Section I: Introduction

When Ajax strives, some Rock’s vast Weight to throw,

The Line too labours, and the Words move slow:

—Essay on Man, Alexander Pope

Prior Definitions

Mathematics

The sum \(\sum\) symbol will be borrowed from mathematics and extended over the domain of poetic objects. This would not present a problem if it were not sometimes necessary to use the \(\sum\) in its mathematical capacity. It will be the convention of the formal system being developed to overload the arguments of the \(\sum\) operation to be defined on numbers as well as syntagmic variables. For that reason, the meaning of the symbol,

\[\sum_i^n x_i\]

Should be attended with the utmost care. When \(x_i\) is a poetic sign, then the summation will be understood to be the aggregation of signs into a poem. If \(x_i\) is a number, then the summation will be understood in its usual arithemtical sense.

Poetics

Given below are existing definitions of poetical devices.

Definition: Feet

• Iamb: One unstressed syllable followed by a stressed syllable.

• Spondee: A stressed syllable followed by a stressed syllable. Employed to slow down the pace of a line.

• Dactyl: A stressed syllable followed by two unstressed syllables. Employed to create a sense of falling or release.

• Trochee: A stressed syllable folowed by an unstressed syllable. Employed to emphasize urgency or directness.

• Anapest: Two unstressed syllables followed by a stressed syllable. Employed to create a sense of building momentum.

• Pyrrchic (Dibrachs): Two unstressed syllables.

• Amphibrach: An unstressed syllable followed by a stressed syllable and then another unstressed syllable.

• Bacchius: An unstressed syllable followed by two stressed syllables.

• Antibacchius: Two stressed syllables followed by an unstressed syllable.

Definition: Lines

• Dimeter: A line with two feet.

• Trimeter: A line with three feet.

• Tetrameter: A line with four feet.

• Pentameter: A line with five feet.

• Hexmeter: A line with six feet.

• Heptameter: A line with seven feet.

• Octameter: A line with eight feet.

• Hendecasyllable: A line consisting of eleven syllables.

Definitions: Stanzas

• Couplet: A stanza with two lines.

• Tercet: A stanza with three lines

• Quatrain: A stanza with four lines

• Cinquain: A stanza with five lines

• Sestet: A stanza with six lines.

• Septet: A stanza with seven lines.

• Octet: A stanza with eight lines.

• Nonet: A stanza with nine lines.

• Decastich: A stanza with ten lines.

• Envoi: A short, concluding stanza.

Formalization

The syntagmic hierarchy, in descending order, is given by,

1. Poems

2. Stanzas

3. Lines

4. Words

5. Syllables

Each layer is composed of elements from the layer beneath it joined together through operations (to be defined shortly).

In this formalization, English letters will be used to represent lines, Greek letters will be used to represent words and Coptic letters will be used to represent syllables.

Note

All symbolic terms will be typeset differently to distinguish them from the level of analysis and definition, and to indicate their nature as “sentences” in the language of syntagmics.

Constants

1. Uppercase English letters (\(A, B, C, ...\) ): Fixed lines.

   1. The uppercase English letter \(S\) is reserved for sets.

2. Uppercase Greek letters (\(\mathrm{A}, \mathrm{B}, \Gamma, ...\)): Fixed words.

3. Uppercase Coptic letters (\(Ⲁ, Ⲃ, Ⲅ, ...\)): Fixed syllables.

4. The lowercase English letter n is reserved for natural numbers.

5. The lowercase Fraktur letter \(\mathfrak{i}\) is reserved for iambs.

6. The lowercase Fraktur letter \(\mathfrak{t}\) is reserved for trochees.

7. The lowercase Fraktur letter \(\mathfrak{s}\) is reserved for spondees.

8. The lowercase Fraktur letter \(\mathfrak{a}\) is reserved for anapests.

9. The lowercase Fraktur letter \(\mathfrak{d}\) is reserved for dactyls.

10. The lowercase Fraktur letter \(\mathfrak{p}\) is reserved for pyrrchis (dibrachs)

11. The empty set \(\varnothing\) is reserved for the pause (caesura).

12. The ampersand \(\text{&}\) represents blank newlines.

Variables

1. Lowercase English letters (\(a, b, c, ...\) ): Indeterminate rhymed lines.

   1. The lowercase English letters \(u, v, w\) are reserved for indeterminate lines, not necessarily rhymed.

   2. The lowercase English letters \(x, y, z\) are reserved for general syntagmic variables (syllables, words, lines, stanzas and poems)

2. The lowercase Greek letters (\(\alpha, \beta, \gamma\)): Indeterminate rhymed words.

   1. The lowercase Greek letter \(\kappa, \lambda, \mu\) are reserved for indetermine words, not necessarily rhymed.

3. The Coptic letters \(ⲣ, ⲡ, Ⲡ\) are reserved for indeterminate syllables. Subscripts are often used with syllabic variables to denote different syllables.

4. The lowercase Fraktur letter \(\mathfrak{x}\) is reserved for indeterminate meters.

5. The lowercase Fraktur letter \(\mathfrak{u}\) is reserved for indeterminate speeds.

6. The lowercase Greek letter \(\varsigma\) is reserved for indeterminate stanzas.

7. The lowercase English letters \(p\) and \(q\) are reserved for indeterminate poems.

Important

Upper English letters are meant to denote particular lines, whereas lowercase English letters are meant to denote indeterminate lines that are related through their rhyme scheme.

Note

The choice of \(ⲡ\) and \(Ⲡ\) to represent syllables mirrors the unstressed and stressed syllables of verses. In other words, \(ⲡ\) is meant to represent indeterminate unstressed syllables, whereas \(Ⲡ\) is meant to represent indeterminate stressed syllables. \(ⲣ\) is used in a more general capacity, to represent stressed or unstressed syllables.

The variables will sometimes be referred to as syntagmic variables, or signs.

Uppercase-lowercase pairs of English letters are understood to be rhymes. The difference in the symbolism is the fixed nature of the denotation. For example, the sign \(A.a.a.A\) denotes one fixed line, a rhyming couplet and then the fixed line again,

The cat on the mat

Got large and fat

So-and-so such that

The cat on the mat

Note that both \(A\) and both instances of \(a\) rhyme in this example. The rhyme structure of a composite sign is encoded through the case of constants and variables. In other words, preemptively using the notation from the next section, \(A \parallel a\), \(B \parallel b\), etc.

The intent behind defining \(p\) and \(q\) as “poetic” variables is to formalize the schema of a certain fixed poetic forms through operations performed on line, word and syllabic variables and constants. “Poetic” variables can be seen as the well-formed formulae that emerge through the calculus that governs the lower levels of the syntagmic hierarchy.

Relations

All syntagmic relations are to be understood as truth values, meaning each expression results in a judgement of truth or falsity.

1. \(y \subset_p x\) (Containment): The sign y is contained in the sign x.

Important

The subscript p is used to differentiate containment from the set relation of “subset”.

The relation of “contains” extends up the levels of the syntagmic hierarchy, capturing each successive level under its umbrella as it moves up each rung of the ladder,

• Words contain syllables

• Lines contain words and syllables

• Stanzas contain lines, words and syllables

Consider the line from Spring and Fall by Gerard Manley Hopkins,

\[x = \text{Though worlds of wanwood leafmeal lie}\]

Then for each word \(\lambda\) in \(\{ \text{Though}, \text{worlds}, ..., \text{lie} \}\),

\[\lambda \subset_p x\]

Similarly, for each syllable \(\rho\) in \(\{ \text{Though}, ... \text{wan}, \text{wood}, ... \text{lie} \}\),

\[\rho \subset_p x\]

2. \(x \parallel y\) (Rhymation): The sign x rhymes with the sign y.

The relation of “rhymes with”, or rhymation, is defined more precisely in Rhymation section.

Operations

This section introduces the primitive operations of syntagimcs.

Important

These are the verbs of the system. They are used to express syntagmic proposition within the system.

In other words, all operations defined in this section are to be understood as object level constructs, in contradistinction to relations like containment or rhymation which are predicated of objects and yield judgements as a result. All syntagmic operations are to be understood as being closed under the domain of signs, meaning each operation will always yield a sign as a result.

1. \(x.y\) (Succession): Successive signs.

2. \(xy\) (Concatenation): Concatenated signs.

3. \(x:y\) (Disjunction): A sign that is either x or y.

4. \(x + y\) (Separation): Separated signs.

5. \(x \circ y\) (Projection) : Sign containing another sign.

6. \(x(y)\) (Appendment): A sign ending in another sign.

7. \((y)x\) (Prependment): A sign beginning with another sign

8. \(x \circ y \,|\, y = z\) (Substitution): Substitute \(z\) for \(y\) in the sign \(x\), where \(x\) contains \(y\), \(y \subset_p x\).

Provisional Notation

1. #x: A lengthened sign.

2. ♭x: A shortened sign.

Virelais require alternating rhymes to shorten and length across stanzas. The signs “#x” and “♭x” are here provisionally offered as a symbolic way of capturing this form. However, further research needs to be done on the exact syntactical rules of these signs.

Separation vs. Succession

To see what is meant by the distinction between separation and succession, let \(x = \text{the fish in the dish}\) and \(y = \text{the dog on a jog}\). Then \(x.y\) means,

the fish in the dish

the dog on a jog

Where as \(x + y\) means,

the fish in the dish

the dog on a jog

From this, it can be see the operation of successions inserts a new line at the end of first line, whereas the operation of separation inserts a new line after the first line and before the second line, to create a blank line between them. In effect, the operation of separation creates stanzas, whereas the operation of succession creates lines within stanzas.

Brackets

Brackets, \([]\), are used to group operations within signs by precedence.

Projection

It is important to clarify that projection is a sign. It is an object within the syntagmic system (or more specifically, an operation which yields an object). It serves a semantic function within the system. This differents from the metalogical nature of containment, which is an expression about the system, i.e. a truth value.

Important

The operation of projection is a sign. The relation of containment is a truth value.

To state “y projects x”, or symbolically,

\[x = x \circ y\]

Can be seen as a form of “poetic factorization”, akin to an arithmetic relation \(9 = 3 \cdot 3\), where one sign is identified as a constituent (or factor) of another. The \(y\) in \(x \circ y\) will sometimes be referred to as a factor of \(x\).

The operation of projection is not commutative,

\[x \circ y \neq y \circ x\]

The sign on the lefthand side \(x\) of a projection \(x \circ y\) is the “larger” sign that contains the “smaller” sign \(y\) on the righthand side. In other words, logically, if \(x\) contains \(y\),

\[[y \subset_p x] \implies [x \circ y = x]\]

However, if \(x\) does not contain \(y\), then \(x \circ y\) is defined to be a caesura, \(\varnothing\), i.e. the absence of a syntagmic variable.

\[[\neg y \subset_p x] \implies [x \circ y = \varnothing]\]

For this reason, \(x \circ y\) can be thought of an indicator variable that returns the first operand if it contains the second operand, and nothing if the first operand does not contain the second operand.

\[[[y \subset_p x] \implies [x = x \circ y]] \lor [x \circ y = \varnothing]\]

In fact, the prior expression can be seen as the logical definition of a factor. To be more precise, a factor \(y\) of a fixed \(x\) is defined as any syntagmic sign that satsifies the open formula given above.

Projection is logically related to appendment and prependment. Note \(y = \text{cat}\) prepends \(x = \text{cat on a mat}\), where as \(z = \text{mat}\) appends \(x\). Both \(z\) and \(y\) project \(x\), as well,

\[x = x \circ y\]

\[x = x \circ z\]

In other words, if a sign prepends or appends another sign, it also projects that sign. Taking the previous two equations and substituting the first into the second,

\[x = [x \circ y] \circ z\]

The brackets are dropped for notationally convenience and it is understood a projection is to be applied starting with the leftmost sign (\(y\)) and moving right to the next projection operand (\(z\)).

\[x = x \circ y \circ z\]

Importantly, projection does not imply prependment or appendment. For example \(t = \text{on}\) projects \(x\), but it does not prepend or append it. In other words, appendment, prependment and projection are logically related as follows,

\[x(y) \implies [x \circ y]\]

And,

\[(y)x \implies [x \circ y]\]

Or more succinctly,

\[[x(y) \lor (y)x] \implies (x \circ y)\]

Important

The converse of this does not hold.

The “zero” property of projection is given by noting that caesuras cannot contain anything but themselves,

\[[\varnothing \cdot y] = \varnothing\]

Which aligns with the definition. In addition, the operation of projection is idempotent,

\[[x \circ y] \circ y = x \circ y\]

The inner term, \(x \circ y\) is guaranteed to be a sign that is either empty or contains \(y\). If it is empty (caesura), then, as noted, projecting it any number times will always result in a caesura. If it contains \(y\), then it will return the very sign that contains \(y\), ensuring \([x \cdot y]\) is well defined.

Shorthand

Shorthand notation is introduced in this section to extend the primitive operations defined in the previous seciton.

1. Summation: The connotation of the \(+\) symbol is leveraged to extend the symbolism to the \(\sum\) symbol. Consider,

\[\sum_1^{n} {a_i}{b_i}{a_i} = a_1.b_1.a_1 + a_2.b_2.a_2 + ... a_n.b_n.a_n\]

This example shows how to represent a poem of arbitrary length composed of tercet stanzas where the first and third lines rhyme.

2. Serialization: A serialization (serialized concatenation) is used in reference to syllables. It simply means the concatenation of a patterned sequence of syllables. Consider,

\[\prod_{i=1}^{n} {ⲡ_i}{Ⲡ_i} = {ⲡ_i}{Ⲡ_i}{ⲡ_i}{Ⲡ_i} ... {ⲡ_n}{Ⲡ_n}\]

This example shows how to represent a line of iambic meter, i.e. sequences of unstressed and then stressed syllables.

3. Exponentiation: An exponent is used as shorthand for excessive succession of rhymes. For example, consider the lines,

   the ball in the bag

   the rip in the rag

   the gig in the gag

   
   

   some dittery dots

   some jittery jots

   these simmering sots.

This can be represented using the operation of succession and the operation of separation with the expression,

\[p = a.a.a + b.b.b\]

Exponentation is used to denote iterated succession. The exponent of a line denotes the numbers of times the rhyme appears. The current example can be expressed,

\[p = a^3 + b^3\]

Scope

The scope of a rhyme is denoted with a bar. Any line variable of the same character that feels under the scope of a bar rhymes, whereas the same variable used outside of the scope of the bar is not required to rhyme with the variable under the bar. An example will help clear this up. Consider the differences that separate the two poetical propositions, \(p\) and \(q\),

\[p = \overline{a.b.a} + \overline{a.b.a}\]

\[q = \overline{a.b.a + a.b.a}\]

In the case of p, the line variable a in the first stanza is not required to rhyme with the line variable a in the second stanza. In the case of q, the line variable a in both the first and second stanza must rhyme. For example, the following values of p and q satisfy these definitions. For p,

the dog is brown

the cat is green.

the fish does drown.

the dog is blue.

the cat is red.

the fish eats you.

Whereas for q,

the dog is brown

the cat is green

the fish does drown

the dog does frown.

the cat is mean.

the fish gets down.

If the bar is omitted from a sign, it is implied to extend over the entire proposition.

Examples

Primitive Operations

\(a.b.a\)

A tercet where the first and third lines rhyme.

\(A.b.A\)

A tercet where the first and third lines are the same.

\(a.b.a + a.b.a\)

Two rhyming tercets.

\(a.b.[b:a]\)

A tercet where the last line rhymes with either the first line or the second line.

Examples

To make clear how shorthand can be leveraged to concisely represent a poetic scheme, some examples are given below.

1. Consider the following poem,

   pippity pop

   slippity slop

   
   

   yippity yap

   kippity cap

This expression can be represented using primitive operations as,

\[p = a.a + b.b\]

Using exponentiation,

\[p = a^2 + b^2\]

Keeping in mind the definition of Scope and applying a summation, this can be further reduced,

\[p = \sum_1^2 \overline{a^2}\]

In general, an arbitrary number of rhyming couplets can be represented,

\[p = \sum_1^n \overline{a^2}\]

Meter

\(\mathfrak{i} = ⲡⲠ\)

The definition of an iamb

\(\mathfrak{t} = Ⲡⲡ\)

The definition of a trochee

\(\mathfrak{s} = ⲠⲠ\)

The definition of a spondee

\(\mathfrak{p} = ⲡⲡ\)

The definition of a pyrrhic

\(\mathfrak{d} = Ⲡⲡⲡ\)

The definition of dactyl

\(\mathfrak{a} = ⲡⲡⲠ\)

The definition of a anapest

Definition: Meters

\(a/\mathfrak{x}_n\) denotes a line in \(\mathfrak{x}\) n-meter.

For example,

\[(a/\mathfrak{i}_4).(b/\mathfrak{i}_3).(a/\mathfrak{i}_4)\]

Refers to a tercet where the first and third line are written in iambic tetrameter, whereas the second line is written in iambic trimeter. In other words,

\[(a/\mathfrak{i}_4) = {\pi_1}{\Pi_1}{\pi_2}{\Pi_2}{\pi_3}{\Pi_3}{\pi_4}{\Pi_4}\]

Note in this example the first and third line rhyme.

The scope of a meter extends to everything contained in the parenthesis it marks. For example,

\[(a.a/\mathfrak{i}_4)\]

Denotes a rhyming couplet where each line is written in iambic tetrameter.

Rhymation

Ending Stress

In order to express the different categories of rhymes that may be used to aggregates lines into a scheme, notation is introduced to accent a sign to indicate its ending stress.

If a sign has no accent mark, then any type of stress satisfies the sign.

Note

Stress accents can affix both lines \(u\) and words \(\lambda\). They do not operate on syllables.

The accented sign will be referred to as a rhyme particle. For instance, \(\hat{x}\) (to be defined immediately) is a rhyme particle. In and of itself, it does not denote a rhyme. It is only in the context of a poetical proposition that it can be said to bear the meaning of a “rhyme”. By writing \(\hat{x}\), all that has been stated is the syllabic form of the sign. In effect, the hat encodes the syllabic form and the vartiable encodes the rhyme scheme.

1. Masculine Stress

A masculine rhyme occurs when the final syllable in two words is stressed and identical phonetically. For example, the following pairs of words are masculine rhymes,

• cat, hat

• bright, light

• despair, compare

A hat is used to denote a masculine ending stress,

\[\hat{x} = x(Ⲡ)\]

2. Feminine Stress

A feminine rhyme occurs when the final syllable in two words is unstressed, and the last two syllables are identical phonetically. For example, the following pairs of words are feminine rhymes,

• mother, another

• flowing, going

A check is used to denote a feminine ending stress,

\[\check{x} = x(Ⲡⲡ)\]

3. Dactylic Stress

A dactylic rhyme occurs when two words ends in identical dactyls. For example, the following pairs of words are dactylic rhymes,

• happily, snappily

• tenderness, slenderness

A dot is used to denote a dactylic ending stress,

\[\dot{x} = x(Ⲡ{ⲡ_1}{ⲡ_2})\]

4. Off Stress

An off rhyme involves imperfect sound correspondence (assonance, consonance, etc.). For example, the following pairs are off rhymes,

• bottle, fiddle (syllabic rhyme)

• hammer, carpenter (weak rhyme)

A tilde is used to denote an off stress,

\[\tilde{x} = [ ... ]\]

Where “…” represents as yet undetermined operation.

Note

Because off-rhymes do not (yet) have a syllabic representation, they are only used within poetical proposition to denote a rhyme. Writing \(\tilde{x}\) has no meaning outside of the poetical proposition, unlike the other forms of rhymes which represent definite syllabic configurations of ending stress.

Shorthand

To avoid unnecessary complexity, the following notations are defined. In the case of masculine rhyme particles,

\[\hat{x.y} = \hat{x}.\hat{y}\]

\[\hat{x + y} = \hat{x} + \hat{y}\]

\[\hat{x:y} = \hat{x}:\hat{y}\]

Similarly for the other types of rhyme particles.

Logical Structure

Now that notation has been introduced to formalize rhyme structure in a poem, the relation of rhymation can be clarified. Rhymation is meant to explicate the relation of “perfect rhymes” within the formal system being developing.

It should first be noted, by definition, that all signs rhyme with themselves,

\[x \parallel x\]

Furthermore, if an arbitary sign \(x\) rhymes with the sign \(y\), then \(y\) rhymes with \(x\), and visa versa,

\[x \parallel y \equiv y \parallel x\]

If two arbitrary signs \(x\) and \(y\) end in the same masculine particle, \(z\), then they rhyme,

\[[x(\hat{z}) \land y(\hat{z})] \implies x \parallel y\]

If two arbitrary signs \(x\) and \(y\) end in the same feminine particle, \(z\), then they rhyme,

\[[x(\check{z}) \land y(\check{z})] \implies x \parallel y\]

If two arbitary signs end in the same dactylic particle, then they rhyme,

\[[x(\dot{z}) \land y(\dot{z})] \implies x \parallel y\]

However, off-rhymes do not imply the relation of rhymation.

If the secondary relations are defined,

• \(\vdash\), Masculine Rhyme: \(x \prec y \equiv [x(\hat{\lambda}) \land y(\hat{\lambda})]\)

• \(\Vdash\), Feminie Rhyme: \(x \Vdash y \equiv [\exists z: [x(\check{z) \land y(\check{z})]]\)

• \(\Vvdash\), Dactylic Rhyme: \(x \Vvdash y \equiv [x(\dot{\lambda}) \land y(\dot{\lambda})]\)

Then, the relation of rhymation can be defined precisely as,

\[x \parallel y \equiv [x [ \vdash \lor \Vdash \lor \vVdash ] y]\]

Where the righthand logical sum, \([ \vdash \lor \Vdash \lor \Vvdash ]\), is shorthand for one of the three relations obtaining between \(x\) and \(y\).

Definitions

With the primitive foundations of syntagmics laid, definitions are now given for quantities of

Lengths

A poetic sign has many different notions of “length” beyond the purely linguistic lengths of a sentence. A sentence, as it is conceived in the fields of formal linguistic, can be broken into sequences of characters, words or phonemes (among other categorizations). A poetic sign possesses these notions of length as a result of its embodiment in the medium of language, but it also possesses dimensions of length over and above the lengths prescribed by syntax, semantics and pragmatics. These concepts of length are derived from the structure of poetic signs and represent a space orthogonal to conventional formal linguistics where the semantics of poems are encoded. These different, but interrelated notions of length, are listed directly below and then defined,

• Stanza Length of a Poem

• Line Length of a Poem

• Line Length of a Stanza

• Syllable Length of a Line

• Syllable Length of a Stanza

• Syllable Length of a Poem

Primitive Lengths

“Primitive” lengths are the immediately measureable quantities of a poem.

Stanza Length of a Poem

Let \(p\) be an arbitrary poem with stanzas \(\varsigma_i\). The stanza length of poem \(p\), denoted \(l(p \mid \varsigma)\), is the natural number \(n\) that satisfies,

\[l_{\varsigma}(p) = n \equiv p = \sum_1^n \varsigma_i\]

Line Length of a Stanza

Let \(\varsigma\) be an arbitrary stanza with lines \(u\). The line length of \(\varsigma\), denoted \(l(\varsigma \mid u)\), is the natural number \(n\) that satisfies,

\[l(\varsigma \mid u) = n \equiv \varsigma = u^n\]

Syllable Length of a Line

Let \(u\) be an arbitrary line with syllables \(\rho_i\). The syllable length of \(u\), denoted \(l(u \mid \rho)\), is the natural number \(n\) that satisfies,

\[l(u \mid \rho) = n \equiv u = \prod_1^n \rho_i\]

In effect, the stanza length of a poem is defined as the number times the operation of separation has been applied to stanzas to create a poem, the line length of a stanza is defined as the number of times succession has been applied to lines to construct a stanza, the syllable length is the number of times concatenation has been applied to the syllables to construct a line.

Note

The definition of a length in a level of the syntagmic hierarchy is given in terms of the level directly below it.

The notation, \(l(p \mid \varsigma)\), \(l(\varsigma \mid u)\) and \(l(u \mid \rho)\) is meant to invoke the concept of “conditioning” from Bayesian analysis. Each type of length is relative to the particular formal term within a syntagmic sign that falls to the right the \(\mid\) marker.

Derivative Lengths

There are several other concepts of length that are derived directly from these definitions, illustrating how these “basic” units of syntagmic length interconnect to form more abstract notions of length.

Line Length of a Poem

Let \(p\) be an arbitrary poem with stanzas \(\varsigma_i\). Let each \(\varsigma_i\) have lines \(u_j\). The line length of \(p\), denoted \(l(p \mid u)\) is defined as,

\[l(p \mid u) = \sum_1^{l(p \mid \varsigma)} l(\varsigma \mid u)\]

Important

\(l(\varsigma \mid u)\) is a number! Therefore, the \(\sum\) that appears in the previous definition is an arithmetical sum. Recall the \(\sum\) symbol is overloaded. It may be benefit the reader to treat the preceding as a definition in the metalanguage of syntagmics, rather than its object language, where the \(\sum\) symbol is used as a semantic construct.

This definition captures the common sense notion that the number of lines in a poem is equal to the sum of the number of lines in each stanza.

Syllable Length of a Stanza

Let \(\varsigma\) be an arbitrary stanza. Let each \(\varsigma\) have lines \(u_i\). Let each line \(u_i\) have syllables \(\rho_j\). The syllable length of \(\varsigma\), denoted \(l(\varsigma \mid \rho)\) is defined as,

\[l(\varsigma \mid \rho) = \sum_1^{l(\varsigma \mid u)} l(u_j \mid \rho)\]

Once again, this captures the idea the number of syllables in a stanza is equal to the sum of the number of syllables in each line of the stanza.

There are two ways to define the syllable length of a poem. It can either be defined using the line length of a poem and syllable length of a line, or it can be defined using the stanza length of a poem and the syllable length of a stanza. Whichever definition is selected, the alternative not selected will become a theorem of the formal system as a consequence of the definitions of length. For the current purposes, the first alternative is selected.

Syllable Length of a Poem

Let \(p\) be an arbitrary poem with stanzas \(\varsigma_i\). Let each \(\varsigma_i\) have lines \(u_j\). Let each line \(u_j\) have syllables \(\rho_k\) The syllable length of \(p\), denoted \(l(p \mid \rho)\) is defined as,

\[l(p \mid \rho) = \sum_1^{l(p \mid u)} l(u \mid \rho)\]

In the previous three definition, the “condition” of the summation limit becomes the summand’s length, while the “condition” of the summand becomes the “condition” of the result. This is directly analogous to dimensional analysis in fields of science, where the units of two quantities must cancel out in order for the result to be unitless. This can be viewed a type of a “poetic dimensional analysis”.

Speed

This document opened with a quote by Alexander Pope that illustrates a phonological phenomenon that is often employed poetically for effect: sentences with clusters of stressed syllables in sequence have the psychological effect of appearing “slow”, as opposed to anapestic or dactylic rhythms which are often associated with “galloping” or “rapid” paces. In other words, there is a correlation between the perceived “speed” of a poem and its use syllabic stresses.

The notion of syntagmic speed is intended to explicate the psychological phenomenon illustrated by Pope and make it conducive to analysis. In making this definition, an important tool for the statistical analysis of poems will be introduced as a result.

First note that syllables are either stressed or unstressed, but not both. Therefore, the total number of syllables in a sign \(x\) is equal to the number of unstressed syllables \(ⲡ\) in \(x\) plus the number of stressed syllables \(Ⲡ\) in \(x\). Introducing the following notation,

• \(l(x \mid Ⲡ)\): The number of stressed syllables in sign \(x\)

• \(l(x \mid ⲡ)\): The number of unstressed syllables in sign \(x\)

It follows logically from the definitions of syllabic length,

\[l(x \mid \rho) = l(x \mid Ⲡ) + l(x \mid ⲡ)\]

With this in mind, the notion of “poetic speed” is formally defined as the “density” of stressed syllables in a sign.

Speed

Let \(x\) be a syntagmic sign such that \(l(x \mid \rho) > 0\). The speed of \(x\), denoted \(\mathfrak{u}(x)\), is defined as,

\[\mathfrak{u}(x) = \frac{l(x \mid Ⲡ)}{l(x \mid \rho)}\]