Rhyme Math (Applying mathematical tools to words that rhyme)

2024-06-23T00:00:00-04:00

notes

The standard understanding over the past almost 40 years is that this word is syllabified [ɪn.vɪ.zə.bl̩] or [ɪn.vɪ.zə<.>b<.>l̩], where “<.>b<.>l̩” is a clunky way of writing ambisyllabic b.

I don’t care that I can’t perfectly algorithmically represent all of the natural language map to the system automatically. But I hope the system is robust enough and I think it is to be able to represent all of the edge cases. If this language is that expressive, then I’m very happy.

Perfect rhymes form a category proof

Rhyme Math (Applying mathematical tools to words that rhyme)

Overview

This blog post will share a formalization on rhymes I’ve been tinkering with. The first part will be accessible to anybody who knows what a rhyme is and can do a little bit of arithmetic.

The second part will become more math heavy. But keep in mind the only thing I’m talking about is two words rhyming. We will look at some of the ways I like to relate rhymes to each other based off of “I don’t know I’m making it up”.

Lastly, I will bring in some set theory and play around with the properties of the system that I’ve created. Ultimately I think that specific sets of rhymes form categories and I would like to prove this. I may not be able to on this blog post but, I hope that gives you an idea of the direction that I’m trying to go in.

Good luck me!

What is a rhyme?

According to Wikipedia’s definition of a rhyme:

Two words rhyme if their final stressed vowel and all following sounds are identical.

It’s not really the best quote but I’m sure we can all agree that “time” rhymes with “mime”. “Sign” rhymes with “time”. “Find” rhymes with “dine”. And all of those words that rhymed, rhyme with this “line”

For rhymes, there are many categories, But for us, we will consider three, Below I will list them,- in descending hierarchy

–Rhyme (the general kind) –Perfect rhyme –Identical rhyme

I’m choosing these three because this system’s mine. And also because I didn’t feel like learning every individual type of rhyme so I’ve decided on the bare bones definition that a word rhymes with another word simply if they share at least one syllable. This way you have a general type, but you can impose further constraints if you so choose. I argue that this generalizes the structure and allows for more interoperability and more complex types if so chosen.

Enough of my dumb enough justifications.

Below I will define all three types of rhymes; 1st in English and then in mathematical terminology

Go over operations that preserve the rhyming relation #### rhymes >These are your regular old words that you rhyme all the time; and you don’t mind that they do. So since I know that you knew, lets continue through.

As stated before a word rhymes with another word if they share at least one syllable. We’re gonna kinda kinda have to abstract a simple and look at and idealized syllabus. And the other issue we’re going to run into is that as long as the syllables have a similar sound they will be considered as rhyming

This means vowel or consonant. So this means “book” and “like” rhyme. They both share the K sound even though the morpheme is different(And one is more plosive than the other). They still share the consonant. It’s just a very very weak relationship. Consider some examples of below:

Examples “Book like” “book bike”

Counter Example “Book lube” “book river”

Although These two words may not be great rhymes for the sake of the system they will be considered as such. They’re just very very weakly related to each other. (footnote:1) And you can see in the counter example that they do roll off the tongue easier than the counter examples.

(footnote:2)

Perfect Rhyme

Lifted directly from Wikipedia:

Perfect rhyme Also called full rhyme, exact rhyme, or true rhyme — is a form of rhyme between two words or phrases, satisfying the following conditions:

I think this definition is perfectly fine. As I mainly putting it in here to show that it’s just an extension of the general case of rhyme. You could go and take any of the categories of rhyme and use them like this.

Identical Rhyme

One might think a perfect rhyme would be one in which all syllables match up but As we saw in the last definition that’s not the case. But when it is that is called identical rhyme.

example:

()

This is also a special case of rhyming. But it’s the case when two different words share the exact same number of syllables and syllables sounds. In the case of rhyming Math, we would say that all identical rhymes are equivalent. More on what that means a little bit below.

And I would also note all homophones are equal and are identical rhymes.

Representing Rhymes And their relationships Mathematically

Let’s continue to look at rhymes more mathematically and explore the relationships between words that rhyme.

formally: Two words rhyme (()): If they share at least one common morpheme, syllable, or phoneme, they rhyme.

For example, if we let ( w_1 ) and ( w_2 ) represent two words that rhyme, you could write:

[ w_1 w_2  () ] or [ {/m/} {/m/} ] where (/m/) represents a given syllable, syllables, phonemes, or morphemes of a word. >Throughout the post I will differ between using IPA and English syllables it’s just to demonstrate that this is extensible in those directions.

Or more simply, if the context is clear:

[ w_1 w_2 ]

“Word one rhymes with word two”

Examples:

Let’s take a look at some simple examples:

  1. For words “cat” and “hat”: [ {/at/} {/at/} ]

  2. For words “sing” and “ring”: [ {/ing/} {/ing/} ]

  3. Using IPA notation:

    • For words “beat” and “feet”: [ {/iːt/} {/iːt/} ]

    • For words “phone” and “bone”: [ {/oʊn/} {/oʊn/} ]

  4. Varying length morphemes:

    • For words “running” and “stunning”: [ {/unning/} {/unning/} ]

    • For words “education” and “vaccination”: [ {/eɪʃən/} {/eɪʃən/} ]

  5. Common morphemes (partial rhymes):

    • For words “walking” and “talking”: [ {/ɔːkɪŋ/} {/ɔːkɪŋ/} ]

    • For words “reaction” and “attraction”: [ {/action/} {/action/} ]

In summary:

This mathematical representation allows us to analyze and categorize rhymes more systematically.

More Relationships between words that rhyme.

OK now that we have some kind of idea of how we’re going to mathematically talk about rhymes let’s go over The other two types of rhymes starting with Identical Rhymes. #### Identical Rhymes

Identical Rhymes ((=)): Two words are identical rhymes if and only if they share the same number of syllables and their syllables sound the same.

For example, if we let ( w_1 ) and ( w_2 ) represent two words that rhyme, you could write:

[ w_1 = w_2  () ] or [ {1/m/} = {2/m/} ] where (/m/) Is the exact same syllable, syllables, or phonemes of a word. Which will equal all of the syllables in the word.

Note the difference in notation where ((=)) is not Equivalent to (())

Properties of Identical Rhymes:

If the rhyme between (w_1) and (w_2) is identical, then: [ w_1 = w_2 ]

Sum of the Syllables or the Morphemes:

The sum of the syllables or morphemes in the rhyme relationship between (P) and (Q) will be even.

Formally: [ |P = Q| = ]

Where the vertical bars ( || ) denote the count or magnitude of the morphemes or syllables.

Examples in LaTeX:
Example 1: Identical Rhymes
Example 2: Identical Rhymes
Example 3: Identical Rhymes (Multi-syllable)
Example 4: Identical Rhymes (Multi-syllable)

Why is this? Because identical rhymes must have the same amount of morphemes or syllables and two even numbers added together is an even numbe and two odd numbers at it together is even as well.

Perfect Rhyme:

If two words Share all the same morphemes except the prefix then they are Perfect Rhyme(()).

Perfect Rhyme (()): Two words share all morphemes except for the beginning.

For example, if we let ( w_1 ) and ( w_2 ) represent two words that rhyme, you could write:

[ w_1 w_2  () ] or [ {/m/} {/m/} ] where (/m/) represents a given syllable, syllables, phonemes, or morphemes of a word.

Here’s an example with a perfect rhyme: [ {/æt/} {/æt/} ]

Properties of Perfect Rhyme:

All perfect rhymes are rhymes

Properties of the relationships

Now that we’ve defined and looked at 3 types of relationships between rhyming words, Let’s look at some properties.

Properties of Rhyme:

Note Notice that the vowels in [ _{/ɪd/}] and

[ _{/ɪð/} ] are the same vowel ɪ, but the consonants (d and ð) differ.

Cardinality and magnitude

In the case of rhyming math I am going to define magnitude only on the relationships defined above such that they are the number of syllables, or morphemes that the words share.

Cardinality is defined as the number of syllables that the words have. Cardinalidty is an important property of Rhyme.

if p () p then |p () p| > (|p| + |q|) /2.

Examples:

Let’s illustrate this concept with a few examples:

  1. Perfect Rhymes (Identical Number of Morphemes):
    • Hat and Cat: [ {} {} ] -Cat and Hat:
      • Perfect Rhyme
      • Rhyme
      • |cat| = |hat| = 1
      • |hat| = |cat| = 1
      • |hat + cat| = 2
      • |hat = cat| = 1 🤔
      • |hat () cat| = 1
      • if |w=w’| then |w| = |w’|
      • if |w| = |w’| then |w=w’|
      • if w=w’ then w () w’
    • Light and Night: [ {} {} ]
  2. Rhymes (Differing Number of Morphemes):
    • Installing and Talking: [ {} {} ] -Installing and Talking:
      • Rhyme
      • |Installing| != |Talking|
      • |Talking| != |Installing|
      • |Installing| = 3
      • |Talking| = 2
      • |Installing + Talking| = 5
      • |Installing = Talking| = 0 🤔
      • |Installing () Talking| = 2
    • Sing and Singing: [ {} {} ] -Sing and Singing:
      • Rhyme
      • |sing| != |singing|
      • |singing| != |sing|
      • |sing| = 1
      • |singing| = 2
      • |sing + singing| = 3
      • |sing = singing| = 0 🤔
      • |sing () singing| = 1 🤔😤🤬
  3. Identical Rhymes (All Morphemes Shared):
    • Sun and Son: [ {} = {} ] -son and sun:
      • Rhyme
      • perfect rhyme ?
      • Identical Rhyme
      • |son| = |sun| = 1
      • |sun| = |son| = 1
      • |son| = 1
      • |sun| = 1
      • |son + sun| = 2
      • |son = sun| = 1
      • |son () sun| = 1
    • Book and Book: [ {} = {} ]

And notice here how for rhymes to be identical they must have the same exact number of morphemes and the same morphemes themselves.

Every rhyme is identical to itself which also means that you get for free is the identity element. Which I will formally state below:

DEF 0.0.1 Identity Element:

[ r_{id}(w) -> w ]

[ r_{id}() -> ]

In summary:

I understand, let’s correct and expand the examples to ensure all relevant morphemes are accounted for in the rhymes.

Sets of rhymes

Now let’s consider sets of these rhymes since we’ve defined them.

I’m going to try and explain similar statements in as many different ways as possible:

In plain English “What are all the words that rhyme with book?”

In slightly more math terms,

Consider the sets of all words ( ) that rhyme with ( ).

And in more of a set theory definition:

[ R() = { w w } ]

or

[ r() = { w w } ]

or

[ r() = { w _{/ook/} } ]

I’m using the fancy notation ({/ook/}) for the set of words that rhyme with “book”. Which consequently means : ({/ook/} )

And if you want some real hierarchy: (_{/ook/} )

notes

Consider a word as a cross product of N variables from the set of English phonemes.

Standard distant function is Levine Stein distance

               baɪk     taɪk. laɪk

But let’s consider bike and tyke. like Notice How Levine Steen distance is 1. But notice how because B and T are both plosives there’s a stronger resonance between them. So I don’t necessarily think that the distance should be one.

I wonder if I can do the combinatorics of articulation places like that guy did the combinatorics of dancing?

It might be worth trying to see if certain classes from the ipa or phonemes are similar besides just strict rhyming. in strict rhyming consideration you might not consider the plosives.

OK by necessity the rhyming operator can only shift the rhyme so far.

The relationship between the words “worm” and “warm” exemplifies a phenomenon known as vowel harmony or vowel assimilation. Despite having different vowel sounds (/ɔː/ in “warm” and /ɜː/ in “worm”), the consonants and the overall mouth shape remain quite similar. This similarity is due to the fact that both vowels are produced with a similar mouth position and shape, particularly in the back of the mouth.

Vowel Harmony or Assimilation:

Phonetic Similarity:

Linguistic Implications:

This relationship between “worm” and “warm” illustrates how phonetic similarities can exist despite differences in specific vowel articulations, highlighting the complexity and richness of linguistic sound systems.

FAQ And other notes

For a while I’ve been trying to come up with some mathematical rhyme theory I stumbled across this blog post which immensely helped me and allowed me to dig deeper and get a turning point so thank you extremely much Mister Lambda.

footnote 1

This is extremely relevant if you decide to define the mathematics on phonenemes or morphemes. For when you go one step below syllables to a granular level you’ll get much more matches right? Just by virtue of more pieces are used to construct the word. Generally, From my shallow research, morphemes seem to be the same size as a syllable that seems to be the general unit but I’m not a linguist or a poet or a smart person.

footnote 2

And if you still don’t believe me, here’s an example from Wikipedia:

“This little piggy stayed (at) home…this little piggy had none.”

I would never have thought these two words would rhyme. Even though the nasal consonants differ because one is M and one is N. They sound phonetically so similar, that the Voicing counts as rhyme! You might think using I P a would give you a better picture of this: “həʊm nʌn” And no it doesn’t. To me there’s nothing in those symbols that suggest they rhyme. Whereas the English spelling actually does. Even though they’re pretty much pronounced differently. this is extremely fascinating!

This is why I have decided on the definition being a such. It would cut out too many relationships otherwise.

Other types of rhyme.

Here’s some other more specific types of rhyme if you’re interested in their poetical and traditional implications. I will still use some notation because this is my blog post.

  1. Perfect Rhyme:
    • Words rhyme exactly, including the stressed vowel and all sounds that follow. [ ] [ ]
  2. Slant Rhyme (or Near Rhyme):
    • Words have similar sounds, but the stressed vowels and/or endings are not identical. [ ] [ ]
  3. Eye Rhyme:
    • Words look like they should rhyme due to their spelling, but they do not rhyme when spoken aloud. [ ]
  4. Identical Rhyme:
    • Words are identical in sound and spelling. [ = ]
  5. Internal Rhyme:
    • Rhyme occurs within a single line of verse, often enhancing rhythm and emphasis. [ ]
  6. Masculine Rhyme:
    • A rhyme that occurs on a single stressed syllable at the end of lines. [ ]
  7. Feminine Rhyme:
    • A rhyme involving two or more syllables, with stress on a syllable other than the last. [ ]
  8. Compound Rhyme:
    • A rhyme involving two or more words. [ ]

Why isn’t it called rap math?

Rap is rhythm applied to poetry(rhyme). We haven’t yet taken the cross product of rhyme and rhythm. 😉