Relating the Formal Characteristics of the Sonnet:
A Theory of Centred Form
by
Kevin J. M. Keane
June, 2011
© 2011 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 characterised 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 8
1.2.1 Concepts, Definitions and Rules 8 1.2.2 Step-by-step Description of Array Model Development 17 1.2.3 Assessment 25 1.2.4 Conclusion 31
Part 2 Modelling Sonnet Traditions
2.0 Introduction 32
2.1 Early Italian Tradition 32
2.1.1 Simplified Rhyme Schemes 32 2.1.2 Centre Array Derivation and Array Model Development 37 2.1.3 Assessment 52 2.1.4 Conclusion and Outlook 52
2.2 Petrarchan Tradition 54
2.2.1 Simplified Rhyme Schemes 54 2.2.2 Centre Matrix 58 2.2.3 Step-by Step Description of Array Model Development 59 2.2.4 Assessment 73 2.2.5 Conclusion 74
2.3 Pleadean Tradition 76
2.3.1 Simplified Rhyme Schemes 76 2.3.2 Pleadean 1: Centre Arrays and Array Models 77 2.3.3 Assessment: Pleadean 1 Array Models 79 2.3.4 Pleadean 1: Centre Sequence and Triangle Models 82 2.3.5 Pleadean 2: Array Models 86 2.3.6 Assessment: Pleadean 2 Array Models 88 2.3.7 Pleadean 1 & 2: Sequence Models 88 2.3.8 Conclusion 100
(cntd.)
iv
Contents Page
Page Part 2 Modelling Sonnet Traditions (cntd.)
2.4 Shakespearean Tradition 101
2.4.0 Introduction 101 2.4.1 Simplified Rhyme Scheme 101 2.4.2 Centre Array 101 2.4.3 Simple Array and Triangle Models 102 2.4.4 Complex Array Model: Step-by-step Description 107 2.4.5 Assessment 113 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 127
3.1 Memorial Day: The Unmaking of a Sonnet 127
3.1.1 Sonnet Cycle Centre Matrix: Sonnet 8 127 3.1.2 Rules for Array Development 129 3.1.3 Memorial Day Array Model: Step-by-step Description 130 3.1.4 Assessment 135 3.1.5 Link between Sonnet Pattern and Sonnet Writing 136 3.1.6 The Problem of Aggregation 138 3.1.7 Conclusion 139
Part 4
4.0 General Conclusion 140
References 142
Appendices
A Early Italian Model: Unsuitability of Two-Type, Three Element Arrays 145 B Early Italian Model: Centre Array Derivation 148 C Petrarchan Model: Centre Matrix Derivation 162 D Pleadean Models: Centre Array Derivation 171 E Shakespearean Model: Centre Array Derivation 174 F Memorial Day: The Unmaking of a Sonnet, Poems 185
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 16
7 Initial Leftwards versus Rightwards Development 24
8 Working Array Model 30
9 Working Model (WM) vs. ‘Simplified’ Early Italian (EI) Rhyme Scheme: Comparison of End-Array Element Changes 35
10 Unsuitability of Two-Element Centre Array 38
11 Unsuitability of Two-Element Centre Array: Change of Directionality at Array 6 39
12 Unsuitability of Three-Element Centre Array: Leftwards & Rightwards Development 40
13 Unsuitability of 4:1 Distribution of Centre Array Elements 41
14 Redundancy in a 4:1 Distribution of Centre Array Elements: Leftwards Development with Directionality Change in Array 6 42
15 Early Italian Model: Centre Array Candidates 1 43
16 Early Italian Model: Centre Array Candidates 2 44
17 Development of Centre Array through Arrays 5 & 11 46
18 Development of Arrays 4–1 and 12–15 47
19 Continuous Redundancy in Array Development 48
20 Early Italian Fifteen Array Models ‘b b a b a’ and ‘b a b b a’ 49
(cntd.)
vi
List of Tables
Table Page
21 Identical 14-Array Sub-Models 50
22 Early Italian Array Models 51
23 Petrarchan Model: Overlapping Paired and Alternating Rhyme Equivalents 1 54
24 Petrarchan Model: Overlapping Paired and Alternating Rhyme Equivalents 2 55
25 Petrarchan Flow Pattern 1 56
26 Petrarchan Flow Pattern 2: Sub-Patterns (a), (b), (c) & (d) 57
27 Petrarchan Centre Matrix 58
28 Petrarchan Array Models 71
29 Identical Petrarchan Array and Triangle Models 72
30 Pleadean 1 & 2: Comparison of Simplified Rhyme Schemes 76
31 Pleadean 1: Simplified Rhyme Scheme with Symmetry 76
32 Pleadean 2: Simplified Rhyme Scheme without Symmetry 77
33 Pleadean 1: Array Models 78
34 Pleadean 1: Array Series Redundancy 80
35 Distribution of Four Stresses, (x), in a Four-Element Array 81
36 Pleadean 1: Accommodation of Conventional Rhyme Scheme 81
37 Pleadean 1: Triangle Models’ Centre Sequence 82
38 Pleadean 2: Array Models 87
39 Pleadean 1: Sequence Model: Leftwards Directionality 90
40 Pleadean 2: Sequence Model: Leftwards Directionality 91
41 Pleadean 2: Sequence Model: Chirality 94
42 Pleadean 1: Sequence Model: Chirality between Model Halves 96
(cntd.)
vii
List of Tables
Table Page
43 Pleadean 1: Sequence Model: Symmetry within Model Halves 96
44 Pleadean 1: Sequence Model: Leftwards and Rightwards Directionality 98
45 Pleadean 2: Sequence Model: Leftwards and Rightwards Directionality 99
46 Simple Shakespearean Array Model 103
47 Simple Shakespearean Triangle Model: Binary Expansion 104–106
4 8 Identical Simple Shakespearean Array and Triangle Models 107
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 128
57 Memorial Day: Centre Matrix Buildup 128
58 Memorial Day: Array Model 134
59 Memorial Day: Array 8 as Volta Equivalent 135
60 Centre Matrix: Distribution of Key Vowels 1 137
61 Centre Matrix: Distribution of Key Vowels 2 137
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
formal characteristics relate to each other, but also how they arise and thus, perhaps,
!2
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 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 may be thought of as 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;
2. five models of classic sonnet forms from the scuola siciliana,
Petrarchan, Pleadean, from which two, and Shakespearean traditions based on the
principle of centred form to show how the claim works in theory;
!3
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
It struck me that these words succinctly captured what for me was the
sonnet’s generally sombre tone. There was also the development of a concrete
representation of death in “all girded up with white” to a more abstract
The Arden Shakespeare. Ed. Duncan-Jones, K. 2005. 2
!4
representation of it 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 text also bore reading linearly, that
is, conventionally, from top left to bottom right. Moreover, it could also be made
sense of, with some grammatical tolerance, when read from bottom to top. These
observations appeared to me to lend support to the idea of a centred origin to at least
one sonnet for they showed that writing about a centre could create meaning.
However, expecting that I was probably making too much of a glance, I decided to
check further. Sonnet 13 seemed to reveal the same pattern:
beauty
after
issue
honour
These words appeared to me to develop the ‘persuasion to marry’ theme
persistent in this group of sonnets (Ellrodt, 1986, p. 38) and to do so with the same
movement from concrete to abstract representation seen in sonnet 12. The evidence
of sonnet 14, however, cast doubt on any idea of a centred origin to its writing:
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 followed with:
!5
men
youthful sap
brave state
rich in youth
I started to think that my speculation was probably mistaken. 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 in lines 7 and 8 that could subsequently be
developed into a sonnet. Yet how was I to try out this idea? Where was the evidence
for it to be collected? There certainly appeared to be little in the scholarly literature
to suggest that the sonnet 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 possible coming from a Sicilian variety of the arabic zajal
(1915, p. 494). Wilkins then later viewed a Sicilian form of the strambotto, the
!6
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). Stephen Burt and David Mickics (2010), for their part,
emphasize the contribution of the “scientific advances of Islamic North Africa along
with the chivalric habits and troubadour poetry of southern France” as an influence
on 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 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 Petrarch’s use of patterning in his sonnet writing, a sestina is not after
all a sonnet, yet its presence in the same collection with his sonnets made the
likelihood of an equivalent underlying pattern for the sonnet seem more likely. 3
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 in Singapore in June, 2010.
!7
Furthermore, the presuppositions behind the creation of the sonnet being
unknown, there was no compelling reason to assume that the sonnet was conceived
with the formal characteristics noted by Hobsbaum in mind. It was therefore still an
open question 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.
Finally, a methodological advantage of postulating the existence of a single
ordering principle to account for the sonnet’s formal characteristics also presented
itself, namely, 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 were 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 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 I mean the 4
property of recurring at regular intervals. By way of illustrating this concept
For background, see: Weisstein, Eric W. "Cyclic Number." From MathWorld–A Wolfram 4 Web Resource. (http://mathworld.wolfram.com/CyclicNumber.html)
!8
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 even in their sum, but in the way the regular development of digit
pairs forms a revolving pattern as they cycle about each other from one fraction to
the next. It is just such a patterning mechanism that will be applied in the models to
develop the equivalents of the sonnet’s formal characteristics from a simple array of
elements.
However, before considering this idea of patterning further, for the sake of
clarity of exposition in the discussion to follow, the zeroes before the decimal points
in the preceding table are dropped in Table 2 below, as are the decimal points, the
addition sign and the sum total. Moreover, the digit pairs of each decimal fraction
are arranged into three separate columns with line numbers and letters added for
ease of reference; lastly, arrows highlight the movement of the digit pairs from one
array to the next.
!9
Table 2 Array Elements a b c
1. 14 28 57 ↙ ↙ 2. 28 57 14 ↙ ↙ 3. 57 14 28
Regarding terms, to distinguish patterning as a concept from the process by
which it is achieved, the building-up of arrays and concomitant changes in the posi-
tions of digit pairs from array to array will be termed development and the digit pairs
elements. Thus, patterning is defined here 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, is
determined by two 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
!10
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 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
!11
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.
Now, if the elements and their order in arrays 1 and 4 are identical, but their
away flows are not, then the away flows from array 4 to array 5 must be different
from the away flows of the previously developed arrays, that is, a change in direc-
tionality 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 understand why this cannot be the case here
requires a consideration of the role that innovation and redundancy are deemed to
play within 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 through array repetition. Array development therefore consists of two
phases, an initial innovative phase developed from a start array, and a second redun-
dancy 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
!12
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
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
!13
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 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:
!14
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.
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
!15
writing of a sonnet may start 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 falsifiable explanation 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:
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
!16
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.
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.
!17
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.
Step 4 arrays 5 & 11
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 cyclical
!18
identity with the centre array, Rule 2 is applied, and directionality in away flows
changes from left to right.
Step 5 arrays 4 & 12
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.
!19
Step 6 arrays 3 & 13
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 right-
wards direction.
!20
Step 7 arrays 2 & 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
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.
!21
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.
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
!22
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. As its
development has been described in detail, however, the model with leftwards
development will be adopted for the purposes of discussion.
!23
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.
!24
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, 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
!25
Rule 3 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, 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 must continue from array 2 to develop array 1 for
only then are its towards flows complete and the conditions necessary for
redundancy, according to Rule 3, fulfilled.
Is it certain, however, 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, were development to
continue, an innovative array might be developed at the fifteenth, or fifteen-
hundredth 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, or
fifteen-hundredth, array cannot be innovative. Alternatively put, if there can be no
!26
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 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 deduction 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 a 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
!27
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 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 equival-
ently 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 indeed.
!28
I shall deal with this problem of the exhaustion of inexhaustibility for now
by ignoring it and limit the inquiry to the five rhyme schemes identified by Kircher
(1979) as representative of the Italian, French and English language sonnet tradi-
tions. 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 ling-
uistic variety of sonnet traditions selected for this inquiry: Any fewer might be
construed as too weak a test for the claim, any more as perhaps 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, the flow lines between the
equivalents of octave and sestet and quatrains and tercets have been removed and
the spaces between them widened slightly.
!29
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
!30
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.
!31
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 Shake-
spearean 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 what shall be
!32
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 quatr-
ains 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 mun-
dane, 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.
!33
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. In this table,
change or lack of change from one end-array element to the next in the working
model (WM) is compared with change or lack of change within the simplified
rhyme scheme of the Early Italian (EI) sonnet developed above.
!34
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
6. b change change b
7. a change change a
8. c start change b
9. a change change a
10. b change change b
11. c change change a
12. b change change b
13. a 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. From these findings, it can be
!35
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 func-
tions 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
!36
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 place-
holder 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.
!37
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
The alternative of changing directionality at array 6 to avoid redundancy
only postpones redundancy until array 3, itself a symbolic and cyclical repetition of
!38
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 altern-
ating 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 direc-
tionality 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 the centre array, ‘a b b’ or ‘b b a’, produce
similar results when developed, as may be seen in Appendix A.
!39
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
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
!40
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 ‘b a a a a’. Neither of these alternatives will do, as,
by simple inspection of Table 13, alternation of placeholders beyond the develop-
ment 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
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:
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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 distribu-
tion 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 mathematical rule for
combinations. This rule gives the number of possible centre arrays as ten, which 2
confirms the ten arrays, presumed exhaustive in the ordering of their elements, listed
in Table 15, below:
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.
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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 are
contiguous. There are, therefore, only five centre arrays, listed in Table 16, that
might still satisfy all five sonnet conditions:
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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 are excluded: arrays ii.), iii.), vi.), with rightwards and 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
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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.
Leftwards development from the centre array ‘b a b b a’ creates the follow-
ing first three pairs of arrays, 7–5 and 9–11, as shown in Table 17 below:
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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 devel-
opment of arrays 4–1 and 12–15 completes the model, as shown in Table 18 below:
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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 further
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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 can be 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.
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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.
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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.
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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
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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 accom-
modate 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 applying
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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.
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2.2 Petrarchan Tradition
2.2.1 Simplified Rhyme Schemes
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
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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.
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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:
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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
10. b a b a b a b a b a b a b a b a
11. a b a b a b a b a b a b a b a b
12. b a b a b a b a b a b a b a b a
13. a b a b a b a b a b a b a b a b
14. b a b a b a b a b a b a b a b a
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,
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whilst the bolded 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
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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-
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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 left blank deliberately.)
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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
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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.
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Petrarchan sonnet pattern: Development of Sub-Pattern (a)
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.
Step 4 Arrays 4 and 11
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.
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Petrarchan sonnet pattern: Development of Sub-Pattern (a)
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.
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Step 5 Arrays 3 and 12
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.
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Petrarchan sonnet pattern: Development of Sub-Pattern (a)
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.
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Step 6 Arrays 2 and 13
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.
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Petrarchan sonnet pattern: Development of Sub-Pattern (a)
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.
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Step 7 Arrays 1 and 14
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.
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Petrarchan sonnet pattern: Development of Sub-Pattern (a)
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
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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
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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
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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.
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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
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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.
!75
2.3 Pleadean Tradition
2.3.1 Simplified Rhyme Schemes
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 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
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
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.
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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
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
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 downwards to the left for the ‘a a b b’ and to the right for the ‘a b b a’ centre
array;
Rule 2. Directionality changes in array 11;
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Rule 3. Series redevelopment of the first four arrays in upwards, and
symmetrical 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
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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.
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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
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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 tétramètre 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, not unexpectedly, 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. 532
! 81
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, applying to it the same development rules. 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
triangle model’s construction to follow, a second rule for ending development is
unnecessary:
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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
(p. 77). As shall be seen below, the triangle model can only be fully developed if this
end-array element is included in the centre sequence.
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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 3
highlight the parts of the sequences that correspond to the Pleadean 1 array models.
p. 723
! 84
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 in Table 33 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
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these directionality 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.
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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
! 87
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
! 88
array and triangle models are not broad enough in the sense that they do not provide
enough, what might be called, ‘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.
! 89
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.
(...)
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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.
(...)
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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 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)
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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
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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. They do, however, coincide if folded once, with array 1 acting as the axis of
symmetry. 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-
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hand side of the model. Once again, the simplified rhyme schemes overlap in the
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.
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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
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This analysis of the equivalents of the two Pleadean sonnet traditions under
consideration shows that whilst 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. 4
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. 754
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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.
...
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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.
...
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2.3.8 Conclusion
The Pleadean sonnet traditions are seen here as a means to 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.
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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 apply-
ing 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.
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
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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 pro-
cedure, 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.
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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.
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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
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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
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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
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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
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be constructed from the simple triangle model described above. 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.
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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.
(The remainder of this page is deliberately left blank.)
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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.
(The remainder of this page is deliberately left blank.)
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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 array
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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.
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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
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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 ele-
ments 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 alternat-
ing, the condition for the Shakespearean rhyme scheme is also satisfied. As to the
volta equivalent, the point of greatest contrastive development before the disag-
gregation 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 characteristics of the Shakespearean
!114
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:
Rule 1. Arrays develop simultaneously downwards and upwards to
the right from arrays 1 & 15, respectively, of the simple
triangle model;
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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.
Step 2 Arrays 2 & 14
1. a b a b b ↘ ↘ ↘ ↘ 2. b a b a b
...
14. b a b a b ↗ ↗ ↗ ↗ 15. a b a b b
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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
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.
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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.
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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 immaterial difference
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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
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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, thus mitigating, but not eliminating, 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 1
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 trad-
ition 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.
p. 531
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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 model’s
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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 incom-
plete is reflected accordingly in the rest of the discussion by foregoing the
directionality change in array 2 of the Early Italian array models. 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.
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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 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.
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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
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
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of their developments and the arrays in which changes in directionality are intro-
duced. 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-super-2
posable 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. 3
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 provided by a
brief introduction to centred writing by way of the author’s sonnet cycle, Memorial
Day: the Unmaking of a Sonnet.
p.1182
p. 473
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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
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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
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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.
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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.
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Step 3 Arrays 5 & 10
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.
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Step 4 Arrays 4 & 11
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.
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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.
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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
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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
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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 the memory of reading a vocalic analysis of Verlaine’s Il 2 pleure dans mon coeur (Chiss, Filliolet et Maingeneau, 1977, II, pp. 123–124 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.
!136
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.
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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
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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.
!139
Part 4
4.0 General Conclusion
The 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 devel-
oped from, the sonnet’s centre. The balance of evidence presented in the inquiry sup-
ports 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, evidence is furnished by way of theo-
retical models showing how complex equivalents of the formal characteristics of
five sonnet traditions, the so-called Early Italian, the Petrarchan, two Pleadean and
the Shakespearean, are 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, equi-
valents 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 array development.
Finally, the equivalent of fourteen line sonnet length is effected by the limit between
!140
innovation and redundancy in array development. To mitigate the risk of error and
bias in the evidence presented, besides the array model, a triangle model is devel-
oped to independently cross-check array model results. It is also shown that the
array and triangle models are situated within a broader pattern of binary expansion
having 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 under-
standing of other sonnet traditions. The establishment of symmetrical and chiral
properties in the models also provides 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, I think, 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.
!141
References
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!144
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
leftwards rightwards
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
!145
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.
!146
Table A1.3: Centre Array ‘b b a’: Leftwards and Rightwards, Upwards Development
leftwards rightwards
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.
!147
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
(p. 44).
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
!148
mechanism does develop. However, when in later models this redundancy
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’
leftwards rightwards
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.
!149
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,
!150
as occurs in arrays 6-5 and 10-11 of both leftwards and rightwards development, a
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’
leftwards rightwards
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.
!151
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
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.
!152
Table B4.1 Centre Array: vi.) ‘b a b a b’
leftwards rightwards
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.
!153
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
!154
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.
!155
Table B5.1 Centre Array: vii.) ‘b a b b a’
leftwards rightwards
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.
!156
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
!157
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.
!158
Table B6.1 Centre Array: ix.) ‘b b a b a’
leftwards rightwards
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.
!159
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
!160
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
!161
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:
!162
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
!163
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,
!164
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 (pp. 91-92).
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.
!165
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.)
leftwards rightwards
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
!166
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.)
leftwards rightwards
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.
!167
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
!168
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.)
leftwards rightwards
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.
!169
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.
!170
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 (p.75) 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.
!171
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.
!172
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.
!173
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
!174
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.
!175
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
!176
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’
leftwards rightwards
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.
!177
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’
leftwards rightwards
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
!178
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
!179
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’
leftwards rightwards
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.
!180
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
!181
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’
leftwards rightwards
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.
!182
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
!183
E7.0 Solutions
There is, thus, only one centre array with placeholders that can
accommodate the equivalents of the Shakespearean formal characteristics: ‘a b b a b'
with leftwards and rightwards directionality.
!184
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
!185
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
!186
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?
!187
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’.
!188
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.
!189
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.
!190
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.
!191
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.
!192
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?
!193
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.
!194
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.
!195
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.
!196
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.
!197
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?
!198
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.
!199
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?
!200
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
!201