Frequencies

This page covers the frequencies of fragments that you find in phonemes and phonological constraints. When generating sounds and text, how frequently should certain things be used? For example, if a word is being generated with consonant inventory "pbɣʂʐɟkgqʕβt" and the rule calls for a consonant, how often should /p/ be chosen vs /b/ vs … vs /β/ vs /t/? By default, generation follows the sinusoidal distribution pattern, but this can be customized.

Distributions

Uniform

The most basic frequency distribution is a uniform one.

Sinusoidal

The sinusoidal distribution is based on \( \cos{x} |_{0}^{\pi/2} \). Given \( k \) possible elements, the starting position on the curve for the \( n \)th element is given by \( p(n) \), with ending position at \( p(n+1) \):

\[ p(n) = \frac{n\pi}{2k} \]

The probability of choosing the \( n \)th element is given by the integral from \( p(n) \) to \( p(n+1) \):

\[ \int_{p(n)}^{p(n+1)} \cos x = \left[ \sin x \right]_{p(n)}^{p(n+1)} \]

As an example, the consonant inventory "pbɣʂʐɟkg" would have the distribution:

It might seem extreme that p appears approximate 10 times more frequently than g, but most languages follow such extreme patterns. This is why Scrabble rewards you 10 times the points for playing a “Q” than an “S.”

Customizing Cosine

The distribution \( f \) can be customized over its \( a \) parameter:

\[ f(x; a) = \cos x + a |_{0}^{\pi/2} \]

{
  "consonants": {
    "values": "pbɣʂʐɟkgqʕβt",
    "distribution": {
      "curve": "cosine",
      "a": 0.2
    }
  }
}

This can even out the extremes of the later-appearing values by giving them a bit of extra weight. In the table below, you can see the effect setting the value of \( a \) (seen in the header) has on the probability distribution of the inventory.

In my experience, setting \( a \) to anything above \( 0.2 \) leads to fairly monotonous languages. There is not a lot of character in the difference between a sound appearing 8.6% of the time versus 7.9%. Natural languages frequently have rarely-used productions in them.

Custom

For tighter control of generation, you can specify each element with a custom weight.

{
  "consonants": [
    { "value": "ɴ", "weight": 30 },
    { "value": "f", "weight": 20 },
    { "value": "ʝ", "weight": 10 },
    { "value": "θ", "weight":  5 },
    { "value": "b", "weight":  5 }
  ]
}

For each element \( x \), the probability is that element’s weight divided by the total of all weights:

\[ f(x) = \frac{weight(x)}{\sum_{i \in C} weight(i)} \]

The JSON specification from above gives the probabilities:

Rule Interference

Since the same phoneme is not allowed to be generated twice in a row, the constraint rules can interfere with the observed frequencies in generated patterns. This is most likely to occur in VV phonotactic patterns, but can occur in any pattern pulling from a repeating pool.

As an example, consider the vowel inventory:

{
  "vowels": [
    { "value": "ɘ", "weight": 50 },
    { "value": "ɑ", "weight": 15 },
    { "value": "i", "weight":  5 }
  ]
}

The pattern VV will generate /ɘɑ/ 3 times more than it will generate /ɘi/ because the weights without /ɘ/ mean there is a 75%/25% distribution for the remainder. What might be more disturbing is that /ɑɘ/ will be generated 10 times more than /ɑi/.

This is not usually a problem, since there are usually enough sounds in your inventory to prevent anyone from noticing. However, it can be an issue when languages have a limited inventory or cases where your phonological constraints are too restrictive. Note that these interact; there aren’t that many nasals, so VDN is more restrictive if your consonant set only has /n/ and /ɴ/.