a b 9. a b b a b ↙ ↙ ↙ ↙ ↘ ↘ ↘ ↘ 10. a b a b b 10. b a b b a ↘ ↘ ↘ ↘ 11. 11. a b a b b ↘ ↘ ↘ ↘ 12. 12. b a b a b

13. 13.

!183

Once again, the change in directionality does not develop the equivalent of

placeholders which support the rhyming couplet that closes the Shakespearean

sonnet. This centre array is therefore also eliminated.

Table E6.2 Centre Array: ix.) ‘b b a b a’

rightwards

1. b a b a b ↖↖ ↖ ↖ 2. b b a b a ↖↖ ↖ ↖ 3. a b b a b ↖↖ ↖ ↖ 4. b a b b a ↖↖ ↖ ↖ 5. a b a b b ↗ ↗ ↗ ↗ 6. b a b b a ↗ ↗ ↗ ↗ 7. a b b a b ↗ ↗ ↗ ↗ 8. b b a b a ↘ ↘ ↘ ↘ 9. a b b a b ↘ ↘ ↘ ↘ 10. b a b b a ↘ ↘ ↘ ↘ 11. a b a b b ↙ ↙ ↙ ↙ 12. b a b b a ↙ ↙ ↙ ↙ 13. a b b a b ↙ ↙ ↙ ↙ 14. b b a b a ↙ ↙ ↙ ↙ 15. b a b a b

!184

E7.0 Solutions

There is, thus, only one centre array with placeholders that can accom-

modate the equivalents of the Shakespearean formal characteristics: ‘a b b a b' with

leftwards and rightwards directionality.

!185

Appendix F

Memorial Day: The Unmaking of a Sonnet, Poems

F0 Preludium 180

F1 Sonnet 1 181

F2 Sonnet 2 182

F3 Sonnet 3 183

F4 Sonnet 4 184

F5 Sonnet 5 185

F6 Sonnet 6 186

F7 Sonnet 7 187

F8 Sonnet 8 188

F9 Sonnet 9 189

F10 Sonnet 10 190

F11 Sonnet 11 191

F12 Sonnet 12 192

F13 Sonnet 13 193

F14 Sonnet 14 194

F15 Postludium 195

!186

F0 Preludium

how fey how free the mitered pages mild do sheen

that drill and flay and wield the timbal railing day

a dewy spinning glass strewn with darts of rhyme

the water’s curtain glide and salt shimmer sand

like a fine thin veil across a faint thin smile

too soon to get over you but not too late

worn out by care your loss my only lair

my hushed chafed dear heart no self-pity no rage

on gloam paling light soul blent and fey

your dark toned breath and spark lit laugh

smooth and spiral fade and chiral as beneath the gaping wind

be free to speak and act to feel be brave be sure

no use their suck of power to keep the state insane

a force that aspired to choose the cause of truth not death

!187

F1 1

My story begins with you, your war, your truth and dreams.

For you, all were irony and hurt, all bitter salt,

whilst I found love, and laughed and sighed and soared and knew

that all my life was you, and all we’d have was mine.

I knew you laws of history, how they moved and stirred, and formed

a child of ardent heart and mind, clawed yet couth,

a force that aspired to choose the cause of truth not death,

to pursue the course of light, not the quietening of a breath.

Yet how be free to choose when life’s but fortune’s die?

And, say, what kindly rule thwarts indifference, cold or cruel?

How lose a love that’s life itself and, parted, still be whole?

For what wounded life can be restored that roots can somehow bind?

What guide or truth that’s sought could find such a law, or doom?

And how should I engage my mind, with your soul, still restless, strewn?

!188

F2 2

What lord, what art or science can furnish me a chart

to plan my journey’s course, to lead a worthy life?

What faults and harms to chide? What simple joys to treasure?

What life to form and want? What life to weigh and measure?

Forbear, and live the motto: no aim, no goal, too high!

To soar, to strain, to dream, endure, survive, defy;

be free to speak and act, to feel, be brave, be sure,

and back the weak and needy, live life to strive for more.

With work and family rich, with love and ease alive,

a share of bumps and flaws, and storms and wars besides,

yet wise, not short on fact: No peace, no calm assured,

so defend the cause that made you, and spend for freedom’s sword.

And when the journey’s over, when freedom’s sword has swung,

we’ll have a brand new motto: ‘e pluribus a gun’.

!189

F3 3

In your way for a change, I took the chance to get

a taste of your air and peer into your sea.

The grain of salt in your talk sent a wave through

my voice that lent a savorous edge to our play,

while your look of crashed surf and shell-slushed sand,

your flair of beach fires, your spiced dusky hands,

your dark toned breath and spark lit laugh,

sent a tide through my mind that washed over the past.

Your stature could be said of the handsome devilled kind,

square-splayed trained in the shoulder, legs long-boned, spill defined,

you stand tallish raked with a gaze mainly floored.

Your nose is a tad too flat, your mouth a spit too dry,

your hair maniacally brown, your chin somewhat awry;

your smile, all grace, your mind, all board.

!190

F4 4

When you debate, you flake: you put bland points blandly,

scoff, then mumble, fumble and crumble away:

you rail like a dove coos, rant like a quail woos,

demur like a finch fainting, and move like a drake quaking:

baiting you is never bracing, simply dull.

And yet, when I stay my tongue and weigh your claims

my hushed, chafed, dear heart, no self-pity, no rage

against the crush of vain fate, no wasteful rush

to sell your chary soul for the prize of a gushing maid

or a fluttering metal stage and worthy patrons’ games,

more a care to put your faith in pressing ways

to do some good shapes your touchingly selfless traits.

Why, then debate life and fame and death and reward?

Rather brave a certain pain and defend the gains of love.

!191

F5 5

Once again you caught my mood, played with it, drew it out,

applauded it, and blew it away. I learnt to sail with you,

surge through waves of doubt with you, scourge gales

with flails of laughter with you, yet still remain my own.

In town, last June, surprised when you called and stayed to brood,

and then said we ought to wed in May, I knew I was lost.

Too soon to get over you, but not too late,

I sought my sunbench haven on the coast, far from ruin.

Then, one day, walking along the shore, safe from you, I thought,

there you stood, staid and fraught, all doom, all wrought, afraid.

The most I could do was praise your hat, and await your gloom.

We stayed throughout the summer, we came back throughout the war.

How long would you stay? How long remain away?

For you, I fled my doubts, for you, I fled my pain.

!192

F6 6

When I once look over the shingle scuffed rock

across the summit ridge to all the immense certainty

that comes from simple awe at wonders born of light,

the sea’s vast opal floor and silent surging vault,

the port’s sudden lilt among the tripping, stumbling

ruts of dipping coast, and, tucked along beside

the water’s curtain glide and salt shimmer sand,

drips of coral sun that glint the dusty shore,

then must I not forsake Thought’s discordant rhyme,

and court the subtle will and solace of Time’s work?

Or, if I sought as such, mind a purer cause,

and take to finding laws just short of sin?

Yet, what’s the truth to find when doubt has lost its worth?

Why, the truth of simple awe at wonders born of light.

!193

F7 7

The ship of painted greed: its thrilled, discretive sway,

its play of being right and wayward, certain ease

that deals and plies and preys on each disabled mind,

to run, too vain or weak, to fight, too meek or guiled.

The sail of teaming pride, its strained insipid breeze,

its spleen of flitter flame and cheerful, baleful spite

that drill and flay and wield the timbal railing day

to sleek and slighting rain that cleaves away the sky.

Yet, stay to see the night, its shade of siren trees,

its fields of latticed sheen and cliffs of searing gray,

its seas of faience flayed and beads of plated light

that say to me defy, disdain the mind unfree.

And when the chains are freed, and when the free proclaim

all greed and pride for slain, what ship, what sail, what main?

!194

F8 8

I left to seek the light, and reading by our tree,

I wished you there with me, that I might be again

the reason why you’d stayed; no need to fade and die,

to tease you pain with rye, to hide your fear, your face.

As if to please itself, a splintered beam of day

bade me desire to stay, to see how may delight

how fey, how free, the mitered pages mild do sheen

and shimmer and sway with leaves of silent beechen gray,

then fade in gleam and fly away to bide unseen

beneath the stillen shade, between the ageless skies.

This sight of traced serene, all choirs of reeling baize,

remained with me a while to brave my knotted grief.

It seemed both tribute paid and scene to praise a life,

a light that played a breeze, a life that eased a breath.

!195

F9 9

The news was still and dark. The few who lived were silent,

mired in calm and spew. I that knew your heart,

what part of you should I remark, could I renew?

Your youth? Your will? Your charm? In truth, no part at all.

For what is this muse’s dance, this tinny, tuneless prance,

but sparks of music spied and psalmed and sighed through

a dewy, spinning glass strewn with darts of rhyme,

from chitter chatter hewn; a mirror’s ruse of shards

that grew in time apart and knew nor chart, nor chime.

And yet, past, true life, ardor lives in few

and dies in far fewer. Why should you then pass

unsung, uncried, for cause this frugal art of mine?

But what balm, what use in this? Nostalgia’s filmy slew?

Sweet pity’s chant. Just this: a wish that beauty last.

!196

F10 10

The shadows mind your name now, your warmth I’ll save inside.

Our window panes are bathed in rill of tallow rain

that stain the darkening sills like wet, white wax

down a steeped stepped stair. Of an evening time,

lightness claims the hearth with rifts of hatch that drape

the air, so fragments of things attain a sifted calm,

like a fine thin veil across a faint thin smile,

or as a braid of hair, tied up, all grays and silver clasps.

Is it too vain then to ask how I shall fade?

At a slow, slighted pace? Or fast, no waste of spite?

Or as grace in taking flight? Or frayed, a plight of harms?

Ask, I may. Escape, I shan’t. Decide, I cannot say.

Yet, when my life is waning, and when the dark rains in,

I’ll braid a veil of lightness, and save your warmth within.

!197

F11 11

How brave we are we both: two graves for homes and hearts.

I dress yours with swathes of asters and sprays of rose.

For mine, I take a small shock of flowers each week

and place it on our shore, in faith with fortunes past.

And yet, how death is coveted by the snares of cloying fate:

a spate of cavilling knocks and petty blows, then scarred,

worn out by care, your loss my only lair.

How hard to make a sense of half so many woes.

Each way seems barred and cold, all gates and frosty paths

along my days and hours. No carriage waits

to bear me off, no gaze, no passing rain, no thought.

A maze unto myself, I am lost where I am found,

ever still where I am bound, I mourn myself before,

before the graves of war, before the aster shore.

!198

F12 12

Burnt leaves of grass, scorched by victory’s blaze,

betrayed by myth’s allure, effaced by torture’s shade.

Drifts of coursing seed, lines of tined wood,

cool lanes winding through sunny miles of rape.

A wide field, mossy stiles between ivy-faced boles

and crackled tilled folds, haystack straw scents

on gloam paling light, soul blent and fey:

the sublime warmth of nature’s life spending stole.

Why, then, risk being scorched again when life is so fraught?

Why rake time burdening my pain with other’s strife?

So that nature’s crackled tilled folds should defray my mortal ills?

Were your mind not stolen, your gifts would find a way,

and in finding a way, you’d lessen others’ strife.

And in lessening others’ strife, less burned, I’d find life?

!199

F13 13

A day of blue, storm light: pageant streams of sun

strew the sky whilst stave of fluted rain ply

my eyes and face as if to shrew my mind of duping plaints,

of harbored guilt and burdens borne without merit.

A trace of cool, gray light plays between the surface

film of a creviced pool; indigo, pewter rays

smooth and spiral, fade and chiral, as beneath the gaping wind

a child looses a kite, a petrel scours the ocean.

Sometimes now I find shades of our summers

in a flurry of laughter, a fall of light, or spill of waves

that remind me love endures though love is lost.

I’m not sorry you desired me, dared my truth and dreams,

yet I rue the life that’s wasted, rue the graves of spite.

So, I slight the spite of rulers with life and ruse and light.

!200

F14 14

The doom of failing lords, their rule of corvine flair,

not waived their warring crimes, not safe from human courts,

no use their craven power to loose a sword of flame

to scorch the truth that names the fraud of their salute.

The swords of rancorous doom that slay with ravin scorn,

not excused their inane slaughter, not freed from moral blame,

no use their suck of power to keep the state insane,

to chain the norms of truth to a painted raven tower.

Yet, who will bear the cost to slew such raucous plague?

And, say, what law allows to save by foreign force?

What sore of wounds remains once foreign hands withdrawn?

To pursue the human course, to prove no life in vain,

fails to better cruelty, when cruelty’s but hate’s game.

So, forgive the cruel their hatred? Forgive, and end hate’s reign?

!201

F15 Postludium

and shimmer and sway with leaves of silent beechen gray

to sleek and slighting rain that cleaves away the sky

from chitter chatter hewn a mirror’s ruse of shards

drips of coral sun that glint the dusty shore

or as a braid of hair tied up all grays and silver clasps

I sought my sunbench haven on the coast far from ruin

how hard to make a sense of half so many woes

against the crush of vain fate no wasteful rush

the sublime warmth of nature’s life spending stole

sent a tide through my mind that washed over the past

a child looses a kite a petrel scours the ocean

and back the weak and needy live life to strive for more

to chain the norms of truth to a painted raven tower

to pursue the course of light not the quietening of a breath

!202

Documents

Relating the Formal Characteristics of the Sonnet: A ... · Relating the Formal Characteristics of the ... Sonnet Writing 137 3.1.6 The Problem of Aggregation ... following formal