Section II: Rhymation

Section II: Rhymation#

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.

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 phonetically identical. 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,

\[\hat{x} \equiv \exists Ⲡ: x \sim Ⲡ\]

  1. Feminine Stress

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

  • mother, another

  • flowing, going

A check is used to denote a feminine ending stress,

\[\check{x} \equiv \exists Ⲡ,ⲡ: x \sim Ⲡⲡ\]

  1. 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,

\[\dot{x} \equiv \exists Ⲡ, {ⲡ_1}, {ⲡ_2}: x \sim Ⲡ{ⲡ_1}{ⲡ_2}\]

  1. 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,

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

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.

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,

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

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

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

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

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

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

If the secondary relations are defined,

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

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

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

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

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