Relating the Formal Characteristics of the Sonnet:
A Theory of Centred Form
by
Kevin J. M. Keane
October, 2015
© 2011, 2015 (revised) Kevin J M Keane
All rights reserved.
Kevin J M Keane
Abstract
Relating the Formal Characteristics of the Sonnet: A Theory of Centred Form
The sonnet is characterized formally by the separated categories of number of lines,
stanzaic form, volta, isometry and rhyme scheme. This inquiry sets out instead to
uncover and weigh evidence for the claim that a sonnet unfolds from its centre to
form a pattern in which its formal characteristics inhere. This idea is referred to as
the theory of centred form. Theoretical evidence is provided by the construction of a
working model from first principles and the subsequent modelling of the formal
characteristics of five classic sonnet traditions. From simplified rhyme schemes,
centre arrays and two-array centre matrices of four and five elements are deduced
and tested by developing them into array models. In each of the models presented,
equivalents of the sonnet's formal characteristics unfold from the model's centre: the
equivalent of isometry results from the development of a fixed array of elements; the
equivalent of the volta is deemed to occur at the point of greatest contrast between
directionality flows in the models; the equivalent of stanzaic form results from
changes in directionality; rhyme scheme equivalents result from cyclicity in array
development; and the equivalent of fourteen line sonnet length in the models is
effected by the limit between array innovation and redundancy. To mitigate the risk
of error and bias in the array models, a second type of model is developed
independently of them to act as a cross-check on their results. Finally, practical
evidence for the claim is furnished in the centred form sonnet cycle, Memorial Day:
the Unmaking of a Sonnet. The balance of evidence strongly supports the claim: A
simple binary pattern unfolding from the equivalent of the sonnet's centre relates
equivalents of the sonnet’s so-called formal characteristics and, in so doing,
suspends the boundary between reflective thought and creative writing.
iii
Contents Page
Page
Part 1
The Sonnet as Centred Form
1.0
Introduction
1
1.1
Rationale for the Inquiry
4
1.2
Working Model
9
1.2.1
1.2.2
1.2.3
1.2.4
Concepts, Definitions and Rules
Step-by-step Description of Array Model Development
Assessment
Conclusion
Part 2
Modelling Sonnet Traditions
2.0
Introduction
33
2.1
Early Italian Tradition
33
2.1.1
2.1.2
2.1.3
2.1.4
Simplified Rhyme Schemes
Centre Array Derivation and Array Model Development
Assessment
Conclusion and Outlook
33
38
53
53
2.2
Petrarchan Tradition
55
2.2.1
2.2.2
2.2.3
2.2.4
2.2.5
Simplified Rhyme Schemes
Centre Matrix
Step-by Step Description of Array Model Development
Assessment
Conclusion
55
59
60
74
75
2.3
Pleadean Tradition
77
2.3.1
2.3.2
2.3.3
2.3.4
2.3.5
2.3.6
2.3.7
2.3.8
Simplified Rhyme Schemes
Pleadean 1: Centre Arrays and Array Models
Assessment: Pleadean 1 Array Models
Pleadean 1: Centre Sequence and Triangle Models
Pleadean 2: Array Models
Assessment: Pleadean 2 Array Models
Pleadean 1 & 2: Sequence Models
Conclusion
9
18
26
32
77
78
80
83
87
89
89
101
(cntd.)
iv
Contents Page
Page
Part 2
Modelling Sonnet Traditions (cntd.)
2.4
Shakespearean Tradition
102
2.4.0
2.4.1
2.4.2
2.4.3
2.4.4
2.4.5
2.4.6
2.4.7
Introduction
Simplified Rhyme Scheme
Centre Array
Simple Array and Triangle Models
Complex Array Model: Step-by-step Description
Assessment
Complex Triangle Model: Step-by-Step Description
Conclusion and Retrospective
102
102
103
103
108
114
115
121
Part 3
Centred Writing
3.0
Introduction
128
3.1
Memorial Day: The Unmaking of a Sonnet
128
3.1.1
3.1.2
3.1.3
3.1.4
3.1.5
3.1.6
3.1.7
Sonnet Cycle Centre Matrix: Sonnet 8
Rules for Array Development
Memorial Day Array Model: Step-by-step Description
Assessment
Link between Sonnet Pattern and Sonnet Writing
The Problem of Aggregation
Conclusion
128
130
131
136
137
139
140
Summary and Outlook
141
Part 4
4.0
References
142
Appendices
A
B
C
D
E
Early Italian Model: Unsuitability of Two-Type,
Three Element Arrays
Early Italian Model: Centre Array Derivation
Petrarchan Model: Centre Matrix Derivation
Pleadean Models: Centre Array Derivation
Shakespearean Model: Centre Array Derivation
146
149
163
172
175
F
Memorial Day: The Unmaking of a Sonnet, Poems
186
v
List of Tables
Table
Page
1
Cyclicity
2
Array Elements
10
3
Individuation of Array Elements
11
4
Directionality Change
12
5
Array Innovation and Redundancy
13
6
Summary of Array Development
17
7
Initial Leftwards versus Rightwards Development
25
8
Working Array Model
31
9
Working Model (WM) vs. ‘Simplified’ Early Italian (EI)
Rhyme Scheme: Comparison of End-Array Element Changes
36
10
Unsuitability of Two-Element Centre Array
39
11
Unsuitability of Two-Element Centre Array:
Change of Directionality at Array 6
40
Unsuitability of Three-Element Centre Array:
Leftwards & Rightwards Development
41
13
Unsuitability of 4:1 Distribution of Centre Array Elements
42
14
Redundancy in a 4:1 Distribution of Centre Array Elements:
Leftwards Development with Directionality Change in Array 6
43
15
Early Italian Model: Centre Array Candidates 1
44
16
Early Italian Model: Centre Array Candidates 2
45
17
Development of Centre Array through Arrays 5 & 11
47
18
Development of Arrays 4–1 and 12–15
48
19
Continuous Redundancy in Array Development
49
20
Early Italian 15-Array Models ‘b b a b a’ and ‘b a b b a’
50
12
9
(cntd.)
vi
List of Tables
Table
Page
21
Identical 14-Array Sub-Models
51
22
Early Italian Array Models
52
23
Petrarchan Model: Overlapping Paired
and Alternating Rhyme Equivalents 1
55
Petrarchan Model: Overlapping Embracing
and Alternating Rhyme Equivalents 2
56
25
Petrarchan Flow Pattern 1
57
26
Petrarchan Flow Pattern 2: Sub-Patterns (a), (b), (c) & (d)
58
27
Petrarchan Centre Matrix
59
28
Petrarchan Array Models
72
29
Identical Petrarchan Array and Triangle Models
73
30
Pleadean 1 & 2: Comparison of Simplified Rhyme Schemes
77
31
Pleadean 1: Simplified Rhyme Scheme with Symmetry
77
32
Pleadean 2: Simplified Rhyme Scheme without Symmetry
78
33
Pleadean 1: Array Models
79
34
Pleadean 1: Array Series Redundancy
81
35
Distribution of Four Stresses, (x), in a Four-Element Array
82
36
Pleadean 1: Accommodation of Conventional Rhyme Scheme
82
37
Pleadean 1: Triangle Models’ Centre Sequence
83
38
Pleadean 2: Array Models
88
39
Pleadean 1: Sequence Model: Leftwards Directionality
91
40
Pleadean 2: Sequence Model: Leftwards Directionality
92
41
Pleadean 2: Sequence Model: Chirality
95
42
Pleadean 1: Sequence Model: Chirality between Model Halves
97
24
(cntd.)
vii
List of Tables
Table
Page
43
Pleadean 1: Sequence Model: Symmetry within Model Halves
97
44
Pleadean 1: Sequence Model: Leftwards and Rightwards
Directionality
99
45
Pleadean 2: Sequence Model: Leftwards and Rightwards
Directionality
100
46
Simple Shakespearean Array Model
104
47
Simple Shakespearean Triangle Model: Binary Expansion
48
Identical Simple Shakespearean Array and Triangle Models
108
49
Identical Shakespearean Sub-Models
112
50
Complex Shakespearean Array Models:
Leftwards and Rightwards Developments
113
51
Shakespearean Complex Array and Triangle Models
120
52
Related Early Italian and Shakespearean Models 1
122
53
EIM (RHS) and Shakespearean Models: Shared Centre Sequence 123
54
EIM (LHS) and Shakespearean Models: Shared Centre Sequence 124
55
Related Early Italian and Shakespearean Models 2
125
56
Memorial Day: Centre Matrix: Internal Elements
129
57
Memorial Day: Centre Matrix Buildup
129
58
Memorial Day: Array Model
135
59
Memorial Day: Array 8 as Volta Equivalent
136
60
Centre Matrix: Distribution of Key Vowels 1
138
61
Centre Matrix: Distribution of Key Vowels 2
138
viii
105–107
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.1 This result
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
1
The first two sentences of this paper are an homage to the mathematician John Nash.
!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 sixstress lines? How might the number of stresses in a line be tied to a pre-set rhyme
scheme?
Whilst much scholarly effort has been invested in analysing and
interpret-ing sonnets by considering the overall contribution made by the sonnet’s
formal characteristics taken separately, much less attention has been paid to
considering the separate contribution made by its formal characteristics taken as a
whole. This is perhaps not surprising as such an approach presupposes that the
sonnet’s formal characteristics are but aspects of an ordering principle that is hard
to discern.
From the positing of such a principle, however, follows the inference of
a cohesive, underlying sonnet pattern in which the sonnet’s formal characteristics
inhere. If such a sonnet pattern could be found, it might reveal not only how the
!2
formal characteristics relate to each other, but also how they arise and thus,
perhaps, contribute to a better understanding of why the sonnet has proved so
popular for so long across so many cultures. Yet, does such a sonnet pattern exist
and, if so, what sort of pattern might it be? How, moreover, is it to be found? The
cost of not finding answers to these questions is considerable as it means
admitting that the form that has produced no small amount of Western
civilization’s best writing and thought over the past eight centuries remains not
only inexplicable, but inexplicably inexplicable.
I shall claim a possible answer to these questions based on the idea that
the formal characteristics of the sonnet are byproducts of a pattern of elements
originating in and unfolding from the sonnet’s centre. In other words, I shall seek
to support the claim that the sonnet’s formal characteristics are better understood
by considering that they all begin at the centre of the sonnet and are developed
from it in a pattern that manifests all the characteristics noted by Hobsbaum.
In sympathy with the view that considers poetic form not as object to be
exploited, but as possibility to be discovered (Lennard, 1996, p. 25), I shall refer
to this idea as the theory of centred form, and sonnet writing based on it as centred
writing. To show that this claim is reasonable by providing good evidence for it,
that is, evidence both sound and sufficient (Turabian, 2007, p. 60), I shall offer:
1.
a rule-based development of a sonnet working model from a centre
array of three elements;
!3
2.
five models of classic sonnet forms from the scuola siciliana, Petrarchan, Pleadean, from which two, and Shakespearean traditions based on
the principle of centred form to show how the claim works in theory;
3.
my own centred form model and sonnet cycle to show how centred
sonnet writing works in practice.
For clarity’s sake, I should like to emphasize that my desired aim in
undertaking this inquiry is to change the way in which the sonnet at the formal
level is understood and represented today. My main aim is not to provide a means
to analyse or interpret the work of other poets. If some of the ideas presented here
do help in the better appreciation of their work, that would, naturally, be very
gratifying. However, that is not my chief concern. This inquiry has been
undertaken because I want to know whether and, if so, how the formal elements of
the sonnet relate to each other in order that readers might have a deeper
appreciation of the sonnet’s beauty and so find more pleasure in their reading.
1.1
Rationale for the Inquiry
The idea of centred form and centred writing occurred to me when,
happening to glance at a copy of Shakespeare’s Sonnets2 lying open on a table one
day, a few words from the middle of Sonnet 12 caught my eye:
I see
all girded up
with white
the wastes
2
The Arden Shakespeare. Ed. Duncan-Jones, K. 2005.
!4
It struck me that these words succinctly captured what I considered to be
the sonnet’s generally sombre tone. There was also the development of an imaged
representation of death in the two phrases “all girded up with white” to its more
abstract representation in “I see the wastes”, a binary construct typical of sonnet
writing. As a consequence, I wondered, speculatively enough, to be sure, whether
the writing of the sonnet might have started at this point. The words made sense
when read linearly, of course, that is, conventionally, from top left to bottom right.
However, they could also be made sense of, with some grammatical tolerance,
when read from bottom to top. These observations encouraged me in my
speculation since they suggested that writing from the centre could create
meaning or, at least, start to create meaning. Expecting, though, that I was
probably making too much of a glance, I decided to check further. Words from the
middle of equivalent lines, that is, lines 5, 7, 8, and 10, in Sonnet 13 seemed to
reveal the same pattern:
beauty
after
issue
honour
The diction here appears to develop the ‘persuasion to marry’ theme
regarded as persistent in this group of sonnets (Ellrodt, 1986, p. 38) and to do so
with the same movement from imaged to abstract representation seen in sonnet
12. The words from the middle of equivalent lines in sonnet 14, however,
disabused me:
!5
fortune
princes
I
in them
Even broadly construed, “princes I fortune in them” would hardly bear
interepretation as a continuation of the ‘persuasion to marry’ motif. The step from
concrete to abstract representation was also missing. When sonnet 15 offered:
men
youthful sap
brave state
rich in youth
I started to think that my speculation was probably just that. After a quick glance
through the remaining sonnets, I became quite convinced of it. Here, for example,
is the diction from the centre of sonnet 154:
votary
general
virgin
took heat
There appeared then to be little evidence from the sonnets as a whole to
support the notion that the writing of a sonnet might start with verses 7 and 8.
However, the idea that it might be possible to write a sonnet from its centre
nevertheless took hold for it seemed plausible in at least two instances that some
kind of pattern was being established that might then be developed into a sonnet.
!6
Yet how was I to try out this idea? Where was the evidence for it to be collected?
There appeared to be little in the scholarly literature on the sonnet to suggest that
it was designed to be written from its centre. On the contrary, Ernest Hatch
Wilkins (1915, 1959), cited by Borgstedt (2009, pp. 120-121), at first supported
the hypothesis that the sonnet’s octave was inspired by one of the earliest Italian
verse forms, the eight verse, single stanza Sicilian strambotto, with the sestet possibly coming from a Sicilian variety of the arabic zajal (1915, p. 494). Wilkins
then later viewed a Sicilian form of the strambotto, the canzuna, as the source of
the octave and the sestet as “a wonderfully appropriate conclusion” (1959, p. 39)
devised by the probable inventor of the form, Giacomo da Lentino. Jost (1989)
favours instead the romano-provençal and sicilian-arabic spheres as the primary
influence on the sonnet's genesis (p. 39). Oppenheimer (1989) radically extends
these opinions by suggesting that “the sonnet’s peculiar fourteen-line structure...is
traceable to Plato’s Timaeus, with its mathematical description of the architecture
of the human soul and of heaven” (p. 3). Kemp (2002) prefers to divide theories
of the sonnet’s origins between the Provençal canzone and the strambotto (p. 46).,
whilst Stephen Burt and David Mickics (2010) emphasize the contribution of the
“scientific advances of Islamic North Africa along with the chivalric habits and
troubadour poetry of southern France” as influencing the sonnet’s origins ( p. 6).
I do not propose to weigh here the relative merits of these opinions as
there is evidence to be found within the sonnet tradition itself for the relevance of
patterning to sonnet writing. Francesco Petrarca, perhaps the most celebrated
sonnet writer of the Italian Renaissance, was undoubtedly familiar with the poetic
potential of patterning as the sestinas in his Canzoniere reveal a close familiarity
!7
with the interlacing retrogradatio cruciata pattern of Arnault Danièl’s emblematic
sestina ‘Lo ferm voler’ (Shapiro, 1980; Spanos, 1978). This might only be
circumstancial evidence for the use of patterning in his sonnet writing, a sestina is
not after all a sonnet, yet the presence along with sonnets in Petrarch’s collection
made the likelihood of an equivalent underlying pattern for the sonnet more
plausible.3
Furthermore, the presuppositions behind the creation of the sonnet being
unknown, there was no compelling reason to assume that the sonnet was
conceived with any or all of the formal characteristics noted by Hobsbaum in
mind. It was still an open question therefore whether the formal characteristics
themselves were no more than a scholarly fata morgana: various useful,
stimulating, even necessary, categories for sonnet analysis and interpretation, yet
also, possibly, an anachronistic distraction and hindrance to a more cohesive
appreciation of the genre.
A methodological advantage of assuming a single ordering principle to
account for the sonnet’s formal characteristics was its inference that the formal
characteristics were inextricably linked. This meant that starting at any one of
them would lead to all of the others, greatly simplifying the work involved: If
such a connection could be found, it would be a strong indication of a common
sonnet ordering principle; if not, it would make it more likely that a sonnet pattern
3
I should like to thank the American author, poet, drama and literary critic, Richard Lord,
for raising the question of a possible connection between sestina and sonnet patterns at
the book launch of Memorial Day: the unmaking of a sonnet at BooksActually in
Singapore in June, 2010.
!8
did not exist, or that the approach adopted was inadequate, in either case putting a
quick end to the inquiry.
1.2
Working Model
1.2.1
Concepts, Definitions and Rules
I should like to begin the discussion of the working model by showing
how ‘parts’ may together form a ‘whole’ that in turn becomes a part of a greater
whole. To illustrate this idea, I shall introduce the concept of cyclicity,4 by which
I mean the property of recurring at regular intervals. By way of illustrating this
concept consider, as shown in Table 1 below, the cyclic number 142,857 in its
decimal fraction form, 0.142857, along with two of its variations:
Table 1
Cyclicity
+
0. 142857
≈ 1:7
0. 285714
≈ 2:7
0. 571428
≈ 4:7
≈ 1. 000000
≈ 7:7
The primary sonnet interest here lies neither in the numeric value of the
fractions, nor in their sum, but in the way the regular development of digit pairs
forms a spiral pattern as they cycle about each other from one fraction to the next.
It is just such a patterning mechanism that I shall apply in the models to develop
the equivalents of the sonnet’s formal characteristics from a simple array.
4
For background, see: Weisstein, Eric W. "Cyclic Number." From MathWorld–A Wolfram
Web Resource. (http://mathworld.wolfram.com/CyclicNumber.html)
!9
This idea of a patterning mechanism is illustrated in table 2 below using
the example from Table 1. I have dropped the zeroes and the decimal points, the
addition sign and the sum total and, in addition, have arranged the digit pairs of
each decimal fraction into three separate columns with line numbers and letters
added for ease of reference; lastly, arrows highlight the axial movement of the
digit pairs from one array to the next.
Table 2
Array Elements
a
b
c
1.
14
28
57
2.
28
3.
57
↙
↙
57
14
↙
↙
14
28
To distinguish patterning as a concept from the process by which it is achieved, I
shall term the building-up of arrays and concomitant changes in the positions of
digit pairs from array to array development, the digit pairs elements and define
patterning as the rule-based development of arrays.
It is apparent from Table 2 that development is a consequence of three
conditions: first, the ordering of elements into an array in line 1; second, the
cyclicity of the elements, that is, the regular recurrence of the elements about each
other and, finally, the direction of cyclicity from one array to the next. Now, the
relative positions of array elements being fixed in line 1 and cyclicity assumed as
a latent property of the array, development depends, therefore, on the direction of
cyclicity from the first, or ‘start’, array. This initial directionality, as I shall call it,
!10
is clearly determined by two simple choices, first, the choice between an
‘upwards’ or ‘downwards’ development from the start array and, second, the
choice between a ‘right’ or ‘left’ shifting of the digit pairs from one array to the
next. Development in the example above may thus be seen upon inspection to be
‘down’ and ‘left’, or ‘downwards left’. To avoid confusion as to whether
directionality is to the left or right, it is helpful to take a cue from the development
of the middle element in each array: If the middle element in column ‘b’ develops
to the left, the other elements in the array do so as well; if to the right, the other
elements follow suit.
One consequence of directionality is the individuation of array elements.
Consider array 4, in Table 3, the result of a continuation of ‘downwards left’
development:
Table 3
Individuation of Array Elements
a
1.
14
2.
28
3.
4.
b
↙
28
↙
57
14
↙
57 14 28
↙ ↙
14 28 57
↙
57
c
Array 4 has the same order of elements as array 1, yet they are not identical as
their respective cyclical properties differ. To see this, assume that each element
has both a symbolic property, here represented by two digits, and cyclical
properties of one or both of the following two types, a flow towards another
element and, additionally, or, alternatively, a flow away from itself, hereinafter
!11
termed towards flow and away flow, respectively. These two cyclical properties
are represented in Table 3 by the same arrows for, evidently, the away flow from
the perspective of one element is the towards flow of the corresponding element
in the subsequent array developed. For example, the element at ‘b1’ has a
downwards left away flow, as does every element in the first array, and each of
these away flows, from the perspective of the elements in the second array, is a
towards flow. Thus, array 4 and array 1, despite having the same elements in the
same order, are not identical as their respective cyclical properties differ, the
former having towards flows, the latter none. Thus, it results that an array element
is only defined when both its symbolic and cyclical properties have been
determined.
If development is continued the away flows from array 4 to array 5 repeat
array 2. To avoid this repetition there is a change in directionality of development
from downwards left to downwards right follows, as shown in Table 4:
Table 4
Directionality Change
4.
14
5.
57
↘
28
14
↘
57
28
Yet, is this not arbitrary? Why should arrays not repeat themselves directionally as well as symbolically? To see why this is not the case here requires
considering the role that innovation and redundancy are deemed to play in the
model. Let the model aim to develop a maximum of innovation and a minimum of
redundancy in array development. Furthermore, let innovation be understood as
the continuous creation of unique arrays and redundancy as the loss of uniqueness
!12
through array repetition. Array development therefore consists of two phases, an
initial innovative phase developed from a start array, and a second redundancy
phase, which, as redundancy is to be minimized, when it occurs, marks the end of
array development and the completion of the model.
Now, given these assumptions and definitions, if a maximum of
innovation is to be achieved and redundancy minimized, then a repetition in the
symbolic prop-erties of an array, that is, the recurrence of two arrays having the
same digit pairs in the same order must, wherever possible, be counterbalanced by
innovation in its cyclical properties. Thus, in the development of array 5,
innovation takes the form of directional change from left to right, as shown in
Table 5:
Table 5
Array Innovation and Redundancy
1.
2.
3.
14
28
57
4.
14
5.
57
28
↙
↙
↙
↘
57
14
28
14
57
↙
↙
↙
↘
14
28
57
28
Before turning these ideas into operational rules for array development,
in order to establish a link between the model’s elements and the sonnet’s formal
characteristics, it is necessary to draw one final distinction between the symbolic
and cyclical properties of array elements on the one hand and their placeholder
!13
function on the other. Every element in the working model is deemed to have a
dual function, first, a combined symbolic and cyclical function to create difference
vis-à-vis other array elements and thus provide for the generation of innovative
arrays and, second, a placeholder function that creates a distinct pattern of flows
between symbolically similar array elements throughout the model. These
functions are not, incidentally, affected by the type of element used in the model.
As long as difference between elements is established, other element types are
permissible. Thus, just as the elements ‘14’, ‘28’ and ‘57’ occupy the first array, so
might in principle any other group of three elements, such as musical notes,
colours, letters, or indeed any combination of elements from these or any other
symbolic category, for it is the possession of a differentiable property that is the
condition for its inclusion as an element in the model, and decidedly not the type
of element per se.
This placeholder function of array elements is central to the main claim
of the inquiry as it is the final pattern of placeholders and the flows between them
that create several of the equivalents of the formal characteristics of the sonnet. In
other words, these equivalents shall be seen to be different aspects of a particular
distribu-tion of placeholders throughout the model and, crucially, of the
directionality and changes in directionality of the flows between them that result.
With distinctions between innovation and redundancy, and elements and
placeholders, drawn, three rules for array development may now be defined.
These rules, the result of numerous tests of possible array developments, are based
on the principle of centred form, in which the maximizing of array innovation and
minimizing of array redundancy plays the key role in limiting the number of
!14
arrays in the model, just as the pattern of flows between placeholders determines
the emergence of the equivalents of isometry, stanzaic form, volta and rhyme
scheme.
However, before turning to the rules for array development, it seems
appropriate now to start to attempt to formalize terms in order to show how the
working model relates to the sonnet. The complete set of arrays to be developed
shall, therefore, be referred to as the model; the initial array from which the other
arrays are developed as the centre array; a line, in this case, of three elements as
an array, and any one digit pair, as noted earlier, as an array element, or simply
element. In a perforce adumbrated way, given the still early stage of the inquiry,
these terms are deemed to correspond to the following aspects of a written sonnet:
model
=
sonnet
array
=
sonnet line, or verse
centre array
=
eighth sonnet line, or verse
element
=
word, or word group
The working model has three rules for array development:
Rule 1.
From the centre array, arrays develop alternately upwards
and downwards, in each direction either to the left or right,
but not to the left and right, nor to the right and left;
Rule 2.
Symbolic repetition of the centre array causes a change in the
direction of development;
Rule 3.
Symbolic repetition of any array and cyclical repetition of its
towards flows halts development and completes the model.
!15
Having defined operational rules for development of the working model,
it is also now not inappropriate to describe the ordering principle underlying it. It
was mentioned in the discussion of directionality that development may be
upwards as well as downwards from the start array. This is just another way of
saying that arrays develop in opposite directions from a ‘middle’, or ‘centre’, thus
connecting the sonnet’s formal characteristics, as presupposed. This principle of
centred form may be described as follows: A sonnet unfolds from its centre
towards its beginning and end to form a pattern in which its formal
characteristics inhere.
A corollary of the centred form principle is its implication of a
fundamental distinction between the manner of reading a sonnet and the writing of
it. Whereas the former is customarily linear, the principle of centred form suggests
that the writing of a sonnet may begin with the creation of a centre and then
continue outwards towards the sonnet’s start and finish while maintaining the
linear legibility of the lines so created. This is not to say, of course, that the
writing of any sonnet starts, or must start, from its centre, that would be absurd; it
is to claim, however, that such writing can elicit the sonnet’s formal characteristics
and, in so doing, offers a testable theory for their presence.
To make it easier to follow the detailed description of the working
model’s construction that follows, Table 6 presents a summary of array
development. Beginning with the centre array, array 8, on the left of the table,
arrays develop alternately towards the top and bottom of the model to end with
array 1:
!16
Table 6
Summary of Array Development
step
1
array
array
array
8
↗
2
3
4
5
6
7
7
6
5
4
3
2
↓
↓
↓
↓
↓
↓
9⤴
10 ⤴ 11 ⤴ 12 ⤴ 13 ⤴ 14
8
1
↗
Development consists of eight steps resulting in a model of fourteen
arrays. Step 1 represents the choice of centre array, the equivalent of line 8 in a
fourteen-line sonnet. Then follow six steps in which symbolically identical, but
cyclically dissimilar, pairs of arrays are developed alternately towards the top and
bottom of the model. The eighth and final step develops array 1, the equivalent of
line 1 in a sonnet. As array 1 is symbolically and cyclically a repeat of array 7,
redundancy is introduced into the model and, according to Rule 3, completes it.
To show how this summary description of the model works in detail, a step-bystep description of the development of the arrays that create the working model
now follows.
!17
1.2.2
Step-by-step Description of Array Model Development
Step 1
centre array
8.
14
28
57
The centre array consists, as remarked above, of three mutually distinguishable
symbolic elements with cyclical properties. It is developed alternately both
upwards and downwards, in both cases initially either to the left or right, and has
only ‘away flows’, but no ‘towards flows’ as the centre array forms the starting
point from which all other arrays are developed and, as such, ‘towards flows’ for
it are undefined.
Step 2
arrays 7 & 9
7.
8.
9.
28
57 14
↖ ↖
14 28 57
↙ ↙
28 57 14
As Rule 1 permits a choice between leftwards and rightwards development from
the centre array, let development start leftwards to create arrays 7, then 9. These
two arrays are identical symbolically, but differ in their directionality, array 7
having upwards left and array 9 downwards left directionality.
!18
Step 3
arrays 6 & 10
6.
7.
8.
9.
10.
57
14 28
↖ ↖
28 57 14
↖ ↖
14 28 57
↙ ↙
28 57 14
↙ ↙
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.
6.
7.
8.
9.
10.
11.
14
28
57
↖ ↖
57 14 28
↖ ↖
28 57 14
↖ ↖
14 28 57
↙ ↙
28 57 14
↙ ↙
57 14 28
↙ ↙
14 28 57
The development of arrays 5 and 11 in Step 4 results in symbolic, but not cyclical
identity with array 8 for, unlike arrays 5 and 11, the centre array elements have no
towards flows. Neither are arrays 5 and 11 identical for, although their towards
flows are both leftwards, the directionality of the former is upwards from the
centre array, that of the latter, downwards. Therefore, due to symbolic, but not
!19
cyclical identity with the centre array, Rule 2 is applied, and directionality in away
flows changes from left to right.
Step 5
arrays 4 & 12
4.
5.
6.
7.
8.
9.
10.
11.
12.
57
14 28
↗ ↗
14 28 57
↖ ↖
57 14 28
↖ ↖
28 57 14
↖ ↖
14 28 57
↙
↙
28 57 14
↙ ↙
57 14 28
↙ ↙
14 28 57
↘ ↘
57 14 28
In Step 5, arrays 4 and 12 are symbolically the same as arrays 6 and 10, but, as
their towards flows are cyclically different, development continues.
!20
Step 6
arrays 3 & 13
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
28
57 14
↗ ↗
57 14 28
↗ ↗
14 28 57
↖ ↖
57 14 28
↖ ↖
28 57 14
↖ ↖
14 28 57
↙ ↙
28 57 14
↙ ↙
57 14 28
↙ ↙
14 28 57
↘ ↘
57 14 28
↘ ↘
28 57 14
In Step 6, lack of either repetition of the centre array or both symbolic and
cyclical repetition of any previous arrays means development continues in the
same rightwards direction.
!21
Step 7
arrays 2 & 14
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
14
28 57
↗ ↗
28 57 14
↗ ↗
57 14 28
↗ ↗
14 28 57
↖ ↖
57 14 28
↖ ↖
28 57 14
↖ ↖
14 28 57
↙ ↙
28 57 14
↙ ↙
57 14 28
↙ ↙
14 28 57
↘ ↘
57 14 28
↘ ↘
28 57 14
↘ ↘
14 28 57
In Step 7, array 8, the centre array, re-emerges symbolically in arrays 2 and 14.
Now, although the centre array is developed here for the fourth and fifth times,
there is as yet no repetition of its cyclical properties: The centre array itself has no
towards flows and, whilst arrays 5 and 11 have leftward towards flows, arrays 2
and 14 have rightward towards flows. As there is thus no repetition of both the
symbolic and cyclical properties of the centre array, development continues with,
however, according to Rule 2, a change in directionality from right to left.
!22
Step 8
array 1
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
28
57 14
↖ ↖
14 28 57
↗ ↗
28 57 14
↗ ↗
57 14 28
↗ ↗
14 28 57
↖ ↖
57 14 28
↖ ↖
28 57 14
↖ ↖
14 28 57
↙ ↙
28 57 14
↙
↙
57 14 28
↙ ↙
14 28 57
↘ ↘
57 14 28
↘ ↘
28 57 14
↘
↘
14 28 57
The development of array 1 in Step 8 results in the symbolic and cyclical
repetition of the towards flows of array 7, thus introducing redundancy into the
model. Further development is halted, according to Rule 3, and the model
complete. Full development has thus taken eight steps, resulting in a model of
fourteen arrays.
!23
What model is created, however, if development from the centre array is
initially rightwards instead of leftwards? The alternative developments are juxtaposed in Table 7 further below. Symbolically, the centre array ‘14 28 57’ recurs in
the same arrays in both models, namely, in arrays 5 and 11, and 2 and 14. The
other two arrays ‘28 57 14’ and ‘57 14 28’ substitute for each other, that is, where
‘28 57 14’ occurs in the leftwards model, it is replaced by ‘57 14 28’ in the
rightwards model, and vice versa. Cyclically, the models’ flows are mirrored.
Table 7 below underscores the symmetry within and between these two
models. The differences between them are not, however, inconsequential. When,
in the second part of the inquiry, it comes to the attempt to model a number of
traditional sonnet forms, it shall be seen that the initial choice between leftwards
and rightwards development has a bearing on the type of rhyme scheme that
results. For the moment, however, the choice between the two models is
indifferent and as its development has been described in detail the model with
leftwards development will be adopted for the purposes of the discussion to
follow.
!24
Table 7
Initial Leftwards versus Rightwards Development
leftwards
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
28
rightwards
57
14
↖ ↖
14 28 57
↗ ↗
28 57 14
↗ ↗
57 14 28
↗ ↗
14 28 57
↖ ↖
57 14 28
↖ ↖
28 57 14
↖ ↖
14 28 57
↙ ↙
28 57 14
↙ ↙
57 14 28
↙ ↙
14 28 57
↘ ↘
57 14 28
↘ ↘
28 57 14
↘ ↘
14 28 57
57
14 28
↗ ↗
14 28 57
↖ ↖
57 14 28
↖ ↖
28 57 14
↖ ↖
14 28 57
↗ ↗
28 57 14
↗ ↗
57 14 28
↗ ↗
14 28 57
↘ ↘
57 14 28
↘ ↘
28 57 14
↘ ↘
14 28 57
↙ ↙
28 57 14
↙ ↙
57 14 28
↙ ↙
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 threeelement centre array up to and including its end-array having been defined and
stepwise illustrated.
!25
1.2.3
Assessment
How well, then, does the model describe the sonnet’s formal
characteristics? Might it not be objected, for example, that the equivalent of the
sonnet’s fourteen lines is accounted for inadequately by the theory of innovation
and redundancy? Surely a maximum of innovation and a minimum of redundancy
implies an absolute amount of the former and an absence of the latter, resulting in
a model of thirteen, and not fourteen, arrays? This objection is based on the
misapprehension that innovation and redun-dancy in the working model are
mutually exclusive, whereas they are, on the contrary, mutually dependent. This is
necessarily so, as the cyclical properties of the elements in the centre array mean
that there cannot be infinite innovation in array development due to the limited
number of possible combinations of the symbolic and cyclical properties of the
elements themselves. Innovation in the working model is, as may be seen upon
inspection of Table 7, limited to six arrays upwards and six downwards away from
the centre.
If innovation is, then, limited, and redundancy preprogrammed, is there
not all the more reason for the number of arrays in the model to be limited to the
thirteen innovative arrays with the redundant fourteenth array omitted? The
difficulty with halting development after the thirteenth array is that, from the
perspective of development within the model, it is not certain whether innovation
continues into the fourteenth array, or not. The only way to find out is to continue
development until symbolic and cyclical repetition occur, as defined by Rule 3.
The difficulty with repetition on the other hand is, of course, that it has to repeat
itself to be repetitive. The development of the fourteenth array, by causing Rule 3
!26
to be invoked and completing the model, addresses these two difficulties by
simultaneously ending both innovation and redundancy, thus maximizing the
former as it minimizes the latter.
Even allowing for this, however, does not array 2 and its away flows
represent symbolic and cyclical repetition of array 8 and, this being so, introduce
redundancy into the model thus restricting the total number of arrays to thirteen?
Rule 3 says that it is not repetition in the away flows, but in the towards flows of
an array, together with symbolic repetition, that creates redundancy and halts
development. As an array’s towards flows are the same as the away flows from
the preceding array, this argument might appear specious. However, an array’s
towards flows mark the incipient development of an array, whilst away flows
show that an array has already been developed or, as in the case of the centre
array, assumed. Hence, development continues from array 2 to develop the
towards flows of array 1 and fulfil the conditions necessary for redundancy.
Is it certain, though, that innovation, seen from the perspective of
development within the model, is at an end after thirteen arrays just because one
redundant array has been developed? Is it not possible that an innovative array
might be developed at the fifteenth array, or later still? Let it be assumed that such
innovation occurs. Now, for innovation to occur it must develop from an array that
is innovative. However, development has resulted in an array that is redundant.
Hence, any further development at the fifteenth array, or later, cannot be
innovative. Alternatively put, if there can be no innovation from redundancy, then
from redundancy, there can only be more redundancy. This implies that
redundancy is not simply repetition of the same arrays, but accretion of
!27
supplementary arrays that share the same symbolic and cyclical properties as
previously developed redundant arrays. In other words, any upward and
downward extension of the working models is continuous away from the centre.
From this it may be further deduced that there is no formal closing of the circle in
the model: The principle of centred form described here assumes that
development in the working model would continue indefinitely in opposite
directions away from its centre were it not restricted by the principle of innovation
and redundancy to fourteen arrays.
How, then, does the model account for the characteristic division of the
sonnet into octave and sestet, with their respective subsequent divisions into
quatrains and tercets? Furthermore, what of the volta and isometry? How are they
related by the principle of centred form? The traditional stanzaic structure of the
sonnet may be thought of hierarchically as a division of fourteen lines into two
major parts, the octave and sestet, followed by two further divisions separating the
octave and sestet into quatrains and tercets, respectively. These three divisions
then correspond to the points in the working model where a contrast in
directionality produces innovation. The directionality of the two away flows of
array 8 contrast with each other, whilst in the upwards and downwards
development of the model, the away flows of arrays 5 and 11 contrast in
directionality with respect to their towards flows. Array 8, having the relatively
starker contrast between its flows due to its lack of towards flows is accordingly
deemed to represent the volta and mark the major stanzaic division between
octave and sestet, leaving arrays 5 and 11 to mark the divisions between quatrains
and tercets. Isometry refers to the constant number of prosodic markers, or
!28
stresses, per line of verse. In the working model, each array has three elements,
each of which is deemed to have the same number of stresses as the other two.
Hence, every array throughout the model has the same number of stresses making
the model isometric.
What, then, of the change in directionality between arrays 2 and 1? To
which stanzaic division does it correspond? How, moreover, is the stanzaic form
of the Shakespearean sonnet with its three quatrains and final couplet to be
satisfied? As noted above, it is only the directionality change introduced by the
symbolic repetition of the centre array when it produces innovation that is
pertinent to the traditional stanzaic form of the sonnet. The change in
directionality between arrays 2 and 1 leads not to innovation, but to redundancy
and is, therefore, as irrelevant to stanzaic form, as it is pertinent to sonnet length.
The stanzaic form of the Shakespearean sonnet, as shall be seen below, results
from a different centre array. In this respect, the working model does not, indeed
cannot, account for it.
Whilst the model so far appears to represent adequately the sonnet equivalents of length, stanzaic form, volta and isometric verses, its limitations become
apparent when the question of accommodating the sonnet traditions’ rhyme
schemes arises. This difficulty appears daunting when considered broadly for, as
Lennard (1996, pp. 25–26) has calculated, and Queneau (1961) has demonstrated,
the number of potential rhyme schemes in a sonnet is very large. I shall deal with
this problem of the exhaustion of inexhaustibility by ignoring it for the moment
and limit the inquiry to the five rhyme schemes identified by Kircher (1979) as
representative of the Italian, French and English language sonnet traditions.
!29
According to Bermann (1988) in her study of the sonnets of Petrarch, Shakespeare
and Baudelaire, these poets’ sonnets reflect “the lyric’s enormous potential for
difference” (p. 2), a view that may serve to justify the number and linguistic
variety of sonnet traditions selected for this inquiry. I think any fewer would be
too weak a test for the claim, any more, I hope, superfluous.
Before turning to these traditions, the numerical elements used to
construct the working model with leftwards directionality are replaced in Table 8
below by variables qua lower case roman letters to represent any elements
fulfilling the symbolic and cyclical conditions given in the definition of array
elements above. The 14, 28 and 57 of the centre array, array 8, are thus
represented from now on by the variables a, b and c, respectively. To better
highlight how the contrasting flow patterns of the working model reflect the
equivalent of sonnet stanzaic form, in the presentation of the final working model
in Table 8 below, I have removed the flow lines between the equivalents of octave
and sestet and quatrains and tercets and widened the spaces between them slightly.
!30
Table 8
Working Array Model
Working Array Model
b
a
b
c
a
c
b
a
↖
↗
↗
↖
↖
↖
b
c
a
c
b
a
c
b
c
a
b
a
c
b
↖
↗
↗
↖
↖
↖
c
↙
↙
↘
↘
a
b
a
c
b
!31
a
c
a
b
c
b
a
c
a
↙
↙
↘
↘
b
c
b
a
c
1.2.4
Conclusion
How can a model consisting of numbers and variables have anything
trenchant to say about poetic form? Whilst a sonnet is always, arguably, made of
words, words are necessary in a model only insofar as they help elucidate the
subject matter of the inquiry, in this case, the idea that a sonnet unfolds from its
centre to form a pattern in which its formal characteristics inhere. For such an
undertaking, symbols as numbers to illustrate, and variables as letters to
generalize, suffice. This is inevitably so for the model presupposes that the
relationship between its elements and the formal characteristics of the sonnet is
constitutively the same, namely, patterned.
!32
Part 2: Modelling Sonnet Traditions
2.0
Introduction
In this second part of the inquiry, the principle of centred form as
developed in the working model in Part 1 is applied in the construction of array
models of the Early Italian, Petrarchan, Pleadean and Shakespearean sonnet
traditions. After the problem of the multiplicity of rhyme schemes is addressed in
the discussion of the Early Italian tradition, concern about the risk of error and
bias in model development leads to the search for an alternative, independently
constructed centred form model to cross-check array model results. The
theoretical basis for this second model, to be termed triangle model due to its
origins in, and geometric similarity with, a binary expansion, is consolidated at
the start of the Shakespearean section. The Shakespearean triangle model’s
subsequent construction, a comparison between its results and those of the
Shakespearean array model and an analysis of the relatedness of the
Shakespearean and Early Italian models close the second part of the inquiry.
2.1
Early Italian Tradition
2.1.1
Simplified Rhyme Schemes
Kircher (p. 414) notes two rhyme schemes as characteristic of the Italian
sonnet, namely, either alternating or embracing rhymes in the octave with two
variations in the sestet:
a b a b / a b a b or a b b a / a b b a , c d c / d c d or c d e / c d e.
As the embracing, or arching, rhyme in the octave is characteristic of the
Petrarchan and Pleadean models described below, discussion here is limited to
!33
what shall be termed the Early Italian sonnet, dating from the scuola siciliana of
the early 13th century, with its alternating rhyme in the octave and two variations
in the sestet:
a b a b a b a b , c d c d c d or c d e c d e .1
Several facts about the formal characteristics of the Early Italian sonnet
may of course be gleaned from these conventional rhyme schemes inter alia that
the sonnets are fourteen lines in length, that they have a two quatrain, two tercet
stanzaic form and that the rhyme schemes themselves have two pairs of rhymes in
the quatrains and either two ternary rhymes or three pairs of rhymes in the sestet.
Yet, the rhyme schemes also reveal something else that at first glance appears
quite mundane, but shall prove helpful to the inquiry, namely, that from one verse
to the next the end rhyme always changes. From ‘a’ in the first verse to ‘b’ in the
second, change; from ‘b’ in the second to ‘a’ in the third, change; from ‘a’ in the
third to ‘b’ in the fourth, change, and so on. Generalizing this observation, the two
variations in the sestet merge so that the rhyme scheme shows continuous change
throughout:
a b a b
a b a b
a b a
b a b
This array relates to the working model in the following manner: Each ‘a’
and ‘b’ in the array corresponds to its equivalent end-array element in each of the
working model’s arrays. The final, that is, rightmost, element of each array in the
working model has, therefore, one more function than the other elements: a
symbolic and cyclical function, a placeholder function and the function of
representing the rhyme scheme in terms of ‘change’ and ‘no change’. If the spaces
Thirty of the thirty-one sonnets Wilkins (1915, p. 83) recognises as belonging to the
group of earliest sonnets have these rhyme schemes.
1
!34
marking stanzaic division into quatrains and tercets in the array above are now
removed, the following simple alternating array results:
a b a b a b a b a b a b a b
This simplified rhyme scheme shows that it is possible for a simple array
of two alternating letters to hide complex information about the number of lines,
stanzaic form and rhyme scheme of a traditional sonnet form and that the rhyme
scheme of the Early Italian sonnet may be understood not only in terms of ternary
and paired rhymes, but also in terms of ‘change’ and ‘no change’. The question
that now naturally arises is how well the simplified Early Italian rhyme scheme is
described by the working model.
Consider the ‘change / no change’ column in Table 9 below situated
between the working model (WM) and the Early Italian model (EI). In this table,
change or lack of change from one end-array element to the next in the working
model is compared with change or lack of change within the simplified rhyme
scheme of the Early Italian sonnet developed above.
!35
Table 9
Working Model (WM) vs. ‘Simplified’ Early Italian (EI) Rhyme Scheme:
Comparison of End-Array Element Changes
array/line
WM
change
/
no change
EI
1.
a
change
start
a
2.
c
change
change
b
3.
a
change
change
a
4.
b
change
change
b
5.
c
change
change
a
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.
!36
From these findings, it can be deduced, in terms of ‘change’ or ‘no change’, that
the working and Early Italian models’ simplified rhyme schemes are identical,
with both providing a basis for a traditional alternating rhyme scheme consisting
of paired rhymes in the octave and ternary or paired rhymes in the sestet.
It might be objected here that it is not a conventional rhyme scheme that
is being compared with the working model's end-array elements, but a simplified
Early Italian rhyme scheme, implying that end rhymes are not being compared at
all. This objection stems, though, from a conflation of the placeholder and
symbolic functions of the elements in the working model and Early Italian
simplified rhyme scheme. The Early Italian simplified rhyme scheme highlights
the placeholder function of the conventional Early Italian rhyme schemes, thus
making a comparison with the working model’s end-array placeholders possible
and appropriate. In other words, it is not end rhymes that are being compared in
Table 9, but their placeholders. The equivalents of the octave and sestet
placeholders may still of course be filled with paired rhymes and paired or ternary
rhymes, respectively: Change from one end-array to the next at the placeholder
level not only does not preclude, it corresponds with an alternating rhyme scheme
at the symbolic level. The end rhymes of the Early Italian sonnet are indeed,
therefore, being compared with the end-array elements of the working model, but
as placeholders, not as rhymes.
To summarize the discussion so far, the rhyme schemes noted by Kircher,
and here characterized as Early Italian, are:
&
a b a b
a b a b
c d c d c d
a b a b
a b a b
c d e c d e
!37
From each of these, in simplified form, the following alternating rhyme scheme
was derived:
a b a b a b a b a b a b a b
This same simplified, alternating rhyme scheme equivalent was also seen
to be derivable from the working model, from which it follows that the working
model satisfactorily describes the rhyme scheme of the Early Italian sonnet at the
placeholder level. From this same simplified rhyme scheme, it may also be
inferred that two types of element, ‘a’ and ‘b’, suffice to construct the Early Italian
model. The question now is how to decide on the correct number, mix and order
of these two types of element for the model’s centre array.
2.1.2
Centre Array Derivation and Array Model Development
In the following, the derivation of the Early Italian model’s centre array
is presented in detail, that is, exemplarily for the Petrarchan, Pleadean and Shakespearean models.
As an approach to answering questions concerning the makeup of the
Early Italian model’s centre array, let the five characteristics of the sonnet noted
by Hobsbaum henceforth be considered as five conditions needing to be satisfied
simultaneously by any model. Let this stringency furthermore be extended to the
types and number of elements in the centre array by applying the following rule:
as few types and number of elements as possible, as many of either as necessary
to satisfy all five sonnet conditions. Additionally, let types take precedence over
number: A solution with fewer types of element and a greater number of
individual elements is hence preferable to a solution with more types and fewer
elements. Let this principle be called the principle of economy.
!38
The methodological advantage of this approach is that it simplifies the
selection of centre arrays that potentially satisfy all five sonnet conditions. For
example, if two types of element, ‘a’ and ‘b’, are indeed sufficient to construct the
Early Italian Model, the development of a centre array consisting of only one of
each of these is clearly insufficient: Two-element arrays, whilst able to represent
an alternating rhyme scheme, produce, in working model terms, precipitate
redundant arrays and thus fail to satisfy the conditions of sonnet length and
stanzaic form. The unsuitability of such centre arrays may be seen in Table 10, in
which the symbolic and cyclical repetition of array 7 occurring in array 5 leads to
redundancy and halts development before the completion of fourteen arrays. For
the purposes of exposition, only upward development from the centre array
through array 5 is shown. In addition, as ‘a b’ is a transposition of ‘b a’, and may,
therefore, stand in lieu of it, only the development of the array ‘a b’ is included.
Furthermore, as leftwards development in a two-array model is tantamount to
rightwards development, only leftwards development is shown in the examples
below.
Table 10
Unsuitability of Two-Element Centre Array
5.
b
6.
a
7.
b
8.
a
↖
↖
↖
!39
a
b
a
b
= array 7
The alternative of changing directionality at array 6 to avoid redundancy
only postpones redundancy until array 3, itself a symbolic and cyclical repetition
of array 5, as may be seen in Table 11 below. The possibilities for the satisfactory
development of a two-element array being exhausted, it is concluded that a twoelement 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.
4.
5.
b
a
b
6.
a
7.
b
8.
a
a
↗
↗
↗
↖
↖
= array 5
b
a
b
a
b
Similarly, in a centre array numbering three elements of two element
types, for example, ‘b a b’, formal sonnet conditions are no closer to being
fulfilled for development necessarily results in the impossibility of
accommodating an alternating rhyme scheme. Table 12, below, shows that in a
leftwards development of the array ‘b a b’, for example, consecutive end-array
elements are immediately produced in array 7. In a rightwards development,
without a change in directionality, similar end-array elements are repeated in the
consecutive arrays 6 and 5. Changing directionality at array 6 to avoid redundancy
results in repeating end-array elements in arrays 3 and 4, failing to satisfy the
alternating rhyme condition. The other two possible three-element distributions of
!40
the centre array, ‘a b b’ or ‘b b a’, produce similar results when developed, as may
be seen in Appendix A.
Table 12
Unsuitability of Three-Element Centre Array:
Leftwards & Rightwards Development
i)
ii)
Leftwards Development
7.
a
8.
b
a
↖
b
b
Rightwards Development
5.
6.
7.
8.
iii)
↖
b
b
a
b
b
a
↗
↗
↗
b
b
a
b
↗
↗
↗
b
a
b
Rightwards Development with Change in Directionality in Array 6
3.
a
4.
b
5.
b
6.
a
7.
8.
b
b
↖
↖
↖
↗
↗
!41
b
a
b
b
b
a
↖
↖
↖
↗
↗
b
b
a
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
scheme is not possible for inevitably, as just remarked, the same two types of
element must follow each other by the end of the third array developed. An even
distribution of elements between element types, ‘a b a b’ or ‘a a b b’, for example,
has the same disadvantage as a two-element ‘a b’ array: It can alternate, but it
cannot be stanzaic.
A centre array of two element types and five elements can, however, as
shall be seen below, satisfy all five sonnet conditions due to the sufficient number
of combinatorial possibilities inherent in its symbolic and cyclical properties. To
see this, let distributions that are clearly unsuitable first be excluded. Consider, for
example, a distribution of 4:1 elements divided between two element types, as in,
for example, the arrays ‘a b b b b’ or ‘a a a a b’. Neither of these alternatives will
do, as, by simple inspection of Table 13, alternation of placeholders beyond the
development of array 6 is impossible:
Table 13
Unsuitability of 4:1 Distribution of Centre Array Elements
leftwards development
5.
6.
7.
8.
rightwards development
b b a b b
↖ ↖ ↖↖
b b b a b
↖ ↖ ↖↖
b b b b a
↖ ↖↖ ↖
a b b b b
b a b b b
↗↗ ↗↗
a b b b b
!42
A change in directionality at array 6 does not help as it leads to a breakdown in
the alternation of end-array elements in array 3, as shown in Table 14, below:
Table 14
Redundancy in a 4:1 Distribution of Centre Array Elements:
Leftwards Development with Directionality Change in Array 6
3.
4.
5.
b
a
b
6.
b
7.
b
8.
a
a
↗
↗
↗
↖
↖
b
b
b
b
b
b
↗
↗
↗
↖
↖
b
↗
b
↗
b
b
b
↗
↖
b
↖
b
b
a
b
b
b
↗
↗
↗
↖
↖
b
a
b
a
b
It follows that the mix of element types and their number in the centre array, is in
the ratio 3:2, either three ‘a’s and two ‘b’s, or three ‘b’s and two ‘a’s. As ‘a a b b b’
is a transposition of ‘b b a a a’ and does not affect the pattern of placeholders in
the model, nothing is lost by choosing one distribution over the other. Let the
distribution then be three ‘b’s and two ‘a’s. Now, the pattern each element traces
during development depends only on the rules for array development, which are a
priori the same for all elements. That is, as far as the ordering of elements within
the centre array is concerned, it is not necessary to distinguish between elements
within each element type: One a or b is as good as any other. Hence, the number
of ways that elements in the centre array may be ordered is governed by the
!43
mathematical rule for combinations.2 This rule gives the number of possible
centre arrays as ten, which confirms the ten arrays, presumed exhaustive in the
ordering of their elements, listed in Table 15, below:
Table 15
Early Italian Model: Centre Array Candidates 1
i.)
a
a
b
b
b
ii.)
a
b
a
b
b
iii.)
a
b
b
a
b
iv.)
a
b
b
b
a
v.)
b
a
a
b
b
vi.)
b
a
b
a
b
vii.)
b
a
b
b
a
viii.)
b
b
a
a
b
ix.)
b
b
a
b
a
x.)
b
b
b
a
a
As remarked above, any centre array with a contiguity of three similar
elements is unsuitable because it compromises alternation of end-array elements
in a fourteen array model therefore ruling out the equivalent of an alternating
rhyme scheme. For this reason, arrays i.), iv.) and x.) do not pass muster. Slightly
less obviously, perhaps, but for the same reason, neither do the arrays v.) and
viii.): Due to the elements’ cyclical properties, the three ‘b’ elements in both cases
Assuming C (n, k) for two element types, b and a, and five elements distributed in the
ratio 3:2, there are in all, (5!/(5-2)!(2)! = 120/12 = 10 combinations of elements.
2
!44
are contiguous. There are, therefore, only five centre arrays, listed in Table 16,
that might still satisfy all five sonnet conditions:
Table 16
Early Italian Model: Centre Array Candidates 2
ii.)
a
b
a
b
b
iii.)
a
b
b
a
b
vi.)
b
a
b
a
b
vii.)
b
a
b
b
a
ix.)
b
b
a
b
a
Combined with the choice of either leftwards or rightwards directionality,
there are, then, in all twice five, or ten, candidate centre arrays. Now, it was seen
in the discussion of the working model that it was changes in directionality that
represent the equivalents of stanzaic form and lead to redundancy and model
completion. From this finding, two further criteria for excluding candidate centre
arrays follow. First, if to achieve the equivalent of alternation in end-array
elements a directionality change is needed in an array other than the fourth or fifth
and eleventh or twelfth arrays, then that candidate is unsuitable for it cannot
satisfy the Early Italian sonnet’s two quatrain, two tercet stanzaic form condition.
Second, if a candidate’s centre array is not redeveloped in array 2, then it also fails
for there is no mechanism to develop a redundant array in array 1, complete the
model in fourteen arrays and satisfy the sonnet condition for number of lines.
With the application of these additional criteria, eight of the ten
remaining candidates may be excluded: arrays ii.), iii.), vi.), with rightwards and
!45
leftwards, vii.) with rightwards and ix.) with leftwards development: Either they
do not provide alternation in end-arrays, or their centre array is not redeveloped in
array 2. Detailed workings in support of this conclusion may be found in
Appendix B. Hence, there are only two arrays that meet the criteria established so
far: vii.), ‘b a b b a’ with leftwards, and ix.) ‘b b a b a’ with rightwards,
development.
The rules for development of the ‘b a b b a’ array are similar to those for
the working model. The differences are as follows. First, array pairs rather than
single arrays are developed simultaneously and not alternately, resulting initially
in models of fifteen rather than fourteen arrays; second, arrays are developed only
with leftwards directionality from the centre array rather than with either
leftwards or rightwards directionality; finally, in addition to the centre array,
development of the array ‘b a b a b’ also leads to directionality change. Including
a rule for model completion that maximises array innovation and minimises array
redundancy, there are, then, three rules for array development in all:
Rule 1.
From the centre array, arrays develop simultaneously
upwards and downwards to the left;
Rule 2.
Symbolic repetition of the centre array or the array
‘b a b a b’ causes a change in directionality;
Rule 3.
Symbolic repetition of any array and cyclical repetition of
its towards flows halts development and completes the
model.
!46
Leftwards development from the centre array ‘b a b b a’ creates the
following first three pairs of arrays, 7–5 and 9–11, as shown in Table 17 below:
Table 17
Development of Centre Array through Arrays 5 & 11
5.
b
6.
b
7.
a
8.
b
9.
10.
11.
a
b
b
↖
↖
↖
↙
↙
↙
a
b
b
a
b
b
a
↖
↖
↖
↙
↙
↙
b
a
b
b
b
a
b
↖
↖
↖
↙
↙
↙
a
b
a
b
a
b
a
↖
↖
↖
↙
↙
↙
b
a
b
a
b
a
b
According to Rule 2, a change in directionality is introduced in arrays 5
and 11 to avoid the development of successive identical end-array elements. The
development of arrays 4–1 and 12–15 completes the model, as shown in Table 18
below:
!47
Table 18
Development of Arrays 4–1 and 12–15
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
a b b a b
↖↖↖↖
b a b b a
↗↗ ↗↗
a b b a b
↗↗ ↗↗
b b a b a
↗↗ ↗↗
b a b a b
↖↖↖↖
b b a b a
↖↖ ↖↖
a b b a b
↖↖ ↖ ↖
b a b b a
↙ ↙↙ ↙
a b b a b
↙↙ ↙↙
b b a b a
↙↙↙↙
b a b a b
↘↘↘↘
b b a b a
↘↘ ↘↘
a b b a b
↘↘ ↘↘
b a b b a
↘↘ ↘↘
a b b a b
The last pair of arrays developed, arrays 1 and 15, repeats arrays 7 and 9 symbolically and their towards flows cyclically. According to Rule 3, development is,
therefore, halted and the model complete. Development stops at this point for the
same reason as in the working model: the maximization of array innovation and
the minimization of array redundancy. As may be seen in Table 19 below, any
!48
further development of arrays results only in the development of additional
redundant arrays. These redundant arrays then repeat indefinitely for, as argued in
the first part of the inquiry, from redundant arrays only more redundant arrays are
developed.
Table 19
Continuous Redundancy in Array Development
-1.
b
1.
a
↖
b
b
↖
a
b
↖
b
a
↖
a
=
6.
b
=
7.
b
=
9.
a
= 10.
...
15.
16.
a
b
b
↙
b
b
↙
a
a
↙
b
↙
The second centre array presumed to satisfy all five Early Italian sonnet
conditions is, as noted above, the array ‘b b a b a’ with rightwards development.
In this model's construction, development of the array ‘a b a b b’, as well as the
centre array, leads to a change in directionality. The two complete array models
are juxtaposed in Table 20 below.
!49
Table 20
Early Italian Fifteen Array Models ‘b b a b a’ and ‘b a b b a’
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
‘b b a b a’
‘b a b b a’
a b b a b
↗ ↗↗ ↗
b b a b a
↖↖ ↖ ↖
a b b a b
↖↖↖↖
b a b b a
↖↖↖↖
a b a b b
↗↗ ↗ ↗
b a b b a
↗↗ ↗ ↗
a b b a b
↗↗ ↗ ↗
b b a b a
↘↘↘↘
a b b a b
↘↘↘↘
b a b b a
↘↘↘↘
a b a b b
↙↙ ↙↙
b a b b a
↙ ↙↙↙
a b b a b
↙ ↙ ↙↙
b b a b a
↘ ↘↘ ↘
a b b a b
a b b a b
↖ ↖↖ ↖
b a b b a
↗ ↗ ↗↗
a b b a b
↗ ↗ ↗↗
b b a b a
↗ ↗ ↗↗
b a b a b
↖↖↖↖
b b a b a
↖↖↖↖
a b b a b
↖↖↖↖
b a b b a
↙↙↙↙
a b b a b
↙↙↙↙
b b a b a
↙↙↙↙
b a b a b
↘ ↘↘ ↘
b b a b a
↘↘↘↘
a b b a b
↘ ↘↘ ↘
b a b b a
↙↙↙ ↙
a b b a b
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
Table 21 below shows, using the example of the model on the right in the
table above, that each of these fifteen array models comprises two identical
fourteen array sub-models. That is, when the arrays 15–2 are read from bottom to
top, they are identical symbolically and cyclically with arrays 1–14.
!50
Table 21
Identical 14-Array Sub-Models
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
a b b a b
↖↖↖↖
b a b b a
↗↗ ↗ ↗
a b b a b
↗↗↗ ↗
b b a b a
↗↗ ↗↗
b a b a b
↖↖ ↖↖
b b a b a
↖ ↖↖ ↖
a b b a b
↖↖ ↖ ↖
b a b b a
↙ ↙↙ ↙
a b b a b
↙ ↙ ↙↙
b b a b a
↙ ↙ ↙↙
b a b a b
↘ ↘↘↘
b b a b a
↘ ↘↘↘
a b b a b
↘ ↘↘↘
b a b b a
a b b a b
↖↖ ↖ ↖
b a b b a
↗↗↗↗
a b b a b
↗↗ ↗↗
b b a b a
↗ ↗↗ ↗
b a b a b
↖↖↖↖
b b a b a
↖↖↖↖
a b b a b
↖↖↖↖
b a b b a
↙ ↙ ↙↙
a b b a b
↙↙↙↙
b b a b a
↙↙↙↙
b a b a b
↘ ↘↘↘
b b a b a
↘↘↘↘
a b b a b
↘↘↘↘
b a b b a
15.
14.
13.
12.
11.
10.
9.
8.
7.
6.
5.
4.
3.
2.
The final fourteen array Early Italian models developed from the centre
arrays ‘b b a b a’ and ‘b a b b a’ are shown in Table 22 below. They are presented
in a form profiling model equivalents of Early Italian sonnet conditions to make it
easier to follow the subsequent discussion. Array numbering has been removed in
the table, as have the arrows between the equivalents of the sonnet's stanzaic
divisions.
!51
Table 22
Early Italian Array Models
Early Italian Array Models
a b b a b
↗ ↗ ↗↗
b b a b a
↖ ↖ ↖↖
a b b a b
↖ ↖ ↖↖
b a b b a
a b b a b
↖ ↖ ↖↖
b a b b a
↗ ↗ ↗↗
a b b a b
↗ ↗ ↗↗
b b a b a
a b a b b
↗↗↗↗
b a b b a
↗ ↗↗↗
a b b a b
↗ ↗ ↗↗
b b a b a
b a b a b
↖ ↖ ↖↖
b b a b a
↖↖↖↖
a b b a b
↖ ↖ ↖↖
b a b b a
a b b a b
↘↘↘↘
b a b b a
↘↘↘↘
a b a b b
a b b a b
↙↙ ↙ ↙
b b a b a
↙↙ ↙ ↙
b a b a b
b a b b a
↙↙↙↙
a b b a b
↙↙↙↙
b b a b a
b b a b a
↘↘↘↘
a b b a b
↘ ↘↘↘
b a b b a
!52
2.1.3
Assessment
How well, then, do these models describe the equivalents of the Early
Italian sonnet’s formal characteristics? The fourteen arrays of the model are
deemed equivalent to the fourteen lines of the Early Italian sonnet. The contrast in
away flows from array 8 marks the equivalent of the division of the sonnet into
octave and sestet just as the changes in directionality at arrays 5 and 11 represent
the equivalent division of the octave into quatrains and the sestet into tercets,
respectively. By dint of the contrast in flow directionality in the model being
greatest at array 8, this array is deemed to represent the equivalent of the volta.
Each array having five elements fulfils the condition for isometry. These five
elements, as noted in the first part of the inquiry, may represent five prosodic
markers, or stresses, and can thus also accommodate the accentuation rules of the
endecasyllabi sciolti, or free hendecasyllables of the standard Italian verse line in
its verso tronco, piano or sdrucciolo forms of ten, eleven or twelve syllables,
respectively. Finally, alternating end-array elements throughout the models are
deemed equivalent to the sonnet’s alternating rhyme scheme. The models thus
appear to describe satisfactorily equivalents of the Early Italian sonnet tradition.
2.1.4
Conclusion and Outlook
The problem posed by the multiplicity of potential sonnet rhymes schemes
for model construction raised at the end of Part 1 is addressed by considering the
formal characteristics of the Early Italian sonnet tradition as conditions to be
satisfied simultaneously. This has the effect of drastically reducing the number of
simplified rhyme schemes that might satisfy all sonnet conditions. By then
!53
applying a principle of economy to this general condition, from a corpus of ten
candidate arrays, two centre arrays that are presumed to satisfy all conditions are
elicited by first excluding those arrays that clearly do not. The two centre arrays
are then developed into array models. Encouragingly, both models appear to
describe satisfactorily key characteristics of the Early Italian sonnet, thus
providing support for the claim and making it reasonable to want to seek more
corroborative evidence in the modelling of other sonnet traditions. However, the
somewhat Procrustean approach to defining array development rules to satisfy
sonnet conditions raises concerns about error and bias in the final models’ results.
Moreover, why are there two array models that satisfy sonnet conditions, rather
than one? Although their mirrored flows suggest that they could be complete parts
of a broader pattern, what that pattern might be is as yet unclear. Therefore, in
search of not only more evidence in support of the claim by way of array models
that satisfy the sonnet conditions of other traditions, but also a means to mitigate
the risk of error and bias in model results, as well as a broader pattern that might
relate the two Early Italian array models developed above, the inquiry now turns
to a consideration of the Petrarchan sonnet tradition.
!54
2.2
Petrarchan Tradition
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
a
13 14 1 2 3 4 5 6 7 8 9 10 11 12 13 14
b
∣a
∣
a b b a a b b a b a b a b a
paired
⟵⟶ alternating
∣
b a b a b a b a
!55
1
∣
a
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 alternation 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
a
13 14
b
a
∣
1 2 3 4 5 6 7 8 9 10 11 12 13 14
1
a b b a a b b a b a b a b a
embracing
⟵ ⟶
alternating
∣a
a b b a a b b a b a b a b a
Again each rhyme type equivalent is developed as far as possible outwards from the
simplified rhyme scheme’s centre, resulting in a balance of eight elements in each.
These findings suggest that, besides the so-called asymmetry between the
embracing and alternating rhymes of a conventional Petrarchan rhyme scheme, there
is also a simpler, underlying bilateral symmetry that evolves from its centre. To
underscore this symmetry, notwithstanding the risk of momentarily getting ahead of
the discussion, the pattern traced by the flows between the Petrarchan array models’
placeholders is presented in Tables 25 and 26 below.
!56
Table 25
Petrarchan Flow Pattern 1
This flow pattern results from superposing the halves of the Petrarchan
array model and is composed, therefore, not of fourteen single arrays, but of seven
array pairs. The pattern may be disaggregated, as in Table 26 below, into four
similar, mirrored sub-patterns:
!57
Table 26
Petrarchan Flow Pattern 2: Sub-Patterns (a), (b), (c) & (d)
(a)
(b)
(c)
(d)
1.
b b a a
b b a a
b b a a
b b a a
2.
b a a b
b a a b
b a a b
b a a b
3.
a a b b
a a b b
a a b b
a a b b
4.
a b b a
a b b a
a b b a
a b b a
5.
b b a a
b b a a
b b a a
b b a a
6.
b a a b
b a a b
b a a b
b a a b
7.
a a b b
a a b b
a a b b
a a b b
8.
b a b a
b a b a
b a b a
b a b a
9.
a b a b
a b a b
a b a b
a b a b
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,
!58
whilst the italicised and underlined black and blue letters in the tables show how the
development of arrays and flow-lines pace each other.
2.2.2
Centre Matrix
Regarding the construction of the Petrarchan array model, as a single array
cannot at the same time develop the equivalents of different rhyme types, embracing
and alternating rhymes, for instance, the model has two ‘centre arrays’. These two
arrays shall be termed centre matrix to avoid possible confusion between the terms
‘centre array’ and ‘centre arrays’. Not only are towards flows for each array in the
centre matrix undefined, as the centre matrix develops different rhyme types, neither
are flows between its arrays. The number, types and mix of elements in the centre
matrix are derived in the same way as for the Early Italian model. A discussion of
the derivation of the Petrarchan centre matrix may be found in Appendix C. Two
centre matrices comprising arrays (7) and (8) are presumed to satisfy the conditions
of the Petrarchan sonnet, first, ‘(7) a a b b, (8) b a b a’ with rightwards and, second,
‘(7) a b b a, (8) a b a b’ with leftwards directionality. The former is shown in Table
27 below and is developed exemplarily for the latter with the difference in directionality between them being taken into account subsequently.
Table 27
Petrarchan Centre Matrix
Centre Matrix: Arrays 7 & 8
7.
a
a
b
b
8.
b
a
b
a
!59
As there are no changes in directionality in the Petrarchan model, it has one
less development rule than those for the working and Early Italian models. That is,
there is one rule for the start and one for the finish of development:
Rule 1.
From the centre matrix, arrays develop simultaneously upwards 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 subpatterns 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-bystep description of the development of both the array model and Sub-Pattern (a),
which serves as proxy for the other sub-patterns.
2.2.3
Step-by-Step Description of Array Model Development
Step 1
Centre Matrix: Arrays 7 & 8
7.
a
a
b
b
8.
b
a
b
a
The derivation of the centre matrix is discussed in Appendix C, as noted
above. The start elements for the development of the flow-lines in Sub-Pattern (a)
are highlighted in arrays 7 and 8. Arrays 6 and 9 and the initial flow lines of Sub-
!60
Pattern (a) that are developed in Step 2 along with their description are placed
together below to help make their relationship clear at the outset.
(The remainder of this page is deliberately blank to better presents the tables and
figures that follow.)
!61
Step 2
arrays 6 and 9
6.
a
b
7.
a
8.
b
9.
a
↗
↘
a
a
b
a
↗
↘
b
b
a
b
↗
↘
b
a
b
Development starts in opposite directions rightwards from the centre
matrix, according to Rule 1. The underlined elements correspond to the origin and
initial development of the sub-pattern.
Petrarchan sonnet pattern: Development of Sub-Pattern (a)
1.
*
*
8.
2.
*
*
9.
3.
*
*
*
*
10.
4.
*
*
*
*
11.
5.
*
*
*
*
12.
6.
*
*
13.
7.
*
*
14.
With the development of the second array pair in the model, the flow lines
of Sub-Pattern (a) start to emerge between placeholders. The asterisks refer to the
placeholders resulting from the direct translation of one half of the model onto the
other. The numbering on each side of the pattern serves to show that array 1 overlays
!62
array 8, as array 2 does array 9, and so forth. Thus, the blue line in the top half
represents the flow from the highlighted element in array 8 to the highlighted
element in array 9, while the black line in the bottom half represents the flow from
the first element in array 7 to the second element in array 6. In the sonnet pattern
diagrams to follow, these lines continue to accompany the development of arrays
throughout the model.
Step 3
Arrays 5 and 10
5.
6.
b
b
7.
a
8.
b
9.
a
10.
b
a
b
↗
↗
↘
↘
↗
a
↗
a
a
b
↘
a
↘
a
b
b
a
b
a
↗
↗
↘
b
b
a
b
↘
a
With no repetition of array pairs, development continues rightwards,
according to Rule 1, to develop the new array pair, 5 &10.
!63
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.
5.
6.
a
b
b
7.
a
8.
b
9.
a
10.
b
11.
a
b
↗
↗
↗
↘
↘
↘
↗
b
↗
a
↗
a
a
b
a
b
↘
a
b
↘
↘
a
a
b
b
a
b
a
↗
↗
↗
↘
a
b
b
a
b
↘
a
↘
b
In step 4, the next array pair, 4 & 11, is developed, according to Rule 1.
!64
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.
!65
Step 5
Arrays 3 and 12
3.
4.
5.
6.
a
a
b
b
7.
a
8.
b
9.
a
10.
b
11.
a
12.
b
a
↗
↗
↗
↗
↘
↘
↘
↘
b
↗
b
↗
b
↗
a
↗
a
a
b
a
b
a
↘
↘
↘
↘
b
a
a
b
b
a
b
a
b
b
↗
↗
↗
↗
↘
↘
↘
↘
a
a
b
b
a
b
a
b
a
In Step 5 the array pair, 3 & 12, is developed. Array 3 repeats array 7
symbolically, but not cyclically as array 7 has no towards flows. Development,
therefore, continues, according to Rule 1.
!66
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. !67
Step 6
Arrays 2 and 13
2.
3.
4.
5.
6.
a
b
a
a
b
b
7.
a
8.
b
9.
a
10.
b
11.
a
12.
b
13.
a
↗
↗
↗
↗
↗
↘
↘
↘
↘
↘
a
b
a
↗
↗
↗
b
↗
a
a
a
b
a
b
a
b
↗
↘
↘
↘
↘
↘
b
b
a
a
b
b
a
b
a
b
a
b
↗
↗
↗
↗
↗
↘
↘
↘
↘
↘
b
a
a
b
b
a
b
a
b
a
b
The array pair 2 & 13 repeats arrays 6 & 9 symbolically and their towards
flows cyclically, thus fulfilling the first part of Rule 2’s condition for redundancy.
!68
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.
!69
Step 7
Arrays 1 and 14
1.
2.
3.
4.
5.
6.
b
b
a
a
b
b
7.
a
8.
b
9.
a
10.
b
11.
a
12.
b
13.
a
14.
b
a
b
↗
↗
↗
↗
↗
↗
↘
↘
↘
↘
↘
↘
a
a
b
b
a
a
a
b
a
b
a
b
a
↗
↗
↗
↗
↗
↗
↘
↘
↘
↘
↘
↘
a
b
b
a
a
b
b
a
b
a
b
a
b
a
↗
↗
↗
↗
↗
↗
↘
↘
↘
↘
↘
↘
b
b
a
a
b
b
a
b
a
b
a
b
a
The array pair, 1 & 14, repeats the array pair 5 & 10 symbolically and its
towards flows cyclically. The two consecutive array pairs, 2 & 13 and 1 & 14, thus
trigger Rule 2, halt development and complete the model. !70
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
!71
being their initial directionalities. The two completed Petrarchan array models are,
therefore, placed side by side in Table 28 in such a way as to highlight the model
equivalents of the Petrarchan tradition's sonnet characteristics.
Table 28
Petrarchan Array Models
Petrarchan Array Models
(II)
(I)
b a a b
↖↖↖
b b a a
↖↖↖
a b b a
↖↖↖
a a b b
b b a a
↗↗↗
b a a b
↗↗ ↗
a a b b
↗↗ ↗
a b b a
b a a b
↖↖↖
b b a a
↖↖↖
a b b a
b b a a
↗↗ ↗
b a a b
↗↗ ↗
a a b b
a b a b
b a b a
b a b a
↙↙↙
a b a b
↙↙↙
b a b a
a b a b
↘↘↘
b a b a
↘↘↘
a 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
!72
Results identical to these two array models may be developed alternatively
by binary expansion, as shown in Table 29 below:
Table 29
Identical Petrarchan Array and Triangle Models
b b
b b a b
b b a a a b
7. ↗
b b a a b
b b a a
↗↗↗↗
b b a a b
↗↗↗↗
b a a b b
↗↗↗↗
a a b b a
↗ ↗↗↗
a b b a a
↗↗↗↗
b b a a b
↗↗↗↗
b a a b b
b a a b
↖↖↖↖
b b a a b
↖↖↖ ↖
a b b a a
↖ ↖↖↖
a a b b a
↖↖↖↖
b a a b b
↖↖↖↖
b b a a b
↖↖↖↖
a b b a a
8. ↘
b a b a b a b a b a
↘ ↘↘↘
b a b a b a b a b
↘ ↘↘↘
b a b a b a b a
↘↘↘↘
b a b a b a b
↘↘↘↘
b a b a b a
↘↘↘↘
b a b a b
↘↘↘↘
b a b a
a b a b a
↙↙↙↙
b a b a b
↙↙↙ ↙
a b a b a
↙↙↙ ↙
b a b a b
↙ ↙↙↙
a b a b a
↙↙↙↙
b a b a b
↙↙↙↙
a b a b
b
b b
b b a
b b a a
(I)
b a b b a b
b a a b
b b
!73
b
a b
a a b
b a a b
b b a a b ↖7.
b a b a b
a b a b
b a b
a b
b
↙8.
(II)
The rightwards and leftwards expansions are termed ‘binary’ as the individual 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 towards the arrays of the centre matrix being undefined, array 8 is the first array to
show a change in flow directionality in a conventional, linear reading of the model.
!74
In order to mark the equivalent of the division of the sonnet into octave and sestet,
this change in directionality is not made explicit in Table 28, however. It may, nonetheless, be straightforwardly deduced by taking into consideration the change in the
ordering of elements from array 8 to array 9. Subsequent division into quatrain and
tercet equivalents is quasi-translational for the quatrains, and rotational for the
tercets. The quatrain equivalents may be discerned by the identical arrays that make
up the first three lines of each. That the ordering of the elements in the final arrays
of the quatrain equivalents differs is due to the need for alternating elements in array
8 of the centre matrix. The tercets are defined by 180 degree rotational symmetry
about a centre lying between arrays 11 and 12, which maps each element of one
tercet directly onto its counterpart in the other. Equivalents of both Petrarchan rhyme
schemes are accommodated by the order of end-array elements in the model: The
end-array elements of arrays 9–14 allow for equivalents of either the two ternary
rhymes or three rhyme pairs of the Petrarchan sonnet noted by Kircher, which, in
terms of the simplified rhyme scheme, are both alternating, and the end-array
elements of arrays 1–8 may be seen upon inspection to accommodate equivalents of
embracing rhyme pairs. The Petrarchan array models thus appear, on balance, to
satisfy the formal conditions of the Petrarchan sonnet traditions under discussion.
2.2.5
Conclusion
The Petrarchan array models show how the principle of centred form can
relate equivalents of the formal characteristics of the Petrarchan sonnet, while the
triangle models help mitigate, but not entirely eliminate, the risk of error and bias in
array model results by providing an independent means to cross-check them. To this
!75
extent, the Petrarchan array and triangle models provide further evidence in support
of the claim. However, the value of the evidence is weakened by the absence of
development rules for, and lack of testing of, the triangle models. With the aim of
addressing these weaknesses, the inquiry now turns to the Pleadean sonnet tradition.
!76
2.3
Pleadean Tradition
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 cd
e e d
a b b a
a b b a
c cd
e d e .1
Transforming both into simplified form and referring to them, somewhat
chicly, as the Pleadean 1 and 2 traditions, respectively, brings even more to the fore,
as shown in Table 30, the single difference that separates them, namely, of course,
the reversed order of their final two end rhymes:
Table 30
Pleadean 1 & 2: Comparison of Simplified Rhyme Schemes
Pleadean 1
a b b a a b b a b b a b b a
Pleadean 2
a b b a a b b a b b a b a b
It may also be seen that the Pleadean 1 array comprises two mirrored subarrays, as shown in Table 31. The inserted bars show how the two arrays are related:
Table 31
Pleadean 1: Simplified Rhyme Scheme with Symmetry
Line:
Pleadean 1
1 2 3 4 5 6 7 8 9 10 11 12 13 14
a b b a a b b a b | b a b b a
Mirrored Sub-Arrays:
b a a b b a b | b a b b a a b
3 4 5 6 7 8 9 10 11 12 13 14 1 2
A conventional rhyme scheme representation emphasizing the so-called
asymmetry of the octave–sestet relationship is, therefore, to the extent that it
masks this symmetry, something of a trompe-l’œil.
Just under four of five Pleadean sonnets make use of these rhyme schemes in a ratio
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.
1
!77
The simplified Pleadean 2 rhyme scheme shows no such symmetry. If the
symmetrical principle above is extended as far as possible within it, the mirrored
sub-arrays shown in Table 32 result. It can be seen that the pairs of sixth and seventh
elements, developed from the centre and underlined in the table, do not correspond:
Table 32
Pleadean 2: Simplified Rhyme Scheme without Symmetry
Line:
1 2 3 4
5 6 7 8 9 10 11 12 13 14
Pleadean 2
a b b a | a b b a b b a b a b
Mirrored Sub-Arrays
b a b a b b a | a b b a b b a
12 13 14 1 2 3 4
5 6 7 8 9 10 11
What might appear as a poetic bagatelle, a difference in the order of two
final end rhymes between two conventional rhyme schemes, distinguishes, as shall
be evidenced below with the development of the Pleadean models, two quite different 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 formal 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 developmental 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;
!78
Rule 2. Directionality changes in array 11;
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
↖↖↖
a b b a
↖↖↖
a a b b
b b a a
↗↗↗
b a a b
↗↗↗
a a b b
↗↗↗
a b b a
↖↖↖
↗↗↗
b a a b
↖↖↖
b b a a
↖↖↖
a b b a
↖↖↖
a a b b
b b a a
↗↗↗
b a a b
↗↗↗
a a b b
↗↗↗
a b b a
↙↙↙
↘↘↘
a b b a
↙↙↙
b b a a
↙↙↙
b a a b
a a b b
↘↘↘
b a a b
↘↘↘
b b a a
↘↘↘
↙↙↙
b b a a
↘↘↘
a b b a
↘↘↘
a a b b
b a a b
↙↙↙
a a b b
↙↙↙
a b b a
!79
2.3.3
Assessment: Pleadean 1 Array Models
Do, then, these models satisfy the formal conditions of the Pleadean 1
sonnet? The models' fourteen arrays are equivalent to the traditional Pleadean sonnet
length of fourteen lines. Development is halted after fourteen arrays not because of
the development of a single redundant array, there is categorically a repetition of
array 6 in array 2 in both, but because, with the development of array 1, the potential
for the innovative symmetrical development of array series is exhausted. To see
this, consider that the array series 8–5 is repeated symbolically and cyclically in the
series 4–1 as is the series 8–10 in the series 12–14, when reflected in array 11.
Extending development beyond the models results in the repetition of these series,
as shown in Table 34 below, using the ‘a b b a’ centre array model as exemplary for
both models. Strictly speaking, array 4 is not a repetition of array 8, which is why it
is bracketed in the column to the right of the table: The former has towards flows
that for the centre array are undefined. This may be regarded as a limitation of the
model. It is also clear from Table 34, however, that the next array developed
upwards is a repetition of the symbolic and cyclical properties of array 4, creating
duplication of the series in arrays 7–4. The model thus represents, with the
aforementioned limitation, a maximum of array series innovation and a minimum of
array series redundancy.
!80
Table 34
Pleadean 1: Array Series Redundancy
-4.
b b a a
=
1.
-3.
b a a b
=
2.
-2.
a a b b
=
3.
-1.
a b b a
=
4.
1.
b b a a
=
5.
2.
b a a b
=
6.
3.
a a b b
=
7.
4.
a b b a
= (8), -1.
5.
b b a a
6.
b a a b
7.
a a b b
8.
a b b a
9.
a a b b
10.
b a a b
11.
b b a a
12.
b a a b
= 10.
13.
a a b b
=
9.
14.
a b b a
=
8.
15.
b b a a
= 11.
16.
a b b a
= 14.
17.
a a b b
= 13.
18.
b a a b
= 12.
19.
b b a a
= 11.
The equivalent of the volta is deemed to occur at the point of greatest
contrastive flows, that is in array 8. As to the equivalents of stanzaic form, the
!81
octave is separated into quartets and the sestet into tercets by the identity of arrays in
the first and reflective identity of arrays about array 11 in the second. The isometry
condition is satisfied by the constant four elements per array, which also accommodates the equivalent of the usual four accents of the four measured tetrameter of
the preferred twelve syllable Pleadean verse, the alexandrine, as shown in Table 35.
Table 35
Distribution of Four Stresses, (x), in a Four-Element Array
1
2
3
4
– – x
– – x
– – x
– – x
Finally, as may be seen in Table 36, the model’s simplified rhyme scheme
accommodates, of course, as this was after all the starting point for the analysis, the
end rhymes of the Pleadean 1 rhyme scheme noted by Kircher:
Table 36
Pleadean 1: Accommodation of Conventional Rhyme Scheme
Pleadean Sonnet 1 Rhyme Scheme
a b b a
a b b a
c c d
e e d
Pleadean 1 Simplified Rhyme Scheme
a b b a
a b b a
b b a
b b a
The model, then, is able to relate equivalents of the formal characteristics of
the Pleadean 1 tradition. Yet how reliable are its results? In the conclusion to the
discussion of the Early Italian tradition,2 it was suggested that the symmetry between 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
2
p. 54
!82
test whether a similar triangle model is capable of relating the two Pleadean 1 array
models shown in Table 33. The purpose of such a model, it may be recalled, is to
show whether and, if so, how, independently of the array models, the formal characteristics of a particular sonnet tradition might be developed and related. If a Pleadean
triangle model could achieve this, it would serve to cross-check the array models’
results. How then is such a model to be constructed for the Pleadean 1 tradition?
2.3.4
Pleadean 1: Centre Sequence and Triangle Models
One approach would be to take the Pleadean 1 simplified rhyme scheme as
a centre sequence for development into a triangle model by, first, substituting it for
the Petrarchan centre sequence that led to the development of the Petrarchan triangle
models and, second, by applying the same development rules to it. As noted earlier,
and as shown in Table 37, the Pleadean 1 centre sequence comprises two mirrored
sub-arrays. Let the bar representing the line of symmetry between them serve as a
point of orientation in the discussion to follow.
Table 37
Pleadean 1: Triangle Models’ Centre Sequence
b a a b b a b
|
b a b b a a b
Let the same developmental rule as for the Petrarchan triangle model now
be applied to this centre sequence, that is, let development continue without a
change in directionality from the outer elements towards the centre. As development
ends when elements meet in the centre, as shown in the stepwise description of the !83
triangle model’s construction to follow, a second rule for ending development is
unnecessary:
Rule 1
From its centre sequence, elements to the left of the line of
symmetry are developed upwards and downwards to the
right, those to its right are developed upwards and
downwards to the left.
A step-by-step description of the development of the triangle models now
follows. As it is a matter of testing whether the models might relate the two
previously developed Pleadean 1 array models, sequences shall be developed only
from the point in the centre sequence where the centre arrays of these models,
underlined in Step 1 below, are located.
Step 1
Centre Sequence with Array Model Centre Arrays
b a a b b a b | b a b b a a b (b)
The bracketed element on the right signifies that this element falls outside
the centre sequence in Table 37. However, here, as in the development of the
Petrarchan triangle model, although it was not made explicit at the time for purposes
of exposition, it is assumed that the centre sequence develops infinitely away from
the centre thus allowing for the inclusion of the next element in the sequence. That
the element ‘b’ is indeed the next element is inferred from the rightmost ‘b’ element
of the second array of the Pleadean 1 array model, counting from the top, in Table
33. As shall be seen below, the triangle model can only be fully developed if this
end-array element is included in the centre sequence.
!84
Step 2
Development of Arrays 7–5 and 9–11
5.
b b a a
b a a b
6.
b a a b
b b a a
7.
a a b b
a b b a
10.
b a a b
b b a a
11.
b b a a
b a a b
8.
9.
↗↗↗
↖↖↖
... a b b a a b b a b | b a b b a a b b a a b b ...
↘↘↘
↙↙↙
a a b b
a b b a
Pairs of new sequences are developed in binary fashion by the successive
division and distribution of individual elements entering from the left and right of
the centre sequence. With the development of each new sequence, the element
closest to the centre of the sequence exits the models. The triangular shape resulting
from development, familiar from the Petrarchan triangle model, is omitted here to
highlight the parts of the sequences that correspond to the Pleadean 1 array models.
!85
Step 3
Development of Arrays 4–1 and 12–14
1.
b b a a
b a a b
2.
b a a b
b b a a
3.
a a b b
a b b a
4.
a b b a
a a b b
5.
b b a a
b a a b
6.
b a a b
b b a a
7.
8.
9.
a a b b
a b b a
... ↗↗↗↗↗↗↗
↖↖↖↖↖↖↖ ...
... 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 ...
... ↘↘↘↘↘↘↘
↙↙↙↙↙↙↙ ...
a a b b
a b b a
10.
b a a b
b b a a
11.
b b a a
b a a b
12.
(a b b a)
(a a b b)
13.
(a a b b)
(a b b a)
14.
(b a a b)
(b b a a)
The models complete, the Pleadean triangle models’ equivalents of arrays
12–14 of the array models are bracketed because they do not correspond to the
results of the Pleadean 1 array models developed above. That is, the triangle model
for the Petrarchan tradition is not transferable to the Pleadean. The reason is due of
course to the directionality change in array 11 of the Pleadean 1 array models, a
development unknown in the Petrarchan models. How, then, are these directionality
!86
changes to be understood and reproduced independently so as to support the
Pleadean 1 array model results? This question is considered now in the discussion of
the second Pleadean tradition.
2.3.5
Pleadean 2: Array Models
The array model for the Pleadean 2 tradition is more complex than that for
the Pleadean 1 in that not one, but three directionality changes are needed in its final
six arrays to satisfy sonnet conditions. As discussed in Appendix D, the centre arrays
for both Pleadean traditions are the same, and their rules for array development
differ only insofar as extra directionality changes need to be taken into account. This
being the case, the array models for the second Pleadean tradition are presented
complete in Table 38 below.
!87
Table 38
Pleadean 2: Array Models
Pleadean 2 Array Models
b a a b
↖↖↖
b b a a
↖↖↖
a b b a
↖↖↖
a a b b
↖↖↖
b a a b
↖↖↖
b b a a
↖↖↖
a b b a
↖↖↖
a a b b
↙↙↙
a b b a
↙↙↙
b b a a
↙↙↙
b a a b
↘ ↘↘
b b a a
↙↙↙
b a a b
↘↘↘
b b a a
b b a a
↗↗↗
b a a b
↗↗↗
a a b b
↗↗↗
a b b a
↗↗↗
b b a a
↗↗↗
b a a b
↗↗↗
a a b b
↗↗↗
a b b a
↘↘↘
a a b b
↘↘↘
b a a b
↘↘↘
b b a a
↙↙↙
b a a b
↘↘↘
b b a a
↙↙↙
b a a b
!88
2.3.6
Assessment: Pleadean 2 Array Models
The equivalent of sonnet length is determined by the same considerations of
array series innovation and redundancy as for the Pleadean 1 array model. The same
is also the case for the equivalents of the volta, isometry and rhyme scheme. The
equivalent of stanzaic form is represented by repetition of arrays in the quartet
equivalents and, differently from the Pleadean 1 model, a contrast between the
constant directionality of the flows in the first tercet equivalent and their constant
variability in the second.
2.3.7
Pleadean 1 & 2: Sequence Models
How then are the results of the Pleadean array models to be accounted for
independently? Given that the Petrarchan triangle model proved inadequate due to
the directionality change introduced in the lower half of the Pleadean 1 model, how
much more inadequate would it prove in dealing with the three directionality
changes of the Pleadean 2 model? Is there, then, another way of defining the triangle
model’s developmental rules that might prove more satisfactory?
One possibility would be to think of the Pleadean 1 and 2 triangle models’
centre sequences as developing according to the same cyclicity principle applied in
the working, Early Italian and Petrarchan array models, wherein development in one
half of the model is mirrored by development in the other. Developing the Pleadean
1 and 2 centre sequences according to this principle, however, cannot account for the
changes in directionality of the Pleadean models as identical sequences would be
created in both halves of the model whereas directionality changes occur only in the
Pleadean models’ lower halves. It would seem then that the patterns created by the
!89
array and triangle models are not broad enough in the sense that they do not provide
enough, what might be thought of as, ‘patterned data’ to allow for the emergence of
an intelligible difference that might account for the directionality changes in the
Pleadean models.
In order to try out the idea of creating a broader pattern, let the Pleadean 1
and 2 centre sequences instead be developed cyclically into what shall be termed
sequence models with a constant fourteen elements per sequence throughout, as
opposed to the four elements per array in the array models and the constantly
diminishing number of elements per sequence in the triangle models. To avoid a
possible confusion of terms, let the centre sequences of the triangle models be called
start sequences when used to develop sequence models. The development rule for
the sequence models is uncomplicated: The start sequences for both the Pleadean 1
and 2 sequence models are developed leftwards without any change in directionality.
This rule results in the creation of models of twenty-seven sequences, as shown in
Tables 39 and 40 below.
!90
Table 39
Pleadean 1: Sequence Model: Leftwards Directionality
(...)
(b a a b b a b b a b b a a b)
(b b a a b b a b b a b b a a)
(a b b a a b b a b b a b b a)
a
b
b
a
b
b
a
b
b
a
a
b
b
a
b
b
a
a
b
b
a
b
b
a
b
b
a
a
a
b
b
a
b
b
a
b
b
a
a
b
b
b
a
a
b
b
a
b
b
a
b
b
a
a
b
a
a
b
b
a
b
b
a
b
b
a
a
b
a
a
b
b
a
b
b
a
b
b
a
a
b
b
b
a
a
b
b
a
b
b
a
b
b
a
a
a
b
b
a
b
b
a
b
b
a
a
b
b
a
b
b
a
a
b
b
a
b
b
a
b
b
a
b
b
a
b
b
a
b
b
a
a
b
b
a
a
a
b
b
a
a
b
b
a
b
b
a
b
b
b
a
b
b
a
b
b
a
a
b
b
a
a
b
a
a
b
b
a
a
b
b
a
b
b
a
b
a
b
b
a
b
b
a
a
b
b
a
a
b
b
b
a
a
b
b
a
a
b
b
a
b
b
a
b
b
a
b
b
a
a
b
b
a
a
b
b
a
b
b
a
a
b
b
a
a
b
b
a
b
b
b
a
b
b
a
a
b
b
a
a
b
b
a
b
a
b
b
a
a
b
b
a
a
b
b
a
b
a
b
b
a
a
b
b
a
a
b
b
a
b
b
b
a
b
b
a
a
b
b
a
a
b
b
a
b
b
a
a
b
b
a
a
b
b
a
b
b
a
b
b
a
b
b
a
a
b
b
a
a
b
b
b
a
a
b
b
a
a
b
b
a
b
b
a
b
a
b
b
a
b
b
a
a
b
b
a
a
b
a
a
b
b
a
a
b
b
a
b
b
a
b
=
=
=
b
b
a
b
b
a
b
b
a
a
b
b
a
a
a
b
b
a
a
b
b
a
b
b
a
b
b
(a b b a a b b a b b a b b a)
(b b a a b b a b b a b b a a)
(b a a b b a b b a b b a a b)
3.
2.
1.
14.
13.
12.
11.
10.
9.
8.
7.
6.
5.
4.
3.
2.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
=
=
=
1.
2.
3.
(...)
!91
↖
*
↙
Table 40
Pleadean 2: Sequence Model: Leftwards Directionality
(...)
(b a a b b a b b a b a b a b)
(b b a a b b a b b a b a b a)
(a b b a a b b a b b a b a b)
b
a
b
a
b
b
a
b
b
a
a
b
b
a
b
b
a
a
b
b
a
b
b
a
b
a
b
a
b
a
b
a
b
b
a
b
b
a
a
b
b
b
a
a
b
b
a
b
b
a
b
a
b
a
b
a
b
a
b
a
b
b
a
b
b
a
a
b
a
a
b
b
a
b
b
a
b
a
b
a
b
b
b
a
b
a
b
a
b
b
a
b
b
a
a
a
b
b
a
b
b
a
b
a
b
a
b
b
a
b
b
a
b
a
b
a
b
b
a
b
b
a
b
b
a
b
b
a
b
a
b
a
b
b
a
a
a
b
b
a
b
a
b
a
b
b
a
b
b
b
a
b
b
a
b
a
b
a
b
b
a
a
b
a
a
b
b
a
b
a
b
a
b
b
a
b
a
b
b
a
b
a
b
a
b
b
a
a
b
b
b
a
a
b
b
a
b
a
b
a
b
b
a
b
b
a
b
a
b
a
b
b
a
a
b
b
a
b
b
a
a
b
b
a
b
a
b
a
b
b
b
a
b
a
b
a
b
b
a
a
b
b
a
b
a
b
b
a
a
b
b
a
b
a
b
a
b
a
b
a
b
a
b
b
a
a
b
b
a
b
b
b
a
b
b
a
a
b
b
a
b
a
b
a
b
a
b
a
b
b
a
a
b
b
a
b
b
a
b
b
a
b
b
a
a
b
b
a
b
a
b
a
b
a
b
b
a
a
b
b
a
b
b
a
b
a
b
b
a
b
b
a
a
b
b
a
b
a
b
a
b
b
a
a
b
b
a
b
b
a
b
=
=
=
a
b
a
b
b
a
b
b
a
a
b
b
a
b
a
b
b
a
a
b
b
a
b
b
a
b
a
(a b b a a b b a b b a b a b)
(b b a a b b a b b a b a b a)
(b a a b b a b b a b a b a b)
3
2.
1.
14.
13.
12.
11.
10.
9.
8.
7.
6.
5.
4.
3.
2.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
=
=
=
1.
2.
3.
(...)
!92
↖
↙
*
These sequence models comprise two continuous leftward developments
upwards and downwards from the model’s centre. That is, differently from the
Pleadean array models, the Pleadean sequence models are constructed without any
change in directionality. This being the case, directionality changes in the Pleadean
array models may be understood grosso modo as resulting from the developmental
compromises required by a centre array of only four elements needing to maintain
cyclical coherency while at the same time developing equivalents of the tradition’s
formal characteristics. If this roughly describes directionality changes within the
array models, what, however, of the specific differences in directionality changes
between them? How is the single change in directionality in the Pleadean 1 model
versus the three in the Pleadean 2 to be made sense of? Is there, for instance, a
common principle that relates them? Moreover, could such a principle help describe
more precisely the directionality changes within the array models as well, thus
providing independent support for their results?
It is with these questions that discussion returns to the two different structural principles mentioned at the outset of the discussion on the Pleadean tradition:
the principles of symmetry and chirality. Wehrli (2008), drawing on Nakahara
(2003) and Kelvin (1893), describes the difference between these two ideas:
Chirality is an attribute of symmetry. A figure is called symmetrical when there
exists a non-identical congruent isomorphism of itself....An object without any
non-identical congruent image is chiral....Chirality, as I choose to understand
the term, is possible in spaces with any number of dimensions. (p. 61)
!93
Symmetry, therefore, unlike chirality, requires that one object coincide perfectly
with another. To describe chirality more fully, Wehrli paraphrases Kelvin, to whom
he ascribes the introduction of the term into the natural sciences:
Chirality means handedness. Our right hand is the mirror image of the left.
Although both hands are isometric, they cannot be brought to coincidence with
each other, i.e., perfectly aligned if one was placed on top of the other. So they
are different from each other, although they are the same metrically. We say
they are chiral. A third hand, which is likewise isometrical to the right and left
hand and which nevertheless cannot be aligned with either in the same way,
does not exist. For every hand there is one, and only one, counterpart opposing
handedness. An object is chiral, when it has a mirror image which is not
identical with it. (p. 60)
Table 40 above shows how chirality originates and evolves within the
sequence model of the Pleadean 2 tradition. The simplified rhyme scheme in Table
32, as noted, serves as the model’s start sequence. It is developed, as seen, upwards
and downwards with leftwards directionality. It also serves as a proxy for rightwards
directionality as each is just the reverse of the other. Arrows on the right-hand side
of the model indicate the constant directionality involved throughout. The model
comprises two groups of fourteen sequences with the start sequence serving as both
first sequence to the fourteen array model developed in the upper half and first
sequence to its mirror image in the lower half. That two models are developed is due
of course to the binary development of the start sequence. The bracketed sequences
and ellipses serve to show that the sequences 1–14 repeat themselves indefinitely
upwards and downwards beyond the model’s limits. Now, the only two simplified
rhyme schemes of the Pleadean 2 tradition developed within the model are in bold in
its leftmost column. The simplified rhyme schemes overlap at the underlined first
!94
element of the start sequence. That is, the underlined element forms the first element
of the simplified rhyme scheme developed in the lower half, and the first element of
the simplified rhyme scheme developed in the upper half of the model. Each half of
the model mirrors the other, as remarked above, yet, if the simplified rhyme schemes
developed in the model are superposed, they do not coincide, as shown in Table 41
below. The simplified rhyme schemes in Table 40 are, therefore, by the definitions
of symmetry and chirality given above, chiral in one dimension and symmetrical in
three.
Table 41
Pleadean 2: Sequence Model: Chirality
lower half
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
upper half
a
b
b
a
a
b
b
a
b
b
a
b
a
b
b
a
b
a
b
b
a
b
b
a
a
b
b
a
14.
13.
12.
11.
10.
9.
8.
7.
6.
5.
4.
3.
2.
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 lefthand side of the model. Once again, the simplified rhyme schemes overlap in the
!95
underlined elements in the start sequence. There is, however, a difference in the
order of the elements between the two Pleadean 1 simplified rhyme schemes developed in this sequence model. The leftmost rhyme scheme is ordered, so to speak,
from within to without, that is, from 1–14 upwards and 1–14 downwards, whilst the
rightmost is ordered from without to within, that is, from the top and bottom of the
model, from 14–1 in both cases. Now, as in the Pleadean 2 sequence model, each
half of the simplified rhyme schemes developed in the Pleadean 1 model mirrors the
other, coinciding if folded, not coinciding if superposed. As in the Pleadean 2
sequence model, therefore, the two halves are chiral in one dimension and
symmetrical in three. Differently from the Pleadean 2 model, however, the rhyme
schemes within each of the top and bottom halves of the Pleadean 1 model are also
symmetrical in two dimensions: Each rhyme scheme may be rotated about the
midpoints lying between them to map onto the other. This chirality between, and
symmetry within, the two halves of the Pleadean 1 sequence model is shown
respectively in Tables 42 and 43 below. In Table 42, the developed simplified rhyme
scheme on the left-hand side in Table 39 serves also to exemplify chirality for the
alternative development to its right, while in Table 43 below, the lower halves of
both developments in Table 39 stand as proxy for those in the upper half. The
rotational sign between rows 7 and 8 is deemed to signal rotation in either direction.
!96
Table 42
Pleadean 1: Sequence Model: Chirality between Model Halves
lower half
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
upper half
a
b
b
a
a
b
b
a
b
b
a
b
b
a
a
b
b
a
b
b
a
b
b
a
a
b
b
a
14.
13.
12.
11.
10.
9.
8.
7.
6.
5.
4.
3.
2.
1.
Table 43
Pleadean 1: Sequence Model: Symmetry within Model Halves
lower half
1.
2.
3.
4.
5.
6.
7.
a
b
b
a
a
b
b
8.
9.
10.
11.
12.
13.
14.
a
b
b
a
b
b
a
!97
↺
a
b
b
a
b
b
a
b
b
a
a
b
b
a
This analysis of the equivalents of the two Pleadean sonnet traditions under
consideration shows that while the relationship between the bottom and top halves
of each model is chiral and symmetrical, the development within the Pleadean 2
sequence model is uniquely so as its simplified rhyme scheme is developed only
once in each of the model’s halves. Its development twice within each half of the
Pleadean 1 sequence model, on the other hand, allows for symmetry in an additional
dimension, as well as chirality, to be included in the one model. Thus, between them,
these two models, taking the two simplified Pleadean rhyme schemes as their
starting point, permit with maximum economy a minimal representation of chirality
and symmetry, qualities traceable back to the one difference in the conventional
representation of their rhyme schemes highlighted in Table 30 above.3
It perhaps needs emphasizing that exclusively leftwards or rightwards
binary development from a start sequence or centre array is inherently chiral, and
that it is these specific directionalities that are able to bring out the symmetrical and
chiral aspects of the simplified rhyme schemes. To underscore this point, the two
other possible initial developments from the Pleadean start sequences, upwards left
and downwards right, and downwards left and upwards right, result merely in the
repeated symmetrical development of the Pleadean’s simplified rhymes schemes.
This is shown by the elements in bold in Tables 44 and 45 below which are
developed from the Pleadean 1 and 2 start sequences. In the tables, upwards left and
downwards right development, being simply the reverse of an upwards right and
downwards left development, stands as a proxy for it.
3
p. 77
!98
Table 44
Pleadean 1: Sequence Model: Leftwards and Rightwards Directionality
...
(b a a b b a b b a b b a a b)
(b b a a b b a b b a b b a a)
(a b b a a b b a b b a b b a)
a
b
b
a
b
b
a
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
b
b
a
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
b
b
a
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
b
b
a
b
b
a
a
b
b
a
a
b
b
a
b
b
a
b
b
a
=
=
=
a a b b a b b a b b
b a a b b a b b a b
b b a a b b a b b a
a
a
b
b
a
b
b
a
b
b
a
a
b
b
a
a
b
b
a
b
b
a
b
b
b
a
a
b
b
a
b
b
a
b
b
a
a
b
b
a
a
b
b
a
b
b
a
b
b
b
a
a
b
b
a
b
b
a
b
b
a
a
b
b
a
a
b
b
a
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
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
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
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
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
b
b
a
a
b
b
a
a
b
b
a
b
b
a
b
b
a
a
b
b
a
a
14.
13.
12.
11.
10.
9.
8.
7.
6.
5.
4.
3.
2. ↖
1.
*
2.
↘
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
b
b
a
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 b a b b a b b a)
(b a b b a a b b a b b a b b)
(b b a b b a a b b a b b a b)
3.
2.
1.
=
=
=
1.
2.
3.
...
!99
Table 45
Pleadean 2: Sequence Model: Leftwards and Rightwards Directionality
...
(b a a b b a b b a b a b a b)
(b b a a b b a b b a b a b a)
(a b b a a b b a b b a b a b)
b
a
b
a
b
b
a
b
b
a
a
b
b
a
b
a
b
a
b
b
a
b
b
a
a
b
b
a
b
a
b
a
b
b
a
b
b
a
a
b
b
a
b
a
b
a
b
b
a
b
b
a
a
b
b
a
b
a
b
a
b
b
a
b
b
a
a
b
b
a
b
a
b
a
b
b
a
b
b
a
a
b
b
a
b
a
b
a
b
b
a
b
b
a
a
b
b
a
b
a
b
a
b
b
a
b
b
a
a
b
b
a
b
a
b
a
b
b
a
b
b
a
a
b
b
a
b
a
b
a
b
b
a
b
b
a
a
b
b
a
b
a
b
a
b
b
a
b
b
a
a
b
b
a
b
a
b
a
b
b
a
b
b
a
a
b
b
a
b
a
b
a
b
b
a
b
b
a
a
b
b
a
b
a
b
a
b
b
a
b
b
a
a
b
b
a
b
a
b
a
b
b
a
b
b
a
a
b
b
a
b
a
b
a
b
b
a
b
b
a
a
b
b
a
b
a
b
a
b
b
a
b
b
a
a
b
b
a
b
a
b
a
b
b
a
b
b
a
a
b
b
a
b
a
b
a
b
b
a
b
b
a
a
b
b
a
b
a
b
a
b
b
a
b
b
a
a
b
b
a
b
a
b
a
b
b
a
b
b
a
a
b
b
a
b
a
b
a
b
b
a
b
b
a
a
b
b
a
b
a
b
a
b
b
a
b
b
a
a
b
b
a
b
a
b
a
b
b
a
b
b
a
a
b
b
a
b
a
b
a
b
b
a
b
b
a
a
b
b
a
b
= 3.
= 2.
= 1.
a
b
a
b
b
a
b
b
a
a
b
b
a
b
a
b
a
b
b
a
b
b
a
a
b
b
a
(a b b a a b b a b b a b a b)
(b a b b a a b b a b b a b a)
(a b a b b a a b b a b b a b)
14.
13.
12.
11.
10.
9.
8.
7.
6.
5.
4.
3.
2. ↖
1.
*
2.
↘
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
=
=
=
1.
2.
3.
...
!100
2.3.8
Conclusion
The Pleadean sonnet traditions bring symmetry and chirality within what
Seamus Heaney (1980) in his essay Feeling into Words, cited by Vendler (1998, p.
8), refers to as the jurisdiction of poetic form: The sequence models with leftwards
directionality reveal the latent symmetrical and chiral properties of the conventional
Pleadean rhyme schemes. The two Pleadean traditions considered separately are
therefore only partially appreciated, rather more so when viewed as counterparts.
The only equivalents of the sonnet’s formal characteristics to be developed
in the sequence models are, however, the rhyme scheme and sonnet length. The
Pleadean sequence and array models, therefore, only partly coincide, and it still
remains to be shown how array model results might be fully developed independently when directionality changes are involved. To seek an answer to this question,
discussion now turns to the final sonnet tradition to be considered in the inquiry, the
Shakespearean.
!101
2.4
Shakespearean Tradition
2.4.0
Introduction
In seeking an approach that relates equivalents of the formal
characteristics of the Shakespearean sonnet independently of the array models, I
shall adopt a disarmingly straightforward principle from the field called by GellMann ‘plectics’, namely, that complexity evolves from simplicity (Gell-Mann,
1996, p. 3). In applying this principle, I shall term the array models discussed so
far complex to the extent that they involve directionality changes. I shall then
infer that these complex models are derived from simpler models without
directionality changes. From this, I shall further infer that these simpler models
are part of a more general binary expansion pattern, the origin and evolution of
which offers an alternative approach, independent of the array models, to the
modelling of the Shakespearean sonnet form.
2.4.1
Simplified Rhyme Scheme
Kircher (p. 415) notes the following rhyme scheme for the
Shakespearean sonnet:
a b a b
c d c d
e f e f
g g
In simplified form, but retaining stanzaic markers for the moment, this
gives the equivalent of an alternating rhyme scheme with a final rhymed couplet:
a b a b
a b a b
a b a b
a a
The absence of contrasting embracing and alternating rhyme equivalents suggests
that a centre array rather than a centre matrix suffices to develop the array model.
!102
2.4.2
Centre Array
Analysis and testing, following the same procedure as in previous
models, indicates the array ‘a b b a b’ with leftwards and rightwards directionality
as a centre array presumed to satisfy the formal conditions of the Shakespearean
sonnet. The detailed workings may be referred to in Appendix E. However, rather
than, as with previous models, straightaway defining rules to develop this centre
array, in order to seek an independent approach based on the principle of
complexity evolving from simplicity, let a simple array model first be constructed
by developing the centre array with no changes in directionality.
2.4.3
Simple Array and Triangle Models
The resulting simple array model appears in Table 46 below.
Immediately following it in Table 47, a second, alternative model to be termed
simple triangle model is developed without directionality changes. Its centre
sequence is composed of the simple array model's leftmost array elements and
centre sequence. It can be seen from the juxtaposing of the two models in Table 48
below that their results are, not unexpectedly, identical. Now, as it is in principle
possible to reverse this procedure, that is, to derive the simple array model from
the simple triangle model, it is deduced that both are identical aspects of the same
binary expansion, differing only in their alternative methods of development.
!103
Table 46
Simple Shakespearean Array Model
✶
↗
↘
a b a b b
↗↗↗↗
b a b b a
↗↗↗↗
a b b a b
↗↗↗↗
b b a b a
↗↗↗↗
b a b a b
↗↗↗↗
a b a b b
↗↗↗↗
b a b b a
↗↗↗↗
a b b a b
↘ ↘ ↘↘
b a b b a
↘ ↘ ↘↘
a b a b b
↘ ↘ ↘↘
b a b a b
↘ ↘ ↘↘
b b a b a
↘ ↘ ↘↘
a b b a b
↘ ↘ ↘↘
b a b b a
↘ ↘ ↘↘
a b a b b
!104
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
Table 47
Simple Shakespearean Triangle Model: Binary Expansion
Steps 1–9. (Steps 10–11 are overleaf).
1.
5.
8.
a
2. a b
a
a
a
a b
a b a
a b
a
a
a b
a b a
a b
a
a
b
a
b
a
b
a
a
b
a
b
a
b
a
a
a b
3. a b a
a b
a
a
b
a
b
b
b
a
b
a
a
b
a
b
b
b
a
b
a
6.
a
b
a
b
b
a
b
b
a
b
a
a
b
a
b
b
a
b
a
b
b
a
b
a
a
b
a
b
b
a
b
a
b
a
b
b
a
b
a
a
a b
a b a
a b
a
a
b
a
b
a
b
a
a
b
a
b
b
b
a
b
a
a
b
a
b
b
a
b
b
a
b
a
a
a b
a b a
a b
a
9.
!105
a
a b
4. a b a
a b
a
7.
a
b
a
b
a
b
a
a
b
a
b
b
b
a
b
a
a
a b
a b a
a b
a
a
b
a
b
a
b
a
a
b
a
b
b
a
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
a
b
a
b
b
a
b
b
a
b
a
a
b
a
b
b
a
b
a
b
b
a
b
a
a
b
a
b
b
b
b
b
a
a
b
a
b
b
a
b
b
a
b
a
a
b
a
b
a
b
a
a
b
a
b
b
a
b
a
b
b
a
b
a
Table 47 (cntd.)
Simple Shakespearean Triangle Model: Binary Expansion.
Steps 10–11. (Step 12 is overleaf).
10.
a
a b
a b a
a b
a
a
b
a
b
a
b
a
a
b
a
b
b
b
a
b
a
a
b
a
b
b
a
b
b
a
b
a
a
b
a
b
b
a
b
a
b
b
a
b
a
a
b
a
b
b
a
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
a
b
a
b
b
a
b
a
b
b
b
a
b
a
b
b
a
b
a
11.
!106
a
a b
a b a
a b
a
a
b
a
b
a
b
a
a
b
a
b
b
b
a
b
a
a
b
a
b
b
a
b
b
a
b
a
a
b
a
b
b
a
b
a
b
b
a
b
a
a
b
a
b
b
a
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
a
b
a
b
b
a
b
a
b
b
b
a
b
a
b
b
a
b
a
a
b
a
b
b
a
b
a
b
b
a
b
b
a
b
a
b
b
a
b
a
Table 47 (cntd.)
Simple Shakespearean Triangle Model: Binary Expansion.
Step 12.
a
a b
a b a
a b a b
a
a b
a b a
a b
a
a
b
a
b
a
b
a
a
b
a
b
b
b
a
b
a
a
b
a
b
b
a
b
b
a
b
a
a
b
a
b
b
a
b
a
b
b
a
b
a
a
b
a
b
b
a
b
a
b
a
b
b
a
b
a
b
a
b
b
a
b
a
b
a
b
a
b
b
a
b
a
b
b
a
b
a
b
b
b
a
b
a
b
b
a
b
b
a
b
a
b
b
a
b
b
a
b
a
b
b
b
a
b
a
b
b
a
b
a
b
b
a
b
a
b
a b a b
a b a
a b
a
!107
Simple Triangle Model
a
b
a
b
b
a
b
a
b
a
b
b
a
b
a
b
a
b
b
a
b
a
b
a
b
a
b
b
a
b
a
b
b
a
b
a
b
b
b
a
b
a
b
b
a
b
b
a
b
a
b
b
a
b
b
a
b
a
b
b
b
a
b
a
b
b
a
b
a
b
b
a
b
a
b
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
Table 48
Identical Simple Shakespearean Triangle and Array Models
Array Model
(Table 46)
✶
↗
↘
a
b
a
b
b
a
b
a
b
a
b
b
a
b
a
b
a
b
b
a
b
a
b
a
b
a
b
b
a
b
a
b
b
a
b
a
b
b
b
a
b
a
b
b
a
b
b
a
b
a
b
b
a
b
b
a
b
a
b
b
b
a
b
a
b
b
a
b
a
b
b
a
b
a
b
Triangle Model
(Table 47)
a
b
a
b
b
a
b
a
b
a
b
b
a
b
a
b
a
b
b
a
b
a
b
a
b
a
b
b
a
b
a
b
b
a
b
a
b
b
b
a
b
a
b
b
a
b
b
a
b
a
b
b
a
b
b
a
b
a
b
b
b
a
b
a
b
b
a
b
a
b
b
a
b
a
b
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
Let these simple Shakespearean array and triangle models without directionality changes now be distinguished from the more complex Shakespearean
array and triangle models with directionality changes.
2.4.4
Complex Array Model: Step-by-Step Description
Having shown that the simple Shakespearean array and triangle models
are identical, to demonstrate two independent approaches to the construction of a
complex Shakespearean model two conditions need to be satisfied. First, a
complex array model, developed from a centre array, must be able to relate and
describe equivalents of the formal characteristics of the Shakespearean sonnet.
Second, a complex triangle model that is identical to the complex array model
must be able to be constructed from the simple triangle model described above.
!108
Assuming these conditions fulfilled, it follows, first, that the complex triangle
model also relates and describes equivalents of the formal characteristics of the
Shakespearean sonnet and, second, that both the complex array and triangle
models may be considered identical aspects of the same binary expansion, only
developed independently. In this way, two independent approaches to the
construction of a Shakespearean model with directionality changes may be
demonstrated.
Turning to the demonstration itself, to construct a complex array model
that relates the equivalents of the formal characteristics of the Shakespearean
sonnet, once again understood as five conditions to be satisfied simultaneously,
there are three rules for development of the centre array, ‘a b b a b’:
Rule 1.
From the centre array, arrays develop simultaneously
upwards and downwards to the right;
Rule 2.
Development of the array ‘a b a b b’ causes a change in
directionality from right to left;
Rule 3.
Symbolic repetition of successive end-array elements halts
development and completes the model.
There now follows a step-by-step description of the application of these
rules. For clarity of presentation, the development with leftwards directionality is
omitted. Its final model is shown alongside the model with rightwards
development in Table 50 further below.
!109
Steps 1 & 2
Development of arrays 7 & 8 and 6 & 10
6.
7.
8.
9.
10.
a b a b b
↗↗↗↗
b a b b a
↗↗↗↗
a b b a b
↘↘↘↘
b a b b a
↘↘↘↘
a b a b b
Development begins, according to Rule 1, with the simultaneous creation
of array pairs upwards and downwards to the right from array 8, the centre array.
Step 3
Development of arrays 5 & 11
5.
6.
7.
8.
9.
10.
11.
b a b b a
↖↖↖↖
a b a b b
↗↗↗↗
b a b b a
↗↗↗↗
a b b a b
↘↘↘↘
b a b b a
↘↘↘↘
a b a b b
↙↙↙↙
b a b b a
With the development of the array ‘a b a b b’ in arrays 6 & 10, directionality changes from right to left, according to Rule 2.
!110
Steps 4, 5, 6 & 7:Development of arrays 4–1 & 12–15
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
a b a b b
↖↖↖↖
b a b a b
↖↖↖↖
b b a b a
↖↖↖↖
a b b a b
↖↖↖↖
b a b b a
↖↖↖↖
a b a b b
↗↗↗↗
b a b b a
↗↗↗↗
a b b a b
↘↘↘↘
b a b b a
↘↘↘↘
a b a b b
↙↙↙↙
b a b b a
↙↙↙↙
a b b a b
↙↙↙↙
b b a b a
↙↙↙↙
b a b a b
↙↙↙↙
a b a b b
Development continues leftwards. However, with the repetition of successive end-array elements in Arrays 2 & 1 and 14 & 15, array development is
halted and the model complete, according to Rule 3. As in the previous models,
this model has fifteen arrays as it comprises not one, but two identical fourteen
!111
array sub-models, arrays 2–15 and 14–1. Juxtaposing these sub-models, as shown
in Table 49, makes their identity manifest:
Table 49
Identical Shakespearean Sub-Models
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
b a b a b
↖↖↖↖
b b a b a
↖↖↖↖
a b b a b
↖↖↖↖
b a b b a
↖↖↖↖
a b a b b
↗↗↗↗
b a b b a
↗↗↗↗
a b b a b
↘↘↘↘
b a b b a
↘↘↘↘
a b a b b
↙↙↙↙
b a b b a
↙↙↙↙
a b b a b
↙↙↙↙
b b a b a
↙↙↙↙
b a b a b
↙↙↙↙
a b a b b
b a b a b
↖↖↖↖
b b a b a
↖↖↖↖
a b b a b
↖↖↖↖
b a b b a
↖↖↖↖
a b a b b
↗↗↗↗
b a b b a
↗↗↗↗
a b b a b
↘↘↘↘
b a b b a
↘↘↘↘
a b a b b
↙↙↙↙
b a b b a
↙↙↙↙
a b b a b
↙↙↙↙
b b a b a
↙↙↙↙
b a b a b
↙↙↙↙
a b a b b
14.
13.
12.
11.
10.
9.
8.
7.
6.
5.
4.
3.
2.
1.
Combining the two sub-models gives the complex Shakespearean array
model of fourteen arrays, as shown on the right-hand side of Table 50 below. On
the left-hand side, in its completed form, is its counterpart with leftwards
development.
!112
Table 50
Complex Shakespearean Array Models: Leftwards and Rightwards Developments
a b a b b
↗↗↗↗
b a b b a
↗↗↗↗
a b b a b
↗↗↗↗
b b a b a
b a b a b
↖↖↖↖
b b a b a
↖↖↖↖
a b b a b
↖↖↖↖
b a b b a
b a b a b
↖↖↖↖
b b a b a
↖↖↖↖
a b b a b
↙↙↙↙
b b a b a
a b a b b
↗ ↗ ↗↗
b a b b a
↗↗ ↗↗
a b b a b
↘ ↘ ↘↘
b a b b a
b a b a b
↘↘↘↘
b b a b a
↘↘↘↘
a b b a b
↘↘↘↘
b a b b a
a b a b b
↙↙↙↙
b a b b a
↙↙↙↙
a b b a b
↙↙↙↙
b b a b a
a b a b b
↘↘↘↘
b a b a b
b a b a b
↙ ↙ ↙↙
a b a b b
!113
2.4.5
Assessment
Are, then, the formal characteristics of the Shakespearean sonnet
described by this array model? The fourteen arrays are deemed equivalent to the
sonnet length condition of fourteen lines. Redundancy is triggered by successive
end-array elements with the same symbolic and cyclical properties. Changes in
directionality at arrays 5 and 9 divide the first twelve arrays into three groups of
four arrays equivalent to the three quatrains of a Shakespearean sonnet, whilst the
final two arrays with similar end-array elements represent the equivalent of the
couplet, thus fulfilling the condition for stanzaic form. With all other end-array
elements alternating, the condition for the Shakespearean rhyme scheme is also
satisfied. As to the volta equivalent, the point of greatest contrastive development
before the disaggregation and combination of the sub-models to create the final
complex array models is at the centre array, array 8, as may be seen in Steps 4, 5,
6 & 7 above. However, in the final complex array model array 8 becomes array 7,
so that by the criterion of greatest contrastive flows, the volta equivalent does not
occur in line 8, but in line 7. This may appear to be a limitation of the model,
though, when read linearly, that is, conventionally from top to bottom, array 8 is
the first array in the model developed after the point of greatest contrastive flows.
As such, it is deemed to represent the equivalent of the volta. With the five
elements in each array able to accommodate the equivalent of the five stresses of
the dominant Shakespearean sonnet metre, the iambic pentameter, the isometry
condition is also satisfied. It seems, on balance, therefore, reasonable to conclude
that the models satisfactorily describe and relate equivalents of the formal
!114
characteristics of the Shakespearean sonnet, fulfilling the first condition for the
construction of independent complex models.
2.4.6
Complex Triangle Model: Step-by-step Description
As remarked above, fulfilment of the second condition requires, first, that
a complex triangle model's results be the same as those of its complex array model
counterpart and, second, that the complex triangle model be developed from its
simple triangle model. Only then will it have been shown that the same complex
models can be developed independently both from within a binary expansion and
from a centre array.
To show how a simple triangle model may be developed into a complex
triangle model with results identical to those of the complex array model, let
arrays 1 & 15 of the simple triangle model shown in Table 47 above be the
starting point for its development. It is, of course, possible to construct the
complex triangle model by beginning development from the centre sequence of
the simple triangle model. However, to make it easier to see the changes in
directionality in the complex triangle model and to emphasize its independent
approach to producing the same results as the complex array model, I have chosen
to start from arrays 1 and 15 of the simple triangle model. That these arrays
represent the start of the centre sequence of the simple triangle model, may be
seen in Step 12 of the triangle model development in Table 47. There are three
rules for development of arrays 1 &15:
!115
Rule 1.
Arrays develop simultaneously downwards and upwards
to the right from arrays 1 & 15, respectively, of the simple
triangle model;
Rule 2.
Symbolic repetition of arrays 1 & 15 leads to a change in
directionality;
Rule 3.
When the centre array, array 8, is developed, development
is halted and the model complete.
A stepwise construction of the complex triangle model now follows.
Step 1
Arrays 1 & 15
1.
a b a b b
15.
a b a b b
a b a b b
As just remarked, the underlined array on the left represents the first five
elements of the simple triangle model’s centre sequence, whilst arrays 1 & 15
represent these elements’ development within the simple triangle model, as shown
in the models on the left- and right-hand sides of Table 47, respectively. As there
can have been no changes in directionality at this point in the model’s
development, arrays 1 & 15 of both the simple and complex triangle models are of
course identical.
!116
Step 2
Arrays 2 & 14
1.
2.
…
14.
15.
a b a b b
↘↘↘↘
b a b a b
b a b a b
↗↗↗↗
a b a b b
In Step 2, the final elements in arrays 1 and 15, reading from the left,
become the first in the next arrays to be developed, the first becomes the second,
the second the third, and so forth, according to the principle of cyclicity.
Step 3
Arrays 3–6 & 13–10
1.
2.
3.
4.
5.
6.
a b a b b
↘↘↘↘
b a b a b
↘↘↘↘
b b a b a
↘↘↘↘
a b b a b
↘↘↘↘
b a b b a
↘↘↘↘
a b a b b
...
10.
11.
12.
13.
14.
15.
!117
a b a b b
↗↗↗↗
b a b b a
↗↗↗↗
a b b a b
↗↗↗↗
b b a b a
↗↗↗↗
b a b a b
↗↗↗↗
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.
Step 4
Arrays 7 & 9
1.
2.
3.
4.
5.
6.
7.
a b a b b
↘↘↘↘
b a b a b
↘↘↘↘
b b a b a
↘↘↘↘
a b b a b
↘↘↘↘
b a b b a
↘↘↘↘
a b a b b
↙↙↙↙
b a b b a
8.
9.
10.
11.
12.
13.
14.
15.
b a b b a
↖↖↖↖
a b a b b
↗↗↗↗
b a b b a
↗↗↗↗
a b b a b
↗↗↗↗
b b a b a
↗↗↗↗
b a b a b
↗↗↗↗
a b a b b
The array ‘a b a b b’ not, of course, being reproduced in arrays 7 and 9,
development continues leftwards, according to Rule 2.
!118
Step 5
Array 8
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
a b a b b
↘↘↘↘
b a b a b
↘↘↘↘
b b a b a
↘↘↘↘
a b b a b
↘↘↘↘
b a b b a
↘↘↘↘
a b a b b
↙↙↙↙
b a b b a
↙↙↙↙
a b b a b
↖↖↖↖
b a b b a
↖↖↖↖
a b a b b
↗↗↗↗
b a b b a
↗↗↗↗
a b b a b
↗↗↗↗
b b a b a
↗↗↗↗
b a b a b
↗↗↗↗
a b a b b
With the development of the centre array, the model is complete,
according to Rule 3. The final model comprises two sub-models, arrays 2–15 and
14–1, each of which represents the final complex triangle model. In Table 51
below, the complex array model from the right-hand side of Table 50 is compared
with the complex triangle model and seen to be identical with it, the only
!119
immaterial difference between them being that their flows move in opposite
directions due to their developments’ different points of departure.
Table 51
Shakespearean Complex Array and Triangle Models
Complex Array Model
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
Complex Triangle Model
b a b a b
↖ ↖ ↖↖
b b a b a
↖ ↖ ↖↖
a b b a b
↖ ↖ ↖↖
b a b b a
↖ ↖ ↖↖
a b a b b
↗↗↗↗
b a b b a
↗↗↗↗
a b b a b
↘↘↘↘
b a b b a
↘↘↘↘
a b a b b
↙↙↙↙
b a b b a
↙↙↙↙
a b b a b
↙↙↙↙
b b a b a
↙↙↙↙
b a b a b
↙↙↙↙
a b a b b
b a b a b
↘↘↘↘
b b a b a
↘↘↘↘
a b b a b
↘↘↘↘
b a b b a
↘↘↘↘
a b a b b
↙↙↙↙
b a b b a
↙↙↙↙
a b b a b
↖↖↖↖
b a b b a
↖↖↖↖
a b a b b
↗↗↗↗
b a b b a
↗↗↗↗
a b b a b
↗↗↗↗
b b a b a
↗↗↗↗
b a b a b
↗↗↗↗
a b a b b
!120
2.4.7
Conclusion and Retrospective
It has been shown to be theoretically possible within a binary expansion
to transform a simple triangle model into a complex triangle model identical to the
complex array model developed from a centre array. As the complex array model
is able to describe satisfactorily how equivalents of the formal characteristics of
the Shakespearean sonnet are related by the principle of centred form, it follows
that the complex triangle model also does so. Furthermore, as each model is
developed independently from different starting points, they provide a means to
cross-check each other’s results and mitigate the risk of error and bias in their
results. By this means, the complex Shakespearean models’ results are presumed
more robust.
The two ideas, first, of a sonnet pattern developing as part of a binary
expansion and, second, complexity evolving from simplicity also help to clarify
the relationship between the two Early Italian array models (EIM). As discussed
above, their mirrored flows, as seen in Table 20, suggested that the models might
be part of a broader pattern. As their elements were by inspection clearly not
symbolically mirrored, however, the question arose as to just how they were
related, how their results might be corroborated and whether one was a better
representation of its tradition than the other. To seek answers to these questions,
let arrays 5 through 11 of the Early Italian model (RHS) of Table 20 now be
reversed so that, leaving aside arrays 1 and 15 for the moment, the model has no
directionality changes. What, as a result, amounts to a Simple Early Italian (RHS)
array model is seen to be identical with the Simple Shakespearean array and
triangle models, as shown below in Table 52.
!121
Table 52
Related Early Italian and Shakespearean Models 1
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
EIM (RHS)
EIM (RHS)
arrays 5–11
reversed
Simple Shakespearean
Array & Triangle
Models
a b b a b
↖ ↖ ↖↖
b a b b a
↗ ↗ ↗↗
a b b a b
↗ ↗ ↗↗
b b a b a
a b b a b
↖ ↖ ↖↖
b a b b a
↗ ↗ ↗↗
a b b a b
↗ ↗ ↗↗
b b a b a
a b a b b
↗ ↗ ↗↗
b a b b a
↗ ↗ ↗↗
a b b a b
↗ ↗ ↗↗
b b a b a
b a b a b
↖ ↖ ↖↖
b b a b a
↖↖↖↖
a b b a b
↖ ↖ ↖↖
b a b b a
b a b a b
↗ ↗ ↗↗
a b a b b
↗ ↗ ↗↗
b a b b a
↗ ↗ ↗↗
a b b a b
b a b a b
↗ ↗ ↗↗
a b a b b
↗ ↗ ↗↗
b a b b a
↗ ↗ ↗↗
a b b a b
a b b a b
↙↙ ↙ ↙
b b a b a
↙↙ ↙ ↙
b a b a b
b a b b a
↘↘ ↘ ↘
a b a b b
↘↘ ↘ ↘
b a b a b
b a b b a
↘↘ ↘ ↘
a b a b b
↘↘ ↘ ↘
b a b a b
b b a b a
↘↘ ↘↘
a b b a b
↘ ↘ ↘↘
b a b b a
↙↙ ↙ ↙
a b b a b
b b a b a
↘↘ ↘↘
a b b a b
↘ ↘ ↘↘
b a b b a
↙↙ ↙ ↙
a b b a b
b b a b a
↘↘ ↘ ↘
a b b a b
↘↘ ↘ ↘
b a b b a
↘↘ ↘ ↘
a b a b b
It follows, ignoring arrays 1 and 15 for a moment longer, that the simple
RHS Early Italian and Shakespearean array and triangle models share the same
centre array in the same centre sequence. The elements of the Early Italian
!122
model’s centre sequence may be deduced from its simple array model, that is, the
model in the middle of Table 52 above, by letting the initial elements of its arrays
7–2 or 9–14, ‘b a b b a b’, be arranged in order to the left of its centre array, ‘a b b
a b’. This permits, as shown in Table 53, a comparison of the Italian (RHS) and
Shakespearean centre sequences, a comparison which shows them to be almost
identical.
Table 53
EIM (RHS) and Shakespearean Models: Shared Centre Sequence
() b a b b a b a b b a b
EIM (RHS)
a b a b b a b a b b a b
Shakespearean
The lack of identity is due to the bracketed, missing element on the lefthand 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 Shakespearean models, which in turn is due to the differing assumptions underlying the
models’ redundancy rules: The Early Italian models’ rules are based on the
working model’s rules that were constructed within the logic of a fourteen-array
model, whereas the Shakespearean model’s rules are developed within the logic of
fifteen-array array and triangle models. Now, due to its two independent
approaches to model construction, the results of the Shakespearean model are
presumed to be more error-resistant than those of the Early Italian models. Its
results shall therefore be preferred. The desirable quality in a working model of
being productively incomplete is reflected accordingly in the rest of the discussion
by foregoing the directionality change in array 2 of the Early Italian array models.
!123
This leads to the development of the array ‘a b a b b’ in its arrays 1 and 15 and
identity with the simple Shakespearean array and triangle models.
Turning now to the other Early Italian array model on the left-hand side
(LHS) of Table 20, as it shares mirrored flows with the RHS model and as the
RHS model can be related to the simple Shakespearean models, it follows that all
three models share the same centre sequence. The different centre array of the
EIM (LHS) model suggests merely that it starts development at a different point
along the centre sequence than the other models. This is shown in Table 54.
Table 54
EIM (LHS) and Shakespearean Models: Shared Centre Sequence
a b a b b a b a b b a b a
a b a b b a b a b b a b
EIM (LHS)
Shakespearean
The only point within the original centre sequence of the EIM (LHS)
from which the centre array ‘b b a b a’ might be developed is that beginning at the
fourth element from the left-hand side. The underlined version of the EIM (LHS)
centre array is preferred for comparative purposes, however, as it is in principle
the same, and more clearly shows the relationship between the Early Italian LHS
model and the simple, Shakespearean and Early Italian RHS models. The detailed
relationship is shown in Table 55 below. As the LHS model is displaced by one
element with respect to the Shakespearean and the Early Italian RHS models, in
the table’s second column the final element in each of its arrays is moved to the
head of the array to compensate. These elements are underlined in array 1, by way
of example. When the away flows of the model’s arrays 5–2 and 11–14 are then
!124
reversed, to undo its complexity, so to speak, the Early Italian LHS model is seen
to be identical with the simple Shakespearean and Early Italian RHS models.
Table 55
Related Early Italian and Shakespearean Models 2
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
EIM (LHS)
EIM (LHS)
final element
heads array
EIM (LHS)
mirrored
Simple
Shakespearean &
EIM (RHS)
b a b a b
↖ ↖ ↖↖
b b a b a
↖ ↖ ↖↖
a b b a b
↖ ↖ ↖↖
b a b b a
b b a b a
↖ ↖ ↖↖
a b b a b
↖ ↖ ↖↖
b a b b a
↖ ↖ ↖↖
a b b a b
a b a b b
↗↗↗↗
b a b b a
↗ ↗ ↗↗
a b b a b
↗ ↗ ↗↗
b b a b a
a b a b b
↗ ↗ ↗↗
b a b b a
↗ ↗ ↗↗
a b b a b
↗ ↗ ↗↗
b b a b a
a b a b b
↗ ↗ ↗↗
b a b b a
↗ ↗ ↗↗
a b b a b
↗ ↗ ↗↗
b b a b a
b a b a b
↗ ↗ ↗↗
a b a b b
↗ ↗ ↗↗
b a b b a
↗ ↗ ↗↗
a b b a b
b a b a b
↗ ↗ ↗↗
a b a b b
↗ ↗ ↗↗
b a b b a
↗ ↗ ↗↗
a b b a b
b a b a b
↗ ↗ ↗↗
a b a b b
↗ ↗ ↗↗
b a b b a
↗ ↗ ↗↗
a b b a b
a b b a b
↘↘ ↘ ↘
b a b b a
↘↘ ↘ ↘
a b a b b
b a b b a
↘↘ ↘ ↘
a b a b b
↘↘↘↘
b a b a b
b a b b a
↘↘↘↘
a b a b b
↘↘↘↘
b a b a b
b a b b a
↘↘ ↘ ↘
a b a b b
↘↘ ↘ ↘
b a b a b
b a b b a
↙↙↙ ↙
a b b a b
↙↙↙ ↙
b b a b a
↙↙↙ ↙
b a b a b
a b a b b
↙↙↙↙
b a b b a
↙↙ ↙ ↙
a b b a b
↙ ↙↙ ↙
b b a b a
b b a b a
↘↘ ↘↘
a b b a b
↘ ↘↘↘
b a b b a
↘↘ ↘ ↘
a b a b b
b b a b a
↘↘↘↘
a b b a b
↘↘ ↘ ↘
b a b b a
↘↘ ↘ ↘
a b a b b
!125
It may be concluded, therefore, first, that the Early Italian and Shakespearean array models are related by the same centre sequence and only differ in
their final construction due to differences in starting points, the initial
directionality of their developments and the arrays in which changes in directionality are introduced. Second, the results of both Early Italian models are, after
adjustments in arrays 1 and 15, corroborated by the simple Shakespearean array
and triangle models. Third, as may be seen perhaps most clearly in Step 5 of the
development of the complex Shakespearean triangle model, arrays 7–1 and 9–15
are non-superposable yet foldable mirror images of each other about array 8. The
model's halves are therefore chiral in one dimension and symmetrical in three, a
finding which also holds for the Early Italian models, as perhaps most easily
observable in Table 18. Differently from the Pleadean models, however, these
chiral and symmetrical properties are immanent within the sonnet patterns
themselves, and are masked only to the extent of their transformation from
fifteen- into fourteen-array models. It seems reasonable to conclude, therefore,
that both symmetry and chirality inhere in the Early Italian and Shakespearean
sonnet patterns. Finally, neither Early Italian array model is a better representation
of the Early Italian tradition than the other as each simply represents different yet
related developments from a common centre sequence.
If the principle of centred form is thus able to describe and relate equivalents of the formal characteristics of the sonnet within and across sonnet traditions
in theory, it still remains to be shown whether it can be applied in practice. In the
third part of the inquiry that now follows, practical evidence for the claim is
!126
provided by a brief introduction to centred writing by way of the author’s sonnet
cycle, Memorial Day: the Unmaking of a Sonnet.
!127
Part 3: Centred Writing
3.0
Introduction
That this part of the inquiry does not start out from another traditional
sonnet form might seem presumptuous. It does serve to show, however, that creative
use may be made of traditional sonnet theory. The sonnet pattern underpinning
Memorial Day: the Unmaking of a Sonnet was not derived from the formal characteristics 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:
1
The poems may be found in Appendix F.
!128
Table 56
Memorial Day: Centre Matrix: Internal Elements
7.
8.
c
a
↖↗
↙↘
b
c
The arrows indicate that the elements in the upper and lower halves of the
model are to unfold in diagonally opposite directions away from its centre. The idea
behind this arrangement is to establish, from the outset, a simple, continuous cycle
of tension and resolution between elements. This tension, represented, prosaically
enough, by inequality in the number of different element types, here, two ‘c’s, one
‘a’ and one ‘b’, is resolved initially as the elements reach numerical equality in the
centre matrix, as shown in Table 57 below. Once set, these initial conditions ensure a
recurrence of tension and resolution between elements throughout the sonnet cycle.
Resolution is ultimately achieved with the development of the cycle's final sonnet
pattern, as described in the discussion of aggregation in section 3.1.6 further below.
Table 57
Memorial Day: Centre Matrix Buildup
Step 1: Internal Elements
Frequency and Distribution
7.
c a
a
b
c
8.
b c
1
1
2
!129
Step 2: Intermediary Elements
Frequency and Distribution
7.
b c a c
a
b
c
8.
a b c b
2
3
3
Step 3: External Elements
Frequency and Distribution
7.
a b c a c b
a
b
c
8.
c a b c b a
4
4
4
The rules for development of the centre matrix shown in Step 3 and a stepby-step description of the array model’s construction now follow.
3.1.2
Rules for Array Development
There are three rules for array development in the Memorial Day model:
Rule 1.
The halves of the centre matrix’s array pairs develop
simultaneously upwards and downwards in four diagonally opposite directions away
from the centre matrix;
Rule 2.
Arrays develop without any change in directionality;
Rule 3.
When a series of three consecutive arrays, symbolically and
cyclically with regard to its flows towards, is repeated,
redundancy enters the model, development is halted and the model complete.
A step-by-step description of the model’s construction now follows.
!130
3.1.3
Memorial Day Array Model: Step-by-step Description
Step 1
Centre Matrix
7.
a b c
8.
c a b
↖↗
↙↘
a c b
c b a
The rationale for the centre matrix is discussed above.
Step 2
Arrays 6 & 9
6.
7.
8.
9.
b c a b a c
↖↖ ↗ ↗
a b c a c b
c a b c b a
↙↙
↘↘
a b c a c b
Development begins, according to Rule 1, with each half of the centre
matrix’s array pairs being simultaneously developed upwards and downwards
diagonally away from the centre matrix to create arrays 6 and 9.
!131
Step 3
Arrays 5 & 10
5.
6.
7.
8.
9.
10.
c a b
↖↖
b c a
↖↖
a b c
c b a
↗↗
b a c
↗↗
a c b
c a b
↙↙
a b c
↙↙
b c a
c b a
↘↘
a c b
↘↘
b a c
As there is, as yet, of course, no series repetition, development continues
according to Rule 2.
!132
Step 4
Arrays 4 & 11
4.
5.
6.
7.
8.
9.
10.
11.
a b c
↖↖
c a b
↖↖
b c a
↖↖
a b c
a c b
↗↗
c b a
↗↗
b a c
↗↗
a c b
c a b
↙↙
a b c
↙↙
b c a
↙↙
c a b
c b a
↘↘
a c b
↘↘
b a c
↘↘
c b a
The first series of three symbolically and, with respect to flows towards,
cyclically identical arrays, 6–4 and 9–11, is now complete. Development continues
according to Rule 2.
!133
Step 5
Arrays 3–1 & 12–14
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
a b c
↖↖
c a b
↖↖
b c a
↖↖
a b c
↖↖
c a b
↖↖
b c a
↖↖
a b c
a c b
↗↗
c b a
↗↗
b a c
↗↗
a c b
↗↗
c b a
↗↗
b a c
↗↗
a c b
c a b
↙↙
a b c
↙↙
b c a
↙↙
c a b
↙↙
a b c
↙↙
b c a
↙↙
c a b
c b a
↘↘
a c b
↘↘
b a c
↘↘
c b a
↘↘
a c b
↘↘
b a c
↘↘
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, development is halted and the model complete, according to Rule 3. The final array model is
presented in Table 58 below in a manner emphasizing how centred development
creates the equivalent of a traditional two quatrain, two tercet sonnet form.
!134
Table 58
Memorial Day: Array Model
Memorial Day Array Model
a b c
↖↖
c a b
↖↖
b c a
↖↖
a b c
a c b
↗↗
c b a
↗↗
b a c
↗↗
a c b
c a b
↖↖
b c a
↖↖
a b c
c b a
↗↗
b a c
↗↗
a c b
c a b c b a
↙↙ ↘↘
a b c
↙↙
b c a
↙↙
c a b
a c b
↘↘
b a c
↘↘
c b a
a b c
↙↙
b c a
↙↙
c a b
a c b
↘↘
b a c
↘↘
c b a
!135
3.1.4
Assessment
The constant number of elements per array throughout the model represents
the equivalent of isometry. The volta equivalent is deemed to occur at the point of
greatest contrast between flows. In a linear reading of the model, as opposed to its
centred form construction, in which flows between arrays 7 & 8 are undefined, this
occurs in array 8, as shown in Table 59.
Table 59
Memorial Day: Array 8 as Volta Equivalent
7.
8.
9.
a b c
↘↘
c a b
↙↙
a b c
a c b
↙↙
c b a
↘↘
a c b
It might be argued, however, that array 9 is the better equivalent of the volta
as, in a linear reading, its towards and away flows contrast with the flows of arrays
1–7, whilst towards flows are undefined for array 8. As there is no contradiction
between this argument and Hobsbaum’s observation, noted at the start of the inquiry,
that the volta is “usually situated at the end of the octave or the beginning of the
sestet”, array 9, situated at the beginning of the sestet equivalent, is deemed to
represent the volta.
With array 9 marking the equivalent of the division into octave and sestet,
the model in Table 58 may be seen to represent the traditional Italian and French,
two quatrain, two tercet form: Overlapping, quasi-embracing rhyme pair equivalents
delimit the quatrains, just as identical groups of three arrays do the tercets.
Although, given that there is change from one end-array element to the next, the
equivalent of an alternating rhyme scheme is accommodated by the model, a
!136
principle that distributes key vowels and accentuation throughout the model’s arrays
is preferred, as described in the next section. Finally, the equivalent of fourteen line
sonnet length is determined by the limit between innovation and redundancy in array
series, as described in Rule 3. The model thus represents the sonnet characteristics
noted by Hobsbaum at the outset of the inquiry. As there is no doubt about the
design of the array model, there is no need for its results to be corroborated independently 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.2 The cycle begins with the
three words: silent, pages and leaves.3 In Table 60 below, the bottom right ‘internal’
‘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 respectively among the remaining elements in the matrix, as shown in Table 61 also below.
2
This idea occurred to me after remembering reading a vocalic analysis of Verlaine’s Il
pleure dans mon coeur (Chiss, Filliolet et Maingeneau, 1977, II, pp. 123–124) that
presented itself during a later reading of Sylvester’s idea of Phonetic Syzygy (1870, p.
11).
3
These words were occasioned by a fall of light on the pages of a book I was reading one
Sunday in May 2009 in the garden of the Isabella Stewart Gardner Museum in Boston,
MA, U.S.A.
!137
Table 60
Centre Matrix: Distribution of Key Vowels 1
Centre Matrix: Internal Elements
7.
8.
c
b
key vowels
a
c
a
/eɪ/
b
/iː/
c
/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
8.
c a b
c b a
/aɪ./
/eɪ/
leaves silent
/aɪ./
/iː/
/iː/
/eɪ/
This schema then helped prompt the opening lines of the cycle, the lines 7 & 8 of
sonnet 8:
7.
how fey, how free, the mitered pages mild do sheen
8.
and shimmer and sway with leaves of silent beechen gray,
The key vowel sounds and their accompanying accentuation were then
distributed line by line throughout the remainder of the sonnet’s arrays as the writing
of the poem unfolded. The rule I adopted was not to stick rigidly to a particular
vowel sound, but rather to stray either by a very little or quite a lot therefrom
depending on the poetic possibilities that presented themselves. In Sonnet 8, about
two-thirds of the groups of three elements comprising each array conform to the key
vowel distribution described above.
!138
3.1.6
The Problem of Aggregation
One final aspect of Memorial Day: the Unmaking of a Sonnet lends support
to the claim of a centred form to sonnet writing: The method applied to combine
individual sonnets into a sonnet cycle. This problem is referred to by Spiller (1997)
as the problem of aggregation. In discussing the return to vogue of sonnet sequences
in nineteenth century English and American literature, he describes the difficulty as
follows:
There is, as we shall see, plenty of scope for originality. However, one problem
always presents itself, in any age or place, because of the nature of the sonnet:
the problem of aggregation into a whole of items that are also meaningful
separately–a difficulty no other genre, in prose or verse, presents. (p. 20)
In Memorial Day, I address this problem by taking alternating pairs of
arrays from the central sonnet, Sonnet 8, for use as the centre matrices of the
subsequent sonnets in the cycle. Thus, as arrays 7 and 8 form the centre matrix of
sonnet 8, so sonnet 8’s arrays 6 and 7 form the centre matrix of sonnet 7, sonnet 8’s
arrays 8 and 9, the centre matrix of sonnet 9, and so forth, until the final sonnet to be
written, sonnet 1, avails itself for its centre matrix of arrays 14 and 1 from sonnet 8.
Apart from providing cohesiveness between the sonnet patterns in the cycle, this
approach has the formal advantage, with regard to the central sonnet, of exhausting
the principle of unfolding from the centre.
This order of writing the sonnets is reflected in the cycle’s preludium and
postludium. The top line of the preludium is the seventh line of the first sonnet to be
written, sonnet 8, as the top line of the postludium is the eighth line of the same
sonnet; the second line of the preludium is the seventh line of sonnet 7, the second
!139
sonnet to be written, as the second line of the postludium is the eighth line of sonnet
7, and so forth throughout. Accordingly, the last lines of the preludium and postludium are the seventh and eighth lines respectively of the last sonnet written for the
cycle, sonnet 1. In the pre- and postludiums, the lines from the individual sonnets
are reproduced without their punctuation.
3.1.7
Conclusion
This brief introduction to centred writing, the marshalling of elements into
a centre matrix, the construction of an array model, the equivalents of the sonnet’s
formal characteristics within the model, the link between sonnet pattern and sonnet
diction, the description of how individual sonnets are aggregated into a sonnet cycle
and the fact of the written sonnet cycle itself constitutes the practical evidence in
support of the claim. The theoretical and practical evidence for the claim furnished,
the inquiry now closes with a general conclusion.
!140
Part 4
Summary and Outlook
This inquiry sets out to uncover and weigh evidence for the claim that a
sonnet unfolds from its centre to form a pattern in which its formal characteristics
inhere. That is, it seeks to try out the idea that the formal characteristics of the
sonnet might be better understood not as a list of somehow-connected, empirical
categories, but as complex byproducts of a simple pattern originating in, and
developed from, the sonnet’s centre. The balance of evidence presented supports
the claim. The initial evidence provided in Part 1 is naturally tenuous as it consists
of a working hypothesis model constructed from first principles. In Part 2, by
applying and extending these principles, however, evidence is furnished by way of
theoretical models showing how complex equivalents of the formal characteristics of five sonnet traditions, the so-called Early Italian, the Petrarchan, two
Pleadean and the Shakespearean, can be related by the principle of centred form.
Finally, in Part 3, practical evidence for the claim is provided by the centred form
sonnet cycle Memorial Day: the Unmaking of a Sonnet. In each of the models
presented, equivalents of the sonnet’s formal characteristics unfold from the
models’ centre: the equivalent of isometry results from the development of a fixed
array of elements; the equivalent of the volta is deemed to occur at the point of
starkest contrast in directionality flows between arrays; the equivalent of stanzaic
form results from symmetries in the pattern of flows between placeholders in the
models, just as rhyme scheme equivalents, for their part, result from cyclicity in
!141
array development; and, finally, the equivalent of fourteen line sonnet length is
effected by the limit between innovation and redundancy in array development. To
mitigate the risk of error and bias in the evidence presented, in addition to the
array model, a triangle model is developed to independently cross-check array
model results. It is also shown that the array and triangle models were situated
within a broader pattern of binary expansion with symmetrical and, in some
traditions, chiral properties. These expansions, furthermore, hold out the prospect
of an independent basis for the comparison of sonnets across traditions as is seen,
for example, in the relatedness of the Early Italian and Shakespearean models.
Research questions raised as a result of the inquiry might turn on whether the
theory is supported by evidence from the sonnet corpus itself and perhaps the
degree to which the theory helps further our understanding of other sonnet
traditions. The establishment of symmetrical and chiral properties in the models
also suggests potential for interdisciplinary research. Additionally, hard questions
regarding the nature of the relationship between reading and writing are also
raised. These questions resolve themselves broadly into a set of three distinctions:
1.
structure
/
pattern
2.
separated category form
/
centred form
3.
linear reading
/
centred writing
The sonnet is as much chastised as it praised for being hard, old and elitist. The
findings and conclusions of this inquiry reveal it to be instead a simple, enabling
pattern for reflective thought and creative writing for anyone, anywhere.
!142
References
Bermann, Sandra L. (1988). The sonnet over time: A study in the sonnets of Petrarch,
Shakespeare, and Baudelaire. (University of North Carolina studies in comparative
literature: No. 63) Chapel Hill, NC: University of North Carolina Press.
Borgstedt, Thomas. (2009). Topik des Sonetts: Gattungstheorie und Gattungsgeschichte. [Topology of the sonnet: Theory and history of genre (sic)]. Tübingen: Max
Niemeyer Verlag.
Burt, Stephen and Mikics, David. (2010). The art of the sonnet. Cambridge, MA: Belknap
Press of Harvard University Press.
Chiss, Jean-Louis, Filliolet, Jacques et Maingeneau, Dominique. (2001). Introduction à la
linguistique française t. II: Syntaxe, communication, poétique. [Introduction to French
linguistics, Vol. 2: Syntax, communication, poetics]. Paris: Hachette.
Ellrodt, Robert. (1986). The Cambridge companion to Shakespeare studies.
Cambridge [Cambridgeshire]; New York: Cambridge University Press.
Gell-Mann, Murray. (1996). Let's call it plectics. John Wiley and Sons, Inc.:
Complexity Vol.1, no. 5.
Heaney, Seamus. (1980). Preoccupations: selected prose. 1968-1978. London; Boston:
Faber and Faber.
Hobsbaum, Philip. (1996). Metre, rhythm and verse form. London; New York, NY:
Routledge.
Jost, François. (1989). Le sonnet de Pétrarque à Baudelaire: modes et modulations. [The
sonnet from Petrarch to Baudelaire: modes and modulations]. Berne; Francfort-s. Main;
New York, NY; Paris: Lang.
!143
Kelvin, William Thomson, Baron. (1904). Robert Boyle Lecture, delivered before the Oxford
University Junior Scientific Club on Tuesday, May 16, 1893. London: Clay.
Kemp, Friedhelm. (2002). Das europäische Sonett. Band 1. [The European sonnet. Vol. 1].
Göttingen, BRD: Wallstein Verlag.
Kircher, Hartmut. (1979). Deutsche Sonette. [German sonnets]. Stuttgart, BRD: Reclam.
Lennard, John. (1996). The poetry handbook: a guide to reading poetry for pleasure and
practical criticism. Oxford; New York, NY: Oxford University Press.
Nakahara, Mikio. (2003). Geometry, topology and physics. Bristol; Philadelphia: Institute
of Physics Publishing. 2nd ed.
Olmsted, Everett Ward. (1897). The sonnet in French literature and the development of the
French sonnet form. (Doctoral dissertation, Cornell University). Ithaca, N.Y.
Oppenheimer, Paul. (1989). The birth of the modern mind: self, consciousness, and the
invention of the sonnet. New York, NY: Oxford University Press.
Queneau, Raymond. (1961). Cent mille milliards de poèmes. [One hundred million million
poems]. Paris: Gallimard.
Shapiro, Marianne. (1980). Hieroglyph of time: The Petrarchan sestina. Minneapolis:
University of Minnesota Press.
Spanos, Margaret. (1978). The sestina: An exploration of the dynamics of poetic structure.
Speculum, V. 53, pp. 545-557.
Spiller, Michael R. G. (1997). The sonnet sequence: a study of it's strategies. New York,
NY: Twayne Publishers.
Sylvester, J. J. (1870). The laws of verse or principles of versification exemplified in
metrical translations. London: Longmans, Green, and Co.
!144
Turabian, Kate L. (2007). A manual for writers of research papers, theses, and
dissertations: Chicago style for students and researchers / Kate L. Turabian: revised by
Wayne C. Booth, Gregory G. Colomb, Joseph M. Williams and University of Chicago Press
editorial staff. 7th ed. Chicago, IL and London: The University of Chicago Press.
Vendler, Helen H. (1998). Seamus Heaney. Cambridge, MA: Harvard University Press.
Wehrli, Hans. (2008). Metaphysics: Chirality as the basic principle of physics. Zürich:
Author. Retrieved from http://www.hanswehrli.ch/en/buch_1.htm January, 2011.
Wilkins, Ernest Hatch. (1915). The invention of the sonnet. Modern Philology, V. 13, no. 8,
December 1915, pp. 463-494.
————————— (1959). The invention of the sonnet and other studies in Italian
literature. Rome: Edizioni di Storia e Litteratura.
!145
Appendix A
Early Italian Model: Unsuitability of Two-Type, Three-Element Arrays
A1.0 Two-Type, Three Element Arrays
As may be seen by inspection of Tables A1.1, A1.2 and A1.3, neither of the
possible remaining two type, three element arrays, ‘a b b’ nor ‘b b a’, produces a
pattern of continuous ‘change’ in end-array elements, that is, neither can accommodate 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
↖
6.
b
a
b
a
b
6.
a
7.
↖
b
↖
8.
5.
↖
↖
7.
b
b
↖
b
a
↗
b
8.
!146
a
b
↗
b
b
Appendix A (cntd.)
Table A1.2
Centre Array ‘a b b’: Leftwards Development: Directionality Change in Array 6
leftwards
1.
2.
3.
b
a
↗
4.
a
↗
b
↗
5.
b
↗
b
a
b
b
↖
b
↖
8.
a
↗
↖
7.
b
b
↗
6.
b
a
a
↖
b
b
Leftwards development for the centre array ‘a b b’, as shown in Table A1.1, produces
successive identical placeholders in arrays 6 and 5. Changing directionality in array 6
to avoid this, shown in Table A1.2, only postpones its redevelopment until arrays 4
and 3. In rightwards development, this occurs immediately in arrays 8 and 7. Thus, in
both cases, it is not possible for end-array placeholders to accommodate alternating
rhyme schemes equivalents.
!147
Table A1.3:
Centre Array ‘b b a’: Leftwards and Rightwards, Upwards Development
leftwards
rightwards
1.
1.
2.
2.
3.
3.
4.
4.
5.
5.
6.
a
b
↖
7.
b
b
6.
b
↖
a
↖
8.
b
↗
b
7.
a
↖
b
a
↗
b
↗
a
8.
b
b
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, threeelement centre arrays is, therefore, suitable for satisfying the rhyme scheme
condition of the Early Italian tradition.
!148
Appendix B
Early Italian Model: Centre Array Derivation
B1.0 Centre Array Candidates
In Table B1.1 are listed the Early Italian centre array candidates from Table 16.
Table B1.1
Early Italian Model: Centre Array Candidates
ii.)
a b a b b
iii.)
a b b a b
vi.)
b a b a b
vii.)
b a b b a
ix).
b b a b a
B2.0 Centre Array: ii.) ‘a b a b b’
In Table B2.1 below, it may be seen that rightwards development from the
centre array ‘a b a b b’ leads immediately to three identical end-array elements,
making the equivalent of the alternating rhyme scheme required by the Early Italian
tradition impossible. This may be seen in the development on the right hand side of
the table. Leftwards development of the centre array leads to the same result in
arrays 4 and 3, and 12 and 13. A change in directionality in arrays 4 and 12 to try to
circumvent this, as shown in Table B2.2, does not produce the necessary redundancy
mechanism, that is, repetition of the centre array in arrays 2 and 14. Now, if a
change in directionality is introduced in arrays 5 and 11, a suitable redundancy
mechanism does develop. However, when in later models this redundancy
!149
mechanism becomes superfluous (p. 120 ff.), the centre array no longer fulfils the
alternating rhyme scheme condition for the Early Italian tradition. The array ‘a b a b
b’ is thus excluded as a possible candidate for the Early Italian model.
Table B2.1
Centre Array: ii.) ‘a b a b b’
leftwards
rightwards
1.
1.
2.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
a b a b b
↖↖ ↖ ↖
b a b a b
↖↖ ↖ ↖
b b a b a
↖↖ ↖ ↖
a b b a b
↖↖ ↖ ↖
b a b b a
↖↖ ↖ ↖
a b a b b
↙↙↙↙
b a b b a
↙↙↙↙
a b b a b
↙↙↙↙
b b a b a
↙↙↙↙
b a b a b
↙↙↙↙
a b a b b
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
14.
15.
15.
!150
b a b a b
↗↗↗↗
a b a b b
↘↘↘↘
b a b a b
Table B2.2
Centre Array: ii.) ‘a b a b b’ No Redundancy Mechanism in Arrays 2 and 14
leftwards
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
a b b a b
↗↗↗↗
b b a b a
↗↗↗↗
b a b a b
↖↖ ↖ ↖
b b a b a
↖↖ ↖ ↖
a b b a b
↖↖ ↖ ↖
b a b b a
↖↖ ↖ ↖
a b a b b
↙↙↙↙
b a b b a
↙↙↙↙
a b b a b
↙↙↙↙
b b a b a
↙↙↙↙
b a b a b
↘↘↘↘
b b a b a
↘↘↘↘
a b b a b
15.
B3.0 Centre Array: iii). ‘a b b a b’
Table B3.1 shows leftwards and rightwards development from the centre array
‘a b b a b’. In order to avoid the rupture of an alternating rhyme scheme equivalent,
as occurs in arrays 6-5 and 10-11 of both leftwards and rightwards development, a
!151
change in directionality has to be introduced in the sixth and tenth arrays. This,
however, denies the possibility of satisfying the two quatrain, two tercet stanzaic
form condition for the Early Italian tradition. Thus, the array ‘a b b a b’ is also
excluded as a centre array candidate for the Early Italian model.
Table B3.1
Centre Array: iii.) ‘a b b a b’
leftwards
rightwards
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
1.
2.
3.
4.
a b a b b
↖↖ ↖ ↖
b a b a b
↖↖ ↖ ↖
b b a b a
↖↖ ↖ ↖
a b b a b
↙↙↙↙
b b a b a
↙↙↙↙
b a b a b
↙↙↙↙
a b a b b
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
b a b a b
↗↗↗↗
a b a b b
↗↗↗↗
b a b b a
↗↗↗↗
a b b a b
↘↘↘↘
b a b b a
↘↘↘↘
a b a b b
↘↘↘↘
b a b a b
12.
13.
14.
15.
B4.0 Centre Array: vi). ‘b a b a b’
In Table B4.1 below, it can be seen that leftwards development of the centre
array ‘b a b a b’ immediately fails the alternating rhyme equivalent condition. In
!152
rightwards development, a change in directionality is necessary in arrays 4 and 12
as, by inspection, continued development in the same direction leads to the disruption 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, however, when this redundancy mechanism becomes superfluous due to the subsequent
development of a a triangle model within a binary expansion, the centre array 'b a b
a b' no longer fulfils the alternating rhyme scheme condition for the Early Italian
tradition. Thus, the ‘b a b a b’ centre array is also excluded as a centre array
candidate for the Early Italian model.
!153
Table B4.1
Centre Array: vi.) ‘b a b a b’
leftwards
rightwards
1.
1.
2.
2.
3.
3.
4.
4.
5.
5.
6.
6.
7.
8.
9.
a b a b b
↖↖ ↖ ↖
b a b a b
↙↙↙↙
a b a b b
7.
8.
9.
10.
10.
11.
11.
12.
12.
13.
13.
14.
14.
15.
15.
!154
b a b a b
↗↗↗↗
a b a b b
↗↗↗↗
b a b b a
↗↗↗↗
a b b a b
↗↗↗↗
b b a b a
↗↗↗↗
b a b a b
↘↘↘↘
b b a b a
↘↘↘↘
a b b a b
↘↘↘↘
b a b b a
↘↘↘↘
a b a b b
↘↘↘↘
b a b a b
Table B4.2
Centre Array: vi). ‘b a b a b’
No Redundancy Mechanism in Arrays 2 and 14
rightwards
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
!155
b b a b a
↖↖ ↖ ↖
a b b a b
↖↖ ↖ ↖
b a b b a
↖↖ ↖ ↖
a b a b b
↗↗↗↗
b a b b a
↗↗↗↗
a b b a b
↗↗↗↗
b b a b a
↗↗↗↗
b a b a b
↘↘↘↘
b b a b a
↘↘↘↘
a b b a b
↘↘↘↘
b a b b a
↘↘↘↘
a b a b b
↙↙↙↙
b a b b a
↙↙↙↙
a b b a b
↙↙↙↙
b b a b a
B5.0 Centre Array: vii.) ‘b a b b a’
As shown in Table B5.1 below, rightwards development of the centre array ‘b
a b b a’ produces successive identical elements in arrays 7 & 6 and 9 & 10, so that
this candidate is unsuitable for developing the alternating rhyme scheme equivalent
of the Early Italian sonnet. A change in direction at array 7 fails to develop the
equivalent of stanzaic form, which requires that directionality changes mark the
transitions between stanzas. In the leftwards development of the centre array,
however, a change of direction at arrays 5 and 11 does develop an appropriate
redundancy mechanism in arrays 2 and 14 with the redevelopment of the centre
array, as shown in Table B5.2 further below. Thus, the array ‘b a b b a’ with
leftwards directionality satisfies the conditions of the Early Italian sonnet.
!156
Table B5.1
Centre Array: vii.) ‘b a b b a’
leftwards
rightwards
1.
1.
2.
2.
3.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
a b a b b
↖↖ ↖ ↖
b a b a b
↖↖ ↖ ↖
b b a b a
↖↖ ↖ ↖
a b b a b
↖↖ ↖ ↖
b a b b a
↙↙↙↙
a b b a b
↙↙↙↙
b b a b a
↙↙↙↙
b a b a b
↙↙↙↙
a b a b b
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
13.
14.
14.
15.
15.
!157
b a b a b
↗↗↗↗
a b a b b
↗↗↗↗
b a b b a
↘↘↘↘
a b a b b
↘↘↘↘
b a b a b
Table B5.2
Centre Array: vii.) ‘b a b b a’
leftwards
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
a b b a b
↖↖ ↖ ↖
b a b b a
↗↗↗↗
a b b a b
↗↗↗↗
b b a b a
↗↗↗↗
b a b a b
↖↖ ↖ ↖
b b a b a
↖↖ ↖ ↖
a b b a b
↖↖ ↖ ↖
b a b b a
↙↙↙↙
a b b a b
↙↙↙↙
b b a b a
↙↙↙↙
b a b a b
↘↘↘↘
b b a b a
↘↘↘↘
a b b a b
↘↘↘↘
b a b b a
↙↙↙↙
a b b a b
!158
B6.0 Centre Array: ix.) ‘b b a b a’
Table B6.1 below shows that leftwards development of the ‘b b a b a’ centre
array almost immediately leads to an unsuitable development in end-array elements.
However, rightwards development, with a change in directionality in arrays 5 and
11, as shown in Table B6.2, redevelops the centre array in arrays 2 and 14 leading to
redundancy and the fulfillment of the number of lines, and all other, conditions for
the Early Italian tradition.
!159
Table B6.1
Centre Array: ix.) ‘b b a b a’
leftwards
rightwards
1.
1.
2.
2.
3.
3.
4.
4.
5.
5.
6.
7.
8.
9.
10.
a b a b b
↖↖ ↖ ↖
b a b a b
↖↖ ↖ ↖
b b a b a
↙↙↙↙
b a b a b
↙↙↙↙
a b a b b
6.
7.
8.
9.
10.
11.
11.
12.
12.
13.
13.
14.
14.
15.
15.
!160
b a b a b
↗↗↗↗
a b a b b
↗↗↗↗
b a b b a
↗↗↗↗
a b b a b
↗↗↗↗
b b a b a
↘↘↘↘
a b b a b
↘↘↘↘
b a b b a
↘↘↘↘
a b a b b
↘↘↘↘
b a b a b
Table B6.2
Centre Array: ix). ‘b b a b a’
rightwards
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
!161
a b b a b
↗↗↗↗
b b a b a
↖↖ ↖ ↖
a b b a b
↖↖ ↖ ↖
b a b b a
↖↖ ↖ ↖
a b a b b
↗↗↗↗
b a b b a
↗↗↗↗
a b b a b
↗↗↗↗
b b a b a
↘↘↘↘
a b b a b
↘↘↘↘
b a b b a
↘↘↘↘
a b a b b
↙↙↙↙
b a b b a
↙↙↙↙
a b b a b
↙↙↙↙
b b a b a
↘↘↘↘
a b b a b
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.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
a b b a b
↗↗↗↗
b b a b a
↖↖ ↖ ↖
a b b a b
↖↖ ↖ ↖
b a b b a
↖↖ ↖ ↖
a b a b b
↗↗↗↗
b a b b a
↗↗↗↗
a b b a b
↗↗↗↗
b b a b a
↘↘↘↘
a b b a b
↘↘↘↘
b a b b a
↘↘↘↘
a b a b b
↙↙↙↙
b a b b a
↙↙↙↙
a b b a b
↙↙↙↙
b b a b a
↘↘↘↘
a b b a b
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
!162
a b b a b
↖↖ ↖ ↖
b a b b a
↗↗↗↗
a b b a b
↗↗↗↗
b b a b a
↗↗↗↗
b a b a b
↖↖ ↖ ↖
b b a b a
↖↖ ↖ ↖
a b b a b
↖↖ ↖ ↖
b a b b a
↙↙↙↙
a b b a b
↙↙↙↙
b b a b a
↙↙↙↙
b a b a b
↘↘↘↘
b b a b a
↘↘↘↘
a b b a b
↘↘↘↘
b a b b a
↙↙↙↙
a b b a b
Appendix C
Petrarchan Model: Centre Matrix Derivation
C1.0
Petrarchan Centre Array Candidates
As the Petrarchan simplified rhyme scheme shows an of embracing and
paired rhymes in its upper half and alternating end rhymes in its lower, it is assumed
that a centre matrix of two arrays, each composed of four elements of two element
types, ‘a’ and ‘b’, are necessary to satisfy Petrarchan sonnet conditions. There are
thus, as listed in Table A5.3.1.1, C (n,k), that is, 4!/ (4-2)!2! = 24/4 = 6, candidate
arrays in all:
Table C1.1:
Petrarchan Centre Matrix Candidates 1
i.)
a a b b
ii.)
a b a b
iii.)
a b b a
iv.)
b b a a
v.)
b a b a
vi.)
b a a b
As only ii.) and v). can represent the continuous change in end-array elements to be
found in the lower half of the Petrarchan sonnet, only they might represent array 8.
As any of the remaining four arrays might represent array 7, there are in all eight
possible centre matrices, as shown in Table C1.2 below:
!163
Table C1.2
Petrarchan Centre Matrix Candidates 2
i.)
a a b b
a b a b
ii.)
a a b b
b a b a
iii.)
a b b a
a b a b
iv.)
a b b a
b a b a
v.)
b b a a
a b a b
vi.)
b b a a
b a b a
vii.)
b a a b
a b a b
viii.)
b a a b
b a b a
From the simplified rhyme scheme, shown in Table C1.3 below, it can be
seen that from the end-array in array 7 to the end-array in array 8 there is change
from ‘b’ to ‘a’ which by transposition is equivalent to ‘a’ to ‘b’. Thus, the candidate
!164
matrices i.), iv.), vi.) and vii.) may immediately be excluded as their end-array
elements show no change from one array to the next.
Table C1.3
Petrarchan Simplified Rhyme Scheme
1 2 3 4 5 6 7 8 9 10 11 12 13 14
a b b a a b b a b a b a b a
The remaining candidates are shown in Table C1.4.
Table C1.4
Petrarchan Centre Array Candidates 3
ii.)
a a b b
b a b a
iii.)
a b b a
a b a b
v.)
b b a a
a b a b
viii)
b a a b
b a b a
C2.0
Directionality: Symmetry and Chirality
In Tables C3.1–4 further below, each of the final candidate matrices is
tested with leftwards and rightwards directionality only. Leftwards and rightwards,
!165
and rightwards and leftwards developments are excluded for the reasons given in the
brief discussion of symmetry and chirality which now follows. A detailed
description of symmetry and chirality is found in the Pleadean section .
In each of the matrices’ lower arrays there is change from one element to
the next. Whether they are developed rightwards or leftwards therefore produces,
symbolically, the same result. For example, the array ‘b a b a’ from the matrix ii.)
above when developed leftwards creates the array ‘a b a b’, when developed
rightwards the symbolically identical array ‘a b a b’. As a result, for every array that
satisfies Petrarchan sonnet conditions in the model’s upper half, there are two
solutions in the lower.
If development of the lower arrays is symbolically the same irrespective of
the directionality of development, the flows between placeholders differ, depending
on whether development is either rightwards or leftwards. The flows in the lower
half of the model will, therefore, either be symmetrical with regard to the flows in
the upper half, or chiral. If they are symmetrical, then the flows in the two halves of
the model coincide and there is no perception of the spatial relationships in the form
of geometrical shapes shown in Tables 25 and 26, no mechanism for introducing
redundancy into the model and, consequently, failure to fulfill the sonnet’s number
of lines condition. Therefore, only developments that allow for chirality between the
flows in the model’s halves is are considered. As such, only models where the
directionality of development is either rightwards or leftwards, but not rightwards
and leftwards or leftwards and rightwards, are tested for only they result in chiral
relationships between the flows in each of the models’ halves.
!166
C3.0
Testing of Centre Matrices
In Table C3.1 below, leftwards development of the centre matrix candidate
ii.) does not create the equivalent of an embracing rhyme. Rightwards development,
however, is satisfactory, as may be seen by inspection.
Table C3.1
Development: Centre Matrix ii.)
leftwards
rightwards
1.
1.
2.
2.
3.
3.
4.
4.
5.
5.
6.
7.
8.
9.
a b b a
↖↖↖
a a b b
6.
7.
b a b a
↙↙↙
a b a b
8.
9.
10.
10.
11.
11.
12.
12.
13.
13.
14.
14.
15.
15.
!167
b b a a
↗↗↗
b a a b
↗↗↗
a a b b
↗↗↗
a b b a
↗↗↗
b b a a
↗↗↗
b a a b
↗↗↗
a a b b
b a b a
↘↘↘
a b a b
↘↘↘
b a b a
↘↘↘
a b a b
↘↘↘
b a b a
↘↘↘
a b a b
↘↘↘
b a b a
↘↘↘
a b a b
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
1.
2.
3.
4.
5.
6.
rightwards
b a a b
↖↖↖
b b a a
↖↖↖
a b b a
↖↖↖
a a b b
↖↖↖
b a a b
↖↖↖
b b a a
1.
2.
3.
4.
5.
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.
10.
11.
12.
13.
14.
15.
↙↙↙
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
15.
!168
Leftwards development of the centre matrix v.) in Table C3.3 creates no equivalent
to an embracing rhyme. Rightwards development is satisfactory.
Table C3.3
Development: Centre Matrix v.)
leftwards
rightwards
1.
1.
2.
2.
3.
3.
4.
4.
5.
6.
5.
b a a b
6.
↖↖↖
a a b b
↗↗↗
a b b a
↗↗↗
b b a a
↗↗↗
b a a b
↗↗↗
a a b b
↗↗↗
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.
10.
10.
11.
11.
12.
12.
13.
13.
14.
14.
15
15.
!169
↘↘↘
b a b a
↘↘↘
a b a b
↘↘↘
b a b a
↘↘↘
a b a b
↘↘↘
b a b a
↘↘↘
a b a b
↘↘↘
b a b a
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
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15
rightwards
a b b a
↖↖↖
a a b b
↖↖↖
b a a b
↖↖↖
b b a a
↖↖↖
a b b a
↖↖↖
a a b b
↖↖↖
b a a b
1.
2.
3.
4.
5.
6.
b a b a
↙↙↙
a b a b
↙↙↙
b a b a
↙↙↙
a b a b
↙↙↙
b a b a
↙↙↙
a b a b
↙↙↙
b a b a
↙↙↙
a b a b
7.
b b a a
↗↗↗
b a a b
8.
b a b a
9.
10.
11.
12.
13.
14.
15.
!170
C4.0
Solutions
There are, therefore, four solutions:
and
array ii.)
rightwards
array iii.)
leftwards
array v.)
rightwards
array viii.)
leftwards
As arrays ii.) and v.), and iii.) and viii.) are transpositions, either of each
may satisfy. Array ii.) with rightwards and array iii.) with leftwards directionality are
chosen for the models themselves.
!171
Appendix D
Pleadean Models: Centre Array Derivation
D1.0
Pleadean Centre Array Derivation
As the Pleadean 1 and 2 simplified rhyme schemes in Table 30 show
predominantly embracing rhymes, as in the upper half of the Petrarchan model, it is
assumed that a four element, two element type centre array offers the best means of
developing equivalents of the Pleadean formal characteristics for both traditions. For
this mix there are thus for each tradition C (n,k), that is, 4!/ (4-2)!2! = 24/4 = 6
centre array candidates, as listed in Table C1.1. They are incidentally the same
candidate arrays as for the Petrarchan centre matrix (Appendix C).
Table D1.1:
Pleadean Centre Array Candidates 1
i.)
a a b b
ii.)
a b b a
iii.)
a b a b
iv.)
b b a a
v.)
b a a b
vi.)
b a b a
Arrays i.), ii.) and iii.) being transpositions of iv.), v.) and vi.) either of the two
groups of three may represent the other. Let the first group be chosen, as in Table
D1.2 below.
!172
Table D1.2
Pleadean Centre Array Candidates 2
i.)
a a b b
ii.)
a b b a
iii.)
a b a b
The array ‘a b a b’ can only develop alternating arrays and may therefore be
eliminated. As shown then in Table D1.3, there are thus just two candidates times
two directionalities, that is four final centre array candidates in all when the choice
between leftwards and rightwards directionality from the centre array is taken into
account,
Table D1.3
Pleadean Centre Array Candidates 3
i.)
a a b b
leftwards
ii.)
a a b b
rightwards
iii.)
a b b a
leftwards
iv.)
a b b a
rightwards
Array ii.), a a b b with rightwards development from array 8, immediately
develops three similar end-array elements, as shown in Table D1.4 below, thus fails
the Pleadean 1 rhyme scheme test and is eliminated as a candidate.
!173
Table D1.4
Unsuitability of Array ii.)
7.
b a a b
8.
a a b b
9.
b a a b
↗
✶
↘
The same fate befalls array iii), a b b a with leftwards development, as
shown in Table D1.5. It is, therefore, also eliminated.
Table D1.5
Unsuitability of Array iii.)
7.
b b a a
8.
a b b a
9.
b b a a
↖
✶
↙
D2.0
Solutions
This leaves just two candidates, array i.), ‘a a b b’ with leftwards and array
iv.), ‘a b b a’ with rightwards development.
!174
Appendix E
Shakespearean Model: Centre Array Derivation
E1.0
Shakespearean Centre Array Candidates
From the discussions of the previous models’ centre arrays and matrices
(Appendices B, C and D) and the analysis of the Shakespearean simplified rhyme
scheme, it is assumed that the development of a centre array of five elements with
two element types suffices to render equivalents to the formal characteristics of the
Shakespearean sonnet. As this is the same assumption made for the Early Italian
model, the same logic applies to the selection of centre array candidates for the
Shakespearean model. The list of candidate arrays is therefore the same as that in
Table 16 for the Early Italian model. The list is shown in Table E1.1.
Table E1.1
Shakespearean Centre Array Candidates
ii.)
a b a b b
iii.)
a b b a b
vi.)
b a b a b
vii.)
b a b b a
ix.)
b b a b a
With either leftwards or rightwards development there are thus five times two
candidates. From the discussion in the appendix to the Early Italian centre arrays,
two of them may be eliminated as their development immediately results in lack of
alternation of end-array elements: ‘a b a b b’ with rightwards and ‘b a b a b’ with
!175
leftwards development. There now follows the test results for the remaining
candidate arrays.
E2.0
Centre Array: ii.) ‘a b a b b’
As may be seen in the model on the left of Table E2.1 below, leftwards
development requires a change in directionality in the fourth and twelfth arrays if
alternation of end-array elements is to be maintained. This change, as may be seen in
the right hand model, does not create the equivalent of a rhyming couplet. It is
therefore eliminated as a candidate.
!176
Table E2.1
Centre Array: ii.) ‘a b a b b’
leftwards
leftwards
1.
1.
2.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
.
13.
a b a b b
↖↖ ↖ ↖
b a b a b
↖↖ ↖ ↖
b b a b a
↖↖ ↖ ↖
a b b a b
↖↖ ↖ ↖
b a b b a
↖↖ ↖ ↖
a b a b b
↙↙↙↙
b a b b a
↙↙↙↙
a b b a b
↙↙↙↙
b b a b a
↙↙↙↙
b a b a b
↙↙↙↙
a b a b b
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
14.
15.
15.
!177
b a b b a
↗↗↗↗
a b b a b
↗↗↗↗
b b a b a
↗↗↗↗
b a b a b
↖↖ ↖ ↖
b b a b a
↖↖ ↖ ↖
a b b a b
↖↖ ↖ ↖
b a b b a
↖↖ ↖ ↖
a b a b b
↙↙↙↙
b a b b a
↙↙↙↙
a b b a b
↙↙↙↙
b b a b a
↙↙↙↙
b a b a b
↘↘↘↘
b b a b a
↘↘↘↘
a b b a b
↘↘↘↘
b a b b a
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.
6.
7.
8.
9.
10.
11.
a b a b b
↖↖ ↖ ↖
b a b a b
↖↖ ↖ ↖
b b a b a
↖↖ ↖ ↖
a b b a b
↙↙↙↙
b b a b a
↙↙↙↙
b a b a b
↙↙↙↙
a b a b b
5.
6.
7.
8.
9.
10.
11.
12.
12.
13.
13.
14.
14.
15.
15.
!178
b a b a b
↗↗↗↗
a b a b b
↗↗↗↗
b a b b a
↗↗↗↗
a b b a b
↘↘↘↘
b a b b a
↘↘↘↘
a b a b b
↘↘↘↘
b a b a b
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
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
rightwards
b a b a b
↗↗↗↗
a b a b b
↗↗↗↗
b a b b a
↗↗↗↗
a b b a b
1.
2.
3.
4.
↗↗↗↗
b b a b a
↗↗↗↗
b a b a b
↖↖ ↖ ↖
b b a b a
↖↖ ↖ ↖
a b b a b
↙↙↙↙
b b a b a
↙↙↙↙
b a b a b
↘↘↘↘
b b a b a
↘↘↘↘
a b b a b
↘↘↘↘
b a b b a
↘↘↘↘
a b a b b
↘↘↘↘
b a b a b
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
!179
a b a b b
↖↖↖↖
b a b a b
↖↖↖↖
b b a b a
↖↖↖↖
a b b a b
↖↖↖↖
b a b b a
↖↖↖↖
a b a b b
↗↗↗↗
b a b b a
↗↗↗↗
a b b a b
↘↘↘↘
b a b b a
↘↘↘↘
a b a b b
↙↙ ↙ ↙
b a b b a
↙↙ ↙ ↙
a b b a b
↙↙↙ ↙
b b a b a
↙↙ ↙ ↙
b a b a b
↙↙ ↙ ↙
a b a b b
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.
2.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
b a b a b
↗↗↗↗
a b a b b
↗↗↗↗
b a b b a
↗↗↗↗
a b b a b
↗↗↗↗
b b a b a
↗↗↗↗
b a b a b
↘↘↘↘
b b a b a
↘↘↘↘
a b b a b
↘↘↘↘
b a b b a
↘↘↘↘
a b a b b
↘↘↘↘
b a b a b
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
14.
15.
15.
!180
b b a b a
↖↖↖↖
a b b a b
↖↖↖↖
b a b b a
↖↖↖↖
a b a b b
↗↗↗↗
b a b b a
↗↗↗↗
a b b a b
↗↗↗↗
b b a b a
↗↗↗↗
b a b a b
↘↘↘↘
b b a b a
↘↘↘↘
a b b a b
↘↘↘↘
b a b b a
↘↘↘↘
a b a b b
↙↙↙↙
b a b b a
↙↙↙↙
a b b a b
↙↙↙↙
b b a b a
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.
5.
6.
7.
8.
9.
10.
11.
12.
a b a b b
↖↖ ↖ ↖
b a b a b
↖↖ ↖ ↖
b b a b a
↖↖ ↖ ↖
a b b a b
↖↖ ↖ ↖
b a b b a
↙↙↙↙
a b b a b
↙↙↙↙
b b a b a
↙↙↙↙
b a b a b
↙↙↙↙
a b a b b
4.
5.
6.
7.
8.
9.
10.
b a b a b
↗↗↗↗
a b a b b
↗↗↗↗
b a b b a
↘↘↘↘
a b a b b
↘↘↘↘
b a b a b
11.
12.
13.
13.
14.
14.
15.
15.
!181
The result in the change in directionality is the development of a model
which does not develop placeholders that accommodate the requisite equivalent of
the final couplet. This centre array is therefore also eliminated from the list of
possible centre array candidates.
Table E5.2
Centre Array: vii.) ‘b a b b a’
leftwards
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
a b a b b
↗↗↗↗
b a b b a
↗↗↗↗
a b b a b
↗↗↗↗
b b a b a
↗↗↗↗
b a b a b
↖↖ ↖ ↖
b b a b a
↖↖ ↖ ↖
a b b a b
↖↖ ↖ ↖
b a b b a
↙↙↙↙
a b b a b
↙↙↙↙
b b a b a
↙↙↙↙
b a b a b
↘↘↘↘
b b a b a
↘↘↘↘
a b b a b
↘↘↘↘
b a b b a
↘↘↘↘
a b a b b
!182
E6.0
Centre Array: ix.) ‘b b a b a’
In table E6.1 below, the model with leftwards directionality develops
similar end-array elements in arrays 7 and 6, and 9 and 10, marring the development
of the equivalent of Shakespearean stanzaic form. This centre array is therefore
excluded. Rightwards development requires change in directionality in arrays 5 and
11 which is shown in Table E6.2 below.
Table E6.1
Centre Array: ix.) ‘b b a b a’
leftwards
rightwards
1.
1.
2.
2.
3.
3.
4.
4.
5.
5.
6.
7.
8.
9.
10.
a b a b b
↖↖ ↖ ↖
b a b a b
↖↖ ↖ ↖
b b a b a
↙↙↙↙
b a b a b
↙↙↙↙
a b a b b
6.
7.
8.
9.
10.
11.
11.
12.
12.
13.
13.
!183
b a b a b
↗↗↗↗
a b a b b
↗↗↗↗
b a b b a
↗↗↗↗
a b b a b
↗↗↗↗
b b a b a
↘↘↘↘
a b b a b
↘↘↘↘
b a b b a
↘↘↘↘
a b a b b
↘↘↘↘
b a b a b
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.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
b a b a b
↖↖ ↖ ↖
b b a b a
↖↖ ↖ ↖
a b b a b
↖↖ ↖ ↖
b a b b a
↖↖ ↖ ↖
a b a b b
↗↗↗↗
b a b b a
↗↗↗↗
a b b a b
↗↗↗↗
b b a b a
↘↘↘↘
a b b a b
↘↘↘↘
b a b b a
↘↘↘↘
a b a b b
↙↙↙↙
b a b b a
↙↙↙↙
a b b a b
↙↙↙↙
b b a b a
↙↙↙↙
b a b a b
!184
E7.0
Solutions
There is, thus, only one centre array with placeholders that can accom-
modate the equivalents of the Shakespearean formal characteristics: ‘a b b a b' with
leftwards and rightwards directionality.
!185
Appendix F
Memorial Day: The Unmaking of a Sonnet, Poems
F0
Preludium
180
F1
Sonnet 1
181
F2
Sonnet 2
182
F3
Sonnet 3
183
F4
Sonnet 4
184
F5
Sonnet 5
185
F6
Sonnet 6
186
F7
Sonnet 7
187
F8
Sonnet 8
188
F9
Sonnet 9
189
F10
Sonnet 10
190
F11
Sonnet 11
191
F12
Sonnet 12
192
F13
Sonnet 13
193
F14
Sonnet 14
194
F15
Postludium
195
!186
F0
Preludium
how fey how free the mitered pages mild do sheen
that drill and flay and wield the timbal railing day
a dewy spinning glass strewn with darts of rhyme
the water’s curtain glide and salt shimmer sand
like a fine thin veil across a faint thin smile
too soon to get over you but not too late
worn out by care your loss my only lair
my hushed chafed dear heart no self-pity no rage
on gloam paling light soul blent and fey
your dark toned breath and spark lit laugh
smooth and spiral fade and chiral as beneath the gaping wind
be free to speak and act to feel be brave be sure
no use their suck of power to keep the state insane
a force that aspired to choose the cause of truth not death
!187
F1
1
My story begins with you, your war, your truth and dreams.
For you, all were irony and hurt, all bitter salt,
whilst I found love, and laughed and sighed and soared and knew
that all my life was you, and all we’d have was mine.
I knew you laws of history, how they moved and stirred, and formed
a child of ardent heart and mind, clawed yet couth,
a force that aspired to choose the cause of truth not death,
to pursue the course of light, not the quietening of a breath.
Yet how be free to choose when life’s but fortune’s die?
And, say, what kindly rule thwarts indifference, cold or cruel?
How lose a love that’s life itself and, parted, still be whole?
For what wounded life can be restored that roots can somehow bind?
What guide or truth that’s sought could find such a law, or doom?
And how should I engage my mind, with your soul, still restless, strewn?
!188
F2
2
What lord, what art or science can furnish me a chart
to plan my journey’s course, to lead a worthy life?
What faults and harms to chide? What simple joys to treasure?
What life to form and want? What life to weigh and measure?
Forbear, and live the motto: no aim, no goal, too high!
To soar, to strain, to dream, endure, survive, defy;
be free to speak and act, to feel, be brave, be sure,
and back the weak and needy, live life to strive for more.
With work and family rich, with love and ease alive,
a share of bumps and flaws, and storms and wars besides,
yet wise, not short on fact: No peace, no calm assured,
so defend the cause that made you, and spend for freedom’s sword.
And when the journey’s over, when freedom’s sword has swung,
we’ll have a brand new motto: ‘e pluribus a gun’.
!189
F3
3
In your way for a change, I took the chance to get
a taste of your air and peer into your sea.
The grain of salt in your talk sent a wave through
my voice that lent a savorous edge to our play,
while your look of crashed surf and shell-slushed sand,
your flair of beach fires, your spiced dusky hands,
your dark toned breath and spark lit laugh,
sent a tide through my mind that washed over the past.
Your stature could be said of the handsome devilled kind,
square-splayed trained in the shoulder, legs long-boned, spill defined,
you stand tallish raked with a gaze mainly floored.
Your nose is a tad too flat, your mouth a spit too dry,
your hair maniacally brown, your chin somewhat awry;
your smile, all grace, your mind, all board.
!190
F4
4
When you debate, you flake: you put bland points blandly,
scoff, then mumble, fumble and crumble away:
you rail like a dove coos, rant like a quail woos,
demur like a finch fainting, and move like a drake quaking:
baiting you is never bracing, simply dull.
And yet, when I stay my tongue and weigh your claims
my hushed, chafed, dear heart, no self-pity, no rage
against the crush of vain fate, no wasteful rush
to sell your chary soul for the prize of a gushing maid
or a fluttering metal stage and worthy patrons’ games,
more a care to put your faith in pressing ways
to do some good shapes your touchingly selfless traits.
Why, then debate life and fame and death and reward?
Rather brave a certain pain and defend the gains of love.
!191
F5
5
Once again you caught my mood, played with it, drew it out,
applauded it, and blew it away. I learnt to sail with you,
surge through waves of doubt with you, scourge gales
with flails of laughter with you, yet still remain my own.
In town, last June, surprised when you called and stayed to brood,
and then said we ought to wed in May, I knew I was lost.
Too soon to get over you, but not too late,
I sought my sunbench haven on the coast, far from ruin.
Then, one day, walking along the shore, safe from you, I thought,
there you stood, staid and fraught, all doom, all wrought, afraid.
The most I could do was praise your hat, and await your gloom.
We stayed throughout the summer, we came back throughout the war.
How long would you stay? How long remain away?
For you, I fled my doubts, for you, I fled my pain.
!192
F6
6
When I once look over the shingle scuffed rock
across the summit ridge to all the immense certainty
that comes from simple awe at wonders born of light,
the sea’s vast opal floor and silent surging vault,
the port’s sudden lilt among the tripping, stumbling
ruts of dipping coast, and, tucked along beside
the water’s curtain glide and salt shimmer sand,
drips of coral sun that glint the dusty shore,
then must I not forsake Thought’s discordant rhyme,
and court the subtle will and solace of Time’s work?
Or, if I sought as such, mind a purer cause,
and take to finding laws just short of sin?
Yet, what’s the truth to find when doubt has lost its worth?
Why, the truth of simple awe at wonders born of light.
!193
F7
7
The ship of painted greed: its thrilled, discretive sway,
its play of being right and wayward, certain ease
that deals and plies and preys on each disabled mind,
to run, too vain or weak, to fight, too meek or guiled.
The sail of teaming pride, its strained insipid breeze,
its spleen of flitter flame and cheerful, baleful spite
that drill and flay and wield the timbal railing day
to sleek and slighting rain that cleaves away the sky.
Yet, stay to see the night, its shade of siren trees,
its fields of latticed sheen and cliffs of searing gray,
its seas of faience flayed and beads of plated light
that say to me defy, disdain the mind unfree.
And when the chains are freed, and when the free proclaim
all greed and pride for slain, what ship, what sail, what main?
!194
F8
8
I left to seek the light, and reading by our tree,
I wished you there with me, that I might be again
the reason why you’d stayed; no need to fade and die,
to tease you pain with rye, to hide your fear, your face.
As if to please itself, a splintered beam of day
bade me desire to stay, to see how may delight
how fey, how free, the mitered pages mild do sheen
and shimmer and sway with leaves of silent beechen gray,
then fade in gleam and fly away to bide unseen
beneath the stillen shade, between the ageless skies.
This sight of traced serene, all choirs of reeling baize,
remained with me a while to brave my knotted grief.
It seemed both tribute paid and scene to praise a life,
a light that played a breeze, a life that eased a breath.
!195
F9
9
The news was still and dark. The few who lived were silent,
mired in calm and spew. I that knew your heart,
what part of you should I remark, could I renew?
Your youth? Your will? Your charm? In truth, no part at all.
For what is this muse’s dance, this tinny, tuneless prance,
but sparks of music spied and psalmed and sighed through
a dewy, spinning glass strewn with darts of rhyme,
from chitter chatter hewn; a mirror’s ruse of shards
that grew in time apart and knew nor chart, nor chime.
And yet, past, true life, ardor lives in few
and dies in far fewer. Why should you then pass
unsung, uncried, for cause this frugal art of mine?
But what balm, what use in this? Nostalgia’s filmy slew?
Sweet pity’s chant. Just this: a wish that beauty last.
!196
F10
10
The shadows mind your name now, your warmth I’ll save inside.
Our window panes are bathed in rill of tallow rain
that stain the darkening sills like wet, white wax
down a steeped stepped stair. Of an evening time,
lightness claims the hearth with rifts of hatch that drape
the air, so fragments of things attain a sifted calm,
like a fine thin veil across a faint thin smile,
or as a braid of hair, tied up, all grays and silver clasps.
Is it too vain then to ask how I shall fade?
At a slow, slighted pace? Or fast, no waste of spite?
Or as grace in taking flight? Or frayed, a plight of harms?
Ask, I may. Escape, I shan’t. Decide, I cannot say.
Yet, when my life is waning, and when the dark rains in,
I’ll braid a veil of lightness, and save your warmth within.
!197
F11
11
How brave we are we both: two graves for homes and hearts.
I dress yours with swathes of asters and sprays of rose.
For mine, I take a small shock of flowers each week
and place it on our shore, in faith with fortunes past.
And yet, how death is coveted by the snares of cloying fate:
a spate of cavilling knocks and petty blows, then scarred,
worn out by care, your loss my only lair.
How hard to make a sense of half so many woes.
Each way seems barred and cold, all gates and frosty paths
along my days and hours. No carriage waits
to bear me off, no gaze, no passing rain, no thought.
A maze unto myself, I am lost where I am found,
ever still where I am bound, I mourn myself before,
before the graves of war, before the aster shore.
!198
F12
12
Burnt leaves of grass, scorched by victory’s blaze,
betrayed by myth’s allure, effaced by torture’s shade.
Drifts of coursing seed, lines of tined wood,
cool lanes winding through sunny miles of rape.
A wide field, mossy stiles between ivy-faced boles
and crackled tilled folds, haystack straw scents
on gloam paling light, soul blent and fey:
the sublime warmth of nature’s life spending stole.
Why, then, risk being scorched again when life is so fraught?
Why rake time burdening my pain with other’s strife?
So that nature’s crackled tilled folds should defray my mortal ills?
Were your mind not stolen, your gifts would find a way,
and in finding a way, you’d lessen others’ strife.
And in lessening others’ strife, less burned, I’d find life?
!199
F13
13
A day of blue, storm light: pageant streams of sun
strew the sky whilst stave of fluted rain ply
my eyes and face as if to shrew my mind of duping plaints,
of harbored guilt and burdens borne without merit.
A trace of cool, gray light plays between the surface
film of a creviced pool; indigo, pewter rays
smooth and spiral, fade and chiral, as beneath the gaping wind
a child looses a kite, a petrel scours the ocean.
Sometimes now I find shades of our summers
in a flurry of laughter, a fall of light, or spill of waves
that remind me love endures though love is lost.
I’m not sorry you desired me, dared my truth and dreams,
yet I rue the life that’s wasted, rue the graves of spite.
So, I slight the spite of rulers with life and ruse and light.
!200
F14
14
The doom of failing lords, their rule of corvine flair,
not waived their warring crimes, not safe from human courts,
no use their craven power to loose a sword of flame
to scorch the truth that names the fraud of their salute.
The swords of rancorous doom that slay with ravin scorn,
not excused their inane slaughter, not freed from moral blame,
no use their suck of power to keep the state insane,
to chain the norms of truth to a painted raven tower.
Yet, who will bear the cost to slew such raucous plague?
And, say, what law allows to save by foreign force?
What sore of wounds remains once foreign hands withdrawn?
To pursue the human course, to prove no life in vain,
fails to better cruelty, when cruelty’s but hate’s game.
So, forgive the cruel their hatred? Forgive, and end hate’s reign?
!201
F15
Postludium
and shimmer and sway with leaves of silent beechen gray
to sleek and slighting rain that cleaves away the sky
from chitter chatter hewn a mirror’s ruse of shards
drips of coral sun that glint the dusty shore
or as a braid of hair tied up all grays and silver clasps
I sought my sunbench haven on the coast far from ruin
how hard to make a sense of half so many woes
against the crush of vain fate no wasteful rush
the sublime warmth of nature’s life spending stole
sent a tide through my mind that washed over the past
a child looses a kite a petrel scours the ocean
and back the weak and needy live life to strive for more
to chain the norms of truth to a painted raven tower
to pursue the course of light not the quietening of a breath
!202